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# $f(T)$ cosmology via Noether symmetry
K. Atazadeh1 and F. Darabi1,2
1Department of Physics, Azarbaijan University of Tarbiat Moallem, Tabriz
53741-161, Iran
2Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha
55134-441, Iran email: atazadeh@azaruniv.eduemail: f.darabi@azaruniv.edu
###### Abstract
We consider Noether symmetry approach to find out exact cosmological solutions
in $f(T)$-gravity. Instead of taking into account phenomenological models, we
apply the Noether symmetry to the $f(T)$ gravity. As a result, the presence of
such symmetries selects viable models and allow to solve the equations of
motion. We show that the generated $f(T)$ leads to a power law expansion for
the cosmological scale factor.
## 1 Introduction
The expansion of the universe is currently undergoing a period of acceleration
which is directly measured from observations such as Type Ia Supernovae [1],
[2] cosmic microwave background (CMB) radiation [3, 4], large scale structure
[5], baryon acoustic oscillations [6], and weak lensing [7]. There are two
remarkable approaches to explain the late time acceleration of the universe:
One is to introduce some unknown matters called “dark energy” in the framework
of general relativity (for a review on dark energy, see, e.g., [8, 9]). The
other is to modify the gravitational theory, e.g., $f(R)$ gravity [10, 11].
To extend gravity beyond general relativity, “teleparallelism" could be
considered by using the Weitzenböck connection, which has no curvature but
torsion, rather than the curvature defined by the Levi-Civita connection [12],
[13]. This approach was also taken by Einstein [14]. The teleparallel
Lagrangian density described by the torsion scalar $T$ has been promoted to a
function of $T$, $i.e.$, $f(T)$, in order to account for the late time cosmic
acceleration [15] as well as inflation [16]. This concept is similar to the
idea of $f(R)$ gravity. Various aspects of $f(T)$ gravity have been examined
in the literature [17, 18, 19, 20, 21]. In particular, the presence of extra
degrees of freedom and the violation of local Lorentz invariance as well as
the existence of non-trivial frames for $f(T)$ gravity have been noted [21].
Evidently, more studies on $f(T)$ gravity are needed to see if the theory is a
viable one. For a comprehensive review of the teleparallel gravity, the reader
is referred to [22].
In this paper, we consider a flat FRW space-time in the framework of the
metric formalism of $f(T)$ gravity. Following [23], we set up an effective
Lagrangian in which the scale factor $a$ and torsion scalar $T$ play the role
of independent dynamical variables. This Lagrangian is constructed in such a
way that its variation with respect to $a$ and $T$ yields the correct
equations of motion as that of an action with a generic $f(T)$ mentioned
above. The form of the function $f(T)$ appearing in the modified action is
then found by demanding that the Lagrangian admits the desired Noether
symmetry [24, 25]. For a study of the Noether symmetry in $f(R)$ cosmology see
[26]. Similarly, we shall see that by demanding the Noether symmetry as a
feature of the Lagrangian of the cosmological model under consideration, we
can obtain the explicit form of the function $f(T)$. Since the existence of a
symmetry results in a constant of motion, we can integrate the field equations
which would then lead to a power law expansion of the universe.
## 2 $f(T)$ gravity and cosmology
To consider teleparallelism, one employs the orthonormal tetrad components
$e_{A}(x^{\mu})$, where an index $A$ runs over $0,1,2,3$ to the tangent space
at each point $x^{\mu}$ of the manifold. Their relation to the metric
$g_{\mu\nu}$ is given by
$g_{\mu\nu}=\eta_{AB}e^{A}_{\mu}e^{B}_{\nu}\,,$ (1)
where $\mu$ and $\nu$ are coordinate indices on the manifold and also run over
$0,1,2,3$, and $e_{A}^{\mu}$ forms the tangent vector on the tangent space
over which the metric $\eta_{AB}$ is defined.
Instead of using the torsionless Levi-Civita connection in General Relativity,
we use the curvatureless Weitzenböck connection in Teleparallelism [12], whose
non-null torsion $T^{\rho}_{\verb| |\mu\nu}$ and contorsion
$K^{\mu\nu}_{\verb| |\rho}$ are defined by
$\displaystyle T^{\rho}_{\verb| |\mu\nu}\equiv
e^{\rho}_{A}\left(\partial_{\mu}e^{A}_{\nu}-\partial_{\nu}e^{A}_{\mu}\right)\,,$
(2) $\displaystyle K^{\mu\nu}_{\verb|
|\rho}\equiv-\frac{1}{2}\left(T^{\mu\nu}_{\verb| |\rho}-T^{\nu\mu}_{\verb|
|\rho}-T_{\rho}^{\verb| |\mu\nu}\right)\,,$ (3)
respectively. Moreover, instead of the Ricci scalar $R$ for the Lagrangian
density in general relativity, the teleparallel Lagrangian density is
described by the torsion scalar $T$ as follows
$T\equiv S_{\rho}^{\verb| |\mu\nu}T^{\rho}_{\verb| |\mu\nu}\,,$ (4)
where
$S_{\rho}^{\verb| |\mu\nu}\equiv\frac{1}{2}\left(K^{\mu\nu}_{\verb|
|\rho}+\delta^{\mu}_{\rho}\ T^{\alpha\nu}_{\verb|
|\alpha}-\delta^{\nu}_{\rho}\ T^{\alpha\mu}_{\verb| |\alpha}\right)\,.$ (5)
The modified teleparallel action for $f(T)$ gravity is given by
$I=\int d^{4}x|e|f(T)\,,$ (6)
where $|e|=\det\left(e^{A}_{\mu}\right)=\sqrt{-g}$ and the units has been
chosen so that $c=16\pi G=1$. Note that in action (6), we have overlooked any
matter contribution in the action. Varying the action in Eq. (6) with respect
to the vierbein vector field $e_{A}^{\mu}$, we obtain the equation [15]
$\frac{1}{e}\partial_{\mu}\left(eS_{A}^{\verb|
|\mu\nu}\right)f_{T}-e_{A}^{\lambda}T^{\rho}_{\verb|
|\mu\lambda}S_{\rho}^{\verb| |\nu\mu}f_{T}+S_{A}^{\verb|
|\mu\nu}\partial_{\mu}\left(T\right)f_{{}_{TT}}+\frac{1}{4}e_{A}^{\nu}f=0\,,$
(7)
where a subscript $T$ denotes differentiation with respect to $T$. We assume
the four-dimensional flat Friedmann-Lemaître-Robertson-Walker (FLRW) space-
time with the metric,
$ds^{2}=h_{\alpha\beta}dx^{\alpha}dx^{\beta}+\tilde{r}^{2}d\Omega^{2}\,,$ (8)
where $\tilde{r}=a(t)r$, $x^{0}=t$ and $x^{1}=r$ with the two-dimensional
metric $h_{\alpha\beta}={\rm diag}(-1,a^{2}(t))$. Here, $a(t)$ is the scale
factor and $d\Omega^{2}$ is the metric of two-dimensional sphere with unit
radius. In this space-time, $g_{\mu\nu}=\mathrm{diag}(-1,a^{2},a^{2},a^{2})$
and the tetrad components $e^{A}_{\mu}=(1,a,a,a)$ yield the exact value of
torsion scalar
$T=-6H^{2},$ (9)
where $H=\dot{a}/a$ is the Hubble parameter and the dot denotes the time
derivative of $\partial/\partial t$.
In the flat FLRW background, it follows from Eq. (7) that the modified
Friedmann equations are given by [15]
$\displaystyle 12f_{{}_{T}}H^{2}+f=0\,,$ (10)
$\displaystyle\dot{H}=\frac{1}{4T\,f_{{}_{TT}}+2f_{{}_{T}}}\left(-T\,f_{{}_{T}}+\frac{f}{2}\right)\,.$
(11)
It is known that $f(T)$ gravity has first-order gravitational field equation
in derivatives. Similar to $f(R)$ gravity in general relativity where the
gravitational field equation is fourth-order in derivatives, it is important
to investigate the theoretical aspects in order to examine whether $f(T)$
gravity can be a gravitational theory like $f(R)$ gravity.
In order to derive the cosmological equations in a FLRW metric, one can define
a canonical Lagrangian ${\cal L}={\cal L}(a,\dot{a},T,\dot{T})$, where ${\cal
Q}=\\{a,T\\}$ is the configuration space and ${\cal
TQ}=\\{a,\dot{a},T,\dot{T}\\}$ is the related tangent bundle on which ${\cal
L}$ is defined. The variable $a(t)$ and $T(t)$ are the scale factor and the
torsion scalar in the FLRW metric, respectively. One can use the method of the
Lagrange multipliers to set $T$ as a constraint of the dynamics. Selecting the
suitable Lagrange multiplier and integrating by parts, the Lagrangian ${\cal
L}$ becomes canonical. In our case, we have
$I=2\pi^{2}\int
dt\,a^{3}\left\\{f(T)-\lambda\left[T+6\left(\frac{\dot{a}^{2}}{a^{2}}\right)\right]\right\\},$
(12)
where $a$ is the scale factor scaled with respect to today’s value (so that
$a=\tilde{a}/\tilde{a}_{0}$ and $a(t_{0})=1$). This choice for $a$, makes it
dimensionless, whereas $[f]=M^{4}$. It is straightforward to show that, for
$f(T)=-T$, one obtains the usual Friedmann equations.
The variation with respect to $T$ of the action gives $\lambda=f_{T}$.
Therefore the previous action can be rewritten as
$I=2\pi^{2}\int
dt\,a^{3}\left\\{f-f_{T}\left[T+6\left(\frac{\dot{a}^{2}}{a^{2}}\right)\right]\right\\}\,,$
(13)
and then, integrating by parts, the point-like FLRW Lagrangian is obtained
${\cal L}=a^{3}\,(f-f_{T}\,T)-6\,f_{T}\,a\,\dot{a}^{2},$ (14)
which is a canonical function of two coupled fields, $T$ and $a$, both
depending on time $t$. The momenta conjugate to variables $a$ and $T$ are
$p_{a}=\frac{\partial{\cal L}}{\partial\dot{a}}=-12f_{T}a\dot{a}.$ (15)
$p_{T}=\frac{\partial{\cal L}}{\partial\dot{T}}=0.$ (16)
The total energy $E_{\cal L}$ corresponding to the $0,0$-Einstein equation is
$E_{{\cal L}}=-6\,f_{T}\,a\,\dot{a}^{2}-a^{3}\,(f-f_{T}\,T)=0.$ (17)
The equations of motion for $T$ and $a$ are
$a^{3}f_{{}_{TT}}\left(T+6\frac{\dot{a}^{2}}{a^{2}}\right)=0,$ (18)
$-6f_{T}\,H^{2}-12f_{T}\frac{\ddot{a}}{a}=3(f-f_{T}\,T)+12f_{{}_{TT}}\,\dot{T}\,H,$
(19)
respectively. Note that from the equations (17), (18) and (19) we can recover
the equation (10), (9) and (11), respectively. Here we have taken $f_{TT}\neq
0$.
Considering $T$ and $a$ as the variables, we see that $T$ coincides with the
definition of the torsion scalar in the FLRW metric (excluding the case
$f_{{}_{TT}}=0$). Geometrically, this is the Euler constraint of the dynamics.
Furthermore, as we will show below, constraints on the form of the function
$f(T)$ can be achieved by asking for the existence of Noether symmetries. On
the other hand, the existence of the Noether symmetries guarantees the
reduction of dynamics and the eventual solvability of the system.
## 3 The Noether symmetry
Solutions for the dynamics given by the point-like Lagrangian (14) can be
obtained by selecting cyclic variables related to some Noether symmetry [26,
27]. In principle, as a physical criterion, this approach allows one to select
$f(T)$ gravity models which are compatible with the Noether symmetry. Let
$\mathcal{L}(q^{i},\dot{q}^{i})$ be a canonical, non degenerate point-like
Lagrangian so that
$\frac{\partial\mathcal{L}}{\partial U}=0,\hskip
42.67912ptdetH_{ij}\equiv\left\|\frac{\partial^{2}\mathcal{L}}{\partial\dot{q}^{i}\partial\dot{q}^{j}}\right\|\neq
0,$ (20)
where $H_{ij}$ is the Hessian matrix related to the Lagrangian $\mathcal{L}$
and a dot denotes derivative with respect to the affine parameter $U$, namely
the cosmic time t. The Lagrangian in analytic mechanics is of the form
$\mathcal{L}=T(\textbf{q},\dot{\textbf{q}})-V(\textbf{q}),$ (21)
where T and V are the positive definite quadratic ‘kinetic energy’ and
‘potential energy’, respectively. The energy function associated with
$\mathcal{L}$ is:
$E_{\mathcal{L}}\equiv\frac{\partial\mathcal{L}}{\partial\dot{q}^{i}}-\mathcal{L},$
(22)
which is the total energy $T+V$ and the constant of motion. Since our
cosmological problem has a finite number of degrees of freedom, we are going
to consider only point transformations. Any invertible transformation of the
‘generalized positions’ $Q^{i}=Q^{i}(\textbf{q})$ induces a a local
transformation of the ‘generalized velocities’ such that:
$\dot{Q}^{i}(\textbf{q})=\frac{\partial Q^{i}}{\partial q^{j}}\dot{q}^{j};$
(23)
the matrix $\mathcal{J}=\left\|\partial Q^{i}/\partial q^{j}\right\|$ is the
Jacobian of the transformation on the positions, and it is assumed to be non-
zero. A point transformation $Q^{i}=Q^{i}(\textbf{q})$ can depend on a (or
more than one) parameter. As starting point, we can assume that a point
transformation depends on a parameter $\epsilon$, so that
$Q^{i}=Q^{i}(\textbf{q},\epsilon)$, and that it gives rise to a one-parameter
Lie group. For infinitesimal values of $\epsilon$, the transformation is then
generated by a vector field: for instance, $\partial/\partial x$ is a
translation along the $x$ axis while $x(\partial/\partial
y)-y(\partial/\partial x)$ is a rotation around the $z$ axis and so on.
In general, an infinitesimal point transformation is represented by a generic
vector field on $Q$
$\textbf{X}=\alpha^{i}(\textbf{q})\frac{\partial}{\partial q^{i}}.$ (24)
The induced transformation (23) is then represented by
$\textbf{X}^{c}=\alpha^{i}\frac{\partial}{\partial
q^{i}}+\left(\frac{d}{d\lambda}\alpha^{i}(\textbf{q})\right)\frac{\partial}{\partial\dot{q}^{j}}.$
(25)
A function $F(\textbf{q},\dot{\textbf{q}})$ is invariant under the
transformation X if
$L_{X}F\equiv\alpha^{i}(\textbf{q})\frac{\partial F}{\partial
q^{i}}+\left(\frac{d}{d\lambda}\alpha^{i}(\textbf{q})\right)\frac{\partial}{\partial\dot{q}^{j}}F=0,$
(26)
where $L_{X}F$ is the Lie derivative of F. If $L_{X}\mathcal{L}=0$, X is then
a symmetry for the dynamics derived by $\mathcal{L}$. Let us now consider a
Lagrangian $\mathcal{L}$ and its Euler-Lagrange equations
$\frac{d}{d\lambda}\frac{\partial\mathcal{L}}{\partial\dot{q}^{j}}-\frac{\partial\mathcal{L}}{\partial
q^{j}}=0.$ (27)
Contracting (27) with $\alpha^{i}$s leads to
$\alpha^{j}\left(\frac{d}{d\lambda}\frac{\partial\mathcal{L}}{\partial\dot{q}^{j}}-\frac{\partial\mathcal{L}}{\partial
q^{j}}\right)=0.$ (28)
Using the total derivative relation as
$\alpha^{j}\frac{d}{d\lambda}\frac{\partial\mathcal{L}}{\partial\dot{q}^{j}}=\frac{d}{d\lambda}\left(\alpha^{j}\frac{\partial\mathcal{L}}{\partial\dot{q}^{j}}\right)-\left(\frac{d\alpha^{j}}{d\lambda}\right)\frac{\partial\mathcal{L}}{\partial\dot{q}^{j}},$
(29)
we obtain from equation (28) that
$\frac{d}{d\lambda}\left(\alpha^{j}\frac{\partial\mathcal{L}}{\partial\dot{q}^{j}}\right)=L_{X}\mathcal{L}.$
(30)
The immediate consequence is the Noether theorem which states: if
$L_{X}\mathcal{L}=0$, then the function
$\Sigma_{0}=\alpha^{k}\frac{\partial\mathcal{L}}{\partial\dot{q}^{k}},$ (31)
is a constant of motion. Equation (31) can be expressed independently of
coordinates as a contraction of X by a Cartan 1-form
$\theta_{\mathcal{L}}\equiv\frac{\partial\mathcal{L}}{\partial\dot{q}^{j}}dq^{j}.$
(32)
For a generic vector field $\textbf{Y}=y^{i}\partial/\partial x^{i}$, and
1-form $\beta=\beta_{i}dx^{i}$, we have by definition
$i_{Y}\beta=y^{i}\beta_{i}$. Thus equation (31) can be expressed as:
$i_{X}\theta_{\mathcal{L}}=\Sigma_{0}.$ (33)
Through a point transformation, the vector field X becomes:
$\tilde{\textbf{X}}=\left(i_{X}dQ^{k}\right)\frac{\partial}{\partial
Q^{k}}+\left(\frac{d}{d\lambda}\left(i_{X}dQ^{k}\right)\right)\frac{\partial}{\partial\dot{Q}_{k}}.$
(34)
If X is a symmetry and we choose a point transformations such that
$i_{X}dQ^{1}=1;\hskip 34.14322pti_{X}dQ^{i}=0\hskip 28.45274pti\neq 1,$ (35)
we get
$\tilde{\textbf{X}}=\frac{\partial}{\partial Q^{1}};\hskip
34.14322pt\frac{\partial\mathcal{L}}{\partial Q^{1}}=0.$ (36)
Consequently, $Q^{1}$ is a cyclic coordinate, related to conserved quantities,
reducing the dynamics of the system to a manageable one. Furthermore, the
change of coordinates given by (35) is not unique. In general, the solution of
equation (35) is not defined on the whole space, rather, it is local in the
sense explained above.
Following [24, 25], we define the Noether symmetry induced on the model by a
vector field $X$ on the tangent space $T{\cal
Q}=\left(a,T,\dot{a},\dot{T}\right)$ of the configuration space ${\cal
Q}=\left(a,T\right)$ of Lagrangian (14)
$X=\alpha\frac{\partial}{\partial a}+\beta\frac{\partial}{\partial
T}+\frac{d\alpha}{dt}\frac{\partial}{\partial\dot{a}}+\frac{d\beta}{dt}\frac{\partial}{\partial\dot{T}},$
(37)
such that the Lie derivative of the Lagrangian with respect to this vector
field vanishes
$L_{X}{\cal L}=0.$ (38)
In (37), $\alpha$ and $\beta$ are functions of $a$ and $T$ and $\frac{d}{dt}$
represents the Lie derivative along the dynamical vector field, that is,
$\frac{d}{dt}=\dot{a}\frac{\partial}{\partial
a}+\dot{T}\frac{\partial}{\partial T}.$ (39)
It is easy to find the constants of motion corresponding to such a symmetry.
Indeed, equation (38) can be rewritten as
$L_{X}{\cal L}=\left(\alpha\frac{\partial{\cal L}}{\partial
a}+\frac{d\alpha}{dt}\frac{\partial{\cal
L}}{\partial\dot{a}}\right)+\left(\beta\frac{\partial{\cal L}}{\partial
T}+\frac{d\beta}{dt}\frac{\partial{\cal L}}{\partial\dot{T}}\right)=0.$ (40)
Noting that $\frac{\partial{\cal L}}{\partial q}=\frac{dp_{q}}{dt}$, we have
$\left(\alpha\frac{dp_{a}}{dt}+\frac{d\alpha}{dt}p_{a}\right)+\left(\beta\frac{dp_{T}}{dt}+\frac{d\beta}{dt}p_{T}\right)=0,$
(41)
which yields
$\frac{d}{dt}\left(\alpha p_{a}+\beta p_{T}\right)=0.$ (42)
Thus, the constants of motion are
$Q=\alpha p_{a}+\beta p_{T},$ (43)
whereas in $f(T)$ theory we have $p_{T}=0$ and so the corresponding constant
of motion becomes $Q=\alpha p_{a}$.
In order to obtain the functions $\alpha$ and $\beta$ we use equation (40). In
general, this equation gives a polynomial in terms of $\dot{a}^{2}$ and
$\dot{a}\dot{T}$ with coefficients being partial derivatives of $\alpha$ and
$\beta$ with respect to the configuration variables $a$ and $T$. Thus, the
resulting expression is identically equal to zero if and only if these
coefficients are zero. This leads to a system of partial differential
equations for $\alpha$ and $\beta$.
## 4 Noether symmetries in $f(T)$ cosmology
For the existence of a symmetry, we can write the following system of
equations (linear in $\alpha$ and $\beta$),
$f_{T}(\alpha+2a\,\partial_{{}_{a}}\alpha)+a\,f_{TT}\beta=0,$ (44)
$a\,f_{T}\,\partial_{{}_{T}}\alpha=0.$ (45)
which are obtained by setting the coefficients of the terms $\dot{a}^{2}$ and
$\dot{a}\dot{T}$ in $L_{\bf X}{\cal L}=0$ to zero. In order to make $L_{\bf
X}{\cal L}=0$ vanish we will also look for those particular $f$’s which, given
the Euler dynamics, also satisfy the constraint
$3\alpha\,(f-T\,f_{T})-a\,\beta\,T\,f_{TT}=0\,.$ (46)
This procedure is different from the usual Noether symmetry approach, in the
sense that now $L_{\bf X}{\cal L}=0$ will be solved not for all dynamics
(which solve the Euler-Lagrange equations), but only for those $f$ which
allows Euler solutions to solve also the constraint (46). Imposing such a
constraint on the form of $f$ will turn out to be, as we will show, a
sufficient condition to find solutions of the Euler-Lagrange equation which
also possess a constant of motion, i.e. a Noether charge. As we shall see
later, the system (44) and (45) can be solved exactly. Having a non-trivial
solution for $\alpha$ and $\beta$ for this system, one finds a constant of
motion if also the constraint (46) is satisfied. A solution of (44) and (45)
exists if explicit forms of $\alpha$, $\beta$ are found. If, at least one of
them is different from zero, a Noether symmetry exists.
When $f_{T}\neq 0$, from equation (45) we get
$\partial_{{}_{T}}\alpha=0\,\Rightarrow\alpha=\alpha(a).$ (47)
Thus, from this equation it can be seen $\alpha$ only depend on $a$. On the
other hand, we can obtain $\alpha$ from equation (46) as follows
$\alpha(a)=\frac{a\,f_{{}_{TT}}T}{3(f-T\,f_{{}_{T}})}\beta(a,T),$ (48)
If one uses this expression in equation (44) to eliminate $\alpha(a)$, one
obtains
$\frac{\partial\beta}{\partial a}=\frac{-3f}{2a\,T\,f_{{}_{T}}}\beta(a,T).$
(49)
To solve this equation we assume that the function $\beta(a,T)$ can be written
in the form $\beta(a,T)=A(a)B(T)$, where $A$ and $B$ are separate functions of
$a$ and $T$, respectively. Substituting this ansatz for $\beta(a,T)$ into
equation (49), we obtain
$\frac{2a}{A}\frac{dA}{da}=-\frac{3f}{T\,f_{{}_{T}}}.$ (50)
Since the left-hand side of this equation is a function of $a$ only while the
right-hand side is a function of $T$, we should have
$-\frac{3f}{T\,f_{{}_{T}}}=C=\mbox{Const},$ (51)
which results in
$f(T)=f_{{}_{0}}T^{(-\frac{3}{C})}.$ (52)
On the other hand equation (50), with its right hand-side equal to $C$, has
the solution
$\frac{2a}{A}\frac{dA}{da}=C\Rightarrow A(a)=a^{C/2}.$ (53)
Now, using (52) and $\beta(a,T)=a^{C/2}B(T)$ in equation (48), we find
$\alpha(a)=\frac{1}{C}T^{-1}\,a^{\frac{C}{2}+1}B(T).$ (54)
Since $\alpha(a)$ should be a function of $a$ only, from the above expression
for $\alpha(a)$ we can write
$B(T)=T,$ (55)
and thus we obtain
$\beta(a,T)=T\,a^{C/2},\hskip
14.22636pt\alpha(a)=\frac{1}{C}a^{\frac{C}{2}+1}.$ (56)
To obtain the corresponding cosmology resulting from this type of $f(T)$, we
note that the existence of Noether symmetry implies the existence of a
constant of motion $Q=\alpha p_{a}$. Hence, using equation (15) we have
$Q=36\frac{a^{(\frac{C}{2}+2)}\dot{a}\,T^{-(1+\frac{3}{C})}}{C}.$ (57)
On the other hand, by taking $f_{{}_{TT}}\neq 0$ from equation (18) we obtain
$T=-6\frac{\dot{a}^{2}}{a^{2}}.$ (58)
By using equations (57) and (58) we can obtain scale factor as following
$a(t)\sim t^{\frac{-2}{C}}.$ (59)
Therefore, in the context of $f(T)=f_{{}_{0}}T^{-(\frac{3}{C})}$ cosmology,
the universe evolves with a power law expansion. It is seen from (59) that the
condition under which the universe expands is $C<0$. The deceleration
parameter as a function of $C$ is therefore given by
$q(C)=-\left(1+\frac{C}{2}\right)$ (60)
The condition for acceleration is $q(C)<0$, thus we have $C>-2$. As it can be
seen, for $C\rightarrow 0$ we have $q\rightarrow-1$, that is the universe
finally approaches the eternal de Sitter phase with infinite acceleration,
however, the accelerated expansion occurs in $-2<C<0$.
## 5 Dark energy equation of state and age of the Universe in $f(T)$
In this section, we consider the accelerated expansion of the universe in the
context of $f(T)$ theory, without introducing the mysteries fluid the so-
called “dark energy” with a negative equation of state (EOS) parameter $w$.
Let us then start from equation (59) and find the Hubble parameter as a
function of the redshift $z$ as
$H(z)=-\frac{2}{C}H_{0}(1+z)^{-\frac{C}{2}},$ (61)
where $a_{0}/a=1+z$ with $a_{0}$ and $H_{0}$ being the values of the parameter
at the present epoch. It is worth noting that equation (61) is the same as
that derived from the standard Friedmann equation with $w=-2/3$. Now, we may
write the Friedmann equation in a formal fashion which would encapsulate any
modification to the standard Friedmann equation in the last term regardless of
its nature [28] that is
$H^{2}/H^{2}_{0}=\Omega_{m}(1+z)^{3}+\delta H^{2}/H^{2}_{0},$ (62)
where $\Omega_{m}=\rho/\rho_{0c}$, $\rho_{0c}=3H^{2}_{0}$. Also, defining the
effective EOS, denoted by $w_{{}_{\rm eff}}(z)$, as
$w_{{}_{\rm eff}}(z)=-1+\frac{1}{3}\frac{d\ln\delta H^{2}}{d\ln(1+z)},$ (63)
we can calculate $w_{{}_{\rm eff}}(z)$ using equations (61), (62) and (63)
with the result
$w_{{}_{\rm
eff}}=-1+\frac{1}{3}\frac{-\frac{1}{C}(1+z)^{-c}-3\Omega_{m}(1+z)^{3}}{\frac{4}{C^{2}}(1+z)^{-c}-\Omega_{m}(1+z)^{3}}.$
(64)
Figure 1 shows the behavior of the effective EOS parameter, $w_{{}_{\rm
eff}}$, as a function of $z$. As it can be seen, for $C=-1$ ($-2<C\leq-1$) and
$\Omega_{m}=0.33$ we have $w_{{}_{\rm eff}}\lessapprox-1$, which is the
characteristic of one type of dark energy, the so-called phantom and from
equation (64), for $C\rightarrow 0^{-}$, we have $w_{{}_{\rm
eff}}\rightarrow-1$.
Figure 1: The behavior of effective EOS, $w_{{}_{\rm eff}}$, as a function of
$z$. When $C=-1$ and $\Omega_{m}=0.33$, we get a phantom behavior in the
model.
To continue we consider the age of the Universe in $f(T)$ model. Thus, the age
of the matter dominated Universe in FLRW models is given by
$H_{0}t_{0}=\frac{2}{3}\frac{1}{\sqrt{1-\Omega_{m}}}\ln\left[\frac{1+\sqrt{1-\Omega_{m}}}{\sqrt{\Omega_{m}}}\right]$
(65)
where $H_{0}^{-1}=9.8\times 10^{9}h^{-1}$ years and the dimensionless
parameter $h$, according to resent data, is about $0.7$. Hence, in the flat
matter dominated universe with $\Omega_{total}=1$ the age of the universe
would be only 9.3 Gyr, whereas the oldest globular clusters yield an age of
about $\sim 13.5$ Gyr [29]. We obtain the age of the universe for our model
from equations (10), (52) and (59) by taking matter in the Friedmann equations
as follows
$H_{0}t_{0}=\frac{-2}{C}\frac{1}{\sqrt{1-\Omega_{m}}}\ln\left[\frac{1+\sqrt{1-\Omega_{m}}}{\sqrt{\Omega_{m}}}\right]$
(66)
For a flat, matter dominated Universe with $\Omega_{m}\approx 0.3$ and $C=-3$
we have a prediction for the age of the Universe of about $13.2$ Gyr. It seems
that the age of the universe in our model is longer than the FLRW model.
Figure 2 shows the behavior of the dimensionless age parameter, $H_{0}t_{0}$,
as a function of $\Omega_{m}$ for different values of $C$.
Figure 2: $H_{0}t_{0}$ as a function of $\Omega_{m}$ for $C=-1.9$ (solid
line), $C=-3$ (dashed line) and $C=-6$ (dotted line). Figure shows that for a
fixed value of $\Omega_{m}$ the predicted age of the universe is longer for
larger values of $C$.
Note added
Before sending our present work to the Archive (on 13 December 2011), we
became aware of the fact that there already existed a paper arXiv:1112.2270
[gr-qc][30] (sent to the Archive on 10 December 2011), in which the authors
had addressed similar issues. Although the content and results of the present
work overlap with the ones in [30], it is worth noting that our work was
completed independently.
## Acknowledgment
This work has been supported financially by Research Institute for Astronomy
and Astrophysics of Maragha (RIAAM) under research project No. 1/2361.
## 6 Conclusions
In this paper we have studied a generic $f(T)$ cosmological model by using
Noether symmetry approach. We have taken the background geometry as a flat
FLRW metric and derived the general equations of motion in this background.
The phase space has been constructed by taking the scale factor $a$ and
torsion scalar $T$ as two independent dynamical variables. The Lagrangian of
the model in the configuration space spanned by $\left\\{a,T\right\\}$ is so
constructed that its variation with respect to these dynamical variables
yields the correct field equations. The existence of Noether symmetry implies
that the Lie derivative of this Lagrangian with respect to the infinitesimal
generator of the desired symmetry vanishes. By applying this condition to the
Lagrangian of the model, we have obtained the explicit form of the
corresponding $f(T)$ function. We have shown that this form of $f(T)$ results
in a power law expansion for the scale factor and the accelerated expansion
occurs in $-2<C<0$
## References
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* [10] A. De Felice and S. Tsujikawa, Living Rev. Rel. 13 (2010) 3 .
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* [20] T. Wang, Phys. Rev. D 84 (2011) 024042.
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* [22] R. Aldrovandi and J. G. Pereira, An Introduction to Teleparallel Gravity. unpublished coursenotes. www.ift.unesp.br/gcg/tele.pdf
* [23] J. C. C. de Souza and V. Faraoni, Class. Quantum Grav. 24 (2007) 3637, arXiv: 0706.1223.
* [24] M. Demianski, R. de Ritis, C. Rubano and P. Scudellaro, Phys. Rev. D 46 (1992) 1391\.
* [25] B. Vakili, Phys. Lett. B 669 (2008) 206, arXiv:0809.4591.
* [26] S. Capozziello, A. De Felice, JCAP 0808 (2008) 016.
* [27] B. Vakili, N. Khosravi and H. R. Sepangi, Class. Quantum Grav. 24 (2007) 931, arXiv: gr-qc/0701075.
* [28] E. V. Linder and A. Jenkins, Mon. Not. R. Astron. Soc. 346 (2003) 573, arXiv: astroph/ 0305286\.
* [29] L. M. Krauss and B. Chaboyer, Science 299 (2003) 65.
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|
arxiv-papers
| 2011-12-13T09:03:01 |
2024-09-04T02:49:25.215961
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "K. Atazadeh, F. Darabi",
"submitter": "Khedmat Atazadeh",
"url": "https://arxiv.org/abs/1112.2824"
}
|
1112.2912
|
# Wavelet approach to operator-valued Hardy spaces
Guixiang Hong Laboratoire de Mathématiques, Université de Franche-Comté,
25030 Besançon Cedex, France
_E-mail address: guixiang.hong@univ-fcomte.fr_ and Zhi Yin Laboratoire de
Mathématiques, Université de Franche-Comté,
25030 Besançon Cedex, France
_E-mail address: hustyinzhi@163.com_
###### Abstract.
This paper is devoted to the study of operator-valued Hardy spaces via wavelet
method. This approach is parallel to that in noncommutative martingale case.
We show that our Hardy spaces defined by wavelet coincide with those
introduced by Tao Mei via the usual Lusin and Littlewood-Paley square
functions. As a consequence, we give an explicit complete unconditional basis
of the Hardy space $H_{1}(\mathbb{R})$ when $H_{1}(\mathbb{R})$ is equipped
with an appropriate operator space structure.
## 1\. Introduction
In this paper, we exploit Meyer’s wavelet methods to the study of the
operator-valued Hardy spaces. We are motivated by two rapidly developed
fields. The firs one is the theory of noncommutative martingales inequalities.
This theory had been already initiated in the 1970’s. Its modern period of
development has begun with Pisier and Xu’s seminal paper [20] in which the
authors established the noncommutative Burkholder-Gundy inequalities and
Fefferman duality theorem between $H_{1}$ and $BMO$. Since then many classical
results have been successfully transferred to the noncommutative world (see
[11], [14], [15], [1]). In particular, motivated by [9], Mei [15] developed
the theory of Hardy spaces on $\mathbb{R}^{n}$ for operator-valued functions.
Our second motivation is the theory of wavelets founded by Meyer. It is
nowadays well known that this theory is important for many domains, in
particular in harmonic analysis. For instance, it provides powerful tools to
the theory of Calderón-Zygmund singular integral operators. More recently,
Meyer’s wavelet methods were extended to study more sophistical subjects in
harmonic analysis. For example, the authors of [5] exploited the properties of
Meyer’s wavelets to give a characterization of product $BMO$ by commutators;
[17] deals with the estimates of bi-parameter paraproducts.
It is in this spirit that we wish to understand how useful wavelet methods are
for noncommutative analysis. The most natural and possible way would be first
to do this in the semi-commutative case. This is exactly the purpose of the
present paper which could be viewed as the first attempt towards the
development of wavelet techniques for noncommutative analysis.
A wavelet basis of $L_{2}({\mathbb{R}})$ is a complete orthonormal system
$(w_{I})_{I\in\mathcal{D}}$, where $\mathcal{D}$ denotes the collection of all
dyadic intervals in $\mathbb{R}$, $w$ is a Schwartz function satisfying the
properties needed for Meryer’s construction in [16], and
$w_{I}(x)\doteq\frac{1}{|I|^{\frac{1}{2}}}w\big{(}\frac{x-c_{I}}{|I|}\big{)},$
where $c_{I}$ is the center of $I$. The central facts that we will need about
the wavelet basis are the orthogonality between different $w_{I}$’s,
$\|w\|_{L_{2}({\mathbb{R}})}=1$ and the regularity of $w$,
$\max(|w(x)|,|w^{\prime}(x)|)\precsim(1+|x|)^{-m},\quad\forall m\geq 2.$
The analogy between wavelets and dyadic martingales is well known. The key
observation is the following parallelism:
$\sum_{|I|=2^{-n+1}}\langle f,w_{I}\rangle w_{I}\sim df_{n},$
where $df_{n}$ denotes $n$-th dyadic martingale difference of $f$. As dyadic
martingales are much easier to handle, this parallelism explains why wavelet
approach to many problems in harmonic analysis is usually simple and
efficient. On the other hand, it also indicates that martingale methods may be
used to deal with wavelets. With this in mind, we develop the operator-valued
Hardy spaces based on the wavelet methods in the way which is well known in
the noncommutative martingales case. Then we show that our Hardy and BMO
spaces coincide with Mei’s. In other words, we provide another approach, which
is much simpler than Mei’s original one, to recover all the results of [15].
This paper is organized as follows. In section 1, we will give some
preliminaries on noncommutative analysis, the definition of
$\mathcal{H}_{p}({\mathbb{R}},{\mathcal{M}})$ with $1\leq p<\infty$ and
$L_{q}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ with $2<q\leq\infty$ in our
setting. In section 2, we are concerned with three duality results. The most
important one is the noncommutative analogue of the famous Fefferman duality
theorem between $\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})$ and
$\mathcal{BMO}^{c}({\mathbb{R}},{\mathcal{M}})$. The second one is the duality
between $\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ and
$L^{c}_{p^{\prime}}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ with $1<p<2$,
where we need the noncommutative Doob’s inequality, this is why we consider
the case $1<p<2$ independently. The last one is the duality between
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ and
$\mathcal{H}^{c}_{p^{\prime}}({\mathbb{R}},{\mathcal{M}})$ with $1<p<\infty$.
As a corollary of the last two results, we identify
$\mathcal{H}^{c}_{q}({\mathbb{R}},{\mathcal{M}})$ and
$L^{c}_{q}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ with $2<q<\infty$.
Section 3 deals with the interpolation of our Hardy spaces. In the last
section, we show that our Hardy spaces coincide with those of [15]. So, we can
give an explicit completely unconditional basis for the space
$H_{1}({\mathbb{R}})$, when $H_{1}({\mathbb{R}})$ is equipped with an
appropriate operator space structure.
We end this introduction by the convention that throughout the paper the
letter $c$ will denote an absolute positive constant, which may vary from
lines to lines, and $c_{p}$ a positive constant depending only on $p$.
## 2\. Preliminaries
### 2.1. Operator-valued noncommutative $L_{p}$-spaces
Let $\mathcal{M}$ be a von Neumann algebra equipped with a normal semifinite
faithful trace $\tau$ and $S^{+}_{\mathcal{M}}$ be the set of all positive
element $x$ in $\mathcal{M}$ with $\tau(s(x))<\infty$, where $s(x)$ is the
smallest projection $e$ such that $exe=x$. Let $S_{\mathcal{M}}$ be the linear
span of $S^{+}_{\mathcal{M}}$. Then any $x\in S_{\mathcal{M}}$ has finite
trace, and $S_{\mathcal{M}}$ is a $w^{*}$-dense $*$-subalgebra of
$\mathcal{M}$.
Let $1\leq p<\infty$. For any $x\in S_{\mathcal{M}}$, the operator $|x|^{p}$
belongs to $S^{+}_{\mathcal{M}}$ ($|x|=(x^{*}x)^{\frac{1}{2}}$). We define
$\|x\|_{p}=\big{(}\tau(|x|^{p})\big{)}^{\frac{1}{p}},\qquad\forall x\in
S_{\mathcal{M}}.$
One can check that $\|\cdot\|_{p}$ is well defined and is a norm on
$S_{\mathcal{M}}$. The completion of $(S_{\mathcal{M}},\|\cdot\|_{p})$ is
denoted by $L_{p}({\mathcal{M}})$ which is the usual noncommutative $L_{p}$\-
space associated with $({\mathcal{M}},\tau)$. For convenience, we usually set
$L_{\infty}({\mathcal{M}})={\mathcal{M}}$ equipped with the operator norm
$\|\cdot\|_{{\mathcal{M}}}$. The elements of $L_{p}({\mathcal{M}},\tau)$ can
be described as closed densely defined operators on $H$ ($H$ being the Hilbert
space on which ${\mathcal{M}}$ acts). We refer the reader to [21] for more
information on noncommutative $L_{p}$-spaces.
In this paper, we are concerned with three operator-valued noncommutative
$L_{p}$-spaces. The first one is the Hilbert-valued noncommutative space
$L_{p}(\mathcal{M};H^{c})$ (resp. $L_{p}(\mathcal{M};H^{r})$), which is
studied at length in [9]. For this space, we need the following properties. In
the sequel, $p^{\prime}$ will always denote the conjugate index of $p$.
###### Lemma 2.1.
Let $1\leq p<\infty$. Then
(2.1) $(L_{p}(\mathcal{M};H^{c}))^{*}=L_{p^{\prime}}(\mathcal{M};H^{c}).$
Thus, for $f\in L_{p}(\mathcal{M};H^{c})$ and $g\in
L_{p^{\prime}}(\mathcal{M};H^{c})$, we have
$|\tau(\langle
f,g\rangle)|\leq\|f\|_{L_{p}(\mathcal{M};H^{c})}\|g\|_{L_{p^{\prime}}(\mathcal{M};H^{c})},$
where $\langle,\rangle$ denotes the inner product of $H$.
###### Lemma 2.2.
Let $1\leq p_{0}<p<p_{1}\leq\infty$, $0<\theta<1$,
$\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$. Then
(2.2)
$[L_{p_{0}}(\mathcal{M};H^{c}),L_{p_{1}}(\mathcal{M};H^{c})]_{\theta}=L_{p}(\mathcal{M};H^{c}).$
A same equality holds for row spaces.
The second one is the $\ell_{\infty}$-valued noncommutative space
$L_{p}(\mathcal{M};\ell_{\infty})$, which is studied by Pisier [19] for an
injective ${\mathcal{M}}$ and Junge [8] for a general ${\mathcal{M}}$ (see
also [11] and [13] for more properties). About this one, we need the following
property:
###### Lemma 2.3.
Let $1\leq p<\infty$. Then
$(L_{p}(\mathcal{M};\ell_{1}))^{*}=L_{p^{\prime}}(\mathcal{M};\ell_{\infty}).$
Thus, for $x=(x_{n})_{n}\in L_{p}(\mathcal{M};\ell_{1})$ and $y=(y_{n})_{n}\in
L_{p^{\prime}}(\mathcal{M};\ell_{\infty})$, we have
(2.3) $\big{|}\sum_{n\geq
1}\tau(x_{n}y_{n})\big{|}\leq\|x\|_{L_{p}(\mathcal{M};\ell_{1})}\|y\|_{L_{p^{\prime}}(\mathcal{M};\ell_{\infty})}.$
The third one is $L_{p}(\mathcal{M};\ell^{c}_{\infty})$ for $2\leq
p\leq\infty$, which was introduced in [4] and is related with the second one
by
$\|(x_{n})_{n}\|_{L_{p}(\mathcal{M};\ell^{c}_{\infty})}=\|(|x_{n}|^{2})_{n}\|_{L_{\frac{p}{2}}({\mathcal{M}};\ell_{\infty})}.$
And these are normed spaces by the following characterization
$\|(x_{n})_{n}\|_{L_{p}(\mathcal{M};\ell^{c}_{\infty})}=\inf_{x_{n}=y_{n}a}\|(y_{n})\|_{\ell_{\infty}(L_{\infty}({\mathcal{M}}))}\|a\|_{L_{p}({\mathcal{M}})}.$
We need the interpolation results about these spaces (see [18]):
###### Lemma 2.4.
Let $2\leq p_{0}<p<p_{1}\leq\infty$, $0<\theta<1$,
$\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$. Then
(2.4)
$[L_{p_{0}}(\mathcal{M};\ell^{c}_{\infty}),L_{p_{1}}(\mathcal{M};\ell^{c}_{\infty})]_{\theta}=L_{p}(\mathcal{M};\ell^{c}_{\infty}).$
### 2.2. Operator-valued Hardy spaces
In this paper, for simplicity, we denote
$L_{\infty}({\mathbb{R}})\bar{\otimes}{\mathcal{M}}$ by $\mathcal{N}$. As
indicated in the introduction, one can observe that we have the following
operator-valued Calderón identity
(2.5) $f(x)=\sum_{I\in\mathcal{D}}\langle f,w_{I}\rangle w_{I}(x),$
which holds when $f\in L_{2}({\mathcal{N}})$. As in the classical case, for
$f\in S_{{\mathcal{N}}}$, we define the two Littlewood-Paley square functions
as
(2.6) $S_{c}(f)(x)=\Big{(}\sum_{I\in\mathcal{D}}\frac{|\langle
f,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}(x)\Big{)}^{\frac{1}{2}}.$ (2.7)
$S_{r}(f)(x)=\Big{(}\sum_{I\in\mathcal{D}}\frac{|\langle
f^{*},w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}(x)\Big{)}^{\frac{1}{2}}.$
For $1\leq p<\infty$, define
$\|f\|_{\mathcal{H}^{c}_{p}}=\|S_{c}(f)\|_{L_{p}({\mathcal{N}})},$
$\|f\|_{\mathcal{H}^{r}_{p}}=\|S_{r}(f)\|_{L_{p}({\mathcal{N}})}.$
These are norms, which can be seen easily from the space
$L_{p}(\mathcal{N};\ell^{c}_{2}(\mathcal{D}))$. So we define the spaces
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ (resp.
$\mathcal{H}^{r}_{p}({\mathbb{R}},{\mathcal{M}})$) as the completion of
$(S_{\mathcal{N}},\|\cdot\|_{\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})})$
(resp.
$(S_{\mathcal{N}},\|\cdot\|_{\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})})$.
Now, we define the operator-valued Hardy spaces as follows: for $1\leq p<2,$
(2.8)
$\mathcal{H}_{p}({\mathbb{R}},{\mathcal{M}})=\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})+\mathcal{H}^{r}_{p}({\mathbb{R}},{\mathcal{M}})$
with the norm
$\|f\|_{\mathcal{H}_{p}}=\inf\\{\|g\|_{\mathcal{H}^{c}_{p}}+\|h\|_{\mathcal{H}^{r}_{p}}:f=g+h,g\in\mathcal{H}^{c}_{p},h\in\mathcal{H}^{r}_{p}\\}$
and for $2\leq p<\infty$,
(2.9)
$\mathcal{H}_{p}({\mathbb{R}},{\mathcal{M}})=\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})\cap\mathcal{H}^{r}_{p}({\mathbb{R}},{\mathcal{M}})$
with the norm defined as
$\|f\|_{\mathcal{H}_{p}}=\max\\{\|f\|_{\mathcal{H}^{c}_{p}},\|f\|_{\mathcal{H}^{r}_{p}}\\}.$
We can identify $\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ as a
subspace of $L_{p}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$, which is related
with the two maps below.
###### Lemma 2.5.
$\rm(i)$ The embedding map $\Phi$ is defined from
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ to
$L_{p}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$ by
(2.10) $\Phi(f)=\Big{(}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\Big{)}_{I\in\mathcal{D}}.$
$\rm(ii)$ The projection map $\Psi$ is defined from
$L_{2}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$ to
$\mathcal{H}^{c}_{2}({\mathbb{R}},{\mathcal{M}})$ by
(2.11)
$\Psi((g_{I}))=\sum_{I\in\mathcal{D}}\int\frac{g_{I}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}dy\cdot
w_{I}.$
### 2.3. Operator-valued $\mathcal{BMO}$ spaces
For $\varphi\in
L_{\infty}({\mathcal{M}};L^{c}_{2}({\mathbb{R}},\frac{dx}{1+x^{2}}))$, set
(2.12)
$\|\varphi\|_{{\mathcal{BMO}}^{c}}=\sup_{J\in\mathcal{D}}\Big{\|}\big{(}\frac{1}{|J|}\sum_{I\subset
J}|\langle\varphi,w_{I}\rangle|^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{{\mathcal{M}}}$
and
$\|\varphi\|_{\mathcal{BMO}^{r}}=\|\varphi^{*}\|_{\mathcal{BMO}^{c}({\mathbb{R}},{\mathcal{M}})}.$
These are again norms modulo constant functions. Define
${\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}})=\\{\varphi\in
L_{\infty}({\mathcal{M}};L^{c}_{2}({\mathbb{R}},\frac{dx}{1+x^{2}})):\|\varphi\|_{{\mathcal{BMO}}^{c}}<\infty\\}$
and
${\mathcal{BMO}}^{r}({\mathbb{R}},{\mathcal{M}})=\\{\varphi\in
L_{\infty}({\mathcal{M}};L^{r}_{2}({\mathbb{R}},\frac{dx}{1+x^{2}})):\|\varphi\|_{{\mathcal{BMO}}^{r}}<\infty\\}$
Now we define
$\mathcal{BMO}({\mathbb{R}},{\mathcal{M}})=\mathcal{BMO}^{c}({\mathbb{R}},{\mathcal{M}})\cap\mathcal{BMO}^{r}({\mathbb{R}},{\mathcal{M}}).$
As in the martingale case [11], we can also define
$L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ for all $2<p\leq\infty$.
For $\varphi\in
L_{p}({\mathcal{M}};L^{c}_{2}({\mathbb{R}},\frac{dx}{1+x^{2}}))$, set
(2.13)
$\|\varphi\|_{L^{c}_{p}{\mathcal{MO}}}=\Big{\|}(\frac{1}{|I^{x}_{k}|}\sum_{I\subset
I^{x}_{k}}|\langle\varphi,w_{I}\rangle|^{2})_{k}\Big{\|}_{L_{\frac{p}{2}}({\mathcal{N}};\ell_{\infty})}$
and
$\|\varphi\|_{L^{r}_{p}{\mathcal{MO}}}=\|\varphi^{*}\|_{L^{c}_{p}{\mathcal{MO}}},$
where $I^{x}_{k}$ denotes the unique dyadic interval with length $2^{-k+1}$
that containing $x$. We will use the convention adopted in [13] for the norm
in $L_{\frac{p}{2}}({\mathcal{N}};\ell_{\infty}).$ Thus
$\Big{\|}(\frac{1}{|I^{x}_{k}|}\sum_{I\subset
I^{x}_{k}}|\langle\varphi,w_{I}\rangle|^{2})_{k}\Big{\|}_{L_{\frac{p}{2}}({\mathcal{N}};\ell_{\infty})}=\Big{\|}{\sup_{k}}^{+}\frac{1}{|I^{x}_{k}|}\sum_{I\subset
I^{x}_{k}}|\langle\varphi,w_{I}\rangle|^{2}\Big{\|}_{L_{\frac{p}{2}({\mathcal{N}})}}$
These are norms, which can be seen from the Banach spaces
$L_{p}(\mathcal{N}\bar{\otimes}B(\ell_{2}(\mathcal{D}));\ell^{c}_{\infty})$.
Again, we can define
$L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})=\\{\varphi\in
L_{p}({\mathcal{M}};L^{c}_{2}({\mathbb{R}},\frac{dx}{1+x^{2}})):\|\varphi\|_{L^{c}_{p}{\mathcal{MO}}}<\infty\\}$
and
$L^{r}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})=\\{\varphi\in
L_{p}({\mathcal{M}};L^{r}_{2}({\mathbb{R}},\frac{dx}{1+x^{2}})):\|\varphi\|_{L^{c}_{r}{\mathcal{MO}}}<\infty\\}$
Define
$L_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})=L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})\cap
L^{r}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}}).$
Note that
$L^{c}_{\infty}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})={\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}})$.
It is easy to check all the spaces we defined here respect to the relevant
norms are Banach spaces.
## 3\. Duality
To prove the first two duality results in this section, we need the following
noncommutative Doob’s inequality from [8].
Let $(\mathcal{E}_{n})_{n}$ be the conditional expectation with respect to a
filtration $({{\mathcal{N}}}_{n})_{n}$ of ${\mathcal{N}}$.
###### Lemma 3.1.
Let $1<p\leq\infty$ and $f\in L_{p}({\mathcal{N}})$. Then
(3.1) $\|{\sup_{n}}^{+}\mathcal{E}_{n}(f)\|_{L_{p}({\mathcal{N}})}\leq
c_{p}\|f\|_{L_{p}({\mathcal{N}})}.$
###### Theorem 3.1.
We have
(3.2)
$(\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}}))^{*}={\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}})$
with equivalent norms. That is, every
$\varphi\in\mathcal{BMO}^{c}({\mathbb{R}},{\mathcal{M}})$ induces a continuous
linear functional $l_{\varphi}$ on
$\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})$ by
(3.3) $l_{\varphi}(f)=\tau\int\varphi^{*}f,\quad\forall f\in S_{\mathcal{N}}.$
Conversely, for every
$l\in(\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}}))^{*}$, there exits a
$\varphi\in\mathcal{BMO}^{c}({\mathbb{R}},{\mathcal{M}})$ such that
$l=l_{\varphi}$. Moreover,
$c^{-1}\|\varphi\|_{\mathcal{BMO}^{c}}\leq\|l_{\varphi}\|_{(\mathcal{H}^{c}_{1})^{*}}\leq
c\|\varphi\|_{\mathcal{BMO}^{c}}$
where $c>0$ is a universal constant.
Similarly, the duality holds between $\mathcal{H}^{r}_{1}$ and
$\mathcal{BMO}^{r}$, between $\mathcal{H}_{1}$ and $\mathcal{BMO}$ with
equivalent norms.
In order to adapt the arguments in the martingale case, we need to define the
truncated square functions for $n\in\mathbb{Z}$,
$S_{c,n}(f)(x)=\Big{(}\sum^{n}_{k=-\infty}\sum_{|I|=2^{-k+1}}\frac{|\langle
f,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}(x)\Big{)}^{\frac{1}{2}}.$
###### Proof.
Since $S_{\mathcal{N}}$ is dense in
$\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})$, by an approximation
argument, we only need to prove the inequality
$|l_{\varphi}(f)|\leq
c\|\varphi\|_{\mathcal{BMO}^{c}}\|f\|_{\mathcal{H}^{c}_{1}}$
for $f\in S_{\mathcal{N}}$. By approximation we may assume that
$S_{c,n}(f)(x)$ is invertible in ${\mathcal{M}}$ for all $x\in{\mathbb{R}}$
and $n\in\mathbb{Z}$. Then we have
$\displaystyle|l_{\varphi}(f)|$ $\displaystyle=|\tau\int\varphi^{*}fdx|$
$\displaystyle=\Big{|}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\langle\varphi,w_{I}\rangle^{*}w_{I}\sum_{|I^{\prime}|=2^{-n+1}}\langle
f,w_{I^{\prime}}\rangle w_{I^{\prime}}dx\Big{|}$
$\displaystyle=\Big{|}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\frac{\langle\varphi,w_{I}\rangle^{*}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\sum_{|I^{\prime}|=2^{-n+1}}\frac{\langle
f,w_{I^{\prime}}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I^{\prime}}dx\Big{|}$
$\displaystyle\leq\sum_{n}\Big{(}\tau\int\big{|}\sum_{|I|=2^{-n+1}}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\big{|}^{2}S^{-1}_{c,n}(f)\Big{)}^{\frac{1}{2}}$
$\displaystyle\qquad\cdot\Big{(}\tau\int\big{|}\sum_{|I|=2^{-n+1}}\frac{\langle\varphi,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\big{|}^{2}{S_{c,n}(f)}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq\Big{(}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\frac{|\langle
f,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}S^{-1}_{c,n}(f)\Big{)}^{\frac{1}{2}}$
$\displaystyle\qquad\cdot\Big{(}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\frac{|\langle\varphi,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}{S_{c,n}(f)}\Big{)}^{\frac{1}{2}}$
$\displaystyle=A\cdot B.$
In the above estimates, the first equality has used the orthogonality of the
$w_{I}$’s on different levels, the second one the orthogonality of the
$w_{I}$’s on the same level and the disjoint of different dyadic $I$’s on the
same level; the first inequality has used the Hölder inequality in Lemma 2.1,
and the second one the Cauchy-Schwarz inequality and the disjointness of
different $I$’s on the same level.
Now, let us estimate $A$:
$\displaystyle\begin{split}A^{2}&=\sum_{n}\tau\int(S^{2}_{c,n}(f)-S^{2}_{c,n-1}(f))S^{-1}_{c,n}(f)\\\
&=\sum_{n}\tau\int(S_{c,n}(f)-S_{c,n-1}(f))(1+S_{c,n-1}(f)S^{-1}_{c,n}(f))\\\
&\leq\sum_{n}\tau\int(S_{c,n}(f)-S_{c,n-1}(f))\|1+S_{c,n-1}(f)S^{-1}_{c,n}(f)\|_{\infty}\\\
&\leq 2\sum_{n}\tau\int(S_{c,n}(f)-S_{c,n-1}(f))\\\
&=2\|f\|_{\mathcal{H}^{c}_{1}}.\end{split}$
For the first inequality, we have used the Hölder inequality and the
positivity of $S_{c,n}(f)-S_{c,n-1}(f)$.
The second term is estimated as follows:
$\displaystyle\begin{split}B^{2}&=\sum_{k}\tau\int(S_{c,k}(f)-S_{c,k-1}(f))\sum_{n\geq
k}\sum_{|I|=2^{-n+1}}\frac{|\langle\varphi,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}\\\
&=\sum_{k}\tau\sum_{j}(S_{c,k}(f)-S_{c,k-1}(f))\int_{I^{j}_{k}}\sum_{n\geq
k}\sum_{|I|=2^{-n+1}}\frac{|\langle\varphi,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}\\\
&=\sum_{k}\tau\sum_{j}\int_{I^{j}_{k}}(S_{c,k}(f)-S_{c,k-1}(f))\frac{1}{|I^{j}_{k}|}\sum_{I\subset
I^{j}_{k}}|\langle\varphi,w_{I}\rangle|^{2}\\\
&\leq\sum_{k}\sum_{j}\tau\int_{I^{j}_{k}}(S_{c,k}(f)-S_{c,k-1}(f))\Big{\|}\frac{1}{|I^{j}_{k}|}\sum_{I\subset
I^{j}_{k}}|\langle\varphi,w_{I}\rangle|^{2}\Big{\|}_{\infty}\\\
&\leq\|\varphi\|^{2}_{{\mathcal{BMO}}^{c}}\sum_{k}\sum_{j}\tau\int_{I^{j}_{k}}(S_{c,k}(f)-S_{c,k-1}(f))\\\
&=\|\varphi\|^{2}_{{\mathcal{BMO}}^{c}}\|f\|_{\mathcal{H}^{c}_{1}}\\\
\end{split}$
The fist equality has used the Fubini theorem, the second one the fact that
$S_{c,k-1}(f)$ and $S_{c,k}(f)$ are constant on the dyadic interval
$I^{j}_{k}=[j2^{-k+1},(j+1)2^{-k+1})$; the first inequality has used the
Hölder inequality and the positivity of $S_{c,n}(f)-S_{c,n-1}(f)$.
Now, let us begin to deal with another direction, i.e. suppose that $l$ is a
bounded linear functional on
$\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})$, we want to find an
operator-valued function $\varphi$ in
${\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}})$, such that $l=l_{\varphi}$
and $l_{\varphi}(f)=\tau\int\varphi^{*}f$ for $f\in
S_{\mathcal{{\mathcal{N}}}}$. By the embedding operator $\Phi$ in (2.10) and
by the Banach-Hahn theorem, $l$ extends to a bounded continuous functional on
$L_{1}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$ of the same norm. Then by the
results in Lemma 2.1 there exists $g=(g_{I})_{I\in\mathcal{D}}$ such that
$\|g\|_{L_{\infty}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))}=\|l\|$, and
$l(f)=\tau\int\sum_{I\in\mathcal{D}}g^{*}_{I}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I},\quad\forall f\in
S_{\mathcal{N}}.$
Now let $\varphi=\Psi(g)$, where $\Psi$ is defined as (2.11). The
orthogonality of the $w_{I}$’s yields
$\displaystyle\begin{split}\big{\|}\sum_{I\subset
J}|\langle\varphi,w_{I}\rangle|^{2}\big{\|}_{{\mathcal{M}}}&=\big{\|}\sum_{I\subset
J}|\int\frac{g_{I}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}|^{2}\big{\|}_{{\mathcal{M}}}\leq\big{\|}\sum_{I\subset
J}\int_{J}|g_{I}|^{2}\big{\|}_{{\mathcal{M}}}\\\
&\leq|J|\big{\|}\sum_{I\subset
J}|g_{I}|^{2}\big{\|}_{L_{\infty}({\mathcal{N}})}\leq|J|\big{\|}(g_{I})_{I}\big{\|}_{L_{\infty}(\mathcal{N};\ell^{c}_{2}(\mathcal{D}))},\\\
\end{split}$
where the first inequality has used the Kadison-Schwartz inequality. Also
thanks to the orthogonality of the $w_{I}$’s, we get
$\displaystyle\begin{split}l(f)&=\tau\int\sum_{I\in\mathcal{D}}g^{*}_{I}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}=\tau\int\varphi^{*}f\\\
\end{split}$
for all $f\in S_{\mathcal{N}}$. Therefore, we complete the proof about
$\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})$ and
$\mathcal{BMO}^{c}({\mathbb{R}},{\mathcal{M}})$. Passing to adjoint, we have
the conclusion concerning $\mathcal{H}^{r}_{1}$ and $\mathcal{BMO}^{r}$.
Finally, by the classical fact that the dual of a sum space is the
intersection space, we obtain the duality between $\mathcal{H}_{1}$ and
$\mathcal{BMO}$. ∎
###### Theorem 3.2.
Let $1<p<2$. We have
(3.4)
$(\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}}))^{*}=L^{c}_{p^{\prime}}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$
with equivalent norms. That is, every $\varphi\in
L^{c}_{p^{\prime}}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ induces a
continuous linear functional $l_{\varphi}$ on
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ by
(3.5) $l_{\varphi}(f)=\tau\int\varphi^{*}f,\quad\forall f\in S_{\mathcal{N}}.$
Conversely, for every
$l\in(\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}}))^{*}$, there exists an
operator-valued function $\varphi\in
L^{c}_{p^{\prime}}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ such that
$l=l_{\varphi}$ and
$c_{p}^{-1}\|\varphi\|_{L^{c}_{p^{\prime}}{\mathcal{MO}}}\leq\|l_{\varphi}\|_{(\mathcal{H}^{c}_{p})^{*}}\leq\sqrt{2}\|\varphi\|_{L^{c}_{p^{\prime}}{\mathcal{MO}}}$
Similarly, the duality holds between $\mathcal{H}^{r}_{p}$ and
$L^{r}_{p^{\prime}}$, between $\mathcal{H}_{p}$ and
$L_{p^{\prime}}{\mathcal{MO}}$ with equivalent norms.
We need the following lemma of [11]. We write it down for the reader’s
convenience but without proof.
###### Lemma 3.2.
Let $s,t$ be two real numbers such that $s<t$ and $0\leq s\leq 1\leq t\leq 2$.
Let $x,y$ be two positive operators such that $x\leq y$ and
$x^{t-s},y^{t-s}\in L_{1}({\mathcal{N}})$. Then
$\tau\int y^{-s/2}(y^{t}-x^{t})y^{-s/2}\leq 2\tau\int
y^{-(s+1-t)/2}(y-x)y^{-(s+1-t)/2}.$
###### Proof.
We need only to prove the first assertion on $\mathcal{H}^{c}_{p}$. Since
$S_{\mathcal{N}}$ is dense in
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$, by an approximation
argument, we only need to prove the inequality
$|l_{\varphi}(f)|\leq
c\|\varphi\|_{L^{c}_{p^{\prime}}{\mathcal{MO}}}\|f\|_{\mathcal{H}^{c}_{p}}$
for $f\in S_{\mathcal{N}}$. By approximation we may assume that
$S_{c,n}(f)(x)$ is invertible in ${\mathcal{M}}$ for all $x\in{\mathbb{R}}$
and $n\in\mathbb{Z}$. By the similar principle in the noncommutative
martingale case as in [11], we have
$\displaystyle\begin{split}|l_{\varphi}(f)|&=|\tau\int\varphi^{*}fdx|\\\
&=\Big{|}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\langle\varphi,w_{I}\rangle^{*}w_{I}\sum_{|I^{\prime}|=2^{-n+1}}\langle
f,w_{I^{\prime}}\rangle w_{I^{\prime}}dx\Big{|}\\\
&=\Big{|}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\frac{\langle\varphi,w_{I}\rangle^{*}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\sum_{|I^{\prime}|=2^{-n+1}}\frac{\langle
f,w_{I^{\prime}}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I^{\prime}}dx\Big{|}\\\
&\leq\sum_{n}\Big{(}\tau\int\big{|}\sum_{|I|=2^{-n+1}}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\big{|}^{2}S^{p-2}_{c,n}(f)\Big{)}^{\frac{1}{2}}\\\
&\qquad\cdot\Big{(}\tau\int\big{|}\sum_{|I|=2^{-n+1}}\frac{\langle\varphi,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\big{|}^{2}{S^{2-p}_{c,n}(f)}\Big{)}^{\frac{1}{2}}\\\
&\leq\Big{(}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\frac{|\langle
f,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}S^{p-2}_{c,n}(f)\Big{)}^{\frac{1}{2}}\\\
&\qquad\cdot\Big{(}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\frac{|\langle\varphi,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}{S^{2-p}_{c,n}(f)}\Big{)}^{\frac{1}{2}}\\\
&=A\cdot B.\\\ \end{split}$
Now we need the above lemma to estimate the first term. Take $s=2-p$ and
$t=2$, the lemma yields
$\displaystyle A^{2}$
$\displaystyle=\sum_{n}\tau\int(S^{2}_{c,n}(f)-S^{2}_{c,n-1}(f))S^{p-2}_{c,n}(f)$
$\displaystyle=\sum_{n}\tau\int
S^{-(2-p)/2}_{c,n}(f)(S^{2}_{c,n}(f)-S^{2}_{c,n-1}(f))S^{-(2-p)/2}_{c,n}(f)$
$\displaystyle\leq 2\sum_{n}\tau\int
S^{-(1-p)/2}_{c,n}(f)(S_{c,n}(f)-S_{c,n-1}(f))S^{-(1-p)/2}_{c,n}(f)$
$\displaystyle=2\sum_{n}\tau\int S_{c,n}(f)-S_{c,n-1}(f)S^{p-1}_{c,n}(f)$
$\displaystyle\leq 2\sum_{n}\tau\int S^{p}_{c,n}(f)-S^{p}_{c,n-1}(f)$
$\displaystyle=2\|f\|^{p}_{\mathcal{H}^{c}_{p}}.$
The last inequality has used two elementary inequalities: $0\leq
S_{c,n-1}(f)\leq S_{c,n}(f)$ implies $S^{p-1}_{c,n-1}(f)\leq S^{p-1}_{c,n}(f)$
for $0<p-1<1$; and
$\tau(S^{p}_{c,n-1}(f))\leq\tau(S^{\frac{1}{2}}_{c,n-1}(f)S^{p-1}_{c,n}(f)S^{\frac{1}{2}}_{c,n-1}(f)).$
The second term can be deduced from the nontrivial duality results in Lemma
2.3 for $1<p<\infty$ as follows.
$\displaystyle\begin{split}B^{2}&=\sum_{k}\tau\int
S^{2-p}_{c,k}(f)-S^{2-p}_{c,k-1}(f)\sum_{n\geq
k}\sum_{|I|=2^{-n+1}}\frac{|\langle\varphi,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}\\\
&=\sum_{k}\tau\sum_{j}S^{2-p}_{c,k}(f)-S^{2-p}_{c,k-1}(f)\int_{I^{j}_{k}}\sum_{n\geq
k}\sum_{|I|=2^{-n+1}}\frac{|\langle\varphi,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}\\\
&=\sum_{k}\tau\sum_{j}\int\mathds{1}_{I^{j}_{k}}(x)S^{2-p}_{c,k}(f)(x)-S^{2-p}_{c,k-1}(f)(x)\frac{1}{|I^{j}_{k}|}\sum_{I\subset
I^{j}_{k}}|\langle\varphi,w_{I}\rangle|^{2}dx\\\ &=\sum_{k}\tau\int
S^{2-p}_{c,k}(f)(x)-S^{2-p}_{c,k-1}(f)(x)\frac{1}{|I^{x}_{k}|}\sum_{I\subset
I^{x}_{k}}|\langle\varphi,w_{I}\rangle|^{2}dx\\\
&\leq\|\sum_{k}S^{2-p}_{c,k}(f)-S^{2-p}_{c,k-1}(f)\|_{L_{({p^{\prime}}/2)^{\prime}}}\Big{\|}\sup_{k}\frac{1}{|I^{x}_{k}|}\sum_{I\subset
I^{x}_{k}}|\langle\varphi,w_{I}\rangle|^{2}\Big{\|}_{L_{{p^{\prime}}/2}}\\\
&=\|\varphi\|^{2}_{L^{c}_{p^{\prime}}{\mathcal{MO}}}\|f\|^{2-p}_{\mathcal{H}^{c}_{p}}\end{split}$
The fist equality has used the Fubini theorem, the second one the fact that
$S_{c,k-1}(f)$ and $S_{c,k}(f)$ are constant on the dyadic intervals with
length $2^{-k+1}$.
For another direction, we can carry out the proof as that in the case $p=1$.
Suppose that $l$ is a bounded linear functional on
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$. By the embedding operator
$\Phi$ and by Hahn-Banach theorem, and the results in Lemma 2.1, we can find
$g=(g_{I})_{I\in\mathcal{D}}$ such that
$\|g\|_{L_{p^{\prime}}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))}=\|l\|$ and
$l(f)=\tau\int\sum_{I\in\mathcal{D}}g^{*}_{I}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I},\forall f\in
S_{\mathcal{N}}.$
Now let $\varphi=\Psi(g)$ defined in (2.11), the orthogonality of the
$w_{I}$’s yields
$\displaystyle\begin{split}\big{\|}{\sup_{n}}^{+}&\frac{1}{|I^{x}_{n}|}\sum_{I\subset
I^{x}_{n}}|\langle\varphi,w_{I}\rangle|^{2}\big{\|}_{L_{p^{\prime}/2}({\mathcal{N}})}\\\
&=\big{\|}{\sup_{n}}^{+}\frac{1}{|I^{x}_{n}|}\sum_{I\subset
I^{x}_{n}}|\int\frac{g_{I}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}|^{2}\big{\|}_{L_{p^{\prime}/2}({\mathcal{N}})}\\\
&\leq\big{\|}{\sup_{n}}^{+}\frac{1}{|I^{x}_{n}|}\sum_{I\subset
I^{x}_{n}}\int_{I^{x}_{n}}|g_{I}|^{2}\big{\|}_{L_{p^{\prime}/2}({\mathcal{N}})}\\\
&\leq\big{\|}{\sup_{n}}^{+}\frac{1}{|I^{x}_{n}|}\int_{I^{x}_{n}}\sum_{I\subset
I^{x}_{n}}|g_{I}|^{2}\big{\|}_{L_{p^{\prime}/2}({\mathcal{N}})}\\\
&\leq\big{\|}{\sup_{n}}^{+}\frac{1}{|I^{x}_{n}|}\int_{I^{x}_{n}}\sum_{I\in\mathcal{D}}|g_{I}|^{2}\big{\|}_{L_{p^{\prime}/2}({\mathcal{N}})}\\\
&\leq
c\big{\|}\sum_{I\in\mathcal{D}}|g_{I}|^{2}\big{\|}_{L_{p^{\prime}/2}({\mathcal{N}})}\\\
&=c\big{\|}(g_{I})_{I}\big{\|}_{L_{p^{\prime}}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))},\\\
\end{split}$
where for the first inequality we have used the Kadison-Schwartz inequality,
and the last inequality is (3.1). Also due to the orthogonality of the
$w_{I}$’s, we get
$\displaystyle\begin{split}l(f)&=\tau\int\sum_{I\in\mathcal{D}}g^{*}_{I}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}=\tau\int\varphi^{*}f,\\\
\end{split}$
for all $f\in S_{\mathcal{N}}$. Therefore, we complete the proof about
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ and
$L^{c}_{p^{\prime}}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$. ∎
Instead of using the noncommutative Doob’s inequality, we will use the
following noncommutative Stein inequality from [20] to prove the duality
between the spaces $\mathcal{H}^{c}_{p}$, $1<p<\infty$.
Let $(\mathcal{E}_{n})_{n}$ be the conditional expectation with respect to a
filtration $({{\mathcal{N}}}_{n})_{n}$ of ${\mathcal{N}}$.
###### Lemma 3.3.
Let $1<p<\infty$ and $a=(a_{n})_{n}\in L_{p}({\mathcal{N}};\ell^{c}_{2})$.
Then there exists a constant depending only on $p$ such that
(3.6)
$\Big{\|}\big{(}\sum_{n}|\mathcal{E}_{n}a_{n}|^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{L_{p}({\mathcal{N}})}\leq
c_{p}\Big{\|}\big{(}\sum_{n}|a_{n}|^{2}\big{)}^{\frac{1}{2}}\Big{\|}_{L_{p}({\mathcal{N}})}.$
###### Theorem 3.3.
For any $1<p<\infty$, we have
(3.7)
$(\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}}))^{*}=\mathcal{H}^{c}_{p^{\prime}}({\mathbb{R}},{\mathcal{M}}),$
###### Proof.
By a similar reason as in the corresponding part of the proof of Theorem 3.1,
we can carry out the following calculation,
$\displaystyle\begin{split}|l_{\varphi}(f)|&=|\tau\int\varphi^{*}fdx|\\\
&=\Big{|}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\langle\varphi,w_{I}\rangle^{*}w_{I}\sum_{|I^{\prime}|=2^{-n+1}}\langle
f,w_{I^{\prime}}\rangle w_{I^{\prime}}dx\Big{|}\\\
&=\Big{|}\sum_{n}\tau\int\sum_{|I|=2^{-n+1}}\frac{\langle\varphi,w_{I}\rangle^{*}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}dx\Big{|}\\\
&\leq\big{\|}\big{(}\sum_{I\in\mathcal{D}}\frac{|\langle
f,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}\big{)}^{\frac{1}{2}}\big{\|}_{L_{p}({\mathbb{R}},{\mathcal{M}})}\cdot\big{\|}\big{(}\sum_{I\in\mathcal{D}}\frac{|\langle\varphi,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}\big{)}^{\frac{1}{2}}\big{\|}_{L_{p^{\prime}}({\mathbb{R}},{\mathcal{M}})}.\\\
\end{split}$
Now, we turn to the proof of the inverse direction. Take a bounded linear
functional $l\in(\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}}))^{*}$, by the
embedding operator $\Phi$ and the Hahn-Banach extension theorem, $l$ extends
to a bounded linear functional on $L_{p}({\mathcal{N}};\ell^{c}_{2})$ with the
same norm. Thus by (2.1), there exists a sequence $g=(g_{I})_{I}$ such that
$\|g\|_{L_{q}({\mathcal{N}};l^{c}_{2}(\mathcal{D}))}=\|l\|$
and
$l(f)=\tau\int\sum_{I\in\mathcal{D}}g^{*}_{p}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I},\forall f\in
S_{\mathcal{N}}.$
Now let $\varphi=\Psi(g)$ where $\Psi$ is defined in (2.11), then applying the
Stein inequality (3.3) to the conditional expectation
$\mathcal{E}_{I}(h)=\sum_{J}\frac{1}{|J|}\int_{J}h(y)dy\cdot\mathds{1}_{J},$
where $J$ is dyadic interval with the same length as $I$, we get
$\displaystyle\begin{split}\|\varphi\|_{\mathcal{H}^{c}_{p^{\prime}}({\mathbb{R}},{\mathcal{M}})}&=\|\big{(}\sum_{I\in\mathcal{D}}|\frac{1}{|I|}\int_{I}g_{I}dy\cdot\mathds{1}_{I}|^{2}\big{)}^{\frac{1}{2}}\|_{L_{p^{\prime}}({\mathcal{N}})}\\\
&\leq\|\big{(}\sum_{I\in\mathcal{D}}|\mathcal{E}_{I}(g_{I})|^{2}\big{)}^{\frac{1}{2}}\|_{L_{p^{\prime}}({\mathcal{N}})}\\\
&\leq
c_{p^{\prime}}\|\big{(}\sum_{I\in\mathcal{D}}|g_{I}|^{2}\big{)}^{\frac{1}{2}}\|_{L_{p^{\prime}}({\mathcal{N}})}.\\\
\end{split}$
By the orthogonality of the $w_{I}$’s, we have
$\displaystyle\begin{split}l(f)=\tau\int\sum_{I\in\mathcal{D}}g^{*}_{I}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}=\tau\int\varphi^{*}f,\\\
\end{split}$
for all $f\in S_{\mathcal{N}}$. ∎
From the proof of the second part of Theorem 3.1, Theorem 3.2 and Theorem 3.3,
we state the boundedness of $\Psi$ as a corollary.
###### Corollary 3.1.
$\rm(i)$ Let $1<p<\infty$, $\Psi$ is a projection map from
$L_{p}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$ onto
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ if we identify the latter as
a subspace of the former.
$\rm(ii)$ Let $2<p\leq\infty$, $\Psi$ is also a bounded map from
$L_{p}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$ to
$L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$.
Theorem 3.2 and Theorem 3.3 immediately imply the following corollary:
###### Corollary 3.2.
Let $2<p<\infty$. Then
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})=L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}}),\quad\forall
2<p<\infty$
with equivalent norms.
However, for the part
$L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})\subset\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$,
we can give another proof. The idea is essentially similar to that in [15],
the good news is that in our case, the argument seems very elegant. Now we
give the detailed proof.
###### Proof.
Our tent space is defined as
$\displaystyle T^{c}_{p}=\Big{\\{}f=\\{f_{I}\\}_{I}\in
L_{p}({\mathcal{M}};\ell^{c}_{2}(\mathcal{D})):\quad\tau\int\big{(}\sum_{I\in\mathcal{D}}\frac{f^{2}_{I}}{|I|}\mathds{1}_{I}\big{)}^{\frac{p}{2}}<\infty\Big{\\}}$
We claim that every $\varphi\in
L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ induces a bounded linear
functional on $T^{c}_{p^{\prime}}$,
$\displaystyle
l_{\varphi}(f)=\tau\int\sum_{I\in\mathcal{D}}\frac{\langle\varphi,w_{I}\rangle^{*}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\frac{f_{I}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}dx$
and
$\|l_{\varphi}\|\leq\|\varphi\|_{L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})}$.
The proof is just the copy of the proof of the first part in the last theorem.
Now $T^{c}_{p^{\prime}}$ is naturally embedded into
$L_{p^{\prime}}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$ by
$(f_{I})_{I}\rightarrow(\frac{f_{I}}{|I|^{\frac{1}{2}}}\mathds{1}_{I})_{I}$.
So by the Hahn-Banach extension theorem, $l_{\varphi}$ extends to an bounded
linear functional on $L_{p^{\prime}}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$
with the same norm. Then by the duality between
$(L_{p^{\prime}}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D})))^{*}=L_{p}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D})).$
there exists a unique $h=(h_{I})_{I}$ such that
$\|h\|_{L_{p}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))}\leq\|l_{\varphi}\|$
and for $f=(f_{I})_{I}\in T^{c}_{p^{\prime}}$,
$\displaystyle
l_{\varphi}(f)=\tau\int\sum_{I\in\mathcal{D}}h^{*}_{I}\frac{f_{I}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}dx$
So we get
$\frac{\langle\varphi,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}=h_{I},$
thus,
$\displaystyle\begin{split}\|\varphi\|_{\mathcal{H}^{c}_{p}}&=\Big{\|}\big{(}\sum_{I\in\mathcal{D}}\frac{\langle\varphi,w_{I}\rangle^{*}}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\big{)}^{\frac{1}{2}}\Big{\|}_{L_{p}({\mathcal{N}})}\\\
&=\|h_{I}\|_{L_{p}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))}\leq\|l_{\varphi}\|\end{split}$
∎
## 4\. Interpolation
This section is devoted to the interpolation of our wavelet Hardy spaces. The
interpolation results below will be needed in the next section to compare our
Hardy spaces with those of Mei.
###### Lemma 4.1.
Let $1<p_{0}<p<p_{1}<\infty$, we have
(4.1)
$[\mathcal{H}^{c}_{p_{0}}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}^{c}_{p_{1}}({\mathbb{R}},{\mathcal{M}})]_{\theta}=\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$
with equivalent norms, where $\theta$ satisfies
$\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}$.
###### Proof.
The embedding map $\Phi$ yields
$[\mathcal{H}^{c}_{p_{0}},\mathcal{H}^{c}_{p_{1}}]_{\theta}\subset\mathcal{H}^{c}_{p}.$
On the other hand, it is the boundedness of the projection map $\Psi$ from
$L_{p}({\mathcal{N}};\ell^{c}_{2}(\mathcal{D}))$ to
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$ stated in Corollary 3.1 that
yields the inverse direction. ∎
###### Theorem 4.1.
Let $1\leq q<p<\infty$, we have
(4.2)
$[\mathcal{BMO}^{c}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}^{c}_{q}({\mathbb{R}},{\mathcal{M}})]_{\frac{q}{p}}=\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})$
with equivalent norms.
###### Proof.
We will prove the theorem by a general strategy as appeared in [18].
Step 1: We prove the conclusion for $2<q<p<\infty$:
(4.3)
$[{\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}^{c}_{q}({\mathbb{R}},{\mathcal{M}})]_{\frac{q}{p}}=\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}}).$
The identity can be seen easily from the following two inclusions. On one
hand, the operator $\Phi$ which in (2.10), together with (2.2) yields
$[\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}^{c}_{q^{\prime}}({\mathbb{R}},{\mathcal{M}})]_{\frac{q}{p}}\subset\mathcal{H}^{c}_{p^{\prime}}({\mathbb{R}},{\mathcal{M}}).$
Then by duality and Corollary 3.2, we have
(4.4)
$L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})\subset[{\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}}),L^{c}_{q}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})]_{\frac{q}{p}}.$
On the other hand, the operator $\mathcal{T}$ identifying
$L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ as a subspace of
$L_{p}(L_{\infty}(\mathcal{N}\bar{\otimes}B(\ell_{2}(\mathcal{D}));\ell^{c}_{\infty})$
defined by
(4.5) $\mathcal{T}(\varphi)={\langle
f,w_{I}\rangle}{|I^{t}_{k}|^{-\frac{1}{2}}}\mathds{1}_{I\subset
I^{t}_{k}}(I)\otimes e_{I,1},$
together with Lemma 2.4 yields
(4.6)
$[{\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}}),L^{c}_{q}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})]_{\frac{q}{p}}\subset
L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}}).$
Step 2: we prove the conclusion for $1<q<p<\infty$. This step can be divided
into two substeps.
Substep 21: $p>2$. Let $p<s<\infty$. By Step 1, we have
$[{\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})]_{\frac{p}{s}}=\mathcal{H}^{c}_{s}({\mathbb{R}},{\mathcal{M}}).$
On the other hand, by Theorem 4.1, we have
$[\mathcal{H}^{c}_{q},\mathcal{H}^{c}_{s}]_{\theta}=\mathcal{H}^{c}_{p},$
where(and in the rest of the paper) $\theta$ denote the interpolation
parameter. Then Wolff’s interpolation theorem yields the result.
Substep 22: $p\leq 2$. Let $s>2$, then by Substep 21, we have
$[{\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})]_{\frac{p}{s}}=\mathcal{H}^{c}_{s}({\mathbb{R}},{\mathcal{M}}).$
Then together with Lemma 4.1, Wolff’s interpolation theorem yields the result.
Step 3: we prove the conclusion for $1=q<p<\infty$. Take $s>\max(p,2)$. By
Step 2 and duality [2, Theorem 4.3.1], we get
$[\mathcal{H}^{c}_{1},\mathcal{H}^{c}_{s}]_{\theta}=\mathcal{H}^{c}_{p}.$
Then together with Step 2, Wolff’s interpolation yields the conclusion. ∎
###### Remark 4.1.
If one can directly prove Lemma 4.1 for $p_{0}=1$, we can prove the above
theorem without the help of
$L^{c}_{p}{\mathcal{MO}}({\mathbb{R}},{\mathcal{M}})$ for $2<p<\infty$ as
carried out in [1], where one needs an auxiliary space.
###### Theorem 4.2.
For $1<p<\infty$, we have
$\mathcal{H}_{p}({\mathbb{R}},{\mathcal{M}})=L_{p}({\mathcal{N}})$
with equivalent norms.
###### Proof.
There are several ways to prove this result. One can prove it by the strategy
in [20] together with Stein’s inequality (3.3). Here, we just use the fact
that $L_{p}({\mathcal{M}})$ with $1<p<\infty$ is a UMD space and our
$(w_{I})_{I}$ is an complete orthonormal basis. So by Theorem 3.8 in [7], we
have
$\|f\|_{L_{p}({\mathcal{N}})}\simeq\Big{(}\mathbb{E}\Big{\|}\sum_{I\in\mathcal{D}}\varepsilon_{I}\frac{\langle
f,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\Big{\|}^{p}_{L_{p}({\mathcal{N}})}\Big{)}^{\frac{1}{p}}.$
Then we complete the proof for $2\leq p<\infty$ by Khintchine’s inequalities.
Now, let us prove the case $1<p<2$. Let
$f\in\mathcal{H}_{p}({\mathbb{R}},{\mathcal{M}})$, then for any $\epsilon>0$,
by the definition of $\mathcal{H}_{p}({\mathbb{R}},{\mathcal{M}})$, there
exists a decomposition $f=f_{c}+f_{r}$ such that
$\|f_{c}\|_{\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})}+\|f_{r}\|_{\mathcal{H}^{r}_{p}({\mathbb{R}},{\mathcal{M}})}\leq\|f\|_{\mathcal{H}_{p}({\mathbb{R}},{\mathcal{M}})}+\epsilon.$
Take any $g\in L_{p^{\prime}}(\mathcal{N})$, by the results for
$p^{\prime}>2$, the operator-valued Calderón identity (2.5) yields
$\displaystyle|\tau\int gf^{*}|$
$\displaystyle=|\sum_{I\in\mathcal{D}}\tau\int\frac{\langle
g,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\cdot\frac{\langle
f^{*},w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}|$
$\displaystyle\leq|\sum_{I\in\mathcal{D}}\tau\int\frac{\langle
g,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\cdot\frac{\langle
f_{c}^{*},w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}|$
$\displaystyle\qquad+|\sum_{I\in\mathcal{D}}\tau\int\frac{\langle
g,w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}\cdot\frac{\langle
f_{r}^{*},w_{I}\rangle}{|I|^{\frac{1}{2}}}\mathds{1}_{I}|$
$\displaystyle\leq\|S_{c}(g)\|_{L_{p^{\prime}}(\mathcal{N})}\|S_{c}(f_{c})\|_{L_{p}(\mathcal{N})}+|S_{r}(g)\|_{L_{p^{\prime}}(\mathcal{N})}\|S_{r}(f_{r})\|_{L_{p}(\mathcal{N})}$
$\displaystyle\leq
c_{p^{\prime}}\|g\|_{L_{p^{\prime}}}(\|f\|_{\mathcal{H}_{p}({\mathbb{R}},{\mathcal{M}})}+\epsilon).$
Taking $\sup$ and let $\epsilon\rightarrow 0$, we get the required result.
Finally, we prove the inverse inequality. Let $f\in L_{p}(\mathcal{N})$, by
duality, we can find two sequences of functions $(F_{c,I})_{I}\in
L_{p}(\mathcal{N};\ell^{c}_{2}(\mathcal{D}))$ and $(F_{r,I})_{I}\in
L_{p}(\mathcal{N};\ell^{r}_{2}(\mathcal{D}))$ such that
$F_{c,I}+F_{r,I}=\langle f,w_{I}\rangle|I|^{-\frac{1}{2}}\mathds{1}_{I}$ and
$\|(F_{c,I})_{I}\|_{L_{p}(\mathcal{N};\ell^{c}_{2}(\mathcal{D}))}+\|(F_{r,I})_{I}\|_{L_{p}(\mathcal{N};\ell^{r}_{2}(\mathcal{D}))}\leq\|f\|_{L_{p}(\mathcal{N})}.$
Let $f_{c}=\Psi(({F_{c,I}})_{I})$ and $f_{r}=\Psi(({F_{r,I}})_{I})$, by
identity (2.5), we have $f=f_{c}+f_{r}$. On the other hand, by the Stein
inequality (3.3), we have
$\|f_{c}\|_{\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})}\leq\|(F_{c,I})_{I}\|_{L_{p}(\mathcal{N};\ell^{c}_{2}(\mathcal{D}))}$
and
$\|f_{r}\|_{\mathcal{H}^{r}_{p}({\mathbb{R}},{\mathcal{M}})}\leq\|(F_{r,I})_{I}\|_{L_{p}(\mathcal{N};\ell^{r}_{2}(\mathcal{D}))}$.
So we have found the desired decomposition of $f$. ∎
###### Theorem 4.3.
The following results hold with equivalent norms:
$\rm(i)$ Let $1\leq q<p<\infty$, we have
(4.7)
$[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),L_{q}({\mathcal{N}})]_{\frac{q}{p}}=L_{p}({\mathcal{N}}).$
$\rm(ii)$ Let $1<q<p\leq\infty$, we have
(4.8)
$[\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}}),L_{p}({\mathcal{N}})]_{\frac{p^{\prime}}{q^{\prime}}}=L_{q}({\mathcal{N}}).$
$\rm(iii)$ Let $1<p<\infty$, we have
(4.9)
$[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}})]_{\frac{1}{p}}=L_{p}({\mathcal{N}}).$
In order to prove this theorem, we need the following result from the theory
of interpolation. We formulate it here without proof.
###### Lemma 4.2.
Let $A_{0},B_{0},A_{1},B_{1}$ be four Banach spaces satisfying the property
needed for interpolation. Then
$[A_{0}+B_{0},A_{1}+B_{1}]_{\theta}\supset[A_{0},A_{1}]_{\theta}+[B_{0},B_{1}]_{\theta}$
and
$[A_{0}\cap B_{0},A_{1}\cap
B_{1}]_{\theta}\subset[A_{0},A_{1}]_{\theta}\cap[B_{0},B_{1}]_{\theta}.$
###### Proof.
$\rm(i)$ We also exploit the similar but different strategy with that in the
proof of Theorem 4.1.
Step 1: we prove the results for $2\leq q<p<\infty$. By Theorem 4.2, Theorem
4.1 and the lemma, we have
$[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),L_{q}({\mathcal{N}})]_{\frac{q}{p}}\subset
L_{p}({\mathcal{N}}).$
The inverse direction follows from
$L_{\infty}({\mathcal{N}})\subset\mathcal{BMO}({\mathbb{R}},{\mathcal{M}})$,
$\displaystyle\begin{split}L_{p}({\mathcal{N}})&=[L_{\infty}({\mathcal{N}}),L_{q}({\mathcal{N}})]_{\frac{q}{p}}\\\
&\subset[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),L_{q}({\mathcal{N}})]_{\frac{q}{p}}\\\
\end{split}$
Step 2: we prove the results for $1\leq q<2\leq p<\infty$. By Step 1, we have
$[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),L_{2}({\mathcal{N}})]_{\frac{2}{p}}=L_{p}({\mathcal{N}}).$
Together with
$L_{2}({\mathcal{N}})=[L_{p}({\mathcal{N}}),L_{q}({\mathcal{N}})]_{\theta},$
Wolff’s interpolation yields the conclusion.
Step 3: we prove the results for $1\leq q<p<2$. By Step 2, we have
$[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),L_{p}({\mathcal{N}})]_{\frac{p}{2}}=L_{2}({\mathcal{N}}).$
Together with
$L_{p}({\mathcal{N}})=[L_{2}({\mathcal{N}}),L_{q}({\mathcal{N}})]_{\theta},$
Wolff’s interpolation yields the conclusion.
$\rm(ii)$ The results for $1<q<p<\infty$ can be immediately proved by duality
and the partial results in $(i)$. For $p=\infty$, take $q<s<\infty$, then by
Wolff’s argument, we get the conclusion.
$\rm(iii)$ First, we prove conclusion for $p<2$. Then by $\rm(i)$ and
$\rm(ii)$, we have
$[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),L_{p}({\mathcal{N}})]_{\frac{p}{p^{\prime}}}=L_{p^{\prime}}({\mathcal{N}})$
and
$[\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}}),L_{p^{\prime}}({\mathcal{N}})]_{\frac{p}{p^{\prime}}}=L_{p}({\mathcal{N}}).$
Therefore, we end with Wolff’s argument. Second, the proof for $p>2$ is the
same. At last, when $p=2$, we can take $s>2$, by the results for $p\neq 2$ and
reiteration theorem in [2, Theorem 4.6.1], we get
$\displaystyle\begin{split}L_{2}&=[L_{s},L_{s^{\prime}}]_{\theta}=[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}})]_{\frac{1}{s}},\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}})]_{\frac{1}{s^{\prime}}}]_{\theta}\\\
&=[\mathcal{BMO}({\mathbb{R}},{\mathcal{M}}),\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}})]_{\theta}.\\\
\end{split}$
∎
## 5\. Comparison with Mei’s results
We denote the column Hardy space by $H^{c}_{p}({\mathbb{R}},{\mathcal{M}})$
and the bounded mean oscillation space by
$BMO^{c}({\mathbb{R}},{\mathcal{M}})$ in [15]. We have the following result.
###### Theorem 5.1.
We have
${\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}})=BMO^{c}({\mathbb{R}},{\mathcal{M}})$
with equivalent norms. Similar results holds for the row spaces. Consequently,
${\mathcal{BMO}}({\mathbb{R}},{\mathcal{M}})={\mathcal{BMO}}({\mathbb{R}},{\mathcal{M}})$
with equivalent norms.
The theorem can be easily seen from the corresponding
$BMO({\mathbb{R}},H)$-spaces. However, we can exploit the idea of [7] to prove
our ${\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}})$ also coincide with that
defined by the mean oscillation $BMO({\mathbb{R}},H)$.
###### Proof.
${\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}})\subset
BMO^{c}({\mathbb{R}},{\mathcal{M}}).$ Let
$\varphi\in\mathcal{BMO}_{c}(\mathbb{R},\mathcal{M})$. As in [7], fix a finite
interval $I\subset\mathbb{R}$, and consider the collections of dyadic
intervals
1. (1)
$\mathcal{D}_{1}:=\\{J\in\mathcal{D};2|J|>|I|\\}$’
2. (2)
$\mathcal{D}_{2}:=\\{J\in\mathcal{D};2|J|\leq|I|,2J\cap 2I=\emptyset\\}$,
3. (3)
$\mathcal{D}_{3}:=\\{J\in\mathcal{D};2|J|\leq|I|,2J\cap 2I\neq\emptyset\\}$.
Let $a_{J}=\langle\varphi,\omega_{J}\rangle$, then we have a priori formal
series
$\varphi_{1}(x)=\sum_{J\in\mathcal{D}_{1}}a_{J}[\omega_{J}(x)-\omega_{J}(c_{I})],\varphi_{i}(x)=\sum_{J\in\mathcal{D}_{i}}a_{J}\omega_{J}(x),i=2,3,$
where $c_{I}$ is the center of the interval $I$. Denote
$\varphi_{I}=\varphi_{1}+\varphi_{2}+\varphi_{3}$, by a similar discussion in
[7], we only need to prove:
$\|\frac{1}{|I|}\int_{I}|\varphi_{I}(x)|^{2}dx\|_{\mathcal{M}}<\infty.$
By scaling we can assume:
$\sup_{I}\frac{1}{|I|}\|\sum_{J\subset I}|a_{J}|^{2}\|=1.$
Then we have the obvious bound for individual terms
$\|a_{J}\|\leq|J|^{\frac{1}{2}}$.
Estimates for $\varphi_{1}$:
$\begin{split}\|\frac{1}{|I|}\int_{I}|\varphi_{1}(x)|^{2}dx\|&\leq\frac{1}{|I|}(\sum_{J\in\mathcal{D}_{1}}\|a_{J}\||\omega_{J}(x)-\omega_{J}(c_{I})|)^{2}dx\\\
&\leq
c\frac{1}{|I|}\int_{I}[\sum_{J\in\mathcal{D}_{1}}|J|^{\frac{1}{2}}|I||J|^{-\frac{3}{2}}(1+\frac{dist(I,J)}{|J|})^{-2}]^{2}dx\\\
&=c[\sum_{j=0}^{\infty}\sum_{|J|\in(2^{j-1},2^{j}]|I|}|I||J|^{-1}(1+\frac{dist(I,J)}{|J|})^{-2}]^{2}<\infty.\end{split}$
Estimates for $\varphi_{2}$:
$\begin{split}\|\frac{1}{|I|}\int_{I}|\varphi_{2}(x)|^{2}dx\|&\leq\frac{1}{|I|}\int_{I}\|\sum_{\mathcal{D}_{2}}a_{J}\omega_{J}(x)\|^{2}dx\\\
&\leq\frac{1}{|I|}\int_{I}(\sum_{\mathcal{D}_{2}}\|a_{J}\||\omega_{J}(x)|)^{2}dx\\\
&\leq
c\frac{1}{|I|}\int_{I}[\sum_{\mathcal{D}_{2}}|J|^{\frac{1}{2}}|J|^{-\frac{1}{2}}(\frac{dist(I,J)}{|J|})^{-2}]^{2}dx\\\
&=c[\sum_{j=1}^{\infty}\sum_{|J|\in(2^{-j-1},2^{-j})|I|,dist(I,J)>2^{-1}|I|}(\frac{dist(I,J)}{|J|})^{-2}]^{2}<\infty.\end{split}$
Estimates for $\varphi_{3}$:
$\begin{split}\|\frac{1}{|I|}\int_{I}|\varphi_{3}(x)|^{2}dx\|&\leq\frac{1}{|I|}\|\sum_{J\in\mathcal{D}_{3}}|a_{J}|^{2}\|\leq\frac{1}{|I|}\|\sum_{J\subset
4I}|a_{J}|^{2}\|<\infty\end{split}$
Hence we deduce that:
$\|\int_{I}|\varphi_{I}(x)|^{2}dx\|_{\mathcal{M}}\leq
c\sum_{i=1}^{3}\|\int_{I}|\varphi_{i}(x)|^{2}dx\|_{\mathcal{M}}\leq c|I|$
Now we turn to the proof of inverse direction
$BMO^{c}({\mathbb{R}},{\mathcal{M}})\subset{\mathcal{BMO}}^{c}({\mathbb{R}},{\mathcal{M}}).$
Let $\varphi\in BMO^{c}(\mathbb{R},\mathcal{M})$. The proof is very similar to
that in Mei’s work [15]. For any dyadic interval $I\subset\mathbb{R}$, write
$\varphi=\varphi_{1}+\varphi_{2}+\varphi_{3}$, where
$\varphi_{1}=(\varphi-\varphi_{2I})\chi_{2I},\varphi_{2}=(\varphi-\varphi_{2I})\chi_{2I^{c}},\varphi_{3}=\varphi_{2I}$.
Thus
$\begin{split}\sum_{J\subset I}|\langle\varphi,\omega_{J}\rangle|^{2}\leq
2(\sum_{J\subset I}|\langle\varphi_{1},\omega_{J}\rangle|^{2}+\sum_{J\subset
I}|\langle\varphi_{2},\omega_{J}\rangle|^{2})\end{split}$
Estimates for $\varphi_{1}$:
$\begin{split}\|\sum_{J\subset
I}|\langle\varphi_{1},\omega_{J}\rangle|^{2}\|\leq\|\int|\varphi_{1}(x)|^{2}dx\|\leq
c\|\int_{2I}|\varphi-\varphi_{2I}|^{2}\|\leq c|I|\end{split}$
Estimates for $\varphi_{2}$:
$\begin{split}\|\sum_{J\subset
I}|\langle\varphi_{2},\omega_{J}\rangle|^{2}\|&=\|\sum_{J\subset
I}|\sum_{k=1}^{\infty}\int_{2^{k+1}I/2^{k}I}\varphi_{2}\omega_{J}dx|^{2}\|\\\
&\leq\|\sum_{J\subset
I}(\sum_{k=1}^{\infty}\frac{1}{2^{2k}}\int_{2^{k+1}I/2^{k}I}|\varphi_{2}|^{2})(\sum_{k=1}^{\infty}2^{2k}\int_{2^{k+1}I/2^{k}I}|\omega_{J}|^{2})\|\\\
&\leq
c(\sum_{k=1}^{\infty}\frac{1}{2^{2k}}\|\int_{2^{k+1}I}|\varphi-\varphi_{2I}|^{2}\|)\\\
&\qquad\qquad(\sum_{J\subset
I}\sum_{k=1}^{\infty}2^{2k}\int_{2^{k+1}I/2^{k}I}|\omega_{J}|^{2})\\\ &\leq
c|I|\|\varphi\|^{2}_{{\mathcal{BMO}}_{c}}\sum_{j=0}^{\infty}2^{j}\sum_{k=1}^{\infty}\int_{2^{k+1}I/2^{k}I}2^{2k}\frac{|2^{-j}I|^{3}}{|2^{k}I|^{4}}\\\
&\leq c|I|\end{split}$
Therefore $\|\sum_{J\subset I}|\langle\varphi,\omega_{J}\rangle|^{2}\|\leq
c|I|$, which completes our proof. ∎
Combined with Theorem 3.2 and Theorem 4.1, we have the following corollary
###### Corollary 5.1.
For $1\leq p<\infty$, we have
$\mathcal{H}^{c}_{p}({\mathbb{R}},{\mathcal{M}})=H^{c}_{p}({\mathbb{R}},{\mathcal{M}}).$
Similar results hold for $\mathcal{H}^{r}_{p}$ and ${H}^{r}_{p}$, and
$\mathcal{H}_{p}$ and ${H}_{p}$.
If ${\mathcal{M}}=\mathbb{C},$ $\mathcal{H}_{1}({\mathbb{R}},\mathbb{C})$ is
just the usual Hardy space $H_{1}({\mathbb{R}})$ on $\mathbb{R}.$
$H_{1}({\mathbb{R}})$ also has the following characterization:
$H_{1}({\mathbb{R}})=\\{f\in L_{1}({\mathbb{R}}):H(f)\in
L_{1}({\mathbb{R}})\\},$
where $H$ is the Hilbert transform. For any $f\in H_{1}({\mathbb{R}})$,
$\|f\|_{H_{1}({\mathbb{R}})}\approx\|f\|_{L_{1}({\mathbb{R}})}+\|H(f)\|_{L_{1}({\mathbb{R}})}.$
Thus $H_{1}({\mathbb{R}})$ can be viewed as a subspace of
$L_{1}({\mathbb{R}})\oplus_{1}L_{1}({\mathbb{R}})$. The latter direct sum has
its natural operator structure as an $L_{1}$ space. This induces an operator
space structure on $H_{1}({\mathbb{R}}).$ Although $(w_{I})_{I\in\mathcal{D}}$
is a unconditional basis of $H_{1}({\mathbb{R}})$, Ricard [22] (see also [23])
proved that $H_{1}({\mathbb{R}})$ does not have complete unconditional basis.
However, in noncommutative analysis, one can introduce another natural
operator space structure on $H_{1}({\mathbb{R}})$ as follows:
$S_{1}(H_{1}({\mathbb{R}}))=\mathcal{H}_{1}({\mathbb{R}},B(\ell_{2})),$ where
$S_{1}$ is the trace class on $\ell_{2}.$ Then we have the following result.
Note that Ricard [23] obtained a similar result using Hilbert space
techniques.
###### Corollary 5.2.
The complete orthogonal systems $(w_{I})_{I\in\mathcal{D}}$ of
$L_{2}({\mathbb{R}})$ is a completely unconditional basis for
$H_{1}({\mathbb{R}})$ if we define the operator space structure imposed on
$H_{1}({\mathbb{R}})$ by
$\mathcal{S}_{1}(H_{1}({\mathbb{R}}))=\mathcal{H}_{1}({\mathbb{R}},B(\ell_{2}))$.
###### Proof.
Fix a finite subset $\mathcal{I}\subset\mathcal{D}$. Let
$T_{\varepsilon}f\doteq\sum_{I\in\mathcal{I}}\varepsilon_{I}\langle
f,w_{I}\rangle w_{I}$, where $\varepsilon_{I}=\pm 1$. By the definition of
$\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})$, the orthogonality of
$(w_{I})_{I\in\mathcal{D}}$ yields immediately that
$\displaystyle\begin{split}\|T_{\varepsilon}f\|_{\mathcal{H}^{c}_{1}}&=\Big{\|}\Big{(}\sum_{I\in\mathcal{I}}\frac{|\langle
f,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}(x)\Big{)}^{\frac{1}{2}}\Big{\|}_{{L_{1}({\mathcal{N}})}}\\\
&\leq\Big{\|}\Big{(}\sum_{I\in\mathcal{D}}\frac{|\langle
f,w_{I}\rangle|^{2}}{|I|}\mathds{1}_{I}(x)\Big{)}^{\frac{1}{2}}\Big{\|}_{{L_{1}({\mathcal{N}})}}=\|f\|_{\mathcal{H}^{c}_{1}}\\\
\end{split}$
Similarly, the above inequality holds for
$\mathcal{H}^{r}_{1}({\mathbb{R}},{\mathcal{M}})$. Now, let
$f\in\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}})$, then for any $\epsilon>0$,
there exists a decomposition $f=g+h$ such that
$\|g\|_{\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})}+\|h\|_{\mathcal{H}^{r}_{1}({\mathbb{R}},{\mathcal{M}})}\leq\|f\|_{\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}})}+\epsilon.$
Therefore
$\displaystyle\begin{split}\|T_{\varepsilon}f\|_{\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}})}&\leq\|T_{\varepsilon}g\|_{\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})}+\|T_{\varepsilon}h\|_{\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})}\\\
&\leq\|g\|_{\mathcal{H}^{c}_{1}({\mathbb{R}},{\mathcal{M}})}+\|h\|_{\mathcal{H}^{r}_{1}({\mathbb{R}},{\mathcal{M}})}\leq\|f\|_{\mathcal{H}_{1}({\mathbb{R}},{\mathcal{M}})}+\epsilon.\\\
\end{split}$
Let $\epsilon\rightarrow 0$, we get the result. ∎
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* [24] T. Wolff, A note on Interpolation Spaces, Harmonic Analysis, Lec. Notes in Math., 1568, Spring-Verlag, Berlin-Heidelberg-New York, 1994.
|
arxiv-papers
| 2011-12-13T15:04:11 |
2024-09-04T02:49:25.225637
|
{
"license": "Public Domain",
"authors": "Guixiang Hong and Zhi Yin",
"submitter": "Guixiang Hong",
"url": "https://arxiv.org/abs/1112.2912"
}
|
1112.3051
|
Fermilab Lattice and MILC Collaborations
# $B$\- and $D$-meson decay constants from three-flavor lattice QCD
A. Bazavov Physics Department, Brookhaven National Laboratory, Upton, NY, USA
C. Bernard Department of Physics, Washington University, St. Louis, Missouri,
USA C.M. Bouchard Physics Department, University of Illinois, Urbana,
Illinois, USA Fermi National Accelerator Laboratory, Batavia, Illinois, USA
Department of Physics, The Ohio State University, Columbus, OH, USA C. DeTar
Physics Department, University of Utah, Salt Lake City, Utah, USA M. Di
Pierro School of Computing, DePaul University, Chicago, Illinois, USA A.X.
El-Khadra Physics Department, University of Illinois, Urbana, Illinois, USA
R.T. Evans Physics Department, University of Illinois, Urbana, Illinois, USA
E.D. Freeland Physics Department, University of Illinois, Urbana, Illinois,
USA Department of Physics, Benedictine University, Lisle, Illinois, 60532,
USA E. Gámiz Fermi National Accelerator Laboratory, Batavia, Illinois, USA
CAFPE and Depto. de Física Teórica y del Cosmos, Universidad de Granada,
Granada, Spain Steven Gottlieb Department of Physics, Indiana University,
Bloomington, Indiana, USA U.M. Heller American Physical Society, Ridge, New
York, USA J.E. Hetrick Physics Department, University of the Pacific,
Stockton, California, USA R. Jain Physics Department, University of
Illinois, Urbana, Illinois, USA A.S. Kronfeld Fermi National Accelerator
Laboratory, Batavia, Illinois, USA J. Laiho SUPA, School of Physics and
Astronomy, University of Glasgow, Glasgow, UK L. Levkova Physics Department,
University of Utah, Salt Lake City, Utah, USA P.B. Mackenzie Fermi National
Accelerator Laboratory, Batavia, Illinois, USA E.T. Neil Fermi National
Accelerator Laboratory, Batavia, Illinois, USA M.B. Oktay Physics
Department, University of Utah, Salt Lake City, Utah, USA J.N. Simone Fermi
National Accelerator Laboratory, Batavia, Illinois, USA R. Sugar Department
of Physics, University of California, Santa Barbara, California, USA D.
Toussaint Department of Physics, University of Arizona, Tucson, Arizona, USA
R.S. Van de Water ruthv@bnl.gov Physics Department, Brookhaven National
Laboratory, Upton, NY, USA
###### Abstract
We calculate the leptonic decay constants of $B_{(s)}$ and $D_{(s)}$ mesons in
lattice QCD using staggered light quarks and Fermilab bottom and charm quarks.
We compute the heavy-light meson correlation functions on the MILC asqtad-
improved staggered gauge configurations which include the effects of three
light dynamical sea quarks. We simulate with several values of the light
valence- and sea-quark masses (down to $\sim m_{s}/10$) and at three lattice
spacings ($a\approx$ 0.15, 0.12, and 0.09 fm) and extrapolate to the physical
up and down quark masses and the continuum using expressions derived in heavy-
light meson staggered chiral perturbation theory. We renormalize the heavy-
light axial current using a mostly nonperturbative method such that only a
small correction to unity must be computed in lattice perturbation theory and
higher-order terms are expected to be small. We obtain $f_{B^{+}}=196.9(8.9)$
MeV, $f_{B_{s}}=242.0(9.5)$ MeV, $f_{D^{+}}=218.9(11.3)$ MeV,
$f_{D_{s}}=260.1(10.8)$ MeV, and the $\mathrm{SU}(3)$ flavor-breaking ratios
$f_{B_{s}}/f_{B}=1.229(26)$ and $f_{D_{s}}/f_{D}=1.188(25)$, where the numbers
in parentheses are the total statistical and systematic uncertainties added in
quadrature.
Lattice QCD, leptonic decays of mesons, chiral perturbation theory
###### pacs:
12.38.Gc, 13.20.Fc, 13.20.He
††preprint: FERMILAB-PUB-11/651-T
## I Introduction
Leptonic decays of $B$ and $D$ mesons, in which the hadron annihilates weakly
to a $W$ boson, are important probes of heavy-to-light quark flavor-changing
interactions. When combined with a nonperturbative lattice QCD calculation of
the heavy-light pseudoscalar meson decay constant, $f_{B}$ or $f_{D}$, a
precise experimental measurement of the leptonic decay width allows the
determination of the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix
element $|V_{ub}|$ or $|V_{cd}|$. Conversely, if the relevant CKM matrix
element is known from an independent process such as semileptonic decay or
from CKM-unitarity constraints, a comparison of the decay constant from
lattice QCD simulations with that measured by experiment provides a
straightforward test of the Standard Model. As the lattice and experimental
determinations become more precise, this test will become more sensitive and
may ultimately reveal, through the appearance of a discrepancy, the presence
of new physics in the quark flavor sector.
Improved determinations of the $B$ meson decay constant $f_{B}$ are of
particular importance given the current, approximately 3-$\sigma$ tension in
the CKM unitarity triangle that may indicate the presence of new physics in
$B_{d}$-mixing or $B\to\tau\nu$ decay Lenz:2010gu ; Lunghi:2010gv ;
Laiho:2011nz . The experimental uncertainty in the branching fraction
${\mathcal{B}}(B\to\tau\nu)$ is at present $\sim 30\%$ Hara:2010dk ; :2010rt ,
but this error is expected to be reduced to $\sim 10\%$ by Belle II at KEK-$B$
in as little as five or six years Masuzawa:2010zz ; Iijima:HINTS09 , at which
point even modest improvements in the determination of $f_{B}$ will
significantly help constrain the apex of the CKM unitarity triangle and
isolate the source of new physics Lunghi:2009ke .
Because leptonic decays are “gold-plated” processes in numerical lattice QCD
simulations (they have a single stable hadron in the initial state and no
hadrons in the final state Davies:2003ik ), they can be determined accurately
using present lattice methods. Currently all realistic lattice calculations of
$f_{D_{(s)}}$ and $f_{B_{(s)}}$ that include the effects of three light
dynamical quarks use staggered lattice fermions Susskind:1976jm ;
Sharatchandra:1981si for the up, down, and strange quarks. Because staggered
fermions are computationally cheaper than other lattice fermion formulations,
they allow for QCD simulations with dynamical quarks as light as $0.05m_{s}$,
several lattice spacings, down to $a\approx 0.045$ fm, large physical volumes,
and high statistics. This enables lattice determinations of many light-light
and heavy-light meson quantities with controlled systematic uncertainties. The
results of staggered lattice calculations are largely in excellent numerical
agreement with experimental results Davies:2003ik . This includes both
postdictions, such as the pion and kaon decay constants Aubin:2004fs , and
predictions, as in the case of the $B_{c}$ meson mass Allison:2004be . Such
successes give confidence that further calculations using the same methods are
reliable. This is essential if lattice QCD calculations of hadronic weak
matrix elements are to be used to test the Standard Model and search for new
physics.
The staggered dynamical quark simulations used here employ the fourth-root
procedure (“rooting”) for eliminating unwanted extra quark degrees of freedom
that arise from lattice fermion doubling. The rooting method is not standard
quantum field theory, and at nonzero lattice spacing it leads to violations of
unitarity Prelovsek:2005rf ; Bernard:2006zw ; Bernard:2007qf ; Aubin:2008wk
that can be considered nonlocal Bernard:2006ee . Nevertheless, there are
strong arguments Shamir:2004zc ; Shamir:2006nj that the desired local,
unitary theory of QCD is reproduced by the rooted staggered lattice theory in
the continuum limit. Further, one can show Bernard:2006zw ; Bernard:2007ma
that the unitarity-violating lattice artifacts in the pseudo-Goldstone boson
sector can be described, and hence removed, using rooted staggered chiral
perturbation theory (rS$\chi$PT), which is a low-energy effective description
of the rooted staggered lattice theory Lee:1999zxa ; Aubin:2003mg ;
Sharpe:2004is . When coupled with other analytical and numerical evidence (see
Refs. Sharpe:2006re ; Kronfeld:2007ek ; Golterman:2008gt ; Bazavov:2009bb for
reviews and Ref. Donald:2011if for a recent study), this gives us confidence
that the rooting procedure is valid. Indeed, the validity of the rooted
staggered lattice simulations is of critical importance to flavor physics
phenomenology, since a majority of the unquenched, three-flavor lattice
results for hadronic weak matrix elements used to determine CKM matrix
elements and as inputs to constraints on the CKM unitarity triangle come from
such simulations Laiho:2009eu .
In this paper, we present new results for the leptonic decay constants of
heavy-light mesons containing bottom and charm quarks. We use the “2+1” flavor
asqtad-improved gauge configurations made publicly-available by the MILC
Collaboration Bernard:2001av . These ensembles include the effects of three
light, dynamical sea-quark flavors: one with mass $m_{h}$ near $m_{s}$ (the
physical strange-quark mass) and the other two with mass $m_{l}$ as small as
$0.1m_{h}$. We generate light valence quarks for the $B$ and $D$ mesons using
the same staggered action as in the sea sector, and generate heavy bottom and
charm quarks using the clover action Sheikholeslami:1985ij with the Fermilab
interpretation ElKhadra:1996mp . Because the Fermilab method uses knowledge of
the heavy-quark limit of QCD to systematically eliminate heavy-quark
discretization errors, exploiting ideas of Symanzik Symanzik:1983dc ;
Symanzik:1983gh and of heavy-quark effective theory (HQET) Kronfeld:2000ck ;
Harada:2001fi ; Harada:2001fj , it is well-suited for both bottom and charm
quarks. We simulate with many values for the light up/down quark mass (the
mass of our lightest pion in both the sea and valence sectors is $\approx 250$
MeV), and at three lattice spacings ranging from $a\approx 0.09$ fm to
$a\approx 0.15$ fm. We then extrapolate our numerical lattice data to the
physical up and down quark masses and continuum guided by expressions derived
in staggered chiral perturbation theory for heavy-light mesons (HMS$\chi$PT)
Aubin:2005aq ; Laiho:2005ue ; Aubin:2007mc .
We renormalize the heavy-light axial current with a mostly nonperturbative
approach, computing the flavor-diagonal (heavy-heavy and light-light)
renormalization factors nonperturbatively and then calculate the remaining
flavor off-diagonal correction factor ($\rho_{A^{4}_{Qq}}$) in lattice
perturbation theory ElKhadra:2001rv ; Harada:2001fi ; ElKhadra:2007qe . This
procedure has the advantage that $\rho_{A^{4}_{Qq}}$ is close to unity.
Furthermore, tadpole diagrams cancel in the ratio needed to obtain
$\rho_{A^{4}_{Qq}}$, thereby improving the convergence of the perturbative
series. Empirically, the size of the 1-loop contribution to
$\rho_{A^{4}_{Qq}}$ is found to be small.
Our results for the charmed-meson decay constants improve upon our published
results for $f_{D}$ and $f_{D_{s}}$ in Ref. Aubin:2005ar in several ways. The
coarsest lattices used in this work have a smaller lattice spacing ($a\approx
0.15$ fm) than those used in our previous work ($a\approx 0.18$ fm). The
number of configurations in the two most chiral ensembles with $a\approx 0.12$
fm has been increased, approximately by factors of 1.4 (sea $m_{l}=0.1m_{h}$)
and 1.7 (sea $m_{l}=0.14m_{h}$). We have added new data on a new $a\approx
0.09$ fm sea-quark ensemble with a light quark mass of $0.1m_{h}$. We now
obtain our results from a combined analysis of our entire data set (all
partially-quenched mass combinations and lattice spacings). Furthermore, we
now compute the bottom meson decay constants $f_{B}$ and $f_{B_{s}}$. We have
presented reports on this project at several conferences Bernard:2006zz ;
Bernard:2007zz ; Bernard:2009wr ; Bazavov:2009ii ; Simone:2010zz ; in our
final analysis of this data set we also improve upon bottom and charm quark
mass-tuning, with increased statistics and a more sophisticated analysis of
heavy-quark discretization effects.
This paper is organized as follows. In Sec. II, we present an overview of the
calculation, including the gluon and light-quark actions used in generating
the gauge configurations and the light- and heavy-quark actions used in
constructing the heavy-light meson correlators. We also introduce the mostly
nonperturbative method for matching the lattice heavy-light current to the
continuum, and the treatment of heavy-quark discretization errors from the
Fermilab action within our chiral-continuum extrapolation. Next, in Sec. III,
we describe the details of our numerical simulations and we present the
parameters used, such as the light-quark masses and lattice spacings. We also
describe the procedure for tuning the hopping parameter in the clover action
so that it corresponds to $b$ and $c$ quarks. In Sec. IV, we define the two-
point correlation functions used to extract the decay constant at each value
of the light-quark mass and lattice spacing. We use two different fitting
procedures to obtain the decay constants that differ in their treatment of the
statistical errors, choice of fit ranges and number of states, and choice of
input correlators. We include the difference between the two in our estimate
of the fitting systematic uncertainty. Next, we present the numerical details
of the calculation of the heavy-light axial-current renormalization factor in
Section V. Putting the results of the two previous sections together, in Sec.
VI, we extrapolate the renormalized decay constant data at unphysical quark
masses and nonzero lattice spacing to the physical light quark masses and zero
lattice spacing using HMS$\chi$PT. In Sec. VII, we estimate the contributions
of the various systematic uncertainties to the decay constants, discussing
each item in our error budget separately. We present the final results for the
decay constants in Sec. VIII, and compare them to other lattice QCD
calculations and to experiment. We describe the impact of our results for
current flavor physics phenomenology and then conclude by discussing the
ongoing improvements to our calculations, and their future impact on searches
for new physics in the quark flavor sector.
Appendix A applies HQET to the Fermilab action to obtain explicit expressions
for heavy-quark discretization effects. Appendix B contains the complete set
of fit results for the heavy-light pseudoscalar meson mass and renormalized
decay constant for all combinations of sea-quark mass, light valence-quark
mass, and heavy-quark mass used in the chiral-continuum extrapolation. These
results will be included as an EPAPS attachment upon publication.
## II Methodology
The decay rate for a charged pseudoscalar meson $H$ (with flavor content $Q$
and $\bar{q}$) to leptons is, in the Standard Model,
$\Gamma(H\to\ell\nu)=\frac{M_{H}}{8\pi}f_{H}^{2}\left|G_{F}V^{*}_{Qq}m_{\ell}\right|^{2}\left(1-\frac{m_{\ell}^{2}}{M_{H}^{2}}\right)^{2},$
(1)
where $M_{H}$ is the mass of the meson $H$, $G_{F}$ is the Fermi constant, and
$V_{Qq}$ is the pertinent element of the CKM matrix. The decay constant
$f_{H}$ parameterizes the pseudoscalar-to-vacuum matrix element of the axial
vector current,
$\left\langle 0|\mathcal{A}^{\mu}|H(p)\right\rangle=ip^{\mu}f_{H},$ (2)
where $p^{\mu}$ is the 4-momentum of the pseudoscalar meson. The flavor
contents of the associated vector current and CKM matrix element are given in
Table 1.
Table 1: Flavor content of the axial vector current and associated CKM matrix element. $H$ | $\mathcal{A}^{\mu}$ | $V$
---|---|---
$D$ | $\bar{d}\gamma^{\mu}\gamma^{5}c$ | $V^{*}_{cd}$
$D_{s}$ | $\bar{s}\gamma^{\mu}\gamma^{5}c$ | $V^{*}_{cs}$
$B$ | $\bar{b}\gamma^{\mu}\gamma^{5}u$ | $V_{ub}$
$B_{s}$ | $\bar{b}\gamma^{\mu}\gamma^{5}s$ | —
Note that the neutral $B_{s}$ decays to a charged lepton pair with an
amplitude proportional to $f_{B_{s}}$; hence the CKM factor in the decay rate
involves more than one CKM matrix element. Because this process is loop-
suppressed in the Standard Model, it is potentially sensitive to new physics
effects. These formulas hold for all pseudoscalar mesons; in the normalization
convention used here, $f_{\pi}(|V_{ud}|/0.97425)=130.41\pm
0.20~{}\textrm{MeV}$ Rosner:2010ak .
In Eq. (2), the 1-particle state assumes the relativistic normalization
convention. For mesons containing a heavy quark, however, it is more
convenient to pull out factors of $M_{H}$ to ensure a smooth $M_{H}\to\infty$
limit:
$\left\langle
0|\mathcal{A}^{\mu}|H(p)\right\rangle(M_{H})^{-1/2}=i(p^{\mu}/M_{H})\phi_{H}.$
(3)
In lattice QCD, the normalization of states on the left-hand side falls out of
correlation functions more naturally. Thus, most of our analysis, including
error analysis, focuses on $\phi_{H}$. We then obtain
$f_{H}=\phi_{H}/\sqrt{M_{H}}$ using the experimentally measured value of the
meson mass Nakamura:2010zzi .
To compute the decay constants with lattice gauge theory, we must choose a
discretization for the heavy quark, the light quark, and the gluons. As in
previous work Aubin:2004ej ; Aubin:2005ar ; Bernard:2008dn ; Bailey:2008wp ;
Bernard:2010fr , we choose the Fermilab method for heavy quarks
ElKhadra:1996mp and staggered quarks with the asqtad action Lepage:1998vj
for the light (valence) quark. The gauge action is Symanzik improved, with
couplings chosen to remove order $\alpha_{s}a^{2}$ errors from gluon loops
Luscher:1985zq , but not those from quark loops Hao:2007iz (which became
available only after the gauge-field generation was well underway).
For heavy bottom and charm quarks, we use the Sheikholeslami-Wohlert (SW)
clover action Sheikholeslami:1985ij with the Fermilab interpretation
ElKhadra:1996mp , which connects to the continuum limit as $am_{Q}\to 0$. This
is an extension of the Wilson action Wilson:1975id , which retains the Wilson
action’s smooth limit as $am_{Q}\to\infty$ and also remains well behaved for
$m_{Q}a\approx 1$. Because this lattice action respects heavy-quark spin-
flavor symmetry, one can apply HQET to organize the discretization effects. In
essence, one uses HQET to develop the $1/m_{Q}$ expansion both for continuum
QCD and for lattice gauge theory (LGT) Kronfeld:2000ck ; Harada:2001fi ;
Harada:2001fj . Discretization effects are then captured order-by-order in the
heavy-quark expansion by the difference of the short-distance coefficients in
the descriptions of QCD and LGT. Thus, in principle, the lattice heavy-quark
action can be improved to arbitrarily high orders in $1/m_{Q}$ by adjusting a
sufficiently large number of parameters in the lattice action. (See Ref.
Oktay:2008ex for details at dimension 6 and 7. In principle, the adjustment
can be done nonperturbatively, such as in the scheme of Ref. Lin:2006ur .) In
practice, we tune the hopping parameter $\kappa$ and the clover coefficient
$c_{\mathrm{SW}}$ of the SW action, to remove discretization effects through
order $1/m_{Q}$ in the heavy-quark expansion.
The HQET analysis of cutoff effects could be applied to any lattice action
with heavy-quark symmetry, such as the action of lattice NRQCD Lepage:1992tx .
In the latter case, it is simply a different perspective on the usual approach
to lattice NRQCD, which derives the heavy-quark Lagrangian formally, and then
replaces derivatives with difference operators. A key feature of the Wilson,
SW, Fermilab and OK Oktay:2008ex actions is their well-behaved continuum
limit, which is especially important for charm. For $m_{Q}a<1$, one can
analyze the cutoff effects in a complementary way with the Symanzik effective
action Symanzik:1983dc ; Symanzik:1983gh . This two-pronged attack shows that
the difference of short-distance coefficients, mentioned above, vanishes as a
suitable power of lattice spacing $a$. In this paper, we shall use our
knowledge of this behavior to constrain heavy-quark discretization effects in
several steps of our analysis. See Secs. III.2, VI, and Appendix A for
details.
The lattice and continuum currents are related by a matching factor
$Z_{A^{\mu}}$ Harada:2001fi :
$Z_{A^{\mu}}A^{\mu}\doteq\mathcal{A}^{\mu}+\mathrm{O}\left(\alpha_{s}a\Lambda
f_{i}(m_{Q}a)\right)+\mathrm{O}\left(a^{2}\Lambda^{2}f_{j}(m_{Q}a)\right),$
(4)
where $\doteq$ denotes equality of matrix elements, and the functions
$f_{i,j}$ that depend on $m_{Q}a$ stem from the difference in the HQET short-
distance coefficients. In the Fermilab method, they remain of order 1 for all
values of $m_{Q}a$ ElKhadra:1996mp ; Oktay:2008ex , and they are given
explicitly in Appendix A. In this work, we compute $Z_{A^{\mu}}$ mostly
nonperturbatively ElKhadra:2001rv and partly in one-loop perturbation theory.
As shown in the analysis of Ref. Harada:2001fi , many of the Feynman diagrams
in the perturbative expansion of $Z_{A^{4}_{Qq}}$ are common or similar to
those in the flavor-conserving renormalization factors $Z_{V^{4}_{QQ}}$ and
$Z_{V^{4}_{qq}}$, which can be computed nonperturbatively. Therefore, we
define $\rho_{A^{4}_{Qq}}$ by
$Z_{A^{4}_{Qq}}=\rho_{A^{4}_{Qq}}\sqrt{Z_{V^{4}_{qq}}Z_{V^{4}_{QQ}}},$ (5)
evaluating only $\rho_{A^{4}_{Qq}}$ in lattice perturbation theory.
The flavor-conserving factors account for most of the value of the heavy-light
renormalization factor $Z_{A^{4}_{Qq}}$. They are obtained by enforcing the
normalization condition, at zero momentum transfer,
$1=Z_{V^{4}_{qq}}\langle H_{q}|V^{4}_{qq}|H_{q}\rangle,$ (6)
where $H_{q}$ is a hadron containing a single quark of flavor $q$, and
$V^{\mu}_{qq}$ is the lattice version of the degenerate vector current. This
condition holds for all discretizations and quark masses and, hence, the heavy
quark (i.e., $Z_{V^{4}_{QQ}}$) as well. The remaining correction factor
$\rho_{A^{4}_{Qq}}$ is close to unity due to the cancellation of most of the
radiative corrections including tadpole graphs. Although such cancellations
have only been explicitly shown at 1-loop in lattice perturbation theory
Harada:2001fi ; ElKhadra:2007qe , we expect similar cancellations to persist
at higher orders. Therefore, the perturbative truncation error in the heavy-
light renormalization factor is subdominant.
## III Lattice Simulation Details
### III.1 Parameters
Table 2: The MILC three-flavor lattices and valence asqtad quark masses used in this work. All of the valence masses were used in version II of the correlator fits (Sec IV.3), while only the ones in bold print were used in version I (Sec IV.2). $\approx a\;[\textrm{fm}]$ | $am_{h}$ | $~{}am_{l}$ | $u_{0}$ | $r_{1}/a$ | $n_{\rm conf}$$\times\kern-3.99994pt$ | $n_{\rm src}$ | valence $am_{q}$
---|---|---|---|---|---|---|---
$0.09$ | $0.031$ | $0.0031$ | $0.8779$ | $3.69$ | $435$$\times\kern-3.99994pt$ | $4$ | ${\bf 0.0031},0.0037,0.0042,{\bf 0.0044},0.0052,{\bf 0.0062},$
| | | | | | ${\bf 0.0087},{\bf 0.0124},{\bf 0.0186},{\bf 0.0272},{\bf 0.031}$
| | $0.0062$ | $0.8782$ | $3.70$ | $557$$\times\kern-3.99994pt$ | $4$ | ${\bf 0.0031},0.0037,{\bf 0.0044},0.0052,{\bf 0.0062},$
| | | | | | ${\bf 0.0087},{\bf 0.0124},{\bf 0.0186},{\bf 0.0272},{\bf 0.031}$
| | $0.0124$ | $0.8788$ | $3.72$ | $518$$\times\kern-3.99994pt$ | $4$ | ${\bf 0.0031},{\bf 0.0042},{\bf 0.0062},{\bf 0.0087},{\bf 0.0124},$
| | | | | | ${\bf 0.0186},{\bf 0.0272},{\bf 0.031}$
$0.12$ | $0.05$ | $0.005$ | $0.8678$ | $2.64$ | $678$$\times\kern-3.99994pt$ | $4$ | ${\bf 0.005},0.006,{\bf 0.007},0.0084,{\bf 0.01},0.012,{\bf 0.014},$
| | | | | | $0.017,{\bf 0.02},0.024,{\bf 0.03},{\bf 0.0415}$
| | $0.007$ | $0.8678$ | $2.63$ | $833$$\times\kern-3.99994pt$ | $4$ | ${\bf 0.005},0.006,{\bf 0.007},0.0084,{\bf 0.01},0.012,{\bf 0.014},$
| | | | | | $0.017,{\bf 0.02},0.024,{\bf 0.03},{\bf 0.0415}$
| | $0.01$ | $0.8677$ | $2.62$ | $592$$\times\kern-3.99994pt$ | $4$ | ${\bf 0.005},0.006,{\bf 0.007},0.0084,{\bf 0.01},0.012,{\bf 0.014},$
| | | | | | $0.017,{\bf 0.02},0.024,{\bf 0.03},{\bf 0.0415}$
| | $0.02$ | $0.8688$ | $2.65$ | $460$$\times\kern-3.99994pt$ | $4$ | ${\bf 0.005},0.006,{\bf 0.007},0.0084,{\bf 0.01},0.012,{\bf 0.014},$
| | | | | | $0.017,{\bf 0.02},0.024,{\bf 0.03},{\bf 0.0415}$
| | $0.03$ | $0.8696$ | $2.66$ | $549$$\times\kern-3.99994pt$ | $4$ | ${\bf 0.005},0.006,{\bf 0.007},0.0084,{\bf 0.01},0.012,{\bf 0.014},$
| | | | | | $0.017,{\bf 0.02},0.024,{\bf 0.03},{\bf 0.0415}$
$0.15$ | $0.0484$ | $0.0097$ | $0.8604$ | $2.13$ | $631$$\times\kern-3.99994pt$ | $4$ | ${\bf 0.0048},{\bf 0.007},{\bf 0.0097},0.013,{\bf 0.0194},0.0242,$
| | | | | | ${\bf 0.029},0.0387,{\bf 0.0484}$
| | $0.0194$ | $0.8609$ | $2.13$ | $631$$\times\kern-3.99994pt$ | $4$ | ${\bf 0.0048},{\bf 0.007},{\bf 0.0097},0.013,{\bf 0.0194},0.0242,$
| | | | | | ${\bf 0.029},0.0387,{\bf 0.0484}$
| | $0.029$ | $0.8614$ | $2.13$ | $576$$\times\kern-3.99994pt$ | $4$ | ${\bf 0.0048},{\bf 0.007},{\bf 0.0097},0.013,{\bf 0.0194},0.0242,$
| | | | | | ${\bf 0.029},0.0387,{\bf 0.0484}$
Table 2 lists the subset of the ensembles of lattice gauge fields generated by
the MILC Collaboration Bazavov:2009bb used in this analysis. We now describe
each entry in the table.
We analyze data at three lattice spacings: $a\approx 0.15$ fm, $a\approx 0.12$
fm, and $a\approx 0.09$ fm. The ensembles contain 2+1 flavors of sea quarks,
using the asqtad-improved staggered action Lepage:1998vj , and the square
(fourth) root of the staggered determinant for the two degenerate light sea
quarks (one strange sea quark). The sea contains one flavor with mass $m_{h}$
close to the physical strange quark mass and two degenerate lighter flavors of
mass $m_{l}$. The tadpole improvement factor $u_{0}$ is a parameter of the
gauge and asqtad staggered (sea) quark action and is determined from the
fourth root of the average plaquette. We calculate the two-point correlation
functions on each ensemble from an average over four different time sources.
The relative lattice scale is determined by calculating $r_{1}/a$ on each
ensemble, where $r_{1}$ is related to the force between static quarks,
$r_{1}^{2}F(r_{1})=1.0$ Sommer:1993ce ; Bernard:2000gd . Table 2 lists
$r_{1}/a$ values for each of the ensembles that result from fitting the
calculated $r_{1}/a$ to a smooth function Allton:1996kr , as explained in Eqs.
(115) and (116) of Ref. Bazavov:2009bb .
In order to fix the absolute lattice scale, one must compute a physical
quantity which can be compared directly to experiment. The combination of the
PDG’s value of $f_{\pi}$ with MILC’s 2009 determination of $r_{1}f_{\pi}$
Bazavov:2009fk yields $r_{1}=0.3117(6)({}^{+12}_{-31})$ fm. From an average
of three methods for scale setting, including one based on $\Upsilon$
splittings, the HPQCD collaboration obtains $r_{1}=0.3133(23)(3)$ fm
Davies:2009tsa , consistent with MILC. Symmetrizing MILC’s error range gives
$r_{1}=0.3108(21)$ fm, and a straightforward average with the HPQCD result
then yields $r_{1}=0.3120(16)$ fm. This average omits likely correlations, due
to the use of MILC sea-quark configurations by both groups. Conservatively
assuming a 100% correlation, we inflate the error to $0.0022$ fm. Finally, for
convenience, we also shift the central value slightly, back to the 2009 MILC
central value. We thus take $r_{1}=0.3117(22)$ fm in this paper.
The complete list of light (asqtad) valence quark masses $m_{q}$ simulated in
this analysis is also given in Table 2. The mass values are selected to be
roughly logarithmically spaced, but to also include the set of light sea quark
masses simulated at each lattice spacing. We use a multimass solver to compute
the valence quark propagators. The marginal numerical cost of including masses
heavier than our lightest $m_{q}\sim 0.1m_{s}$ is small and logarithmic
spacing is designed to constrain the chiral logarithms.
Table 3: Table of clover coefficients and $\kappa$ values for charm and bottom used in heavy-light two-point simulations. | | | $\kappa_{sim}$
---|---|---|---
$\approx a$ [fm] | $am_{l}/am_{h}$ | $c_{\mathrm{SW}}$ | charm | bottom
0.09 | 0.0031/0.031 | 1.478 | 0.127 | 0.0923
| 0.0062/0.031 | 1.476 | |
| 0.0124/0.031 | 1.473 | |
0.12 | 0.005/0.05 | 1.72 | 0.122 | 0.086
| 0.007/0.05 | 1.72 | |
| 0.01/0.05 | 1.72 | |
| 0.02/0.05 | 1.72 | |
| 0.03/0.05 | 1.72 | |
0.15 | 0.0097/0.0484 | 1.570 | 0.122 | 0.076
| 0.0194/0.0484 | 1.567 | |
| 0.0290/0.0484 | 1.565 | |
In Table 3, we show the coefficient of the Sheikholeslami-Wohlert term
$c_{\mathrm{SW}}$ of the clover action and the $\kappa$ values used to compute
heavy-light two-point functions. The coefficient of the clover term is set to
the tadpole-improved tree-level value $c_{\mathrm{SW}}=u_{0}^{-3}$. For the
$a\approx 0.09$ and $0.15$ ensembles the tadpole coefficient is taken from the
average plaquette. We note, however, that at lattice spacing $a\approx
0.12\,\textrm{fm}$ the tadpole coefficient $u_{0}$ appearing in both the
valence asqtad action and the heavy quark clover action is taken from the
average of the Landau link evaluated on the $am_{l}/am_{h}=0.01/0.05$
ensemble. Hence, in our $a\approx 0.12~{}\textrm{fm}$ lattice data there is a
mismatch between light valence and sea quark mass definitions. As discussed in
Sec. VII, this (inadvertent) choice leads to a small error in the decay
constants. We have remedied this mismatch by using the plaquette $u_{0}$
everywhere in new runs started while this analysis was underway.
The charm and bottom kappa values listed in Table 3 are based on our initial
kappa tuning analysis using about one fourth of our final statistics. We then
used a larger data set to refine our determination of the $\kappa$ values
corresponding to bottom and charm as described in the next subsection. We
adjust our data post-facto to correspond to tuned values of
$\kappa_{\mathrm{c}}$ and $\kappa_{\mathrm{b}}$ using the measured value of
the derivative $\delta\phi/\delta\kappa$.
### III.2 Input quark masses $m_{c}$ and $m_{b}$
Our method for tuning $\kappa$ for charm and bottom quarks closely follows
that of Ref. Bernard:2010fr , where further details can be found. We start
with the dispersion relation for a heavy particle on the lattice
ElKhadra:1996mp
$E^{2}(\bm{p})=M_{1}^{2}+\frac{M_{1}}{M_{2}}\bm{p}^{2}+\frac{1}{4}A_{4}\,(a\bm{p}^{2})^{2}+\frac{1}{3}A_{4^{\prime}}a^{2}\sum_{j=1}^{3}|p_{j}|^{4}+\ldots,$
(7)
where
$M_{1}\equiv E(\bm{0})$ (8)
is called the rest mass, and the kinetic mass is given by
$M_{2}^{-1}\equiv\left.\frac{\partial E(\bm{p})}{\partial
p_{j}^{2}}\right|_{\bm{p}=\bm{0}}.$ (9)
These meson masses differ from corresponding quark masses, $m_{1}$ and
$m_{2}$, by binding-energy effects. The bare mass or, equivalently, the
hopping parameter $\kappa$ must be adjusted so that these masses reproduce an
experimental charmed or $b$-flavored meson mass. When they differ, as they do
when $m_{Q}a\not\ll 1$, one must choose. Decay constants are unaffected by the
heavy-quark rest mass $m_{1}$ Kronfeld:2000ck , so it does not make sense to
adjust the bare mass to $M_{1}$. We therefore focus on $M_{2}$, adjusting
$\kappa$ to the strange pseudoscalars $D_{s}$ and $B_{s}$, both because the
signal degrades for lighter spectator masses and because this avoids
introducing an unnecessary systematic uncertainty due to a chiral
extrapolation.
The first step is to compute the correlator $C_{2}^{(S_{1}S_{2})}(t,\bm{p})$
in Eq. (19) (below) for several 3-momenta $\bm{p}$ and several values of
$\kappa$ and light quark mass, bracketing charm and bottom, and strange,
respectively. We use all momenta such that $|\bm{p}|\leq 4\pi/L$. Second, we
fit the time dependence of the multichannel correlation matrix
$C_{2}^{(S_{1}S_{2})}$ to a sum of exponentials—including the usual staggered-
fermion oscillating terms—and extract the ground state energy $aE(\bm{p})$ by
minimization of an augmented $\chi^{2}$ Bernard:2010fr ; Lepage:2001ym ;
Morningstar:2001je . Third, we fit the energies to the dispersion relation
given in Eq. (7), through $\mathrm{O}(p_{i}^{4})$. The output of this fit is
$aM_{1}$, $M_{1}/M_{2}$, $A_{4}$, and $A_{4^{\prime}}$, all as functions of
$\kappa$. Fourth, we form $M_{2}(\kappa)$ from the first two fit outputs and
$r_{1}/a$, propagating the error with a single-elimination jackknife. Finally,
we obtain our tuned $\kappa_{\mathrm{c}}$ and $\kappa_{\mathrm{b}}$ by
interpolating in $\kappa$ so that $M_{2}(\kappa)$ matches the experimentally
known $D_{s}$ and $B_{s}$ masses. The $\kappa$ values used to compute $M_{2}$
are listed in Table 4. For each of the lattice spacings listed, we used the
ensemble with light-to-strange sea-quark mass ratio $am_{l}/am_{h}=0.2$. The
resulting tuned values of $\kappa_{\mathrm{c}}$ and $\kappa_{\mathrm{b}}$ are
shown with errors in Table 5.
Table 4: Hopping-parameter values used to compute the dispersion relation. | | $\kappa_{Q}$
---|---|---
$\approx a$ [fm] | $n_{\textrm{conf}}\times n_{\textrm{src}}$ | charm | bottom
0.09 | $1912\times 4$ | 0.1240, 0.1255, 0.1270 | 0.090, 0.092, 0.094
0.12 | $592\times 4$ | 0.114, 0.117, 0.119, 0.122, 0.124 | 0.074, 0.086, 0.093, 0.106
0.15 | $631\times 8$ | 0.100, 0.115, 0.122, 0.125 | 0.070, 0.076, 0.080, 0.090
Table 5: Hopping parameter values $\kappa_{\mathrm{c}}$ and $\kappa_{\mathrm{b}}$ corresponding to charm and bottom. The outputs of the tuning are labeled $\kappa_{\mathrm{tuned}}$, where the first error is from statistics and the second is from $r_{1}$, which enters through matching to the experimentally-measured $D_{s}$ and $B_{s}$ meson masses. The derivative $d\phi/d\kappa$ is used to correct the values of $\phi$ obtained with the simulated values $\kappa_{\mathrm{sim}}$ listed in Table 3 to the tuned values given below. | charm | bottom
---|---|---
$\approx a$ [fm] | $\kappa_{\mathrm{tuned}}$ | $d\phi/d\kappa$ | $\kappa_{\mathrm{tuned}}$ | $d\phi/d\kappa$
0.09 | 0.12691(18)(13) | $-21$. | 66 | 0.0959(13)(3) | $-7$. | 41
0.12 | 0.12136(37)(19) | $-18$. | 23 | 0.0856(19)(3) | $-6$. | 82
0.15 | 0.12093(36)(24) | $-15$. | 40 | 0.0788(11)(3) | $-6$. | 07
We constrain the coefficients $A_{4}$ and $A_{4^{\prime}}$ with Gaussian
priors derived from the HQET theory of cutoff effects, adding the contribution
of the priors to the $\chi^{2}$ in the minimization procedure Lepage:2001ym ;
Morningstar:2001je . (In principle, we could include such priors for $M_{1}$
and $M_{1}/M_{2}$ too, but in practice we take priors so wide that these fit
parameters are solely data-driven.) Neglecting binding energies, we have exact
tree-level expressions for $a_{4}$ and $a_{4^{\prime}}$, the quark analogs of
$A_{4}$ and $A_{4^{\prime}}$. The differences $A_{4}-a_{4}^{[0]}$ and
$A_{4^{\prime}}-a_{4^{\prime}}^{[0]}$ stem from both perturbative and
nonperturbative effects. The asymptotics of the former can be estimated along
the lines of Appendix A.3, and the latter can be deduced following the methods
of Refs. Kronfeld:2000ck ; Kronfeld:1996uy . Briefly, we constrain
$A_{n}(\kappa)$, $n\in\\{4,4^{\prime}\\}$, to a Gaussian with central value
$a^{[0]}_{n}(m_{0}a)+\alpha_{s}a^{[1]}_{n}(m_{0}a)+\bar{\Lambda}a\,A^{\prime}_{n}(m_{0}a).$
(10)
Here $a^{[0]}_{n}$ is the exact tree-level contribution, $a^{[1]}_{n}$ is an
estimate of the one-loop contribution, and $A^{\prime}_{n}$ is an expression
for the binding-energy contribution. The width of the Gaussian is determined
by combining in quadrature the chosen widths of the separate contributions, as
outlined in Appendix A.3.
The details of the $\kappa_{\mathrm{c}}$ and $\kappa_{\mathrm{b}}$
determination differ from that of Ref. Bernard:2010fr in two respects. First,
we use the pseudoscalar meson masses rather than the spin average of
pseudoscalar and vector meson masses, leading to a modest reduction of the
statistical error. Second, we include the quartic terms in Eq. (7), allowing
us to fold discretization effects directly into the dispersion-relation fit.
Although we consider these two changes improvements, the change in the tuned
$\kappa$ values as compared to Ref. Bernard:2010fr stems primarily from the
substantial increase in statistics on key ensembles.
The computations of the correlation functions needed to extract $\phi_{D}$ and
$\phi_{B}$ have been carried out using the fiducial values listed in Table 3.
These simulation $\kappa$’s were obtained near the beginning of the project,
but while the runs were in progress, we redetermined the hopping parameters
utilizing increased statistics and reflecting an updated value of $r_{1}$
Bazavov:2009fk . The resulting improved determinations of
$\kappa_{\mathrm{c}}$ and $\kappa_{\mathrm{b}}$ differ slightly from the
simulation values. In order to adjust $\phi$ from the simulated value
$\kappa_{\mathrm{sim}}$ to the tuned value $\kappa_{\mathrm{tuned}}$, we write
$\phi_{\rm tuned}=\phi_{\rm
sim}+\frac{d\phi}{d\kappa}(\kappa_{\mathrm{tuned}}-\kappa_{\mathrm{sim}}),$
(11)
where the derivatives $d\phi/d\kappa$ listed in Table 5 are obtained from
tuning runs with nearby $\kappa$ values. As explained in Sec. VII, these
derivatives are also used to propagate to the decay constants the statistical
and scale uncertainties on $\kappa_{\mathrm{tuned}}$ listed in Table 5.
## IV Two-point correlator fits
We obtain the unrenormalized decay amplitude for every combination of heavy-
quark mass, light-quark mass, and sea-quark ensemble by fitting the heavy-
light meson two-point correlation functions, described in Sec. IV.1. We use
two independent fitting procedures, which we refer to as “Analysis I” and
“Analysis II”. These procedures differ in several respects. In Analysis I, we
use a jackknife procedure for estimating errors, while in Analysis II, we use
a bootstrap procedure. The two analyses also differ in their methods for
handling autocorrelations in the data and in their choices of fit ranges,
priors for masses and amplitudes, and numbers of states included. In the end,
we use Analysis I (Sec. IV.2) to obtain central values, and use differences
from fits with different distance ranges and/or number of states included, and
from Analysis II (Sec. IV.3) to estimate the systematic error due to choices
made in the fit procedure.
### IV.1 Lattice correlators
The lattice axial-vector current is given by
$A^{4}_{a}(x)=[\bar{\Psi}(x)\gamma^{4}\gamma^{5}\Omega(x)]_{a}\chi(x),$ (12)
where $\chi(x)$ is the one-component field appearing in the staggered action,
and
$\Omega(x)=\gamma_{1}^{x_{1}/a}\gamma_{2}^{x_{2}/a}\gamma_{3}^{x_{3}/a}\gamma_{4}^{x_{4}/a}$
is the transformation connecting naive and staggered fields Kawamoto:1981hw .
The heavy-quark field $\Psi$ is a four-component (Dirac) spinor field, and the
remaining free Dirac index is interpreted as a taste label.
To remove tree-level discretization errors in the lattice axial current, the
heavy-quark field $\Psi$ is “rotated”:
$\Psi=[1+ad_{1}\bm{\gamma}\cdot\bm{D}]\psi,$ (13)
where $\psi$ is the field appearing in the clover action. Tree-level
improvement is obtained when
$d_{1}=\frac{1}{2+m_{0}a}-\frac{1}{2(1+m_{0}a)},$ (14)
where
$m_{0}a=\frac{1}{u_{0}}\left(\frac{1}{2\kappa}-\frac{1}{2\kappa_{\mathrm{crit}}}\right)$
(15)
is the tapdole-improved bare mass. The critical hopping parameter
$\kappa_{\mathrm{crit}}$ is the one for which the clover-clover pion mass
vanishes.
As usual in lattice gauge theory, we obtain the matrix element in (3) from
two-point correlation functions. We introduce pseudoscalar operators
$\mathcal{O}_{a}^{(S)}(x)=\sum_{y}[\bar{\psi}(y)S(y,x)\gamma^{5}\Omega(x)]_{a}\chi(x),$
(16)
depending on a “smearing” function $S$. In this work, we use two functions,
the local (or unsmeared) source $S(x,y)=\delta_{xy}$, and the smeared source
(in Coulomb gauge)
$S(x,y)=\delta_{x_{4}y_{4}}S(\bm{x}-\bm{y}),$ (17)
where $S(\bm{r})$ is the 1$S$ solution of the Richardson potential for the
quarkonium systems Richardson:1978bt . We obtain $S(\bm{x}-\bm{y})$ by scaling
the radial Richardson wavefunction to lattice units, interpolating it to
lattice sites, and then using it as the spatial source for the heavy-quark
propagators Menscher:2005kj .
We introduce two-point correlation functions
$\displaystyle\Phi_{2}^{(S)}(t)$ $\displaystyle=$
$\displaystyle\sum_{a=1}^{4}\sum_{\bm{x}}\left\langle{A^{4}_{a}}^{\dagger}(t,\bm{x})\mathcal{O}_{a}^{(S)}(0)\right\rangle,$
(18) $\displaystyle C_{2}^{(S_{1}S_{2})}(t,\bm{p})$ $\displaystyle=$
$\displaystyle\sum_{a=1}^{4}\sum_{\bm{x}}e^{i\bm{p}\cdot\bm{x}}\left\langle{\mathcal{O}_{a}^{(S_{1})}}^{\dagger}(t,\bm{x})\mathcal{O}_{a}^{(S_{2})}(0)\right\rangle,$
(19)
where $\langle\bullet\rangle$ now represents the ensemble average. For large
time separations, $\Phi_{2}^{(S)}$ is proportional to the matrix element
$\phi_{H}$, and the proportionality is determined from
$C_{2}^{(SS)}(t,\bm{0})$. Each two-point function is constructed from a
staggered quark propagator with local ($\delta$) sources and sinks. We compute
$C_{2}$ functions for all (four) combinations $S_{1},S_{2}=\delta$ and 1S,
requiring heavy clover quark propagators with all combinations of 1S smeared
and local sources and sinks. Only the local sink clover propagators are needed
to compute the $\Phi_{2}$ functions. With the sum over tastes in Eqs. (18) and
(19), the correlation functions $\Phi_{2}$ and $C_{2}$ can also be cast in a
heavy-naive formalism Wingate:2002fh .
The staggered light quarks in the axial-current and pseudoscalar two-point
correlation functions lead to the presence of opposite-parity states that
oscillate in time as $(-1)^{t}$. Hence the two-point functions take the
following forms:
$\displaystyle\Phi_{2}^{(S)}(t)$ $\displaystyle=$ $\displaystyle
A_{\Phi}^{(S)}\left(e^{-Mt}+e^{-M(T-t)}\right)+\widetilde{A}_{\Phi}^{(S)}\left(-1\right)^{t}\left(e^{-\widetilde{M}t}+e^{-\widetilde{M}(T-t)}\right)$
(20) $\displaystyle+$ $\displaystyle
A_{\Phi}^{\prime\,(S)}\left(e^{-M^{\prime}t}+e^{-M^{\prime}(T-t)}\right)+\ldots,$
$\displaystyle C_{2}^{(S_{1}S_{2})}(t,\vec{p}=0)$ $\displaystyle=$
$\displaystyle
A^{(S_{1})}A^{(S_{2})}\left(e^{-Mt}+e^{-M(T-t)}\right)+\widetilde{A}^{(S_{1})}\widetilde{A}^{(S_{2})}\left(-1\right)^{t}\left(e^{-\widetilde{M}t}+e^{-\widetilde{M}(T-t)}\right)$
(21) $\displaystyle+$ $\displaystyle
A^{\prime\,(S_{1})}A^{\prime\,(S_{2})}\left(e^{-M^{\prime}t}+e^{-M^{\prime}(T-t)}\right)+\ldots,$
where a prime denotes a standard excited state of the same parity and a tilde
denotes the mass or amplitude of an opposite-parity state. The oscillating
behavior is visible throughout the entire lattice temporal extent, and must be
included in fits to extract the ground-state mass and amplitudes.
We then obtain the renormalized decay amplitude in lattice units from the
ratio
$a^{3/2}\phi_{H}=\sqrt{2}\frac{Z_{A^{4}_{Qq}}A_{\Phi}^{(S)}}{A^{(S)}},$ (22)
where $A_{\Phi}^{(S)}$ and $A^{(S)}$ are the amplitudes of the ground state
exponentials defined in Eqs. (20) and (21), and the renormalization factor
$Z_{A^{4}_{Qq}}$ is discussed in Sec. V.
### IV.2 Analysis I
Our primary analysis of two-point correlation functions $\Phi_{2}^{(S)}$ and
$C_{2}^{(S_{1}S_{2})}$—“Analysis I”—proceeds as follows. The amplitudes
$A_{\Phi}^{(S)}$ and $A^{(S)}$ in Eq. (22) are determined from fits to
multiple correlators using the full data correlation matrix. In Analysis I, we
start by fitting combinations A, B, C and D in Table 6.
Table 6: Combinations of two-point functions that can be used to extract $a^{3/2}\phi_{H}$. All combinations of two and three correlators are shown. Additional combinations of four or more correlators are not enumerated. two-point | fit combination
---|---
function | A | B | C | D | E | F
$\Phi_{2}^{(1S)}(t)$ | $\bullet$ | | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$
$\Phi_{2}^{(\delta)}(t)$ | | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$
$C_{2}^{(1S,1S)}(t)$ | $\bullet$ | | | | $\bullet$ | $\bullet$
$C_{2}^{(\delta,\delta)}(t)$ | | $\bullet$ | | | $\bullet$ |
$C_{2}^{(\delta,1S)}(t)$ | | | $\bullet$ | | | $\bullet$
$C_{2}^{(1S,\delta)}(t)$ | | | | $\bullet$ | |
We find combination A, which uses the axial-current correlator with a $1S$
smeared source and the pseudoscalar correlator with a $1S$ smeared source and
sink, to be suitable. The extra complexity of combinations of three
correlators (C and D) give little benefit, and the errors from combination A
are somewhat smaller than those from combination B.
For fits to charm-light meson correlators, we include just one simple
exponential (the desired state) and one oscillating exponential, which we call
a “1+1 state fit”. We choose the minimum distance, $t_{\textit{min}}$, such
that contributions from excited states are small compared to our statistical
errors. Because we fit two propagators simultaneously while imposing the
constraint that the masses be equal, this is a six parameter fit: two
amplitudes for each propagator and a common mass for each of the simple and
oscillating exponentials. To help stabilize the fit, the amplitudes and mass
of the oscillating state are weakly constrained by Gaussian priors, which are
incorporated as additional terms in $\chi^{2}$ in the fitting procedure
Lepage:2001ym ; Morningstar:2001je . The central values for these priors are
determined by a trial fit where the prior for the opposite parity mass is set
to $500\pm 250$ MeV above the ground state mass and the amplitudes are
unconstrained. Then the jackknife fits use central values for the opposite
parity state amplitudes and mass determined by the trial fit, with widths
(Gaussian) that are typically three to ten times the error estimates on these
parameters, so in the end the priors make a negligible contribution to
$\chi^{2}$. (Although 500 MeV is a reasonable guess for the mass gap to the
first excited state of the meson, we actually expect that this excited state
in the fit approximates the contributions of a number of physical states,
likely including both single and multiparticle channels.) Empirically, the
width of the prior is made narrow enough to insure that the fits converge to
sensible values. We propagate the uncertainties in the correlator fits to the
subsequent chiral-continuum extrapolation with a jackknife procedure. In the
jackknife resamples, we center the priors at the values found in the fit to
the full ensemble, again with widths that are typically three to ten times the
error estimates on these parameters.
The bottom meson correlators fall off much more rapidly with $t$, so it is
difficult to take a large enough minimum distance to insure that excited state
contributions are negligible. Therefore we use a fit with two simple
exponentials and one oscillating exponential or a “2+1 state fit”. The mass of
this excited state is also weakly constrained by priors in the same way that
the opposite parity mass is, except that the width of the prior on the excited
state mass is set to 200 MeV.
Figure 1 shows the heavy-light pseudoscalar mass as a function of the minimum
time used in the fit. The left-hand plots show sample fits to bottom
correlators, while the right-hand plots show sample fits to charm correlators.
We select fitting ranges to give reasonable fits for all sea-quark ensembles
and all valence-quark masses. We quantify the goodness-of-fit with the “$p$
value” Nakamura:2010zzi , which is the probability that a fit with this number
of degrees of freedom would have a $\chi^{2}$ larger than this value. Table 7
gives the fit ranges for charm-light and bottom-light correlators on the three
lattice spacings, both for the fits used for the central values and for
alternate fits used in estimating systematic errors from choices of fit
parameters. The meson masses, $a^{3/2}\phi_{H}$ values, and $p$ values for the
data set used in Analysis I are tabulated in Appendix B.
Figure 1: Ground-state rest mass $M_{H}$ versus minimum distance $t_{\textit{min}}$ included in the fit. For each lattice spacing, we show an ensemble with the dynamical light-quark mass $m_{l}$ in the middle of the range. Similarly, we show correlators with a valence quark mass $m_{V}$ in the center of the ranges used. The top two panels are at $a\approx 0.15$ fm, the middle two at $a\approx 0.12$ fm and the bottom two at $a\approx 0.09$ fm. In each row the left panel shows results for charm and the right-panel shows results for bottom. The size of each plot symbol is proportional to the $p$ value (confidence level) of the fit, with the symbol size in the legends of the upper right panel corresponding to $p=50\%$. The red octagons are for fits including one state of each parity (“1+1 fits”) and the blue squares are for fits including an excited state of the same parity as the ground state (“2+1 fits”). In each panel, the arrow indicates the fit that is used in Sec. VI. Table 7: Numbers of states and time ranges used in two-point Analysis I. In the number of states, “1+1” means one simple exponential and one oscillating state (opposite parity). The fits in columns two through five were used for the central values, while the fits in columns six through nine were used in estimating systematic errors from the choice of fit ranges (see Sec. VII.3). | central fits | alternate fits
---|---|---
| charm | bottom | charm | bottom
$\approx a$ [fm] | $n_{\textrm{states}}$ | t range | $n_{\textrm{states}}$ | t range | $n_{\textrm{states}}$ | t range | $n_{\textrm{states}}$ | t range
0.15 | 1+1 | 11–23 | 2+1 | 4–20 | 1+1 | 12–23 | 1+1 | 9–20
0.12 | 1+1 | 14–31 | 2+1 | 5–22 | 1+1 | 16–31 | 1+1 | 12–22
0.09 | 1+1 | 21–47 | 2+1 | 7–30 | 1+1 | 23–47 | 1+1 | 16–30
The decay amplitude $a^{3/2}\phi_{H}$ is highly correlated among different
light valence-quark masses on the same ensemble. To propagate the correlations
among the different valence masses to the subsequent chiral-continuum
extrapolation, in Sec. VI, we use a single-elimination jackknife procedure to
estimate the covariance matrix of $a^{3/2}\phi_{H}$ for the selected valence
quark masses. This is done by computing the covariance matrix of the single
elimination jackknife samples, and multiplying by $\left(N-1\right)^{2}$,
where $N$ is the number of configurations in the ensemble. In fact, when all
valence quark masses are kept, the covariance matrices are close enough to
singular to be unmanageable. This reflects the fact that the correlators for
intermediate valence masses can be very accurately predicted from the
correlators for nearby masses, so some of the correlators provide very little
new information. Therefore, we omit some valence quark masses, using only
those set in bold in Table 2.
We use a single elimination jackknife rather than an omit-$J$ jackknife
because a large number of samples are needed to compute a reliable covariance
matrix. Successive configurations in the ensemble are not independent,
however, so we must take autocorrelations into account. We do so by repeating
the calculation after first blocking the data by a factor of four. (This block
size of four is determined from tests on the $a\approx 0.12$ fm lattices using
fit Analysis II, for which it gives a reasonable compromise between
suppressing autocorrelations and leaving enough data points for the
statistical analysis.) We then compute, for each valence-quark mass $i$, the
ratio $R_{i}$ of the diagonal element of the covariance matrix with a block
size of four to the same element of the unblocked covariance matrix:
$R_{i}=\sigma_{ii}^{(4)}/\sigma_{ii},$ (23)
where the superscript denotes the block size. The rescaled covariance matrix
for $a^{3/2}\phi_{H}$ is given by
$C_{ij}^{(4)}=C_{ij}\sqrt{R_{i}R_{j}},$ (24)
which preserves the eigenvalue structure of the covariance matrix, whereas
simply using the covariance matrix of the blocked data would be more likely to
produce spurious small eigenvalues. The rescaling factors $R_{i}$ themselves
have errors, and in many cases turn out to be less than one. In such cases, we
do not replace the $R_{i}$ by one, despite the fact that this would likely be
a better estimate of the individual $R_{i}$. Doing so would yield a covariance
matrix with a bias toward larger errors, and could produce misleading
estimates of goodness-of-fit in the later analysis.
Finally, we combine the covariance matrices from all of the individual
ensembles into larger covariance matrices, one each for the charm and bottom
$a^{3/2}\phi_{H}$. Since different ensembles are statistically independent,
these large covariance matrices are block diagonal, with each block containing
the correlations among different light valence-quark masses on a single sea-
quark ensemble.
### IV.3 Analysis II
Analysis II is a second, independent analysis of the two-point correlators
that uses the bootstrap method to propagate correlated errors from the two-
point analysis through to the chiral extrapolations. In Analysis II, we block
average the two-point correlator data over four sequential configurations
(which themselves are spaced by more than four trajectories) before the
analysis. In the bootstrap procedure, we resample the data (with replacement),
taking the number of sampled configurations to be equal to the number of
blocked configurations in each bootstrap ensemble. For each bootstrap
ensemble, we recompute the covariance matrix. During the bootstrap process, we
randomly draw from a gaussian distribution new prior mean values of each
constrained parameter belonging to an excited state while keeping the widths
fixed. The ground state parameters are given loose priors so that the fitted
values are determined by the data. To help stabilize the fits, the ground
state prior means are not randomized in the bootstrap. Prior values for the
energy splittings are taken from a chiral quark model calculation for the $D$
and $B$ meson systems Pierro:2001uq . Prior widths are taken to be about
$200\,\textrm{MeV}$ for excited states. Excited state amplitudes
$\log(A^{(S)})$ have a prior width $\sigma_{\log A}=2$.
On each gauge ensemble, the same sequence of gauge resamplings is taken for
all valence $m_{q}$ to preserve correlations among $a^{3/2}\phi_{H}$ values.
Our final results are based upon 4,000 bootstrap replications of the data. We
use the central values of $a^{3/2}\phi_{H}$ from the fits to the entire
ensemble in the chiral-continuum extrapolation, and use the bootstrap values
to obtain the covariance matrix.
To optimize the determination of $a^{3/2}\phi_{H}$, we compare simultaneous
fits of up to six two-point functions; the various combinations of up to four
functions are listed in Table 6. At a minimum, one axial-current correlator
must be paired with one propagator (combinations A or B in Table 6) to extract
$a^{3/2}\phi_{H}$. Combination A, using smeared operators, is used in Analysis
I, described above. Because fits of four or more two-point functions over a
wide time range can lead to a poorly determined data covariance matrix having
large rank relative to the number of available configurations, we focus on
combinations having two or three correlators. Unlike combination A,
combination B does not take advantage of smeared sources and the ratio does
not show convincing plateaus over the range of times with decent signal to
noise. Comparing combination C to D, the smeared source in C is less noisy
than the smeared sink in D.
Given these considerations, for fits to charm correlators, we use two-point
function combination C to obtain $a^{3/2}\phi_{H}$ which uses both of the
axial current functions. We look for stability of the ground-state mass and
amplitude when varying $t_{\textit{min}}$, $t_{\textit{max}}$, and the number
of excited states included in the fit. We also compare fit results from other
combinations of correlators to check that we have isolated the correct ground-
state energy and matrix element. Our final results come from fits accounting
for two pseudoscalar states and two (oscillating) opposite-parity states.
For fits to bottom correlators, we use combination B for our final results;
this is the same set used in Analysis I. Combination C gives fits with rather
low confidence levels for the $B$ meson and tends to result in larger errors
for $a^{3/2}\phi_{H}$. Again, we examine fits varying the time range; we also
try fits with up to three pseudoscalar states plus three oscillating opposite
parity states. We use these fits and fits to alternate combinations of two-
point correlators as a consistency check.
The fit results from the two different analyses are consistent with each other
for most cases, but there are a few cases where they differ by a standard
deviation or more (see Figure 11). The $a^{3/2}\phi_{H}$ results from the two
analyses are propagated through the chiral-continuum extrapolations in Secs.
VI.2 and VI.3. The resulting differences in the extrapolated results in turn
provide the basis for our systematic error analysis due to fit choices given
in Sec. VII.3.
## V Heavy-light current matching
In this section, we discuss in more detail the ingredients of Eq. (5), which
allow a “mostly nonperturbative” matching procedure ElKhadra:2001rv .
### V.1 Perturbative calculation of $\rho_{A^{4}_{Qq}}$
The perturbative expansion of $\rho_{A^{4}_{Qq}}$ can be written as
$\rho_{A^{4}_{Qq}}=1+\alpha_{s}(q^{*})\rho_{A^{4}}^{[1]}(m_{Q}a,m_{q}a)+\ldots.$
(25)
where $\alpha_{s}$ is the strong coupling and $\rho_{A^{4}}^{[1]}$ is the one-
loop coefficient. The one-loop coefficient is calculated in Ref.
ElKhadra:2007qe using lattice perturbation theory, where we see explicitly
that $\rho_{A^{4}}^{[1]}$ is small because most of the one-loop corrections
cancel. The self-energy contributions cancel exactly (to all orders, in fact),
and, in practice, we are in a region where
$\rho_{A^{4}}^{[1]}(m_{Q}a,m_{q}a)$, viewed as a function of $m_{Q}a$, has two
zeroes. Therefore the renormalization factor $\rho_{A^{4}_{Qq}}$ is close to
unity for both bottom and charm.
The perturbative calculation of $\rho_{A^{4}_{Qq}}$ in Eq. (25) proceeds as
follows. We use $\alpha_{s}(q^{*})$ defined in the $V$ scheme Lepage:1992xa
as determined in Ref. Mason:2005zx , and take $q^{*}=2/a$, which is close to
the optimal choice of Refs. Lepage:1992xa ; Hornbostel:2002af for a wide
range of quark masses. The one-loop coefficients $\rho_{A^{4}}^{[1]}$ are
computed for light-quark masses $am_{q}=0.001,0.01,0.04$ to cover the range
used in this analysis. From these we obtain $\rho_{A^{4}_{Qq}}$ at other
light-quark masses by linear interpolation in $am_{q}$. For illustration,
Table 8 lists $\rho_{A^{4}_{bq}}$ and $\rho_{A^{4}_{cq}}$ evaluated at the
light valence mass $am_{q}=0.01$ for the eleven sea-quark ensembles used in
this work. Note that the sea-quark mass dependence is indirect, via the
plaquette used to determine $\alpha_{s}(q^{*})$. The dependence on the light-
quark mass in the current is very mild: for bottom, $\rho_{A^{4}_{bq}}$
changes with $am_{q}$ by $0.07$–$0.2$%, depending on lattice spacing, and for
charm, $\rho_{A^{4}_{cq}}$ changes by around $0.1$%. On the fine ensembles,
the $am_{q}$ dependence is almost as large as the total one-loop correction
because the overall cancellation, especially in $\rho_{A^{4}_{cq}}$, is so
fortuitously good.
Table 8: The perturbative correction factor $\rho_{A^{4}_{Qq}}$ for the heavy-light current $A^{4}$ at the simulated charm and bottom heavy quark $\kappa$ values given in Table 3 and at $am_{q}=0.01$ for the different sea-quark ensembles. The statistical errors associated with the numerical integration are negligible. $\approx a$ [fm] | $am_{l}$/ | $am_{h}$ | $\rho_{A^{4}_{bq}}$ | $\rho_{A^{4}_{cq}}$
---|---|---|---|---
0.09 | 0.0031/ | 0.031 | 1.0026 | 1.0000
| 0.0062/ | 0.031 | 1.0026 | 1.0000
| 0.0124/ | 0.031 | 1.0026 | 1.0000
0.12 | 0.005/ | 0.05 | 1.0081 | 0.9959
| 0.007/ | 0.05 | 1.0081 | 0.9959
| 0.010/ | 0.05 | 1.0081 | 0.9959
| 0.020/ | 0.05 | 1.0080 | 0.9960
| 0.030/ | 0.05 | 1.0079 | 0.9961
0.15 | 0.0097/ | 0.0484 | 1.0270 | 0.9937
| 0.0194/ | 0.0484 | 1.0267 | 0.9938
| 0.0290/ | 0.0484 | 1.0265 | 0.9938
### V.2 Nonperturbative computation of $Z_{V^{4}_{qq}}$and $Z_{V^{4}_{QQ}}$
The nonperturbative part of the matching factor $Z_{A^{4}_{Qq}}$ is obtained
from the temporal components of the clover-clover and staggered-staggered
vector currents. At zero-momentum transfer, the (correctly normalized) vector
current simply counts flavor-number, so it is possible to obtain $Z_{V^{4}}$
nonperturbatively for any discretization and any mass ElKhadra:2001rv .
For the staggered-staggered current, we compute
$C_{3}^{(S_{1}S_{2})}(t_{2},0,t_{1})=\sum_{ab}\sum_{\bm{x},\bm{y}}\left\langle\mathcal{O}_{a}^{(S_{1})}(t_{2},\bm{y})V^{4}_{ab}(0){\mathcal{O}_{b}^{(S_{2})}}^{\dagger}(t_{1},\bm{x})\right\rangle,$
(26)
where, as in Eq. (16), $\mathcal{O}_{a}^{(S)}$ is a smeared or local clover-
staggered meson operator with mass chosen to optimize the signal, and
$V^{4}_{ab}(x)=\bar{\chi}(x)[\Omega^{\dagger}(x)\gamma^{4}\Omega(x)]_{ab}\chi(x)$
(27)
is the temporal component of the staggered-staggered vector current. The
three-point functions $C_{3}$ are computed from the same staggered quarks used
for the clover-staggered two point functions. The staggered quark is
transformed into an improved naive quark by applying the $\Omega$ matrix; this
naive quark at time $t_{1}$ is then used as the source term when computing the
charm propagator. We smear the source at $t_{1}$ so that $S_{1}=S_{2}$.
We compute $Z_{V^{4}_{qq}}$using a $D_{q}$ meson [cf. Eq. (16)], which
provides a good signal. The three-point function
$C_{3}^{(S_{1}S_{2})}(t_{2},0,t_{1})$ contains states of both the desired and
the opposite parity, with the latter carrying oscillating $(-1)^{t}$
dependence. We construct $C_{3}^{(S_{1}S_{2})}(t_{2},0,t_{1})$ with local
sources $S_{1}=S_{2}=\delta$ and compute it at multiple even and odd values of
$t_{1}$ and $t_{2}$ in order to disentangle the ground-state amplitude from
the other contributions.
Within the time range $t_{1}<0<t_{2}$ and in the limit of large separations,
$|t_{1}|,t_{2}\gg a$,
$\displaystyle C_{3}^{(\delta,\delta)}(t_{2},0,t_{1})$ $\displaystyle=$
$\displaystyle Z_{V^{4}_{qq}}^{-1}A^{2}\exp\left(-E(t_{2}-t_{1})\right)$ (28)
$\displaystyle+$ $\displaystyle
Z^{\prime}AB\left[(-1)^{t_{1}}\exp\left(E^{\prime}t_{1}-Et_{2}\right)+(-1)^{t_{2}}\exp\left(Et_{1}-E^{\prime}t_{2}\right)\right]$
$\displaystyle+$ $\displaystyle
Z^{\prime\prime}B^{2}(-1)^{(t_{1}+t_{2})}\exp\left(-E^{\prime}(t_{2}-t_{1})\right)+\ldots,$
neglecting contributions from excited states. We extract $Z_{V^{4}_{qq}}$ from
a minimum $\chi^{2}$ fit to the three-point function using the right-hand side
of Eq. (28) as the model function. The fit is linear in the free parameters
$Z_{V^{4}_{qq}}^{-1}$, $Z^{\prime\prime}$ and $Z^{\prime}$, while we fix the
ground-state energies $E$ and $E^{\prime}$, and the operator overlaps $A$ and
$B$ to the values determined by fitting the two-point function
$C_{2}^{(\delta)}(t,\bm{0})$. We use a single-elimination jackknife procedure
to compute the data covariance matrix.
Table 9 presents our results for $Z_{V^{4}_{qq}}$ on the ensembles used in
this work.
Table 9: Light-light vector current renormalization factor $Z_{V^{4}_{qq}}$. Values in bold are used in computing the heavy-light current renormalizations. With our conventions, the tree-level value of $Z_{V^{4}_{qq}}$ is 2. A colon is used to represent the range of time values included in the fit. $\approx a$ [fm] | $am_{l}$/ | $am_{h}$ | $n_{\textrm{conf}}$ | $-t_{1}$ | $t_{2}$ | $am_{q}$ | $Z_{V^{4}_{qq}}$
---|---|---|---|---|---|---|---
0.09 | 0.0124/ | 0.031 | 518 | 23:12 | 11:13 | 0.0272 | 1.868(49)
| | | 23:12 | 11:13 | 0.0124 | 1.883(69)
0.12 | 0.01/ | 0.05 | 592 | 15:9 | 7:11 | 0.03 | 1.853(45)
| 0.007/ | 0.05 | 523 | 20:7 | 7:12 | 0.03 | 1.882(56)
0.15 | 0.0097/ | 0.0484 | 631 | 20:5 | 4:12 | 0.0484 | 1.704(34)
| | | 20:5 | 4:12 | 0.029 | 1.709(40)
| | | 20:5 | 4:12 | 0.0242 | 1.711(42)
| | | 20:5 | 4:12 | 0.0194 | 1.707(45)
| | | 20:5 | 4:12 | 0.0097 | 1.662(55)
The three-point functions for the $Z_{V^{4}_{qq}}$ calculation are generated
at a single source time, $t_{\mathrm{src}}=0$ (instead of the four used for
two-point functions $\Phi_{2}^{(S)}$ and $C_{2}^{(S_{1}S_{2})}$). At $a\approx
0.12~{}\textrm{fm}$ we have results at two values of the sea quark masses
which are consistent within errors. At $a\approx 0.09$ and
$0.15~{}\textrm{fm}$ we have results for several values of $m_{q}$. We do not
see evidence for a dependence upon $m_{q}$ with current statistics. The
errors, however, increase at smaller quark mass. Hence, we use the
$Z_{V^{4}_{qq}}$ corresponding to $m_{q}\sim m_{s}$ in Eq. (22). In the table,
they are set in bold.
For the clover-clover current, we compute
$\widetilde{C}_{3}^{(S_{1}S_{2})}(t_{2},t_{1},0)=\sum_{\bm{x},\bm{y}}\left\langle{\widetilde{\mathcal{O}}^{(S_{1})\dagger}}(t_{2},\bm{y})V^{4}_{QQ}(t_{1},\bm{x})\widetilde{\mathcal{O}}^{(S_{2})}(0)\right\rangle,$
(29)
where
$V^{4}_{QQ}(x)=\bar{\Psi}(x)\gamma^{4}\Psi(x)$ (30)
is the temporal component of the (rotated) clover-clover vector current. The
clover-clover bilinear
$\widetilde{\mathcal{O}}^{(S)}(x)=\sum_{y}\bar{\psi}(y)S(x,y)\gamma^{5}s(x)$
(31)
consists of a heavy-quark field corresponding to charm or bottom, as the case
may be, and a light clover-quark field $s$ with mass chosen to provide a good
signal. At large time separations, these three-point functions are
proportional to $Z_{V^{4}_{QQ}}^{-1}$, with the proportionality coming from
$\widetilde{C}_{2}^{(S_{1}S_{2})}(t)=\sum_{\bm{x}}\left\langle\widetilde{\mathcal{O}}^{(S_{1})\dagger}(t,\bm{x})\widetilde{\mathcal{O}}^{(S_{2})}(0)\right\rangle.$
(32)
We compute $Z_{V^{4}_{QQ}}$using a $\bar{Q}s$ meson, where the strange quark
is simulated with the clover action to circumvent oscillating opposite-parity
states [cf. Eq. (31)]. We restrict our calculation of $\widetilde{C}_{2,3}$ to
$S=S_{1}=S_{2}$ using both local and 1S smearing functions. The function
$\widetilde{C}_{2}$ combines a local-local clover quark with mass around
$m_{s}$ and a heavy clover quark propagator with source and sink $S$. The
function $\widetilde{C}_{3}$ requires the same heavy- and light-quark
propagators as needed in $\widetilde{C}_{2}$. An additional heavy-quark
propagator originating from $t_{2}$ has as its source the light quark
propagator restricted to $t_{2}$, multiplied by $\gamma_{5}$ and convolved
with smearing function $S$.
Table 10: Heavy-heavy vector current renormalization factor $Z_{V^{4}_{QQ}}$computed at several $\kappa$ values, covering the charm and bottom quark masses, for three lattice spacings. $\approx a$ [fm] | $am_{l}/am_{h}$ | $n_{\textrm{conf}}\times n_{\textrm{src}}$ | $\kappa_{Q}$ | $Z_{V^{4}_{QQ}}$
---|---|---|---|---
0.09 | $0.0062/0.031$ | 1912$\,\times\,$2 | 0.1283 | 0.2749(4)
| | | 0.127 | 0.2830(4)
| | | 0.110 | 0.3856(6)
| | | 0.0950 | 0.4730(8)
| | | 0.0931 | 0.4840(9)
0.12 | $0.007/0.05$ | 2110$\,\times\,$2 | 0.124 | 0.2899(4)
| | | 0.122 | 0.3028(4)
| | | 0.116 | 0.3410(5)
| | | 0.098 | 0.4507(7)
| | | 0.086 | 0.5209(10)
| | | 0.074 | 0.5894(15)
0.15 | $0.0194/0.0484$ | 631$\,\times\,$2 | 0.122 | 0.3195(14)
| | | 0.118 | 0.3440(16)
| | | 0.088 | 0.5195(48)
| | | 0.076 | 0.5898(81)
In Eq. (31), we use a random color wall source with three dilutions for both
the heavy and light spectator quarks that originate from $t=0$. We generate
two- and three-point functions for both local-local and smeared-smeared
source-sink combinations where the smearing is applied to the heavy quark. We
compute the 2- and 3-point functions at several values of $\kappa$ spanning a
range from around the charm quark to the bottom quark. We determine
$Z_{V^{4}_{QQ}}$ from a fit to the plateaus in the jackknifed ratio of the
three-point and two-point functions. Our results are summarized in Table 10.
In order to properly normalize the derivative $d\phi/d\kappa_{Q}$ (see Eq.
(11)), we need values of $Z_{V^{4}_{QQ}}$ at $\kappa$ values other than those
used in the $Z_{V^{4}_{QQ}}$ simulations. We therefore fit the simulation
results to the interpolating quartic polynomial
$Z_{V^{4}_{QQ}}(\kappa)=1+\sum_{j=1}^{4}c_{j}\kappa^{j}$ (33)
which reproduces the infinite mass limit $Z_{V^{4}_{QQ}}\to 1$. Our codes
employ the hopping parameter version of the action; so, at tree level
$c_{1}=-6u_{0}$ and for $j>1$, $c_{j}=0$. We constrain the interpolation
parameters to the tree-level values taking $\sigma_{j}=4$ as the widths. Table
11 shows values for $Z_{V^{4}_{QQ}}$ interpolated to the nominal charm and
bottom $\kappa_{\mathrm{sim}}$ used in our decay constant runs.
Table 11: Heavy-heavy vector current renormalization factor $Z_{V^{4}_{QQ}}$ corresponding to the charm and bottom $\kappa_{\rm sim}$ values used in the decay constant simulations. | charm | bottom
---|---|---
$\approx a$ [fm] | $\kappa_{Q}$ | $Z_{V^{4}_{QQ}}$ | $\kappa_{Q}$ | $Z_{V^{4}_{QQ}}$
0.09 | 0.127 | 0.2829(4) | 0.0923 | 0.4891(9)
0.12 | 0.122 | 0.3029(4) | 0.086 | 0.5216(10)
0.15 | 0.122 | 0.3199(14) | 0.076 | 0.5868(81)
Figure 2 plots the data in Table 10 together with the interpolation of Eq.
(33).
Figure 2: Plot of $Z_{V^{4}_{QQ}}/(1-6u_{0}\kappa)$ vs. $m_{0}a/(1+m_{0}a)$
for the three lattice spacings.
To aid perturbative intuition, the values of $Z_{V^{4}_{QQ}}$ in the figure
are scaled by the tree-level expression $1-6u_{0}\kappa$; the relation between
$\kappa$ and $m_{0}a/(1+m_{0}a)$ can be inferred from Eq. (15).
## VI Chiral and continuum extrapolation
In this section, we present the combined chiral and continuum extrapolations
used to obtain the physical values of the $B_{(s)}$ and $D_{(s)}$ meson decay
constants. We first discuss the use of $SU(3)$ chiral perturbation theory for
heavy-light mesons in Sec. VI.1, giving the formulas used for the chiral fits
and describing our method for incorporating heavy-quark and light-quark
discretization effects into the extrapolation. We then show the chiral fits
for the $D$ system in Sec. VI.2, and for the $B$ system in Sec. VI.3.
### VI.1 Chiral Perturbation Theory framework
The errors introduced by the chiral and (light-quark) continuum extrapolations
are controlled with rooted staggered chiral perturbation theory (rS$\chi$PT)
Lee:1999zxa ; Aubin:2003mg applied to heavy-light mesons. In Ref.
Aubin:2005aq , the heavy-light decay constant was calculated to one-loop in
rS$\chi$PT at leading order in the heavy-quark expansion [$(1/M_{H})^{0}$],
where $M_{H}$ is a generic heavy-light meson mass. A replica trick is used in
rS$\chi$PT to take into account the effect of the fourth root of the staggered
determinant Bernard:2006zw ; Bernard:2007ma .
In addition to using the form calculated in Ref. Aubin:2005aq , we also use a
chiral fit form that includes, in the loops, the effects of hyperfine
splittings (e.g., $M_{B}^{*}-M_{B}$) and flavor splittings (e.g.,
$M_{B_{s}}-M_{B}$). These splittings are $\sim\\!100$ MeV, and so not much
smaller than $M_{\pi}$, despite the fact that they are formally of order
$1/M_{H}$. Since the lightest pseudoscalar meson masses in our simulations are
$\sim 225$ MeV, it is not immediately obvious that including the splittings is
necessary or useful. Their inclusion is motivated, first of all, by the
observation of Arndt and Lin Arndt:2004bg that finite-volume effects in the
one-loop diagrams can be substantially larger with the splittings present.
This is mainly due to the fact that accidental cancellations in finite volume
effects between different diagrams at $(1/M_{H})^{0}$ disappear once
splittings are included. As described below, it is not difficult to include
the splitting effects into the calculation of Ref. Aubin:2005aq . We also
discuss the extent to which including the splittings, but not other effects
that could occur at order $1/M_{H}$, is a systematic approximation. In
practice, we do fits both including and omitting the splittings, and use the
difference as one estimate of the chiral extrapolation error. For central
values, we include the splittings, because this yields a more conservative
estimate of finite-volume effects.
With staggered quarks, the (squared) pseudoscalar meson masses are
$M^{2}_{ab,\xi}=B_{0}(m_{a}+m_{b})+a^{2}\Delta_{\xi},$ (34)
where $m_{a}$ and $m_{b}$ are quark masses, $B_{0}$ is a parameter of
$\chi$PT, and the representation of the meson under the taste symmetry group
is labeled by $\xi=P,A,T,V,I$ Lee:1999zxa . The exact non-singlet chiral
symmetry of staggered quarks as $m_{a},m_{b}\to 0$ ensures that
$\Delta_{P}=0$. All of these pseudoscalars appear in the “pion” cloud around
the heavy-light meson in the simulation, and all of them therefore affect the
decay constant.
Working at leading order [$(1/M_{H})^{0}$] in the heavy-quark expansion and at
one loop, or next-to-leading order (NLO), in the chiral expansion, the
rS$\chi$PT expression for the decay constant with light valence quark $q$
takes the form Aubin:2005aq
$\displaystyle\phi_{H_{q}}=\phi_{H}^{0}\Bigg{[}$ $\displaystyle
1+\frac{1}{16\pi^{2}f^{2}}\frac{1+3g_{\pi}^{2}}{2}\Biggl{\\{}-\frac{1}{16}\sum_{e,\xi}\ell(M_{eq,\xi}^{2})$
(35) $\displaystyle{}-\frac{1}{3}\sum_{j\in{\cal
M}_{I}^{(2,q)}}\frac{\partial}{\partial M^{2}_{Q,I}}\left[R^{[2,2]}_{j}({\cal
M}_{I}^{(2,q)};\mu^{(2)}_{I})\ell(M_{j}^{2})\right]$
$\displaystyle{}-\biggl{(}a^{2}\delta^{\prime}_{V}\sum_{j\in\hat{\cal
M}_{V}^{(3,q)}}\frac{\partial}{\partial
M^{2}_{Q,V}}\left[R^{[3,2]}_{j}(\hat{\cal
M}_{V}^{(3,q)};\mu^{(2)}_{V})\ell(M_{j}^{2})\right]+[V\to
A]\biggr{)}\Biggr{\\}}$ $\displaystyle{}+p(m_{q},m_{l},m_{h},a^{2})\Bigg{]},$
where $m_{q}$ is the light valence-quark mass, $e$ runs over the sea quarks,
the lighter two of which have masses $m_{l}$, and the heavier, $m_{h}$.111The
physical values of the average up-down quark mass and of the strange-quark
mass are denoted by $\hat{m}=(m_{u}+m_{d})/2$ and $m_{s}$, respectively. The
parameter $\phi_{H}^{0}$ is independent of the light masses, and $p$ is an
analytic function. We fit the charm and bottom systems separately, so
$\phi_{H}^{0}$ depends, in practice, on the heavy-quark mass. The meson mass
$M_{Q,\xi}$ is similar to $M_{ab,\xi}$ in Eq. (34), but constructed from a
valence quark-antiquark, $q\bar{q}$. The light-meson decay constant $f\approx
f_{\pi}\cong 130.4$ MeV and the $H$-$H^{*}$-$\pi$ coupling $g_{\pi}$ controls
the size of the one-loop effects. Taste-violating hairpin diagrams, which
arise only at non-zero lattice spacing, are parameterized by
$\delta^{\prime}_{A}$ and $\delta^{\prime}_{V}$. The residue functions
$R^{[n,k]}_{j}(\\{M\\},\\{\mu\\})$ are defined in Ref. Aubin:2003mg . Chiral
logarithms are written in terms of the functions $\ell(M^{2})$ Bernard:2001yj
:
$\displaystyle\ell(M^{2})$ $\displaystyle=$ $\displaystyle
M^{2}\ln\frac{M^{2}}{\Lambda_{\chi}^{2}}\qquad{\rm[infinite\ volume]},$ (36)
$\displaystyle\ell(M^{2})$ $\displaystyle=$ $\displaystyle
M^{2}\left(\ln\frac{M^{2}}{\Lambda_{\chi}^{2}}+\delta_{1}(ML)\right)\qquad{\rm[spatial\
volume}\ L^{3}{\rm]},$ (37) $\displaystyle\delta_{1}(ML)$
$\displaystyle\equiv$
$\displaystyle\frac{4}{ML}\sum_{\bm{r}\neq\bm{0}}\frac{K_{1}(|\bm{r}|ML)}{|\bm{r}|}.$
(38)
Here $\Lambda_{\chi}$ is the chiral scale, $K_{1}$ the Bessel function of
imaginary argument, and $\bm{r}$ any non-zero three-vector with integer
components. The mass sets in the residue functions of Eq. (35) are
$\displaystyle\mu^{(2)}$ $\displaystyle=$
$\displaystyle\\{M^{2}_{U},M^{2}_{S}\\},$ (39) $\displaystyle{\cal M}^{(2,q)}$
$\displaystyle=$ $\displaystyle\\{M_{Q}^{2},M_{\eta}^{2}\\},$ (40)
$\displaystyle\hat{\cal M}^{(3,q)}$ $\displaystyle=$
$\displaystyle\\{M_{Q}^{2},M_{\eta}^{2},M_{\eta^{\prime}}^{2}\\},$ (41)
where $M_{U}$ ($M_{S}$) is the mass of the pseudoscalar $l\bar{l}$
($h\bar{h}$) meson.
The salient feature of the chiral extrapolation of $\phi_{H_{q}}$ is that the
chiral logs have a characteristic curvature as $m_{q}\to 0$ Kronfeld:2002ab .
At non-zero lattice spacing, the presence of the additive splittings
$a^{2}\Delta_{\xi}$ in the meson masses reduces the curvature of the chiral
logarithms. The characteristic curvature returns, however, as the continuum
limit is approached.
To combine data from several lattice spacings into one chiral extrapolation,
it is necessary to convert lattice units to (some sort of) physical units. As
mentioned in Sec. III.1, we convert in two steps, first by canceling lattice
units with the appropriate power of $r_{1}/a$. In particular, pseudoscalar
meson masses [cf. Eq. (34)] become
$r_{1}^{2}M^{2}_{ab,\xi}=(r_{1}/a)^{2}(aM_{ab,\xi})^{2}$, and the decay
constant [cf. Eq. (35)] becomes
$r_{1}^{3/2}\phi_{H}=(r_{1}/a)^{3/2}(a^{3/2}\phi_{H})$, with $a^{3/2}\phi_{H}$
determined from Analyses I or II (cf. Sec. IV). Strictly speaking, one must
take the quark mass dependence of $r_{1}$ into account, either separately or
by modifying the right-hand side of Eq. (35) accordingly. At the present level
of accuracy, we ignore this subtlety, canceling units ensemble-by-ensemble
with the computed $r_{1}/a$. Since $r_{1}$ is expected to depend smoothly on
$m_{l}$ and $m_{h}$, we are unlikely to introduce an uncontrolled error into
the extrapolated decay constants. (After completing the chiral-continuum
extrapolation in $r_{1}$ units, we then use $r_{1}=0.3117(22)$ fm (cf. Sec.
III.1) to convert to MeV.)
To quantify the size of NLO (and higher) corrections to $\chi$PT, it is useful
to define dimensionless parameters $x_{q}$, $x_{l}$ and $x_{h}$ proportional
to the quark masses $m_{q}$, $m_{l}$ and $m_{h}$:
$x_{q,l,h}\equiv\frac{(r_{1}B_{0})(r_{1}/a)(2am_{q,l,h})}{8\pi^{2}f_{\pi}^{2}r_{1}^{2}}\
.$
Since the splittings $a^{2}\Delta_{\xi}$ are added to the quark mass terms in
Eq. (34), it is similarly useful to define
$\displaystyle x_{\Delta_{\xi}}$ $\displaystyle\equiv$
$\displaystyle\frac{r_{1}^{2}a^{2}\Delta_{\xi}}{8\pi^{2}f_{\pi}^{2}r_{1}^{2}},$
(42) $\displaystyle x_{\bar{\Delta}}$ $\displaystyle\equiv$
$\displaystyle\frac{r_{1}^{2}a^{2}\bar{\Delta}}{8\pi^{2}f_{\pi}^{2}r_{1}^{2}},$
(43)
where $\bar{\Delta}$ is the average pion splitting
$\bar{\Delta}={\textstyle\frac{1}{16}}(\Delta_{P}+4\Delta_{A}+6\Delta_{T}+4\Delta_{V}+\Delta_{I}).$
(44)
The $x_{i}$ are in “natural” units for $\chi$PT, in the sense that one expects
that chiral corrections, when written as series in the $x_{i}$, have
coefficients [or low-energy constants (LECs)] that are of order 1.
We then take the analytic function $p$ in Eq. (35) to have the following form
at NLO
$L_{\rm val}(x_{q}+x_{\Delta_{\rm val}})+L_{\rm
sea}(2x_{l}+x_{h}+3x_{\Delta_{\rm
sea}})+L_{a}\frac{a^{2}}{16\pi^{2}f_{\pi}^{2}r_{1}^{4}},$ (45)
where $L_{\rm val}$, $L_{\rm sea}$ and $L_{a}$ are quark-mass-independent LECs
that we fit from our data, and we define
$\displaystyle x_{\Delta_{\rm val}}$ $\displaystyle\equiv$
$\displaystyle\frac{9}{5}x_{\bar{\Delta}}-\frac{4}{5}x_{\Delta_{I}},$ (46)
$\displaystyle x_{\Delta_{\rm sea}}$ $\displaystyle\equiv$
$\displaystyle\frac{9}{11}x_{\bar{\Delta}}+\frac{2}{11}x_{\Delta_{I}},$ (47)
The low-energy constants $L_{\rm val}$, $L_{\rm sea}$ and $L_{a}$ depend
implicitly on the chiral scale $\Lambda_{\chi}$, so that the complete
expression, Eq. (35), is independent of $\Lambda_{\chi}$. As in Ref.
Aubin:2005aq , we choose to include the $a^{2}$ dependent terms
$x_{\Delta_{\rm sea}}$ and $x_{\Delta_{\rm val}}$ in the coefficients of
$L_{\rm val}$ and $L_{\rm sea}$ so that these coefficients represent those
combinations of meson masses that arise naturally under a change of
$\Lambda_{\chi}$ in the chiral logarithms.
The LEC $L_{a}$ arises from analytic taste-violating effects; it serves as a
counterterm to absorb changes proportional to the taste-violating hairpins
$\delta^{\prime}_{A}$ and $\delta^{\prime}_{V}$ under a change in chiral
scale. As such, we take the $a^{2}$ coefficient of $L_{a}$ in Eq. (45) to vary
with lattice spacing like $x_{\Delta_{\rm val}}$. As long as $L_{a}$ then
appears as an independent fit parameter, the introduction of the
$x_{\Delta_{\rm sea}}$ and $x_{\Delta_{\rm val}}$ terms in the coefficients of
$L_{\rm val}$ and $L_{\rm sea}$ in Eq. (45) has a negligible effect on the
results from the chiral fits. However, we find that the introduction of these
terms significantly reduces the magnitude of $L_{a}$; in other words, most of
the discretization error from the light quarks appears to be due to the
$a^{2}$ dependence of the light meson masses in the chiral loops. We leave
$L_{\rm val}$ , $L_{\rm sea}$ and $L_{a}$ unconstrained in the fits that
determine central values; their size is of $\mathrm{O}(1)$ as expected (and is
in fact $\leq 0.6$).
In the region of the strange-quark mass, the data for the decay constants show
some curvature, and at least some quadratic terms in the quark masses (NNLO
effects) must in general be added in order to obtain acceptable ($p>0.01$)
fits. There are four such LECs, giving a NNLO contribution to $p$ of the form
$Q_{1}x_{q}^{2}+Q_{2}(2x_{l}+x_{h})^{2}+Q_{3}x_{q}(2x_{l}+x_{h})+Q_{4}(2x_{l}^{2}+x_{h}^{2}).$
(48)
Fits omitting the $Q_{1}$ and $Q_{3}$ terms give poor confidence levels and
are rejected; adding the $Q_{2}$ and $Q_{4}$ terms does not change the fit
results much, but increases over-all errors by up to 30%. To be conservative,
we include all four terms in fits for central values; other acceptable fits
(for example, fixing $Q_{2}$ or $Q_{4}$ or both to zero) are included among
the alternatives used to estimate the systematic error of the chiral
extrapolation.
For the central-value fits, the $Q_{i}$ are mildly constrained by Gaussian
priors with central value 0 and width 0.5, since that is roughly the expected
size in natural units. After fitting, the posterior values satisfy
$|Q_{i}|\leq 0.5$, and $Q_{1}$ and $Q_{3}$ have errors $\approx 0.05$ (much
less than the prior width), indicating that they are constrained by the data.
$Q_{2}$ and $Q_{4}$ have errors $\sim 0.5$, indicating that they are largely
constrained by the priors. Changing the prior widths for the $Q_{i}$ to 1.0
has a negligible effect on central values and errors of the decay constants,
although the posterior $Q_{2}$ and $Q_{4}$ typically increase in size and
error, as expected.
While the chiral form introduced so far gives acceptable simultaneous fits to
our data from all available lattice spacings, we still need to estimate the
size of heavy-quark and generic light-quark discretization errors. Following
the Bayesian approach advocated in Refs. Lepage:2001ym ; Morningstar:2001je ,
we add constrained lattice-spacing-dependent terms to the fit function until
the statistical errors of the results cease to increase appreciably. For the
heavy quark, we take up to six such terms, $f_{E}(m_{0}a)$, $f_{X}(m_{0}a)$,
$f_{Y}(m_{0}a)$, $f_{B}(m_{0}a)$, $f_{3}(m_{0}a)$, and $f_{2}(m_{0}a)$, where
$m_{0}$ is the heavy quark bare mass. Details about the origin and form of
these six functions are given in Appendix A. These functions estimate
fractional (not absolute) errors, and as such are included within the square
brackets in Eq. (35) (or its equivalent, Eq. (53) below). The first three are
$\mathrm{O}(a^{2})$ corrections and are added to the fit with coefficients
$z_{i}\,(a\Lambda)^{2}$, $i\in\\{E,X,Y\\}$, where $\Lambda$ is a scale
characteristic of the heavy-quark expansion, and the $z_{i}$ are parameters
with prior value 0 and prior width 1 (for $f_{Y}$) or $\sqrt{2}$ (for $f_{E}$
and $f_{X}$, since they each appear twice in the analysis of Appendix A). The
next two terms are $\mathrm{O}(\alpha_{s}a)$ corrections, added with
coefficients $z_{i}\,\alpha_{s}a\Lambda$, $i\in\\{B,3\\}$, with $z_{i}$ taken
to have prior value 0 and prior width 1 (for $f_{B}$) or $\sqrt{2}$ (for
$f_{3}$, again because it appears twice). The final term arises from the
propagation to the decay constants of heavy-quark errors in the tuning of the
heavy-quark hopping parameter, $\kappa$. It comes in with coefficient
$z_{2}\,(a\Lambda)^{3}$, with $z_{2}$ having prior value 0 and prior width 1.
We take a large value $\Lambda=700$ MeV, which provides conservatively wide
priors, especially for the first five terms. Once one of each of the first two
types of terms is added, the errors already reach $\sim\\!80\%$ of their
values with all six added.
Similar terms representing generic light-quark errors, which are not
automatically included in the fit function (unlike taste-violating terms), may
also be added. With the asqtad staggered action, generic discretization
effects are of $\mathrm{O}(\alpha_{s}a^{2})$. We allow the physical LECs
$\phi_{H}^{0}$, $L_{\rm val}$, $L_{\rm sea}$, $Q_{1}$, $Q_{2}$, $Q_{3}$, and
$Q_{4}$, to have small relative variations with lattice spacing with
coefficients $C_{i}\alpha_{s}(a\Lambda)^{2}$, where $i$ stands for any of the
seven physical LECs, $\Lambda$ is again taken to be 700 MeV, and the $C_{i}$
have prior value 0 with prior width 1. This corresponds to a maximum of about
a $3\%$ difference for a given LEC between the $a\approx 0.12$ fm and the
$a\approx 0.09$ fm ensembles. Once several heavy-quark discretization terms
have been introduced, these light-quark terms further increase the total error
of individual decay constants by less than $10\%$. However, the errors on the
decay constant ratios $f_{D_{s}}/f_{D^{+}}$ and $f_{B_{s}}/f_{B^{+}}$ are
significantly increased by light-quark discretization effects, because the
heavy-quark effects on the ratios cancel to first approximation. For our
central values, we include all six heavy-quark and all seven light-quark
terms, so the total error from a given fit should estimate all (taste-
conserving) discretization errors, as well as normal statistical effects. To
estimate “heavy-quark” and “light-quark” discretization effects separately, we
set to zero the light- or heavy-quark discretization terms, respectively, and
then subtract the statistical errors in quadrature. Such separate errors are
not relevant to any final results quoted below, but are included as separate
lines in the error budget for informational purposes.
As mentioned above, our preferred fit form modifies Eq. (35) by including the
effects of hyperfine and flavor splittings of the heavy-light mesons in one-
loop diagrams. We now briefly describe how one may adjust the results of Ref.
Aubin:2005aq to include these splittings. In Eq. (35), the contributions
proportional to $g_{\pi}^{2}$ come from diagrams with internal $H^{*}$
propagators, namely the left-hand diagrams in Fig. 5 of Ref. Aubin:2005aq .
Contributions with no factor of $g_{\pi}^{2}$ come from diagrams with light-
meson tadpoles, namely the right-hand diagrams in Fig. 5 of Ref. Aubin:2005aq
. The latter have no internal heavy-light propagators, so are unaffected by
any heavy-light splittings. The splittings in the former diagrams depend on
whether the light-meson line is connected (Fig. 5a, left, of Ref. Aubin:2005aq
), or disconnected (Fig. 5b, left). In the disconnected case, the $H^{*}$ in
the loop always has the same flavor ($q$) as the external $H_{q}$, so there is
no flavor splitting between the two, only a hyperfine splitting. In the
connected case, the $H^{*}$ in the loop has the flavor of the virtual sea
quark loop (which we labeled by $e$ in Eq. (35)), so there is flavor splitting
with the external $H_{q}$, in addition to the hyperfine splitting.
Let $\Delta^{*}$ be the (lowest-order) hyperfine splitting, and $\delta_{eq}$
be the flavor splitting between a heavy-light meson with light quark of flavor
$e$ and one of flavor $q$. At lowest order, $\delta_{eq}$ is proportional to
the quark-mass difference (or light-meson squared mass difference), which can
be written in terms of a parameter $\lambda_{1}$:
$\delta_{eq}\cong
2\lambda_{1}B_{0}(m_{e}-m_{q})\cong\lambda_{1}(M^{2}_{E}-M^{2}_{Q}),$ (49)
where $M_{E}$ is the mass of an $e\bar{e}$ light meson. Here we have used the
notation of Arndt and Lin Arndt:2004bg and included a factor of $B_{0}$ in
the middle expression; $B_{0}$ is omitted in the notation of Ref. Boyd:1994pa
, Eq. (16), and of Ref. Aubin:2005aq , Eq. (45).
By convention, the mass of the external $H$ is removed in the heavy quark
effective theory, so the mass shell is at $\bm{k}=\bm{0}$, where $\bm{k}$ is
the external three-momentum. When there is no splitting, the internal $H^{*}$
has its pole at the same place, which makes the integrals particularly simple,
giving the chiral log function $\ell(M^{2})$, Eq. (36). If a splitting
$\Delta$ is present, the integrals involve a significantly more complicated
function, which we denote
$J(M,\Delta)=(M^{2}-2\Delta^{2})\log(M^{2}/\Lambda^{2})+2\Delta^{2}-4\Delta^{2}F(M/\Delta)\qquad\textrm{[infinite
volume]}.$ (50)
Here the function $F$ is most simply expressed Stewart:1998ke ;
Becirevic:2003ad
$F(1/x)=\begin{cases}-\frac{\sqrt{1-x^{2}}}{x_{\phantom{g}}}\left[\frac{\pi}{2}-\tan^{-1}\frac{x}{\sqrt{1-x^{2}}}\right],&\text{if
$|x|\leq 1$,}\\\ \frac{\sqrt{x^{2}-1}}{x}\ln(x+\sqrt{x^{2}-1}),&\text{if
$|x|\geq 1$,}\end{cases}$ (51)
which is valid for all $x$.
It is then straightforward to write down the generalization of Eq. (35) to
include splittings. The basic rule is to replace
$\ell(M^{2})\to J(M,\Delta)$ (52)
in the terms proportional to $g_{\pi}^{2}$. It is not hard to show that
$J(M,0)=\ell(M^{2})$, so this replacement is consistent with the original
result neglecting the splittings. In making the replacements, one must choose
the correct value of the splitting $\Delta$ in each term. As mentioned above,
in terms that come from the diagram with a disconnected light-meson
propagator, one must put $\Delta=\Delta^{*}$. But in terms that come from the
diagram with a connected light-meson propagator, one must put
$\Delta=\Delta^{*}+\delta_{eq}$, because the internal heavy-light meson is a
$H^{*}_{e}$, while the external meson is an $H_{q}$. The result for the heavy-
light meson decay amplitude including the splittings is then
$\displaystyle\phi_{H_{q}}=\phi_{H}^{0}\Bigg{[}$ $\displaystyle
1+\frac{1}{16\pi^{2}f^{2}}\frac{1}{2}\Biggl{\\{}-\frac{1}{16}\sum_{e,\Xi}\ell(M_{eq,\Xi}^{2})$
(53) $\displaystyle{}-\frac{1}{3}\sum_{j\in{\cal
M}_{I}^{(2,x)}}\frac{\partial}{\partial M^{2}_{X,I}}\left[R^{[2,2]}_{j}({\cal
M}_{I}^{(2,x)};\mu^{(2)}_{I})\ell(M_{j}^{2})\right]$
$\displaystyle{}-\biggl{(}a^{2}\delta^{\prime}_{V}\sum_{j\in\hat{\cal
M}_{V}^{(3,x)}}\frac{\partial}{\partial
M^{2}_{X,V}}\left[R^{[3,2]}_{j}(\hat{\cal
M}_{V}^{(3,x)};\mu^{(2)}_{V})\ell(M_{j}^{2})\right]+[V\to A]\biggr{)}$
$\displaystyle{}-3g_{\pi}^{2}\frac{1}{16}\sum_{e,\Xi}J(M_{eq,\Xi},\Delta^{*}+\delta_{eq})$
$\displaystyle{}-g_{\pi}^{2}\sum_{j\in{\cal
M}_{I}^{(2,x)}}\frac{\partial}{\partial M^{2}_{X,I}}\left[R^{[2,2]}_{j}({\cal
M}_{I}^{(2,x)};\mu^{(2)}_{I})J(M_{j},\Delta^{*})\right]$
$\displaystyle{}-3g_{\pi}^{2}\biggl{(}a^{2}\delta^{\prime}_{V}\sum_{j\in\hat{\cal
M}_{V}^{(3,x)}}\frac{\partial}{\partial
M^{2}_{X,V}}\left[R^{[3,2]}_{j}(\hat{\cal
M}_{V}^{(3,x)};\mu^{(2)}_{V})J(M_{j},\Delta^{*})\right]+[V\to
A]\biggr{)}\Biggr{\\}}$ $\displaystyle{}+p(m_{q},m_{l},m_{h},a^{2})\Bigg{]}.$
It is also straightforward to include finite-volume effects into Eq. (53). One
simply replaces
$J(M,\Delta)\to J(M,\Delta)+\delta J(M,\Delta,L),$ (54)
where $\delta J(M,\Delta,L)$ is the finite-volume correction in a spatial
volume $L^{3}$. The correction can be written in terms of functions defined in
Refs. Arndt:2004bg ; Aubin:2007mc :
$\delta
J(M,\Delta,L)=\frac{M^{2}}{3}\delta_{1}(ML)-16\pi^{2}\left[\frac{2\Delta}{3}J_{FV}(M,\Delta,L)+\frac{\Delta^{2}-M^{2}}{3}K_{FV}(M,\Delta,L)\right]\
,$ (55)
with
$K_{FV}(M,\Delta,L)\equiv\frac{\partial}{\partial\Delta}J_{FV}(M,\Delta,L),$
(56)
and $\delta_{1}(ML)$ as given in Eq. (38).
Before turning to the fit details and results, we briefly discuss the extent
to which including the splittings as in Eq. (53), and not other possible
$1/M_{H}$ effects, is a systematic improvement on Eq. (35). In fact, in a
parametric sense within the power counting introduced by Boyd and Grinstein
Boyd:1994pa , this is a systematic improvement, as long as we make some
further specifications as to how Eq. (53) should be applied. As we detail
below, however, the power counting of Ref. Boyd:1994pa is only marginally
applicable to our data. For that reason we ultimately fit to both Eq. (53) and
Eq. (35) and take the difference as one measure of the chiral extrapolation
error.
For the following discussion, let $\Delta$ be a generic splitting
($\Delta^{*}$ or $\delta_{eq}$ or a linear combination of the two), and $M$ be
a generic light pseudoscalar mass. The power counting introduced in Ref.
Boyd:1994pa takes
$\frac{\Delta^{2},\;\Delta M,\;M^{2}}{M_{H}}\ll\Delta\sim M.$ (57)
For our data, treating $\Delta$ and $M$ as the same size is not dangerous,
even though $\Delta$ is significantly smaller than our simulation $M$
values—at worst this means that we include some terms unnecessarily. The
condition $M^{2}/M_{H}\ll\Delta$, which is necessary to drop other $1/M_{H}$
contributions as still higher order, is marginally valid, however. For the $D$
system, $M_{K}^{2}/M_{D}\approx 130~{}\textrm{MeV}$, which is roughly of the
same size as $\Delta^{*}$ and $\delta_{sd}$. For the $B$ system,
$M_{K}^{2}/M_{B}\approx 47~{}\textrm{MeV}$, of the same size as $\Delta^{*}$
but somewhat less than $\delta_{sd}$. For the purposes of the chiral
extrapolation, however, what matters is the applicability of the power
counting at the lowest simulated light meson masses, not its applicability at
$M_{K}$.222We assume here that the fit to the data is good over the full mass
range simulated. It is not important for the chiral extrapolation that the fit
be systematic in the region around $M_{K}$, but it must describe the data in
that range so that we can correctly interpolate to the physical kaon mass. For
our lightest simulated pions with mass $\sim M_{K}/2$, we can reduce the left
hand side of the inequality in Eq. (57) by a factor of four, at which point it
becomes reasonably applicable.
Having tentatively accepted the power counting of Eq. (57), it is clear that
$F(M/\Delta)$ in Eq. (50) should be treated as $\mathrm{O}(1)$. Then the
difference between $J(M,\Delta)$ and the chiral logarithm it replaces,
$\ell(M^{2})$ is of the same order as $\ell(M^{2})$ itself, so including the
splittings becomes mandatory at the one-loop order to which we are working.
The next question is whether Eq. (35) includes _all_ effects to this order. As
discussed by Boyd and Grinstein, the key issue is whether operators with two
or more derivatives (two or more powers of residual momentum $\bm{k}$) on the
heavy fields can contribute. Such operators are suppressed by $1/M_{H}$
relative to the leading-order heavy-light Lagrangian, which has a single
derivative. Since we are keeping $\Delta^{*}$, which is also in principle a
$1/M_{H}$ effect, one might worry that such operators could contribute at the
same order. The power counting implies, however, that the relevant diagrams
pick up a factor of $(\Delta,M)/M_{H}$ relative to the terms being kept in Eq.
(53). The reason for the difference is that the explicit extra factor of
$\bm{k}$ turns into $\Delta$ or $M$—the only dimensional constants
available—after integration. In the term that generates the hyperfine
splitting itself, in contrast, the dimensional quantity balanced against
$1/M_{H}$ is $\Lambda$—a heavy-quark QCD scale—rather than $M$. The power
counting in Eq. (57) effectively treats $\Lambda$ as larger than $M$ (so that
$\Delta\sim\Lambda^{2}/M_{H}\sim M$). Similarly, the term that generates the
flavor splittings has a single factor of $m_{q}$ and no residual momentum, and
Eq. (57) effectively takes $m_{q}\sim\bm{k}$ in such terms.
Boyd and Grinstein do find some other contributions at the same order as Eq.
(53), but most come from terms that are simply $\Lambda/M_{H}$ times terms in
the leading-order heavy-light Lagrangian or current, and thus give simply an
overall factor times the result without them. The exceptions are the terms
multiplied by $g_{2}$ in Eq. (15) of Ref. Boyd:1994pa and by $\rho_{2}$ in
Eq. (18) of Ref. Boyd:1994pa . These are operators that have the same
dimension as the original Lagrangian current operators, but that violate
heavy-quark spin symmetry, and therefore give different contributions to the
pseudoscalar and vector meson decay constants at this order. Since we are only
looking at pseudoscalar meson decay constants here, however, and since these
effects are flavor-independent, we can also absorb all of the $1/M_{H}$
effects into (1) the effects of the splittings in the loop, described by Eq.
(53), and (2) an overall factor in front of the full one-loop result.
The overall factor in Eq. (53) is $1/(16\pi^{2}f^{2})$. Since $f$ is not fixed
at one loop, one should in any case allow it to vary over a reasonable range,
which we take to be $f_{\pi}$ to $f_{K}$. We allow such variations even when
we fit to the form without splittings, Eq. (35). The difference between using
$f_{\pi}$ and $f_{K}$ corresponds to a 45% change in the size of the one-loop
coefficient, but produces only a 1 to 3 MeV change in the decay
constants.333Most of the change in the size of the overall coefficient is
compensated by a change in the LECs that come from the fit to our data. We
therefore assume that any further $1/M_{H}$ uncertainty in
$1/(16\pi^{2}f^{2})$ has negligible effects on our results.
Finally, there is a question of whether terms coming from taste violations
contribute something new at the same order in which we include splittings.
Since taste-violating terms in the Lagrangian can enter just like light-quark
masses, this is a possibility in principle. Corresponding to the terms in the
quark masses that generate flavor splittings of heavy-light mesons (cf. Eq.
(45) of Ref. Aubin:2005aq ), there are taste-violating terms given in Eq. (51)
of that paper. Just as for the quark-mass terms, however, we are only
interested here in contributions that change the heavy-light meson mass, not
ones coupling the mesons to pion fields. When the pion fields are set to zero,
all the terms in Eq. (51) of Ref. Aubin:2005aq just give a constant heavy-
light meson mass term proportional to $a^{2}$ that contributes equally to the
$H$ and $H^{*}$ masses of all valence flavors. Terms that produce a hyperfine
splitting would have to also violate heavy quark spin symmetry, and hence be
of order $a^{2}\Lambda/M_{H}$. Similarly, terms that produce flavor splitting
would need to violate flavor symmetry, and hence be of order
$a^{2}m_{q}/\Lambda_{\chi}$. Both such contributions are higher order in our
power-counting. Since there is no splitting, there is no contribution to the
decay constants because the effect will vanish when we put the external $B$ or
$D$ meson on mass shell.
In our chiral fits, we take the physical light-quark masses, as well as the
parameters $B_{0}$, $a^{2}\Delta_{\xi}$, $\delta^{\prime}_{A}$, and
$\delta^{\prime}_{V}$, from the MILC Collaboration’s results of rS$\chi$PT
fits to light pseudoscalars masses and decay constants Aubin:2004fs ;
Bernard:2007ps on ensembles that include lattice spacing $a\approx 0.15$ fm
through $a\approx 0.06$ fm. Table 12 shows the values used.
Table 12: Inputs to our heavy-light chiral fits taken from the MILC Collaboration’s light-meson chiral fits Aubin:2004fs ; Bernard:2007ps . The physical bare-quark masses $m_{u}$, $m_{d}$, $\hat{m}\equiv(m_{u}+m_{d})/2$, and $m_{s}$ are determined by demanding that the charged pion and kaons take their physical masses after the removal of electromagnetic effects. Errors in the masses are due to statistics, chiral extrapolation systematics, scale determination, and (for $m_{d}$ and $m_{u}$) the estimate of electromagnetic effects, respectively. “Continuum” values are found from chiral fits that have been extrapolated to the continuum, but masses are still in units of the “fine” ($a\approx 0.09$ fm) lattice spacing, and with the fine-lattice value of the mass renormalization. Values for $r_{1}^{2}a^{2}\delta^{\prime}_{A}$ and $r_{1}^{2}a^{2}\delta^{\prime}_{V}$ take into account newer MILC analyses Bazavov:2009fk as noted in the text. The light-meson analysis determining these quantities assumes that they scale like the taste-violating splittings $\Delta_{\xi}$ and are larger by a factor of 1.68 on the $0.15~{}$fm lattices than on the $0.12~{}$fm lattices, and smaller by a factor 0.35 on the $0.09~{}$fm lattices than on the $0.12~{}$fm lattices. The statistical and systematic errors on $r_{1}B_{0}$ and $r_{1}^{2}a^{2}\Delta_{\xi}$ are not given here; such errors have negligible effect on the heavy-light decay constants. Quantity | Lattice spacing
---|---
| $a\approx 0.15~{}$fm | $a\approx 0.12~{}$fm | $a\approx 0.09~{}$fm | “continuum”
$am_{s}\times 10^{2}$ | $4.29(1)(8)(6)$ | $3.46(1)(10)(5)$ | $2.53(0)(6)(4)$ | $2.72(1)(7)(4)$
$a\hat{m}\times 10^{3}$ | $1.55(0)(3)(2)$ | $1.25(0)(4)(2)$ | $0.927(2)(27)(13)$ | $0.997(2)(32)(14)$
$am_{d}\times 10^{3}$ | $2.20(0)(4)(3)(5)$ | $1.78(0)(6)(3)(4)$ | $1.31(0)(4)(2)(3)$ | $1.40(0)(5)(2)(3)$
$am_{u}\times 10^{4}$ | $8.96(2)(17)(13)(49)$ | $7.31(2)(23)(10)(40)$ | $5.47(1)(16)(8)(30)$ | $5.90(1)(19)(9)(32)$
$r_{1}B_{0}$ | 6.43 | 6.23 | 6.38 | 6.29
$r_{1}^{2}a^{2}\Delta_{A}$ | $\hphantom{-}0.351$ | $\hphantom{-}0.205$ | $\hphantom{-}0.0706$ | 0
$r_{1}^{2}a^{2}\Delta_{T}$ | $\hphantom{-}0.555$ | $\hphantom{-}0.327$ | $\hphantom{-}0.115$ | 0
$r_{1}^{2}a^{2}\Delta_{V}$ | $\hphantom{-}0.721$ | $\hphantom{-}0.439$ | $\hphantom{-}0.152$ | 0
$r_{1}^{2}a^{2}\Delta_{I}$ | $\hphantom{-}0.897$ | $\hphantom{-}0.537$ | $\hphantom{-}0.206$ | 0
$r_{1}^{2}a^{2}\delta^{\prime}_{A}$ | — | $-0.28(6)$ | — | 0
$r_{1}^{2}a^{2}\delta^{\prime}_{V}$ | — | $\hphantom{-}0.00(7)$ | — | 0
In general, we use older MILC determinations since newer versions, e.g., those
in Ref. Bazavov:2009fk , do not cover the full range of lattice spacings
employed here (but are consistent where they overlap). The exceptions are the
values of the taste-violating hairpin parameters
$r_{1}^{2}a^{2}\delta^{\prime}_{A}$ and $r_{1}^{2}a^{2}\delta^{\prime}_{V}$.
For them, the newer analysis including two-loop chiral logarithms gives larger
systematic errors and a changed sign of the central value of
$r_{1}^{2}a^{2}\delta^{\prime}_{V}$, which has always been consistent with
zero. For these parameters, we therefore use the wider ranges listed in Table
12, which encompasses both types of analyses. For comparison, the results of
the analysis of Ref. Bernard:2007ps were
$r_{1}^{2}a^{2}\delta^{\prime}_{A}=-0.30(1)(4)$ and
$r_{1}^{2}a^{2}\delta^{\prime}_{V}=-0.05(2)(4)$.
In order to fit Eq. (53) to our lattice data, it is also necessary to input
values for the hyperfine splitting $\Delta^{*}$ and for $\lambda_{1}$ in Eq.
(49). For $B$ mesons, we have Nakamura:2010zzi
$\displaystyle\Delta^{*}=M_{B^{*}}-M_{B}$ $\displaystyle\approx$
$\displaystyle 45.8~{}\textrm{MeV},$ (58)
$\displaystyle\delta_{sd}=M_{B_{s}}-M_{B}$ $\displaystyle\approx$
$\displaystyle 87.0~{}\textrm{MeV},$ (59) $\displaystyle\lambda_{1}$
$\displaystyle\approx$ $\displaystyle 0.192~{}\textrm{GeV}^{-1},$ (60)
where we use $M_{E}=M_{S}=0.6858(40)~{}\textrm{GeV}$ Davies:2009tsa and
$M_{Q}=M_{\pi^{0}}\approx 135.0~{}\textrm{MeV}$ to obtain $\lambda_{1}$ from
the experimental data. Similarly, for $D$ mesons, we have
$\displaystyle\Delta^{*}=M_{D_{0}^{*}}-M_{D_{0}}$ $\displaystyle\approx$
$\displaystyle 142.1~{}\textrm{MeV},$ (61)
$\displaystyle\delta_{sd}=M_{D_{s}}-M_{D_{\pm}}$ $\displaystyle\approx$
$\displaystyle 98.9~{}\textrm{MeV},$ (62) $\displaystyle\lambda_{1}$
$\displaystyle\approx$ $\displaystyle 0.219~{}\textrm{GeV}^{-1}.$ (63)
In the chiral fit, we input the relevant physical $\Delta^{*}$ and
$\lambda_{1}$ from either Eqs. (58)–(60) or (61)–(63), and then use Eq. (49)
with the actual $m_{e}$ and $m_{q}$ from each data point, and $B_{0}$ the
slope for a given ensemble, from Table 12. We emphasize here that $B_{0}$
comes from a simple tree-level chiral fit of light meson masses to Eq. (34).
This is adequate for our purposes, since the resulting meson masses are only
used within the one-loop chiral logarithms.
We can now present the actual chiral fits and show how we extract results and
systematic errors from them. Recall that we compute $\phi_{H_{q}}$ for many
combinations of the valence and light sea-quark masses, and at three lattice
spacings: $a\approx$ 0.15, 0.12, and 0.09 fm. We fit all the decay constant
data to the form given either by Eq. (53) or by Eq. (35). One-loop finite-
volume effects are included through Eq. (54) or Eq. (37). There are four
unconstrained free parameters in our fits: the LO parameter $\phi^{0}_{H}$,
and the one-loop LECs $L_{\rm val}$, $L_{\rm sea}$, $L_{a}$ [Eq. (45)]. The
central fit fixes the chiral coupling $f$ at $f_{\pi}$, but a range of
couplings are considered in alternative fits, as described in more detail in
Sec. VII. Similarly, the $H$-$H^{*}$-$\pi$ coupling $g_{\pi}$, which is poorly
constrained by our data, is taken in the range $0.51\pm 0.20$. This
encompasses a range of phenomenological and lattice determinations
Casalbuoni:1996pg ; Stewart:1998ke ; Anastassov:2001cw ; Abada:2002vj ;
Arnesen:2005ez ; Ohki:2008py ; Bulava:2010ej , as discussed in Ref.
Bernard:2008dn . In the central fit, $g_{\pi}$ is held fixed at 0.51, while it
is varied in alternative fits described in Sec. VII. Although changing
$g_{\pi}$ is equivalent to changing $f$ when splittings are omitted [cf. Eq.
(35)], the effects are inequivalent when splittings are included [cf. Eq.
(53)]. This is especially true of the finite-volume effects, for which the
splittings have the potential to produce significant changes Arndt:2004bg .
Some additional parameters constrained by Bayesian priors are also included in
the chiral fits, as discussed above. The taste-violating hairpin parameters
$\delta^{\prime}_{V}$ and $\delta^{\prime}_{A}$ are given by the ranges in
Table 12. In addition, up to six heavy-quark and up to seven light-quark
lattice-spacing dependent terms, are added for investigation of discretization
effects. Except where otherwise noted, all twelve such terms are included in
the fits plotted below: this gives errors that include true statistical errors
plus our estimate of discretization effects from the heavy quarks and generic
(taste non-violating) discretization errors from the light quarks. In
addition, some or all of the (mildly) constrained NNLO LECs,
$Q_{1},\dots,Q_{4}$, are included. Again, unless otherwise noted, the fits
below include all four such parameters; such fits tend to give larger (and
hence more conservative) errors than fits that restrict the number of these
parameters. In total, there are 23 fit parameters in the central fits: the 19
constrained parameters listed in this paragraph, and the 4 unconstrained
parameters listed in the previous paragraph.
### VI.2 Chiral fits and extrapolations for the $D$ system
Figure 3 shows our central chiral fit to $r_{1}^{3/2}\phi_{D^{+}}$ and
$r_{1}^{3/2}\phi_{D_{s}}$.
Figure 3: Central chiral fit for the $D$ system, based on Analysis I of the
fits to 2-point correlators. Only (approximately) unitary points are shown.
Data from ensembles at $a\approx 0.15$ fm, $a\approx 0.12$ fm and $a\approx
0.09$ fm are shown, but the $a\approx 0.15$ fm ensembles are not included in
the fit. The bursts show extrapolated values for $\phi_{D_{s}}$ and
$\phi_{D^{+}}$, with the purely statistical errors in bright red and the
statistical plus discretization errors in darker red. The physical strange-
quark mass corresponds to an abscissa value of $m_{x}\approx 0.1$.
Data from ensembles at $a\approx 0.15$ fm, $a\approx 0.12$ fm and $a\approx
0.09$ fm are shown, but the $a\approx 0.15$ fm ensembles are not included in
the fit. The points and covariance matrix are obtained from Analysis I (Sec.
IV.2) of the two-point functions. For clarity, only the unitary (full QCD)
points are shown for $\phi_{D}$ (and approximately unitary for
$\phi_{D_{s}}$), but the fit is to all the partially-quenched data on the
$a\approx 0.12$ fm and $a\approx 0.09$ fm ensembles. The fit properly takes
into account the covariance of the data; $\chi^{2}/{\rm dof}$ and the $p$
value (goodness of fit) are reasonable, as shown. The points in Fig. 3 are
plotted as a function of mass $m_{x}$, where, for $\phi_{D^{+}}$, the light
valence mass $m_{q}$ and the light sea mass $m_{l}$ are given by
$m_{q}=m_{l}=m_{x}$. For $\phi_{D_{s}}$, only $m_{l}=m_{x}$ varies, while
$m_{q}$ is held fixed at the value $m_{s_{v}}$ near the physical strange mass
$m_{s}$.444On the $a\approx 0.15$ fm ensembles, $m_{s_{v}}$ is equal to the
value of the strange sea quark mass $m_{h}$ ($am_{s_{v}}=0.0484$), but on the
other two ensembles we take it lower than $m_{h}$, because $m_{h}$ has been
chosen somewhat larger than the physical strange mass. In the figure,
$am_{s_{v}}=0.415$ for the $a\approx 0.12$ fm ensembles and $am_{s_{v}}=0.272$
for the $a\approx 0.09$ fm ensembles. In order to be able to compare ensembles
at different lattice spacings, we have adjusted the bare quark masses by the
ratio $Z_{m}/Z^{\mathrm{0.09\,fm}}_{m}$, where $Z_{m}$ is the (one-loop) mass
renormalization constant Aubin:2004ck , and $Z^{\mathrm{0.09\,fm}}_{m}$ is its
value on the $a\approx 0.09$ fm ensembles.
The continuum extrapolation is carried out by taking the fitted parameters and
setting $a^{2}=0$ in all taste-violating terms (parameterized by
$\Delta_{\xi}$, $\delta^{\prime}_{A}$, $\delta^{\prime}_{V}$, and $L_{a}$),
all heavy-quark discretization effects (parameterized by $z_{E}$, $z_{X}$,
$z_{Y}$, $z_{B}$, $z_{3}$, and $z_{2}$) and all generic light-quark
discretization effects (parameterized by $C_{i}$). The red lines (solid for
$\phi_{D^{+}}$, dotted for $\phi_{D_{s}}$) show the effect of extrapolating to
the continuum and setting the strange quark mass (both sea, $m_{h}$, and
valence, $m_{s_{v}}$) to the physical value $m_{s}$.
Finally, the bursts give the result after the chiral extrapolation in the
continuum, i.e., setting $m_{x}=m_{d}$ for $\phi_{D^{+}}$, and $m_{x}=\hat{m}$
for $\phi_{D_{s}}$. The larger, dark red, error bars on the bursts show the
total error from the fit, which includes heavy-quark and generic light-quark
discretization errors using Bayesian priors, as described above. The smaller,
bright red error bars, show purely statistical errors, which are computed by a
fit with all the discretization prior functions turned off. In plotting the
red line for $\phi_{D^{+}}$, the light sea mass is shifted slightly
($m_{l}=m_{x}+\hat{m}-m_{d}$) so that it takes its proper mass when
$m_{x}=m_{d}$. (We neglect isospin violations in the sea.) The small mass
differences between $\hat{m}$ and $m_{d}$ (and the corresponding difference
between $\hat{m}$ and $m_{u}$ for the $B^{+}$) produce changes in $\phi$ that
are much smaller than our current errors, but we include them here with an eye
to future work, where the precision will improve.
The trend of the data for the coarsest lattice spacing ($a\approx 0.15$ fm,
the magenta points in Fig. 3) tends to be rather different than for the finer
lattice spacings, especially for the $D_{s}$, which is why we exclude the
$a\approx 0.15$ fm data from the central fit. This trend is even more
exaggerated for the $B$ system, but with particularly large statistical
errors; see Fig. 6 below. Nevertheless, the effect of including the $a\approx
0.15$ fm points in the fit is a rough indication of the size of discretization
errors. Figure 4 shows what happens to the fit when these points are included:
$\phi_{D^{+}}$ and $\phi_{D_{s}}$ each move up an amount comparable to (but
less than) the size of the larger (dark red) error bars, which represent heavy
and generic light quark discretization errors (as well as statistical errors,
which are smaller). The consistency is reassuring.
Figure 4: Same as Fig. 3, but including points at $a\approx 0.15$ fm in the
chiral-continuum fit.
As discussed in Sec. IV, we also examine Analysis II of the 2-point functions.
Figure 5 shows the effect of using Analysis II in the chiral fits.
Figure 5: Same as Fig. 3, but using Analysis II of the 2-point function.
The differences in the decay constant results between Fig. 3 and Fig. 5 are
included in the decay-constant error budgets as a “fitting error”. Note that
the covariance matrix calculation in Analysis II results in an apparent
underestimate of $\chi^{2}$ (and, consequently, a high apparent $p$ value). We
believe that this stems from binning of the data to remove autocorrelation
effects, which has the disadvantage of reducing the number of samples used to
compute the covariance matrix. It is then difficult to determine small
eigenvalues accurately. Indeed the eigenvalues of the (normalized) correlation
matrix tend to have a lower bound of $\sim 10^{-4}$ to $10^{-3}$ with this
approach, whereas they typically go down to $10^{-5}$ in Analysis I. [Recall
that in Analysis I we keep all samples, and deal with autocorrelation effects
by Eq. (24).] Nevertheless, the difficulty with small eigenvalues explains
only a small fraction of the difference between the results from Analyses I
and II. For example, $f_{D}$ is changed by only 0.2 MeV when we smooth
eigenvalues from Analysis I that are less than $10^{-3}$, following the method
of Ref. Bernard:2002pc . This may be compared to the total difference between
$f_{D}$ in Analyses I and II, which is 1.7 MeV.
### VI.3 Chiral fits and extrapolations for the $B$ system
Results for the $B$ system closely resemble those for the $D$ system in most
respects. One important difference is that the signal-to-noise ratio is worse
for the $B$ system because the mass difference that controls the noise,
$2m_{B}-m_{\eta_{b}}-m_{\pi}$, increases with the mass of the heavy quark
Lepage:TASI . Therefore, the preferred fit in Analysis I for the charm case (1
simple exponential + 1 oscillating exponential at large $t_{\textit{min}}$) is
too noisy here, and we must use fits with an extra excited state and smaller
$t_{\textit{min}}$ (see Sec. IV.2). Consequently, our $B$-system results have
larger statistical errors. On the other hand, heavy-quark discretization
errors are smaller in the $B$ system. In the HQET analysis of discretization
effects they appear in the heavy-quark expansion, which works better for $B$’s
to begin with Oktay:2008ex .
Figure 6 shows, for unitary points only, our central chiral fit for the $B$
system.
Figure 6: Central chiral fit for the $B$ system, with data from Analysis I of
the 2-point functions. Only (approximately) unitary points are shown. Data
from ensembles at $a\approx 0.15$ fm, $a\approx 0.12$ fm, and $a\approx 0.09$
fm are shown, but the $a\approx 0.15$ fm ensembles are not included in the
fit. The bursts show extrapolated values for $\phi_{B_{s}}$ and
$\phi_{B^{+}}$, with the purely statistical errors in bright red and the
statistical plus discretization errors in darker red. The physical strange-
quark mass corresponds to an abscissa value of $m_{x}\approx 0.1$.
This is based on Analysis I of the 2-point functions. As in Fig. 3, the red
lines (solid for $\phi_{B^{+}}$, dotted for $\phi_{B_{s}}$) show the effect of
extrapolation to the continuum and setting the strange quark mass to its
physical value $m_{s}$. For the solid red line, the light sea mass is again
shifted slightly, but now $m_{l}=m_{x}+\hat{m}-m_{u}$, so that it takes its
proper mass when $m_{x}=m_{u}$. The bursts show the final results, and come
from setting $m_{x}=m_{u}$ for $\phi_{B^{+}}$ and $m_{x}=\hat{m}$ for
$\phi_{B_{s}}$. As before, the smaller, bright red, error bars, show purely
statistical errors, and the larger, dark red, error bars come from the fit
with Bayesian priors and include heavy-quark and generic light-quark
discretization errors as well as statistical errors.
In Fig. 6, the $a\approx 0.15$ fm data are both noisy and far from those of
the finer lattice spacings. Therefore, these ensembles are again dropped from
the central fit. Figure 7 shows the effect of including the $a\approx 0.15$ fm
points.
Figure 7: Same as Fig. 6, but including points at $a\approx 0.15$ fm in the
fit.
Note that the resulting continuum-extrapolated line for $\phi_{B_{s}}$ (dotted
red line) now has what appears to be a rather unphysical shape, showing a
significant initial increase as the light sea-quark mass is decreased,
starting at the right side of the graph. Hence, the differences caused by
including the $a\approx 0.15$ fm points is 10 to 20% larger than the dark red
error bars in Fig. 6, and 40 to 60% larger than discretization errors
estimated by removing the statistical errors from the dark red bars. Because
the trend for $a\approx 0.15$ fm is so different from the other spacings, and
because of the unphysical behavior when these points are included in the fit,
we believe this difference overestimates the true discretization error, and we
do not enlarge the errors coming from the fit.
Figure 8 shows the effect of using Analysis II of the correlation functions.
In order to make these comparisons as direct as possible, we first turn off
all the Bayesian discretization terms in the fits.
Figure 8: Same as Fig. 6, but using Analysis II of the 2-point functions.
Compared to the results from Fig. 6, this fit gives a value of $f_{B_{s}}$
about 1 MeV higher and a value of $f_{B^{+}}$ about 2 MeV lower. These
differences are included in our estimate of the fitting errors due to excited
state contamination in Sec. VII.
## VII Estimation of systematic errors
In this section, we present a careful, quantitative accounting for the
uncertainties in our calculation. We consider in turn discretization errors,
fitting errors, errors from inputs $r_{1}$ and quark-mass tuning,
renormalization, and finite-volume effects. Table 13 details our error budget.
Table 13: Total error budget for the heavy-light decay constants. Uncertainties are in MeV for decay constants. The total combines errors in quadrature. The first row includes statistics, heavy-quark discretization errors, and generic light-quark discretization errors, as explained in the text. Errors in parentheses are approximate sub-parts of errors that are computed in combination. Source | $f_{D^{+}}$ (MeV) | $f_{D_{s}}$ (MeV) | $f_{D_{s}}/f_{D^{+}}$ | $f_{B^{+}}$ (MeV) | $f_{B_{s}}$ (MeV) | $f_{B_{s}}/f_{B^{+}}$
---|---|---|---|---|---|---
Statistics $\oplus$ discretization | 9. | 2 | 8. | 9 | 0. | 014 | 5. | 5 | 5. | 1 | 0. | 013
(statistics) | (2. | 3) | (2. | 3) | (0. | 005) | (3. | 6) | (3. | 4) | (0. | 010)
(heavy-quark disc.) | (8. | 2) | (8. | 3) | (0. | 007) | (3. | 7) | (3. | 8) | (0. | 004)
(light-quark disc.) | (2. | 9) | (1. | 5) | (0. | 012) | (2. | 5) | (2. | 1) | (0. | 007)
Chiral extrapolation | 3. | 2 | 2. | 2 | 0. | 014 | 2. | 9 | 2. | 8 | 0. | 014
Two-point functions | 3. | 3 | 1. | 6 | 0. | 013 | 3. | 0 | 4. | 1 | 0. | 015
Scale ($r_{1}$) | 1. | 0 | 1. | 0 | 0. | 001 | 1. | 0 | 1. | 4 | 0. | 001
Light quark masses | 0. | 3 | 1. | 4 | 0. | 005 | 0. | 1 | 1. | 3 | 0. | 006
Heavy quark tuning | 2. | 8 | 2. | 8 | 0. | 003 | 3. | 9 | 3. | 9 | 0. | 005
$u_{0}$ adjustment | 1. | 8 | 2. | 0 | 0. | 001 | 2. | 5 | 2. | 8 | 0. | 001
Finite volume | 0. | 6 | 0. | 0 | 0. | 003 | 0. | 5 | 0. | 1 | 0. | 003
$Z_{V^{4}_{QQ}}$ and $Z_{V^{4}_{qq}}$ | 2. | 8 | 3. | 4 | 0. | 000 | 2. | 6 | 3. | 1 | 0. | 000
Higher-order $\rho_{A_{4}}^{Qq}$ | 1. | 5 | 1. | 8 | 0. | 001 | 1. | 4 | 1. | 7 | 0. | 001
Total Error | 11. | 3 | 10. | 8 | 0. | 025 | 8. | 9 | 9. | 5 | 0. | 026
### VII.1 Heavy-quark and generic light-quark discretization effects
As described in Sec. VI and Appendix A, we parameterize possible heavy-quark
and generic light-quark discretization effects and follow a Bayesian approach
in including such effects in our chiral fitting function. Consequently, the
raw “statistical” error that comes from our fits is not a pure statistical
error but includes an estimate of the errors coming from the discretization
effects. This inclusive error is shown with the dark red error bars in the
plots in Sec. VI, and is listed in the first line of Table 13.
For informational purposes, it is useful to break down this inclusive error
into its component parts, at least approximately. We can see what errors to
expect and, hence, target for improvement in future simulations. In
particular, with our current actions, the light-quark and heavy-quark
discretization errors should behave differently as a function of lattice
spacing, with heavy-quark errors decreasing more slowly as $a$ is reduced. To
extract the pure statistical errors, we rerun the fits with all the Bayesian
discretization terms set to zero. We then find the pure heavy-quark (or pure
light-quark) discretization contributions, by turning back on the heavy-quark
(light-quark) terms, and then subtracting in quadrature the pure statistical
errors from the resulting raw errors. These individual errors are shown in
Table 13 in parentheses. Note that the total error at the bottom of the table
includes the error on the first line, not the sum of the three errors in
parentheses, when these differ. Note also that the discretization errors are
similar to what we would have obtained with less sophisticated power counting.
### VII.2 Chiral extrapolation and taste-violating light-quark discretization
effects
As described in Sec. VI, we modify the chiral fit function in a variety of
ways to estimate the error associated with the chiral extrapolation:
1. $\chi$1.
Set the chiral coupling $f$ to $f_{K}$ instead of $f_{\pi}$.
2. $\chi$2.
Allow the chiral coupling $f$ to be a Bayesian fit parameter, with prior value
$f_{\pi}$ and prior width equal to $f_{K}-f_{\pi}$.
3. $\chi$3.
Replace the $H$-$H^{*}$-$\pi$ coupling $g_{\pi}$ (which is 0.51 in the central
fit) with 0.31 or 0.71, which are the extremes of the range discussed in Sec.
VI.
4. $\chi$4.
Allow $g_{\pi}$ to be a constrained fit parameter, with prior value 0.51 and
prior width 0.20.
5. $\chi$5.
Fix to zero those NNLO analytic terms [$Q_{2}$ and/or $Q_{4}$ in Eq. (48)]
that may be eliminated without making the fit unacceptably poor.
6. $\chi$6.
Use the chiral function without hyperfine and flavor splittings, i.e., use Eq.
(35) instead of Eq. (53).
7. $\chi$7.
Use combinations of modifications $\chi$1 and $\chi$3 or modifications $\chi$2
and $\chi$3. These choices can produce significantly larger deviations since
changes in $g_{\pi}$ have a similar effect on the fit function as changes in
$f$.
These modifications typically change the decay constant by 1–3 MeV, and the
ratios by 1–1.5%. We take the chiral extrapolation error of a given quantity
to be the largest change (of either sign) under the above modifications, and
list it in Table 13. In several cases, ($f_{D^{+}}$, $f_{D_{s}}/f_{D^{+}}$,
and $f_{B_{s}}/f_{B^{+}}$) the largest change comes from modification $\chi$6,
eliminating the heavy-light splittings. The fit without the splittings is
shown for the $D$ system in Fig. 9.
Figure 9: Same as Fig. 3 but omitting heavy-light hyperfine and flavor
splittings in the chiral fit function.
It may be compared to Fig. 3 to see the effects: the curvature at small mass
for $\phi_{D^{+}}$ is slightly greater without the splittings, which results
in a decrease of $f_{D^{+}}$ of 3.2 MeV. Note that the $p$ values of the two
fits are almost identical, so the goodness-of-fit cannot be used to choose one
version of the chiral extrapolation over the other.
Modifications of $f$ and/or $g_{\pi}$ produce the largest changes in the other
quantities, namely $f_{D_{s}}$, $f_{B^{+}}$ and $f_{B_{s}}$. In particular,
putting $f=f_{K}$ and $g_{\pi}=0.31$ results in an increase of +2.9 for
$f_{B^{+}}$ and +2.8 MeV for $f_{B_{s}}$. The modified fit is shown in Fig.
10, and may be compared with Fig. 6 to see the effects of the changes.
Figure 10: Same as Fig. 6 but with $f=f_{K}$ and $g_{\pi}=0.31$ in the chiral
fit function.
Increasing $f$ and decreasing $g_{\pi}$ both suppress the chiral logarithms
[cf. Eq. (53)] and give fit functions with less curvature and smaller slope at
low quark mass.
Since the rS$\chi$PT fit functions in Eqs. (35) and (53) explicitly include
one-loop discretization effects coming from taste violations in the (rooted)
staggered light quark action, the chiral error estimates we describe here
inherently include taste-violating discretization errors. However, it seems
unlikely that the current data can accurately distinguish between such taste-
violating errors of order $\alpha_{s}^{2}a^{2}$ and generic light-quark
discretization effects of order $\alpha_{s}a^{2}$, or even heavy-quark
discretization effects. Indeed, the taste-violating LEC $L_{a}$ [cf. Eq. (45)]
is not well constrained by our fits and is consistent with zero within large
errors. The central fits give
$\displaystyle L_{a}$ $\displaystyle=$ $\displaystyle+0.6\pm 6.5\qquad(D{\rm\
system}),$ (64) $\displaystyle L_{a}$ $\displaystyle=$ $\displaystyle-1.9\pm
8.8\qquad(B{\rm\ system}),$ (65)
where the error is the raw statistical error. (Note that we do not constrain
$L_{a}$ by any prior width.) The errors in $L_{a}$ decrease by about 10% if
Bayesian parameters for generic light-quark errors are removed, and an
additional 10% if the parameters for heavy-quark errors are removed. Thus,
there is “cross talk” between various error sources, making it difficult to
completely distinguish the various types of discretization errors. Future
work, with more and finer lattice spacings, should make a cleaner separation
possible.
### VII.3 Fitting errors
The “fitting errors” are the errors introduced in the analysis of the two-
point correlators. They represent the effects of various choices of fit ranges
and fitting functions, and are an estimate of the systematic effect of the
contamination by excited states. We compare results from the three choices of
two-point fitting (see Sec. IV): Analysis I, Analysis II, and a modified
Analysis I using 1 simple + 1 oscillating state, but values of
$t_{\textit{min}}$ larger than those described in Sec. IV.2.
Some of these differences may, in fact, be due simply to statistical effects,
and hence already included in the statistical error. Figure 11 shows the
differences between values of $\phi_{B_{q}}$ in Analyses I and II, divided by
the average statistical error for each of the common partially quenched data
points.
Figure 11: Difference of $\phi_{B_{q}}$ values from Analyses I and II, divided
by the average statistical error at each of the common valence and sea mass
points. The order along the abscissa is arbitrary.
Only 10 of 74 differences are greater than 1 statistical $\sigma$.
Nevertheless, there appears to be some significant systematic trend in that 46
of 74 points are positive. To be conservative, we take the largest difference
between the Analysis-I fits and the other two fits as the fitting error for
each physical quantity, and list it in Table 13. For $f_{D_{s}}$ and
$f_{B_{s}}$, the difference is largest for chiral fits based on 2-point
Analysis II, while, for the other four quantities, the difference is largest
for the modified Analysis I.
### VII.4 Scale uncertainty
We use the scale $r_{1}=0.3117(22)$ fm to tune the values of the quark masses
and convert the decay constants into physical units (see Sec. III.1). To find
the scale errors on the final results, we shift $r_{1}$ to 0.3139 fm or 0.3095
fm and redo the analysis. Although $\phi_{H}$ scales like $r_{1}^{-3/2}$, the
change in the results under a change in $r_{1}$ is smaller than pure
dimensional analysis would imply, because our estimates of the physical light
masses and the heavy-quark $\kappa_{\mathrm{c}}$ and $\kappa_{\mathrm{b}}$
also shift, producing partially compensating changes in $\phi_{H}$. At
$r_{1}=0.3139$ fm, we shift the light masses in Table 12 upward by the scale
error shown in that table. [The lattice light-quark masses scale like
$r_{1}^{2}$, because they are approximately linear in the squared meson masses
$(r_{1}m_{\pi})^{2}$ and $(r_{1}m_{K})^{2}$.] Similarly, we shift the tuned
$\kappa_{\mathrm{c}}$ and $\kappa_{\mathrm{b}}$ downward by the scale error in
Table 5 because the bare heavy quark mass increases with $r_{1}$. We then
adjust $\phi_{B_{(s)}}$ and $\phi_{D_{(s)}}$ at each lattice spacing using Eq.
(11) and the values of $d\phi/d\kappa$ given in Table 5. Redoing the preferred
chiral fits shown in Figs. 3 and 6, extrapolating to the continuum, and
plugging in the adjusted continuum light quark masses gives the scale error
listed in Table 13.
### VII.5 Light-quark mass determinations
To estimate the error from the light-quark mass determination, we follow a
similar procedure to that in the scale-error case. We shift the continuum
light quark masses in Table 12 by the sum in quadrature of all errors except
scale errors. This includes the statistical errors, the chiral errors and,
where relevant, the electromagnetic errors. We then plug the new masses into
the continuum-extrapolated chiral fits and take the difference from the
central results to give the errors listed in Table 13. The relative direction
of shifts on different masses makes little difference in the size of the
errors on the decay constants $f_{D_{s}}$, $f_{D^{+}}$, $f_{B_{s}}$, and
$f_{B^{+}}$, since they are sensitive primarily to the valence quark masses.
However, it does affect the error of the ratios $f_{D_{s}}/f_{D^{+}}$ and
$f_{B_{s}}/f_{B^{+}}$. The largest effect clearly occurs when the strange mass
is shifted in the opposite direction from the lighter masses. To be
conservative, we take the size of change of the ratios in this case as the
error, but this is almost certainly an overestimate because the statistical
and chiral extrapolation errors on the light quark masses are positively
correlated between the strange mass and the other masses.
Note that the errors from the light-quark masses in Table 13 are much larger
for $f_{D_{s}}$ and $f_{B_{s}}$ than for $f_{D^{+}}$ and $f_{B^{+}}$. That
simply reflects the facts that the decay constants have a nonzero limit when
the quark masses vanish, and that the dependence on the quark masses is
reasonably linear. Thus a given percent error in the strange mass produces a
much larger percent difference in $f_{D_{s}}$ and $f_{B_{s}}$, than the same
percent error in the $d$ or $u$ mass does in $f_{D^{+}}$ and $f_{B^{+}}$.
### VII.6 Bottom and charm quark mass determinations
The propagation of statistical errors in the tuned $\kappa_{\mathrm{c}}$ and
$\kappa_{\mathrm{b}}$ to the decay constants is complicated by the fact that
the independent errors at each lattice spacing affect the final results in a
nontrivial way through the continuum and chiral extrapolations. At each
lattice spacing, we choose 200 gaussian-distributed ensembles of trial
$\kappa$ values with central value equal to the tuned values, and standard
deviation equal to the statistical error, taken from Table 5. For a given
choice of trial $\kappa$ values at each lattice spacing, we produce an
adjusted trial data sample by shifting the $\phi_{H}$ values according to Eq.
(11), but with the trial values replacing the tuned values. We then perform
the complete chiral fit and extrapolation procedure on each of the 200 trial
data sets. The standard deviation over trials of a given decay constant or
decay constant ratio is taken to be the heavy quark tuning error, and is
listed in Table 13.
### VII.7 Tadpole factor ($u_{0}$) adjustment
In order to improve the convergence of lattice perturbation theory, we use
tadpole-improved actions for the gluons, light quarks, and heavy quarks
Lepage:1992xa . For the gluon and sea-quark actions we take the tadpole factor
$u_{0}$ from the average plaquette. On the $a\approx 0.15$ fm and $a\approx
0.09$ fm lattices we use the same choice for the light valence and heavy-quark
actions. On the $a\approx 0.12$ fm lattices, however, we use the tadpole
factor $u_{0}$ taken from the Landau link in the valence-quark action and in
the clover term in the heavy-quark action. This results in a slight mismatch
between the light valence and sea-quark actions on these ensembles, and also
affects the values obtained for the tuned bottom- and charm-quark masses
$\kappa_{b}$ and $\kappa_{c}$. The difference between $u_{0}$ obtained from
the average plaquette and the Landau link is approximately 3–4% on the
$a\approx 0.12$ fm ensembles.
We propagate this difference through the chiral/continuum extrapolation as
follows. First, we compute the heavy-strange meson decay amplitudes
$\phi_{B_{s}}$ and $\phi_{D_{s}}$ with both choices for $u_{0}$ on the
ensemble with $am_{l}/am_{h}=0.01/0.05$, $a\approx 0.12$ fm. For each choice
of $u_{0}$, we compute $\phi_{B_{s}}$ and $\phi_{D_{s}}$ directly at the tuned
values of $\kappa_{b}$ and $\kappa_{c}$, thereby avoiding an interpolation in
$\kappa$. Next, we renormalize the lattice decay amplitudes using the
nonperturbative, flavor-diagonal current renormalization factors
$Z_{V^{4}_{qq}}$ and $Z_{V^{4}_{QQ}}$ obtained for each case. (We neglect the
slight difference in the perturbative correction $\rho_{A^{4}_{Qq}}$.) Then,
we calculate the ratio of the renormalized decay amplitudes, finding no
difference within errors:
$\displaystyle\phi_{c}^{\rm plaquette}/\phi_{c}^{\rm Landau}$ $\displaystyle=$
$\displaystyle 1.005(13),$ (66) $\displaystyle\phi_{b}^{\rm
plaquette}/\phi_{b}^{\rm Landau}$ $\displaystyle=$ $\displaystyle 1.014(20).$
(67)
As expected, the $u_{0}$ dependence from the bare current and renormalization
factors mostly cancels. Finally, we repeat the chiral/continuum extrapolation
shifting $\phi_{c}$ and $\phi_{b}$ on the $a\approx 0.12$ fm ensembles by the
statistical errors reported in Eqs. (66)–(67). We find that these percent-
level errors in $\phi_{c}$ and $\phi_{b}$ lead to approximately 1% errors in
the extrapolated decay constants and approximately 0.1% errors in the decay-
constant ratios. These errors are listed as “$u_{0}$ adjustment” in the error
budget in Table 13.
### VII.8 Heavy-light current renormalization
There are two sources of systematic error in our heavy-light current
renormalization. The first is due to the perturbative calculation of
$\rho_{A^{4}_{Qq}}$ and the second is due to the nonperturbative calculation
of $Z_{V^{4}_{QQ}}$ and $Z_{V^{4}_{qq}}$.
The perturbative calculation of $\rho_{A^{4}_{Qq}}$ has been carried out to
one-loop order. Since $\rho_{A^{4}_{Qq}}$ is defined from a ratio of
renormalization factors [see Eq. (5)], its perturbative corrections are small
by construction. Indeed, as can be seen from the results for
$\rho_{A^{4}_{Qq}}$ given in Table 8, we observe very small corrections. For
bottom they range from $0.3$% at $a\approx 0.09$ fm to $0.8$% at $a\approx
0.12$ fm and $2.8$% at $a\approx 0.15$ fm. For charm they range from less than
$0.08$% at $a\approx 0.09$ fm to $0.4$% at $a\approx 0.12$ fm and $0.6$% at
$a\approx 0.15$ fm. As shown in Ref. ElKhadra:2007qe the perturbative
corrections to the $\rho$-factors for the spatial currents, while still small,
tend to be bigger than those for the temporal currents $A^{4}$ and $V^{4}$. We
therefore estimate the error due to neglecting higher order terms as
$\rho_{V^{1}_{Qq}}^{[1]}\,\alpha_{s}^{2}$. We take $\alpha_{s}$ at $a\approx
0.09$ fm and $\rho_{V^{1}_{Qq}}^{[1]}\approx 0.1$, which is the largest one-
loop coefficient for $\rho_{V^{1}_{Qq}}$ in the mass range $m_{Q}a\leq 3$.
This procedure yields a systematic error of $0.7$%, which we take for both
charm and bottom decay constants.
The decay constant ratios $f_{B_{s}}/f_{B^{+}}$ and $f_{D_{s}}/f_{D^{+}}$
depend on the corresponding ratios of $\rho_{A^{4}_{Qs}}/\rho_{A^{4}_{Qq}}$.
These ratios differ from unity only because of the small variation of the
$\rho_{A^{4}_{Qq}}$ with light valence mass, which is described in Sec. V. We
take the variation of the $\rho_{A^{4}_{Qq}}$ with light valence mass at
$a\approx 0.09$ fm as the error. This yields an error of $0.1\%$ for both
bottom and charm.
The dominant corrections in the heavy-light renormalization factor as defined
in Eq. (5) are due to $Z_{V^{4}_{QQ}}$ and $Z_{V^{4}_{qq}}$ which are
calculated nonperturbatively. The values (and errors) for $Z_{V^{4}_{qq}}$ and
$Z_{V^{4}_{QQ}}$ are listed in Tables 9 and 11, respectively. To obtain the
error in $Z_{V^{4}_{Qq}}=\sqrt{Z_{V^{4}_{qq}}Z_{V^{4}_{QQ}}}$ we add the
statistical errors in $Z_{V^{4}_{qq}}$ and $Z_{V^{4}_{QQ}}$ in quadrature. The
error on $Z_{V^{4}_{Qq}}$ is dominated by the error on $Z_{V^{4}_{qq}}$. The
errors are largest, 1.3%, on the $a\approx 0.09~{}\textrm{fm}$ ensemble and
they are about the same for both charm and bottom on the two finest ensembles
used to obtain our main decay constant results. Hence we use 1.3% as our
estimate for the uncertainty in $Z_{V^{4}_{Qq}}$.
### VII.9 Finite volume effects
To study finite volume effects, we use the chiral fit function with heavy-
light hyperfine and flavor splittings included (Eq. (53)), since the effects
are known to be larger with the splittings than without Arndt:2004bg . The
central fit includes the (one-loop) finite volume corrections, Eq. (54), on
the lattice data, and then takes the infinite volume limit when extracting the
final results for the decay constants. We then take the larger of the
following two values as our estimate of the finite volume error:
1. V1.
The difference between the central result and the result from a chiral fit in
which the finite volume corrections are omitted.
2. V2.
The largest finite volume correction to the relevant data points, as
determined by the central fit. For $\phi_{D^{+}}$ and $\phi_{B^{+}}$, the
“relevant data points” are the ones on each ensemble with the lightest valence
mass, i.e., those closest to the chirally extrapolated point. For
$\phi_{D_{s}}$ and $\phi_{B_{s}}$, the relevant points are the ones on each
ensemble with valence mass closest to $m_{s}$.
Method V1 gives a larger difference for $\phi_{D_{s}}$ and $\phi_{B_{s}}$;
method V2 for $\phi_{D^{+}}$ and $\phi_{B^{+}}$ and the ratios. The resulting
values are shown in Table 13. Note that our choices are conservative because
we correct for the (one-loop) finite volume errors, but nevertheless take the
full size of these effects as our error.
## VIII Results and Conclusions
After adding the error estimates described in the previous section in
quadrature, we obtain:
$\displaystyle f_{B^{+}}$ $\displaystyle=$ $\displaystyle
196.9(8.9)~{}\textrm{MeV},$ (68) $\displaystyle f_{B_{s}}$ $\displaystyle=$
$\displaystyle 242.0(9.5)~{}\textrm{MeV},$ (69) $\displaystyle
f_{B_{s}}/f_{B^{+}}$ $\displaystyle=$ $\displaystyle 1.229(0.026),$ (70)
$\displaystyle f_{D^{+}}$ $\displaystyle=$ $\displaystyle
218.9(11.3)~{}\textrm{MeV},$ (71) $\displaystyle f_{D_{s}}$ $\displaystyle=$
$\displaystyle 260.1(10.8)~{}\textrm{MeV},$ (72) $\displaystyle
f_{D_{s}}/f_{D^{+}}$ $\displaystyle=$ $\displaystyle 1.188(0.025).$ (73)
Since our most reliable method of determining discretization errors combines
them with statistical errors, we do not quote separate statistical and
systematic errors.
Figure 12 shows a comparison of our results for charmed decay constants with
other lattice QCD calculations and with experiment.
Figure 12: Comparison of $f_{D}$ and $f_{D_{s}}$ with other two- and three-
flavor lattice QCD calculations and with experiment. Results shown come from
Refs. Davies:2010ip ; :2011gx ; :2008sq ; Rosner:2010ak ; Alexander:2009ux ;
:2007ws ; Naik:2009tk ; Onyisi:2009th ; Lees:2010qj . The HPQCD $f_{D}$ value
is computed from their update to $f_{D_{s}}$ and their earlier result for the
ratio $f_{D_{s}}/f_{D}$.
Our results agree with the only other three-flavor lattice QCD determination
from the HPQCD collaboration Davies:2010ip , which is obtained with HISQ
staggered valence quarks and asqtad staggered sea quarks. (The difference in
$f_{D_{s}}$ is a bit greater than 1$\sigma$.) They are also consistent with
the two-flavor results of the ETM Collaboration using twisted-mass Wilson
fermions :2011gx , although the ETM error budget does not include an estimate
of the uncertainty due to quenching the strange quark. One can also compare
with “experimental” determinations of $f_{D}$ and $f_{D_{s}}$ if one assumes
CKM unitarity to obtain the matrix elements $|V_{cd}|$ and $|V_{cs}|$. For the
$D$ meson, Rosner and Stone combine CLEO’s measurement of branching fraction
${\mathcal{B}}(D^{+}\to\mu^{+}\nu)$ :2008sq with the latest determination of
$|V_{cd}|$ from the PDG Nakamura:2010zzi to obtain
$f_{D}=206.7(8.9)~{}\textrm{MeV}$ Rosner:2010ak . For the $D_{s}$ meson, they
average CLEO and Belle results for ${\mathcal{B}}(D_{s}^{+}\to\mu^{+}\nu)$
Alexander:2009ux ; :2007ws with CLEO and BABAR results for
${\mathcal{B}}(D_{s}^{+}\to\tau^{+}\nu)$ Alexander:2009ux ; Naik:2009tk ;
Onyisi:2009th ; Lees:2010qj to obtain a combined average for the two decay
channels of $f_{D_{s}}=257.5(6.1)~{}\textrm{MeV}$ Rosner:2010ak . The Heavy
Flavor Averaging Group obtains a similar average,
$f_{D_{s}}=257.3(5.3)~{}\textrm{MeV}$ Asner:2010qj . Our results are
consistent with these values, confirming Standard Model expectations at the
$\sim 5\%$ level.
Figure 13 shows a similar comparison of our results for bottom meson decay
constants with other lattice QCD calculations.
Figure 13: Comparison of $f_{B}$ and $f_{B_{s}}$ with other two- and three-
flavor lattice QCD calculations. Results shown come from Refs. McNeile:2011ng
; Gamiz:2009ku ; :2011gx ; Albertus:2010nm . In the case of $f_{B_{s}}$ HPQCD
has two separate calculations using NRQCD $b$ quarks and using HISQ $b$
quarks; we show both the published NRQCD result (HPQCD ’09) and the more
recent HISQ result (HPQCD ’11) in the plot above.
Our results agree with the published three-flavor determination using NRQCD
$b$-quarks and Asqtad staggered light quarks of the HPQCD collaboration
Gamiz:2009ku , but are only marginally consistent with HPQCD’s more recent
calculation of $f_{B_{s}}$ using HISQ light valence quarks McNeile:2011ng .
Our results are also consistent with the two-flavor results of the ETM
collaboration :2011gx , who use Wilson heavy quarks and interpolate between
the charm-mass region and the static limit to obtain results for bottom.
Further, our result for the ratio $f_{B_{s}}/f_{B}$ also agrees with the
significantly less precise three-flavor determination using static $b$-quarks
and domain-wall light quarks by the RBC and UKQCD Collaborations
Albertus:2010nm .
For the $D$ system the largest uncertainties in our current calculation stem
from heavy-quark discretization, while the chiral extrapolation, the $Z_{V}$
factors, excited states, heavy-quark tuning, and the chiral-continuum
extrapolation play important but subdominant roles. For the $B$ system, heavy-
quark tuning, statistics, and excited states are the sources of the largest
errors, while the $Z_{V}$ factors and the chiral-continuum extrapolation
(incorporating our estimate of heavy-quark discretization effects) are next in
size. Recall that a novel feature of our work is the treatment of heavy-quark
discretization effects, via the functions $f_{i}$ in Eq. (4), and priors
constraining the chiral-continuum fits to follow this form. At tree level, we
have explicit calculations of the mismatch, some of which appeared already in
Ref. ElKhadra:1996mp and all of which are compiled in Ref. Oktay:2008ex .
Beyond the tree level, the continuum and static limits can be used to
constrain the functional form. That said, the theoretical guidance of the
priors cannot be highly effective in an analysis, such as this, with only two
lattice spacings. Indeed, the quoted heavy-quark discretization errors are
similar to less sophisticated power-counting estimates.
While completing this analysis, we have begun runs to generate data that will
address the main sources of uncertainty reported here. The new data set will
contain four times the configurations used here to reduce the statistical
errors in the correlation functions and, thus, directly improve the decay
amplitudes, the determinations of the hopping-parameters $\kappa_{\mathrm{c}}$
and $\kappa_{\mathrm{b}}$, and the renormalization factors $Z_{V^{4}_{qq}}$
and $Z_{V^{4}_{QQ}}$, all of which feed into the decay constant. Our new data
will also encompass two finer lattice spacings of $a\approx 0.06$ fm and
$a\approx 0.045$ fm, in order to explicitly reduce light- and heavy-quark
discretization errors and better control the continuum extrapolation. With
four lattice spacings, our new method of heavy-quark discretization priors
will be put to a more stringent test. The new runs will also include light
valence- and sea-quark masses down to $\sim m_{s}/20$ in order to better
control the chiral extrapolation to the physical $d$ and $u$ quark masses.
In order to reduce errors further, we will have to eliminate the errors from
the matching factors and from quenching the charmed quark. The MILC
Collaboration Bazavov:2010pi is generating ensembles with 2+1+1 flavors of
sea quarks with the HISQ action, with plans to provide a range of lattice
spacings and sea quark masses equal to or more extensive than the 2+1 asqtad
ensembles. Use of the HISQ action for the charm valence quark will allow us to
further reduce many of the uncertainties, and provides the particularly nice
advantage that one can use the local pseudoscalar density without
multiplicative renormalization to obtain the continuum matrix element
Follana:2007uv . In several years, once the full suite of HISQ ensembles with
several sea-quark masses and lattice spacings has been analyzed, we expect to
obtain percent-level errors for both $B$\- and $D$-meson decay constants. This
will enable precise tests of the Standard Model and may help to reveal the
presence of new physics in the quark-flavor sector.
###### Acknowledgements.
We thank David Lin for his finite-volume chiral-log Mathematica code, upon
which our own code is based. Computations for this work were carried out with
resources provided by the USQCD Collaboration, the Argonne Leadership
Computing Facility, the National Energy Research Scientific Computing Center,
and the Los Alamos National Laboratory, which are funded by the Office of
Science of the U.S. Department of Energy; and with resources provided by the
National Institute for Computational Science, the Pittsburgh Supercomputer
Center, the San Diego Supercomputer Center, and the Texas Advanced Computing
Center, which are funded through the National Science Foundation’s
Teragrid/XSEDE Program. This work was supported in part by the U.S. Department
of Energy under Grants No. DE-FC02-06ER41446 (C.D., L.L., M.B.O.), No. DE-
FG02-91ER40661 (S.G.), No. DE-FG02-91ER40677 (C.M.B, R.T.E., E.D.F., E.G.,
R.J., A.X.K.), No. DE-FG02-91ER40628 (C.B), No. DE-FG02-04ER-41298 (D.T.); by
the National Science Foundation under Grants No. PHY-0555243, No. PHY-0757333,
No. PHY-0703296 (C.D., L.L., M.B.O.), No. PHY-0757035 (R.S.), No. PHY-0704171
(J.E.H.); by the URA Visiting Scholars’ program (C.M.B., R.T.E., E.G.,
M.B.O.); by the Fermilab Fellowship in Theoretical Physics (C.M.B.); by the M.
Hildred Blewett Fellowship of the American Physical Society (E.D.F.); and by
the Science and Technology Facilities Council and the Scottish Universities
Physics Alliance (J.L.). This manuscript has been co-authored by employees of
Brookhaven Science Associates, LLC, under Contract No. DE-AC02-98CH10886 with
the U.S. Department of Energy. R.S.V. acknowledges support from BNL via the
Goldhaber Distinguished Fellowship. Fermilab is operated by Fermi Research
Alliance, LLC, under Contract No. DE-AC02-07CH11359 with the United States
Department of Energy.
## Appendix A Heavy-quark Discretization Effects
We are using the heavy-quark Lagrangian as given in ElKhadra:1996mp , with
$\kappa_{t}=\kappa_{s}$ (or, equivalently, $\zeta=1$), $r_{s}=1$, and
$c_{B}=c_{E}=c_{\rm SW}$. This amounts to the Sheikholeslami-Wohlert
Lagrangian Sheikholeslami:1985ij for Wilson fermions Wilson:1975id . The
current has a heavy quark of this type, rotated as in Eq. (13) (cf. Eqs.
(7.8)–(7.10) of Ref. ElKhadra:1996mp ), and a staggered light quark. At the
tree level, the heavy-quark rotation is the same no matter what the other
quark is. The discretization effects are estimated from a (continuum)
effective field theory Kronfeld:2000ck ; Harada:2001fi ; Harada:2001fj , as
shown explicitly for decay constants in Eqs. (8.7)–(8.12) of Ref.
Kronfeld:2000ck .
### A.1 Theory
Both QCD and lattice gauge theory can be described via
$\displaystyle{\cal L}_{\rm QCD}\doteq{\cal L}_{\text{HQET}}$ $\displaystyle=$
$\displaystyle\sum_{i}{\cal C}_{i}^{\rm cont}(m_{Q})\mathcal{O}_{i},$ (74)
$\displaystyle{\cal L}_{\rm LGT}\doteq{\cal L}_{\text{HQET}(m_{0}a)}$
$\displaystyle=$ $\displaystyle\sum_{i}{\cal C}_{i}^{\rm
lat}(m_{Q},m_{0}a)\mathcal{O}_{i},$ (75)
where the ${\cal C}_{i}$ are short-distance coefficients and the
$\mathcal{O}_{i}$ are operators describing the long-distance physics. The
coefficients have dimension $4-\dim\mathcal{O}_{i}$. For lattice gauge theory.
they depend on $m_{0}a$, which is a ratio of short distances $a$ and
$1/m_{Q}$. The effective-theory operators $\mathcal{O}_{i}$ in Eqs. (74) and
(75) are the same.
The error from each term is simply the difference
${\tt error}_{i}=\left|\left[{\cal C}_{i}^{\rm lat}(m_{Q},m_{0}a)-{\cal
C}_{i}^{\rm cont}(m_{Q})\right]\mathcal{O}_{i}\right|.$ (76)
The relative error in our matrix elements can be estimated by setting
$\langle\mathcal{O}_{i}\rangle\sim\Lambda_{\text{QCD}}^{\dim\mathcal{O}_{i}-4}$;
choices for the QCD scale $\Lambda_{\text{QCD}}$ are discussed below. The
coefficient mismatch can be written
${\cal C}_{i}^{\rm lat}(m_{Q},m_{0}a)-{\cal C}_{i}^{\rm
cont}(m_{Q})=a^{\dim\mathcal{O}_{i}-4}f_{i}(m_{0}a).$ (77)
This recovers the usual counting of powers of $a$ (familiar from Symanzik
Symanzik:1983dc ; Symanzik:1983gh ), but maintaining the full $m_{0}a$
dependence. The final expression for the discretization errors is then
${\tt error}_{i}\propto
f_{i}(m_{0}a)(a\Lambda_{\text{QCD}})^{\dim\mathcal{O}_{i}-4}.$ (78)
For Wilson fermions, $\lim_{m_{0}a\to 0}f_{i}=\text{constant}$ (whereas in
lattice NRQCD without fine tuning this is not the case). We have explicit
calculations of the $f_{i}$ for the $O(a)$ and $O(a^{2})$ errors at the tree
level ElKhadra:1996mp ; Oktay:2008ex . The next subsection discusses how to
use them to guide a continuum-limit extrapolation the $O(\alpha_{s}a)$ and
$O(a^{2})$ errors.
Equations (74) and (75) can be generalized to currents. For the axial-vector
current,
$\displaystyle\mathcal{A}^{\mu}$ $\displaystyle\doteq$ $\displaystyle C^{\rm
cont}_{A_{\perp}}(m_{Q})\bar{q}i\gamma_{\perp}^{\mu}\gamma_{5}h_{v}-C^{\rm
cont}_{A_{\parallel}}(m_{Q})v^{\mu}\bar{q}\gamma_{5}h_{v}-\sum_{i}B^{\rm
cont}_{Ai}(m_{Q})\mathcal{Q}^{\mu}_{Ai},$ (79) $\displaystyle A_{\rm
lat}^{\mu}$ $\displaystyle\doteq$ $\displaystyle C^{\rm
lat}_{A_{\perp}}(m_{Q},m_{0}a)\bar{q}i\gamma_{\perp}^{\mu}\gamma_{5}h_{v}-C^{\rm
lat}_{A_{\parallel}}(m_{Q},m_{0}a)v^{\mu}\bar{q}\gamma_{5}h_{v}-\sum_{i}B^{\rm
lat}_{Ai}(m_{Q},m_{0}a)\mathcal{Q}^{\mu}_{Ai},$ (80)
and $\doteq$ again means in the sense of matrix elements. Here $v^{\mu}$
selects the temporal component and $\perp$ the spatial, and the list of
dimension-4 operators $\mathcal{Q}$ can be found in Refs. Harada:2001fi .
The matrix element of the temporal component of the axial-vector current [cf.
Eq. (18)] is normalized by multiplying with $Z_{A^{4}}=C^{\rm
cont}_{A_{\parallel}}/C^{\rm lat}_{A_{\parallel}}$. The current mismatch then
leads to errors
$a^{\dim\mathcal{Q}_{i}-3}f_{i}(m_{0}a)=Z_{A^{4}}B^{\rm lat}_{Ai}-B^{\rm
cont}_{Ai},$ (81)
with the sum running over the two operators $\mathcal{Q}$ that point in the
temporal direction Harada:2001fi .
### A.2 Error Estimation
The total error from heavy-quark discretization effects is then
${\tt error}=\sum_{i}z_{i}\,(a\Lambda_{\rm QCD})^{s_{i}}f_{i}(m_{0}a)$ (82)
where the sum runs over Lagrangian operators $\mathcal{O}_{i}$ of dimension 5
and 6 and current operators $\mathcal{Q}_{i}$ of dimension 4 and 5,
$s_{i}=\dim\mathcal{O}_{i}-4$ or $\dim\mathcal{Q}_{i}-3$, and the $z_{i}$ are
unknown coefficients. The functions $f_{i}$ (summarized below) have been
computed for $\mathrm{O}(a^{2})$ and estimated for $\mathrm{O}(\alpha_{s}a)$.
We omit contributions of order $\alpha_{s}^{l}a^{2}$, whether from extra
operators or from iterating to second order operators with coefficients of
order $\alpha_{s}a$.
In the past, we have taken a very conservative $\Lambda_{\rm QCD}=700~{}{\rm
MeV}$ and assumed a Gaussian distribution for the $z_{i}$ centered on 0 and of
width 1. This amounts to treating the discretization errors as independent and
adding them in quadrature. It also implicitly assumes that the data have
nothing to say about the size or relative importance of the terms.
Here, however, we incorporate these errors into the chiral-continuum
extrapolation, discussed in Sec. VI. This means that the $z_{i}$ are now
constrained fit parameters, with prior constraints discussed in Sec. VI.
The $f_{i}$ are collected next.
#### A.2.1 $\mathrm{O}(a^{2})$ errors
We start with these, because explicit expressions for the functions
$f_{i}(m_{0}a)$ are available. The Lagrangian leads to two bilinears,
$\bar{h}\bm{D}\cdot\bm{E}h$ and
$\bar{h}i\bm{\Sigma}\cdot[\bm{D}\times\bm{E}]h$, and many four-quark
operators. At the tree level the coefficients of all four-quark operators
vanish. At the tree level the coefficients of the two bilinears are the same,
and the mismatch function is
$f_{E}(m_{0}a)=\frac{1}{8m_{E}^{2}a^{2}}-\frac{1}{2(2m_{2}a)^{2}}.$ (83)
Using explicit expressions for $1/m_{2}$ ElKhadra:1996mp and $1/m_{E}^{2}$
Oktay:2008ex , one finds
$f_{E}(m_{0}a)=\frac{1}{2}\left[\frac{c_{E}(1+m_{0}a)-1}{m_{0}a(2+m_{0}a)(1+m_{0}a)}-\frac{1}{4(1+m_{0}a)^{2}}\right].$
(84)
We are using $c_{E}=1$, so
$f_{E}(m_{0}a)=\frac{2+3m_{0}a}{8(2+m_{0}a)(1+m_{0}a)^{2}}.$ (85)
With no further assumptions, this term enters twice independently, so we take
the width of this prior to be $\sqrt{2}$ rather than 1.
The current leads to three more terms with non-zero coefficients,
$\bar{q}\Gamma\bm{D}^{2}h$, $\bar{q}\Gamma i\bm{\Sigma}\cdot\bm{B}h$, and
$\bar{q}\Gamma\bm{\alpha}\cdot\bm{E}h$, which can be deduced from Eq. (A17) of
Ref. ElKhadra:1996mp . Their coefficients can be read off from Eq. (A19). When
$c_{B}=r_{s}$ the first two share the same coefficient
$\displaystyle f_{X}(m_{0}a)$ $\displaystyle=$
$\displaystyle\frac{1}{8m_{X}^{2}a^{2}}-\frac{\zeta
d_{1}(1+m_{0}a)}{m_{0}a(2+m_{0}a)}-\frac{1}{2(2m_{2}a)^{2}},$ (86)
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{1}{(2+m_{0}a)(1+m_{0}a)}+\frac{1}{2(1+m_{0}a)}-\frac{1}{4(1+m_{0}a)^{2}}-\frac{1}{(2+m_{0}a)^{2}}\right],$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{1}{2(1+m_{0}a)}-\left(\frac{m_{0}a}{2(2+m_{0}a)(1+m_{0}a)}\right)^{2}\right],$
where the last term on the second line comes from using the tree-level $d_{1}$
(as we do in the simulations). Because of the two-fold appearance, we again
take the prior width to be $\sqrt{2}$.
For $\bar{q}\Gamma\bm{\alpha}\cdot\bm{E}h$
$\displaystyle f_{Y}(m_{0}a)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{d_{1}}{m_{2}a}-\frac{\zeta(1-c_{E})(1+m_{0}a)}{m_{0}a(2+m_{0}a)}\right],$
(87) $\displaystyle=$
$\displaystyle\frac{2+4m_{0}a+(m_{0}a)^{2}}{4(1+m_{0}a)^{2}(2+m_{0}a)^{2}},$
where the last line reflects the choices made for $c_{E}$ and $d_{1}$.
#### A.2.2 $\mathrm{O}(\alpha_{s}a)$ and $\mathrm{O}(a^{3})$ errors
Here the mismatch functions $f_{i}(m_{0}a)$ start at order $\alpha_{s}$, and
we do not have explicit expressions for them. We take unimproved tree-level
coefficients as a guide to the combinatoric factors and the asymptotic
behavior as $m_{0}a\to 0$ and $m_{0}a\to\infty$.
The Lagrangian leads to two bilinears, the kinetic energy
$\mathcal{O}_{2}=\bar{h}\bm{D}^{2}h$ and the chromomagnetic moment
$\mathcal{O}_{B}=\bar{h}i\bm{\Sigma}\cdot\bm{B}h$. We match the former
nonperturbatively, by identifying the meson’s kinetic mass with the physical
mass; the discretization error $f_{2}$ stems, therefore, from discretization
effects in $M_{2}$.
The computed kinetic meson mass is
$M_{2}=m_{2}(\kappa)+\textrm{continuum binding energy}+\delta M_{2},$ (88)
where Bernard:2010fr
$\delta
M_{2}=\frac{\bar{\Lambda}^{2}}{6m_{Q}}\left[5\left(\frac{m_{2}^{3}}{m_{4}^{3}}-1\right)+4w_{4}(m_{2}a)^{3}\right],$
(89)
and $m_{2}$, $m_{4}$, and $w_{4}$ are functions of $m_{0}a$ and, hence,
$\kappa$. (See Refs. ElKhadra:1996mp ; Oktay:2008ex for explicit
expressions.) Equating $M_{2}$ to a physical meson mass means that we choose
$\kappa$ such that $m_{2}(\kappa)+\delta M_{2}=m_{Q}$, thereby making in
$\phi$ a relative error
$\texttt{error}_{2}=\bar{\Lambda}\left(\frac{1}{2m_{2}}-\frac{1}{2m_{Q}}\right)=\bar{\Lambda}\left(\frac{1}{2m_{Q}-2\delta
M_{2}}-\frac{1}{2m_{Q}}\right)\approx\bar{\Lambda}\frac{\delta
M_{2}}{2m_{Q}^{2}}.$ (90)
The right-most expression is $(a\bar{\Lambda})^{3}\,f_{2}(m_{0}a)$,
$f_{2}=[~{}]/12(m_{2}a)^{3}$, where $[~{}]$ is the bracket in Eq. (89). It is
formally smaller than the other errors considered here—$f_{2}$ is of order 1
for all $m_{0}a$. Numerically, however, it is not much smaller.
At the tree level the chromomagnetic mismatch is
$f^{[0]}_{B}(m_{0}a)=\frac{c_{B}-1}{2(1+m_{0}a)}.$ (91)
This has the right asymptotic behavior in both limits, so our Ansatz for the
one-loop mismatch function is simply
$f_{B}(m_{0}a)=\frac{\alpha_{s}}{2(1+m_{0}a)},$ (92)
and ${\tt error}_{B}$ is this function multiplied by $a\Lambda$. We take
$\alpha_{s}=0.288$ on the $a\approx 0.12$ fm ensembles, which is the value
determined for $\alpha_{V}$ from the plaquette Lepage:1992xa with one-loop
running to scale $q^{*}=2.5/a$. On other ensembles, $\alpha_{s}$ is found by
assuming that the measured average taste splitting goes like
$\alpha_{s}^{2}a^{2}$ (with $a$ determined from $r_{1}/a$). This gives
$\alpha_{s}$ values that track $\alpha_{V}(q^{*}=2.5/a)$ quite well, which is
why we make that $q^{*}$ choice. The results are rather insensitive to the
details here. For example, using $\alpha_{s}=0.325$ on the $a\approx 0.12$ fm
ensembles, which corresponds to $q^{*}=2.0/a$, increases the error estimate by
less than 0.6 MeV for $f_{D^{+}}$, and less than 0.25 MeV for $f_{B^{+}}$.
The current leads to one more term, with tree-level mismatch function
$f^{[0]}_{3}(m_{0}a)=\frac{m_{0}a}{2(2+m_{0}a)(1+m_{0}a)}-d_{1},$ (93)
and the tree-level $d_{1}$ is chosen so that $f_{3}^{[0]}=0$. As with the
mismatch function $f_{B}$, we would like to anticipate $f_{3}^{[1]}$ by
setting $d_{1}^{[1]}=0$ and multiplying the rest with $\alpha_{s}$. But it is
not generic that this vanishes as $m_{0}a\to 0$. Therefore, we take
$f_{3}(m_{0}a)=\frac{\alpha_{s}}{2(2+m_{0}a)},$ (94)
which has the right asymptotic behavior. We take the prior width as
$\sqrt{2}$, because $A^{4}$ has two such corrections Harada:2001fi .
### A.3 Dispersion relation, Eq. (7)
We take a similar approach to the dispersion relation, Eq. (7), with the
difference that we now know the sign of the leading effect.
The tree-level functions are
$\displaystyle a_{4}^{[0]}$ $\displaystyle=$
$\displaystyle\frac{1}{(m_{2}^{[0]}a)^{2}}-\frac{m_{1}^{[0]}a}{(m_{4}^{[0]}a)^{3}},$
(95) $\displaystyle a_{4^{\prime}}^{[0]}$ $\displaystyle=$ $\displaystyle
m_{1}^{[0]}a\,w_{4}^{[0]}.$ (96)
The binding energy enters $A_{4}$ and $A_{4^{\prime}}$ via the meson’s kinetic
energy. Hence, the binding contributions are
$\displaystyle A^{\prime}_{4}$ $\displaystyle=$
$\displaystyle\frac{3m_{1}^{[0]}a}{m_{2}^{[0]}a\,(m_{4}^{[0]}a)^{3}}-\frac{2}{(m_{2}^{[0]}a)^{3}}-\frac{1}{(m_{4}^{[0]}a)^{3}},$
(97) $\displaystyle A^{\prime}_{4^{\prime}}$ $\displaystyle=$ $\displaystyle
w_{4}^{[0]}\left(1-\frac{m_{1}^{[0]}a}{m_{2}^{[0]}a}\right),$ (98)
and in Eq. (10) the binding energy floats within a Gaussian prior described by
$(\bar{\Lambda},\sigma_{\bar{\Lambda}})=(600,400)$ MeV. This choice
conservatively brackets the binding energy of a heavy-strange meson. For the
higher-order perturbative contribution to the coefficients, we take the
Ansätze based on the asymptotic behavior:
$\displaystyle a_{4}^{[1]}$ $\displaystyle=$
$\displaystyle\frac{y_{4}+z_{4}\ln(1+m_{0}a)}{(1+m_{0}a)^{2}},$ (99)
$\displaystyle a_{4^{\prime}}^{[1]}$ $\displaystyle=$
$\displaystyle\frac{y_{4^{\prime}}m_{0}a+z_{4^{\prime}}\ln(1+m_{0}a)}{1+m_{0}a},$
(100)
where the $y$s and $z$s float within Gaussian priors described by
$(y_{4},\sigma_{y_{4}})=(3,5)$, $(z_{4},\sigma_{z_{4}})=(1,2)$,
$(y_{4^{\prime}},\sigma_{y_{4^{\prime}}})=(0,0)$, and
$(z_{4^{\prime}},\sigma_{z_{4^{\prime}}})=(0,2)$. The terms proportional to
$y_{i}$ stem from the $m_{0}a\to 0$ limit, in which the renormalization of
$m_{4}$ must coincide with that of $m_{1}$, and $a_{4}=m_{1}a\,w_{4}$ must
vanish like $m_{0}a$. The terms proportional to $z_{i}$ stem from the
$m_{0}a\to\infty$ limit, where the static limit is obtained. Except for
$y_{4^{\prime}}$, the numerical values have been chosen consistent with one-
loop experience for $m_{1}$ and $m_{2}$ Mertens:1997wx . We have set
$y_{4^{\prime}}\equiv 0$, because at small $m_{0}a$ it is indistinguishable
from the other term in $a_{4^{\prime}}^{[1]}$, and our range of $m_{0}a$ does
not reach far into the region $m_{0}a\gg 1$.
## Appendix B Two point fit results from Analysis I
Table 14: Heavy-light pseudoscalar meson masses and renormalized decay amplitudes obtained from Analysis I fits of the charm correlators at lattice spacing $a\approx 0.09$ fm. $am_{l}/am_{s}$ | $am_{q}$ | $aM_{H}$ | $a^{3/2}\phi_{H}$ | $\chi^{2}/\mathrm{dof}$ | $p$
---|---|---|---|---|---
0.0031/0.031 | 0.0031 | 0.7523(0.0016) | 0.0857(0.0015) | 58/48 | 0.23
0.0031/0.031 | 0.0044 | 0.7553(0.0014) | 0.0873(0.0013) | 56/48 | 0.28
0.0031/0.031 | 0.0062 | 0.7589(0.0011) | 0.0890(0.0011) | 55/48 | 0.33
0.0031/0.031 | 0.0087 | 0.7634(0.0009) | 0.0910(0.0009) | 53/48 | 0.38
0.0031/0.031 | 0.0124 | 0.7699(0.0007) | 0.0936(0.0007) | 53/48 | 0.41
0.0031/0.031 | 0.0186 | 0.7807(0.0005) | 0.0978(0.0006) | 52/48 | 0.44
0.0031/0.031 | 0.0272 | 0.7954(0.0004) | 0.1030(0.0005) | 50/48 | 0.5
0.0031/0.031 | 0.031 | 0.8018(0.0004) | 0.1052(0.0004) | 50/48 | 0.5
0.0062/0.031 | 0.0031 | 0.7541(0.0030) | 0.0875(0.0027) | 56/48 | 0.37
0.0062/0.031 | 0.0044 | 0.7577(0.0023) | 0.0899(0.0021) | 52/48 | 0.49
0.0062/0.031 | 0.0062 | 0.7613(0.0019) | 0.0917(0.0018) | 50/48 | 0.58
0.0062/0.031 | 0.0087 | 0.7654(0.0015) | 0.0933(0.0015) | 58/51 | 0.43
0.0062/0.031 | 0.0124 | 0.7712(0.0012) | 0.0952(0.0012) | 52/48 | 0.48
0.0062/0.031 | 0.0186 | 0.7810(0.0009) | 0.0985(0.0010) | 56/48 | 0.37
0.0062/0.031 | 0.0272 | 0.7952(0.0006) | 0.1032(0.0008) | 59/48 | 0.28
0.0062/0.031 | 0.031 | 0.8015(0.0005) | 0.1052(0.0007) | 60/48 | 0.25
0.0124/0.031 | 0.0031 | 0.7551(0.0038) | 0.0930(0.0036) | 60/48 | 0.27
0.0124/0.031 | 0.0042 | 0.7554(0.0031) | 0.0926(0.0028) | 65/48 | 0.15
0.0124/0.031 | 0.0062 | 0.7574(0.0023) | 0.0929(0.0021) | 65/48 | 0.16
0.0124/0.031 | 0.0087 | 0.7608(0.0017) | 0.0938(0.0015) | 59/48 | 0.28
0.0124/0.031 | 0.0124 | 0.7666(0.0013) | 0.0957(0.0012) | 49/48 | 0.63
0.0124/0.031 | 0.0186 | 0.7766(0.0008) | 0.0991(0.0009) | 42/48 | 0.85
0.0124/0.031 | 0.0272 | 0.7907(0.0006) | 0.1038(0.0007) | 48/48 | 0.64
0.0124/0.031 | 0.031 | 0.7969(0.0005) | 0.1058(0.0006) | 53/48 | 0.47
Table 15: Heavy-light pseudoscalar meson masses and renormalized decay amplitudes obtained from Analysis I fits of the charm correlators at lattice spacing $a\approx 0.12$ fm. $am_{l}/am_{s}$ | $am_{q}$ | $aM_{H}$ | $a^{3/2}\phi_{H}$ | $\chi^{2}/\mathrm{dof}$ | $p$
---|---|---|---|---|---
0.005/0.050 | 0.005 | 0.9943(0.0032) | 0.1436(0.0030) | 30/30 | 0.52
0.005/0.050 | 0.007 | 0.9977(0.0024) | 0.1453(0.0024) | 29/30 | 0.6
0.005/0.050 | 0.01 | 1.0026(0.0018) | 0.1477(0.0019) | 28/30 | 0.64
0.005/0.050 | 0.014 | 1.0090(0.0016) | 0.1508(0.0017) | 28/30 | 0.64
0.005/0.050 | 0.02 | 1.0186(0.0013) | 0.1551(0.0015) | 29/30 | 0.58
0.005/0.050 | 0.03 | 1.0345(0.0010) | 0.1620(0.0012) | 33/30 | 0.42
0.005/0.050 | 0.0415 | 1.0526(0.0008) | 0.1694(0.0010) | 36/30 | 0.27
0.007/0.050 | 0.005 | 0.9948(0.0035) | 0.1442(0.0035) | 17/30 | 0.98
0.007/0.050 | 0.007 | 0.9975(0.0027) | 0.1455(0.0028) | 19/30 | 0.95
0.007/0.050 | 0.01 | 1.0019(0.0021) | 0.1476(0.0021) | 22/30 | 0.89
0.007/0.050 | 0.014 | 1.0081(0.0016) | 0.1504(0.0017) | 24/30 | 0.83
0.007/0.050 | 0.02 | 1.0178(0.0012) | 0.1547(0.0014) | 23/30 | 0.85
0.007/0.050 | 0.03 | 1.0338(0.0009) | 0.1615(0.0010) | 20/30 | 0.94
0.007/0.050 | 0.0415 | 1.0520(0.0007) | 0.1687(0.0008) | 19/30 | 0.95
0.010/0.050 | 0.005 | 0.9958(0.0039) | 0.1461(0.0041) | 15/30 | 0.99
0.010/0.050 | 0.007 | 1.0000(0.0031) | 0.1486(0.0032) | 20/30 | 0.94
0.010/0.050 | 0.01 | 1.0057(0.0024) | 0.1516(0.0026) | 26/30 | 0.75
0.010/0.050 | 0.014 | 1.0126(0.0019) | 0.1549(0.0021) | 29/27 | 0.41
0.010/0.050 | 0.02 | 1.0226(0.0015) | 0.1594(0.0017) | 33/30 | 0.39
0.010/0.050 | 0.03 | 1.0387(0.0011) | 0.1662(0.0014) | 31/30 | 0.5
0.010/0.050 | 0.0415 | 1.0567(0.0008) | 0.1733(0.0011) | 27/30 | 0.68
0.020/0.050 | 0.005 | 0.9942(0.0046) | 0.1537(0.0050) | 49/30 | 0.036
0.020/0.050 | 0.007 | 0.9959(0.0036) | 0.1533(0.0039) | 49/30 | 0.036
0.020/0.050 | 0.01 | 0.9987(0.0027) | 0.1532(0.0031) | 48/30 | 0.051
0.020/0.050 | 0.014 | 1.0037(0.0021) | 0.1543(0.0024) | 45/30 | 0.075
0.020/0.050 | 0.02 | 1.0124(0.0016) | 0.1575(0.0019) | 43/30 | 0.11
0.020/0.050 | 0.03 | 1.0274(0.0011) | 0.1632(0.0014) | 37/30 | 0.27
0.020/0.050 | 0.0415 | 1.0447(0.0009) | 0.1695(0.0012) | 32/30 | 0.48
0.030/0.050 | 0.005 | 0.9830(0.0042) | 0.1475(0.0042) | 33/30 | 0.39
0.030/0.050 | 0.007 | 0.9853(0.0033) | 0.1485(0.0033) | 33/30 | 0.4
0.030/0.050 | 0.01 | 0.9897(0.0025) | 0.1505(0.0025) | 32/30 | 0.47
0.030/0.050 | 0.014 | 0.9960(0.0020) | 0.1534(0.0020) | 31/30 | 0.53
0.030/0.050 | 0.02 | 1.0054(0.0015) | 0.1574(0.0016) | 32/30 | 0.46
0.030/0.050 | 0.03 | 1.0205(0.0011) | 0.1633(0.0012) | 37/30 | 0.27
0.030/0.050 | 0.0415 | 1.0376(0.0009) | 0.1695(0.0010) | 40/30 | 0.15
Table 16: Heavy-light pseudoscalar meson masses and renormalized decay amplitudes obtained from Analysis I fits of the charm correlators at lattice spacing $a\approx 0.15$ fm. $am_{l}/am_{s}$ | $am_{q}$ | $aM_{H}$ | $a^{3/2}\phi_{H}$ | $\chi^{2}/\mathrm{dof}$ | $p$
---|---|---|---|---|---
0.0097/0.0484 | 0.0048 | 1.1659(0.0044) | 0.1979(0.0052) | 20/20 | 0.5
0.0097/0.0484 | 0.007 | 1.1710(0.0034) | 0.2017(0.0040) | 22/20 | 0.37
0.0097/0.0484 | 0.0097 | 1.1768(0.0027) | 0.2054(0.0032) | 25/20 | 0.26
0.0097/0.0484 | 0.0194 | 1.1951(0.0016) | 0.2159(0.0020) | 25/20 | 0.26
0.0097/0.0484 | 0.029 | 1.2117(0.0012) | 0.2242(0.0015) | 20/20 | 0.51
0.0097/0.0484 | 0.0484 | 1.2432(0.0009) | 0.2385(0.0012) | 15/20 | 0.79
0.0194/0.0484 | 0.0048 | 1.1726(0.0046) | 0.2106(0.0052) | 23/20 | 0.35
0.0194/0.0484 | 0.007 | 1.1749(0.0036) | 0.2105(0.0041) | 23/20 | 0.35
0.0194/0.0484 | 0.0097 | 1.1785(0.0028) | 0.2113(0.0031) | 23/20 | 0.32
0.0194/0.0484 | 0.0194 | 1.1935(0.0016) | 0.2174(0.0020) | 30/20 | 0.092
0.0194/0.0484 | 0.029 | 1.2091(0.0013) | 0.2244(0.0016) | 32/20 | 0.055
0.0194/0.0484 | 0.0484 | 1.2400(0.0010) | 0.2381(0.0013) | 27/20 | 0.17
0.0290/0.0484 | 0.0048 | 1.1613(0.0044) | 0.1975(0.0049) | 17/20 | 0.72
0.0290/0.0484 | 0.007 | 1.1660(0.0034) | 0.2010(0.0040) | 18/20 | 0.64
0.0290/0.0484 | 0.0097 | 1.1717(0.0026) | 0.2049(0.0031) | 21/20 | 0.47
0.0290/0.0484 | 0.0194 | 1.1896(0.0015) | 0.2151(0.0019) | 24/20 | 0.3
0.0290/0.0484 | 0.029 | 1.2058(0.0011) | 0.2229(0.0015) | 23/20 | 0.32
0.0290/0.0484 | 0.0484 | 1.2368(0.0008) | 0.2364(0.0011) | 20/20 | 0.49
Table 17: Heavy-light pseudoscalar meson masses and renormalized decay amplitudes obtained from Analysis I fits of the bottom correlators at lattice spacing $a\approx 0.09$ fm. $am_{l}/am_{s}$ | $am_{q}$ | $aM_{H}$ | $a^{3/2}\phi_{H}$ | $\chi^{2}/\mathrm{dof}$ | $p$
---|---|---|---|---|---
0.0031/0.031 | 0.0031 | 1.6509(0.0018) | 0.1359(0.0016) | 41/39 | 0.48
0.0031/0.031 | 0.0044 | 1.6532(0.0016) | 0.1378(0.0015) | 40/39 | 0.51
0.0031/0.031 | 0.0062 | 1.6562(0.0015) | 0.1402(0.0014) | 40/39 | 0.49
0.0031/0.031 | 0.0087 | 1.6601(0.0013) | 0.1433(0.0014) | 42/39 | 0.42
0.0031/0.031 | 0.0124 | 1.6659(0.0012) | 0.1475(0.0013) | 45/39 | 0.31
0.0031/0.031 | 0.0186 | 1.6752(0.0011) | 0.1542(0.0012) | 47/39 | 0.23
0.0031/0.031 | 0.0272 | 1.6879(0.0009) | 0.1628(0.0011) | 49/39 | 0.19
0.0031/0.031 | 0.031 | 1.6934(0.0009) | 0.1664(0.0011) | 49/39 | 0.18
0.0062/0.031 | 0.0031 | 1.6539(0.0046) | 0.1358(0.0051) | 40/39 | 0.56
0.0062/0.031 | 0.0044 | 1.6557(0.0039) | 0.1377(0.0044) | 37/39 | 0.68
0.0062/0.031 | 0.0062 | 1.6584(0.0032) | 0.1402(0.0037) | 34/39 | 0.77
0.0062/0.031 | 0.0087 | 1.6620(0.0027) | 0.1434(0.0031) | 34/39 | 0.8
0.0062/0.031 | 0.0124 | 1.6675(0.0022) | 0.1480(0.0026) | 36/39 | 0.72
0.0062/0.031 | 0.0186 | 1.6767(0.0018) | 0.1550(0.0022) | 41/39 | 0.53
0.0062/0.031 | 0.0272 | 1.6892(0.0014) | 0.1637(0.0019) | 45/39 | 0.37
0.0062/0.031 | 0.031 | 1.6946(0.0014) | 0.1672(0.0018) | 45/39 | 0.35
0.0124/0.031 | 0.0031 | 1.6532(0.0036) | 0.1387(0.0038) | 52/39 | 0.16
0.0124/0.031 | 0.0042 | 1.6550(0.0033) | 0.1407(0.0034) | 48/39 | 0.27
0.0124/0.031 | 0.0062 | 1.6576(0.0030) | 0.1432(0.0031) | 40/39 | 0.55
0.0124/0.031 | 0.0087 | 1.6606(0.0027) | 0.1456(0.0029) | 35/39 | 0.77
0.0124/0.031 | 0.0124 | 1.6650(0.0024) | 0.1488(0.0027) | 33/39 | 0.84
0.0124/0.031 | 0.0186 | 1.6730(0.0019) | 0.1544(0.0023) | 36/39 | 0.73
0.0124/0.031 | 0.0272 | 1.6847(0.0016) | 0.1623(0.0021) | 42/39 | 0.48
0.0124/0.031 | 0.031 | 1.6900(0.0015) | 0.1657(0.0020) | 45/39 | 0.38
Table 18: Heavy-light pseudoscalar meson masses and renormalized decay amplitudes obtained from Analysis I fits of the bottom correlators at lattice spacing $a\approx 0.12$ fm. $am_{l}/am_{s}$ | $am_{q}$ | $aM_{H}$ | $a^{3/2}\phi_{H}$ | $\chi^{2}/\mathrm{dof}$ | $p$
---|---|---|---|---|---
0.005/0.050 | 0.005 | 1.9170(0.0044) | 0.2236(0.0050) | 45/27 | 0.03
0.005/0.050 | 0.007 | 1.9197(0.0039) | 0.2263(0.0046) | 46/27 | 0.022
0.005/0.050 | 0.01 | 1.9235(0.0033) | 0.2300(0.0040) | 46/27 | 0.021
0.005/0.050 | 0.014 | 1.9287(0.0029) | 0.2347(0.0036) | 45/27 | 0.027
0.005/0.050 | 0.02 | 1.9367(0.0024) | 0.2418(0.0031) | 43/27 | 0.046
0.005/0.050 | 0.03 | 1.9503(0.0020) | 0.2532(0.0026) | 39/27 | 0.096
0.005/0.050 | 0.0415 | 1.9657(0.0017) | 0.2654(0.0023) | 36/27 | 0.17
0.007/0.050 | 0.005 | 1.9147(0.0036) | 0.2224(0.0039) | 37/27 | 0.12
0.007/0.050 | 0.007 | 1.9177(0.0033) | 0.2254(0.0037) | 35/27 | 0.17
0.007/0.050 | 0.01 | 1.9219(0.0030) | 0.2292(0.0036) | 34/27 | 0.2
0.007/0.050 | 0.014 | 1.9272(0.0028) | 0.2337(0.0037) | 35/27 | 0.19
0.007/0.050 | 0.02 | 1.9351(0.0026) | 0.2401(0.0037) | 36/27 | 0.15
0.007/0.050 | 0.03 | 1.9485(0.0022) | 0.2508(0.0035) | 38/27 | 0.096
0.007/0.050 | 0.0415 | 1.9638(0.0019) | 0.2628(0.0031) | 40/27 | 0.07
0.010/0.050 | 0.005 | 1.9182(0.0047) | 0.2254(0.0047) | 30/27 | 0.4
0.010/0.050 | 0.007 | 1.9207(0.0041) | 0.2284(0.0042) | 32/27 | 0.29
0.010/0.050 | 0.01 | 1.9250(0.0035) | 0.2328(0.0037) | 36/27 | 0.18
0.010/0.050 | 0.014 | 1.9307(0.0030) | 0.2383(0.0033) | 39/27 | 0.097
0.010/0.050 | 0.02 | 1.9391(0.0025) | 0.2457(0.0028) | 43/27 | 0.048
0.010/0.050 | 0.03 | 1.9527(0.0020) | 0.2569(0.0024) | 47/27 | 0.02
0.010/0.050 | 0.0415 | 1.9682(0.0017) | 0.2689(0.0021) | 51/27 | 0.0092
0.020/0.050 | 0.005 | 1.9136(0.0060) | 0.2278(0.0069) | 33/27 | 0.27
0.020/0.050 | 0.007 | 1.9163(0.0050) | 0.2305(0.0059) | 33/27 | 0.28
0.020/0.050 | 0.01 | 1.9200(0.0042) | 0.2340(0.0050) | 31/27 | 0.36
0.020/0.050 | 0.014 | 1.9249(0.0036) | 0.2381(0.0043) | 29/27 | 0.47
0.020/0.050 | 0.02 | 1.9322(0.0031) | 0.2437(0.0039) | 28/27 | 0.52
0.020/0.050 | 0.03 | 1.9445(0.0027) | 0.2526(0.0038) | 30/27 | 0.42
0.020/0.050 | 0.0415 | 1.9590(0.0025) | 0.2627(0.0039) | 33/27 | 0.3
0.030/0.050 | 0.005 | 1.9030(0.0058) | 0.2196(0.0073) | 38/27 | 0.12
0.030/0.050 | 0.007 | 1.9058(0.0049) | 0.2223(0.0064) | 32/27 | 0.29
0.030/0.050 | 0.01 | 1.9099(0.0041) | 0.2258(0.0056) | 27/27 | 0.56
0.030/0.050 | 0.014 | 1.9155(0.0034) | 0.2306(0.0048) | 23/27 | 0.74
0.030/0.050 | 0.02 | 1.9239(0.0028) | 0.2376(0.0040) | 22/27 | 0.77
0.030/0.050 | 0.03 | 1.9372(0.0022) | 0.2479(0.0034) | 25/27 | 0.64
0.030/0.050 | 0.0415 | 1.9518(0.0019) | 0.2585(0.0032) | 28/27 | 0.49
Table 19: Heavy-light pseudoscalar meson masses and renormalized decay amplitudes obtained from Analysis I fits of the bottom correlators at lattice spacing $a\approx 0.15$ fm. $am_{l}/am_{s}$ | $am_{q}$ | $aM_{H}$ | $a^{3/2}\phi_{H}$ | $\chi^{2}/\mathrm{dof}$ | $p$
---|---|---|---|---|---
0.0097/0.0484 | 0.0048 | 2.2553(0.0071) | 0.3311(0.0115) | 36/25 | 0.097
0.0097/0.0484 | 0.007 | 2.2576(0.0061) | 0.3341(0.0102) | 37/25 | 0.09
0.0097/0.0484 | 0.0097 | 2.2611(0.0052) | 0.3389(0.0089) | 36/25 | 0.1
0.0097/0.0484 | 0.0194 | 2.2757(0.0036) | 0.3568(0.0063) | 34/25 | 0.16
0.0097/0.0484 | 0.029 | 2.2901(0.0030) | 0.3727(0.0053) | 33/25 | 0.16
0.0097/0.0484 | 0.0484 | 2.3175(0.0023) | 0.4002(0.0046) | 35/25 | 0.12
0.0194/0.0484 | 0.0048 | 2.2296(0.0175) | 0.2743(0.0416) | 32/25 | 0.2
0.0194/0.0484 | 0.007 | 2.2349(0.0142) | 0.2823(0.0357) | 34/25 | 0.15
0.0194/0.0484 | 0.0097 | 2.2416(0.0118) | 0.2917(0.0309) | 36/25 | 0.1
0.0194/0.0484 | 0.0194 | 2.2639(0.0072) | 0.3243(0.0202) | 36/25 | 0.1
0.0194/0.0484 | 0.029 | 2.2819(0.0054) | 0.3482(0.0152) | 30/25 | 0.27
0.0194/0.0484 | 0.0484 | 2.3124(0.0038) | 0.3839(0.0109) | 24/25 | 0.59
0.0290/0.0484 | 0.0048 | 2.2402(0.0073) | 0.3101(0.0123) | 29/25 | 0.32
0.0290/0.0484 | 0.007 | 2.2464(0.0061) | 0.3199(0.0104) | 30/25 | 0.28
0.0290/0.0484 | 0.0097 | 2.2524(0.0052) | 0.3289(0.0089) | 31/25 | 0.25
0.0290/0.0484 | 0.0194 | 2.2695(0.0036) | 0.3502(0.0066) | 27/25 | 0.42
0.0290/0.0484 | 0.029 | 2.2847(0.0030) | 0.3665(0.0058) | 21/25 | 0.72
0.0290/0.0484 | 0.0484 | 2.3125(0.0025) | 0.3939(0.0057) | 18/25 | 0.87
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|
arxiv-papers
| 2011-12-13T21:40:19 |
2024-09-04T02:49:25.248725
|
{
"license": "Public Domain",
"authors": "A. Bazavov, C. Bernard, C. M. Bouchard, C. DeTar, M. Di Pierro, A. X.\n El-Khadra, R. T. Evans, E. D. Freeland, E. Gamiz, Steven Gottlieb, U. M.\n Heller, J. E. Hetrick, R. Jain, A. S. Kronfeld, J. Laiho, L. Levkova, P. B.\n Mackenzie, E. T. Neil, M. B. Oktay, J. N. Simone, R. Sugar, D. Toussaint, R.\n S. Van de Water",
"submitter": "Ruth Van de Water",
"url": "https://arxiv.org/abs/1112.3051"
}
|
1112.3056
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2011-205 LHCb-PAPER-2011-031 December 13, 2011
Measurement of the $C\\!P$ violating phase $\phi_{s}$ in $\kern
3.73305pt\overline{\kern-3.73305ptB}{}^{0}_{s}\rightarrow J/\psi f_{0}(980)$
The LHCb Collaboration111Authors are listed on the following pages.
Measurement of mixing-induced $C\\!P$ violation in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays is of prime importance
in probing new physics. So far only the channel $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi$ has been
used. Here we report on a measurement using an LHCb data sample of 0.41 fb-1,
in the $C\\!P$ odd eigenstate $J/\psi f_{0}(980)$, where
$f_{0}(980)\rightarrow\pi^{+}\pi^{-}$. A time dependent fit of the data with
the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ lifetime and the
difference in widths of the heavy and light eigenstates constrained to the
values obtained from $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi$ yields a
value of the $C\\!P$ violating phase of $-0.44\pm 0.44\pm 0.02~{}{\rm\,rad}$,
consistent with the Standard Model expectation.
Keywords: LHC, $C\\!P$ violation, Hadronic $B$ Decays, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson
PACS: 13.25.Hw, 14.40.Nd, 11.30.Er
Submitted to Physics Letters B
The LHCb Collaboration
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Bondar29, W. Bonivento15, S. Borghi47,50, A. Borgia52, T.J.V. Bowcock48, C.
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T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty50, A.
Lai15, D. Lambert46, R.W. Lambert24, E. Lanciotti37, G. Lanfranchi18, C.
Langenbruch11, T. Latham44, C. Lazzeroni55, R. Le Gac6, J. van Leerdam23,
J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J. Lefran$c$cois7, O. Leroy6, T.
Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles48, R. Lindner37, C. Linn11,
B. Liu3, G. Liu37, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-
March38, H. Lu38,3, J. Luisier38, A. Mac Raighne47, F. Machefert7, I.V.
Machikhiliyan4,30, F. Maciuc10, O. Maev29,37, J. Magnin1, S. Malde51, R.M.D.
Mamunur37, G. Manca15,d, G. Mancinelli6, N. Mangiafave43, U. Marconi14, R.
Märki38, J. Marks11, G. Martellotti22, A. Martens8, L. Martin51, A. Martín
Sánchez7, D. Martinez Santos37, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev29, E. Maurice6, B. Maynard52, A. Mazurov16,32,37, G. McGregor50, R.
McNulty12, M. Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S.
Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski25, R. Mountain52, I. Mous23, F. Muheim46, K.
Müller39, R. Muresan28,38, B. Muryn26, B. Muster38, M. Musy35, J. Mylroie-
Smith48, P. Naik42, T. Nakada38, R. Nandakumar45, I. Nasteva1, M. Nedos9, M.
Needham46, N. Neufeld37, C. Nguyen-Mau38,o, M. Nicol7, V. Niess5, N.
Nikitin31, A. Nomerotski51, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen49, K. Pal52, J. Palacios39, A.
Palano13,b, M. Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo47, C.
Parkes50,37, C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49, S.K.
Paterson49, G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos
Alvarez36, A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A.
Petrolini19,i, A. Phan52, E. Picatoste Olloqui35, B. Pie Valls35, B.
Pietrzyk4, T. Pilař44, D. Pinci22, R. Plackett47, S. Playfer46, M. Plo
Casasus36, G. Polok25, A. Poluektov44,33, E. Polycarpo2, D. Popov10, B.
Popovici28, C. Potterat35, A. Powell51, J. Prisciandaro38, V. Pugatch41, A.
Puig Navarro35, W. Qian52, J.H. Rademacker42, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk40, G. Raven24, S. Redford51, M.M. Reid44, A.C. dos Reis1,
S. Ricciardi45, K. Rinnert48, D.A. Roa Romero5, P. Robbe7, E. Rodrigues47,50,
F. Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43, S. Roiser37, V.
Romanovsky34, M. Rosello35,n, J. Rouvinet38, T. Ruf37, H. Ruiz35, G.
Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail47, B. Saitta15,d,
C. Salzmann39, M. Sannino19,i, R. Santacesaria22, C. Santamarina Rios36, R.
Santinelli37, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m,
A. Satta21, M. Savrie16,e, D. Savrina30, P. Schaack49, M. Schiller24, S.
Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt37, O. Schneider38, A.
Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba18,l, M.
Seco36, A. Semennikov30, K. Senderowska26, I. Sepp49, N. Serra39, J. Serrano6,
P. Seyfert11, M. Shapkin34, I. Shapoval40,37, P. Shatalov30, Y. Shcheglov29,
T. Shears48, L. Shekhtman33, O. Shevchenko40, V. Shevchenko30, A. Shires49, R.
Silva Coutinho44, T. Skwarnicki52, A.C. Smith37, N.A. Smith48, E. Smith51,45,
K. Sobczak5, F.J.P. Soler47, A. Solomin42, F. Soomro18, B. Souza De Paula2, B.
Spaan9, A. Sparkes46, P. Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39,
S. Stoica28, S. Stone52,37, B. Storaci23, M. Straticiuc28, U. Straumann39,
V.K. Subbiah37, S. Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26,
S. T’Jampens4, E. Teodorescu28, F. Teubert37, C. Thomas51, E. Thomas37, J. van
Tilburg11, V. Tisserand4, M. Tobin39, S. Topp-Joergensen51, N. Torr51, E.
Tournefier4,49, M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23, M. Ubeda
Garcia37, A. Ukleja27, P. Urquijo52, U. Uwer11, V. Vagnoni14, G. Valenti14, R.
Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis42, M.
Veltri17,g, B. Viaud7, I. Videau7, X. Vilasis-Cardona35,n, J. Visniakov36, A.
Vollhardt39, D. Volyanskyy10, D. Voong42, A. Vorobyev29, H. Voss10, S.
Wandernoth11, J. Wang52, D.R. Ward43, N.K. Watson55, A.D. Webber50, D.
Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G. Wilkinson51, M.P.
Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25, W.
Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Xing52, Z.
Yang3, R. Young46, O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3, L. Zhang52,
W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin37.
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
23Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraców, Poland
26AGH University of Science and Technology, Kraców, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55University of Birmingham, Birmingham, United Kingdom
56Physikalisches 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
An important goal of heavy flavour experiments is to measure the mixing-
induced $C\\!P$ violation phase in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays, $\phi_{s}$. As this
phase is predicted to be small in the Standard Model (SM) [1], new physics can
induce large changes [2]. Here we use the decay mode $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi f_{0}(980)$.
If only the dominant decay diagrams shown in Fig. 1 contribute, then the value
of $\phi_{s}$ using $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi f_{0}(980)$
is the same as that measured using $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi$ decay.
Figure 1: Dominant decay diagrams for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi f_{0}(980)$
or $J/\psi\phi$ decays.
Motivated by a prediction in Ref. [3], LHCb searched for and made the first
observation of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi f_{0}(980)$
decays [4] that was subsequently confirmed by other experiments [5,
*Abazov:2011hv, 7]. Time dependent $C\\!P$ violation can be measured without
an angular analysis, as the final state is a $C\\!P$ eigenstate. From now on
$f_{0}$ will stand only for $f_{0}(980)$.
In the Standard Model, in terms of CKM matrix elements,
$\phi_{s}=-2\arg\left[\frac{V_{ts}V_{tb}^{*}}{V_{cs}V_{cb}^{*}}\right].$ The
equations below are written assuming that there is only one decay amplitude,
ignoring possible small contributions from other diagrams [8,
*Fleischer:2011au]. The decay time evolutions for initial $B_{s}^{0}$ and
$\overline{B}_{s}^{0}$ are [10, *Bigi:2000yz]
$\displaystyle\Gamma\left(\stackrel{{\scriptstyle(\rule[1.39998pt]{3.5pt}{0.06996pt})}}{{B^{0}_{s}}}\rightarrow
J/\psi f_{0}\right)$ $\displaystyle=$ $\displaystyle{\cal
N}e^{-\Gamma_{s}t}\,\Bigg{\\{}e^{\Delta\Gamma_{s}t/2}(1+\cos\phi_{s})+e^{-\Delta\Gamma_{s}t/2}(1-\cos\phi_{s})$
(1)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\pm\sin\phi_{s}\sin\left(\Delta
m_{s}\,t\right)\Bigg{\\}},$
where $\Delta\Gamma_{s}$ is the decay width difference between light and heavy
mass eigenstates, $\Delta\Gamma_{s}=\Gamma_{\rm L}-\Gamma_{\rm H}$. The decay
width $\Gamma_{s}$ is the average of the widths $\Gamma_{\rm L}$ and
$\Gamma_{\rm H}$, and ${\cal N}$ is a time-independent normalization factor.
The plus sign in front of the $\sin\phi_{s}$ term applies to an initial
$\overline{B}^{0}_{s}$ and the minus sign for an initial $B^{0}_{s}$ meson.
The time evolution of the untagged rate is then
$\Gamma\left(B_{s}^{0}\rightarrow J/\psi f_{0}\right)+\Gamma\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
f_{0}\right)={\cal
N}e^{-\Gamma_{s}t}\,\Bigg{\\{}e^{\Delta\Gamma_{s}t/2}(1+\cos\phi_{s})+e^{-\Delta\Gamma_{s}t/2}(1-\cos\phi_{s})\Bigg{\\}}.$
(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
both samples of flavour tagged and untagged decays. Both Eqs. 1 and 2 are
insensitive to the change $\phi_{s}\rightarrow\pi-\phi_{s}$ when
$\Delta\Gamma_{s}\rightarrow-\Delta\Gamma_{s}$.
## 2 Selection requirements
We use a data sample of 0.41 fb-1 collected in 2010 and the first half of 2011
at a centre-of-mass energy of 7 TeV. This analysis is restricted to events
accepted by a $J/\psi\rightarrow\mu^{+}\mu^{-}$ trigger. The LHCb detector and
the track reconstruction are described in Ref. [12]. The detector elements
most important for this analysis are the VELO, a silicon strip device that
surrounds the $pp$ interaction region, and other tracking devices. Two Ring
Imaging Cherenkov (RICH) detectors are used to identify charged hadrons, while
muons are identified using their penetration through iron.
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 (ndof) less than 11. We work in units where
$c=\hbar=1$. Only candidates with dimuon invariant mass between $-$48 MeV to
+43 MeV of the $J/\psi$ mass peak are selected. Pion candidates are selected
if they are inconsistent with having been produced at the primary vertex. The
impact parameter (IP) is 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 $>9$ for each track. For further
consideration particles forming di-pion candidates must be positively
identified in the RICH system, and must have their scalar sum $p_{\rm T}>900$
MeV.
To select $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidates we
further require that the two pions 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 $\chi^{2}$/ndof $<5$, that this
vertex is $>1.5$ mm from the primary, and points to the primary vertex at an
angle not different from its momentum direction by more than 11.8 mrad.
The invariant mass of selected $\mu^{+}\mu^{-}\pi\pi$ combinations, where the
di-muon pair is constrained to have the $J/\psi$ mass, is shown in Fig. 2 for
both opposite-sign and like-sign di-pion combinations, requiring di-pion
invariant masses within 90 MeV of 980 MeV. Here like-sign combinations are
defined as the sum of $\pi^{+}\pi^{+}$ and $\pi^{-}\pi^{-}$ candidates. The
signal shape, the same for both $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $\overline{B}^{0}$, is a
double-Gaussian, where the core Gaussian’s mean and width are allowed to vary,
and the fraction and width ratio for the second Gaussian are fixed to the
values obtained in a separate fit to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi$. The
mean values of both Gaussians are required to be the same. The combinatoric
background is described by an exponential function. Other background
components are $B^{-}\rightarrow J/\psi h^{-}$, where $h^{-}$ can be either a
$K^{-}$ or a $\pi^{-}$ and an additional $\pi^{+}$ is found, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\eta^{\prime}$, $\eta^{\prime}\rightarrow\rho\gamma$, $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi$,
$\phi\rightarrow\pi^{+}\pi^{-}\pi^{0}$, and $\overline{B}^{0}\rightarrow
J/\psi\overline{K}^{*0}$. The shapes for these background sources are taken
from Monte Carlo simulation based on PYTHIA [13] and GEANT-4 [14] with their
normalizations allowed to vary. We performed a simultaneous fit to the
opposite-sign and like-sign di-pion event distributions. There are 1428$\pm$47
signal events within $\pm$20 MeV of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass peak. The background
under the peak in this interval is 467$\pm$11 events, giving a signal purity
of 75%. Importantly, the like-sign di-pion yield at masses higher than the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ gives an excellent
description of the shape and level of the background. Simulation studies have
demonstrated that it also describes the background under the peak.
Figure 2: (a) Invariant mass of $J/\psi\pi^{+}\pi^{-}$ combinations when the
$\pi^{+}\pi^{-}$ pair is required to be within $\pm$90 MeV of the nominal
$f_{0}(980)$ mass. The data have been fitted with a double-Gaussian signal and
several background functions. The thin (red) solid line shows the signal, the
long-dashed (brown) line the combinatoric background, the dashed (green) line
the $B^{-}$ background (mostly at masses above the signal peak), the dotted
(blue) line the $\overline{B}^{0}\rightarrow J/\psi\overline{K}^{*0}$
background, the dash-dot line (purple) the $\overline{B}^{0}\rightarrow
J/\psi\pi^{+}\pi^{-}$ background, the dotted line (black) the sum of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\eta^{\prime}$
and $J/\psi\phi$ backgrounds (barely visible), and the thick-solid (black)
line the total. (b) The mass distribution for like-sign candidates.
The invariant mass of di-pion combinations is shown in Fig. 3 for both
opposite-sign and like-sign di-pion combinations within $\pm$20 MeV of the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidate mass peak. A
large signal is present near the nominal $f_{0}(980)$ mass. Other $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
J/\psi\pi^{+}\pi^{-}$ signal events are present at higher masses. In what
follows we only use events in the $f_{0}$ signal region from 890 to 1070 MeV.
Figure 3: Invariant mass of $\pi^{+}\pi^{-}$ combinations (points) and a fit
to the $\pi^{\pm}\pi^{\pm}$ data (dashed line) for events in the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ signal region. The region
between the vertical arrows contains the events selected for further analysis.
## 3 S-wave content
Since the initial isospin of the $s\overline{s}$ system that produces the two
pions is zero, and since the $G$-parity of the two pions is even, only even
spin is allowed for the $\pi^{+}\pi^{-}$ pair. Since no spin-4 resonances have
been observed below 2 GeV, the angular distributions are described by the
coherent combination of spin-0 and spin-2 resonant decays. We use the helicity
basis and define the decay angles as $\theta_{J/\psi}$, the angle of the
$\mu^{+}$ in the $J/\psi$ rest frame with respect to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ direction, and
$\theta_{f_{0}}$, the angle of the $\pi^{+}$ in the $\pi^{+}\pi^{-}$ rest
frame with respect to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ direction. The spin-0
amplitude is labeled as $A_{00}$, the three spin-2 amplitudes as $A_{2i}$,
$i=-1,0,1$, and $\delta$ is the strong phase between the $A_{20}$ and $A_{00}$
amplitudes.
After integrating over the angle between the two decay planes the joint
angular distribution is given by [15]
$\displaystyle\frac{d\Gamma}{d\cos\theta_{f_{0}}d\cos\theta_{J/\psi}}$
$\displaystyle=$
$\displaystyle\left|A_{00}+\frac{1}{2}A_{20}e^{i\delta}\sqrt{5}\left(3\cos^{2}\theta_{f_{0}}-1\right)\right|^{2}\sin^{2}\theta_{J/\psi}$
(3)
$\displaystyle+\frac{1}{4}\left(\left|A_{21}\right|^{2}+\left|A_{2-1}\right|^{2}\right)\left(15\sin^{2}\theta_{f_{0}}\cos^{2}\theta_{f_{0}}\right)\left(1+\cos^{2}\theta_{J/\psi}\right).$
Since the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ is spinless,
when it decays into a spin-1 $J/\psi$ and a spin-0 $f_{0}$, $\theta_{J/\psi}$
should be distributed as $\sin^{2}\theta_{J/\psi}$ and $\cos\theta_{f_{0}}$
should be uniformly distributed.
The helicity distributions of the opposite-sign data selected with
reconstructed $J/\psi\pi^{+}\pi^{-}$ mass within $\pm$20 MeV of the known
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass and within $\pm$90
MeV of the nominal $f_{0}(980)$ mass, are shown in Fig. 4; the data have been
background subtracted, using the like-sign data, and acceptance corrected
using Monte Carlo simulation. We perform a two-dimensional unbinned angular
fit. The ratio of rates is found to be
$\displaystyle\frac{\left|A_{20}\right|^{2}}{\left|A_{00}\right|^{2}}~{}~{}~{}~{}~{}$
$\displaystyle=$ $\displaystyle(0.1^{+2.6}_{-0.1})\%,$
$\displaystyle\frac{\left|A_{21}\right|^{2}+\left|A_{2-1}\right|^{2}}{\left|A_{00}\right|^{2}}$
$\displaystyle=$ $\displaystyle(0.0^{+1.7}_{-0.0})\%,$ (4)
where the uncertainties are statistical only. The spin-2 amplitudes are
consistent with zero. Note that the $A_{20}$ amplitude corresponds to $C\\!P$
odd final states, and thus would exhibit the same $C\\!P$ violating phase as
the $J/\psi f_{0}$ final state, while the $A_{2\pm 1}$ amplitude can be either
$C\\!P$ odd or even. Thus this sample is taken as pure $C\\!P$ odd.
Figure 4: Efficiency corrected, background subtracted angular distributions in
the $\pi^{+}\pi^{-}$ mass region within $\pm$90 MeV of 980 MeV and within
$\pm$20 MeV of the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass
for (a) $\cos^{2}\theta_{J/\psi}$, and (b) $\cos\theta_{f_{0}}$. The solid
lines show the expectations for a spin-0 object.
## 4 Time resolution and acceptance
The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decay time is
defined here as $t=m\,{\vec{d}\cdot\vec{p}}/{|\vec{p}|^{2}}$, where $m$ is the
reconstructed invariant mass, $\vec{p}$ the momentum and $\vec{d}$ the flight
vector of the candidate $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
from the primary to the secondary vertices. If more than one primary vertex is
found, the one that corresponds to the smallest IP $\chi^{2}$ of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidate is chosen.
The decay time resolution probability distribution function (PDF) is
determined from data using $J/\psi$ detected without any requirement on
detachment from the primary vertex (prompt) plus two oppositely charged
particles from the primary vertex with the same selection criteria as for
$J/\psi f_{0}$ events, except for the IP $\chi^{2}$ requirement. Monte Carlo
simulation shows that the time resolution PDF is well modelled by these
events. Fig. 5 shows the $t$ distribution for our $J/\psi\pi^{+}\pi^{-}$
prompt 2011 data sample. To describe the background time distribution three
components are needed, (i) prompt, (ii) a small long lived background ($f_{\rm
LL1}=2.64\pm 0.10$)% modeled by an exponential decay function, and (iii) an
even smaller component ($f_{\rm LL2}=0.46\pm 0.02$)% from $b$-hadron decay
described by an additional exponential. Each of these are convolved
individually with a triple-Gaussian resolution function with common means,
whose components are listed in Table 1. The overall equivalent time resolution
is $\sigma_{t}$= 38.4 fs.
The functional form for the time dependence is given by
$\displaystyle N(t)$ $\displaystyle=$ $\displaystyle(1-f_{\rm LL1}-f_{\rm
LL2})\cdot 3G+f_{\rm LL1}\left[\frac{1}{\tau_{1}}\exp(-t/\tau_{1})\otimes
3G\right]$ (5) $\displaystyle+f_{\rm
LL2}\cdot\left[1/\tau_{2}\cdot\exp(-t/\tau_{2})\otimes 3G\right].$
The fractions $f_{\rm LL1}$ and $f_{\rm LL2}$ , and their respective lifetimes
$\tau_{1}$ and $\tau_{2}$, are varied in the fit. The parameters of the
triple-Gaussian time resolution, $3G$, are listed in Table 1. The symbol
$\otimes$ indicates a convolution.
Figure 5: Decay time distribution for prompt $J/\psi\pi^{+}\pi^{-}$ events.
The dashed line (red) shows the long lived components, while the solid line
(blue) shows the total.
A decay time acceptance is introduced by the triggering and event selection
requirements. Monte Carlo simulations show that the shape of the decay time
acceptance function is well modelled by
$A(t)=C\frac{\left[a\left(t-t_{0}\right)\right]^{n}}{1+\left[a\left(t-t_{0}\right)\right]^{n}}~{}~{},$
(6)
where $C$ is a normalization constant. Furthermore, the parameter values are
found to be the same for simulated $\overline{B}^{0}\rightarrow
J/\psi\overline{K}^{*0}$ events with $\overline{K}^{*0}\rightarrow
K^{-}\pi^{+}$, as for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi f_{0}$.
Fig. 6(a) shows the $J/\psi\overline{K}^{*0}$ mass distribution in data with
an additional requirement that the kaon candidate be positively identified in
the RICH system, and that the $K^{-}\pi^{+}$ invariant mass be within $\pm$100
MeV of 892 MeV. There are 36881$\pm$208 signal events. The sideband subtracted
decay time distribution is shown in Fig. 6(b) and fit using the above defined
acceptance function gives values of $a=(1.89\pm 0.07)$ ps-1, $n=1.84\pm 0.12$,
$t_{0}=(0.127\pm 0.015)$ ps , and also a value of the $\overline{B}^{0}$
lifetime of 1.510$\pm$0.016 ps, where the error is statistical only. This is
in good agreement with the PDG average of 1.519$\pm$0.007 ps [16].
Figure 6: Distributions for $\overline{B}^{0}\rightarrow
J/\psi\overline{K}^{*0}$ events (a) $\overline{B}^{0}$ candidate mass
distribution and (b) decay time distribution, where the small background has
been subtracted using the $\overline{B}^{0}$ candidate mass sidebands.
Another check is provided by a recent CDF lifetime measurement of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi f_{0}$ of
$1.70^{+0.12}_{-0.11}\pm 0.03$ ps obtained by fitting the data to a single
exponential [7]. Such a fit to our data yields $1.68\pm 0.05$ ps, where the
uncertainty is only statistical.
## 5 Fit strategy
### 5.1 Likelihood function characterization
The selected events are used to maximize a likelihood function
${\cal L}=\prod_{i}^{N}P(m_{i},t_{i},q_{i}),$ (7)
where $m_{i}$ is the reconstructed candidate $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass, $t_{i}$ the decay time,
and $N$ the total number of events. The flavour tag, $q_{i}$, takes values of
+1, $-1$ and 0, respectively, if the signal meson is tagged as $B_{s}^{0}$,
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$, or untagged. The
likelihood contains three components: signal, long-lived (LL) background and
short-lived (SL) background.
For tagged events we have
$\displaystyle P(m_{i},t_{i},q_{i})$ $\displaystyle=$ $\displaystyle N_{\rm
sig}\epsilon_{\rm sig}^{\rm tag}P_{m}^{\rm sig}(m_{i})P_{t}^{\rm
sig}(t_{i},q_{i})$ (8) $\displaystyle+N_{\rm LL}\epsilon_{\rm LL}^{\rm
tag}P_{m}^{\rm bkg}(m_{i})P_{t}^{\rm LL}(t_{i})+N_{\rm SL}\epsilon_{\rm
SL}^{\rm tag}P_{m}^{\rm bkg}(m_{i})P_{t}^{\rm SL}(t_{i}),$
where: (i) $P_{m}^{\rm sig}(m_{i})$ and $P_{m}^{\rm bkg}(m_{i})$ are the PDFs
describing the dependence on reconstructed mass $m_{i}$ for signal and
background events; (ii) $P_{t}^{\rm sig}(t_{i},q_{i})$ is the PDF used to
describe the signal decay rates for the decay time $t_{i}$; (iii) $P_{t}^{\rm
LL}(t_{i})$ is the PDF describing the long-lived background decay rates, and
$P_{t}^{\rm SL}(t_{i})$ describes the short-lived background, both of which do
not depend on the tagging; (iv) $\epsilon^{\rm tag}$ refers to the respective
tagging efficiencies for signal, long-lived and short-lived backgrounds.
For untagged events we have
$\displaystyle P(m_{i},t_{i},0)$ $\displaystyle=N_{\rm sig}(1-\epsilon_{\rm
sig}^{\rm tag})P_{m}^{\rm sig}(m_{i})P_{t}^{\rm sig}(t_{i},0)$ (9)
$\displaystyle+N_{\rm LL}(1-\epsilon_{\rm LL}^{\rm tag})P_{m}^{\rm
bkg}(m_{i})P_{t}^{\rm LL}(t_{i})+N_{\rm SL}(1-\epsilon_{\rm SL}^{\rm
tag})P_{m}^{\rm bkg}(m_{i})P_{t}^{\rm SL}(t_{i}).$
The total yields of the signal and background components are fixed to the
number of events determined from the fit to the mass distributions (see Sec.
2). For both, the PDF is a product which models the invariant mass
distribution and the time-dependent decay rates. The $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass spectrum is described by
a double-Gaussian for the signal and an exponential function for the
background (see Fig. 2). From Eqs. 1 and 2, the decay time function for the
signal is
$R(t,q_{i})\propto
e^{-\Gamma_{s}t}\left\\{\cosh\frac{\Delta\Gamma_{s}t}{2}+\cos\phi_{s}\sinh\frac{\Delta\Gamma_{s}t}{2}-q_{i}D\sin\phi_{s}\sin(\Delta
m_{s}t)\right\\}.$ (10)
The probability of a wrong tag, $\omega$, is included in the dilution factor
$D\equiv(1-2\omega)$ (see Section 5.2).
The signal PDF is taken as a product of the decay time function, $R(t,q_{i})$,
convolved with the triple Gaussian time resolution function multiplied with
the time acceptance function found from $J/\psi K^{*0}$ discussed in Section
4. The background decay time PDFs are determined using the like-sign
$\pi^{\pm}\pi^{\pm}$ combinations. The time distribution of the like-sign
background agrees in both yield and shape with the opposite-sign events in the
upper $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass candidate
sideband 50$-$200 MeV above the mass peak.
The background functions and parameters are listed in Table 1. The short-lived
background component results from combining prompt $J/\psi$ events with a
opposite-sign pion pair that is not rejected by our selection requirements.
The long-lived part constitutes $\approx$85% of the background.
Table 1: The PDFs for the invariant mass and proper time describing the signal and background. $P_{t}^{\rm sig}$ refers to the decay time distribution in Eq. 9 and $A$ is given in Eq. 6. Where two numbers are listed, the first refers to the 2011 data and the second to the 2010 data. If only one number is listed they are the same for both years. The symbol $\hat{t}$ refers to the true time. | $P_{m}$ | $P_{t}$
---|---|---
Signal
| Double-Gaussian ($2G$) | $P_{t}^{\rm sig}(t,q)=R(\hat{t},q)\otimes 3G(t-\hat{t};\mu,\sigma_{1}^{t},\sigma_{2}^{t},\sigma_{3}^{t},f_{2}^{t},f_{3}^{t})$
| $2G(m;m_{0},\sigma_{1},\sigma_{2},f_{2})$ | $\cdot A(t;a,n,t_{0})$
| $m_{0}$= 5366.5(3) MeV | $\mu=-0.0021(1)$ ps, $-0.0011(1)$ ps
| $\sigma_{1}$=8.6(3) MeV | $\sigma_{1}^{t}=0.0300(4)$ ps, $0.0295(5)$ ps
| $\sigma_{2}$=26.8(9) MeV | $\sigma_{2}^{t}/\sigma_{1}^{t}=1.92(4)$, $1.88(3)$
| $f_{2}$= 0.14(2) | $\sigma_{3}^{t}/\sigma_{1}^{t}=14.6(10)$, $14.0(9)$
| | $f_{2}^{t}=0.23(2)$, $0.27(3)$
| | $f_{3}^{t}=0.0136(6)$, $0.0121(7)$
| | $a=1.89(7)$ ps-1, $n=1.84(12)$, $t_{0}=0.127(15)$ ps
Long-lived background
| Exponential | $[e^{-\hat{t}/{\tau^{\rm bkg}}}\otimes 2G(t-\hat{t};\mu,\sigma_{1}^{t},\sigma_{2}^{t},f_{2}^{t})]\cdot A(t;a,n,t_{0})$
| | $\mu=0$
| | $\sigma_{1}^{t}=0.088$ ps
| | $\sigma_{2}^{t}=5.94$ ps
| | $f_{2}^{t}=0.0137$
| | $\tau^{\rm bkg}=0.96$ ps
| | $a=4.44$ ps-1, $n=4.56$, $t_{0}=0$ ps
Short-lived background
| Exponential | $2G(t;\mu,\sigma_{1}^{t},\sigma_{2}^{t},f_{2}^{t})\cdot A(t;a,n,t_{0})$
| | All parameters are the same as for LL background
### 5.2 Flavour tagging
Flavour tagging uses decays of the other $b$ hadron in the event, exploiting
information from several sources including high transverse momentum muons,
electrons and kaons, and the charge of inclusively reconstructed secondary
vertices. The decisions of the four tagging algorithms are individually
calibrated using $B^{-}\rightarrow J/\psi K^{-}$ decays and combined [17]. The
effective tagging performance is characterized by $\epsilon^{\rm tag}_{\rm
sig}D^{2}$, where $\epsilon^{\rm tag}_{\rm sig}$ is the efficiency and $D$ the
dilution. We use a per-candidate analysis that uses both the information of
the tag decision and of the predicted mistag probability to classify and
assign a weight to each event. The PDFs of the predicted mistag are taken from
the side-bands for the background and side-band subtracted data for the
signal.
The calibration procedure uses a linear dependence between the estimated per
event mistag probability $\eta$ and the actual mistag probability $\omega$
given by $\omega=p_{0}+p_{1}\cdot\left(\eta-\langle\eta\rangle\right)$, where
$p_{0}$ and $p_{1}$ are calibration parameters and $\langle\eta\rangle$ is the
average estimated mistag probability as determined from the calibration
sample. In the 2011 data $p_{0}=0.384\pm 0.003\pm 0.009$, $p_{1}=1.037\pm
0.040\pm 0.070$, and $\langle\eta\rangle=0.379$, with similar values in the
2010 sample. In this paper whenever two errors are given, the first is
statistical and the second systematic. Systematic uncertainties are evaluated
by using different channels to perform the calibration including
$\overline{B}^{0}\rightarrow D^{*+}\mu^{-}\overline{\nu}$, $B^{+}\rightarrow
J/\psi K^{+}$ separately from $B^{-}\rightarrow J/\psi K^{-}$, and viewing the
dependence on different data taking periods. For our 2011 sample
$\epsilon^{\rm tag}_{\rm sig}$ is (25.6$\pm$1.3)% providing us with 365$\pm$22
tagged signal events. For signal the mean mistag fraction,
$\langle\eta\rangle$, is 0.375$\pm$0.005, while for background the mean is
0.388$\pm$0.006. After subtracting background using like-sign events, we
determine $D=0.289$ leading to an $\epsilon D^{2}$ of 2.1% [17].
## 6 Results
Several parameters are input as Gaussian constraints in the fit. These include
the LHCb measured value of $\Delta m_{s}=(17.63\pm 0.11\pm 0.02)$ ps-1 [18],
the tagging parameters $p_{0}$ and $p_{1}$, and both the decay width given by
the $J/\psi\phi$ analysis of $\Gamma_{s}=(0.657\pm 0.009\pm 0.008)$ ps-1 and
$\Delta\Gamma_{s}=(0.123\pm 0.029\pm 0.011$) ps-1 [19]; we also include the
correlation of $-0.30$ between $\Gamma_{s}$ and $\Delta\Gamma_{s}$.222The
final fitted values of these parameters are shifted by less than 2% from their
input values. The fit has been validated both with samples generated from PDFs
and with full Monte Carlo simulations.
Fig. 7 shows the difference of log-likelihood value compared to that at the
point with the best fit, as a function of $\phi_{s}$. At each $\phi_{s}$
value, the likelihood function is maximized with respect to all other
parameters. The best fit value is $\phi_{s}=-0.44\pm 0.44$ rad. The projected
decay time distribution is shown in Fig. 8.
Figure 7: Log-likelihood profile of $\phi_{s}$ for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi f_{0}$
events. Figure 8: Decay time distribution from the fit for $J/\psi f_{0}$
candidates. The solid line shows the results of the fit, the dashed line shows
the signal, and the shaded region the background.
## 7 Systematic uncertainties
The systematic errors are small compared to the statistical errors. No
additional uncertainty is needed for errors on $\Delta m_{s}$, $\Gamma_{s}$,
$\Delta\Gamma_{s}$ or flavour tagging, since Gaussian constraints are applied
in the fit. Other uncertainties associated parameters fixed in the fit are
evaluated by changing them by $\pm$1 standard deviation from their nominal
values and determining the change in fit value of $\phi_{s}$. These are listed
in Table 2. An additional uncertainty is included due to the possible $C\\!P$
even D-wave. This has been measured at $(0.0^{+1.7}_{-0.0})$% of the S-wave
and contributes a small error to $\phi_{s}$, +0.007 rad, as determined by
repeating the fit with the mistag rate increased by 1.7%. The asymmetry in
production between $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ is believed to be small, about
1%, and similar to the same asymmetry in $B^{0}$ production which has been
measured by LHCb to be about 1% [20]. The effect of neglecting a 1% production
asymmetry is the same as ignoring a 1% difference in the mistag rate and
causes negligible bias in $\phi_{s}$.
Table 2: Summary of systematic uncertainties. Here $N_{\rm bkg}$ refers to the number of background events, $N_{\rm sig}$ the number of signal, $N_{\eta^{\prime}}$ the number of $\eta^{\prime}$, $\alpha$ the exponential background parameter for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ candidate mass, $N_{\rm LL}/N_{\rm bkg}$ the long-lived background fraction. The Gaussian signal parameters are the mean $m_{0}$, the width $\sigma(m)$; $t_{0}$, $a$ and $n$ are the three parameters in the acceptance time function. The resolution in signal time is given by $\sigma(t)$, and the background lifetime by $\tau_{\rm bkg}$. The final uncertainty is found by adding all the sources in quadrature. Quantity (Q) | $\pm\Delta$Q | $+$Change | $-$Change
---|---|---|---
| | in $\phi_{s}~{}~{}$ | in $\phi_{s}$
$N_{\rm bkg}$ | 10.1 | 0.0025 | $-0.0030$
$N_{\eta^{\prime}}$ | 3.4 | $-0.0001$ | $-0.0001$
$N_{\rm sig}$ | 46.47 | $-0.0030$ | 0.0028
$\alpha$ | $1.7\cdot 10^{-4}$ | $-0.0002$ | $-0.0002$
$N_{\rm LL}/N_{\rm bkg}$ | 0.0238 | 0.0060 | $-0.0063$
$m_{0}$ (MeV) | 0.32 | -0.0003 | 0.0011
$\sigma(m)$ (MeV) | 0.31 | $-0.0026$ | 0.0020
$\tau_{\rm bkg}$ (ps) | 0.05 | $-0.0075$ | 0.0087
$\sigma(t)$ (ps) | 5% | $-0.0024$ | 0.0022
$t_{0}$ (ps) | 0.015 | $0.0060$ | 0.0050
$a$ (ps-1) | 0.07 | $-0.0065$ | $-0.0065$
$n$ | 0.12 | $-0.0089$ | $-0.0089$
$C\\!P$-even D-wave | | $0.0070$ | 0
Total Systematic Error | +0.018 | $-0.017$
## 8 Conclusions
Using 0.41 fb-1 of data collected with the LHCb detector, the decay mode
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi
f_{0}$, $f_{0}\rightarrow\pi^{+}\pi^{-}$ is selected and then used to measure
the $C\\!P$ violating phase, $\phi_{s}$. We perform a time dependent fit of
the data with the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
lifetime and the difference in widths of the heavy and light eigenstates
constrained. Based on the likelihood curve in Fig. 7 we find
$\phi_{s}=-0.44\pm 0.44\pm 0.02~{}{\rm\,rad},$
consistent with the SM value of $-0.0363^{+0.0016}_{-0.0015}$ rad [1].
Assuming the SM , the probability to observe our measured value is 36%. There
is an ambiguous solution with $\phi_{s}\rightarrow\pi-\phi_{s}$ and
$\Delta\Gamma_{s}\rightarrow-\Delta\Gamma_{s}$. The precision of the result
mostly results from using the tagged sample, though the untagged events also
contribute.
LHCb provides an independent measurement of $\phi_{s}=0.15\pm 0.18\pm 0.06$
[19] using the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi\phi$ decay.
Combining these two results, taking into account all correlations by
performing a joint fit, we obtain
$\phi_{s}=0.07\pm 0.17\pm 0.06~{}{\rm rad}~{}~{}{\rm(combined)}.$
This is the most accurate determination of $\phi_{s}$ to date, and is
consistent with the SM prediction.
## 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
| 2011-12-13T21:53:04 |
2024-09-04T02:49:25.270218
|
{
"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 G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis, J.\n Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, D.\n S. Bailey, 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, F.\n Constantin, A. Contu, A. Cook, M. Coombes, G. Corti, 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, F. De Lorenzi, J. M. De Miranda, L. De Paula, P. De Simone, D. Decamp,\n M. 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, L. Est\\`eve, A. Falabella, E.\n Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, 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, N. Gauvin, M.\n Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, G. Haefeli, C.\n Haen, S. 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, P. Koppenburg, A. Kozlinskiy, L. Kravchuk, K. Kreplin,\n M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, T.\n Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles,\n R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E. Lopez\n Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I.\n V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R. M. D. Mamunur,\n G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, 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, R. Messi, S. Miglioranzi, D. A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Nedos, M.\n Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora Goicochea, P.\n Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis,\n M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M.\n Patel, S. K. Paterson, G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A.\n Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.\n L. Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrella, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R. Plackett, S.\n Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B.\n Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig\n Navarro, W. Qian, J. H. Rademacker, B. Rakotomiaramanana, M. S. Rangel, I.\n Raniuk, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, K.\n Rinnert, D. A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez\n Perez, G. J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T.\n Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, C. Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R.\n Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M.\n Savrie, D. Savrina, P. Schaack, M. Schiller, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, A. C. Smith, N. A.\n Smith, E. Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F. Soomro, B. Souza\n 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. Topp-Joergensen, N. Torr, E. Tournefier, M. T.\n Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, P. Urquijo,\n U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S.\n Vecchi, J. J. Velthuis, M. Veltri, B. Viaud, I. Videau, X. Vilasis-Cardona,\n J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, H. Voss, 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. Zavertyaev, F.\n Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, E. Zverev, A.\n Zvyagin",
"submitter": "Sheldon Stone",
"url": "https://arxiv.org/abs/1112.3056"
}
|
1112.3062
|
11institutetext: Miriam Ney 22institutetext: Andreas Schreiber
33institutetext: Simulation and Software Technology, German Aerospace Centre,
Berlin, Cologne, Germany
33email: NeyMiriam@googlemail.com, Andreas.Schreiber@dlr.de 44institutetext:
Guy K. Kloss 55institutetext: School of Computing + Mathematical Sciences,
Auckland University of Technology,
Auckland, New Zealand
55email: Guy.Kloss@aut.ac.nz
# Using Provenance
to support Good Laboratory Practice
in Grid Environments
Miriam Ney 11 Guy K. Kloss 22 Andreas Schreiber 11
###### Abstract
Conducting experiments and documenting results is daily business of
scientists. Good and traceable documentation enables other scientists to
confirm procedures and results for increased credibility. Documentation and
scientific conduct are regulated and termed as “good laboratory practice.”
Laboratory notebooks are used to record each step in conducting an experiment
and processing data. Originally, these notebooks were paper based. Due to
computerised research systems, acquired data became more elaborate, thus
increasing the need for electronic notebooks with data storage, computational
features and reliable electronic documentation. As a new approach to this, a
scientific data management system (DataFinder) is enhanced with features for
traceable documentation. Provenance recording is used to meet requirements of
traceability, and this information can later be queried for further analysis.
DataFinder has further important features for scientific documentation: It
employs a heterogeneous and distributed data storage concept. This enables
access to different types of data storage systems (e. g. Grid data
infrastructure, file servers). In this chapter we describe a number of
building blocks that are available or close to finished development. These
components are intended for assembling an electronic laboratory notebook for
use in Grid environments, while retaining maximal flexibility on usage
scenarios as well as maximal compatibility overlap towards each other. Through
the usage of such a system, provenance can successfully be used to trace the
scientific workflow of preparation, execution, evaluation, interpretation and
archiving of research data. The reliability of research results increases and
the research process remains transparent to remote research partners.
## 1 Introduction
With the “Principles of Good Laboratory Practice and Compliance Monitoring”
the OECD provides research institutes with guidelines and a framework to
ensure good and reliable research. It defines “Good Laboratory Practice” as
_“a quality system concerned with the organisational process and the
conditions under which non-clinical health and environmental safety studies
are planned, performed, monitored, recorded, archived and reported”_ (p. 14 in
OECD_GLP_PrinciplesNo1 ). This definition can be extended to other fields of
research. To prove the quality of research is of relevance for credibility and
reliability in the research community. Next to organisational processes and
environmental guidelines, part of the good laboratory practice is to maintain
a laboratory notebook when conducting experiments.
The scientist documents each step, either taken in the experiment or
afterwards when processing data. Due to computerised research systems,
acquired data increases in volume and becomes more elaborate. This increases
the need to migrate from originally paper-based to electronic notebooks with
data storage, computational features and reliable electronic documentation.
For these purposes suitable data management systems for scientific data are
available.
### 1.1 A Sample Use Case
As an example use case a group of biologists are conducting research. This
task includes the collection of specimen samples in the field. Such samples
may need to be archived physically. The information on these samples must be
present within the laboratory system to refer to it from further related
entries. Information regarding these samples possibly includes the archival
location, information on name, type, date of sampling, etc.
The samples form the basis for further studies in the biological (wet)
laboratories. Researchers in these environments are commonly not computer
scientists, but biologists who just “want to get their research done.” An
electronic laboratory notebook application therefore must be similarly easy to
operate in day-to-day practice like a paper-based notebook. All notes
regarding experimentation on the samples and further derivative stages
(processing, treatments, etc.) must be recorded, and linked to a number of
other artifacts (other specimen, laboratory equipment, substances, etc.).
As a result of this experimentation further artifacts are derived, which need
to be managed. These could be either further physical samples, or information
(data, measurements, digital images, instrument readings, etc.). Along with
these artifacts the team manages documents outlining the project plan,
documents on experimental procedures, etc.
In the end every managed artifact (physical or data) must be linked through a
contiguous, unbroken chain of records, the provenance trail. The biologists in
our sample use case cooperate with researchers from different institutes in
different (geographical) locations. Therefore, the management of all data as
well as provenance must be enabled in distributed environments, physically
linked through the Internet. The teams rely on a common Grid-based
authentication, which is used to authorise principals (users, equipment,
services) across organisational boundaries.
The recorded provenance of all managed artifacts can be used in a variety of
ways. Firstly, it is useful to document and _prove_ proper scientific
procedures and conduct. Beyond this compliance requirement provenance
information can be used in further ways: It enables often previously not
possible (or very tedious) ways of analysis. By querying the present
provenance information, questions can be answered which depend on the recorded
information. These questions may include some of the following:
* •
_Question for origin:_ What artifacts were used in the generation of another
artifact?
* •
_Question for inheritance:_ What artifacts and information were generated
using a given artifact?
* •
_Question for participants:_ What actors (people, devices, applications,
versions of tools, etc.) were employed in the generation of an artifact?
* •
_Question for dependencies:_ Which resources from other projects/processes
have been used in the generation of an artifact?
* •
_Question for progress:_ In what stage of a processing chain is a given
artifact? Has the process the artifact is part of been finalised?
* •
_Question for quality:_ Did the process the artifact is part of reach a
satisfactory conclusion by some given regulations or criteria?
### 1.2 Data Management with the DataFinder
In order find a solution to common data management problems, the German
Aerospace Centre (DLR) – as Germany’s largest research institute – developed
an open source data management application aimed at researchers and engineers:
_DataFinder_ SchlauchSchreiber2007_DataFinder–Scientific ; DataFinderProject .
DataFinder is a distributed data management system. It allows heterogeneous
storage back-ends, meta-data management, flexible extensions to the user
interface and script-based automation. To implement required features for
reliable and auditable electronic documentation provenance technologies can be
used BunemanKhannaEtAl2001_WhyandWhereDataProvenance .
When analysing the data management situation in scientific or research labs,
several problems are noticeable:
* •
Each scientist individually is solely responsible for the data generated and
managing it as deemed fit. Often others cannot access it, and duplication of
effort may occur.
* •
If a scientist leaves the organisation, it is possible that no one understands
the structure of the data left behind. Information can be lost.
* •
Researchers often spend a lot of time searching for data. This waste of time
decreases productivity.
* •
Due to long archiving periods and an increasing data production rate, the data
volume to store increases significantly.
To overcome this situation common in many research institutes, the DLR
facility Simulation and Software Technology has developed the scientific data
management system DataFinder (cf. DataFinderProject ).
#### General Concepts
DataFinder is an open source software written in Python. It uses a server and
a client component. The server component holds data and associated meta-data.
Data and meta-data is aggregated in a shared data repository and accessed and
managed through the client application. Fig. 1 shows the user interface of the
DataFinder, when connected to a shared repository.
Figure 1: User interface of the DataFinder.
It is designed similar to a file manager on common operating systems. The left
hand side presents the local file hierarchy, and the right displays the shared
repository. All data on the server can be augmented with arbitrary meta-data.
Common actions available for both sides are: open, copy, paste, import and
export data. Opening an entry will make an attempt to use the local system’s
default association for a file. These operations are all essential due to the
nature of DataFinder being a data management tool. One must be aware that on
some operations (e. g. copying) provenance related information is not copied
with it. Copying would create a fork in the provenance graph to create a
duplicate of a formerly uniquely referenced artifact. Special treatment to
treat these cases in a way as to extend the graph properly are not in place,
yet.
An advantage of DataFinder is, that an individual data model is configured for
a shared repository, which must be followed by all its users. A data model
defines the structure of collections. Collections can contain (configurable)
allowed data types, that can be inserted into the collection. The data model
also defines a pre-defined meta-data structure for these collections. This
meta-data can be specified to be either optional or mandatory information when
importing a data item. Based on the data model, data can be managed on a
heterogeneous storage system (certain data items stored in different storage
sub-systems, see 1.2). This requires that DataFinder provides the ability to
manage data on different storage systems, under the control of a single user
interface under a single view (even within the same collection).
Lastly, it is the possibility to extend the application with Python scripts.
This enables a user to take advantage of more customised features, such as
tool integration, task automation, etc.
The DataFinder-based system aims at providing many options and to be highly
extensible for many purposes. DataFinder is already in use in different fields
of research. New use cases are identified and extensions implemented
frequently. One of these is the new use case for supporting a good laboratory
practice capable notebook as outlined in this chapter.
#### Distributed Data Storage
One of the key features of DataFinder is the capability to use different
distributed, heterogeneous storage systems (concurrently). A user has the
freedom to store data on different systems, while meta-data for this data can
be kept either on the same or on a different storage system.
Possible data storage options can be accessed for example through: WebDAV,
Subversion, FTP, GridFTP. Other available storage systems possibilities are
Amazon S3 Cloud services as well as a variety of hosted file systems. Meta-
data for systems not capable of providing extensive free-form meta-data is
managed centrally with another system. Such systems then are accessed through
meta-data capable protocols like WebDAV or Subversion. Further storage back-
ends are relatively simple to integrate, due to the highly modular factory
design of the application. This design feature of DataFinder will be further
examined in Sect. 3 for the integration of a distributed Grid data storage
infrastructure.
It must be noted at this point however, that DataFinder is responsible for
maintaining consistently managed data. DataFinder uses these protocols and
systems for this purpose. If data is accessed _without_ using the DataFinder
directly on the server through other clients, data policies may be compromised
(due to different access restrictions), or consistency may be compromised
(with writing access to the storage systems). With certain caution, this can
however be used to integrate other (legacy) systems into the overall concept.
Due to the design of the DataFinder it is further possible to manage physical
(real world) items, such as laboratory analysis samples or offline media (e.
g. video tapes, CDs, DVDs). Physical items can be stored on shelves, or
archived in any other way. These can be valuable artifacts for research, and
the knowledge of their existence as well as their proper management is a
common necessity. Therefore, it is crucial to managed them electronically in a
similar fashion by the same management tools. Doing so enables extensive meta-
queries provided by the DataFinder, taking advantage of utilising the search
capabilities over all managed items in the same way. Furthermore, this enables
to reference them consistently in provenance assertions from within the realm
of the provenance enabled system.
### 1.3 Overview
This chapter ties the link between the existing DataFinder application to
convert it into a tool useful for a good laboratory practice compliant
electronic notebook. It will introduce how DataFinder can be combined with
provenance recording services and (Grid) storage servers to form the back bone
of such a system. The concept of DataFinder is to be a system that can be
customised towards different deployment scenarios, it is to support the
researchers or engineers in _their_ way of working. This includes the
definition of a data storage hierarchy, required meta-data for storage items
and much more, usually alongside with customisation or automation scripts and
customised GUI dialogues. In a similar fashion, DataFinder can be used to
construct an electronic laboratory notebook with provenance recording for good
laboratory practice. Again, to do so one creates the required data models and
customises GUI dialogues to suit the purpose.
The used provenance technologies and their applications are described in Sect.
2. Concepts to integrate Grid technologies for scientific data management are
outlined in Sect. 3. Sect. 4 presents the results of integrating the good
laboratory practice into a provenance system as well as a data management
system. It also provides a solution on how to connect these two system
practically. Finally, the concept of the resulting system of an electronic
laboratory notebook is evaluated.
## 2 Provenance Management
Provenance originates from the Latin word: “provenire” meaning “to come from”
Merriam-Webster2010_OnlineDictionary . It is described as “the place that sth.
originally came from” thus the origin or source of something (cf.
Wehmeier2000_OxfordAdvancedLearnersDict ). It was originally used for art, but
other disciplines adapted it for their objects, such as fossils or documents.
In the field of computer science and data origin it could be defined as:
> “The provenance of a piece of data is the process that led to that piece of
> data.” Moreau2010_FoundationsProvenanceWeb
Based on this understanding, approaches for identifying provenance use cases
for modeling processes and for integrating provenance tracking into
applications are developed. Also, concepts to store and visualise provenance
information are investigated. An overview of the different areas of provenance
gives Fig. 2.
Figure 2: Provenance taxonomy according to
SimmhanPlaleEtAl2005_SurveyDataProvTech .
The figure shows five major areas: _Usage, Subject, Representation, Storage
and Dissemination._ SimmhanPlaleEtAl2005_SurveyDataProvTech gives a detailed
description on each area and their subdivisions. In this application, embedded
provenance tracking in the data management system enables DataFinder to
provide information about the chain of steps or events leading to a data item
as it is. The following list outlines relevant elements of the taxonomy from
Fig. 2 (additionally framed elements):
Use of provenance:
Provenance is used to present _information_ of the origin of the data, but
also to provide _data quality._
Subject of Provenance:
The subject is the _process_ of conducting a study or experiment. It is
focused on documentation. To identify the subject further, the Provenance
Incorporating Methodology (PrIMe, Sect. 4.1) is used.
Provenance Representation:
Provenance information will be represented in an _annotational_ model, based
on the Open Provenance Model (OPM, Sect. 2.1) and it will mainly hold
_syntactic information._
Storing Provenance:
Provenance information will be stored in the prOOst (Sect. 2.2) system (can
also hold additional information).
Provenance Dissemination:
To extract provenance information, the provenance system can be queried using
a graph traversal language (Sect. 2.2).
The main concepts of OPM and the provenance system prOOst are described in the
following sections, whereas PrIMe is discussed in the scope of applying the
technical system to the domain of good laboratory practice in Sect. 4.
### 2.1 OPM – Open Provenance Model
The Open Provenance Model MoreauCliffordEtAl2010_OpenProvenanceModel is the
result of the third “provenance challenge” efforts
SimmhanGrothEtAl2011_ThirdProvChallengeOPM to provide an interchangeable
format between provenance systems. In its core specification, it defines
elements (nodes and edges) to describe the provenance of a process.
Nodes can be _processes, agents/actors_ and _artifacts/data items._ The nodes
can be connected through edges, such as _“used”, “wasUndertakenBy”,
“wasTriggeredBy”, “wasDerivedFrom”_ and _“isBasedOn”._ Each edge is directed,
clearly defining the possible relations within a provenance model. Each node
can be enriched by annotations. Fig. 3 gives an example for conducting
experiments in a biological laboratory and it shows the usage of the model
notation. In the example, a scientist (actor) discovers a biological anomaly
(controls the process of thinking and inspiration). So he starts experimenting
(triggered by the discovery). For it to produce research results (derived from
experimenting), he needs (uses) specimen samples to work on. If the results
show a significant research outcome, a research paper can be written (based
on) the results.
Figure 3: Example a biological study as an OPM model.
### 2.2 Provenance Storage with prOOst
Groth et al. describe in GrothMilesEtAl2005_ArchitectureProvenanceSystems
theoretically the architecture of a provenance system. In
MoreauCliffordEtAl2010_OpenProvenanceModela the representation of a
provenance system is described as follows: A provenance aware application
sends information of interest to the provenance store. From this store
inquiries and information is gathered, and possibly given back to the
application.
To record the information, different approaches have been investigated. In
HollandBraunEtAl2008_ChoosingDataModel four different realisations are
discussed: Relational, XML with XPath, RDF with SPARQL and semi-structured
approaches. They conclude semi-structured approaches to be most promising. In
semi-structured systems, the used technology has no formal structure, but it
provides means of being queried.
This work uses a semi-structured approach for the provenance storage system
_prOOst._ It uses the graph database “Neo4j” Neo4jProject for storage and the
graph traversal language “Gremlin” GremlinProject for querying. Furthermore,
it provides a REST interface to record data into the store, and a web front
end to query the database. The prOOst provenance system was published under
the Apache license in July 2011 on
SourceForge.111http://sourceforge.net/projects/proost/
It is not the first implementation using a graph database for storage
technology. In TylissanakisCotronis2009_DataProvenanceReproducibility this
approach was already successfully tested. Neo4j was chosen as it is a robust,
performant and popular choice for graph storage systems. Additionally it
readily connectible with the suitable Gremlin query system to meet our
requirements. Further discussions on alternative storage or query systems are
outside the scope of this chapter. Further information on the implementation
of OPM model provenance assertions using these systems are described in the
following two sections.
#### Graph Database: Neo4j
> “Neo4j is a graph database, a fully transactional database that stores data
> structured as graphs.” (cf. Neo4jProject )
An advantage of graph databases like Neo4j is that they offer very flexible
storage models, allowing for a rapid development. Neo4j is dually licensed
(AGPLv3 open source and commercial).
Figure 4: OPM example in Neo4j.
Modelling OPM using Neo4j is described in more detail in
Wendel2010_UsingProvenanceSoftwareDev . Fig. 4 shows the previous example
(from Fig. 3) modelled as an OPM graph. Each element is represented by a node
(vertex) in the database. Nodes are indexed according to the Neo4j standard.
The nodes can be annotated with further (OPM specific) information, such as
“process” or “artifact”. Analogously, also the edges connecting the nodes are
indexed and annotated with a label (the OPM relationship).
#### Query Language: Gremlin
“Gremlin is a graph traversal language” GremlinProject . Gremlin already
provides an interface to interact with the Neo4j graph database. The following
example shows its use for querying Neo4j on the example database, searching
for the names (identifiers) of all discoveries of a certain scientistX:
$_g := neo4j:open(’database’)
$scientists := g:key($_g, ’type’, ’scientist’)
$scientistX := g:key($scientists, ’identifier’, ’scientistX’)
$discoveries := $scientistX/inE/inV[@identifier’]
## 3 Distributed, Scientific Data Management
The previous sections have discussed the technical means to manage data on the
user side (DataFinder) and to store and query the provenance information. As
indicated, DataFinder can handle a variety of different data storage servers.
However, to store data and its associated meta-data on the same system, and to
take full advantage of Grid technologies for cross-organisational federated
access, a suitable data storage service has to be chosen. For the example use
case of the team of biologists, federated access management (e. g. through
Shibboleth222http://shibboleth.internet2.edu/) and integration with further
Grid-based resources would be desired (e. g. for resources to compute on
sequenced genome data).
An electronic laboratory notebook system is a data management system, only
with the particular needs towards managing the experimental and laboratory
relevant data in a suitable fashion. This can generally be accomplished by
tweaking a generic storage system for data and (extensive) associated meta-
data towards the use case for supporting good laboratory practice. This
section therefore mainly raises the questions towards the use of such storage
systems in Grid-based environments.
Various ways are possible to envision for making relevant data available to
researchers in distributed teams. Commonly encountered mechanisms in such
(Grid) research environments are based on top of GridFTP (the “classic” Grid
data protocol) or WebDAV (extension to the HTTP protocol). In some
environments more full featured infrastructures, like
iRODS333http://www.irods.org/ have been deployed. One such environment is the
New Zealand based “Data Fabric” – as implemented for the New Zealand eScience
Infrastructure (during the recently concluded BeSTGRID project). iRODS offers
data replication over multiple geographically distributed storage locations,
with one centralised meta-data catalogue. Its data is exposed through the
iRODS native tools and libraries, as well as through WebDAV (using
Davis444WebDAV-iRODS/SRB gateway: http://projects.arcs.org.au/trac/davis/), a
web-based front-end and GridFTP (through the Griffin GridFTP server
ZhangCoddingtonEtAl2010_GriffinForArbitraryDataSources ;
ZhangKlossEtAl_GriffinProject with an iRODS back-end using
Jargon555https://www.irods.org/index.php/Jargon).
### 3.1 Integration of Existing Storage Servers
We are discussing data integration solutions according to the above mentioned
scenario of the New Zealand Data Fabric. From this, slight variations of the
setup can be easily extrapolated.
Three obvious possibilities exist to use this type of infrastructure for
provenance enabled data management and/or as a laboratory notebook system for
distributed environments. For all these, users need to be managed and mapped
between multiple systems, as iRODS introduces its own mandatory user
management. This may only be required for the storage layer, but it does
introduce a redundancy. The options are discussed in the following paragraphs.
The easiest, and directly usable, way is to integrate this Grid Data Fabric as
an external _WebDAV_ data store, using the existing persistence module. Even
though WebDAV is a comprehensive storage solution for the DataFinder for data
and meta-data, this service layer on top of iRODS does not permit the required
WebDAV protocol means to access the meta-data. An additional meta-data server
is required, and therefore potentially multiple incompatible and separate sets
meta-data may exist for the same data item stored. Unfortunately this WebDAV
service does not use the full common Grid credentials for access, but is
limited to MyProxy666Software for managing X.509 Public Key Infrastructure
(PKI) security credentials: http://grid.ncsa.illinois.edu/myproxy/ based
authentication as a work around.
As the next step up, DataFinder can be equipped with a _GridFTP_ back-end in
its persistence layer. Such a module was already available for a previous
version (1.3) of DataFinder, and only requires some porting effort for the
current (2.x) series. Again, GridFTP is only able to access the payload data,
and is not capable to access any relevant meta-data, resulting in the need of
an additional and separate meta-data service. An advantage is that this
solution uses the common Grid credentials for authentication.
Lastly, the development of a native _iRODS_ storage back-end based on the
txIRODS Python bindings777http://code.arcs.org.au/gitorious/txirods is a
possibility. This solution could also use the iRODS meta-data capabilities for
native storage on top of the payload data storage. Unfortunately, this last
solution also requires the use of the native iRODS user credentials for
accessing the repository, as it is completely incompatible to any of the
common Grid authentication procedures.
The above mentioned scenarios can be freely modified, particularly the first
two regarding their underlying storage infrastructure. One could deploy other
storage systems that expose access using WebDAV or GridFTP as service front
ends for simplicity, potentially sacrificing any of the other desired features
of iRODS like cross-site replication.
When sketching out a potential deployment, the above mentioned scenarios did
not strike us as being particularly nice to implement or manage. Several
shortcomings were quite obvious. Firstly, the central meta-data catalogue,
which can turn out to be a bottle neck. Particularly meta-data heavy scenarios
requiring extensive queries on meta-data would suffer due to increased
latencies. Secondly, the iRODS system provides a multitude of features, which
make the system implementation as well as its deployment at times quite
convoluted. A simpler, more straight forward system is often preferred.
Lastly, multiple user management systems can be an issue, particularly if this
includes the burden of mapping between, particularly if they are based on
different concepts. Grid user management is conceptually based on cross-
organisational federation, including virtual organisations (VOs) and
delegation using proxy certificates, which cannot be neatly projected to other
user concepts as employed by iRODS.
### 3.2 Designing an Alternative Storage Concept – MataNui
The idea for an alternative storage solution came up, which is simpler and a
better “Grid citizen.” For performant storage of many or large files inclusive
meta-data, the NoSQL database MongoDB with its driver side file system
implementation “GridFS” seemed like a good choice. A big advantage of this
storage concept is, that MongoDB can perform sharding (horizontal
partitioning) and replication (decentralised storage with cross-site
synchronisation) “out of the box.” Therefore, the only concerns to target were
to provide suitable service front-ends to the storage sub-system, to offer the
capabilities for the required protocols and interfaces to the DataFinder. This
means that research teams can opt for running local server instances
(alternatively to accessing a remote server) for an increase in performance as
well as decrease in latencies. This local storage sub-system also increases
data storage redundancy, which leads to a better fault protection in cases of
server or networking problems. Each storage server individually can be exposed
through different service front-ends, reducing bottle necks. These service
front-ends can be deployed in a site specific manner, reducing the number of
server instances to those required for a site.
This distributed storage concept for data and meta-data, complimented by
individual front-end services in a building block fashion, has been dubbed
“MataNui” Kloss2010_MataNuiBuildingGridDataInfrastructure . The MataNui server
Kloss_MataNuiProject itself provides full access to all content, including
server side query capability and protection through native Grid (proxy)
certificate authentication (X.509 certificates). As the authentication is
based on native Grid means, it is obvious to base the user management on Grid
identities as well, the distinguished names (DN) of the users. MataNui is
based on a REST principle based Web Service (using JSON encoding), and is
therefore easy to access through client side implementations.
Exposing further server side protocols is done by deploying generic servers,
that have been equipped with a storage back-end accessing the MataNui data
structures hosted in the MongoDB/GridFS containers. It was relatively simple
to implement the GridFTP protocol server on the basis of the free and open
Griffin ZhangCoddingtonEtAl2010_GriffinForArbitraryDataSources ;
ZhangKlossEtAl_GriffinProject server. A first beta development level GridFS
back-end is already part of the Griffin code base. Possibly later a WebDAV
front end is going to be implemented, equipping one of the quite full featured
Catacomb888http://catacomb.tigris.org/ or
LimeStone999https://github.com/tolsen/limestone servers with a GridFS back-end
for data and meta-data. Such servers then could also be used to access (and
query) the meta-data through the WebDAV protocol, if the storage back-end
supports this. Lastly, it is even possible to use a file system driver to
mount a remote GridFS into the local Linux/UNIX system. However, access
control to the content is provided through the services on top of the
MongoDB/GridFS server. Therefore, this will likely circumvent any protective
mechanisms. A better solution would be to mount a WebDAV exposed service into
a local machine’s file system hierarchy.
Figure 5: Conceptual links between components in a Grid-based data fabric to
support researchers in distributed environments. The system provides for
decentralised access to geographically distributed data repositories, while
enabling administrators to only expose local storage through service front-
ends as required.
Access through protocols as GridFTP and WebDAV is quite straight forward
through various existing clients in day-to-day use within the eResearch
communities. This is different with the MataNui RESTful service. As outlined
in Sect. 3.1 already, the DataFinder can be quite easily extended towards
providing further persistence back-ends, like a potential iRODS back-end. In a
similar fashion a MataNui REST service client back-end can be implemented. The
big difference being, that it does not require any external modules that are
not well maintained. It can mainly be based on the already available standard
library for HTTP(S) server access, with the addition of suitable cryptography
provider for extended X.509 certificate management. This can be done either by
simply wrapping the OpenSSL command line tool or by using one of the mature
and well maintained libraries such as
pyOpenSSL.101010https://launchpad.net/pyopenssl
This modularity of service front-ends leaves administrators the option to set
up sites with exactly the features required locally. However, in a global
perspective, MataNui enables a new perspective on the functionality of a data
fabric for eResearch. Fig. 5 provides a conceptual overview of how such a
distributed data repository can be structured. Every storage site requires an
instance of MongoDB with GridFS. These are linked with each other into a
replication set (with optional sharding). The storage servers for the
different sites expose the repository through one or more locally hosted
services, such as the MataNui RESTful Web Service, a GridFTP server, etc.
These services can be accessed by clients suitably equipped for the particular
service. Clients, such as the DataFinder, may require an additional
implementation for a particular persistence back-end. Some of these clients
(e. g. DataFinder or a WebDAV client) may be equipped to take advantage of the
full meta-data capabilities of the data fabric, whereas others (e. g. GridFTP
or file system mounted WebDAV) may only access the data content along with
some rudimentary system meta-data (time stamp, size, etc.).
In a scenario like this data and its meta-data can be managed in the
distributed environment through DataFinder. Seamless integration when working
with other Grid resources is unproblematic: All systems share the same type of
credentials, and data can be transferred between Grid systems directly through
GridFTP without the need of being routed through the user’s workstation.
## 4 Results
The following describes the application of the previously discussed
technologies to implement the provenance enabled electronic laboratory
notebook. For this also the data management system DataFinder requires
customisation (through Python script extensions) to suit the users’ needs. It
is enhanced with features to trace documentation.
First the development of the provenance model for good laboratory practice by
means of the PrIMe methodology is described in Sect. 4.1. Required
modifications applied in the DataFinder code are outlined in Sect. 4.2. Sect.
4.3 evaluates the integration of DataFinder for the purpose of use as an
electronic laboratory notebook in a final system. More information on this
evaluation can be found in Ney2011_GLP . Lastly, Sect. 4.4 gives an outlook on
improving DataFinder in its role as an electronic laboratory notebook, as well
as on deploying such an infrastructure fully to Grid environments.
### 4.1 Developing a Provenance Model for Good Laboratory Practice
Munroe et al. MunroeMilesEtAl2006_PrIMe:MethodologyDeveloping developed the
PrIMe methodology to identify parameters for “provenance enabling”
applications. These parameters then can be used to answer provenance
questions. A provenance question usually identifies a scenario, in which
provenance information is needed. Questions relevant for the analysis, are for
example: “Who inserted data item $X$?”, “What data items belong to a report
$X$?” and “ What is the logical successor of data item $X$?”.
Figure 6: Structure of PrIMe approach
MunroeMilesEtAl2006_PrIMe:MethodologyDeveloping .
This approach was modified (in Wendel2010_UsingProvenanceSoftwareDev ), as it
used the older p-assertion protocol (p. 15 in
Wendel2010_UsingProvenanceSoftwareDev and p. 2 in
MunroeMilesEtAl2006_PrIMe:MethodologyDeveloping ) instead of the now more
common Open Provenance Model (OPM) MoreauCliffordEtAl2010_OpenProvenanceModel
. The p-assertion protocol is similar in use to OPM, so the approach can
easily be adapted. The following list describes the three phases of the
adapted PrIMe version in correspondence to the PrIMe structure from Fig. 6:
Phase 1:
“In phase 1 of PrIMe, the kinds of provenance related questions to be answered
about the application must be identified”
MunroeMilesEtAl2006_PrIMe:MethodologyDeveloping (p. 7). First, provenance
_questions_ are determined. Then, corresponding _data items/artifacts_ that
are relevant to the the answer, are investigated.
Phase 2:
_Sub-processes, actors_ and _interactions_ are identified in phase 2. The sub-
processes are part of the adaptation (Step 2.1). Actors generate data items or
control the process. Relations between sub-processes and data items are
defined as interactions (Step 2.2). Actors, processes and interactions are
modeled with OPM.
Phase 3:
The last phase finally adapts a system to the provenance model. In this phase,
the provenance store is populated with information from the application. In
the discussed scenario, this is accomplished via REST requests to the storing
system.
Some exemplary questions that could be relevant in the sample use case have
already been given in Sect. 1.1. After analysing the questions, participating
processes need to be identified. A scientific experiment for which
documentation is provided can be divided into five sub-processes:
1. 1.
Preparation of the experiment, generating a study plan.
2. 2.
Execution of the experiment according to a study plan, generating raw data.
3. 3.
Evaluation of raw data, making them processable for interpretation.
4. 4.
Interpretation of data, publishing it or processing the data further.
5. 5.
Preservation of the data according to regularities.
The very generic nature of these sub-processes is mandated by the OECD
principles of good laboratory practice OECD_GLP_PrinciplesNo1 . Obviously,
researchers can augment each of these with further internal sub-processes as
required by the project or studies undertaken.
Figure 7: OPM for scientific experiment documentation.
These sub-processes are modeled with the Open Provenance Model (OPM). Fig. 7
shows the model in OPM notation for good laboratory practice. The five
rectangles in the figure symbolise the above mentioned sub-processes. Data
items/artifacts are indicated by circles: These are managed by the DataFinder.
Lastly, the octagons represent the actors controlling the processes.
Provenance information is gathered in the data management system on data
import and modification. Then – according to the provenance model – this
information is sent to a provenance storage system (as described in Sect. 2).
### 4.2 Adjustments for Good Laboratory Practice in the DataFinder
To use the DataFinder as a supportive tool for good laboratory practice, a new
data model and Python extensions were developed. The main part of the data
model is presented in Fig. 8.
Figure 8: Laboratory notebook data model used for the DataFinder.
The model is derived through requirements analysis in OECD_GLP_PrinciplesNo1 .
It divides the data into the five major categories according to Sect. 4.1:
Preparation, execution, evaluation, interpretation and archiving. All
experiments pass through theses categories in their five processes. Each
process needs or generates different types of data. Data is aggregated in
(nested) _collections,_ the data repository equivalent of directories in a
file system. Collections representing these processes aggregate data items
belonging to that process. Each collection or element can mandate attached
meta-data (such as type or dates). The data model also provides structural
elements at a higher level of the hierarchy to differentiate between different
studies and experiments. Processes and data items are reflecting the model
structure in Fig. 7. The DataFinder repository structure is defined through
its underlying data model111111The complete data model is described in XML and
available on
https://wiki.sistec.dlr.de/DataFinderOpenSource/LaboratoryNotebook
(implemented according to the OPM model).
In the screen shot of Fig. 1 at the beginning of this chapter, a user is
connected to a shared repository (left side) operating on the described data
model. The user now is required to organise data accumulated according to this
model. For example, a new collection of manuals may only be created within a
parent collection of the “preparation” type. A “preparation” collection can
then be either part of a “study” or “experiment.”
Three further extensions to DataFinder have been developed. They are needed to
support good laboratory practice in DataFinder:
* •
The most important extension is an _observer mechanism,_ listening on _import
events_ into the DataFinder. Upon the import of a new document, it reacts by
prompting with a dialog asking for input items within the system that have
influenced the data item/artifact. After analysing the corresponding process,
the information is recorded in the provenance store.
* •
A second extension supports _evidential archiving._ For this the user can send
an archive to an archiving service, to analyses the credibility of the
archive.121212This is not further discussed, because it is a separate project
in Germany. To provide sufficient information, the user can activate a
specific script extension, which generates an archive composed of information
relevant to the data from information in the provenance store. The user
selects a study report, and the provenance store is queried for all data items
influencing the report for each step.
* •
Lastly, a _digital signing mechanism_ was implemented through an extension,
aiming at increasing credibility of data items through non-repudiation.
### 4.3 Integration Evaluation of an Electronic Laboratory Notebook
Tab. 1 evaluates the DataFinder concepts on the requirements defined in Chap.
3.1 of Ney2011_GLP . It explains how each requirement is integrated into the
DataFinder system.131313The table and its description is adapted from
Ney2011_GLP The table shows that almost all requirements are either already
currently met, are implemented through extensions as described here, or
otherwise currently implemented. As a result, DataFinder can be used as
laboratory notebook, supporting the concepts of good laboratory practice, and
is therefore supportive to scientific working methods.
Requirement | Implemented? | Details
---|---|---
• | Chain of events | yes
with extensions described here | provenance for modeling the use case and storing the information
• | Durability | yes | with extension from this application, but also through former solutions
• | Immediate documentation | under development | a web portal is implemented
• | Genuineness | yes
customisation issue | combination of work flow integration in the DataFinder and the provenance service
• | Protocol style | yes
original | can be added as files to the system
• | Short notes | yes
original | as extra files or meta data to a data item
• | Verifying results | yes (rudimentary) | signing concept and implementation as extension
• | Accessibility | yes
original | open source software
• | Collaboration | yes
original | same shared repository for each user, with similar information
• | Device integration | yes
customisation issue | integration via script API
• | Enabling environmental specialisation | yes
customisation issue | can be customised with scripts and data model
• | Flexible Infrastructure | yes
original | client: platform independent Python application server;
meta data: WebDAV or SVN (extensible);
data: several (extensible)
• | Individual Sorting | partly
under development | customising the view of the repositories is possible (but saving the settings is in planning)
• | Rights/privilege management | yes
under construction | the server supports it on the client side, the integration into DataFinder is currently developed
• | Variety of data formats | yes
original | any data format can be integrated, opening them depends on the users system
• | Searchability | yes
original | full text and meta data search
• | Versioning | yes | SVN as storage back-end is developed to enable versioned meta data and data
Table 1: Implementation of the laboratory notebook requirements into the
DataFinder
### 4.4 Outlook: Improving DataFinder-Based Laboratory Notebook
Of course the systems discussed in this chapter themselves are still research
in progress and under constant development. On the one hand we can already
envision a list of laboratory notebook features for desired improvements. On
the other hand, this DataFinder-based laboratory notebook can be deployed to
Grid environments.
#### Improving Laboratory Notebook Features
After the implementation, the next step is to deploy and integrate the system
not only as a data management system, but as laboratory notebook to suit the
needs of different organisations. For every deployment, customisation through
automation scripts and specialised GUI dialogues need to be performed.
Particularly for the purpose of electronic laboratory notebooks, some generic
and easy integrated note editor widget would be much appreciable for free form
note taking (instead of using external editors and importing the resulting
data files). One a much more specialised level the following future features
are considered to be beneficial to further improve the laboratory notebook
functionality of DataFinder, and therefore meet the requirements of other
deployment scenarios:
_Mobile version of DataFinder:_
A mobile version of the data management system client could ease the
scientist’s documentation efforts when working on-site, away from the
established (office, lab) environment. This way the scientist could augment
data items through notes or add/edit meta-data and data on-the-fly.
Requirement of immediate documentation could be met through this extension.
_Automatic generation of reports:_
For many project leaders it is interesting or important to be kept up-to-date
on the current status of a project or what their team members are currently
doing. For this they can currently only access the data directly. A feature
summarising current reports and gives an intermediate report, could simplify
the check up. This feature was found in the evaluated laboratory notebook
mbllab mbllab-ElektronischesLaborbuch .
_Integrated standard procedures:_
In GLP, a standard procedures defines workflows for specific machines. In the
laboratory notebook mbllab mbllab-ElektronischesLaborbuch these are
integrated and give the user a guideline for actions.
_More elaborate signing and documenting features:_
Scientists discuss results of colleagues. For more collaborative work
situations, DataFinder needs to be enhanced with better features for user
interactions. On the one hand a discussion/commenting mechanism on data items
could be supported, on the other hand a scientist can sign data and leave some
kind of digital identity card. This could be used to reference a list of other
items signed or projects worked on. In the evaluated laboratory software
NoteBookMaker NoteBookMaker , a witness principle with library card is
integrated. Each notebook page contains an area, where a scientist can witness
(authenticate) an entry. After witnessing the data, the information of the
witnessing person’s identity is displayed on the corresponding page. This
witnessing information is then connected to a library card listing personal
information and projects.
_A graphical representation:_
A graphical representation of the provenance information on the server or in
the DataFinder can help to make provenance information visually more
accessible. This integration of provenance data in DataFinder assists a user
in understanding correlations between items.
_Configuration options:_
Selecting a specific provenance or archiving system should be possible. This
could be handled through a new option in the data store’s configuration.
Additionally a dialog prompting for this information needs to be implemented.
#### Migrating the Laboratory Notebook to the Grid
Sect. 3.2 already explains how a data management system suitable for the Grid
can be constructed. The laboratory notebook system is “resting” on top of that
particular data storage system, under support of a provenance store to enable
provenance enabled working schemes. Therefore, the two aspects of an
underlying Grid-based data storage system and of a Grid-enabled provenance
store need to be discussed.
While MongoDB with GridFS is a mature product ready to deploy, the overlaying
service infrastructure for a Grid-enabled data service is not quite as
matured. Currently the Griffin GridFTP server ZhangKlossEtAl_GriffinProject
is in productive deployment both in the Australian as well as the New Zealand
eScience infrastructures. However the GridFS storage back-end already works,
but is still only available in a beta version and needs a little further
completion and testing. The situation is similar with the MataNui RESTful Web
Service front end, which still needs implementation of further query
functionality. Current tests of the two systems have showed that throughput
bottle necks to both services currently seems mostly limited by the throughput
of the underlying disk (RAID) storage system or network interface (giga-bit
ethernet), while the database and service layer implementation is easily
holding up even on a moderately equipped system (CPU and memory).
To access this MataNui infrastructure with the DataFinder at least one of two
things still has to be implemented: The GridFTP data store back-end needs to
be ported from the 1.x line of DataFinder versions, or a MataNui data store
back-end needs to be implemented for the current version. For best results
preferably the latter has got priority on the list of further implementations
to reach this goal. Due to the nature of the service as well as the
persistence abstraction in the 2.x DataFinder versions, this should be
relatively straight forward. This enables DataFinder to completely retire
WebDAV or Subversion as a centralised data server for data content as well as
meta-data, relocating this information completely onto a Grid infrastructure.
In such a setup, DataFinder accesses the MataNui service natively, while all
managed (payload) data can be accessed through GridFTP (Griffin server) for
the purpose of compatibility with other Grid environments. This supports
common usage for example using file staging for Grid job submission. Storage
server side replication ensures seamless usage in geographically distributed
research teams while retaining high throughput and low latencies through the
geographically closest storage server.
The provenance store prOOst currently does not yet support access of its REST
service through Grid authenticated means. Once this is implemented for the
newly releases provenance store, every required service for a Grid-enabled
data service with provenance capabilities, can be accessed using the same
credentials and common Grid access protocols.
## 5 Conclusion
This chapter sketches a scenario of using provenance tracking with DataFinder
to support good laboratory practice and to track relations between stored
documents. In this scenario DataFinder is used in a distributed system
together with a central provenance store. This makes it possible to access and
update data from virtually anywhere with a network connection, while keeping
track of all interactions with data items through recorded provenance
information at any time. When implementing the laboratory notebook, stored
provenance information can be queried to enable the extraction of additional
valuable meta-data information on data items. As a result, provenance is
successfully used to trace typical scientific workflows comprising of
preparation, execution, evaluation, interpretation and archiving of research
data. The reliability – and therefore credibility – of research results is
increased, and assistance to help understand involved processes is provided
for the researcher.
Such a system can be implemented on top of a Grid data infrastructure, as the
described MataNui system. The MataNui service is mostly functioning already,
but still needs integration into DataFinder as a full-featured storage back-
end for data as well as meta-data. Additionally, it is already possible to
expose the data repository to Grid environments directly using the GridFTP
protocol. GridFTP is commonly used for scripts, automation and compatibility
with other Grid enabled tools. The overall MataNui concept has been designed
to be capable of handling files large in number and size, as well as manage
arbitrary amounts of meta-data associated with each data item. It is usable in
distributed projects with a self-replicating, federated data infrastructure.
This federation can drastically improve data access latency and throughput by
connecting to a geographically close service. Through support for server side
queries, meta-data searches can be processed very efficiently by avoiding
transfers of potentially large numbers of data sets to a client. Lastly, the
implementation of MataNui has been undertaken with the vision of it being
robust as well as easy to deploy and use.
## References
* (1) Gremlin graph traversal language Web Site. URL https://github.com/tinkerpop/gremlin/wiki
* (2) mbllab – Das elektronische Laborbuch. URL http://elektronisches-laborbuch.de/
* (3) Neo4j Graph Database Web Site. URL http://neo4j.org/
* (4) NoteBookMaker for PC and Mac, The World Leader in Virtual NoteBooks. URL http://www.notebookmaker.com
* (5) Buneman, P., Khanna, S., Tan, W.C.: Why and Where: A Characterization of Data Provenance. Tech. rep., University of Pennsylvania (2001). Online available: http://repository.upenn.edu/cis_papers/210/
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|
arxiv-papers
| 2011-12-13T22:30:36 |
2024-09-04T02:49:25.279980
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Miriam Ney and Guy K. Kloss and Andreas Schreiber",
"submitter": "Guy Kloss",
"url": "https://arxiv.org/abs/1112.3062"
}
|
1112.3183
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2011-214
LHCb-PAPER-2011-021
Measurement of the $C\\!P$-violating phase $\phi_{s}$
in the decay $B^{0}_{s}\\!\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
The LHCb Collaboration
R. Aaij23, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi42, C. Adrover6, A.
Affolder48, Z. Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J.
Anderson39, R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L.
Arrabito53, A. Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16,
R.J. Barlow50, C. Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10,
Th. Bauer23, A. Bay38, I. Bediaga1, S. Belogurov30, K. Belous34, I.
Belyaev30,37, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson46, J.
Benton42, R. Bernet39, M.-O. Bettler17, M. van Beuzekom23, A. Bien11, S.
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Bondar29, W. Bonivento15, S. Borghi47,50, A. Borgia52, T.J.V. Bowcock48, C.
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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
23Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraców, Poland
26AGH University of Science and Technology, Kraców, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55University of Birmingham, Birmingham, United Kingdom
56Physikalisches 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
We present a measurement of the time-dependent $C\\!P$-violating asymmetry in
$B^{0}_{s}\\!\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ decays,
using data collected with the LHCb detector at the LHC. The decay time
distribution of $B^{0}_{s}\\!\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ is characterized by the decay widths $\Gamma_{\mathrm{H}}$ and
$\Gamma_{\mathrm{L}}$ of the heavy and light mass eigenstates of the
$B^{0}_{s}$-$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ system and
by a $C\\!P$-violating phase $\phi_{s}$. In a sample of about 8500
$B^{0}_{s}\\!\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ events
isolated from $0.37$ $\mbox{\,fb}^{-1}$ of $pp$ collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ we measure
$\phi_{s}\>=\>0.15\>\pm\>0.18\;\text{(stat)}\>\pm\>0.06\;\text{(syst) rad}$.
We also find an average $B^{0}_{s}$ decay width
$\Gamma_{s}\equiv(\Gamma_{\mathrm{L}}+\Gamma_{\mathrm{H}})/2\>=\>0.657\>\pm\>0.009\;\text{(stat)}\>\pm\>0.008\;\text{(syst)
${\rm\,ps^{-1}}$}$ and a decay width difference
$\Delta\Gamma_{s}\equiv\Gamma_{\mathrm{L}}-\Gamma_{\mathrm{H}}\>=\>0.123\>\pm\>0.029\;\text{(stat)}\>\pm\>0.011\;\text{(syst)
${\rm\,ps^{-1}}$}$. Our measurement is insensitive to the transformation
$(\phi_{s},\Delta\Gamma_{s})\mapsto(\pi-\phi_{s},-\Delta\Gamma_{s})$.
To be submitted to Physical Review Letters
In the Standard Model (SM) $C\\!P$ violation arises through a single phase in
the CKM quark mixing matrix Kobayashi:1973fv ; *Cabibbo:1963yz. In neutral $B$
meson decays to a final state which is accessible to both $B$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ mesons, the interference between the
amplitude for the direct decay and the amplitude for decay after oscillation,
leads to a time-dependent $C\\!P$-violating asymmetry between the decay time
distributions of $B$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$
mesons. The decay $B^{0}_{s}\\!\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ allows the measurement of such an asymmetry, which can be
expressed in terms of the decay width difference of the heavy (H) and light
(L) $B^{0}_{s}$ mass eigenstates
$\Delta\Gamma_{s}\equiv\Gamma_{\mathrm{L}}-\Gamma_{\mathrm{H}}$ and a single
phase $\phi_{s}$ Carter:1980hr ; *Carter:1980tk; *Bigi:1981qs; *Bigi:1986vr.
In the SM, the decay width difference is
$\Delta\Gamma_{s}^{\text{SM}}=0.087\pm 0.021$ ${\rm\,ps^{-1}}$ Lenz:2006hd ;
*Badin:2007bv; *Lenz:2011ti, while the phase is predicted to be small,
$\phi_{s}^{\text{SM}}=-2\arg\left(-V_{ts}V_{tb}^{*}/V_{cs}V_{cb}^{*}\right)=-0.036\pm
0.002$ rad Charles:2011va . This value ignores a possible contribution from
sub-leading decay amplitudes Faller:2008gt . Contributions from physics beyond
the SM could lead to much larger values of $\phi_{s}$ phisnewphysics .
In this Letter we present measurements of $\phi_{s}$, $\Delta\Gamma_{s}$ and
the average decay width
$\Gamma_{s}\equiv(\Gamma_{\mathrm{L}}+\Gamma_{\mathrm{H}})/2$. Previous
measurements of these quantities have been reported by the CDF and DØ
collaborations Aaltonen:2007he ; *Abazov:2008fj; *Abazov:2011ry;
*Aaltonen:2011cq. We use an integrated luminosity of $0.37$$\mbox{\,fb}^{-1}$
of $pp$ collision data recorded at a centre-of-mass energy
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ by the LHCb experiment during the
first half of 2011. The LHCb detector is a forward spectrometer at the Large
Hadron Collider and is described in detail in Ref. Alves:2008zz .
We look for $B^{0}_{s}\\!\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
candidates in decays to ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\to\mu^{+}\mu^{-}$ and $\phi\to K^{+}K^{-}$. Events are selected by a
trigger system consisting of a hardware trigger, which selects muon or hadron
candidates with high transverse momentum with respect to the beam direction
($p_{\rm T}$), followed by a two stage software trigger. In the first stage a
simplified event reconstruction is applied. Events are required to either have
two well-identified muons with invariant mass above 2.7
$\mathrm{\,Ge\kern-1.00006ptV}$, or at least one muon or one high-$p_{\rm T}$
track with a large impact parameter to any primary vertex. In the second stage
a full event reconstruction is performed and only events with a muon candidate
pair with invariant mass within $120$ $\mathrm{\,Me\kern-1.00006ptV}$ of the
nominal ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass Nakamura:2010zzi
are retained. We adopt units such that $c=1$ and $\hbar=1$.
For the final event selection muon candidates are required to have $p_{\rm
T}>0.5$ $\mathrm{\,Ge\kern-1.00006ptV}$.
${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 an invariant
mass in the range $3030-3150$ $\mathrm{\,Me\kern-1.00006ptV}$. The latter
corresponds to about eight times the $\mu^{+}\mu^{-}$ invariant mass
resolution and covers part of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ radiative tail. The $\phi$ selection requires two oppositely charged
particles that are identified as kaons, form a common vertex and have an
invariant mass within $\pm 12$ $\mathrm{\,Me\kern-1.00006ptV}$ of the nominal
$\phi$ mass Nakamura:2010zzi . The $p_{\rm T}$ of the $\phi$ candidate is
required to exceed 1 $\mathrm{\,Ge\kern-1.00006ptV}$. The mass window covers
approximately 90% of the $\phi\to K^{+}K^{-}$ lineshape.
We select $B^{0}_{s}$ candidates from combinations of a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and a $\phi$ with invariant
mass $m_{B}$ in the range $5200-5550$ $\mathrm{\,Me\kern-1.00006ptV}$. The
latter is computed with the invariant mass of the $\mu^{+}\mu^{-}$ pair
constrained to the nominal $J/\psi$ mass. The decay time $t$ of the
$B^{0}_{s}$ is obtained from a vertex fit that constrains the
$B^{0}_{s}\to\mu^{+}\mu^{-}K^{+}K^{-}$ candidate to originate from the primary
vertex Hulsbergen:2005pu . The $\chi^{2}$ of the fit, which has $7$ degrees of
freedom, is required to be less than $35$. In the small fraction of events
with more than one candidate, only the candidate with the smallest $\chi^{2}$
is kept. $B^{0}_{s}$ candidates are required to have a decay time within the
range $0.3<t<14.0\;\rm ps$. Applying a lower bound on the decay time
suppresses a large fraction of the prompt combinatorial background whilst
having a small effect on the sensitivity to $\phi_{s}$. From a fit to the
$m_{B}$ distribution, shown in Fig. 1, we extract a signal of $8492\pm 97$
events.
Figure 1: Invariant mass distribution for
$B^{0}_{s}\to\mu^{+}\mu^{-}K^{+}K^{-}$ candidates with the mass of the
$\mu^{+}\mu^{-}$ pair constrained to the nominal $J/\psi$ mass. Curves for
fitted contributions from signal (dashed), background (dotted) and their sum
(solid) are overlaid.
The $B^{0}_{s}\\!\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi{}\rightarrow\mu^{+}\mu^{-}K^{+}K^{-}$ decay proceeds via two
intermediate spin-1 particles (_i.e._ with the $K^{+}K^{-}$ pair in a P-wave).
The final state can be $C\\!P$-even or $C\\!P$-odd depending upon the relative
orbital angular momentum between the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and the $\phi$. The same final state can also be produced with
$K^{+}K^{-}$ pairs with zero relative orbital angular momentum (S-wave)
Stone:2008ak . This S-wave final state is $C\\!P$-odd. In order to measure
$\phi_{s}$ it is necessary to disentangle the $C\\!P$-even and $C\\!P$-odd
components. This is achieved by analysing the distribution of the
reconstructed decay angles $\Omega=(\theta,\psi,\varphi)$ in the transversity
basis Dighe:1995pd ; *Dighe:1998vk; Dunietz:2000cr . In the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rest frame we define a right-
handed coordinate system such that the $x$ axis is parallel to the direction
of the $\phi$ momentum and the $z$ axis is parallel to the cross-product of
the $K^{-}$ and $K^{+}$ momenta. In this frame $\theta$ and $\varphi$ are the
azimuthal and polar angles of the $\mu^{+}$. The angle $\psi$ is the angle
between the $K^{-}$ momentum and the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ momentum in the rest frame of the $\phi$.
We perform an unbinned maximum likelihood fit to the invariant mass $m_{B}$,
the decay time $t$, and the three decay angles $\Omega$. The probability
density function (PDF) used in the fit consists of signal and background
components which include detector resolution and acceptance effects. The PDFs
are factorised into separate components for the mass and for the remaining
observables.
The signal $m_{B}$ distribution is described by two Gaussian functions with a
common mean. The mean and width of the narrow Gaussian are fit parameters. The
fraction of the second Gaussian and its width relative to the narrow Gaussian
are fixed to values obtained from simulated events. The $m_{B}$ distribution
for the combinatorial background is described by an exponential function with
a slope determined by the fit. Possible peaking background from decays with
similar final states such as $B^{0}\\!\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ is found to be negligible from studies using simulated events.
The distribution of the signal decay time and angles is described by a sum of
ten terms, corresponding to the four polarization amplitudes and their
interference terms. Each of these is the product of a time-dependent function
and an angular function Dighe:1995pd ; *Dighe:1998vk
$\frac{\mathrm{d}^{4}\Gamma(B^{0}_{s}\\!\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi)}{\mathrm{d}t\;\mathrm{d}\Omega}\;\propto\;\sum^{10}_{k=1}\>h_{k}(t)\>f_{k}(\Omega)\,.$
(1)
The time-dependent functions $h_{k}(t)$ can be written as
$h_{k}(t)\;=\;N_{k}e^{-\Gamma_{s}t}\>\left[c_{k}\cos(\Delta
m_{s}t)\,+d_{k}\sin(\Delta m_{s}t)\,\right.\\\
\left.+a_{k}\cosh\left(\tfrac{1}{2}\Delta\Gamma_{s}t\right)+b_{k}\sinh\left(\tfrac{1}{2}\Delta\Gamma_{s}t\right)\right].$
(2)
where $\Delta m_{s}{}$ is the $B^{0}_{s}$ oscillation frequency. The
coefficients $N_{k}$ and $a_{k},\ldots,d_{k}$ can be expressed in terms of
$\phi_{s}$ and four complex transversity amplitudes $A_{i}$ at $t=0$. The
label $i$ takes the values $\\{\perp,\parallel,0\\}$ for the three P-wave
amplitudes and S for the S-wave amplitude. In the fit we parameterize each
$A_{i}(0)$ by its magnitude squared $|A_{i}(0)|^{2}$ and its phase
$\delta_{i}$, and adopt the convention $\delta_{0}=0$ and
$\sum|A_{i}(0)|^{2}=1$. For a particle produced in a $B^{0}_{s}$ flavour
eigenstate the coefficients in Eq. 2 and the angular functions $f_{k}(\Omega)$
are then, see Dunietz:2000cr ; Xie:2009fs , given by
$\begin{array}[]{c|c|c|c|c|c|c}k&f_{k}(\theta,\psi,\varphi)&N_{k}&a_{k}&b_{k}&c_{k}&d_{k}\\\
\hline\cr
1&2\,\cos^{2}\psi\left(1-\sin^{2}\theta\cos^{2}\phi\right)&|A_{0}(0)|^{2}&1&-\cos\phi_{s}&0&\sin\phi_{s}\\\
2&\sin^{2}\psi\left(1-\sin^{2}\theta\sin^{2}\phi\right)&|A_{\|}(0)|^{2}&1&-\cos\phi_{s}&0&\sin\phi_{s}\\\
3&\sin^{2}\psi\sin^{2}\theta&|A_{\perp}(0)|^{2}&1&\cos\phi_{s}&0&-\sin\phi_{s}\\\
4&-\sin^{2}\psi\sin
2\theta\sin\phi&|A_{\|}(0)A_{\perp}(0)|&0&-\cos(\delta_{\perp}-\delta_{\|})\sin\phi_{s}&\sin(\delta_{\perp}-\delta_{\|})&-\cos(\delta_{\perp}-\delta_{\|})\cos\phi_{s}\\\
5&\tfrac{1}{2}\sqrt{2}\sin 2\psi\sin^{2}\theta\sin
2\phi&|A_{0}(0)A_{\|}(0)|&\cos(\delta_{\|}-\delta_{0})&-\cos(\delta_{\|}-\delta_{0})\cos\phi_{s}&0&\cos(\delta_{\|}-\delta_{0})\sin\phi_{s}\\\
6&\tfrac{1}{2}\sqrt{2}\sin 2\psi\sin
2\theta\cos\phi&|A_{0}(0)A_{\perp}(0)|&0&-\cos(\delta_{\perp}-\delta_{0})\sin\phi_{s}&\sin(\delta_{\perp}-\delta_{0})&-\cos(\delta_{\perp}-\delta_{0})\cos\phi_{s}\\\
7&\tfrac{2}{3}(1-\sin^{2}\theta\cos^{2}\phi)&|A_{\mathrm{S}}(0)|^{2}&1&\cos\phi_{s}&0&-\sin\phi_{s}\\\
8&\tfrac{1}{3}\sqrt{6}\sin\psi\sin^{2}\theta\sin
2\phi&|A_{\mathrm{S}}(0)A_{\|}(0)|&0&-\sin(\delta_{\|}-\delta_{\mathrm{S}})\sin\phi_{s}&\cos(\delta_{\|}-\delta_{\mathrm{S}})&-\sin(\delta_{\|}-\delta_{\mathrm{S}})\cos\phi_{s}\\\
9&\tfrac{1}{3}\sqrt{6}\sin\psi\sin
2\theta\cos\phi&|A_{\mathrm{S}}(0)A_{\perp}(0)|&\sin(\delta_{\perp}-\delta_{\mathrm{S}})&\sin(\delta_{\perp}-\delta_{\mathrm{S}})\cos\phi_{s}&0&-\sin(\delta_{\perp}-\delta_{\mathrm{S}})\sin\phi_{s}\\\
10&\tfrac{4}{3}\sqrt{3}\cos\psi(1-\sin^{2}\theta\cos^{2}\phi)&|A_{\mathrm{S}}(0)A_{0}(0)|&0&-\sin(\delta_{0}-\delta_{\mathrm{S}})\sin\phi_{s}&\cos(\delta_{0}-\delta_{\mathrm{S}})&-\sin(\delta_{0}-\delta_{\mathrm{S}})\cos\phi_{s}\\\
\end{array}$
We neglect $C\\!P$ violation in mixing and in the decay amplitudes. The
differential decay rates for a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson produced at time $t=0$
are obtained by changing the sign of $\phi_{s}$, $A_{\perp}(0)$ and
$A_{\mathrm{S}}(0)$, or, equivalently, the sign of $c_{k}$ and $d_{k}$ in the
expressions above. The PDF is invariant under the transformation
$(\phi_{s},\Delta\Gamma_{s},\delta_{\|},\delta_{\perp},\delta_{\mathrm{S}})\mapsto(\pi-\phi_{s},-\Delta\Gamma_{s},-\delta_{\|},\pi-\delta_{\perp},-\delta_{\mathrm{S}})$
which gives rise to a two-fold ambiguity in the results.
We have verified that correlations between decay time and decay angles in the
background are small enough to be ignored. Using the data in the $m_{B}$
sidebands, which we define as selected events with $m_{B}$ outside the range
$5311-5411$ $\mathrm{\,Me\kern-1.00006ptV}$, we determine that the background
decay time distribution can be modelled by a sum of two exponential functions.
The lifetime parameters and the relative fraction are determined by the fit.
The decay angle distribution is modelled using a histogram obtained from the
data in the $m_{B}$ sidebands. The normalisation of the background with
respect to the signal is determined by the fit.
The measurement of $\phi_{s}$ requires knowledge of the flavour of the
$B^{0}_{s}$ meson at production. We exploit the following flavour specific
features of the accompanying (non-signal) $b$-hadron decay to tag the
$B^{0}_{s}$ flavour: the charge of a muon or an electron with large transverse
momentum produced by semileptonic decays, the charge of a kaon from a
subsequent charmed hadron decay and the momentum-weighted charge of all tracks
included in the inclusively reconstructed decay vertex. These signatures are
combined using a neural network to estimate a per-event mistag probability,
$\omega$, which is calibrated with data from control channels LHCb-
PAPER-2011-027 . The fraction of tagged events in the signal sample is
$\varepsilon_{\text{tag}}=(24.9\pm 0.5)\%$. The dilution of the $C\\!P$
asymmetry due to the mistag probability is $D=1-2\omega$. The effective
dilution in our signal sample is $D=0.277\pm 0.006~{}\mathrm{(stat)}\pm
0.016~{}\mathrm{(syst)}$, resulting in an effective tagging efficiency of
$\varepsilon_{\text{tag}}D^{2}=(1.91\pm 0.23)\%$. The uncertainty in $\omega$
is taken into account by allowing calibration parameters described in Ref.
LHCb-PAPER-2011-027 to vary in the fit with Gaussian constraints given by
their estimated uncertainties. Both tagged and untagged events are used in the
fit. The untagged events dominate the sensitivity to the lifetimes and
amplitudes.
To account for the decay time resolution, the PDF is convolved with a sum of
three Gaussian functions with a common mean and different widths. Studies on
simulated data have shown that selected prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$ combinations have
nearly identical resolution to signal events. Consequently, we determine the
parameters of the resolution model from a fit to the decay time distribution
of such prompt combinations in the data, after subtracting non-$J/\psi$ events
with the sPlot method Pivk:2004ty using the $\mu^{+}\mu^{-}$ invariant mass
as discriminating variable. The resulting dilution is equivalent to that of a
single Gaussian with a width of 50 fs. The uncertainty on the decay time
resolution is estimated to be 4% by varying the selection of events and by
comparing in the simulation the resolutions obtained for prompt combinations
and $B^{0}_{s}$ signal events. This uncertainty is accounted for by scaling
the widths of the three Gaussians by a common factor of $1.00\pm 0.04$, which
is varied in the fit subject to a Gaussian constraint. In similar fashion the
uncertainty on the mixing frequency is taken into account by varying it within
the constraint imposed by the LHCb measurement $\Delta m_{s}=17.63\pm
0.11~{}\mathrm{(stat)}\pm 0.02~{}\mathrm{(syst)}$ ${\rm\,ps^{-1}}$ LHCB-
PAPER-2011-010 .
The decay time distribution is affected by two acceptance effects. First, the
efficiency decreases approximately linearly with decay time due to
inefficiencies in the reconstruction of tracks far from the central axis of
the detector. This effect is parameterized as $\epsilon(t)\propto(1-\beta t)$
where the factor $\beta=0.016$ ${\rm\,ps^{-1}}$ is determined from simulated
events. Second, a fraction of approximately 14% of the events has been
selected exclusively by a trigger path that exploits large impact parameters
of the decay products, leading to a drop in efficiency at small decay times.
This effect is described by the empirical acceptance function
$\epsilon(t)\;\propto\;(at)^{c}\,/\,[1+(at)^{c}]$, applied only to these
events. The parameters $a$ and $c$ are determined in the fit. As a result, the
events selected with impact parameter cuts do effectively not contribute to
the measurement of $\Gamma_{s}$.
The uncertainty on the reconstructed decay angles is small and is neglected in
the fit. The decay angle acceptance is determined using simulated events. The
deviation from a flat acceptance is due to the LHCb forward geometry and
selection requirements on the momenta of final state particles. The acceptance
varies by less than 5% over the full range for all three angles.
The results of the fit for the main observables are shown in Table 1. The
likelihood profile for $\delta_{\|}$ is not parabolic and we therefore quote
the 68% confidence level (CL) range $3.0<\delta_{\|}<3.5$. The correlation
coefficients for the statistical uncertainties are
$\rho(\Gamma_{s},\Delta\Gamma_{s})=-0.30$, $\rho(\Gamma_{s},\phi_{s})=0.12$
and $\rho(\Delta\Gamma_{s},\phi_{s})=-0.08$. Figure 2 shows the data
distribution for decay time and angles with the projections of the best fit
PDF overlaid. To assess the overall agreement of the PDF with the data we
calculate the goodness of fit based on the point-to-point dissimilarity test
Williams:2010vh . The $p$-value obtained is $0.68$. Figure 3 shows the 68%,
90% and 95% CL contours in the $\Delta\Gamma_{s}$-$\phi_{s}$ plane. These
contours are obtained from the likelihood profile after including systematic
uncertainties, and correspond to decreases in the natural logarithm of the
likelihood, with respect to its maximum, of 1.15, 2.30 and 3.00 respectively.
Table 1: Fit results for the solution with $\Delta\Gamma_{s}>0$ with
statistical and systematic uncertainties.
parameter value $\sigma_{\text{stat.}}$ $\sigma_{\text{syst.}}$ $\Gamma_{s}$
[ps-1] 0.657 0.009 0.008 $\Delta\Gamma_{s}$ [ps-1] 0.123 0.029 0.011
$|A_{\perp}(0)|^{2}$ 0.237 0.015 0.012 $|A_{0}(0)|^{2}$ 0.497 0.013 0.030
$|A_{\mathrm{S}}(0)|^{2}$ 0.042 0.015 0.018 $\delta_{\perp}$ [rad] 2.95 0.37
0.12 $\delta_{\mathrm{S}}$ [rad] 2.98 0.36 0.12 $\phi_{s}$[rad] 0.15 0.18 0.06
Figure 2: Projections for the decay time and transversity angle distributions
for events with $m_{B}$ in a $\pm\,20$ $\mathrm{\,Me\kern-1.00006ptV}$ range
around the $B^{0}_{s}$ mass. The points are the data. The dashed, dotted and
solid lines represent the fitted contributions from signal, background and
their sum. The remaining curves correspond to different contributions to the
signal, namely the $C\\!P$-even P-wave (dashed with single dot), the
$C\\!P$-odd P-wave (dashed with double dot) and the S-wave (dashed with triple
dot).
The sensitivity to $\phi_{s}$ stems mainly from its appearance as the
amplitude of the $\sin(\Delta m_{s}t)$ term in Eq. 1, which is diluted by the
decay time resolution and mistag probability. Systematic uncertainties from
these sources and from the mixing frequency are absorbed in the statistical
uncertainties as explained above. Other systematic uncertainties are
determined as follows, and added in quadrature to give the values shown in
Table 1.
To test our understanding of the decay angle acceptance we compare the
rapidity and momentum distributions of the kaons and muons of selected
$B^{0}_{s}$ candidates in data and simulated events. Only in the kaon momentum
distribution do we observe a significant discrepancy. We reweight the
simulated events to match the data, rederive the acceptance corrections and
assign the resulting difference in the fit result as a systematic uncertainty.
This is the dominant contribution to the systematic uncertainty on all
parameters except $\Gamma_{s}$. The limited size of the simulated event sample
leads to a small additional uncertainty. The systematic uncertainty due to the
background decay angle modelling was found to be negligible by comparing with
a fit where the background was removed statistically using the sPlot method
Pivk:2004ty .
In the fit each $|A_{i}(0)|^{2}$ is constrained to be greater than zero, while
their sum is constrained to unity. This can result in a bias if one or more of
the amplitudes is small. This is the case for the S-wave amplitude, which is
compatible with zero within $3.2$ standard deviations. The resulting biases on
the $|A_{i}(0)|^{2}$ have been determined using simulations to be less than
0.010 and are included as systematic uncertainties.
Finally, a systematic uncertainty of $0.008$ ps-1 was assigned to the
measurement of $\Gamma_{s}$ due to the uncertainty in the decay time
acceptance parameter $\beta$. Other systematic uncertainties, such as those
from the momentum scale and length scale of the detector, were found to be
negligible.
Figure 3: Likelihood confidence regions in the $\Delta\Gamma_{s}$-$\phi_{s}$
plane. The black square and error bar corresponds to the Standard Model
prediction Lenz:2006hd ; *Badin:2007bv; *Lenz:2011ti; Charles:2011va .
In summary, in a sample of $0.37$$\mbox{\,fb}^{-1}$ of $pp$ collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ collected with the LHCb detector we
observe $8492\pm 97$ $B^{0}_{s}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ events with $K^{+}K^{-}$ invariant mass within $\pm\,12$
$\mathrm{\,Me\kern-1.00006ptV}$ of the $\phi$ mass. With these data we perform
the most precise measurements of $\phi_{s}$, $\Delta\Gamma_{s}$ and
$\Gamma_{s}$ in $B^{0}_{s}\\!\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ decays, substantially improving upon previous measurements
Aaltonen:2007he ; *Abazov:2008fj; *Abazov:2011ry; *Aaltonen:2011cq and
providing the first direct evidence for a non-zero value of
$\Delta\Gamma_{s}$. Two solutions with equal likelihood are obtained, related
by the transformation
$(\phi_{s},\Delta\Gamma_{s})\mapsto(\pi-\phi_{s},-\Delta\Gamma_{s})$. The
solution with positive $\Delta\Gamma_{s}$ is
$\begin{array}[]{cclllllll}\phi_{s}&=&0.15&\pm&0.18&\text{(stat)}&\pm&0.06&\text{(syst)
rad},\\\\[4.2679pt]
\Gamma_{s}&=&0.657&\pm&0.009&\text{(stat)}&\pm&0.008&\text{(syst)
${\rm\,ps^{-1}}$},\\\\[4.2679pt]
\Delta\Gamma_{s}&=&0.123&\pm&0.029&\text{(stat)}&\pm&0.011&\text{(syst)
${\rm\,ps^{-1}}$},\end{array}$
and is in agreement with the Standard Model prediction Lenz:2006hd ;
*Badin:2007bv; *Lenz:2011ti; Charles:2011va . Values of $\phi_{s}$ in the
range $0.52<\phi_{s}<2.62$ and $-2.93<\phi_{s}<-0.21$ are excluded at 95%
confidence level. In a future publication we shall differentiate between the
two solutions by exploiting the dependence of the phase difference between the
P-wave and S-wave contributions on the $K^{+}K^{-}$ invariant mass Xie:2009fs
.
## 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
| 2011-12-14T12:19:58 |
2024-09-04T02:49:25.298026
|
{
"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 G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis, J.\n Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, D.\n S. Bailey, 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, F.\n Constantin, A. Contu, A. Cook, M. Coombes, G. Corti, 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, F. De Lorenzi, J. M. De Miranda, L. De Paula, P. De Simone, D. Decamp,\n M. 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, L. Est\\'eve, A. Falabella, E.\n Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, 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, N. Gauvin, M.\n Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, G. Haefeli, C.\n Haen, S. 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, P. Koppenburg, A. Kozlinskiy, L. Kravchuk, K. Kreplin,\n M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, T.\n Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\'evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles,\n R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E. Lopez\n Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I.\n V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R. M. D. Mamunur,\n G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, 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, R. Messi, S. Miglioranzi, D. A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Nedos, M.\n Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora Goicochea, P.\n Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis,\n M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M.\n Patel, S. K. Paterson, G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A.\n Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.\n L. Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrella, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R. Plackett, S.\n Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B.\n Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig\n Navarro, W. Qian, J. H. Rademacker, B. Rakotomiaramanana, M. S. Rangel, I.\n Raniuk, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, K.\n Rinnert, D. A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez\n Perez, G. J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T.\n Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, C. Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R.\n Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M.\n Savrie, D. Savrina, P. Schaack, M. Schiller, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, A. C. Smith, N. A.\n Smith, E. Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F. Soomro, B. Souza\n 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. Topp-Joergensen, N. Torr, E. Tournefier, M. T.\n Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, P. Urquijo,\n U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S.\n Vecchi, J. J. Velthuis, M. Veltri, B. Viaud, I. Videau, X. Vilasis-Cardona,\n J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, H. Voss, 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. Zavertyaev, F.\n Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, E. Zverev, A.\n Zvyagin",
"submitter": "Wouter Hulsbergen",
"url": "https://arxiv.org/abs/1112.3183"
}
|
1112.3187
|
.
# John-Nirenerg inequality and atomic decomposition for noncommutative
martingales
Guixiang Hong∗ Laboratoire de Mathématiques, Université de Franche-Comté,
25030 Besançon Cedex, France
_E-mail address: guixiang.hong@univ-fcomte.fr_ and Tao Mei† Department of
Mathematics, Wayne State University,
656 W. Kirby Detroit, MI 48202. USA
_E-mail address: mei@wayne.edu_
###### Abstract.
In this paper, we study the John-Nirenberg inequality for ${\mathcal{BMO}}$
and the atomic decomposition for $\mathcal{H}_{1}$ of noncommutative
martingales. We first establish a crude version of the column (resp. row)
John-Nirenberg inequality for all $0<p<\infty$. By an extreme point property
of $L_{p}$-space for $0<p\leq 1$, we then obtain a fine version of this
inequality. The latter corresponds exactly to the classical John-Nirenberg
inequality and enables us to obtain an exponential integrability inequality
like in the classical case. These results extend and improve Junge and Musat’s
John-Nirenberg inequality. By duality, we obtain the corresponding $q$-atomic
decomposition for different Hardy spaces $\mathcal{H}_{1}$ for all
$1<q\leq\infty$, which extends the $2$-atomic decomposition previously
obtained by Bekjan et al. Finally, we give a negative answer to a question
posed by Junge and Musat about ${\mathcal{BMO}}$.
* Partially supported by ANR grant 2011-BS01-008-01.
† Partially supported by NSF grant DMS-0901009.
MR(2000) Subject Classification. Primary 46L52, 46L53; Secondary 60G42, 60G46.
Keywords. Noncommutative $L_{p}$-spaces, Hardy spaces and BMO spaces,
Noncommutative martingales, John-Nirenberg inequality, Atomic decomposition
## 1\. Introduction
This paper deals with BMO spaces and atomic decomposition for noncommutative
martingales. The modern period of development of noncommutative martingale
inequalities began with Pisier and Xu’s seminal paper [18] in which they
established the noncommutative Burkholder-Gundy inequalities and Fefferman
duality theorem between $\mathcal{H}_{1}$ and ${\mathcal{BMO}}$. Since then
remarkable progress has been made in the field. We refer, for instance, to
[6], [9], [11], [20] for other noncommutative martingales inequalities, to
[14], [1] for interpolation of noncommutative Hardy spaces and to [16], [17]
for the noncommutative Gundy and Davis decompositions. Let us also mention two
other works that motivate the present paper. The first one is Junge and
Musat’s noncommutative John-Nirenberg theorem [8] and the second the
$2$-atomic decomposition of the Hardy spaces $\mathcal{H}_{1}$ by Bekjan,
Chen, Perrin and Yin [1].
Before describing our main results, we recall the classical John-Nirenberg
inequalities in the martingale theory. Let $(\Omega,\mathcal{F},\mathbb{P})$
be a probability space and $(\mathcal{F}_{n})_{n\geq 0}$ an increasing
sequence of sub-$\sigma$-algebras of $\mathcal{F}$ with the associated
conditional expectations $(\mathbb{E}_{n})_{n\geq 0}$. The $BMO(\Omega)$ space
is defined as the set of all $x\in L_{1}(\Omega)$ with the norm
(1.1)
$\displaystyle\|x\|_{BMO}=\sup_{n}\|\mathbb{E}_{n}|x-x_{n-1}|\|_{\infty}<\infty.$
The classical John-Nirenberg theorem says that there exist two universal
constants $c_{1}$, $c_{2}>0$ such that if $\|x\|_{BMO}<c_{2}$, then
(1.2)
$\displaystyle\sup_{n}\|\mathbb{E}_{n}(e^{c_{1}|x-x_{n-1}|})\|_{\infty}<1.$
This statement is equivalent to the following one: There exists an absolute
constant $c$ such that for all $1\leq p<\infty$,
(1.3)
$\displaystyle\|x\|_{BMO}\leq\sup_{n}\|\mathbb{E}_{n}|x-x_{n-1}|^{p}\|^{\frac{1}{p}}_{\infty}\leq
cp\|x\|_{BMO}.$
A duality argument yields
(1.4) $\displaystyle\|\mathbb{E}_{n}|x-x_{n-1}|^{p}\|^{\frac{1}{p}}_{\infty}$
$\displaystyle=\sup_{b\in L_{\infty}(\mathcal{F}_{n}),\|b\|_{1}\leq
1}\left(\int|x-x_{n-1}|^{p}bd\mathbb{P}\right)^{\frac{1}{p}}$ (1.5)
$\displaystyle=\sup_{b\in L_{\infty}(\mathcal{F}_{n}),\|b\|_{p}\leq
1}\|(x-x_{n-1})b\|_{p}.$
Furthermore, by the extreme point property of $L_{1}(\mathcal{F}_{n})$ and
(1.4), the John-Nirenberg theorem (1.3) can be rewritten as follows
(1.6)
$\displaystyle\|x\|_{BMO}\leq\sup_{n}\sup_{E\in\mathcal{F}_{n}}\frac{1}{{\mathbb{P}(E)}^{1/p}}\|(x-x_{n-1})\mathds{1}_{E}\|_{p}\leq
cp\|x\|_{BMO}.$
Accordingly, (1.2) can be reformulated as: For any $n\geq 1$,
$E\in\mathcal{F}_{n}$ and $\lambda>0$
(1.7)
$\displaystyle\frac{1}{\mathbb{P}(E)}\,\mathbb{P}\big{(}\big{\\{}\omega\in
E\,:\,|x(\omega)-x_{n-1}(\omega)|>\lambda\big{\\}}\big{)}\leq
c_{2}\exp(-c_{1}\lambda/\|x\|_{BMO}).$
Junge and Musat [8] proved a noncommutative version of John-Nirenberg theorem
corresponding to (1.5). To state their result we need fix some notation. Let
${\mathcal{M}}$ be a von Neumann algebra with a normal faithful tracial state
$\tau$. Let $({\mathcal{M}}_{n})_{n\geq 1}$ be an increasing sequence of von
Neumann subalgebras of ${\mathcal{M}}$ such that the union of
${\mathcal{M}}_{n}$’s is $w^{*}$-dense in ${\mathcal{M}}$. Let
$\mathcal{E}_{n}$ be the conditional expectation of ${\mathcal{M}}$ with
respect to ${\mathcal{M}}_{n}$. Define
$\|x\|_{{\mathcal{BMO}}^{c}}=\sup_{n\geq
1}\|\mathcal{E}_{n}|x-x_{n-1}|^{2}\|^{\frac{1}{2}}_{\infty}$
and
${\mathcal{BMO}}({\mathcal{M}})=\\{x\in
L_{1}({\mathcal{M}}):\|x\|_{{\mathcal{BMO}}}<\infty\\}$
with
$\|x\|_{{\mathcal{BMO}}}=\max\\{\|x\|_{{\mathcal{BMO}}^{c}},\|x^{*}\|_{{\mathcal{BMO}}^{c}}\\}.$
Then Junge and Musat’s John-Nirenberg inequality reads as follows: There
exists an absolute constant $c$ such that for all $2\leq p<\infty$,
$\|x\|_{{\mathcal{BMO}}}\leq\mathcal{B}_{p}(x)\leq cp\|x\|_{{\mathcal{BMO}}},$
where
$\displaystyle\mathcal{B}_{p}(x)=\max\\{$
$\displaystyle\sup_{n}\sup_{b\in{\mathcal{M}}_{n},\|b\|_{p}\leq
1}\|(x-x_{n-1})b\|_{p},$
$\displaystyle\sup_{n}\sup_{b\in{\mathcal{M}}_{n},\|b\|_{p}\leq
1,}\|b(x-x_{n-1})\|_{p}\\}.$
However, this theorem does not correspond to the commonly used form of the
classical John-Nirenberg inequality. On the other hand, it does not hold (see
Remark 3.14 for a counterexample) when considering
${\mathcal{BMO}}^{c}({\mathcal{M}})$ or ${\mathcal{BMO}}^{r}({\mathcal{M}})$
separately. The first purpose of this paper is to remedy these aspects of
Junge and Musat’s theorem. The following is one of our main results. We refer
to the next section for all spaces and notations used below.
$\mathcal{P}({\mathcal{M}})$ denotes the set of all projections of
${\mathcal{M}}$.
###### Theorem A.
For $0<p<\infty$, we have
$\alpha^{-1}_{p}\|x\|_{{\mathcal{BMO}}}\leq\mathcal{PB}_{p}(x)\leq\beta_{p}\|x\|_{{\mathcal{BMO}}},$
where
$\displaystyle\mathcal{PB}_{p}(x)=\max\\{$
$\displaystyle\sup_{n}\sup_{e\in\mathcal{P}({\mathcal{M}}_{n})}\|(x-x_{n-1})\frac{e}{(\tau(e))^{1/p}}\|_{p},$
$\displaystyle\sup_{n}\sup_{e\in\mathcal{P}({\mathcal{M}}_{n})}\|\frac{e}{(\tau(e))^{1/p}}(x-x_{n-1})\|_{p}\\}.$
The two constants $\alpha_{p}$ and $\beta_{p}$ have the following properties
1. (i)
$\alpha_{p}=1$ for $2\leq p<\infty$;
2. (ii)
$\alpha_{p}\leq C^{1/p-1/2}$ for $0<p<2$;
3. (iii)
$\beta_{p}\leq cp$ for $2\leq p<\infty$;
4. (iv)
$\beta_{p}=1$ for $0<p<2.$
This result goes beyond Junge/Musat’s result in two aspects. First we extend
their result to all $0<p<\infty$. Second, the $b$’s in the definition of
$\mathcal{B}_{p}(\cdot)$ are reduced to projections $e$’s in
$\mathcal{PB}_{p}(\cdot)$, which corresponds exactly to the form (1.6) in the
classical case. Furthermore, the optimal constants $\beta_{p}$ in Theorem A
enable us to formulate John-Nirenberg inequality that corresponds to the form
(1.7). That is, let $x\in{\mathcal{BMO}}({\mathcal{M}})$, then for all natural
numbers $n\geq 1$, all $e\in\mathcal{P}({\mathcal{M}}_{n})$ and for all
$\lambda>0$, we have
$\frac{1}{\tau(e)}\tau(\mathds{1}_{(\lambda,\infty)}(|(x-x_{n-1})e|)+\mathds{1}_{(\lambda,\infty)}(|e(x-x_{n-1})|))\leq
4\exp(-\frac{c\lambda}{\|x\|_{{\mathcal{BMO}}}})$
with $c$ an absolute constant.
By the essentially same idea, we establish similar results for
${\mathcal{BMO}}^{c}({\mathcal{M}})$ and ${\mathcal{BMO}}^{r}({\mathcal{M}})$
separately, but only with $2\leq p<\infty$ (see Remark 3.9).
We now turn to the second objective of this paper: the atomic decomposition of
different noncommutative Hardy spaces. Let us recall the 2-atomic
decomposition obtained in [1]. An element $a\in L_{1}({\mathcal{M}})$ is said
to be a $(1,2)_{c}$-atom with respect to $({\mathcal{M}}_{n})_{n\geq 1}$, if
there exist $n\geq 1$ and $e\in\mathcal{P}({\mathcal{M}}_{n})$ such that
(i) ${\mathcal{E}}_{n}(a)=0$; (ii)$ae=a$; (iii)
$\|a\|_{2}\leq(\tau(e))^{-1/2}$.
The atomic Hardy space $\mathsf{h}_{1,\rm{at}}^{c}({\mathcal{M}})$ is defined
as the space of all $x\in L_{1}({\mathcal{M}})$, such that the following
$\|\cdot\|_{\mathsf{h}_{1,\rm{at}}^{c}}$ norm is finite,
$\|x\|_{\mathsf{h}_{1,\rm{at}}^{c}}=\|{\mathcal{E}}_{1}x\|_{1}+\inf\sum_{j}|\lambda_{j}|.$
Here the infimum is taken for possible decompositions
$x-{\mathcal{E}}_{1}x=\sum_{j}\lambda_{j}a_{j}$ with
$\lambda_{j}\in{\mathbb{C}}$, $a_{j}$ being $(1,2)_{c}$-atom. It is proved in
[1] that $x\in\mathsf{h}_{1}^{c}({\mathcal{M}})$ if and only if
$x\in\mathsf{h}_{1,\rm{at}}^{c}({\mathcal{M}})$ and
$\|x\|_{\mathsf{h}_{1}^{c}}\simeq\|x\|_{\mathsf{h}_{1,\rm{at}}^{c}}.$
Together with the equivalence
${\mathcal{H}}_{1}^{c}({\mathcal{M}})=\mathsf{h}_{1}^{c}({\mathcal{M}})+\mathsf{h}_{1}^{d}({\mathcal{M}})$,
the authors of [1] also obtained a $2$-atomic decomposition for
${\mathcal{H}}_{1}^{c}({\mathcal{M}})$.
Let us briefly recall the argument used in [1]. The dual space of
$\mathrm{h}_{1,\mathrm{at}}^{c}({\mathcal{M}})$ can be described as
$\Lambda^{c}({\mathcal{M}})=\\{x\in
L_{2}({\mathcal{M}}):\|x\|_{\Lambda^{c}}<\infty\\}$
with
$\|x\|_{\Lambda^{c}}=\max\\{\|{\mathcal{E}}_{1}x\|_{\infty},\quad\sup_{n\geq
1}\sup_{e\in\mathcal{P}_{n}}(\frac{1}{\tau(e)}\tau(e|x-x_{n}|^{2}))^{\frac{1}{2}}\\}.$
Actually, the supremum in the definition above can be taken for all $b\in
L_{1}({\mathcal{M}}_{n})$ since the extreme points of the unit ball of
$L_{1}({{\mathcal{M}}}_{n})$ are all multiples of projections. Therefore,
$\displaystyle\|x\|_{\Lambda^{c}}$ $\displaystyle=$
$\displaystyle\max\\{\|{\mathcal{E}}_{1}x\|_{\infty},\quad\sup_{n\geq
1}\sup_{b\in{\mathcal{M}}_{n}}(\frac{1}{\|b\|_{1}}\tau(b|x-x_{n}|^{2}))^{\frac{1}{2}}\\}$
$\displaystyle=$
$\displaystyle\max\\{\|{\mathcal{E}}_{1}x\|_{\infty},\quad\sup_{n\geq
1}\|{\mathcal{E}}_{n}|x-x_{n}|^{2}\|_{\infty}^{\frac{1}{2}}\\}$
$\displaystyle=$ $\displaystyle\|x\|_{\mathsf{bmo}^{c}}.$
Then the duality
$\mathsf{h}_{1}^{c}({\mathcal{M}})=\mathsf{bmo}^{c}({\mathcal{M}})$ yields
$\mathsf{h}_{1,\rm{at}}^{c}({\mathcal{M}})=\mathsf{h}_{1}^{c}({\mathcal{M}})$.
It is well known in the classical theory that $2$-atoms in the previous atomic
decomposition can be replaced by $q$-atoms for any $1<q\leq\infty$. Let us
recall these atoms in the commutative case. A function $a\in L_{1}(\Omega)$ is
said to be a $q$-atom if there exist $n\geq 1$ and $E\in\mathcal{F}_{n}$ such
that
(i) $\mathbb{E}_{n}a=0$; (ii) $\\{a\neq 0\\}\subset E$; (iii)
$\|a\|_{q}\leq\mathbb{P}(E)^{-1+\frac{1}{q}}.$
We refer to [22] for more information.
The main difficulty to obtain $q$-atomic decompositions in the noncommutative
case is that the key equivalence (1) no longer holds if one replaces the power
indices $2$ by $q^{\prime}\neq 2$, $1\leq q^{\prime}<\infty$. We overcome this
obstacle by Theorem A.
###### Theorem B.
For all $1<q\leq\infty$,
$\mathcal{H}_{1}({\mathcal{M}})=\mathsf{h}^{\mathrm{at}}_{1,q}({\mathcal{M}})$
with equivalent norms. Here $\mathsf{h}^{\mathrm{at}}_{1,q}({\mathcal{M}})$ is
the $q$-atomic Hardy spaces with its atoms defined as: $a\in
L_{1}({\mathcal{M}})$ is said to be a $(1,q)$-atom with respect to
$({\mathcal{M}}_{n})_{n\geq 1}$, if there exist $n\geq 1$ and a projection
$e\in\mathcal{P}({\mathcal{M}}_{n})$ such that
1. (i)
${\mathcal{E}}_{n}(a)=0$;
2. (ii)
$r(a)\leq e$ or $l(a)\leq e$;
3. (iii)
$\|a\|_{q}\leq(\tau(e))^{-\frac{1}{q^{\prime}}}$.
This is exactly the noncommutative analogue of the classical atomic
decomposition. Moreover, applying the conditional version of John-Nirenberg
inequality for ${\mathcal{BMO}}^{c}({\mathcal{M}})$ (resp.
${\mathcal{BMO}}^{r}({\mathcal{M}})$), we get a $q$-atomic decomposition for
$\mathsf{h}^{c}_{1}({\mathcal{M}})$ (resp.
$\mathsf{h}^{r}_{1}({\mathcal{M}})$) with $1<q\leq\infty$ (see Theorem 4.12),
hence recover the $2$-atomic decomposition of [1] mentioned above.
As in the classical case (see e.g. [3]), we also find some applications of our
results. Indeed, the John-Nirenberg inequality and atomic decomposition built
in this paper have been used in [5] to establish $H_{1}\rightarrow L_{1}$
boundedness of noncommutative paraproducts or martingale transforms with
noncommuting symbols or coefficients.
Our paper is organized as follows. Section 2 is on preliminaries and notation.
All the results on John-Nirenberg inequality will be presented in section 3.
Section 4 is devoted to the atomic decomposition of Hardy spaces. In section
5, we answer Junge/Musat’s question in [8] which implies that the John-
Nirenberg inequality in the classical sense does not hold any more in the
noncommutative setting.
In this article, the letter $c$ always denotes an absolute positive constant,
while $C$ an absolute constant bigger than 1. They may vary from lines to
lines.
## 2\. Preliminaries and notations
Throughout this paper, we will work on a von Neumann algebra ${\mathcal{M}}$
with a normal faithful normalized trace $\tau$. For all $0<p\leq\infty$, let
$L_{p}({\mathcal{M}},\tau)$ or simply $L_{p}({\mathcal{M}})$ be the associated
noncommutative $L_{p}$ spaces. For $x\in L_{p}({\mathcal{M}})$ we denote the
right and left supports of $x$ by $r(x)$ and $l(x)$ respectively. $r(x)$
(resp. $l(x)$) is also the least projection $e$ such that $xe=x$ (resp.
$ex=x$). If $x$ is selfadjoint, $r(x)=l(x)$, denoted by $s(x)$. We mainly
refer the reader to [19] for more information on noncommutative $L_{p}$
spaces.
Let us recall some basic notions on noncommutative martingales. Let
$({\mathcal{M}}_{n})_{n\geq 1}$ be an increasing sequence of von Neumann
subalgebras of ${\mathcal{M}}$ such that the union of the
${\mathcal{M}}_{n}$’s is $w^{*}$-dense in ${\mathcal{M}}$. Let
$\mathcal{E}_{n}$ be the conditional expectation of ${\mathcal{M}}$ with
respect to ${\mathcal{M}}_{n}$. A sequence $x=(x_{n})$ in
$L_{1}({\mathcal{M}})$ is called a noncommutative martingale with respect to
$({\mathcal{M}}_{n})_{n\geq 1}$ if $\mathcal{E}_{n}(x_{n+1})=x_{n}$ for every
$n\geq 1$. If in addition, all the $x_{n}$’s are in $L_{p}({\mathcal{M}})$ for
some $1\leq p\leq\infty$, $x$ is called an $L_{p}$-martingale. In this case we
set
$\|x\|_{p}=\sup_{n\geq 1}\|x_{n}\|_{p}.$
If $\|x\|_{p}<\infty$, $x$ is called a bounded $L_{p}$-martingale.
Let $x=(x_{n})$ be a noncommutative martingale with respect to
$({\mathcal{M}}_{n})_{n\geq 1}$. Define $dx_{n}=x_{n}-x_{n-1}$ for $n\geq 1$
with the convention that $x_{0}=0$ and $\mathcal{E}_{0}=\mathcal{E}_{1}$. The
sequence $dx=(dx_{n})_{n}$ is called the martingale difference sequence of
$x$. In the sequel, for any operator $x\in L_{1}({\mathcal{M}})$ we denote
$x_{n}=\mathcal{E}_{n}(x)$ for $n\geq 1$.
The sequence $({\mathcal{M}}_{n})_{n\geq 1}$ will be fixed throughout the
paper. All martingales will be with respect to $({\mathcal{M}}_{n})_{n\geq
1}$. Let $1\leq p<\infty$. Define $\mathcal{H}^{c}_{p}$ (resp.
$\mathcal{H}^{r}_{p}$) as the completion of all finite $L_{p}$-martingales
under the norm $\|x\|_{\mathcal{H}^{c}_{p}}=\|S_{c}(x)\|_{p}$ (resp.
$\|x\|_{\mathcal{H}^{r}_{p}}=\|S_{r}(x)\|_{p}$), where $S_{c}(x)$ and
$S_{r}(x)$ are defined as
$S_{c}(x)=\big{(}\sum_{k\geq 1}|dx_{k}|^{2}\big{)}^{1/2},\quad
S_{r}(x)=S_{c}(x^{*}).$
The noncommutative martingale Hardy spaces $\mathcal{H}_{p}({\mathcal{M}})$
are defined as follows: if $1\leq p<2$,
$\mathcal{H}_{p}({\mathcal{M}})=\mathcal{H}^{c}_{p}({\mathcal{M}})+\mathcal{H}^{r}_{p}({\mathcal{M}})$
equipped with the norm
$\|x\|_{\mathcal{H}_{p}}=\inf_{x=y+z}\\{\|y\|_{\mathcal{H}^{c}_{p}}+\|z\|_{\mathcal{H}^{r}_{p}}\\}.$
When $2\leq p<\infty$,
$\mathcal{H}_{p}({\mathcal{M}})=\mathcal{H}^{c}_{p}({\mathcal{M}})\cap\mathcal{H}^{r}_{p}({\mathcal{M}})$
equipped with the norm
$\|x\|_{\mathcal{H}_{p}}=\max\\{\|x\|_{\mathcal{H}^{c}_{p}},\|x\|_{\mathcal{H}^{r}_{p}}\\}.$
The space ${\mathcal{BMO}}^{c}$ is defined as
${\mathcal{BMO}}^{c}({\mathcal{M}})=\\{x\in
L_{1}({\mathcal{M}}):\|x\|_{{\mathcal{BMO}}^{c}}<\infty\\}$
where
$\|x\|_{{\mathcal{BMO}}^{c}}=\sup_{n\geq
1}\|\mathcal{E}_{n}|x-x_{n-1}|^{2}\|^{1/2}_{\infty},$
and
${\mathcal{BMO}}^{r}({\mathcal{M}})=\\{x:x^{*}\in{\mathcal{BMO}}^{c}({\mathcal{M}})\\}.$
Define
${\mathcal{BMO}}({\mathcal{M}})={\mathcal{BMO}}^{c}({\mathcal{M}})\cap{\mathcal{BMO}}^{r}({\mathcal{M}})$
equipped with the norm
$\|x\|_{{\mathcal{BMO}}}=\max\\{\|x\|_{{\mathcal{BMO}}^{c}},\|x\|_{{\mathcal{BMO}}^{r}}\\}.$
Pisier and Xu [18] proved the two fundamental results:
$\mathcal{H}_{p}({\mathcal{M}})=L_{p}({\mathcal{M}})$ and
$(\mathcal{H}_{1}({\mathcal{M}}))^{*}={\mathcal{BMO}}({\mathcal{M}})$. Their
work triggered a rapid development of the noncommutative martingale theory.
We will also work on the conditional version of Hardy and BMO spaces developed
in [9]. Let $x=(x_{n})_{n\geq 1}$ be a finite martingale in
$L_{2}({\mathcal{M}})$. We set
$s_{c}(x)=\big{(}\sum_{k\geq
1}\mathcal{E}_{k-1}|dx_{k}|^{2}\big{)}^{1/2}\quad\mbox{and}\quad
s_{r}(x)=s_{c}(x^{*}).$
Let $0<p<\infty$. Define $\mathsf{h}^{c}_{p}({\mathcal{M}})$ (resp.
$\mathsf{h}^{r}_{p}({\mathcal{M}})$) as the completion of all finite
$L_{\infty}$-martingales under the (quasi-)norm
$\|x\|_{\mathsf{h}^{c}_{p}}=\|s_{c}(x)\|_{p}$ (resp.
$\|x\|_{\mathsf{h}^{r}_{p}}=\|s_{r}(x)\|_{p}$). Define
$\mathsf{h}^{d}_{p}({\mathcal{M}})$ as the subspace of
$\ell_{p}(L_{p}({\mathcal{M}}))$ consisting of all martingale difference
sequences, where $\ell_{p}(L_{p}({\mathcal{M}}))$ is the space of all
sequences $a=(a_{n})_{n\geq 1}$ in $L_{p}({\mathcal{M}})$ such that
$\|a\|_{\ell_{p}(L_{p}({\mathcal{M}}))}=\big{(}\sum_{n\geq
1}\|a_{n}\|^{p}_{p}\big{)}^{1/p}<\infty$
with the usual modification for $p=\infty$. The noncommutative conditional
martingale Hardy spaces are defined as follows: if $0<p<2$,
$\mathsf{h}_{p}({\mathcal{M}})=\mathsf{h}^{c}_{p}({\mathcal{M}})+\mathsf{h}^{r}_{p}({\mathcal{M}})+\mathsf{h}^{d}_{p}({\mathcal{M}})$
equipped with the (quasi-)norm
$\|x\|_{\mathsf{h}_{p}}=\inf_{x=y+z+w}\\{\|y\|_{\mathsf{h}^{c}_{p}}+\|z\|_{\mathsf{h}^{r}_{p}}+\|w\|_{\mathsf{h}^{d}_{p}}\\}.$
When $2\leq p<\infty$,
$\mathsf{h}_{p}({\mathcal{M}})=\mathsf{h}^{c}_{p}({\mathcal{M}})\cap\mathsf{h}^{r}_{p}({\mathcal{M}})\cap\mathsf{h}^{d}_{p}({\mathcal{M}})$
equipped with the norm
$\|x\|_{\mathsf{h}_{p}}=\max\\{\|x\|_{\mathsf{h}^{c}_{p}},\|x\|_{\mathsf{h}^{r}_{p}},\|x\|_{\mathsf{h}^{d}_{p}}\\}.$
The space $\mathsf{bmo}^{c}$ is defined as
$\mathsf{bmo}^{c}({\mathcal{M}})=\\{x\in
L_{1}({\mathcal{M}}):\|x\|_{\mathsf{bmo}^{c}}<\infty\\}$
where
$\|x\|_{\mathsf{bmo}^{c}}=\max\left\\{\|\mathcal{E}_{1}(x)\|_{\infty},\quad\sup_{n\geq
1}\|\mathcal{E}_{n}|x-x_{n}|^{2}\|^{1/2}_{\infty}\right\\}.$
Let
$\mathsf{bmo}^{r}({\mathcal{M}})=\\{x:x^{*}\in\mathsf{bmo}^{c}({\mathcal{M}})\\}.$
Let $\mathsf{bmo}^{d}({\mathcal{M}})$ be the subspace of
$\ell_{\infty}(L_{\infty}({\mathcal{M}}))$ consisting of all martingale
difference sequences. Note that
$\mathsf{bmo}^{d}({\mathcal{M}})=\mathsf{h}^{d}_{\infty}({\mathcal{M}})$.
Define
$\mathsf{bmo}({\mathcal{M}})=\mathsf{bmo}^{c}({\mathcal{M}})\cap\mathsf{bmo}^{r}({\mathcal{M}})\cap\mathsf{bmo}^{d}({\mathcal{M}})$
equipped with the norm
$\|x\|_{\mathsf{bmo}}=\max\\{\|x\|_{\mathsf{bmo}^{c}},\|x\|_{\mathsf{bmo}^{r}},\|x\|_{\mathsf{bmo}^{d}}\\}.$
We refer to [9], [12], [20], [21], [7], [17] for more information on these
spaces.
## 3\. John-Nirenberg inequality
### 3.1. A crude version
###### Definition 3.1.
For $0<p<\infty$, we define
1. (i)
$\mathsf{bmo}^{c}_{p}({\mathcal{M}})=\big{\\{}x\in
L_{1}({\mathcal{M}}):\|x\|_{\mathsf{bmo}^{c}_{p}}<\infty\big{\\}}$
with
$\|x\|_{\mathsf{bmo}^{c}_{p}}=\max\big{\\{}\|\mathcal{E}_{1}(x)\|_{\infty},\quad\sup_{n}\sup_{a\in{\mathcal{M}}_{n},\|a\|_{p}\leq
1,}\|(x-x_{n})a\|_{\mathsf{h}^{c}_{p}}\big{\\}};$
2. (ii)
$\mathsf{bmo}^{r}_{p}({\mathcal{M}})=\\{x:x^{*}\in\mathsf{bmo}^{c}_{p}({\mathcal{M}})\\};$
3. (iii)
$\mathsf{bmo}_{p}({\mathcal{M}})=\mathsf{bmo}^{c}_{p}({\mathcal{M}})\cap\mathsf{bmo}^{r}_{p}({\mathcal{M}})\cap\mathsf{bmo}^{d}({\mathcal{M}})$
equipped with the (quasi-)norm
$\|x\|_{\mathsf{bmo}_{p}}=\max\\{\|x\|_{\mathsf{bmo}^{c}_{p}},\|x\|_{\mathsf{bmo}_{p}^{r}},\|x\|_{\mathsf{bmo}^{d}}\\}.$
###### Remark 3.2.
When $p=2$, these are exactly the spaces $\mathsf{bmo}^{c}({\mathcal{M}})$,
$\mathsf{bmo}^{r}({\mathcal{M}})$ and $\mathsf{bmo}({\mathcal{M}})$.
Below is our first version of the column (resp. row) John-Nirenberg
inequality.
###### Theorem 3.3.
For all $0<p<\infty$, there exist two constants $\alpha_{p}$ and $\beta_{p}$
such that
$\alpha^{-1}_{p}\|x\|_{\mathsf{bmo}^{c}}\leq\|x\|_{\mathsf{bmo}^{c}_{p}}\leq\beta_{p}\|x\|_{\mathsf{bmo}^{c}},$
with $\alpha_{p}$ and $\beta_{p}$ satisfying
1. (i)
$\alpha_{p}=1$ for $2\leq p<\infty$;
2. (ii)
$\alpha_{p}\leq C^{1/p-1/2}$ for $0<p<2$;
3. (iii)
$\beta_{p}\leq cp$ for $2\leq p<\infty$;
4. (iv)
$\beta_{p}=1$ for $0<p<2.$
The similar inequalities hold for $\|\cdot\|_{\mathsf{bmo}^{r}_{p}}$ and
$\|\cdot\|_{\mathsf{bmo}^{r}}$.
###### Proof.
We only need to prove the column case, since the row case can be done by
replacing $x$ with $x^{*}$. First consider the case $2<p<\infty$. We will show
the following inequalities:
$\|x\|_{\mathsf{bmo}^{c}_{2}}\leq\|x\|_{\mathsf{bmo}^{c}_{p}}\leq
cp\|x\|_{\mathsf{bmo}^{c}_{2}}.$
The left inequality is obtained directly by Hölder’s inequality. In fact,
taking $a\in{\mathcal{M}}_{n}$ with $\|a\|_{2}\leq 1$, there exists a
factorization $a=a_{0}a_{1}$ such that $\|a_{0}\|_{p}=\|a\|^{2/p}_{2}\leq 1$
and $\|a_{1}\|_{{2p}/{(p-2)}}=\|a\|^{(p-2)/p}_{2}\leq 1$, so
$\displaystyle\|(x-x_{n})a\|^{2}_{\mathsf{h}^{c}_{2}}$
$\displaystyle=\tau(a^{*}_{1}a^{*}_{0}s^{2}_{c}(x-x_{n})a_{0}a_{1})$
$\displaystyle\leq\|a^{*}_{1}\|_{\frac{2p}{p-2}}\|a_{0}^{*}s^{2}_{c}(x-x_{n})a_{0}\|_{\frac{p}{2}}\|a_{1}\|_{\frac{2p}{p-2}}$
$\displaystyle\leq\|(x-x_{n})a_{0}\|^{2}_{\mathsf{h}^{c}_{p}}.$
We invoke complex interpolation to prove the right inequality. Fix $n$, let
$b\in{L_{p}({\mathcal{M}}_{n})}$ with $\|b\|_{p}\leq 1$ and
$S=\\{z\in\mathbb{C}:0\leq Rez\leq 1\\}$. Then by interpolation between
$L_{p}$ spaces $L_{p}=(L_{2},L_{\infty})_{\theta}$, there exists an operator-
valued function $B$ which is continuous on $S$ and analytic in the interior of
$S$ such that $B(\theta)=b$ and
$\sup_{t\in\mathbb{R}}\|B(it)\|_{2}\leq
1,\qquad\sup_{t\in\mathbb{R}}\|B(1+it)\|_{{\infty}}\leq 1.$
Define
$f(z)=(x-x_{n})B(z).$
Then on the one hand, by the definition of
$\mathsf{bmo}^{c}_{2}({\mathcal{M}})$, we have
$\|f(it)\|_{\mathsf{h}^{c}_{2}}\leq\|x\|_{\mathsf{bmo}^{c}_{2}}.$
On the other hand, by a simple calculation, we have
$\|f(1+it)\|_{\mathsf{bmo}^{c}_{2}}\leq\|x-x_{n}\|_{\mathsf{bmo}^{c}_{2}}\|B(1+it)\|_{\infty}\leq\|x\|_{\mathsf{bmo}^{c}_{2}}.$
Therefore, by interpolation,
$\|f(\theta)\|_{(\mathsf{h}^{c}_{2},\mathsf{bmo}^{c})_{\theta}}\leq\|x\|_{\sf{bmo}^{c}_{2}}=\|x\|_{\sf{bmo}^{c}}.$
However by [1],
$(\mathsf{h}^{c}_{2},\mathsf{bmo}^{c})_{\theta}\subset\mathsf{h}^{c}_{p}$
with relevant constant majorized by $cp$. We then deduce that
(3.1) $\displaystyle\|f(\theta)\|_{\mathsf{h}^{c}_{p}}\leq
cp\|x\|_{\mathsf{bmo}^{c}},$
hence the desired inequality holds.
For the case $0<p<2$. We show the following inequalities:
$\|x\|_{\mathsf{bmo}^{c}_{p}}\leq\|x\|_{\mathsf{bmo}^{c}_{2}}\leq
C^{1/p-1/2}\|x\|_{\mathsf{bmo}^{c}_{p}}.$
Again, the left inequality is obtained by Hölder’s inequality. It remains to
prove the right one. We choose $2<p_{1}<\infty$ and $0<\theta<1$ such that
${1}/{2}={(1-\theta)}/{p}+{\theta}/{p_{1}}$. Fix $n$, by the definition of
$\mathsf{bmo}^{c}_{p}({\mathcal{M}})$, we can view $x-x_{n}$ as a bounded
operator from $L_{p}({\mathcal{M}}_{n})$ to
$\mathsf{h}^{c}_{p}({\mathcal{M}})$. Then we have the following two
inequalities:
$\|x-x_{n}\|_{L_{p}({\mathcal{M}}_{n})\rightarrow\mathsf{h}^{c}_{p}}\leq\|x\|_{\mathsf{bmo}^{c}_{p}},\qquad\|x-x_{n}\|_{L_{p_{1}}({\mathcal{M}}_{n})\rightarrow\mathsf{h}^{c}_{p_{1}}}\leq\|x\|_{\mathsf{bmo}^{c}_{p_{1}}}.$
Then by interpolation, we get
$\|x-x_{n}\|_{L_{2}({\mathcal{M}}_{n})\rightarrow(\mathsf{h}^{c}_{p},\mathsf{h}^{c}_{p_{1}})_{\theta}}\leq\|x\|^{1-\theta}_{\mathsf{bmo}^{c}_{p}}\|x\|^{\theta}_{\mathsf{bmo}^{c}_{p_{1}}}.$
Now by the trivial contractive inclusion
$(\mathsf{h}^{c}_{p},\mathsf{h}^{c}_{p_{1}})_{\theta}\subset\mathsf{h}^{c}_{2}$,
and the right inequality in the case $2<p_{1}<\infty$, we get
$\|x-x_{n}\|_{L_{2}({\mathcal{M}}_{n})\rightarrow\mathsf{h}^{c}_{2}}\leq
cp_{1}\|x\|^{1-\theta}_{\mathsf{bmo}^{c}_{p}}\|x\|^{\theta}_{\mathsf{bmo}^{c}_{2}}.$
Therefore,
$\|x\|_{\mathsf{bmo}^{c}_{2}}\leq(cp_{1})^{\theta}\|x\|^{1-\theta}_{\mathsf{bmo}^{c}_{p}}\|x\|^{\theta}_{\mathsf{bmo}^{c}_{2}},$
hence
$\|x\|_{\mathsf{bmo}^{c}_{2}}\leq(cp_{1})^{\frac{\theta}{1-\theta}}\|x\|_{\mathsf{bmo}^{c}_{p}}.$
Noting that ${\theta}/({1-\theta})=({1/p-1/2})/({1/2-1/p_{1}})$, we get the
desired estimate by taking $C=(cp_{1})^{1/(1/2-1/p_{1})}$. ∎
###### Remark 3.4.
The constant in (3.1) is optimal. This can be seen as follows. By Lemma 4.3 in
[1], $\mathsf{h}^{c}_{p^{\prime}}({\mathcal{M}})$ embeds into
$(\mathsf{h}^{c}_{2}({\mathcal{M}}),\mathsf{h}^{c}_{1}({\mathcal{M}}))_{\theta}$
with constant independent of $p^{\prime}$. So
$((\mathsf{h}^{c}_{2}({\mathcal{M}}))^{*},(\mathsf{h}^{c}_{1}({\mathcal{M}}))^{*})_{\theta}$
embeds into $(\mathsf{h}^{c}_{p^{\prime}}({\mathcal{M}}))^{*}$ with constant
independent of $p$ by duality. Finally, by the optimal embedding
$(\mathsf{h}^{c}_{p^{\prime}}({\mathcal{M}}))^{*}\subset\mathsf{h}^{c}_{p}({\mathcal{M}})$
with constant $cp$ in [9] and
$\mathsf{bmo}^{c}({\mathcal{M}})\subset(\mathsf{h}^{c}_{1}({\mathcal{M}}))^{*}$
in [17],
$(\mathsf{h}^{c}_{2}({\mathcal{M}}),\mathsf{bmo}^{c}({\mathcal{M}}))_{\theta}$
embeds into $\mathsf{h}^{c}_{p}({\mathcal{M}})$ with optimal constant $cp$.
It is natural to ask whether there is a result similar to Theorem 3.3 for
${\mathcal{BMO}}^{c}$ by replacing $\mathsf{h}^{c}_{p}$ and $x-x_{n}$ in the
definition of $\mathsf{bmo}^{c}_{p}$ by $\mathcal{H}^{c}_{p}$ and $x-x_{n-1}$
respectively. Using the identity
${\mathcal{BMO}}^{c}({\mathcal{M}})\simeq\mathsf{bmo}^{c}({\mathcal{M}})\cap\mathsf{bmo}^{d}({\mathcal{M}})$
proved in [17], we are reduced to deal with the diagonal space
$\mathsf{bmo}^{d}({\mathcal{M}})$. Surprisingly, the result is true only for
$2\leq p<\infty$ (see Remark 3.9).
###### Definition 3.5.
For $1\leq p<\infty$, we define
1. (i)
${\mathcal{BMO}}^{c}_{p}({\mathcal{M}})=\left\\{x\in
L_{1}({\mathcal{M}}):\|x\|_{{\mathcal{BMO}}^{c}_{p}}<\infty\right\\}$
with
$\|x\|_{{\mathcal{BMO}}^{c}_{p}}=\sup_{n}\sup_{a\in{\mathcal{M}}_{n},\|a\|_{p}\leq
1}\|(x-x_{n-1})a\|_{\mathcal{H}^{c}_{p}};$
2. (ii)
${\mathcal{BMO}}^{r}_{p}({\mathcal{M}})=\\{x:x^{*}\in{\mathcal{BMO}}^{c}_{p}({\mathcal{M}})\\};$
3. (iii)
${\mathcal{BMO}}_{p}({\mathcal{M}})={\mathcal{BMO}}^{c}_{p}({\mathcal{M}})\cap{\mathcal{BMO}}^{r}_{p}({\mathcal{M}})$
equipped with the norm
$\|x\|_{{\mathcal{BMO}}_{p}}=\max\\{\|x\|_{{\mathcal{BMO}}^{c}_{p}},\|x\|_{{\mathcal{BMO}}_{p}^{r}}\\}.$
###### Remark 3.6.
For $p=2$, we recover the spaces ${\mathcal{BMO}}^{c}({\mathcal{M}})$,
${\mathcal{BMO}}^{r}({\mathcal{M}})$ and ${\mathcal{BMO}}({\mathcal{M}})$.
The following lemma will alow us to handle with the diagonal space
$\mathsf{bmo}^{d}({\mathcal{M}})$.
###### Lemma 3.7.
For $2\leq p<\infty,$ we have
$cp^{-1}\|b\|_{{\infty}}\leq\sup_{a\in{\mathcal{M}},\|a\|_{p}\leq
1}\|ba\|_{{\mathcal{H}}^{c}_{p}}\leq cp^{\frac{1}{2}}\|b\|_{{\infty}}.$
###### Proof.
Note that $\|\cdot\|_{{\mathcal{H}}^{c}_{p}}\leq cp^{1/2}\|\cdot\|_{p}$ (see
[20], Remark 5.4 as a reference for the constant we use here), we have
$\sup_{a\in{\mathcal{M}},\|a\|_{p}\leq 1}\|ba\|_{{\mathcal{H}}^{c}_{p}}\leq
cp^{\frac{1}{2}}\sup_{a\in{\mathcal{M}},\|a\|_{p}\leq
1}\|ba\|_{p}=cp^{\frac{1}{2}}\|b\|_{\infty}.$
For the first inequality, without loss of generality assume
$\|b\|_{\infty}=1.$ Note that for selfadjoint $x\in$
${{\mathcal{M}}},\|x\|_{p}\leq cp\|x\|_{{\mathcal{H}}_{p}^{c}}$ (see [20],
Remark 5.4). Then
$\displaystyle\|b^{*}\|_{{\infty}}$
$\displaystyle=\sup_{y\in{\mathcal{M}},\|y\|_{2p}\leq 1}\|yb^{*}\|_{{2p}}$
$\displaystyle=\sup_{y\in{\mathcal{M}},\|y\|_{2p}\leq
1}\|b|y|^{2}b^{*}\|_{p}^{\frac{1}{2}}$ $\displaystyle\leq
cp^{\frac{1}{2}}\sup_{y\in{\mathcal{M}},\|y\|_{2p}\leq
1}\|b|y|^{2}b^{*}\|_{{\mathcal{H}}^{c}_{p}}^{\frac{1}{2}}$ $\displaystyle\leq
cp^{\frac{1}{2}}\sup_{a\in{\mathcal{M}},\|a\|_{p}\leq
1}\|ba\|_{{\mathcal{H}}_{p}^{c}}^{\frac{1}{2}}.$
And then $cp^{-1}\|b\|_{{\infty}}\leq\sup_{a\in{\mathcal{M}},\|a\|_{p}\leq
1}\|ba\|_{{\mathcal{H}}^{c}_{p}}.$ ∎
###### Theorem 3.8.
For all $2\leq p<\infty$, we have
${\mathcal{BMO}}^{c}_{p}({\mathcal{M}})={\mathcal{BMO}}^{c}({\mathcal{M}})$
with equivalent norms. More precisely,
$cp^{-1}\|x\|_{{{\mathcal{BMO}}}^{c}}\leq\|x\|_{{{\mathcal{BMO}}}_{p}^{c}}\leq
cp\|x\|_{{{\mathcal{BMO}}}^{c}}.$
Similarly,
${\mathcal{BMO}}^{r}_{p}({\mathcal{M}})={\mathcal{BMO}}^{r}({\mathcal{M}})$
with equivalent norms.
Using the previous lemma and the identity
${\mathcal{BMO}}^{c}({\mathcal{M}})\simeq\mathsf{bmo}^{c}({\mathcal{M}})\cap\mathsf{bmo}^{d}({\mathcal{M}})$,
we can easily deduce Theorem 3.8 from Theorem 3.3. We will however present a
direct proof.
###### Proof.
We only prove the inequalities for the column case, the row case can be dealt
with similarly. By the previous lemma and Hölder’s inequality, we have
$\displaystyle\|\mathcal{E}_{n}\sum_{k=n}^{\infty}|dx_{k}|^{2}\|_{{\infty}}$
$\displaystyle\leq\sup_{b\in{\mathcal{M}}^{+}_{n},\|b\|_{1}\leq
1}\tau\left(\sum_{k=n+1}^{\infty}|dx_{k}|^{2}b\right)+\|x_{n}-x_{n-1}\|^{2}_{{\infty}}$
$\displaystyle\leq\sup_{b\in{\mathcal{M}}^{+}_{n},\|b\|_{1}\leq
1}\tau\left(\sum_{k=n+1}^{\infty}|(dx_{k})b^{\frac{1}{p}}|^{2}b^{\frac{p-2}{p}}\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad+cp^{2}\sup_{a\in{\mathcal{M}}_{n},\|a\|_{p}\leq
1}\|(x_{n}-x_{n-1})a\|^{2}_{{\mathcal{H}}^{c}_{p}}$
$\displaystyle\leq\sup_{b\in{\mathcal{M}}^{+}_{n},\|b\|_{1}\leq
1}\left\|\sum_{k=n+1}^{\infty}|(dx_{k})b^{\frac{1}{p}}|^{2}\right\|_{{\frac{p}{2}}}\left\|b^{\frac{p-2}{p}}\right\|_{{(\frac{p}{2})^{\prime}}}$
$\displaystyle\quad\quad\quad\quad\quad\quad+cp^{2}\sup_{a\in{\mathcal{M}}_{n},\|a\|_{p}\leq
1}\|(x_{n}-x_{n-1})a\|^{2}_{{\mathcal{H}}_{p}^{c}}$
$\displaystyle\leq\sup_{b\in{\mathcal{M}}^{+}_{n},\|b\|_{1}\leq
1}\left\|(x-x_{n})b^{\frac{1}{p}}\right\|^{2}_{{\mathcal{H}}^{c}_{p}}$
$\displaystyle\quad\quad\quad\quad\quad\quad+cp^{2}\sup_{a\in{\mathcal{M}}_{n},\|a\|_{p}\leq
1}\|(x_{n}-x_{n-1})a\|^{2}_{{\mathcal{H}}^{c}_{p}}.$
Then by
$\|\mathcal{E}_{n}x\|_{\mathcal{H}^{c}_{p}}\leq\|x\|_{\mathcal{H}^{c}_{p}}$,
$\|x\|_{{{\mathcal{BMO}}}_{2}^{c}}\leq
cp\sup_{a\in{\mathcal{M}}_{n},\|a\|_{p}\leq
1}\left\|(x-x_{n-1})a\right\|_{{\mathcal{H}}^{c}_{p}}=cp\|x\|_{{{\mathcal{BMO}}}^{c}_{p}}.$
Conversely, by the previous lemma,
$\displaystyle\|x\|_{{{\mathcal{BMO}}}_{p}^{c}}$
$\displaystyle\leq\sup_{n}\sup_{a\in{\mathcal{M}}_{n},\|a\|_{p}\leq
1}\left\|(x-x_{n})a\right\|_{{\mathcal{H}}^{c}_{p}}$
$\displaystyle\quad\quad\quad\quad\quad\quad+\sup_{n}\sup_{a\in{\mathcal{M}}_{n},\|a\|_{p}\leq
1}\|(x_{n}-x_{n-1})a\|_{{\mathcal{H}}^{c}_{p}}$
$\displaystyle\leq\sup_{n}\sup_{a\in{\mathcal{M}}_{n},\|a\|_{p}\leq
1}\left\|(x-x_{n})a\right\|_{{\mathcal{H}}^{c}_{p}}+cp^{\frac{1}{2}}\sup_{n}\|x_{n}-x_{n-1}\|_{{\infty}}$
(3.2) $\displaystyle\leq\sup_{n}\sup_{a\in{\mathcal{M}}_{n},\|a\|_{p}\leq
1}\left\|(dx_{k}a)_{k=n+1}^{\infty}\right\|_{L_{p}(\ell^{c}_{2})}+cp^{\frac{1}{2}}\|x\|_{{{\mathcal{BMO}}}^{c}_{2}}.$
Note that, by the Hahn-Banach theorem and the duality between
${\mathcal{H}}^{c}_{1}({{\mathcal{M}}})$ and
${\mathcal{BMO}}^{c}({{\mathcal{M}}})$, there exists a sequence
$(b_{n})_{n=1}^{\infty}\in L_{\infty}({{\mathcal{M}}};\ell_{2}^{c})$ such that
$\left\|(b_{n})_{n=1}^{\infty}\right\|_{L_{\infty}(\ell^{c}_{2})}=\|x\|_{{{\mathcal{BMO}}}^{c}},\quad
dx_{k}=\mathcal{E}_{k}b_{k}-\mathcal{E}_{k-1}b_{k}.$
Thus by the noncommutative Stein inequality (see [20] for the constant used
below) and Hölder’s inequality,
$\displaystyle\sup_{a\in{\mathcal{M}}_{n},\|a\|_{p}\leq
1}\left\|(dx_{k}a)_{k=n+1}^{\infty}\right\|_{L_{p}(\ell^{c}_{2})}$
$\displaystyle\quad\quad\quad\leq\sup_{a\in{\mathcal{M}}_{n},\|a\|_{p}\leq
1}\left\|(\mathcal{E}_{k}(b_{k}a))_{k=n+1}^{\infty}\right\|_{L_{p}(\ell^{c}_{2})}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\sup_{a\in{\mathcal{M}}_{n},\|a\|_{p}\leq
1}\left\|(\mathcal{E}_{k}b_{k+1}a)_{k=n}^{\infty}\right\|_{L_{p}(\ell^{c}_{2})}$
$\displaystyle\quad\quad\quad\leq cp\sup_{a\in{\mathcal{M}}_{n},\|a\|_{p}\leq
1}\left\|(b_{k}a)_{k=n+1}^{\infty}\right\|_{L_{p}(\ell^{c}_{2})}$
$\displaystyle\quad\quad\quad\leq
cp\left\|\sum_{k=1}^{\infty}|b_{k}|^{2}\right\|_{\infty}^{\frac{1}{2}}=cp\|x\|_{{{\mathcal{BMO}}}_{2}^{c}}.$
Combining this with (3.2) we finish the proof. ∎
###### Remark 3.9.
It is a bit surprising that Theorem 3.8 is actually wrong for any $p<2$.
Indeed, choose a filtration ${\mathcal{M}}_{1}$, ${\mathcal{M}}_{2}$,
${\mathcal{M}}_{3}$,…,${\mathcal{M}}_{n-1}$ and $y\in{\mathcal{M}}_{n-1}$ such
that $\|y\|_{p}=1$ and $\|y\|_{\mathcal{H}_{p}^{c}}=c_{n}>>1$. Let
${\mathcal{M}}_{n}=L_{\infty}(\Omega,{\mathcal{M}}_{n-1})$ with
$\Omega=\\{0,1\\}$ with $\mu\\{1\\}=\mu\\{0\\}=1/2$. We certainly can view
${\mathcal{M}}_{k}$, $k<n$ as the space of constant functions on $\Omega$, so
${\mathcal{M}}_{k}\subset{\mathcal{M}}_{n}$. Let $x=1$ on $\\{0\\}$ and $x=-1$
on $\\{1\\}$ then $x_{n-1}=0$. Let $a=y$ on $\\{0\\}$ and $a=-y$ on $\\{1\\}$.
Then $(x-x_{n-1})a=y$ whose $\mathcal{H}_{p}^{c}$ norm equals $c_{n}$ and
$\|a\|_{p}=1$, so $\|x\|_{{\mathcal{BMO}}^{c}_{p}}\geq c_{n}$. But
$\|x\|_{{\mathcal{BMO}}^{c}_{2}}=1$.
In the rest of this subsection, we turn to Junge/Musat’s type of John-
Nirenberg inequality. In [8], Junge and Musat established the inequality for
$2<p<\infty$ in the state case. Later the second author of the present paper
gave a simple proof for all $1\leq p<\infty$ in the tracial setting (see
[13]). The idea of the proof of Theorem 3.3 can be applied to obtain this
inequality for all $0<p<\infty$ (see Corollary 3.13). We start again with
$\mathsf{bmo}({\mathcal{M}})$.
###### Theorem 3.10.
For all $0<p<\infty$, we have
$\alpha^{-1}_{p}\|x\|_{\mathsf{bmo}}\leq\mathsf{b}_{p}(x)\leq\beta_{p}\|x\|_{\mathsf{bmo}}$
where
$\displaystyle\mathsf{b}_{p}(x)=\max\\{$
$\displaystyle\sup_{n}\|(dx_{n})_{n}\|_{\infty},\
\sup_{n}\sup_{b\in{\mathcal{M}}_{n},\|b\|_{p}\leq 1}\|(x-x_{n})b\|_{p},$
$\displaystyle\sup_{n}\sup_{b\in{\mathcal{M}}_{n},\|b\|_{p}\leq
1}\|b(x-x_{n})\|_{p}\\}.$
The constant $\alpha_{p}$ and $\beta_{p}$ have the same orders as those in
Theorem 3.3.
###### Proof.
We first treat the case $2\leq p<\infty$. For $p=2$, it is trivial. So we can
assume $2<p<\infty$. The inequality
$\|x\|_{\mathsf{bmo}}\leq\mathsf{b}_{p}(x)$
follows from Hölder’s inequality. We will prove the reverse inequality by
interpolation. By a simple calculation, we have the following estimates
$\|(x-x_{n})b\|_{\mathsf{bmo}^{c}}\leq\|x\|_{\mathsf{bmo}^{c}}\|b\|_{{\infty}},$
$\|(x-x_{n})b\|_{\mathsf{bmo}^{r}}\leq\|x\|_{\mathsf{bmo}^{r}}\|b\|_{{\infty}},$
$\|(x-x_{n})b\|_{\mathsf{bmo}^{d}}\leq\|x\|_{\mathsf{bmo}^{d}}\|b\|_{{\infty}}.$
Then it follows that
$\|(x-x_{n})b\|_{\mathsf{bmo}}\leq\|x\|_{\mathsf{bmo}}\|b\|_{{\infty}}.$
On the other hand, it is clear that
$\|(x-x_{n})b\|_{2}=\|(x-x_{n})b\|_{\mathsf{h}^{c}_{2}}\leq\|x\|_{\mathsf{bmo}}\|b\|_{2}.$
Then by the interpolation result of [1], we have
(3.3) $\displaystyle\|(x-x_{n})b\|_{p}$ $\displaystyle\leq
cp\|(x-x_{n})b\|_{(L_{2},\mathsf{bmo})_{\theta}}$ $\displaystyle\leq
cp\|x\|_{\mathsf{bmo}}\|b\|_{p}.$
In the same way, we obtain
$\|b(x-x_{n})\|_{p}\leq cp\|x\|_{\mathsf{bmo}}\|b\|_{p}.$
Thus we prove the assertion.
Now we turn to the case $0<p<2$, by Hölder’s inequality, we obtain the trivial
part
$\mathsf{b}_{p}(x)\leq\mathsf{b}_{2}(x)=\|x\|_{\mathsf{bmo}}.$
Let us prove the inverse one, let $2<p_{1}<\infty$ and $\theta$ be such that
$\frac{1}{2}=\frac{1-\theta}{p}+\frac{\theta}{p_{1}}.$
We view $x-x_{n}$ and $(x-x_{n})^{*}$ as two operators. By interpolation,
$\displaystyle\|(x-x_{n})\|_{L_{2}({\mathcal{M}}_{n})\rightarrow
L_{2}({\mathcal{M}})}$
$\displaystyle\leq\|(x-x_{n})\|^{1-\theta}_{L_{p}({\mathcal{M}}_{n})\rightarrow
L_{p}({\mathcal{M}})}\|(x-x_{n})\|^{\theta}_{L_{p_{1}}({\mathcal{M}}_{n})\rightarrow
L_{p_{1}}({\mathcal{M}})}$
and similarly for $(x-x_{n})^{*}$. By the estimate for $p_{1}>2$, we have
$\mathsf{b}_{2}(x)\leq(cp_{1})^{\theta}\mathsf{b}^{1-\theta}_{p}(x)\mathsf{b}^{\theta}_{2}(x).$
Therefore, we obtain
$\|x\|_{\mathsf{bmo}}\leq(cp_{1})^{\frac{\theta}{1-\theta}}\mathsf{b}_{p}(x)=C^{{1/p}-{1/2}}\mathsf{b}_{p}(x),$
with $C=(cp_{1})^{1/(1/2-1/p_{1})}$. ∎
###### Remark 3.11.
The constant in (3.3) is optimal. This can be seen as follows. By Lemma 4.3 in
[1], $\mathsf{h}^{c}_{p^{\prime}}({\mathcal{M}})$ embeds into
$(\mathsf{h}^{c}_{2}({\mathcal{M}}),\mathsf{h}^{c}_{1}({\mathcal{M}}))_{\theta}$
with constant independent of $p^{\prime}$. So
$\mathsf{h}_{p^{\prime}}({\mathcal{M}})$ embeds into
$(\mathsf{h}_{2}({\mathcal{M}}),\mathsf{h}_{1}({\mathcal{M}}))_{\theta}$ with
constant independent of $p^{\prime}$. Now by Theorem 4.1 in [21],
$L_{p^{\prime}}({\mathcal{M}})$ embeds into
$\mathsf{h}_{p^{\prime}}({\mathcal{M}})$, hence into
$(\mathsf{h}_{2}({\mathcal{M}}),\sf h_{1}({\mathcal{M}}))_{\theta}$ with
optimal constant $c/(p^{\prime}-1)$. Then by duality,
$((\mathsf{h}_{2}({\mathcal{M}}))^{*},(\mathsf{h}_{1}({\mathcal{M}}))^{*})_{\theta}$
embeds into $(L_{p^{\prime}}({\mathcal{M}}))^{*}=L_{p}({\mathcal{M}})$ with
best constant $cp$. At last, by
$\mathsf{bmo}({\mathcal{M}})\subset(\mathsf{h}_{1}({\mathcal{M}}))^{*}$ in
[17], $(\mathsf{h}_{2}({\mathcal{M}}),\mathsf{bmo}({\mathcal{M}}))_{\theta}$
embeds into $L_{p}({\mathcal{M}})$ with optimal constant $cp$.
###### Remark 3.12.
We can directly compare the norms $\|\cdot\|_{\mathsf{bmo}_{p}}$ and
$\mathsf{b}_{p}(\cdot)$ directly for $1<p<\infty$ by using Theorem 3.3.
Let us justify this remark. We first deal with the case $2<p<\infty$. Fix $n$,
for any $b\in{\mathcal{M}}_{n}$ with $\|b\|_{p}\leq 1$, by the noncommutative
Burkholder inequality [9], we have
$\|(x-x_{n})b\|_{\mathsf{h}^{c}_{p}}\leq
cp\|(x-x_{n})b\|_{p},\quad\|b(x-x_{n})\|_{\mathsf{h}^{r}_{p}}\leq
cp\|b(x-x_{n})\|_{p},$
hence
$\|(x-x_{n})b\|_{\mathsf{h}^{c}_{p}},\;\|b(x-x_{n})\|_{\mathsf{h}^{r}_{p}}\leq
cp\mathsf{b}_{p}(x)$
Then by Theorem 3.3,
$\displaystyle\|x\|_{\mathsf{bmo}_{p}}\leq cp\mathsf{b}_{p}(x).$
Another direction can be done by the way in Theorem 3.10,
$\mathsf{b}_{p}(x)\leq cp\|x\|_{\sf{bmo}}\leq cp\|x\|_{\mathsf{bmo}_{p}}.$
For the case $1<p<2$. The trivial part
$\displaystyle\mathsf{b}_{p}(x)\leq c\|x\|_{\mathsf{bmo}_{p}}$
follows from the noncommutative Burkholder inequality in [9]. Now let us prove
the inverse one. Take $b\in{\mathcal{M}}_{n}$ with $\|b\|_{2}\leq 1$. By
Hölder’s inequality, we have
$\displaystyle\|(x-x_{n})b\|^{2}_{2}=\tau(b^{2/{p^{\prime}}}(x-x_{n})^{*}(x-x_{n})b^{2/p})$
$\displaystyle\leq\|b^{2/{p^{\prime}}}(x-x_{n})^{*}\|_{{p^{\prime}}}\|(x-x_{n})b^{2/p}\|_{p}$
and
$\displaystyle\|b(x-x_{n})\|^{2}_{2}=\tau((x-x_{n})^{*}b^{2/{p^{\prime}}}b^{2/p}(x-x_{n}))$
$\displaystyle\leq\|(x-x_{n})^{*}b^{2/{p^{\prime}}}\|_{{p^{\prime}}}\|b^{2/p}(x-x_{n})\|_{p}.$
So by the result in Theorem 3.3 for $2<p^{\prime}<\infty$, we have
$\displaystyle\|b(x-x_{n})\|^{2}_{2},\|(x-x_{n})b\|^{2}_{2}$
$\displaystyle\leq\max\\{\|b^{2/{p^{\prime}}}(x-x_{n})^{*}\|_{{p^{\prime}}},\|(x-x_{n})^{*}b^{2/{p^{\prime}}}\|_{{p^{\prime}}}\\}$
$\displaystyle\quad\cdot\max\\{\|(x-x_{n})b^{2/p}\|_{p},\|b^{2/p}(x-x_{n})\|_{p}\\}$
$\displaystyle\leq
c\|x\|_{\mathsf{bmo}_{p^{\prime}}}\cdot\mathsf{b}_{p}(x)\leq
cp^{\prime}\|x\|_{\mathsf{bmo}_{2}}\cdot\mathsf{b}_{p}(x)$
Then by the definition of $\mathsf{bmo}_{2}({\mathcal{M}})$, we finish the
proof by Theorem 3.3
$\displaystyle\|x\|_{\mathsf{bmo}_{p}}\leq\|x\|_{\mathsf{bmo}_{2}}\leq
cp^{\prime}\mathsf{b}_{p}(x).$
The following corollary extends Junge/Musat’s theorem to all $0<p<\infty$. It
can be proved similarly as Theorem 3.3. However, using the identity
${\mathcal{BMO}}({\mathcal{M}})\simeq\mathsf{bmo}({\mathcal{M}})$ proved in
[17], we give a simpler proof.
###### Corollary 3.13.
For $0<p<\infty$, we have
$\alpha^{-1}_{p}\|x\|_{{\mathcal{BMO}}}\leq\mathcal{B}_{p}(x)\leq\beta_{p}\|x\|_{{\mathcal{BMO}}},$
where
$\displaystyle\mathcal{B}_{p}(x)=\max\\{$
$\displaystyle\sup_{n}\sup_{b\in{\mathcal{M}}_{n},\|b\|_{p}\leq
1}\|(x-x_{n-1})b\|_{p},$
$\displaystyle\sup_{n}\sup_{b\in{\mathcal{M}}_{n},\|b\|_{p}\leq
1}\|b(x-x_{n-1})\|_{p}\\}.$
The constant $\alpha_{p}$ and $\beta_{p}$ have the same orders as those in
Theorem 3.3.
###### Proof.
For $2\leq p<\infty$, it is very easy to get
$\mathcal{B}_{p}(x)\leq\mathrm{b}_{p}(x)\leq cp\|x\|_{\mathsf{bmo}}\leq
cp\|x\|_{\mathcal{BMO}}$
from the triangular inequality
$\|(x-x_{n-1})b\|_{p}\leq\|(x-x_{n})b\|_{p}+\|(x_{n}-x_{n-1})b\|_{p},$
with $b\in{\mathcal{M}}_{n}$ and $\|b\|_{p}\leq 1$. And the rest of the proof
is the same to Theorem 3.10. ∎
###### Remark 3.14.
The following example shows that Junge/Musat’s John-Nirenberg inequality does
not hold for $\mathsf{bmo}^{c}$ or ${\mathcal{BMO}}^{c}$. The example is the
same as the one given in Remark 3.20 of [8]. Let $n$ be a positive integer and
consider the von Neumann algebra
${\mathcal{M}}=L_{\infty}(\mathbb{T})\bar{\otimes}M_{n},$
where $M_{n}$ is the algebra of $n\times n$ matrices with normalized trace.
For $k\geq 1$ let ${\mathcal{F}}_{k}$ be the $\sigma$-algebra generated by
dyadic intervals in $\mathbb{T}$ of length $2^{-k}$. Denote by
${\mathcal{M}}_{k}$ the subalgebra
$L_{\infty}(\mathbb{T},{\mathcal{F}}_{k})\bar{\otimes}M_{n}$ of
${\mathcal{M}}$ and let $\mathcal{E}_{k}=\mathbb{E}_{k}\otimes id_{M_{n}}$ be
the conditional expectation onto ${\mathcal{M}}_{k}$. Let $r_{k}$ be the
$k$-th Rademacher function on $\mathbb{T}$ and consider
$x=\sum^{n}_{k=1}r_{k}\otimes e_{1k}.$
Then $x$ is a martingale relative to the filtration $(\mathcal{M}_{k})_{k\geq
1}$ and the martingale differences are given by $dx_{k}=r_{k}\otimes e_{1k}$.
A simple calculation shows that
$\sup_{m}\|x-x_{m}\|_{p}=(n-1)^{\frac{1}{2}}n^{-\frac{1}{p}},$
while
$\|x\|_{\mathsf{bmo}^{c}}=\sup_{m}\left\|\sum^{n}_{k=m+1}\mathcal{E}_{m}|d_{k}x|^{2}\right\|^{\frac{1}{2}}_{\infty}=1.$
Let $p>2$. Then for any $c>0$, there exists $n\geq 1$ such that
$(n-1)^{1/2}n^{-1/p}>c$. Hence
$\sup_{m}\sup_{b\in{\mathcal{M}}_{m},\|b\|p\leq
1}\|(x-x_{m})b\|_{p}\geq\sup_{m}\|x-x_{m}\|_{p}>>\|x\|_{\mathsf{bmo}^{c}}.$
### 3.2. A fine version
Now we can formulate the fine version of the column (resp. row) John-Nirenberg
inequality.
###### Definition 3.15.
For $0<p<\infty$, we define
$\mathsf{bmo}^{c}_{p,\rm{pr}}({\mathcal{M}})=\big{\\{}x\in
L_{1}({\mathcal{M}}):\|x\|_{\mathsf{bmo}^{c}_{p,\rm{pr}}}<\infty\big{\\}}$
with
$\|x\|_{\mathsf{bmo}^{c}_{p,\rm{pr}}}=\max\big{\\{}\|\mathcal{E}_{1}(x)\|_{\infty},\quad\sup_{n}\sup_{e\in\mathcal{P}({\mathcal{M}}_{n})}\|(x-x_{n})\frac{e}{(\tau(e))^{{1}/{p}}}\|_{\mathsf{h}^{c}_{p}}\big{\\}}.$
Similarly,
$\mathsf{bmo}^{r}_{p,\rm{pr}}({\mathcal{M}})=\\{x:x^{*}\in\mathsf{bmo}^{c}_{p,\rm{pr}}({\mathcal{M}})\\}\;\textrm{
with
}\;\|x\|_{\mathsf{bmo}_{p,\rm{pr}}^{r}}=\|x^{*}\|_{\mathsf{bmo}_{p,\rm{pr}}^{c}}.$
Finally,
$\mathsf{bmo}_{p,\rm{pr}}({\mathcal{M}})=\mathsf{bmo}^{c}_{p,\rm{pr}}({\mathcal{M}})\cap\mathsf{bmo}^{r}_{p,\rm{pr}}({\mathcal{M}})\cap\mathsf{bmo}^{d}({\mathcal{M}})$
equipped with
$\|x\|_{\mathsf{bmo}_{p,\rm{pr}}}=\max\\{\|x\|_{\mathsf{bmo}^{c}_{p,\rm{pr}}},\|x\|_{\mathsf{bmo}_{p,\rm{pr}}^{r}},\|x\|_{\mathsf{bmo}^{d}}\\}.$
The fine version of the column (resp. row) John-Nirenberg inequality is stated
as follows.
###### Theorem 3.16.
For all $0<p<\infty$, we have
$\alpha^{-1}_{p}\|x\|_{\mathsf{bmo}^{c}}\leq\|x\|_{\mathsf{bmo}^{c}_{p,\rm{pr}}}\leq\beta_{p}\|x\|_{\mathsf{bmo}^{c}}.$
The constants $\alpha_{p}$ and $\beta_{p}$ have the same properties as those
in Theorem 3.3. The same inequalities hold for $\|\cdot\|_{\mathsf{bmo}^{r}}$
and $\|\cdot\|_{\mathsf{bmo}^{r}_{p,\rm{pr}}}$.
###### Proof.
We first consider the case $0<p\leq 1$. By Theorem 3.3, the trivial part
$\|x\|_{\mathsf{bmo}^{c}_{p,\mathrm{pr}}}\leq\|x\|_{\mathsf{bmo}^{c}_{p}}\leq\|x\|_{\mathrm{bmo}^{c}}$
follows from the fact that ${e}/{(\tau(e))^{1/p}}\in{\mathcal{M}}_{n}$ and its
$L_{p}$-norm equals 1. Now we turn to the proof of the inverse inequality.
Since any $a\in{\mathcal{M}}_{n}$ with $\|a\|_{p}\leq 1$ can be approximated
by sums $\sum_{k}\lambda_{k}{e_{k}}/{(\tau(e_{k}))^{1/p}}$ with $e_{k}$’s in
${\mathcal{M}}_{n}$ and $\sum_{k}|\lambda_{k}|^{p}\leq 1$. Thus we can assume
that $a$ itself is such a sum. Then
$\displaystyle\|(x-x_{n})a\|^{p}_{\mathsf{h}^{c}_{p}}$
$\displaystyle=\|\sum_{k}\lambda_{k}(x-x_{n})\frac{e_{k}}{(\tau(e_{k}))^{1/p}}\|^{p}_{\mathsf{h}^{c}_{p}}$
$\displaystyle\leq\sum_{k}|\lambda_{k}|^{p}\|(x-x_{n})\frac{e_{k}}{(\tau(e_{k}))^{1/p}}\|^{p}_{\mathsf{h}^{c}_{p}}$
$\displaystyle\leq\sum_{k}|\lambda_{k}|^{p}\|x\|^{p}_{\mathsf{bmo}^{c}_{p,\mathrm{pr}}}\leq\|x\|^{p}_{\mathsf{bmo}^{c}_{p,\mathrm{pr}}}.$
Therefore by Theorem 3.3,
$\|x\|_{\mathsf{bmo}^{c}}\leq C^{1/p-1/2}\|x\|_{\mathsf{bmo}^{c}_{p}}\leq
C^{1/p-1/2}\|x\|_{\mathsf{bmo}^{c}_{p,\rm{pr}}}.$
Now let $1<p<\infty$. Again, because of the fact that
${e}/{(\tau(e))^{1/p}}\in{\mathcal{M}}_{n}$ and its $L_{p}$-norm equals 1, by
Theorem 3.3,
(3.4)
$\displaystyle\|x\|_{\mathsf{bmo}^{c}_{p,\mathrm{pr}}}\leq\|x\|_{\mathsf{bmo}^{c}_{p}}\leq
c_{1}p\|x\|_{\mathsf{bmo}^{c}}.$
We exploit the result for $p=1$ to prove the inverse inequality. By Hölder’s
inequality, we have
$\displaystyle\|x\|_{\mathsf{bmo}^{c}_{1,\mathrm{pr}}}\leq\|x\|_{\mathsf{bmo}^{c}_{p,\mathrm{pr}}}.$
We end the proof by Theorem 3.3 and the result for $p=1$,
$\displaystyle\|x\|_{\mathsf{bmo}^{c}}\leq
C^{1/p-1/2}\|x\|_{\mathsf{bmo}^{c}_{1}}\leq
C^{1/p-1/2}\|x\|_{\mathsf{bmo}^{c}_{1,\mathrm{pr}}}\leq
C^{1/p-1/2}\|x\|_{\mathsf{bmo}^{c}_{p,\mathrm{pr}}}.$
∎
Now we give the distributional form of the John-Nirenberg inequality for
$\mathsf{bmo}^{c}({\mathcal{M}})$ and $\mathsf{bmo}^{r}({\mathcal{M}})$.
###### Theorem 3.17.
Let $x\in\mathsf{bmo}^{c}({\mathcal{M}})$. Then for all natural numbers $n\geq
1$, all $e\in\mathcal{P}({\mathcal{M}}_{n})$ and for all $\lambda>0$, we have
$\frac{1}{\tau(e)}\tau(\mathds{1}_{(\lambda,\infty)}(s_{c}((x-x_{n})e)))\leq
2\exp(-\frac{c\lambda}{\|x\|_{\mathsf{bmo}^{c}}}),$
with $c$ an absolute constant. Here $\mathds{1}_{(\lambda,\infty)}(a)$ denotes
the spectral projection of a positive operator $a$ corresponding to the
interval $(\lambda,\infty)$.
###### Proof.
By homogeneity, we can assume $\|x\|_{\mathsf{bmo}^{c}}=1$. We first deal with
the case $\lambda\geq 2c_{1}$, where $c_{1}$ is the constant in inequality
(3.4). Let $p=\lambda/{(2c_{1})}\geq 1$, by Chebychev’s inequality and Theorem
3.16,
$\displaystyle\tau(\mathds{1}_{(\lambda,\infty)}(s_{c}((x-x_{n})e)))\leq\tau(e)\frac{\|(x-x_{n})e\|^{p}_{\mathsf{h}^{c}_{p}}}{\lambda^{p}}$
$\displaystyle\leq\tau(e)(c_{1}p\lambda^{-1})^{p}=\tau(e)\exp(p\ln(c_{1}p\lambda^{-1}))=\tau(e)\exp(-\frac{\ln
2}{2c_{1}}\lambda).$
When $0<\lambda<2c_{1}$,
$\displaystyle\frac{1}{\tau(e)}\tau(\mathds{1}_{(\lambda,\infty)}(s_{c}((x-x_{n})e)))\leq
1<2\exp(-\frac{\ln 2}{2c_{1}}\lambda).$
Therefore, we obtain the desired result by letting $c={\ln 2}/{(2c_{1})}$. ∎
Based on the crude version of Junge/Musat’s John-Nirenberg inequality in
Theorem 3.10 (resp. Corollary 3.8) for $\sf{bmo}({\mathcal{M}})$ (resp.
$\mathcal{BMO}({\mathcal{M}})$), the argument in the proof of Theorem 3.16 can
be adapted to get the fine version of Junge/Musat’s John-Nirenberg inequality.
###### Corollary 3.18.
For all $0<p<\infty$, we have
$\alpha^{-1}_{p}\|x\|_{\mathsf{bmo}}\leq\mathcal{P}\mathsf{b}_{p}(x)\leq\beta_{p}\|x\|_{\mathsf{bmo}},$
where
$\displaystyle\mathcal{P}\mathsf{b}_{p}(x)=\max\\{$
$\displaystyle\sup_{n}\|(dx_{n})_{n}\|_{{\infty}},\quad\sup_{n}\sup_{e\in{\mathcal{M}}_{n}}\|(x-x_{n})\frac{e}{(\tau(e))^{1/p}}\|_{p},$
$\displaystyle\sup_{n}\sup_{e\in{\mathcal{M}}_{n}}\|\frac{e}{(\tau(e))^{1/p}}(x-x_{n})\|_{p}\\}.$
The constants $\alpha_{p}$ and $\beta_{p}$ have the same orders as those in
Theorem 3.3.
###### Corollary 3.19.
For $0<p<\infty$, we have
$\alpha^{-1}_{p}\|x\|_{{\mathcal{BMO}}}\leq\mathcal{PB}_{p}(x)\leq\beta_{p}\|x\|_{{\mathcal{BMO}}},$
where
$\displaystyle\mathcal{PB}_{p}(x)=\max\\{$
$\displaystyle\sup_{n}\sup_{e\in{\mathcal{M}}_{n}}\|(x-x_{n-1})\frac{e}{(\tau(e))^{1/p}}\|_{p},$
$\displaystyle\sup_{n}\sup_{e\in{\mathcal{M}}_{n}}\|\frac{e}{(\tau(e))^{1/p}}(x-x_{n-1})\|_{p}\\}.$
The constant $\alpha_{p}$ and $\beta_{p}$ have the same orders as those in
Theorem 3.3.
Again, based on Corollary 3.19, by arguments similar to the proof of Thoerem
3.17, we obtain the exponential integrability form of the John-Nirenberg
inequality for ${\mathcal{BMO}}({\mathcal{M}})$.
###### Theorem 3.20.
Let $x\in{\mathcal{BMO}}({\mathcal{M}})$. Then for all natural numbers $n\geq
1$, all $e\in\mathcal{P}({\mathcal{M}}_{n})$ and for all $\lambda>0$, we have
$\frac{1}{\tau(e)}\tau(\mathds{1}_{(\lambda,\infty)}(|(x-x_{n-1})e|)+\mathds{1}_{(\lambda,\infty)}(|e(x-x_{n-1})|))\leq
4\exp(-\frac{c\lambda}{\|x\|_{{\mathcal{BMO}}}})$
with $c$ an absolute constant.
## 4\. atomic decomposition
### 4.1. A crude version of atoms
According to the crude version of the noncommutative John-Nirenberg
inequality, we introduce the following
###### Definition 4.1.
For $1<q\leq\infty$, $a\in L_{1}({\mathcal{M}})$ is said to be a
$(1,q,c)$-atom with respect to $({\mathcal{M}}_{n})_{n\geq 1}$, if there exist
$n\geq 1$ and a factorization $a=yb$ such that
1. (i)
${\mathcal{E}}_{n}(y)=0$;
2. (ii)
$b\in L_{q^{\prime}}({\mathcal{M}}_{n})$ and $\|b\|_{q^{\prime}}\leq 1$;
3. (iii)
$\|y\|_{\mathsf{h}^{c}_{q}}\leq 1$ for $1<q<\infty$;
$\|y\|_{\mathsf{bmo}^{c}}\leq 1$ for $q=\infty$.
Similarly, we define the notion of a $(1,q,r)$-atom with $a=yb$ replaced by
$a=by$.
###### Lemma 4.2.
Let $1<q\leq\infty$. If $a$ is a $(1,q,c)$-atom, then
$\|a\|_{\mathsf{h}^{c}_{1}}\leq 1.$
The analogous inequality holds for $(1,q,r)$-atoms.
###### Proof.
We first deal with the case $1<q<\infty$. By definition, there exists an $n$
such that the $(1,q,c)$-atom $a$ admits a factorization $a=yb$ as in
Definition 4.1. Then
$s^{2}_{c}(a)=b^{*}\sum_{k>n}\mathcal{E}_{k-1}|dy_{k}|^{2}b=b^{*}s^{2}_{c}(y)b.$
Thus by Hölder’s inequality,
$\|a\|_{\mathsf{h}^{c}_{1}}=\|s_{c}(a)\|_{1}\leq\|s_{c}(y)\|_{q}\|b\|_{q^{\prime}}\leq
1.$
For the case $q=\infty$, the calculation is a bit different,
$\displaystyle\|a\|_{\mathsf{h}^{c}_{1}}$
$\displaystyle=\left\|b^{*}s^{2}_{c}(y)b\right\|^{1/2}_{{1/2}}=\tau(\mathcal{E}_{n}(b^{*}s^{2}_{c}(y)b)^{1/2})$
$\displaystyle\leq\tau((\mathcal{E}_{n}(b^{*}s_{c}(y)b))^{1/2})\leq\|\mathcal{E}_{n}(s_{c}(y))\|_{\infty}\|b\|_{1}$
$\displaystyle\leq\left\|y\right\|_{\mathsf{bmo}^{c}}\|b\|_{1}\leq 1.$
We have used the trace preserving property of conditional expectations in the
fourth equality and the operator Jensen inequality in the first inequality.
For the second inequality, we have used the property that
$\mathcal{E}_{n}\cdot\mathcal{E}_{k-1}=\mathcal{E}_{n}$ for all $k>n$ and
Hölder’s inequality. ∎
###### Definition 4.3.
We define $\mathsf{h}^{c}_{1,\mathrm{at}_{q}}({\mathcal{M}})$ as the Banach
space of all $x\in L_{1}({\mathcal{M}})$ which admit a decomposition
$x=\sum_{k}\lambda_{k}a_{k}$, where for each $k$, $a_{k}$ a $(1,q,c)$-atom or
an element in the unit ball of $L_{1}({\mathcal{M}}_{1})$, and
$\lambda_{k}\in\mathbb{C}$ satisfying $\sum_{k}|\lambda_{k}|<\infty$. We equip
this space with the norm
$\|x\|_{\mathsf{h}^{c}_{1,\mathrm{at}_{q}}}=\inf\sum_{k}|\lambda_{k}|,$
where the infimum is taken over all decompositions of $x$ described above.
Similarly, we define $\mathsf{h}^{r}_{1,\mathrm{at}_{q}}({\mathcal{M}})$.
Now, by Lemma 4.2, we have the obvious inclusion
$\mathsf{h}^{c}_{1,\mathrm{at}_{q}}({\mathcal{M}})\subset\mathsf{h}^{c}_{1}({\mathcal{M}})$.
In fact, the two spaces coincide thanks to the following theorem.
###### Theorem 4.4.
For all $1<q\leq\infty$, we have
$\mathsf{h}^{c}_{1}({\mathcal{M}})=\mathsf{h}^{c}_{1,\mathrm{at}_{q}}({\mathcal{M}})$
with equivalent norms. Similarly,
$\mathsf{h}^{r}_{1}({\mathcal{M}})=\mathsf{h}^{r}_{1,\mathrm{at}_{q}}({\mathcal{M}})$
with equivalent norms.
We prove this theorem by duality. We require the following lemmas.
###### Lemma 4.5.
$(\rm{i})$ For all $1<q\leq 2$, $L_{2}({\mathcal{M}})$ densely and
continuously embeds into $\mathsf{h}^{c}_{1,\mathrm{at}_{q}}({\mathcal{M}})$.
$(\rm{ii})$ For all $2<q\leq\infty$, $L_{q}({\mathcal{M}})$ densely and
continuously embeds into $\mathsf{h}^{c}_{1,\mathrm{at}_{q}}({\mathcal{M}})$.
###### Proof.
$(\rm{i})$. For any $x\in L_{2}({\mathcal{M}})$, we decompose it as a linear
combination of two atoms:
$x=\|x-\mathcal{E}_{1}(x)\|_{2}\frac{x-\mathcal{E}_{1}(x)}{\|x-\mathcal{E}_{1}(x)\|_{2}}+\|\mathcal{E}_{1}(x)\|_{2}\frac{\mathcal{E}_{1}(x)}{\|\mathcal{E}_{1}(x)\|_{2}}.$
Indeed, on the one hand, ${\mathcal{E}_{1}(x)}/{\|\mathcal{E}_{1}(x)\|_{2}}\in
L_{2}({\mathcal{M}}_{1})\subset L_{1}({\mathcal{M}}_{1})$ and
$\|\frac{\mathcal{E}_{1}(x)}{\|\mathcal{E}_{1}(x)\|_{2}}\|_{1}=\frac{\|\mathcal{E}_{1}(x)\|_{1}}{\|\mathcal{E}_{1}(x)\|_{2}}\leq
1.$
On the other hand,
$\frac{x-\mathcal{E}_{1}(x)}{\|x-\mathcal{E}_{1}(x)\|_{2}}=\frac{x-\mathcal{E}_{1}(x)}{\|x-\mathcal{E}_{1}(x)\|_{2}}\cdot\mathds{1}\doteq
y\cdot b.$
Clearly, $\mathcal{E}_{1}(y)=0$, $\|b\|_{q^{\prime}}\leq 1$ and
$\|y\|_{\mathsf{h}^{c}_{q}}=\|\frac{x-\mathcal{E}_{1}(x)}{\|x-\mathcal{E}_{1}(x)\|_{2}}\|_{\mathsf{h}^{c}_{q}}\leq\|\frac{x-\mathcal{E}_{1}(x)}{\|x-\mathcal{E}_{1}(x)\|_{2}}\|_{\mathsf{h}^{c}_{2}}\leq
1.$
Thus $x$ is a sum of two atoms and
$\|x\|_{\mathsf{h}^{c}_{1,\mathrm{at}_{q}}}\leq\|x-\mathcal{E}_{1}(x)\|_{2}+\|\mathcal{E}_{1}(x)\|_{2}\leq\sqrt{2}\|x\|_{2}.$
The density is trivial.
$(\rm{ii})$. This case is similar to the previous one. We first deal with the
case $2<q<\infty$. Given $x\in L_{q}({\mathcal{M}})$, we write again:
$x=c_{q}\|x-\mathcal{E}_{1}(x)\|_{q}\frac{x-\mathcal{E}_{1}(x)}{c_{q}\|x-\mathcal{E}_{1}(x)\|_{q}}+\|\mathcal{E}_{1}(x)\|_{q}\frac{\mathcal{E}_{1}(x)}{\|\mathcal{E}_{1}(x)\|_{q}},$
where $c_{q}$ is fixed below. Indeed,
${\mathcal{E}_{1}(x)}/{\|\mathcal{E}_{1}(x)\|_{q}}\in
L_{q}({\mathcal{M}}_{1})\subset L_{1}({\mathcal{M}}_{1})$ and
$\|\frac{\mathcal{E}_{1}(x)}{\|\mathcal{E}_{1}(x)\|_{q}}\|_{1}=\frac{\|\mathcal{E}_{1}(x)\|_{1}}{\|\mathcal{E}_{1}(x)\|_{q}}\leq
1.$
On the other hand,
$\frac{x-\mathcal{E}_{1}(x)}{c_{q}\|x-\mathcal{E}_{1}(x)\|_{q}}=\frac{x-\mathcal{E}_{1}(x)}{c_{q}\|x-\mathcal{E}_{1}(x)\|_{q}}\cdot\mathds{1}\doteq
y\cdot b,$
$\mathcal{E}_{1}(\frac{x-\mathcal{E}_{1}(x)}{c_{q}\|x-\mathcal{E}_{1}(x)\|_{q}})=0,\quad\|b\|_{q^{\prime}}\leq
1$
and the noncommutative Burkholder inequality in [9] yields
$\|y\|_{\mathsf{h}^{c}_{q}}=\|\frac{x-\mathcal{E}_{1}(x)}{c_{q}\|x-\mathcal{E}_{1}(x)\|_{q}}\|_{\mathsf{h}^{c}_{q}}\leq
c_{q}\|\frac{x-\mathcal{E}_{1}(x)}{c_{q}\|x-\mathcal{E}_{1}(x)\|_{q}}\|_{q}\leq
1.$
Therefore,
$\|x\|_{\mathsf{h}^{c}_{1,\mathrm{at}_{q}}}\leq
c_{q}\|x-\mathcal{E}_{1}(x)\|_{q}+\|\mathcal{E}_{1}(x)\|_{q}\leq(2c_{q}+1)\|x\|_{q}.$
The case $q=\infty$ is proved in the same way just by replacing the
noncommutative Burkholder inequality by the trivial fact that
$\|\cdot\|_{\mathsf{bmo}^{c}}\leq\|\cdot\|_{\infty}$. The density is trivial.
∎
###### Lemma 4.6.
Let $1<q<\infty$. Then
$(\mathsf{h}^{c}_{1,\mathrm{at}_{q}}({\mathcal{M}}))^{*}=\mathsf{bmo}^{c}_{q^{\prime}}({\mathcal{M}})$
with equivalent norms. More precisely,
1. (i)
Every $x\in\mathsf{bmo}^{c}_{q^{\prime}}({\mathcal{M}})$ defines a bounded
linear functional on $\mathsf{h}^{c}_{1,\mathrm{at}_{q}}({\mathcal{M}})$ by
(4.1) $\displaystyle\varphi_{x}(a)=\tau(x^{*}a),\forall
a\in(1,q,c)\textrm{-atoms}.$
2. (ii)
Conversely, each
$\varphi\in(\mathsf{h}^{c}_{1,\mathrm{at}_{q}}({\mathcal{M}}))^{*}$ is given
as (4.1) by some $x\in\mathsf{bmo}^{c}_{q^{\prime}}({\mathcal{M}})$.
Similarly,
$(\mathsf{h}^{r}_{1,\mathrm{at}_{q}}({\mathcal{M}}))^{*}=\mathsf{bmo}^{r}_{q^{\prime}}({\mathcal{M}})$
with equivalent norms.
###### Proof.
$\rm(i)$ Let $x\in\mathsf{bmo}^{c}_{q^{\prime}}$, and $a=yb$ where $a$ is a
$(1,q,c)$-atom as in Definition 4.1. Then
$\displaystyle|\tau(x^{*}a)|$ $\displaystyle=|\tau(\mathcal{E}_{n}(x^{*}y)b)|$
$\displaystyle=|\tau(\mathcal{E}_{n}((x^{*}-x^{*}_{n})y)b)|=|\tau(((x-x_{n})b^{*})^{*}y)|.$
Thus, by the duality identity
$\mathsf{h}^{c}_{q}({\mathcal{M}})=(\mathsf{h}^{c}_{q^{\prime}}({\mathcal{M}}))^{*}$
(see [9] for the relevant constants),
$\displaystyle|\tau(x^{*}a)|\leq\left\|(x-x_{n})b^{*}\right\|_{\mathsf{h}^{c}_{q^{\prime}}}\|y\|_{\mathsf{h}^{c}_{q}}$
$\displaystyle\leq\|x\|_{\mathsf{bmo}^{c}_{q^{\prime}}}.$
$\rm(ii)$. Let $\varphi$ be any linear functional on
$\mathsf{h}^{c}_{1,\mathrm{at}_{q}}({\mathcal{M}})$. When $1<q\leq 2$, by
Lemma 4.5 we can find $x\in L_{2}({\mathcal{M}})$ such that
$\varphi(y)=\tau(x^{*}y),\qquad\forall y\in L_{2}({\mathcal{M}}),$
and
$\|\varphi\|=\sup_{y\in L_{2},\|y\|_{\mathsf{h}^{c}_{1,\mathrm{at}_{q}}}\leq
1}|\tau(x^{*}y)|.$
When $2<q<\infty$, by the same Lemma 4.5, we get the same representation of
$\varphi$ with an $x\in L_{q^{\prime}}({\mathcal{M}})$. Then fix $n$ and take
any $b\in{\mathcal{M}}_{n}$ with $\|b\|_{q^{\prime}}\leq 1$. Again, by the
duality
$\mathsf{h}^{c}_{q}({\mathcal{M}})=(\mathsf{h}^{c}_{q^{\prime}}({\mathcal{M}}))^{*}$,
we do the following calculation:
$\displaystyle\|(x-x_{n})b\|_{\mathsf{h}^{c}_{q^{\prime}}}$
$\displaystyle=\sup_{\|y\|_{(\mathsf{h}^{c}_{q^{\prime}})^{*}}\leq
1}|\tau(b^{*}(x^{*}-x^{*}_{n})y)|$
$\displaystyle\leq\sup_{\|y\|_{\mathsf{h}^{c}_{q}}\leq
cq}|\tau(b^{*}(x^{*}-x^{*}_{n})y)|$
$\displaystyle=\sup_{\|y\|_{\mathsf{h}^{c}_{q}}\leq
cq}|\tau((x^{*}-x^{*}_{n})(y-y_{n})b^{*})|$
$\displaystyle=\sup_{\|y\|_{\mathsf{h}^{c}_{q}}\leq
cq}|\tau(x^{*}((y-y_{n})b^{*}))|$ $\displaystyle\leq cq\|\varphi\|$
Here, we have used the fact that $\tau(x-x_{n})=\tau(y-y_{n})=0$ in the second
and third equality respectively. The second inequality is due to the fact that
$(y-y_{n})b^{*}$ is a $(1,q,c)$-atom. ∎
Now we are at a position to prove Theorem 4.4.
###### Proof.
We consider here only the case $1<q<\infty$ and postpone the case $q=\infty$
to the end of the proof of Theorem 4.12 below. We only need to show the
inclusion
$\mathsf{h}^{c}_{1}({\mathcal{M}})\subset\mathsf{h}^{c}_{1,\mathrm{at}_{q}}({\mathcal{M}}).$
Take $x\in\mathsf{h}^{c}_{1,\mathrm{at}_{q}}({\mathcal{M}})$, by Theorem 3.3
and Lemma 4.6, we can conduct the following calculation,
$\displaystyle\|x\|_{\mathsf{h}^{c}_{1,\mathrm{at}_{q}}}$
$\displaystyle=\sup_{\|y\|_{(\mathsf{h}^{c}_{1,\mathrm{at}_{q}})^{*}}\leq
1}|\tau(x^{*}y)|$
$\displaystyle\leq\sup_{\|y\|_{\mathsf{bmo}^{c}_{q^{\prime}}}\leq
cq}|\tau(x^{*}y)|$ $\displaystyle\leq\sup_{\|y\|_{\mathsf{bmo}^{c}}\leq
cq}|\tau(x^{*}y)|\leq cq\|x\|_{\mathsf{h}^{c}_{1}}.$
Then we end the proof with the density of
$\mathsf{h}^{c}_{1,\mathrm{at}_{q}}({\mathcal{M}})$ in
$\mathsf{h}^{c}_{1}({\mathcal{M}})$. ∎
###### Definition 4.7.
We define
$\mathsf{h}_{1,\mathrm{at}_{q}}({\mathcal{M}})=\mathsf{h}^{c}_{1,\mathrm{at}_{q}}({\mathcal{M}})+\mathsf{h}^{r}_{1,\mathrm{at}_{q}}({\mathcal{M}})+\mathsf{h}^{d}_{1}({\mathcal{M}})$
equipped with the sum norm
$\|x\|_{\mathsf{h}_{1,\mathrm{at}_{q}}}=\inf_{x=x_{c}+x_{r}+x_{d}}\\{\|x_{c}\|_{\mathsf{h}^{c}_{1,\mathrm{at}_{q}}}+\|x_{r}\|_{\mathsf{h}^{r}_{1,\mathrm{at}_{q}}}+\|x_{d}\|_{\mathsf{h}^{d}_{1}}\\}.$
Then by Theorem 4.4, we obtain the atomic decomposition of
$\mathsf{h}_{1}({\mathcal{M}})$.
###### Corollary 4.8.
We have
$\mathsf{h}_{1}({\mathcal{M}})=\mathsf{h}_{1,\mathrm{at}_{q}}({\mathcal{M}})$
with equivalent norms.
Combined with Davis’ decomposition presented in [17], the above theorem yields
$\mathcal{H}_{1}({\mathcal{M}})=\mathsf{h}_{1,\mathrm{at}_{q}}({\mathcal{M}})$
with equivalent norms. In other words, we obtain an atomic decomposition for
$\mathcal{H}_{1}({\mathcal{M}})$ too.
### 4.2. A fine version of atoms
###### Definition 4.9.
For $1<q\leq\infty$, $a\in L_{1}({\mathcal{M}})$ is said to be a
$(1,q,c)_{\mathrm{pr}}$-atom with respect to $({\mathcal{M}}_{n})_{n\geq 1}$,
if there exist $n\geq 1$ and a projection $e\in\mathcal{P}({\mathcal{M}}_{n})$
such that
1. (i)
${\mathcal{E}}_{n}(a)=0$;
2. (ii)
$r(a)\leq e$;
3. (iii)
$\|a\|_{\mathsf{h}^{c}_{q}}\leq(\tau(e))^{-\frac{1}{q^{\prime}}}$ for
$1<q<\infty$; $\|a\|_{\mathsf{bmo}^{c}}\leq{(\tau(e))}^{-1}$ for $q=\infty$.
Similarly, we define $(1,q,r)_{\mathrm{pr}}$-atoms with $r(a)$ replaced by
$l(a)$.
###### Remark 4.10.
A $(1,q,c)_{\mathrm{pr}}$-atom $a$ is necessarily a $(1,q,c)$-atom. Indeed, we
can factorize $a$ as $a=yb$ with $y=a(\tau(e))^{{1}/{q^{\prime}}}$ and
$b=e(\tau(e))^{-1/{q^{\prime}}}$.
###### Definition 4.11.
We define $\mathsf{h}^{c}_{1,\mathrm{at}_{q,\mathrm{pr}}}({\mathcal{M}})$ to
be the Banach space of all $x\in L_{1}({\mathcal{M}})$ which admit a
decomposition $x=\sum_{k}\lambda_{k}a_{k}$, where for each $k$, $a_{k}$ is a
$(1,q,c)_{\mathrm{pr}}$-atom or an element in the unit ball of
$L_{1}({\mathcal{M}}_{1})$, and $\lambda_{k}\in\mathbb{C}$ satisfying
$\sum_{k}|\lambda_{k}|<\infty$. We equip this space with the norm
$\|x\|_{\mathsf{h}^{c}_{1,\mathrm{at}_{q,\mathrm{pr}}}}=\inf\sum_{k}|\lambda_{k}|,$
where the infimum is taken over all decompositions of $x$ described above.
Similarly, we define
$\mathsf{h}^{r}_{1,\mathrm{at}_{q,\mathrm{pr}}}({\mathcal{M}})$.
Now, by Remark 4.10 and Lemma 4.4, we have the obvious inclusion
$\mathsf{h}^{c}_{1,\mathrm{at}_{q,\mathrm{pr}}}({\mathcal{M}})\subset\mathsf{h}^{c}_{1}({\mathcal{M}})$.
In fact, the two spaces coincide thanks to the following theorem.
###### Theorem 4.12.
For all $1<q\leq\infty$, we have
$\mathsf{h}^{c}_{1}({\mathcal{M}})=\mathsf{h}^{c}_{1,\mathrm{at}_{q,\mathrm{pr}}}({\mathcal{M}})$
with equivalent norms. Similarly,
$\mathsf{h}^{r}_{1}({\mathcal{M}})=\mathsf{h}^{r}_{1,\mathrm{at}_{q,\mathrm{pr}}}({\mathcal{M}})$
with equivalent norms.
Again, we prove this theorem for $1<q<\infty$ by showing
$(\mathsf{h}^{c}_{1,\mathrm{at}_{q,\mathrm{pr}}}({\mathcal{M}}))^{*}=\mathsf{bmo}^{c}_{q^{\prime},\mathrm{pr}}({\mathcal{M}})$.
The latter duality equality is proved in the same way as Theorem 4.6. We leave
the details to the reader. However by the argument in Theorem 4.6, we can not
prove the theorem in the case $q=\infty$, due to the lack of Riesz
representation. Here we provide another way to do it, which seems new, even in
the commutative case.
Let $\mathcal{P}$ be the set of projections of ${\mathcal{M}}$. Given
$e\in\mathcal{P}$ let
$n_{e}=\min\\{k\;:\;e\in\mathcal{P}(\mathcal{M}_{k})\\}.$
Note that $n_{e}=\infty$ if the set on the right hand side is empty. This case
is of no interest in the discussion below. For a family
$(g_{e})_{e\in\mathcal{P}}\subset\mathsf{bmo}^{c}({\mathcal{M}})$ define
$\|(g_{e})_{e}\|_{L^{\mathcal{P}}_{1}(\mathsf{bmo}^{c})}=\sum_{e\in\mathcal{P}}\tau(e)\|g_{e}\|_{\mathsf{bmo}^{c}}.$
We will consider the Banach space:
$L^{\mathcal{P}}_{1}(\mathsf{bmo}^{c})=\\{(g_{e})_{e}\;:\;g_{e}e=g_{e},\;\mathcal{E}_{n_{e}}g_{e}=0,\;\|(g_{e})_{e}\|_{L^{\mathcal{P}}_{1}(\mathsf{bmo}^{c})}<\infty\\}.$
We will also need the following space consisting of families in
$\mathsf{h}^{c}_{1}({\mathcal{M}})$:
$L^{\mathcal{P}}_{\infty}(\mathsf{h}^{c}_{1})=\\{(f_{e})_{e}\;:\;f_{e}e=f_{e},\;\mathcal{E}_{n_{e}}f_{e}=0,\;\|(f_{e})_{e}\|_{L^{\mathcal{P}}_{\infty}(\mathsf{h}^{c}_{1})}<\infty\\},$
where
$\|(f_{e})_{e}\|_{L^{\mathcal{P}}_{\infty}(\mathsf{h}^{c}_{1})}=\sup_{e\in\mathcal{P}}\frac{1}{\tau(e)}\,\|f_{e}\|_{\mathsf{h}_{1}^{c}}.$
For convenience, we denote $L^{\mathcal{P}}_{1}(\mathsf{bmo}^{c})$ by $X$ and
$L^{\mathcal{P}}_{\infty}(\mathsf{h}^{c}_{1})$ by $Z$. We embed
$\mathsf{bmo}^{c}_{1,\mathrm{pr}}({\mathcal{M}})$ isomorphically into $Z$ via
the following map
$\pi(y)=((y-y_{n_{e}})e)_{e}.$
Set $Y=\pi(\mathsf{bmo}^{c}_{1,\mathrm{pr}}({\mathcal{M}}))$.
###### Lemma 4.13.
With the notation above we have
1. (i)
$Z$ is a subspace of $X^{*}$ with equivalent norms, so is $Y$.
2. (ii)
$Y$ is w*-closed in $X^{*}$.
###### Proof.
(i). Let $(f_{e})_{e}\in Z$, for any $(g_{e})_{e}\in X$, we have
$\displaystyle|\langle(f_{e})_{e},(g_{e})_{e}\rangle|$
$\displaystyle=|\sum_{e}\tau((f_{e})^{*}g_{e})|$
$\displaystyle\leq\sqrt{2}\sum_{e}\|f_{e}\|_{\mathsf{h}^{c}_{1}}\|g_{e}\|_{\mathsf{bmo}^{c}}$
$\displaystyle\leq\sqrt{2}\sup_{e}\frac{1}{\tau(e)}\,\|f_{e}\|_{\mathsf{h}_{1}^{c}}\cdot\sum_{e}\tau(e)\,\|g_{e}\|_{\mathsf{bmo}^{c}}$
$\displaystyle=\sqrt{2}\|(f_{e})_{e}\|_{Z}\|(g_{e})_{e}\|_{X}.$
Thus we get $\|(f_{e})_{e}\|_{X^{*}}\leq\sqrt{2}\|(f_{e})_{e}\|_{Z}.$
We turn to the proof of the inverse inequality. For any $(f_{e})_{e}\in Z$,
fix $e_{0}\in\mathcal{P}$, we have
$\displaystyle\frac{1}{\tau({e_{0}})}\,\|f_{e_{0}}\|_{\mathsf{h}^{c}_{1}}$
$\displaystyle=\sup_{\|g\|_{\mathsf{bmo}^{c}}\leq
1}\frac{1}{\tau({e_{0}})}\,\big{|}\tau((f_{e_{0}})^{*}g)\big{|}$
$\displaystyle=\sup_{\|g\|_{\mathsf{bmo}^{c}}\leq
1}\frac{1}{\tau({e_{0}})}\,\big{|}\tau((f_{e_{0}})^{*}(g-g_{n_{e_{0}}}){e_{0}})\big{|}$
$\displaystyle\leq\sup_{\|(g-g_{n_{e_{0}}}){e_{0}}\|_{\mathsf{bmo}^{c}}\leq
1}\frac{1}{\tau({e_{0}})}\,\big{|}\tau((f_{e_{0}})^{*}(g-g_{n_{e_{0}}}){e_{0}})\big{|}.$
Then we define $(g_{e})_{e}$ as
$g_{e}={(g-g_{n_{e_{0}}}{e_{0}})}/{\tau({e_{0}})}$ if $e={e_{0}}$, otherwise
$g_{e}=0$. Thus
$\displaystyle\frac{1}{\tau({e_{0}})}\,\|f_{e_{0}}\|_{\mathsf{h}^{c}_{1}}\leq\|(f_{e})_{e}\|_{X^{*}}\|(g_{e})_{e}\|_{X}\leq\|(f_{e})_{e}\|_{X^{*}},$
which implies $\|(f_{e})_{e}\|_{Z}\leq\|(f_{e})_{e}\|_{X^{*}}$.
(ii). Since $Y$ is a subspace of $X^{*}$, by Krein and Smulian’s theorem, we
only need to prove that for all $t>0$, $Y\cap B_{t}(X^{*})$ is w*-closed in
$X^{*}$, where $B_{t}(X^{*})$ is the closed ball of $X^{*}$ centered at the
origin and with radius $t$. Take a net
$(y^{\alpha})_{\alpha}\subset\mathsf{bmo}^{c}_{1,\mathrm{pr}}({\mathcal{M}})$
such that $\pi((y^{\alpha})_{\alpha})\subset Y\cap B_{t}(X^{*})$. Hence
$(y^{\alpha})_{\alpha}$ are bounded in
$\mathsf{bmo}^{c}_{1,\mathrm{pr}}({\mathcal{M}})$. Suppose that,
(4.2)
$\displaystyle\langle\pi(y^{\alpha}),(g_{e})_{e}\rangle\rightarrow\langle\xi,(g_{e})_{e}\rangle,\quad\quad\forall(g_{e})_{e}\in
X,$
for some $\xi\in B_{t}(X^{*})$. We will show that $\xi\in Y$, which will
complete the proof. We need two facts. The first one is that
$\mathsf{bmo}^{c}_{1,\mathrm{pr}}({\mathcal{M}})$ is a dual space by Theorem
3.16, so its unit ball is w*-compact. Therefore, the bounded net
$(y^{\alpha})_{\alpha}$ in $\mathsf{bmo}^{c}_{1,\mathrm{pr}}({\mathcal{M}})$
admits a $w^{*}$-cluster point $y$. Without loss of generality, we assume that
$(y^{\alpha})_{\alpha}$ converges to $y$ in the $w^{*}$-topology:
(4.3) $\displaystyle\langle y^{\alpha},x\rangle\rightarrow\langle
y,x\rangle,\quad\quad\forall x\in\mathsf{h}^{c}_{1}({\mathcal{M}}).$
The second fact is that for any $(g_{e})_{e}\in X$, the sum $\sum_{e}g_{e}$ is
absolutely summable in $\mathsf{h}^{c}_{1}({\mathcal{M}})$. Indeed, by Lemma
4.2
$\displaystyle\sum_{e}\|g_{e}\|_{\mathsf{h}^{c}_{1}}\leq\sum_{e}\tau(e)\|g_{e}\|_{\mathsf{bmo}^{c}}=\|(g_{e})_{e}\|_{X}.$
Therefore, for any $(g_{e})_{e}\in X$, we have
$\displaystyle\langle\pi(y^{\alpha}),(g_{e})_{e}\rangle$
$\displaystyle=\sum_{e}\tau(((y^{\alpha}_{e}-y^{\alpha}_{n_{e}})e)^{*}g_{e})$
$\displaystyle=\tau((y^{\alpha})^{*}\sum_{e}g_{e})$
Combining 4.2 and 4.3, we deduce that $\xi=\pi(y)\in Y$, as desired. ∎
We can now prove Theorem 4.12 in the case of $q=\infty$.
###### Proof.
Let $Y_{\perp}$ be the preannihilator of $Y$ in $X^{*}$:
$Y_{\perp}=\\{(g_{e})_{e}\in X\;:\;\langle\pi(y),(g_{e})_{e}\rangle=0,\forall
y\in\mathsf{bmo}^{c}_{1,\mathrm{pr}}({\mathcal{M}})\\}.$
Then by the bipolar theorem
$Y\simeq(X/Y_{\perp})^{*}.$
Using the second fact in the proof of the previous lemma, we get
$\displaystyle Y_{\perp}$ $\displaystyle=\\{(g_{e})_{e}\in
X\;:\;\tau(y^{*}\sum_{e}g_{e})=0,\forall
y\in\mathsf{bmo}^{c}_{1,\mathrm{pr}}({\mathcal{M}})\\}$
$\displaystyle=\\{(g_{e})_{e}\in
X\;:\;\sum_{e}g_{e}=0\;\mathrm{in}\;\mathsf{h}^{c}_{1}({\mathcal{M}})\\}.$
Then for $(g_{e})_{e}\in X/{Y_{\perp}}$, let
$g=\sum_{e\in\mathcal{P}}g_{e}.$
Then
$\displaystyle\|(g_{e})_{e}\|_{X/Y_{\perp}}$
$\displaystyle=\inf\\{\sum_{e}\tau(e)\,\|(g^{\prime}_{e})_{e}\|_{\mathsf{bmo}^{c}}\,:\,g=\sum_{e}g^{\prime}_{e},\;(g^{\prime}_{e})_{e}\in
X\\}$
$\displaystyle=\inf\\{\sum_{e}|\lambda_{e}|\,:\,g=\sum_{e}\lambda_{e}a_{e},\,(\lambda_{e}a_{e})_{e}\in
X,\,\|a_{e}\|_{\mathsf{bmo}^{c}}\leq\frac{1}{\tau(e)}\\}$
$\displaystyle=\|g\|_{\mathsf{h}^{c}_{1,\mathrm{at}_{\infty,\mathrm{pr}}}}.$
Consequently, for any
$x\in\mathsf{h}^{c}_{1,\mathrm{at}_{\infty,\mathrm{pr}}}({\mathcal{M}})$ and
any decomposition $x=\sum_{e}\lambda_{e}a_{e}$,
$\displaystyle\|x\|_{\mathsf{h}^{c}_{1,\mathrm{at}_{\infty,\mathrm{pr}}}}$
$\displaystyle=\|(\lambda_{e}a_{e})_{e}\|_{X/Y_{\perp}}$
$\displaystyle=\|(\lambda_{e}a_{e})_{e}\|_{Y^{*}}$
$\displaystyle=\sup_{y\in\mathsf{bmo}^{c}_{1,\mathrm{pr}},\|\pi(y)\|_{Y}\leq
1}|\langle(\lambda_{e}a_{e}),\pi(y)\rangle|$
$\displaystyle\leq\sup_{\|y\|_{\mathsf{bmo}^{c}}\leq
c}|\tau((\sum_{e}\lambda_{e}a_{e})^{*}y)|\leq c\|x\|_{\mathsf{h}^{c}_{1}}.$
Therefore, combined with Lemma 4.2 and Remark 4.10, the density of
$\mathsf{h}^{c}_{1,\mathrm{at}_{\infty,\mathrm{pr}}}({\mathcal{M}})$ in
$\mathsf{h}^{c}_{1}({\mathcal{M}})$ (due to Lemma 4.5) yields the desired
duality identity
$\mathsf{h}^{c}_{1,\mathrm{at}_{\infty,\mathrm{pr}}}({\mathcal{M}})=\mathsf{h}^{c}_{1}({\mathcal{M}})$.
∎
Let us return back to the unsettled case $q=\infty$ in the proof of Theorem
4.4. Since a fine atom is necessarily a crude atom, we get
$\mathsf{h}^{c}_{1}({\mathcal{M}})\subset\mathsf{h}^{c}_{1,\mathrm{at}_{\infty}}({\mathcal{M}})$,
hence
$\mathsf{h}^{c}_{1}({\mathcal{M}})=\mathsf{h}^{c}_{1,\mathrm{at}_{\infty}}({\mathcal{M}})$
with equivalent norms due to Lemma 4.2. Thus Theorem 4.4 is completely proved.
###### Definition 4.14.
We define
$\mathsf{h}_{1,\mathrm{at}_{q,\mathrm{pr}}}({\mathcal{M}})=\mathsf{h}^{c}_{1,\mathrm{at}_{q,\mathrm{pr}}}({\mathcal{M}})+\mathsf{h}^{r}_{1,\mathrm{at}_{q,\mathrm{pr}}}({\mathcal{M}})+\mathsf{h}^{d}_{1}({\mathcal{M}})$
equipped with the sum norm
$\|x\|_{\mathsf{h}_{1,\mathrm{at}_{q,\mathrm{pr}}}}=\inf_{x=x_{c}+x_{r}+x_{d}}\\{\|x_{c}\|_{\mathsf{h}^{c}_{1,\mathrm{at}_{q,\mathrm{pr}}}}+\|x_{r}\|_{\mathsf{h}^{r}_{1,\mathrm{at}_{q,\mathrm{pr}}}}+\|x_{d}\|_{\mathsf{h}^{d}_{1}}\\}.$
Then by Theorem 4.12 and Perrin’s noncommutative Davis decomposition (see
[17]), we get the atomic decomposition of $\mathsf{h}_{1}({\mathcal{M}})$ and
$\mathcal{H}_{1}({\mathcal{M}})$.
###### Corollary 4.15.
We have
$\mathcal{H}_{1}({\mathcal{M}})=\mathsf{h}_{1}({\mathcal{M}})=\mathsf{h}_{1,\mathrm{at}_{q,\mathrm{pr}}}({\mathcal{M}}),$
for any $1<q\leq\infty$, with equivalent norms.
However, using Corollary 3.18, we can obtain another kind of atomic
decomposition for $\mathsf{h}_{1}({\mathcal{M}})$ or
$\mathcal{H}_{1}({\mathcal{M}})$, which is exactly the noncommutative analogue
of the classical case.
###### Definition 4.16.
For $1<q\leq\infty$, $a\in L_{1}({\mathcal{M}})$ is said to be a $(1,q)$-atom
with respect to $({\mathcal{M}}_{n})_{n\geq 1}$, if there exist $n\geq 1$ and
a projection $e\in\mathcal{P}({\mathcal{M}}_{n})$ such that
1. (i)
${\mathcal{E}}_{n}(a)=0$;
2. (ii)
$r(a)\leq e$ or $l(a)\leq e$;
3. (iii)
$\|a\|_{q}\leq(\tau(e))^{-\frac{1}{q^{\prime}}}$.
###### Definition 4.17.
We define $\mathsf{h}^{\mathrm{at}}_{1,q}({\mathcal{M}})$ as the Banach space
of all $x\in L_{1}({\mathcal{M}})$ which admit a decomposition
$x=y+\sum_{k}\lambda_{k}a_{k}$, where for each $k$, $a_{k}$ is a $(1,q)$-atom
or an element in the unit ball of $L_{1}({\mathcal{M}}_{1})$,
$\lambda_{k}\in\mathbb{C}$ satisfying $\sum_{k}|\lambda_{k}|<\infty$, and
where the martingale differences of $y$ satisfy $\sum_{j\geq
1}\|dy_{j}\|_{1}<\infty$. We equip this space with the norm
$\|x\|_{\mathsf{h}^{\mathrm{at}}_{1,q}}=\inf\big{\\{}\sum_{j}\|dy_{j}\|_{1}+\sum_{k}|\lambda_{k}|\big{\\}},$
where the infimum is taken over all decompositions of $x$ as above.
###### Lemma 4.18.
If $a$ is a $(1,q)$-atom, then
$\|a\|_{\mathsf{h}_{1}}\leq\frac{cq}{q-1}.$
###### Proof.
Without loss of generality, suppose $a$ is a $(1,q)$-atom with $r(a)\leq e$.
We apply Corollary 3.18 and the duality
$(\mathsf{h}_{1}({\mathcal{M}}))^{*}=\mathsf{bmo}({\mathcal{M}})$.
$\displaystyle\|a\|_{\mathsf{h}_{1}}$ $\displaystyle\leq
c\sup_{\|x\|_{\mathsf{bmo}}\leq 1}\tau(x^{*}a)$
$\displaystyle=c\sup_{\|x\|_{\mathsf{bmo}}\leq 1}\tau((x-x_{n})^{*}a)$
$\displaystyle=c\sup_{\|x\|_{\mathsf{bmo}}\leq 1}\tau(((x-x_{n})e)^{*}a)$
$\displaystyle\leq c\|a\|_{q}\|(x-x_{n})e\|_{q^{\prime}}\leq cq^{\prime}.$
∎
###### Theorem 4.19.
For all $1<q\leq\infty$, we have
$\mathcal{H}_{1}({\mathcal{M}})=\mathsf{h}_{1}({\mathcal{M}})=\mathsf{h}^{\mathrm{at}}_{1,q}({\mathcal{M}})$
with equivalent norms.
By Lemma 4.18, Corollary 3.18 and using arguments similar to those in the
proof of Theorem 4.4, we can prove the theorem for the case $1<q<\infty$. For
the case $q=\infty$, we use the argument in Theorem 4.12. Instead of
$L^{\mathcal{P}}_{1}(\mathsf{bmo}^{c})$ and
$L^{\mathcal{P}}_{\infty}(\mathsf{h}^{c}_{1})$, we consider the following two
spaces:
$\displaystyle L^{\mathcal{P}}_{1}(L_{\infty})$
$\displaystyle=\big{\\{}(g_{e})_{e}\;:\;g_{e}e=g_{e}\,\mathrm{or}\,eg_{e}=g_{e},\mathcal{E}_{n_{e}}g_{e}=0,\,\|(g_{e})_{e}\|_{L^{\mathcal{P}}_{1}(L_{\infty})}<\infty\big{\\}},$
$\displaystyle L^{\mathcal{P}}_{\infty}(L_{1})$
$\displaystyle=\big{\\{}(f_{e})_{e}\;:\;f_{e}e=f_{e}\,\mathrm{or}\,ef_{e}=f_{e},\,\mathcal{E}_{n_{e}}f_{e}=0,\,\|(f_{e})_{e}\|_{L^{\mathcal{P}}_{\infty}(L_{1})}<\infty\big{\\}},$
where
$\displaystyle\|(g_{e})_{e}\|_{L^{\mathcal{P}}_{1}(L_{\infty})}$
$\displaystyle=\sum_{e}\tau(e)\,\|g_{e}\|_{{\infty}},$
$\displaystyle\|(f_{e})_{e}\|_{L^{\mathcal{P}}_{\infty}(L_{1})}$
$\displaystyle=\max\big{\\{}\sup_{e}\frac{1}{\tau(e)}\,\|f_{e}e\|_{1},\;\sup_{e}\frac{1}{\tau(e)}\,\|ef_{e}\|_{1}\big{\\}}.$
Then by Lemma 4.18 and Corollary 3.18, we get the announced results. We leave
the details to the reader.
###### Remark 4.20.
The part of this paper on the crude versions of the John-Nirenberg
inequalities and atomic decomposition can be easily extended to the type III
case with minor modifications.
## 5\. An open question of Junge and Musat
It is an open question asked in [8] (on page 136) that given $2<p<\infty$,
whether there exists a constant $c_{p}$ such that
(5.1)
$\sup_{k}\|\mathcal{E}_{k}|x-\mathcal{E}_{k-1}x|^{p}\|_{\infty}^{\frac{1}{p}}\leq
c_{p}\|x\|_{{\mathcal{BMO}}}?$
It is easy to see that the answer is negative for matrix-valued functions with
irregular filtration. In the following, we show that the answer is negative
even for matrix-valued dyadic martingales. Recall that Remark 3.14 already
shows that the answer is negative if one considers the column norm
$\|\cdot\|_{{\mathcal{BMO}}^{c}}$ alone on the right hand side.
Let ${\mathcal{M}}$ and ${\mathcal{M}}_{k}$ be as in Remark 3.14. We consider
this special case and show that the best constant $c_{p}(n)\;$such that (5.1)
holds is bigger than $c(\log(n+1))^{1/p}$ for all $p\geq 3.$ Let $b$ be an
$M_{n}$-valued function on ${\mathbb{T}}$. We need the so-called “sweep”
function of $b$
$S(b)=\sum_{k=1}^{\infty}|db_{k}|^{2}.$
Note that it is just the square of the usual square function. Matrix-valued
sweep functions have been studied in [2], [4], [13] etc. It is proved in [13]
that the best constant $c_{n}$ such that
(5.2) $\|S(b)\|_{{\mathcal{BMO}}^{c}}\leq c_{n}\|b\|_{\infty}^{2}$
is $c(\log(n+1))^{2}$. A similar result had been proved previously by Blasco
and Pott (see [2]) by considering $\|b\|_{{\mathcal{BMO}}^{c}}^{2}$ on the
right side of (5.2).
###### Lemma 5.1.
Assume $\|f\|_{{\mathcal{BMO}}^{c}}\leq
c(n)\sup_{k}\|\mathcal{E}_{k}|f-\mathcal{E}_{k-1}f\||_{\infty}$ for any
selfadjoint $f$. Then $c(n)\geq c(\log(n+1))^{2}.$
###### Proof.
Under the assumption, we have
$\displaystyle\|S(b)\|_{{\mathcal{BMO}}^{c}}$ $\displaystyle\leq
c(n)\sup_{m}\|\mathcal{E}_{m}|S(b)-\mathcal{E}_{m-1}S(b)|dt\|_{\infty}$
$\displaystyle=c(n)\sup_{m}\Big{\|}\mathcal{E}_{m}\big{|}\sum_{k=1}^{\infty}|db_{k}|^{2}-\mathcal{E}_{m-1}\sum_{k=1}^{\infty}|db_{k}|^{2}\big{|}\Big{\|}_{\infty}$
$\displaystyle=c(n)\sup_{m}\Big{\|}\mathcal{E}_{m}\big{|}\sum_{k=m}^{\infty}|db_{k}|^{2}-\mathcal{E}_{m-1}\sum_{k=m}^{\infty}|db_{k}|^{2}\big{|}\Big{\|}_{\infty}.$
Let $x=\sum_{k=m}^{\infty}|db_{k}|^{2}$ and
$y=\mathcal{E}_{m-1}\sum_{k=m}^{\infty}|db_{k}|^{2}$. By the convexity of
$|\cdot|^{2}$, we get
$\big{|}\frac{x-y}{2}\big{|}^{2}\leq\frac{|x|^{2}+|y|^{2}}{2}\leq\frac{|x|^{2}+\|y\|_{\infty}^{2}\mathds{1}}{2}\leq\frac{(|x|+\|y\|_{\infty}\mathds{1})^{2}}{2}.$
Then by Löwner-Heinz’s inequality,
$\big{|}\frac{x-y}{2}\big{|}\leq\frac{|x|+\|y\|_{\infty}\mathds{1}}{\sqrt{2}}.$
Thus by the triangle inequality, we have
$\displaystyle\|S(b)\|_{{\mathcal{BMO}}^{c}}$ $\displaystyle\leq
2c(n)\sup_{m}\left\|\mathcal{E}_{m}x+\|y\|_{\infty}\mathds{1}\right\|_{\infty}$
$\displaystyle=2c(n)\sup_{m}\|\mathcal{E}_{m}|b-\mathcal{E}_{m-1}b|^{2}\|_{\infty}+2c(n)\|\mathcal{E}_{m-1}|b-\mathcal{E}_{m-1}b|^{2}\|_{\infty}$
$\displaystyle\leq
2c(n)\|b\|_{{\mathcal{BMO}}^{c}}^{2}+2c(n)\|\mathcal{E}_{m}|b-\mathcal{E}_{m-1}b|^{2}\|_{\infty}$
$\displaystyle\leq 4c(n)\|b\|_{{\mathcal{BMO}}^{c}}^{2}.$
We then get $c(n)\geq c(\log(n+1))^{2}$ by (5.2). ∎
###### Lemma 5.2.
Let $0<p<\infty$ and $\mathcal{E}_{m}$ be the conditional expectation from
${{\mathcal{M}}}$ onto ${{\mathcal{M}}}_{m}$, we have
$\|\mathcal{E}_{m}|x|^{\frac{p+1}{2}}\|_{\infty}\leq\|\mathcal{E}_{m}|x|^{p}\|_{\infty}^{\frac{1}{2}}\|\mathcal{E}_{m}|x\||_{\infty}^{\frac{1}{2}}.$
###### Proof.
By Hölder’s inequality, we get
$\displaystyle\|\mathcal{E}_{m}|x|^{\frac{p+1}{2}}\|_{\infty}$
$\displaystyle=$ $\displaystyle\sup_{\|a\|_{L_{1}^{+}({{\mathcal{M}}}_{m})\leq
1}}\tau(\mathcal{E}_{m}|x|^{\frac{p+1}{2}}a)$ $\displaystyle=$
$\displaystyle\sup_{\|a\|_{L_{1}^{+}({{\mathcal{M}}}_{m})\leq
1}}\tau(a^{\frac{1}{2}}|x|^{\frac{p}{2}}|x|^{\frac{1}{2}}a^{\frac{1}{2}})$
$\displaystyle\leq$
$\displaystyle\sup_{\|a\|_{L_{1}^{+}({{\mathcal{M}}}_{m})\leq
1}}(\tau(a|x|^{p}))^{\frac{1}{2}}(\tau(a|x|))^{\frac{1}{2}}$ $\displaystyle=$
$\displaystyle\|\mathcal{E}_{m}|x|^{p}\|_{\infty}^{\frac{1}{2}}\|\mathcal{E}_{m}|x\||_{\infty}^{\frac{1}{2}}.$
∎
###### Theorem 5.3.
Suppose
$\sup_{k}\|\mathcal{E}_{k}|f-\mathcal{E}_{k-1}f|^{p}\|_{\infty}^{1/p}\leq
c_{p}(n)\|f\|_{{\mathcal{BMO}}}$ for some $p\geq 3.$ Then
$c_{p}(n)\geq c(\log(n+1))^{\frac{2}{p}}.$
###### Proof.
Fix a selfadjoint $M_{n}$-valued function $b.$ By the operator Jensen
inequality and Lemma 5.2, for $p\geq 3,$
$\displaystyle\|b\|_{{\mathcal{BMO}}}^{2}$ $\displaystyle=$
$\displaystyle\sup_{m}\|\mathcal{E}_{m}|b-\mathcal{E}_{m-1}b|^{2}\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\sup_{m}\|\mathcal{E}_{m}|b-\mathcal{E}_{m-1}b|^{\frac{p+1}{2}}\|_{\infty}^{\frac{4}{p+1}}$
$\displaystyle\leq$
$\displaystyle\sup_{m}\|\mathcal{E}_{m}|b-\mathcal{E}_{m-1}b|^{p}\|_{\infty}^{\frac{2}{p+1}}\sup_{m}\|\mathcal{E}_{m}|b-\mathcal{E}_{m-1}b\||_{\infty}^{\frac{2}{p+1}}$
$\displaystyle\leq$
$\displaystyle(c_{p}(n)\|b\|_{{\mathcal{BMO}}})^{\frac{2p}{p+1}}\sup_{m}\|\mathcal{E}_{m}|b-\mathcal{E}_{m-1}b\||_{\infty}^{\frac{2}{p+1}}.$
Then
$\|b\|_{{\mathcal{BMO}}}\leq(c_{p}(n))^{p}\sup_{m}\|\mathcal{E}_{m}|b-\mathcal{E}_{m-1}b\||_{\infty}.$
By Lemma 5.1, we get
$(c_{p}(n))^{p}\geq c(\log(n+1))^{2}.$
∎
From Theorem 5.3, we get a negative answer for the open question by letting
$n\rightarrow\infty.$
Acknowledgments. The authors are grateful to Quanhua Xu for his helpful
discussions which led to a big improvement of the paper.
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|
arxiv-papers
| 2011-12-14T12:31:34 |
2024-09-04T02:49:25.308103
|
{
"license": "Public Domain",
"authors": "Guixiang Hong and Tao Mei",
"submitter": "Guixiang Hong",
"url": "https://arxiv.org/abs/1112.3187"
}
|
1112.3417
|
arxiv-papers
| 2011-12-15T04:00:24 |
2024-09-04T02:49:25.345791
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jian-Jun Yao, Wenwei Ge, Yaodong Yang, Yanxi Li, Jiefang Li, Peter\n Finkel and D.Viehland",
"submitter": "Jianjun Yao",
"url": "https://arxiv.org/abs/1112.3417"
}
|
|
1112.3515
|
# Differential branching fraction and angular analysis of the decay
$\bm{B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}}$
(LHCb collaboration) R. Aaij23, C. Abellan Beteta35,n, B. Adeva36, M.
Adinolfi42, C. Adrover6, A. Affolder48, Z. Ajaltouni5, J. Albrecht37, F.
Alessio37, M. Alexander47, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves
Jr22, S. Amato2, Y. Amhis38, J. Anderson39, R.B. Appleby50, O. Aquines
Gutierrez10, F. Archilli18,37, L. Arrabito53, A. Artamonov 34, M. Artuso52,37,
E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back44, D.S. Bailey50, V.
Balagura30,37, W. Baldini16, R.J. Barlow50, C. Barschel37, S. Barsuk7, W.
Barter43, A. Bates47, C. Bauer10, Th. Bauer23, A. Bay38, I. Bediaga1, S.
Belogurov30, K. Belous34, I. Belyaev30,37, E. Ben-Haim8, M. Benayoun8, G.
Bencivenni18, S. Benson46, J. Benton42, R. Bernet39, M.-O. Bettler17, M. van
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Bjørnstad50, T. Blake37, F. Blanc38, C. Blanks49, J. Blouw11, S. Blusk52, A.
Bobrov33, V. Bocci22, A. Bondar33, N. Bondar29, W. Bonivento15, S.
Borghi47,50, A. Borgia52, T.J.V. Bowcock48, C. Bozzi16, T. Brambach9, J. van
den Brand24, J. Bressieux38, D. Brett50, M. Britsch10, T. Britton52, N.H.
Brook42, H. Brown48, A. Büchler-Germann39, I. Burducea28, A. Bursche39, J.
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Camboni35, P. Campana18,37, A. Carbone14, G. Carboni21,k, R. Cardinale19,i,37,
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Cauet9, M. Charles51, Ph. Charpentier37, N. Chiapolini39, K. Ciba37, X. Cid
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M.M. Reid44, A.C. dos Reis1, S. Ricciardi45, K. Rinnert48, D.A. Roa Romero5,
P. Robbe7, E. Rodrigues47,50, F. Rodrigues2, P. Rodriguez Perez36, G.J.
Rogers43, S. Roiser37, V. Romanovsky34, M. Rosello35,n, J. Rouvinet38, T.
Ruf37, H. Ruiz35, G. Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P.
Sail47, B. Saitta15,d, C. Salzmann39, M. Sannino19,i, R. Santacesaria22, C.
Santamarina Rios36, R. Santinelli37, E. Santovetti21,k, M. Sapunov6, A.
Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina30, P.
Schaack49, M. Schiller24, S. Schleich9, M. Schlupp9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B.
Sciascia18, A. Sciubba18,l, M. Seco36, A. Semennikov30, K. Senderowska26, I.
Sepp49, N. Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval40,37,
P. Shatalov30, Y. Shcheglov29, T. Shears48, L. Shekhtman33, O. Shevchenko40,
V. Shevchenko30, A. Shires49, R. Silva Coutinho44, T. Skwarnicki52, A.C.
Smith37, N.A. Smith48, E. Smith51,45, K. Sobczak5, F.J.P. Soler47, A.
Solomin42, F. Soomro18, B. Souza De Paula2, B. Spaan9, A. Sparkes46, P.
Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S.
Stone52,37, B. Storaci23, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, S.
Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26, S. T’Jampens4, E.
Teodorescu28, F. Teubert37, C. Thomas51, E. Thomas37, J. van Tilburg11, V.
Tisserand4, M. Tobin39, S. Topp-Joergensen51, N. Torr51, E. Tournefier4,49,
M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23, M. Ubeda Garcia37, A. Ukleja27,
P. Urquijo52, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez35, P.
Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis42, M. Veltri17,g, B. Viaud7, I.
Videau7, X. Vilasis-Cardona35,n, J. Visniakov36, A. Vollhardt39, D.
Volyanskyy10, D. Voong42, A. Vorobyev29, H. Voss10, S. Wandernoth11, J.
Wang52, D.R. Ward43, N.K. Watson55, A.D. Webber50, D. Websdale49, M.
Whitehead44, D. Wiedner11, L. Wiggers23, G. Wilkinson51, M.P. Williams44,45,
M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25, W. Witzeling37, S.A.
Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Xing52, Z. Yang3, R. Young46,
O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3, L. Zhang52, W.C. Zhang12, Y.
Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin37.
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
23Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraców, Poland
26AGH University of Science and Technology, Kraców, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55University of Birmingham, Birmingham, United Kingdom
56Physikalisches 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
###### Abstract
The angular distributions and the partial branching fraction of the decay
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ are studied using an integrated
luminosity of $0.37\mbox{\,fb}^{-1}$ of data collected with the LHCb detector.
The forward-backward asymmetry of the muons, $A_{\mathrm{FB}}$, the fraction
of longitudinal polarisation, $F_{\mathrm{L}}$, and the partial branching
fraction, $\mathrm{d}{\cal B}/\mathrm{d}q^{2}$, are determined as a function
of the dimuon invariant mass. The measurements are in good agreement with the
Standard Model predictions and are the most precise to date. In the dimuon
invariant mass squared range
$1.00-6.00\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$, the results are
$A_{\mathrm{FB}}=-0.06\,^{+0.13}_{-0.14}\pm 0.04$, $F_{\mathrm{L}}=0.55\pm
0.10\pm 0.03$ and $\mathrm{d}{\cal B}/\mathrm{d}q^{2}=(0.42\pm 0.06\pm
0.03)\times 10^{-7}c^{4}/\mathrm{\,Ge\kern-1.00006ptV}^{2}$. In each case, the
first error is statistical and the second systematic.
_Published in Physical Review Letters 108, 181806 (2012)_
###### pacs:
11.30.Fs, 13.20.He, 13.35.Hb
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
| |
---|---|---
| | LHCb-PAPER-2011-020
| | CERN-PH-EP-2011-211
The process $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ is a flavour changing
neutral current decay. In the Standard Model (SM) such decays are suppressed,
as they can only proceed via loop processes involving electroweak penguin or
box diagrams. As-yet undiscovered particles could give additional
contributions with comparable amplitudes, and the decay is therefore a
sensitive probe of new phenomena. A number of angular observables in
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ decays can be theoretically
predicted with good control of the relevant form factor uncertainties. These
include the forward-backward asymmetry of the muons, $A_{\mathrm{FB}}$, and
the fraction of longitudinal polarisation, $F_{\mathrm{L}}$, as functions of
the dimuon invariant mass squared, $q^{2}$ Kruger:1999xa . These observables
have previously been measured by the BaBar, Belle, and CDF experiments
Aubert:2008ju ; *PhysRevLett.103.171801; *Aaltonen:2011ja. A more precise
determination of $A_{\mathrm{FB}}$ is of particular interest as, in the
$1.00<q^{2}<6.00\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ region, previous
measurements favour an asymmetry with the opposite sign to that expected in
the SM. If confirmed, this would be an unequivocal sign of phenomena not
described by the SM. This letter presents the most precise measurements of
$A_{\mathrm{FB}}$, $F_{\mathrm{L}}$ and the partial branching fraction,
$\mathrm{d}{\cal B}/\mathrm{d}q^{2}$, to date. The data used for this analysis
were taken with the LHCb detector at CERN during 2011 and correspond to an
integrated luminosity of $0.37\mbox{\,fb}^{-1}$. The $K^{*0}$ is reconstructed
through its decay into the $K^{+}\pi^{-}$ final state.
The LHCb detector Alves:2008zz is a single-arm spectrometer designed to study
$b$-hadron decays. A silicon strip vertex detector positioned around the
interaction region is used to measure the trajectory of charged particles and
allows the reconstruction of the primary proton-proton interactions and the
displaced secondary vertices characteristic of $B$-meson decays. A dipole
magnetic field and further charged particle tracking stations allow momenta in
the range $5<p<100{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to be determined with
a precision of $\delta p/p=0.4$–$0.6\%$. The experiment has an acceptance for
charged particles with pseudorapidity between 2 and 5. Two ring imaging
Cherenkov (RICH) detectors allow kaons to be separated from pions or muons
over a momentum range $2<p<100{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Muons are
identified on the basis of the number of hits in detectors interleaved with an
iron muon filter.
The $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ angular distribution is
governed by six $q^{2}$-dependent transversity amplitudes. The decay can be
described by $q^{2}$ and the three angles $\theta_{l},~{}\theta_{K},~{}\phi$.
For the $B^{0}$ ($\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$),
$\theta_{l}$ is the angle between the $\mu^{+}$ ($\mu^{-}$) and the opposite
of the $B^{0}$ ($\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) direction
in the dimuon rest frame, $\theta_{K}$ the angle between the kaon and the
direction opposite to the $B$ meson in the $K^{*0}$ rest frame, and $\phi$ the
angle between the $\mu^{+}\mu^{-}$ and $K^{+}\pi^{-}$ decay planes in the $B$
rest frame. The inclusion of charge conjugate modes is implied throughout this
letter. At a given $q^{2}$, neglecting the muon mass, the normalised partial
differential width integrated over $\theta_{K}$ and $\phi$ is
$\displaystyle\frac{1}{\Gamma}\frac{\mathrm{d}^{2}\Gamma}{\mathrm{d}\cos\theta_{l}\,\mathrm{d}q^{2}}$
$\displaystyle=$
$\displaystyle\frac{3}{4}{F_{\mathrm{L}}}(1-\cos^{2}\theta_{l})+$ (1)
$\displaystyle\frac{3}{8}(1-{F_{\mathrm{L}}})(1+\cos^{2}\theta_{l})+{A_{\mathrm{FB}}}\cos\theta_{l}$
and integrated over $\theta_{l}$ and $\phi$ it is
$\displaystyle\frac{1}{\Gamma}\frac{\mathrm{d}^{2}\Gamma}{\mathrm{d}\cos\theta_{K}\,\mathrm{d}q^{2}}$
$\displaystyle=$ $\displaystyle\frac{3}{2}{F_{\mathrm{L}}}\cos^{2}\theta_{K}+$
(2) $\displaystyle\frac{3}{4}(1-{F_{\mathrm{L}}})(1-\cos^{2}\theta_{K}).$
These expressions do not include any broad S-wave contribution to the
$B^{0}\rightarrow K^{+}\pi^{-}\mu^{+}\mu^{-}$ decay and any contribution from
low mass tails of higher $K^{*0}$ resonances. These contributions are assumed
to be small and are neglected in the rest of the analysis.
Signal candidates are isolated from the background using a set of selection
criteria which are detailed below. An event-by-event weight is then used to
correct for the bias induced by the reconstruction, trigger and selection
criteria. In order to extract $A_{\mathrm{FB}}$ and $F_{\mathrm{L}}$,
simultaneous fits are made to the $K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass
distribution and the angular distributions. The partial branching fraction is
measured by comparing the efficiency corrected yield of $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ decays to the yield of
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$, where
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$.
Candidate $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ events are first required
to pass a hardware trigger which selects muons with a transverse momentum,
$\mbox{$p_{\rm T}$}>1.48{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In the
subsequent software trigger, at least one of the final state particles is
required to have both $\mbox{$p_{\rm
T}$}>0.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and impact parameter
$>100~{}\mu$m with respect to all of the primary proton-proton interaction
vertices in the event Gligorov:1300771 . Finally, the tracks of two or more of
the final state particles are required to form a vertex which is significantly
displaced from the primary vertices in the event hlt2toponote .
In the final event selection, candidates with $K^{+}\pi^{-}\mu^{+}\mu^{-}$
invariant mass in the range
$5100<m_{K^{+}\pi^{-}\mu^{+}\mu^{-}}<5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
and $K^{+}\pi^{-}$ invariant mass in the range
$792<m_{K^{+}\pi^{-}}<992{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are
accepted. Two types of backgrounds are then considered: combinatorial
backgrounds, where the particles selected do not come from a single $b$-hadron
decay; and peaking backgrounds, where a single decay is selected but with some
of the particle types mis-identified. In addition, the decays
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
$B^{0}\\!\rightarrow\psi(2S)K^{*0}$, where
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu},\psi(2S)\rightarrow\mu^{+}\mu^{-}$, are removed by rejecting events
with dimuon invariant mass, $m_{\mu^{+}\mu^{-}}$, in the range
$2946<m_{\mu^{+}\mu^{-}}<3176\mathrm{\,Me\kern-1.00006ptV}/c^{2}$ or
$3586<m_{\mu^{+}\mu^{-}}<3776\mathrm{\,Me\kern-1.00006ptV}/c^{2}$.
The combinatorial background, which is smoothly distributed in the
reconstructed $K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass, is reduced using a
Boosted Decision Tree (BDT). The BDT uses information about the event
kinematics, vertex and track quality, impact parameter and particle
identification information from the RICH and muon detectors. The variables
that are used in the BDT are chosen so as to induce the minimum possible
distortion in the angular and $q^{2}$ distributions. For example, no
additional requirement is made on the $p_{\rm T}$ of both of the muons as, at
low $q^{2}$, this would remove a large proportion of events with
$|\cos{\theta_{l}}|\sim 1$. The BDT is trained entirely on data, using samples
that are independent of that which is used to make the measurements: triggered
and fully reconstructed
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ events
are used as a proxy for the signal decay, and events from the upper
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ mass sideband
($5350<m_{K^{+}\pi^{-}\mu^{+}\mu^{-}}<5600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
are used as a background sample. The lower mass sideband is not used, as it
contains background events formed from partially reconstructed $B$ decays.
These events make a negligible contribution in the signal region and have
properties different from the combinatorial background which is the dominant
background in this region.
A cut is made on the BDT output in order to optimise the sensitivity to
$A_{\mathrm{FB}}$ averaged over all $q^{2}$. The selected sample has a signal-
to-background ratio of three to one.
Peaking backgrounds from $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ (where
$\phi\rightarrow K^{+}K^{-}$),
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
$B^{0}\\!\rightarrow\psi(2S)K^{*0}$ are considered and reduced with a set of
vetoes. In each case, for the decay to be a potential signal candidate, at
least one particle needs to be misidentified. For example,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ events
where a kaon or pion is swapped for one of the muons, peak around the nominal
$B^{0}$ mass and evade the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ veto
described above. Vetoes for each of these backgrounds are formed by changing
the relevant particle mass hypotheses and recomputing the invariant masses,
and by making use of the particle identification information. In order to
avoid having a strongly peaking contribution to the $\cos{\theta_{K}}$ angular
distribution in the upper mass sideband, $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ candidates are removed. Events with $K^{+}\mu^{+}\mu^{-}$
invariant mass within $60{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the
nominal $B^{+}$ mass are rejected. The vetoes for all of these peaking
backgrounds remove a negligible amount of signal.
After the application of the BDT cut and the above vetoes, a fit is made to
the $K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass distribution in the entire
accepted mass range (see Fig. 1). A double-Gaussian distribution is used for
the signal mass shape and an exponential function for the background. The
signal shape is fixed from data using a fit to the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ mass
peak. In the full $q^{2}$ range, in a signal mass window of $\pm
50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ($\pm 2.5\sigma$) around the
measured $B^{0}$ mass, the fit gives an estimate of $337\pm 21$ signal events
with a background of $97\pm 6$ events.
Figure 1: $K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass distribution after the
application of the full selection as data points with the fit overlaid. The
signal component is the green (light) line, the background the red (dashed)
line and the full distribution the blue (dark) line.
The residual peaking background is estimated using simulated events. As
detailed below, the accuracy of the simulation is verified by comparing the
particle (mis-) identification probabilities with those derived from control
channels selected from the data. The residual peaking backgrounds are reduced
to a level of $6.1$ events, i.e. $1.8\%$ of the 337 observed signal events.
The backgrounds from $B^{0}_{s}\\!\rightarrow\phi\mu^{+}\mu^{-}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ decays
do not give rise to any forward-backward asymmetry and are ignored. However,
in addition to the above backgrounds, $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ decays with the kaon and pion swapped give rise to a
0.7% contribution. The change in the sign of the particle which is taken to be
the kaon results in a $B^{0}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) being reconstructed as a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ ($B^{0}$), therefore changing the
sign of $A_{\mathrm{FB}}$ for the candidate. This misidentification is
accounted for in the fit for the angular observables.
The selected $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$ candidates are
weighted in order to correct for the effects of the reconstruction, trigger
and selection. The weights are derived from simulated $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$ events and are normalised such that the average weight
is one. In order to be independent of the physics model used in the
simulation, the weights are computed based on $\cos{\theta_{K}}$,
$\cos{\theta_{l}}$ and $q^{2}$ on an event-by-event basis. The variation of
detector efficiency with the $\phi$ angle is small and ignoring this variation
does not bias the measurements. Only events with
$0.10<q^{2}<19.00\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ are analysed.
Owing to the relatively unbiased selection, 89% of events have weights between
0.7 and 1.3, and only 3% of events have a weight above 2. The distortions in
the distributions of $\cos{\theta_{K}}$, $\cos{\theta_{l}}$ and $q^{2}$ that
are induced originate from two main sources. Firstly, in order to pass through
the iron muon filter and give hits in the muon stations, tracks must have at
least 3${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ momentum. At low $q^{2}$ this
removes events with $|\cos{\theta_{l}}|\sim 1$. This effect stems from the
geometry of the LHCb detector and is therefore relatively easy to model.
Secondly, events with $\cos{\theta_{K}}\sim 1$, and hence a slow pion, are
removed both by the pion reconstruction and by the impact parameter
requirements used in the trigger and BDT selection.
A number of control samples are used to verify the simulation quality and to
correct for differences with respect to the data. The reproduction of the
$B^{0}$ momentum and pseudorapidity distributions is verified using
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
decays. These decays are also used to check that the simulation reproduces the
measured properties of selected events. The hadron and muon
(mis-)identification probabilities are adjusted using decays where the tested
particle type can be determined without the use of the particle identification
algorithms. A tag and probe approach with
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ decays
is used to isolate a clean sample of genuine muons. The decay
$D^{*+}\rightarrow D^{0}\pi^{+}$, where $D^{0}\rightarrow K^{-}\pi^{+}$, is
used to give an unambiguous source of kaons and pions. The statistical
precision with which it is possible to make the data/simulation comparison
gives rise to a systematic uncertainty in the weights which is evaluated
below.
The observables $A_{\mathrm{FB}}$ and $F_{\mathrm{L}}$ are extracted in bins
of $q^{2}$. In each bin, a simultaneous fit to the
$K^{+}\pi^{-}\mu^{+}\mu^{-}$ invariant mass distribution and the
$\cos{\theta_{K}}$ and $\cos{\theta_{l}}$ distributions is performed. The
angular distributions are fitted in both the signal mass window and in the
upper mass sideband which determines the background parameters. The angular
distributions for the signal are given by Eqs. 1 and 2 and a second order
polynomial in $\cos{\theta_{K}}$ and in $\cos{\theta_{l}}$ is used for the
background.
In order to obtain a positive probability density function over the entire
angular range, Eqs. 1 and 2 imply that the conditions
$\left|A_{\mathrm{FB}}\right|\leq\frac{3}{4}(1-F_{\mathrm{L}})$ and
$0<F_{\mathrm{L}}<1$ must be satisfied. To account for this, the maximum
likelihood values for $A_{\mathrm{FB}}$ and $F_{\mathrm{L}}$ are extracted by
performing a profile-likelihood scan over the allowed range. The uncertainty
on the central value of $A_{\mathrm{FB}}$ and $F_{\mathrm{L}}$ is calculated
by integrating the probability density extracted from the likelihood, assuming
a flat prior in $A_{\mathrm{FB}}$ and $F_{\mathrm{L}}$, inside the allowed
range. This gives an (asymmetric) 68% confidence interval.
The partial branching fraction is measured in each of the $q^{2}$ bins from a
fit to the efficiency corrected $K^{+}\pi^{-}\mu^{+}\mu^{-}$ mass spectrum.
The efficiencies are determined relative to the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ decay
which is used as a normalisation mode.
The event weighting and fitting procedure is validated by fitting the angular
distribution of $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ events, where the physics parameters are known from previous
measurements Aubert:2007hz . The product of the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$
branching fractions is $\sim 75$ times larger than the branching fraction of
$B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$, allowing a precise test of the
procedure to be made. Fitting the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
angular distribution, weighted according to the event-by-event procedure
described above, yields values for $F_{\mathrm{L}}$ and $A_{\mathrm{FB}}$ in
good agreement with those found previously.
For $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$, the fit results for
$A_{\mathrm{FB}}$, $F_{\mathrm{L}}$ and $\mathrm{d}{\cal B}/\mathrm{d}q^{2}$
are shown in Fig. 2 and are tabulated together with the signal and background
yields in Table 1. The fit projections are given in the appendix. Signal
candidates are observed in each $q^{2}$ bin with more than $5\sigma$
significance. The compatibility of the fits and the data are assessed using a
binned $\chi^{2}$ test and all fits are found to be of good quality. The
measurements in all three quantities are more precise than those of previous
experiments and are in good agreement with the SM predictions. The predictions
are taken from Ref. Bobeth:2011gi . In the low $q^{2}$ region they rely on the
factorisation approach Beneke:2001at , which loses accuracy when approaching
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ resonance; in the high
$q^{2}$ region, an operator product expansion in the inverse $b$-quark mass,
$1/m_{b}$, and in $1/\sqrt{q^{2}}$ is used Grinstein:2004vb , which is only
valid above the open charm threshold. In both regions the form factor
calculations are taken from Ref. Ball:2004rg and a dimensional estimate is
made on the uncertainty from expansion corrections Egede:2008uy .
Figure 2: $A_{\mathrm{FB}}$, $F_{\mathrm{L}}$ and $\mathrm{d}{\cal
B}/\mathrm{d}q^{2}$ as a function of $q^{2}$. The SM prediction is given by
the cyan (light) band, and this prediction rate-averaged across the $q^{2}$
bins is indicated by the purple (dark) regions. No SM prediction is shown for
the region between the two regimes in which the theoretical calculations are
made (see text).
In the $1.00<q^{2}<6.00\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ region, the
fit gives $A_{\mathrm{FB}}=-0.06\,^{+0.13}_{-0.14}\pm 0.04$,
$F_{\mathrm{L}}=0.55\pm 0.10\pm 0.03$ and $\mathrm{d}{\cal
B}/\mathrm{d}q^{2}=(0.42\pm 0.06\pm 0.03)\times
10^{-7}c^{4}/\mathrm{\,Ge\kern-1.00006ptV}^{2}$, where the first error is
statistical and the second systematic. The theoretical predictions in the same
$q^{2}$ range are $A_{\mathrm{FB}}=-0.04\pm 0.03$,
$F_{\mathrm{L}}=0.74\,^{+0.06}_{-0.07}$ and $\mathrm{d}{\cal
B}/\mathrm{d}q^{2}=(0.50\,^{+0.11}_{-0.10})\times
10^{-7}c^{4}/\mathrm{\,Ge\kern-1.00006ptV}^{2}$. The LHCb $A_{\mathrm{FB}}$
measurement is a factor $1.5-2.0$ more precise than previous measurements from
the Belle, CDF and BaBar collaborations Aubert:2008ju ;
*PhysRevLett.103.171801; *Aaltonen:2011ja which are, respectively,
$A_{\mathrm{FB}}=0.26^{+0.27}_{-0.30}\pm 0.07$,
$A_{\mathrm{FB}}=0.29^{+0.20}_{-0.23}\pm 0.07$ and, for
$q^{2}<6.25\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$,
$A_{\mathrm{FB}}=0.24^{+0.18}_{-0.23}\pm 0.05$. The positive value of
$A_{\mathrm{FB}}$ preferred in the
$1.00<q^{2}<6.00\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$ range in these
previous measurements is not favoured by the LHCb data. The previous
measurements of $F_{\mathrm{L}}$ in the same $q^{2}$ regions are
$F_{\mathrm{L}}=0.67\pm 0.23\pm 0.05$ (Belle),
$F_{\mathrm{L}}=0.69^{+0.19}_{-0.21}\pm 0.08$ (CDF) and
$F_{\mathrm{L}}=0.35\pm 0.16\pm 0.04$ (BaBar). These are in good agreement
with the LHCb result.
Table 1: Central values with statistical and systematic uncertainties for
$A_{\mathrm{FB}}$, $F_{\mathrm{L}}$ and $\mathrm{d}{\cal B}/\mathrm{d}q^{2}$
as a function of $q^{2}$. The $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$
signal and background yields in the $\pm
50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ signal mass window with their
statistical uncertainties are also indicated, together with the statistical
significance of the signal peak that is observed.
$q^{2}$ | $A_{\mathrm{FB}}$ | $F_{\mathrm{L}}$ | $\mathrm{d}{\cal B}/\mathrm{d}q^{2}$ | Signal | Background | Significance
---|---|---|---|---|---|---
$(\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4})$ | | | $(\times 10^{-7}~{}c^{4}/\mathrm{\,Ge\kern-1.00006ptV}^{2})$ | yield | yield | ($\sigma$)
$0.10<q^{2}<2.00$ | $-0.15\pm 0.20\pm 0.06$ | $0.00~\,{}^{+\,0.13}_{-\,0.00}\,\pm 0.02$ | $0.61\pm 0.12\pm 0.06$ | $48.6\pm 8.1$ | $16.2\pm 2.3\phantom{0}$ | 08.6
$2.00<q^{2}<4.30$ | $\phantom{-}0.05~\,{}^{+\,0.16}_{-\,0.20}\,\pm 0.04$ | $0.77\pm 0.15\pm 0.03$ | $0.34\pm 0.09\pm 0.02$ | $26.5\pm 6.5$ | $15.7\pm 2.2\phantom{0}$ | 05.4
$4.30<q^{2}<8.68$ | $\phantom{-}0.27~\,{}^{+\,0.06}_{-\,0.08}\,\pm 0.02$ | $0.60~\,{}^{+\,0.06}_{-\,0.07}\,\pm 0.01$ | $0.69\pm 0.08\pm 0.05$ | $104.7\pm 11.9$ | $31.7\pm 3.3\phantom{0}$ | 12.4
$10.09<q^{2}<12.86$ | $\phantom{-}0.27~\,{}^{+\,0.11}_{-\,0.13}\,\pm 0.02$ | $0.41\pm 0.11\pm 0.03$ | $0.55\pm 0.09\pm 0.07$ | $62.2\pm 9.2$ | $20.4\pm 2.6\phantom{0}$ | 09.6
$14.18<q^{2}<16.00$ | $\phantom{-}0.47~\,{}^{+\,0.06}_{-\,0.08}\,\pm 0.03$ | $0.37\pm 0.09\pm 0.05$ | $0.63\pm 0.11\pm 0.05$ | $44.2\pm 7.0$ | $4.2\pm 1.3$ | 10.2
$16.00<q^{2}<19.00$ | $\phantom{-}0.16~\,{}^{+\,0.11}_{-\,0.13}\,\pm 0.06$ | $0.26~\,{}^{+\,0.10}_{-\,0.08}\,\pm 0.03$ | $0.50\pm 0.08\pm 0.05$ | $53.4\pm 8.1$ | $7.0\pm 1.7$ | 09.8
$1.00<q^{2}<6.00$ | $-0.06~\,{}^{+\,0.13}_{-\,0.14}\,\pm 0.04$ | $0.55\pm 0.10\pm 0.03$ | $0.42\pm 0.06\pm 0.03$ | $\phantom{0}76.5\pm 10.6$ | $33.1\pm 3.2\phantom{0}$ | 09.9
For the determination of $A_{\mathrm{FB}}$ and $F_{\mathrm{L}}$, the dominant
systematic uncertainties arise from the event-by-event weights which are
extracted from simulated events, and from the model used to describe the
angular distribution of the background. The uncertainty on the event-by-event
weights is evaluated by fluctuating these weights within their statistical
uncertainties and repeating the fitting procedure. The uncertainty from the
background model which is used is estimated by changing this model to one
which uses binned templates from the upper mass sideband rather than a
polynomial parameterisation.
The dominant systematic errors for the determination of $\mathrm{d}{\cal
B}/\mathrm{d}q^{2}$ arise from the uncertainties on the particle
identification and track reconstruction efficiencies. These efficiencies are
extracted from control channels and are limited by the relevant sample sizes.
The systematic uncertainty is estimated by fluctuating the efficiencies within
the relevant uncertainties and repeating the fitting procedure. An additional
systematic uncertainty of $\sim 4\%$ arises from the uncertainty in the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$
branching fractions Nakamura:2010zzi .
The total systematic error on each of $A_{\mathrm{FB}}$ and $F_{\mathrm{L}}$
($\mathrm{d}{\cal B}/\mathrm{d}q^{2}$) is typically $\sim 30\%$ (50%) of the
statistical error, and hence adds $\sim 4\%$ ($\sim 11\%$) to the total
uncertainty.
In summary, using $0.37\mbox{\,fb}^{-1}$ of data taken with the LHCb detector
during 2011, $A_{\mathrm{FB}}$, $F_{\mathrm{L}}$ and $\mathrm{d}{\cal
B}/\mathrm{d}q^{2}$ have been determined for the decay $B^{0}\\!\rightarrow
K^{*0}\mu^{+}\mu^{-}$. These are the most precise measurements of these
quantities to-date. All three observables show good agreement with the SM
predictions.
## 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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Appendix
The following fit projections are published as EPAPS material.
Figure 3: Fit projections for $m_{K\pi\mu\mu}$, $\cos{\theta_{l}}$ and
$\cos{\theta_{K}}$ for the $q^{2}$ bins: $0.10<q^{2}<2.00$, $2.00<q^{2}<4.30$
and $4.30<q^{2}<8.68\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$.
Figure 4: Fit projections for $m_{K\pi\mu\mu}$, $\cos{\theta_{l}}$ and
$\cos{\theta_{K}}$ for the $q^{2}$ bins: $10.09<q^{2}<12.86$,
$14.18<q^{2}<16.00$ and
$16.00<q^{2}<19.00\mathrm{\,Ge\kern-1.00006ptV}^{2}/c^{4}$.
|
arxiv-papers
| 2011-12-15T14:16:58 |
2024-09-04T02:49:25.354838
|
{
"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 G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis, J.\n Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, D.\n S. Bailey, 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. Blouw,\n S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi,\n A. 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, F.\n Constantin, A. Contu, A. Cook, M. Coombes, G. Corti, 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, F. De Lorenzi, J. M. De Miranda, L. De Paula, P. De Simone, D. Decamp,\n M. 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, L. Est\\`eve, A. Falabella, E.\n Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, 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, N. Gauvin, M.\n Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, G. Haefeli, C.\n Haen, S. 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, P. Koppenburg, A. Kozlinskiy, L. Kravchuk, K. Kreplin,\n M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, T.\n Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles,\n R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E. Lopez\n Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I.\n V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R. M. D. Mamunur,\n G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, 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, R. Messi, S. Miglioranzi, D. A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Nedos, M.\n Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora Goicochea, P.\n Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis,\n M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M.\n Patel, S. K. Paterson, G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A.\n Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.\n L. Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrella, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R. Plackett, S.\n Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B.\n Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig\n Navarro, W. Qian, J. H. Rademacker, B. Rakotomiaramanana, M. S. Rangel, I.\n Raniuk, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, K.\n Rinnert, D. A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez\n Perez, G. J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T.\n Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, C. Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R.\n Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M.\n Savrie, D. Savrina, P. Schaack, M. Schiller, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, A. C. Smith, N. A.\n Smith, E. Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F. Soomro, B. Souza\n 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. Topp-Joergensen, N. Torr, E. Tournefier, M. T.\n Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, P. Urquijo,\n U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S.\n Vecchi, J. J. Velthuis, M. Veltri, B. Viaud, I. Videau, X. Vilasis-Cardona,\n J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, H. Voss, 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. Zavertyaev, F.\n Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, E. Zverev, A.\n Zvyagin",
"submitter": "Mitesh Patel",
"url": "https://arxiv.org/abs/1112.3515"
}
|
1112.3544
|
arxiv-papers
| 2011-12-15T15:51:32 |
2024-09-04T02:49:25.363334
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Abner C.-Y. Huang, Leslie Y Chen, Kuo-Chen Wei, Kai Wang, Chiung-Yin\n Huang, Danielle Yi, Chuan Yi Tang, David J. Galas, Leroy E. Hood",
"submitter": "Abner Chih Yi Huang",
"url": "https://arxiv.org/abs/1112.3544"
}
|
|
1112.3703
|
# Some generalizations of Calabi compactness theorem
Bruno Bianchini Luciano Mari Marco Rigoli
###### Abstract
111Mathematic subject classification 2010: primary 53C20; secondary 34C10.
Keywords: compactness, Myers’ type theorems, oscillation, positioning of
zeros.
In this paper we obtain generalized Calabi-type compactness criteria for
complete Riemannian manifolds that allow the presence of negative amounts of
Ricci curvature. These, in turn, can be rephrased as new conditions for the
positivity, for the existence of a first zero and for the nonoscillatory-
oscillatory behaviour of a solution $g(t)$ of $g^{\prime\prime}+Kg=0$,
subjected to the initial condition $g(0)=0$, $g^{\prime}(0)=1$. A unified
approach for this ODE, based on the notion of critical curve, is presented.
With the aid of suitable examples, we show that our new criteria are sharp
and, even for $K\geq 0$, in borderline cases they improve on previous works of
Calabi, Hille-Nehari and Moore.
Fortaleza
dedicated to Gervasio Colares for his $80^{\mathrm{th}}$ birthday
$\begin{array}[]{cc}\begin{array}[]{c}\text{Dipartimento di Matematica Pura e
Applicata}\\\ \text{Universit\\`{a} degli Studi di Padova}\\\ \text{Via
Trieste 63}\\\ \text{I-35121 Padova, ITALY}\\\ \text{e-mail:
bianchini@dmsa.unipd.it}\end{array}&\qquad\begin{array}[]{c}\text{Dipartimento
di Matematica}\\\ \text{Universit\\`{a} degli Studi di Milano}\\\ \text{Via
Saldini 50}\\\ \text{I-20133 Milano, ITALY}\\\ \text{e.mail:
lucio.mari@libero.it, marco.rigoli@unimi.it}\end{array}\end{array}$
## 1 Basic comparison and Myers type compactness result
Hereafter, we consider a connected, complete Riemannian manifold
$(M,\langle\,,\,\rangle)$, and a chosen reference origin $o\in M$. Let
$D_{o}=M\backslash(\left\\{o\right\\}\cup\mathrm{cut}(o))$ be the maximal
domain of normal coordinates centered at $o$, and denote with $r(x)$ the
distance function from $o$. The classical Bonnet-Myers theorem, showing the
compactness of $M$ under the condition
$\mathrm{Ricc}\geq(m-1)B^{2}\langle\,,\,\rangle$ (1.1)
for some $B>0$, can be proved as a consequence of the Laplacian comparison
theorem. Indeed, let us recall the following generalized form of this latter.
###### Theorem 1 (Theorem 2.4 of [16]).
Let $M$ be as above. Assume that the radial Ricci curvature satisfies
$\mathrm{Ricc}(\nabla r,\nabla r)(x)\geq-(m-1)G(r(x))\qquad\text{on }\ M,$
(1.2)
for some function $G\in C^{0}(\mathbb{R}^{+}_{0})$, and let $g\in
C^{2}(\mathbb{R}^{+}_{0})$ be a solution of
$\left\\{\begin{array}[]{l}g^{\prime\prime}-Gg\geq 0\\\\[5.69046pt]
g(0)=0,\quad g^{\prime}(0)=1.\end{array}\right.$ (1.3)
Let $(0,R_{0})$ (possibly $R_{0}=+\infty$) be the maximal interval where $g$
is positive. Then,
$D_{o}\subset B_{R_{0}}$ (1.4)
and the inequality
$\Delta r(x)\leq(m-1)\frac{g^{\prime}(r(x))}{g(r(x))}$ (1.5)
holds pointwise on $D_{o}$ and weakly on $M$.
Suppose the validity of (1.1) so that $G(t)=-B^{2}$. A simple checking shows
that $g(t)=B^{-1}\sin(Bt)$ solves (1.3). Its first positive zero is at
$2\pi/B$. Then (1.4) gives that $\overline{D_{o}}\equiv M$ is bounded. Since
$M$ is closed, the Hopf-Rinow theorem implies that $M$ is compact. In fact, we
have also shown that $\mathrm{diam}(M)\leq 2\pi/B$, but since (1.1) is
indipendent of the origin $o$ we can improve the above to the sharp estimate
$\mathrm{diam}(M)\leq\pi/B$.
Cleary the key point of our proof lies in the validity of the inclusion
$D_{o}\subset B_{R_{o}}$. The way to prove this latter is as follows. Suppose
to have shown (1.5) on $D_{o}\cap B_{R_{o}}$
A computation in normal coordinates gives
$\Delta r=\frac{\partial}{\partial r}\log\sqrt{\tilde{g}(r,\theta)},$
where $\tilde{g}(r,\theta)$ is the determinant of the metric in this
coordinate system. Thus, (1.5) on $D_{o}\cap B_{R_{0}}$ reads
$\frac{\partial}{\partial
r}\log\sqrt{\tilde{g}(r,\theta)}\leq(m-1)\frac{g^{\prime}(r)}{g(r)}.$ (1.6)
Fix the unit vector $\theta$ and let $\gamma_{\theta}$ be the unit speed
geodesic emanating from $o$ with $\dot{\gamma}_{\theta}(o)=\theta$.
$\gamma_{\theta}$ will stop to be minimizing after the first cut point
attained at $t=c(\theta)>0$. With $\epsilon>0$ sufficiently small, we
integrate (1.6) on $[\epsilon,{\rm min}\\{c(\theta),R_{o}\\}]$, we let
$\epsilon\to 0^{+}$ and we use the asymptotic behaviours in $0$ to get
$\sqrt{\tilde{g}(r,\theta)}\leq g(r)^{m-1},$
Since $\tilde{g}(r,\theta)>0$ on $D_{o}$, we have $R_{0}\geq c(\theta)$, that
is, $D_{o}\subset B_{R_{0}}$.
However, by a result of M. Morse, a complete manifold $M$ is compact if and
only if each unit speed geodesic $\gamma_{\theta}$ emanating from some fixed
origin $o$ ceases to be a segment i.e. length minimizing, for a value
$c(t_{o})$ of its parameter $t$ which is finite. Thus, the above reasoning
appears to be slightly redundant, in the sense that it provides a bound
$R_{0}$ which is independent of the considered unit speed geodesic from $o$.
This motivates the following result of Galloway [8].
###### Theorem 2.
Let $(M,\langle\,,\,\rangle)$ be a complete Riemannian manifold of dimension
$m\geq 2$. Assume that, for some origin $o$ and for every unit speed geodesic
$\gamma:\mathbb{R}^{+}_{0}\rightarrow M$ emanating from $o$, the solution $g$
of
$\left\\{\begin{array}[]{l}g^{\prime\prime}+\dfrac{\mathrm{Ricc}(\gamma^{\prime},\gamma^{\prime})(t)}{m-1}g=0,\\\\[11.38092pt]
g(0)=0,\quad g^{\prime}(0)=1\end{array}\right.$ (1.7)
has a first positive zero. Then, $M$ is compact with finite fundamental group.
###### Proof.
Let $r_{0}>0$ be the first positive zero of $g$ solution of (1.7). Multiply
the equation in (1.7) by $g$, integrate by parts and use the initial
conditions to get
$\int_{0}^{r_{0}}(g^{\prime})^{2}-\int_{0}^{r_{0}}\frac{\rm{Ricc}(\dot{\gamma},\dot{\gamma})}{m-1}g^{2}=0$
(1.8)
By Rayleigh characterization, this means that the operator
$L=\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}+\frac{\rm{Ricc}(\dot{\gamma},\dot{\gamma})}{m-1}$
satisfies
$\lambda_{1}^{L}([0,r_{0}])\leq 0,$
and by monotonicity of eigenvalue
$\lambda_{1}^{L}([0,r])<0\qquad\forall\ r>r_{0}.$
But $L$ is the stability operator for the geodesic $\gamma$, and on $[0,T]$
$\gamma$ is minimizing only if
$\lambda_{1}^{L}([0,T])\geq 0.$
Thus if the value $c(\gamma)$ gives the cut-point di $o$ along $\gamma$ it
must be $c(\gamma)\leq r_{0}$. By Morse result $M$ is compact. The same
procedure can also be applied to the Riemannian universal covering
$\widetilde{M}\rightarrow M$, showing that $\widetilde{M}$ is compact and thus
that $\Pi_{1}(M)$ is finite. ∎
If we ignore that $L$ is the stability operator for the unit speed geodesic
$\gamma$ we can proceed with the following analytic alternative proof.
Let $p\in D_{o}$, and let $\gamma:[0,r(p)]\rightarrow M$ be the minimizing
geodesic from $o$ to $p$ so that $r(\gamma(t))=t$ and $\nabla
r\circ\gamma=\dot{\gamma}$ for $t\in[0,r(p)]$. We fix a local orthonormal
coframe $\\{\theta^{i}\\}$ to perform computations. Here $1\leq i,j,\ldots\leq
m$ and we use Einstein summation convention. Then for the distance function
$r$ on $D_{o}$ we have
$\mathrm{d}r=r_{i}\theta^{i},$
and Gauss lemma writes
$r_{i}r_{i}\equiv 1.$ (1.9)
Taking covariant derivative of (1.9) we obtain
$r_{ij}r_{i}=0$ (1.10)
that is,
$\mathrm{Hess}\,r(\nabla r,\cdot)=0.$ (1.11)
Covariant differentiation of (1.10) yields
$r_{ijk}r_{i}+r_{ij}r_{ik}=0.$ (1.12)
From the simmetry $r_{ij}=r_{ji}$ we deduce that $r_{ijk}=r_{jik}$, and by the
Ricci commutation rules
$r_{ijk}=r_{ikj}+r_{t}R_{tijk}$
$R_{tijk}$ the components of the Riemann tensor. Using this in (1.12) we get
$0=r_{ijk}r_{i}+r_{ij}r_{ik}=r_{jik}r_{i}+r_{ij}r_{ik}=r_{jki}r_{i}+r_{t}R^{t}_{jik}r_{i}+r_{ij}r_{ik}.$
Thus, tracing with respect to $j$ and $k$
$r_{i}r_{kki}+r_{t}r_{i}R_{ti}+r_{ik}r_{ik}=0,$
with $R_{ti}$ the components of the Ricci tensor. In other words
$\left<\nabla\Delta r,\nabla r\right>+{\rm Ricc}\left(\nabla r,\nabla
r\right)+\left|{\rm Hess}(r)\right|^{2}=0$
Computing along $\gamma$
$\frac{\mathrm{d}}{\mathrm{d}t}\left(\Delta r\circ\gamma\right)+\left|{\rm
Hess}(r)\right|^{2}+{\rm Ricc}\left(\nabla r,\nabla r\right)=0$
on $[0,r(p)]$. Using (1.11) and Newton’s inequality, we have
$\left|{\rm Hess}(r)\right|^{2}\geq\frac{(\Delta r)^{2}}{m-1},$
and setting $\varphi(t)=\Delta r\circ\gamma(t)$ from the above we obtain
$\frac{\mathrm{d}}{\mathrm{d}t}\varphi(t)+\frac{\varphi(t)^{2}}{m-1}+\mathrm{Ricc}\left(\nabla
r,\nabla r\right)\leq 0$ (1.13)
on $[0,r(p)]$. Furthermore, it is well known that
$\Delta r=\frac{m-1}{r}+o(1)\qquad{\rm as}\ r\to 0^{+}$
Hence, since $\gamma$ is minimizing
$\frac{1}{m-1}\varphi(t)=\frac{1}{(r\circ\gamma)(t)}+o(1)=\frac{1}{t}+o(1)\qquad{\rm
as}\ t\to 0^{+}$ (1.14)
Defining
$u(t)=t\exp\left\\{\int_{0}^{t}\left(\frac{\varphi(s)}{m-1}-\frac{1}{s}\right)\mathrm{d}s\right\\}$
on $[0,r(p)]$, $u$ is well defined because of (1.14) and a computation using
(1.13) gives
$\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}u+\frac{{\rm
Ricc}(\dot{\gamma},\dot{\gamma})}{m-1}u\leq 0$ (1.15)
Let now $h$ be any $C^{1}([0,r(p)])$ function such that $h(0)=0=h(r(p))$.
Since $u>0$ on $(0,r(p)]$ the function $h^{2}u^{\prime}/u$ is well defined on
$(0,r(p)]$. Differentiating, using (1.15) and Young inequality we get
$\frac{\mathrm{d}}{\mathrm{d}t}\left(h^{2}\frac{u^{\prime}}{u}\right)\leq-\frac{{\rm
Ricc(\dot{\gamma},\dot{\gamma})}}{m-1}h^{2}-h^{2}\left(\frac{u^{\prime}}{u}\right)^{2}+2hh^{\prime}\frac{u^{\prime}}{u}\leq-\frac{{\rm
Ricc(\dot{\gamma},\dot{\gamma})}}{m-1}h^{2}+(h^{\prime})^{2}$
Fix $\epsilon>0$ sufficiently small. Integration of the above on
$[\epsilon,r(p)]$ gives
$-h^{2}(\epsilon)\frac{u^{\prime}(\epsilon)}{u(\epsilon)}\leq\int_{\epsilon}^{r(p)}(h^{\prime})^{2}-\int_{\epsilon}^{r(p)}\frac{{\rm
Ricc(\dot{\gamma},\dot{\gamma})}}{m-1}h^{2}$
Since $h(\epsilon)=A\epsilon+o(1)$, for $\epsilon\to 0^{+}$ where
$A\in\mathbb{R}$, letting $\epsilon\to 0^{+}$ we obtain
$\int_{0}^{r(p)}(h^{\prime})^{2}-\int_{0}^{r(p)}\frac{{\rm
Ricc(\dot{\gamma},\dot{\gamma})}}{m-1}h^{2}\geq 0$ (1.16)
This contradicts (1.8) unless $r(p)\leq r_{0}$.
Thus we have reduced the compactness problem for the complete manifold $M$ to
the problem of the existence of a first zero for solutions of the Cauchy
problem
$\left\\{\begin{array}[]{l}g^{\prime\prime}+K(t)g=0\qquad\text{on
}\mathbb{R}^{+}\\\\[5.69046pt] g(0)=0,\quad
g^{\prime}(0)=1.\end{array}\right.$ (CP)
where in our geometric application
$K(t)=K_{\gamma}(t)=\frac{{\rm Ricc(\dot{\gamma},\dot{\gamma})}}{m-1}(t)$
(1.17)
We observe that the existence of a first zero is also ”a posteriori”
guaranteed via an oscillation result for the same equation, and that uniform
upper estimate for the positioning of the first zero yields a diameter
estimate. In this perspective the original result of Calabi can be stated as
follows (see also Theorem 3.11 of [2]).
###### Theorem 3 (Theorems 1 and 2 of [4]).
Let $M$ be as above, and assume that $\mathrm{Ricc}\geq 0$ on $M$. Suppose
that for each unit speed geodesic $\gamma$ emanating from $o$ there exist
$0<a<b$, possibly depending on $\gamma$, such that
$\int_{a}^{b}\sqrt{\frac{\mathrm{Ricc}(\gamma^{\prime},\gamma^{\prime})(s)}{m-1}}\mathrm{d}s>\left\\{\left(1+\frac{1}{2}\log\frac{b}{a}\right)^{2}-1\right\\}^{1/2}.$
(1.18)
Then, $M$ is compact and has finite fundamental group. In particular, this
holds provided that
$\limsup_{t\rightarrow+\infty}\left(\int_{1}^{t}\sqrt{\frac{\mathrm{Ricc}(\gamma^{\prime},\gamma^{\prime})(s)}{m-1}}\mathrm{d}s-\frac{1}{2}\log
t\right)=+\infty.$ (1.19)
###### Remark 1.
_As a matter of fact, under the assumption $\mathrm{Ricc}\geq 0$ on $M$,
(1.19) gives an oscillation result for (CP)._
In Calabi result the requirement $\mathrm{Ricc}\geq 0$ is essential. We stress
that (1.18) is, to the best of our knowledge, the first instance of a
condition in finite form for the existence of a first zero, that is, a
condition involving the potential $K$ only in a compact interval $[a,b]$. One
of the main purpose of the present paper is to extend the result even when
Ricci is negative somewhere. It shall be observed that the problem of
obtaining Myers type compactness theorems under the presence of a suitably
small amount of negative Ricci curvature has already been a flourishing field
of research, for which we refer the reader to [21], [6], [18] and the
references therein. However, the techniques employed in these papers are of
various nature and neither of them relies on oscillation type results for a
linear ODE, nor it gives explicit bounds for the amount of negative curvature
allowed. Indeed, it should be pointed out that the method in [21] via Jacobi
fields is not distant from our approach. A much closely related result is the
recent [13], where the case $\mathrm{Ricc}\geq-B^{2}$ is analyzed.
## 2 The role of the critical curve
As we will see shortly, in order to extend Calabi result, we shall deal with a
slightly different ODE. In particular, we are concerned with the following
problems:
* i)
study the existence of a first zero of solutions $z(r)$ of
$\left\\{\begin{array}[]{l}(v(r)z^{\prime}(r))^{\prime}+A(r)v(r)z(r)=0\qquad\text{on
}\ \mathbb{R}^{+}\\\\[5.69046pt] z(0^{+})=z_{0}>0,\end{array}\right.$ (2.1)
with $A(t)\geq 0$, $v(t)>0$ on $\mathbb{R}^{+}$;
* ii)
give an upper bound for the positioning of the first zero of $z$;
* iii)
study the oscillatory behavior of (2.1);
* iv)
extend the obtained result when $A(r)$ changes sign.
Towards these aims we introduce the ”critical curve” $\chi(r)$ relative to
(2.1) or to the next Cauchy problem
$\left\\{\begin{array}[]{l}(v(r)z^{\prime}(r))^{\prime}+A(r)v(r)z(r)=0\qquad\text{on
}\ [r_{0},+\infty),\quad r_{0}>0\\\\[5.69046pt]
z(r_{0}^{+})=z_{0}\in\mathbb{R},\end{array}\right.$ (2.2)
To do this we require the assumptions
$0\leq v(r)\in
L^{\infty}_{\mathrm{loc}}(\mathbb{R}_{0}^{+}),\qquad\frac{1}{v(r)}\in
L^{\infty}_{\mathrm{loc}}(\mathbb{R}^{+}),\qquad\lim_{r\rightarrow
0^{+}}v(r)=0$ (V1)
(the last equation request is intended on a rapresentative of $v$) and the
integrability condition
$\frac{1}{v(r)}\in L^{1}(+\infty).$ (V${}_{\text{L1}}$)
We set
$\chi(r)=\left\\{2v(r)\int_{r}^{+\infty}\frac{\mathrm{d}s}{v(s)}\right\\}^{-2}=\left\\{\left(-\dfrac{1}{2}\log\int_{r}^{+\infty}\dfrac{\mathrm{d}s}{v(s)}\right)^{\prime}\right\\}^{2}$
(2.3)
Fix $0<R<r$, from the definition di $\chi$ we deduce
$\int^{r}_{R}\sqrt{\chi(s)}\mathrm{d}s=\frac{1}{2}\log\left\\{\left(\int_{R}^{+\infty}\frac{\mathrm{d}s}{v(s)}\right)\Big{/}\left(\int_{r}^{+\infty}\frac{\mathrm{d}s}{v(s)}\right)\right\\}\qquad\forall\
0<R<r,$ (2.4)
Thus letting $r\rightarrow+\infty$, we obtain
$\sqrt{\chi(r)}\not\in L^{1}(+\infty)$ (2.5)
It is worth to stress that the function $\chi$ only depends on the weight $v$,
not on $A$. Note that, although (CP) can be thought as a version of (2.1) with
$v\equiv 1$, assumptions (V1), (V${}_{\text{L1}}$) are not satisfied. Thus,
the next main Theorem 4 below cannot be directly applied to (CP).
The study of the Cauchy problem (2.1) turns out to be extremely useful in a
number of different geometric problems, not only those described in this
paper. For instance, a mainstream application of it is to derive spectral
estimates for stationary Schrödinger tipe operators via radialization
techniques. In this case, the role of $v$ is played by the volume growth of
geodesic spheres centered at $o$, for which (V1) is the highest regularity
that we can in general guarantee. However, since there are natural upper and
lower bounds coming from the Laplacian comparison theorems, it is worth to
relate the critical curve with that of, say, an upper bound for $v$. More
precisely, for $f$ satisfying
$\displaystyle f\in L^{\infty}_{\rm
loc}(\mathbb{R}_{0}^{+}),\qquad\frac{1}{f}\in L^{\infty}_{\rm
loc}(\mathbb{R}^{+}),\qquad 0\leq v\leq f\quad{\rm on}\ \mathbb{R}_{0}^{+}$
(F1) $\frac{1}{f}\in L^{1}(+\infty)$ (F${}_{\text{L1}}$)
we shall compare $\chi(r)$ with the critical curve $\chi_{f}(r)$ defined again
via (2.3). We observe that, for any positive constant $c$,
$\chi_{cf}=\chi_{f}$. This suggests that, in general, $v\leq f$ does not imply
$\chi\leq\chi_{f}$. To recover this property we need a more stringent relation
between $v$ and $f$.
###### Proposition 1 (Proposition 4.13 of [2]).
Let $v,f$ satisfy (V1), (V${}_{\text{L1}}$) on some interval
$I=(r_{0},+\infty)\subset\mathbb{R}^{+}$. Then,
* (i)
If $v/f$ is non-increasing on $I$, $\chi(r)\leq\chi_{f}(r)$ on $I$;
* (ii)
If $v/f$ is non-decreasing on $I$, $\chi(r)\geq\chi_{f}(r)$ on $I$;
In the case $v(r)=\mathrm{vol}(\partial B_{r})$, the above proposition fits
well with the Bishop-Gromov comparison theorem for volumes ([16], Theorem
2.14). The interested reader may consult Chapter 4 of [2], where the authors
give a detailed discussion on the critical curve, together with estimates on
$\chi$ when $v(r)=\mathrm{vol}(\partial B_{r})$, explicit examples, and many
applications. For instance, the deep relationship between $\chi(r)$ and
optimal weights for Hardy inequalities is discussed. Since, as we will see, in
dealing with Calabi-type compactness results the role of $v$ will be played by
some suitable weight which has no direct relation with volumes, we shall not
pursue this line of argument any further.
We now list the assumptions under which we will treat either of the Cauchy
problems (2.1) or (2.2).
$v(r)\int_{r}^{a}\frac{\mathrm{d}s}{v(s)};\qquad\frac{1}{v(r)}\int_{o}^{r}v(s)\mathrm{d}s\
\in L^{\infty}([0,a])$ (V2)
for some $a\in\mathbb{R}^{+}$.
$\frac{1}{v(r)}\int_{0}^{r}v(s)\mathrm{d}s=o(1)\qquad{\rm as}\quad r\to 0^{+}$
(V3) $A(r)\in L^{\infty}_{\rm loc}(\mathbb{R}_{0}^{+})$ (A1)
Conditions
* 1.
(A1), (V1), (V2) and (V3) guarantee the existence of a solution
$z\in\mathrm{Lip}_{\mathrm{loc}}(\mathbb{R}^{+}_{0})$ of (2.1)
* 2.
(A1), (V1) and (V2) its uniqueness
* 3.
(A1), (V1) the fact that each solution $z\not\equiv 0$ has isolated zeros, if
any.
Note that (V2) and (V3) are automatically satisfied if $v(r)$ is non-
decreasing in a neighbourhood of $0$.
The following theorem summarizes some of the results obtained in [3].
###### Theorem 4.
Let (A1), (V1), (F1), (V${}_{\text{L1}}$) be met, and let
$z\in\mathrm{Lip}_{\mathrm{loc}}(\mathbb{R}_{0}^{+})$ be a solution of
$\left\\{\begin{array}[]{l}(v(r)z^{\prime}(r))^{\prime}+A(r)v(r)z(r)=0\qquad\text{on
}\ \mathbb{R}^{+},\\\\[5.69046pt] z(0^{+})=z_{0}>0.\end{array}\right.$ (2.6)
Then,
* (1)
[_Theorem 5.2 of[3]_] If $A(r)\leq\chi(r)$ on $\mathbb{R}^{+}$, then $z>0$ on
$\mathbb{R}^{+}$. Furthermore, there exists $r_{1}>0$ and a constant
$C=C(r_{1})>0$ such that
$z(r)\geq\displaystyle-C\sqrt{\int_{r}^{+\infty}\dfrac{\mathrm{d}s}{f(s)}}\log\int_{r}^{+\infty}\dfrac{\mathrm{d}s}{f(s)}\qquad\text{on
}[r_{1},+\infty).$ (2.7)
* (2)
[_Corollary 5.4 of[3]_] If $A(r)\leq\chi(r)$ on $[r_{0},+\infty)$, for some
$r_{0}>0$, then $z$ is nonoscillatory, that is, it has only finitely many
zeroes (if any).
* (3)
[_Corollary 6.3 of[3]_] If $A\geq 0$ on $\mathbb{R}^{+}$, $A\not\equiv 0$ and
there exist $r>R>0$ such that $A\not\equiv 0$ on $[0,R]$ and
$\begin{array}[]{l}\displaystyle\int_{R}^{r}\left(\sqrt{A(s)}-\sqrt{\chi_{f}(s)}\right)\mathrm{d}s>-\dfrac{1}{2}\left(\log\int_{0}^{R}A(s)v(s)\mathrm{d}s+\log\int_{R}^{+\infty}\frac{\mathrm{d}s}{f(s)}\right)\end{array}$
(2.8)
then $z$ has a first zero. Moreover, this is attained on $(0,\overline{R}]$,
where $\overline{R}>0$ is the unique real number satisfying
$\int^{r}_{R}{\sqrt{A(s)}\mathrm{d}s}=-\frac{1}{2}\log\int_{0}^{R}A(s)v(s)\mathrm{d}s-\frac{1}{2}\log\int_{r}^{\overline{R}}{\frac{\mathrm{d}s}{f(s)}}$
(2.9)
* (4)
[_Theorem 6.6 of[3]_] If $A\geq 0$ on $\mathbb{R}^{+}$ and, for some (hence
any) $R>0$ such that $A\not\equiv 0$ on $[0,R]$,
$\limsup_{r\rightarrow+\infty}\int_{R}^{r}\left(\sqrt{A(s)}-\sqrt{\chi_{f}(s)}\right)\mathrm{d}s=+\infty$
(2.10)
then $z$ is oscillatory, that is, it has infinitely many zeroes.
###### Remark 2.
_In fact, for $(2)$ and $(4)$ to hold, it is enough that $z$ solves the Cauchy
problem only on $[r_{0},+\infty)$, for some $r_{0}>0$ and for some initial
condition $z(r_{0})$, $(vz^{\prime})(r_{0})$._
It is worth to make some observations on the conditions in the above theorem.
* -
In $(1)$, $A\leq\chi$ cannot be replaced with $A\leq\chi_{f}$. The reason is
that, as already observed, no relations between $\chi$ and $\chi_{f}$ can be
deduced from the sole requirement $v\leq f$ in (F1). However, note that
$\chi_{f}$ appears both in (2.8) and in (2.10). This is due to the technique
developed for $(3)$ and $(4)$, which is different from that used for $(1)$ and
$(2)$.
* -
The lower bound (2.7) is sharp. Indeed, it can be showed that if $z$ is
positive on $\mathbb{R}^{+}$ and $A\geq\chi$ on some $[r_{0},+\infty)$, then
necessarily $z$ is bounded from above by the quantity on the RHS of (2.7), for
some $C>0$.
* -
The right hand side of (2.8) is independent both of $r$ and of the behavior of
$A$ after $R$. Therefore, the left hand side of (2.8) represents how much must
$A$ exceed a critical curve modelled on $f$ in the compact region $[R,r]$ in
order to have a first zero for $z$, and it only depends on the behavior of $A$
and $f$ before $R$ (the first addendum of the RHS), and on the growth of $f$
after $R$. This is conceptually simpler than Calabi compactness condition,
where the role of $a,b$ is balanced between the two sides of (1.18).
###### Remark 3.
_The assumptions in $(3)$ and $(4)$ can be weakened. Indeed, it is enough that
$z$ solves the inequality $(vz^{\prime})^{\prime}+Avz\leq 0$ on
$\mathbb{R}^{+}$, and that its initial condition satisfies_
$\frac{vz^{\prime}}{z}(0^{+})=0.$
_Note that sufficiently mild singularities of $z$ as $r\rightarrow 0^{+}$ are
allowed, depending on the order of zero of $v(r)$ at $0$._
###### Remark 4.
_Using ( 2.4) we see that (2.10) can be equivalently expressed as_
$\limsup_{r\to+\infty}\left\\{\int_{R}^{r}\sqrt{A(s)}+\frac{1}{2}\log\int_{r}^{+\infty}\frac{\mathrm{d}s}{f(s)}\right\\}=+\infty.$
(2.11)
The similarity between (2.11) and (1.19) is evident. Indeed, as a first
application of Theorem 4 let us show that Calabi condition (1.19) implies that
the solution of (CP), with
$K(t)=\frac{{\mathrm{Ricc}}(\dot{\gamma},\dot{\gamma})}{m-1}(t)\geq 0,$ (2.12)
is oscillatory.
Indeed, choose any $v$ satisfying (V1), (V2), (V3) and $v^{-1}\in
L^{1}(+\infty)\backslash L^{1}(0^{+})$, for instance $v(r)=r^{m-1}$ for some
$m\geq 3$. Let $r=r(t)$ be the inverse function of
$t(r)=\left(\int_{r}^{+\infty}\frac{\mathrm{d}s}{v(s)}\right)^{-1}$ (2.13)
and define
$z(r)=\frac{g(t(r))}{t(r)}$ (2.14)
Then $z$ solves
$\left\\{\begin{array}[]{l}\displaystyle(vz^{\prime})^{\prime}+\frac{K(t(r))t^{4}(r)}{v^{2}(r)}v(r)z=0\qquad{\rm
on}\ \mathbb{R}^{+}\\\\[11.38092pt]
z(0)=1\qquad(vz^{\prime})(0)=0\end{array}\right.$ (2.15)
where now differentiation is with respect to the variable $r$. If (2.11) holds
with $f=v$ and
$A(r)=\frac{K(t(r))t^{4}(r)}{v^{2}(r)}\geq 0,$
then $z$ oscillates and so does $g$. A change of variables shows that (2.11)
is exactly (1.19).
The literature on the qualitative properties of solutions of (CP) is enormous,
and considerable steps towards the comprehension of the matter have been made
throughout all of the $20^{\mathrm{th}}$ century. In particular, a number of
sharp oscillatory and nonoscillatory conditions for $g$ have been found. Here,
we only quote two of the finest. The first is the so-called Hille-Nehari
criterion, see [19], p.45 and [10], Theorem $5$ and Corollary $1$.
###### Theorem 5.
Let $K\in C^{0}(\mathbb{R})\cap L^{1}(+\infty)$ be non-negative, and consider
a solution $g$ of $g^{\prime\prime}+Kg=0$. Denote with $k(t)$, $k_{*}$ and
$k^{*}$ respectively the quantities
$k(t)=t\int_{t}^{+\infty}K(s)\mathrm{d}s,\qquad
k_{*}=\liminf_{t\rightarrow+\infty}k(t),\qquad
k^{*}=\limsup_{t\rightarrow+\infty}k(t).$
We have:
* -
if $g$ is nonoscillatory, then necessarily $k_{*}\leq 1/4$ and $k^{*}\leq 1$;
* -
if $k(t)\leq 1/4$ for $t$ large enough, in particular if $k^{*}<1/4$, then $g$
is nonoscillatory.
As a consequence, $k_{*}>1/4$ is a sufficient condition for $g$ to be
oscillatory.
###### Remark 5.
_If $K\not\in L^{1}(+\infty)$, the result applies with $k_{*}=k^{*}=+\infty$,
and $g$ is thus oscillatory. This case is due to W.B. Fite [7]._
###### Remark 6.
_Improving on an old criterion of Kneser, it can be showed (see[2],
Proposition 2.23) that if $k(t)\leq 1/4$ on the whole $\mathbb{R}^{+}$, then
the solution $g$ of (CP) is positive and increasing on $\mathbb{R}^{+}$._
###### Remark 7.
_Hille-Nehari criterion detects the oscillation of $g$ when $K(t)\geq
B^{2}/(1+t^{2})$ on $\mathbb{R}^{+}$, for some $B>1/2$. In a geometrical
context, this particular case has been investigated in [5], where the authors
have also obtained upper bounds for the first zero of $g$ solving (CP)._
###### Remark 8.
_For every $B\in[0,1/2]$, the Cauchy problem associated to the Euler equation_
$\left\\{\begin{array}[]{l}g^{\prime\prime}+\dfrac{B^{2}}{(1+t)^{2}}g=0,\\\\[11.38092pt]
g(0)=0,\quad g^{\prime}(0)=1,\end{array}\right.$
_has the explicit, positive solution_
$g(s)=\left\\{\begin{array}[]{ll}\sqrt{1+t}\log(1+t)&\quad\text{if }\
B=1/2;\\\\[8.5359pt]
\displaystyle\frac{1}{\sqrt{1-4B^{2}}}\Big{(}(1+t)^{B^{\prime\prime}}-(1+t)^{1-B^{\prime\prime}}\Big{)}&\quad\text{if
}\ B\in[0,1/2),\end{array}\right.$
_where_
$B^{\prime\prime}=\frac{1+\sqrt{1-4B^{2}}}{2}\in(1/2,1]$
_(see[19], p.45). For $B=1/2$, this example shows that Hille-Nehari criterion
is sharp._
When $k_{*}=k^{*}=1/4$, Hille-Nehari criterion cannot grasp the behaviour of
$g$. As we shall see, combining $(2)$ and $(4)$ of Theorem 4 in an iterative
way, we can construct sharper and sharper oscillation and nonoscillation
criteria that can detect the behaviour of $g$ even in some cases when the
Hille-Nehari theorem fails to give information.
The second result we quote allows sign-changing potentials $K$ and is due to
R. Moore (see [14], Theorem 2)
###### Theorem 6.
Let $K\in C^{0}(\mathbb{R})$. Each solution $g$ of $g^{\prime\prime}+Kg=0$ is
oscillatory provided that, for some $\lambda\in[0,1)$, there exists
$\lim_{t\rightarrow+\infty}\int_{0}^{t}s^{\lambda}K(s)\mathrm{d}s=+\infty,$
(2.16)
###### Remark 9.
_Setting $\lambda=0$ in Moore statement we recover a result of W. Ambrose [1]
and A. Wintner [20] (one can also consult [9], Corollaries 3.5 and 3.6 for a
different proof and a generalization). Remark 8 shows that in Moore result the
interval of the parameter $\lambda$ cannot be extended to $[0,1]$. Thus, Euler
equation suggests that, when restricted to the case $K\geq 0$, Moore criterion
is somehow weaker than that of Hille-Nehari._
Another observation on Moore result is that, although sharp from many points
of view, it requires that the negative part of $K$ be, loosely speaking,
globally smaller than the positive part. This is the essence of the existence
of the limit in (2.16). One of our goal in the next section will be to obtain
an oscillation criterion that allows $K$ to have a relevant negative part.
Furthermore, with the aid of (2.8), we will also find a condition in finite
form for the existence of a first zero that allows $K$ to be negative
somewhere. As far as we know, there is still no result in this direction
besides some very recent work of P. Mastrolia, G. Veronelli and M. Rimoldi,
which we recall here for the sake of completeness.
###### Theorem 7 (Theorem 5 of [13]).
Suppose that $K\in L^{\infty}(\mathbb{R}^{+}_{0})$ satisfies $K\geq-B^{2}$,
for some $B\geq 0$, and let $g$ be a solution of (CP). Suppose that there
exist $0<a<b$ and $\lambda\neq 1$ for which either
$\int_{a}^{b}sK_{\gamma}(s)\mathrm{d}s>B\left\\{b+a\frac{e^{2Ba}+1}{e^{2Ba}-1}\right\\}+\frac{1}{4}\log\left(\frac{b}{a}\right)$
(2.17)
or
$\int_{a}^{b}s^{\lambda}K_{\gamma}(s)\mathrm{d}s>B\left\\{b^{\lambda}+a^{\lambda}\frac{e^{2Ba}+1}{e^{2Ba}-1}\right\\}+\frac{\lambda^{2}}{4(1-\lambda)}\left\\{a^{\lambda-1}-b^{\lambda-1}\right\\}$
(2.18)
holds (if $B=0$, this has to be intended in a limit sense). Then, $g$ has a
first zero.
###### Remark 10.
_The case $B=0$ of the above result is due to Z. Nehari, see [15], p.432 (8),
with an entirely different proof. We point out that, in [13], the authors also
give an upper bound for the position of the first zero._
## 3 Extensions of Calabi compactness criterion
We shall now deal with (2.1) under the further assumption that $A$ is possibly
negative. Hereafter, we require the validity of (A1), (V1), (V2), (V3), (F1) .
Let $z\in\mathrm{Lip}_{\mathrm{loc}}(\mathbb{R}^{+}_{0})$ be a solution of
$\left\\{\begin{array}[]{l}(vz^{\prime})^{\prime}+Avz=0\qquad\text{on }\
\mathbb{R}^{+},\\\\[5.69046pt] z(0^{+})=z_{0}>0,\end{array}\right.$ (3.1)
or of the analogous problem on $[r_{0},+\infty)$.
Choose a function $W\in L^{\infty}_{\mathrm{loc}}(\mathbb{R}^{+}_{0})$ such
that
$W\geq 0\quad\text{a.e. on }\mathbb{R}^{+},\qquad W+A\geq 0\quad\text{a.e. on
}\ \mathbb{R}^{+}.$ (3.2)
For instance, $W$ can be taken to be the negative part of $A$. To apply the
results of the previous section, we need to produce, starting from (3.1) and
$W$, a solution $\widetilde{z}$ of a linear ODE of the type
$(\bar{v}\widetilde{z}^{\prime})^{\prime}+\bar{A}\bar{v}\widetilde{z}=0$, for
some new volume function $\bar{v}$ and some $\bar{A}\geq 0$. Towards this
purpose, consider a solution
$w(r)\in\mathrm{Lip}_{\mathrm{loc}}(\mathbb{R}^{+}_{0})$ of
$\left\\{\begin{array}[]{l}(vw^{\prime})^{\prime}-Wvw\geq 0\qquad\mathrm{on\
}\mathbb{R}^{+}\\\\[5.69046pt] w(0^{+})=w_{0}>0.\end{array}\right.$ (3.3)
Note that from
$(vw^{\prime})^{\prime}\geq Wvw$
we deduce $w^{\prime}\geq 0$ a.e., hence $w$ has a positive essential infimum
on $\mathbb{R}_{0}^{+}$. Therefore, the function $\widetilde{z}=z/w$ is well
defined on $\mathbb{R}_{0}^{+}$ and solves
$\left\\{\begin{array}[]{l}\big{(}[vw^{2}]\widetilde{z}^{\prime}\big{)}^{\prime}+\big{(}A+W\big{)}[vw^{2}]\widetilde{z}\leq
0\qquad\text{on }\ \mathbb{R}^{+}\\\\[5.69046pt]
\widetilde{z}(0)=z_{0}/w_{0}>0,\end{array}\right.$ (3.4)
As observed in Remark 3, the inequality sign in (3.4) is irrelevant for the
proofs of $(3)$, $(4)$ of Theorem 4. In this way, $(3)$ and $(4)$ can be
extended to cover sign-changing potentials by simply replacing $A$ with $A+W$,
$v$ with $vw^{2}$ and $f$ with $fw^{2}$. The main problem therefore shifts to
the search of explicit solutions $w$ of (3.3), once $v$ and $W$ are given.
Up to taking some care when dealing with the initial condition, the same
procedure can be carried on even when $v\equiv 1$. In this case, we are able
to provide an explicit form for $w$ when the potential $W$ is a polynomial.
This leads to the following theorem (see Theorem 6.41 of [2]). In the
statement below, we denote with $I_{\nu}$ is the positive Bessel function of
order $\nu$.
###### Theorem 8 (Compactness with sign-changing curvature).
Let $(M,\langle\,,\,\rangle)$ be a complete m-dimensional Riemannian manifold.
For each unit speed geodesic $\gamma$ emanating from a fixed origin $o$,
define
$K_{\gamma}(t)=\frac{\mathrm{Ricc}(\gamma^{\prime},\gamma^{\prime})(t)}{m-1}.$
Assume that one of the following set of assumptions is met.
* $(i)$
The function $K_{\gamma}(t)$ satisfies
$K_{\gamma}(t)\geq-B^{2}\big{(}1+t^{2}\big{)}^{\alpha/2}\qquad\text{on }\
\mathbb{R}^{+},$
for some $B>0$ and $\alpha\geq-2$ possibly depending on $\gamma$. Having set
$0\leq A_{\gamma}(t)=K_{\gamma}(t)+B^{2}\big{(}1+t^{2}\big{)}^{\alpha/2},$
suppose also that, for some $0<S<t$ such that $A_{\gamma}\not\equiv 0$ on
$[0,S]$,
$\begin{array}[]{l}\displaystyle\int_{S}^{t}\left(\sqrt{A_{\gamma}(\sigma)}-\sqrt{\chi_{w^{2}}(\sigma)}\right)\mathrm{d}\sigma\\\\[14.22636pt]
\qquad\qquad\qquad>\displaystyle-\frac{1}{2}\left(\log\int_{0}^{S}A_{\gamma}(\sigma)w^{2}(\sigma)\mathrm{d}\sigma+\log\int_{S}^{+\infty}\frac{\mathrm{d}\sigma}{w^{2}(\sigma)}\right),\end{array}$
(3.5)
where
$w(t)=\left\\{\begin{array}[]{ll}\displaystyle\sinh\left(\frac{2B}{2+\alpha}\left[(1+t)^{1+\frac{\alpha}{2}}-1\right]\right)&\quad\text{if
}\ \alpha\geq 0;\\\\[8.5359pt] \displaystyle
t^{1/2}I_{\frac{1}{2+\alpha}}\left(\frac{2B}{2+\alpha}t^{1+\frac{\alpha}{2}}\right)&\quad\text{if
}\ \alpha\in(-2,0);\\\\[8.5359pt] \displaystyle t^{B^{\prime}}&\quad\text{if
}\ \alpha=-2,\end{array}\right.$ (3.6)
and $B^{\prime}=(1+\sqrt{1+4B^{2}})/2$.
* $(ii)$
The function $K_{\gamma}(t)$ satisfies
$K_{\gamma}(t)\geq\frac{B^{2}}{(1+t)^{2}}\qquad\text{on }\ \mathbb{R}^{+},$
for some $B\in[0,1/2]$ possibly depending on $\gamma$. Having set
$0\leq A_{\gamma}(t)=K_{\gamma}(t)-\frac{B^{2}}{(1+t)^{2}},$
suppose also that, for some $0<S<t$ such that $A_{\gamma}\not\equiv 0$ on
$[0,S]$, inequality (3.5) holds with
$w(t)=\left\\{\begin{array}[]{ll}(1+t)^{B^{\prime\prime}}-(1+t)^{1-B^{\prime\prime}}&\quad\text{if
}\ B\in[0,1/2);\\\\[8.5359pt] \sqrt{1+t}\log(1+t)&\quad\text{if }\
B=1/2,\end{array}\right.$ (3.7)
and $B^{\prime\prime}=(1+\sqrt{1-4B^{2}})/2$.
Then, $M$ is compact and has finite fundamental group.
###### Remark 11.
_Note that, both for ( 3.6) and for (3.7), the critical curve related to
$w^{2}$ exists since $1/w^{2}\in L^{1}(+\infty)$._
###### Proof.
By Theorem 2, it is enough to prove that, for every $\gamma$ issuing from $o$,
the solution $g$ of
$\left\\{\begin{array}[]{l}g^{\prime\prime}+K_{\gamma}(t)g=0\\\\[5.69046pt]
g(0)=0,\quad g^{\prime}(0)=1\end{array}\right.$ (3.8)
has a first zero.
(i) A straightforward computation shows that the function $w$ in (3.6) is a
positive solution of
$w^{\prime\prime}-B^{2}(1+t^{2})^{\alpha/2}w\geq 0\qquad\text{on
}\mathbb{R}^{+}$
whose initial condition, in the cases $\alpha\in(-2,0)$ and $\alpha\geq 0$, is
$w(0)=0,\qquad w^{\prime}(0)=C>0.$ (3.9)
Consider $\widetilde{z}=g/w$. Then, $\widetilde{z}$ solves
$(w^{2}\widetilde{z}^{\prime})^{\prime}+A_{\gamma}w^{2}\widetilde{z}\leq
0\qquad\text{on }\mathbb{R}^{+}.$ (3.10)
In order to apply $(3)$ of Theorem 4 to the differential inequality (3.10), we
shall make use of Remark 3. From (3.9), in each of the cases of (3.6) we
obtain
$\frac{w^{2}\widetilde{z}^{\prime}}{\widetilde{z}}(0^{+})=\left(w^{2}\frac{g^{\prime}}{g}-ww^{\prime}\right)(0^{+})=0.$
(3.11)
We can thus apply $(3)$ of Theorem 4, and (3.5) implies that $\widetilde{z}$
(hence $g$) has a first zero. Case $(ii)$ is analogous. Indeed, by Remark 8,
$w$ in (3.7) is a solution of the Cauchy problem
$\left\\{\begin{array}[]{l}\displaystyle
w^{\prime\prime}+\frac{B^{2}}{(1+t)^{2}}w=0\\\\[11.38092pt] g(0)=0,\quad
g^{\prime}(0)=C>0.\end{array}\right.$
∎
###### Remark 12.
_We recall that, by ( 2.4), inequality (3.5) is equivalent to the somehow
simpler_
$\int_{S}^{t}\sqrt{A_{\gamma}(\sigma)}\mathrm{d}\sigma>-\frac{1}{2}\left(\log\int_{0}^{S}A_{\gamma}(\sigma)w^{2}(\sigma)\mathrm{d}\sigma+\log\int_{t}^{+\infty}\frac{\mathrm{d}\sigma}{w^{2}(\sigma)}\right).$
(3.12)
_However, ( 3.5) put in evidence that the RHS does not depend on $t$, as
opposed to conditions like (1.18) and (2.18) where both $a$ and $b$ appear in
the LHS as well as in the RHS. Furthermore, although somehow complicated,
(3.5) is entirely explicit once we are able to compute the critical curve
related to $w^{2}$. In general, this can only be done numerically, but in some
cases a closed expression can be given. For instance, this is so for $m=3$,
$B=1/2$ in (3.7), for $B=0$ in (3.7) and for $\alpha=0,-2$ in (3.6):_
$\int_{t}^{+\infty}\frac{\mathrm{d}\sigma}{w^{2}(\sigma)}=\left\\{\begin{array}[]{ll}\displaystyle\frac{t^{-\sqrt{1+4B^{2}}}}{\sqrt{1+4B^{2}}}&\quad\text{for
}\eqref{wespo},\ \alpha=-2\text{ and for }B=0;\\\\[17.07182pt] \displaystyle
B^{-1}\big{[}\mathrm{coth}(Bt)-1\big{]}&\quad\text{for }\eqref{wespo},\
\alpha=0;\\\\[8.5359pt] \displaystyle\frac{1}{\log(1+t)}&\quad\text{for
}\eqref{wpoli},\ B=1/2,\ m=3.\end{array}\right.$
_Therefore, in the case $B=0$, (3.12) reads_
$\int_{S}^{t}\sqrt{K_{\gamma}(\sigma)}\mathrm{d}\sigma>-\frac{1}{2}\left(\log\int_{0}^{S}\sigma^{2}K_{\gamma}(\sigma)\mathrm{d}\sigma-\log
t\right),$
_that should be compared to ( 1.18), while, for $\alpha=0$, (3.12) becomes_
$\int_{S}^{t}\sqrt{K_{\gamma}(\sigma)+B^{2}}\mathrm{d}\sigma>-\frac{1}{2}\left(\log\int_{0}^{S}K_{\gamma}(\sigma)\sinh^{2}(B\sigma)\mathrm{d}\sigma+\log\frac{\coth(Bt)-1}{B}\right),$
_that should be compared to ( 2.17) and (2.18)._
Easier expressions can be obtained when considering oscillatory conditions. We
state the result in analytic form.
###### Theorem 9 (Generalized Calabi criterion).
Let $K\in L^{\infty}_{\mathrm{loc}}(\mathbb{R}_{0}^{+})$, and let $g\not\equiv
0$ be a solution of $g^{\prime\prime}+Kg=0$. Then, $g$ oscillates in each of
the following cases:
* $(1)$
$K$ satisfies
$K(t)\geq-B^{2}t^{\alpha}\qquad\text{when }\ t>t_{0},$ (3.13)
for some $B>0$, $\alpha\geq-2$ and $t_{0}>0$, and the following conditions
hold:
$\begin{array}[]{l}\text{for }\
\alpha=-2,\quad\displaystyle\limsup_{t\rightarrow+\infty}\left(\int_{t_{0}}^{t}\sqrt{K(\sigma)+\frac{B^{2}}{\sigma^{2}}}\mathrm{d}\sigma-\frac{\sqrt{1+4B^{2}}}{2}\log
t\right)=+\infty;\\\\[14.22636pt] \text{for }\
\alpha>-2,\quad\displaystyle\limsup_{t\rightarrow+\infty}\left(\int_{t_{0}}^{t}\sqrt{K(\sigma)+B^{2}\sigma^{\alpha}}\mathrm{d}\sigma-\frac{2B}{\alpha+2}t^{\frac{\alpha}{2}+1}\right)=+\infty.\end{array}$
(3.14)
* $(2)$
$K$ satisfies
$K(t)\geq\frac{B^{2}}{t^{2}}\qquad\text{when }\ t>t_{0},$ (3.15)
for some $B\in[0,1/2]$, $t_{0}>0$, and the following conditions hold:
$\begin{array}[]{l}\text{for }\
B<\frac{1}{2},\quad\displaystyle\limsup_{t\rightarrow+\infty}\left(\int_{t_{0}}^{t}\sqrt{K(\sigma)-\frac{B^{2}}{\sigma^{2}}}\mathrm{d}\sigma-\frac{\sqrt{1-4B^{2}}}{2}\log
t\right)=+\infty;\\\\[14.22636pt] \text{for }\
B=\frac{1}{2},\quad\displaystyle\limsup_{t\rightarrow+\infty}\left(\int_{t_{0}}^{t}\sqrt{K(\sigma)-\frac{1}{4\sigma^{2}}}\mathrm{d}\sigma-\frac{1}{2}\log\log
t\right)=+\infty;\end{array}$ (3.16)
###### Proof.
$(1)$. The equation $w^{\prime\prime}-B^{2}t^{\alpha}w=0$ on, say,
$[1,+\infty)$ has the particular positive solution
$\begin{array}[]{ll}w(t)=\displaystyle\sqrt{t}I_{\frac{1}{2+\alpha}}\left(\frac{2B}{2+\alpha}t^{1+\frac{\alpha}{2}}\right)&\quad\text{if
}\ \alpha>-2;\\\\[14.22636pt] w(t)=\displaystyle t^{B^{\prime}},\quad
B^{\prime}=\frac{1+\sqrt{1+4B^{2}}}{2}&\quad\text{if }\ \alpha=-2,\end{array}$
(3.17)
where $I_{\nu}(t)$ is the Bessel function of order $\nu$. From
$\qquad I_{\nu}(t)=\frac{e^{t}}{\sqrt{2\pi t}}(1+o(1))\qquad\text{as }\
t\rightarrow+\infty$
(see [12], p. 102), in both cases $\alpha=-2$ and $\alpha>-2$ we deduce that
$1/w^{2}\in L^{1}(+\infty)$. Moreover,
$\int_{t}^{+\infty}\frac{\mathrm{d}\sigma}{w^{2}(\sigma)}\sim\left\\{\begin{array}[]{ll}C\exp\left(-\frac{4B}{2+\alpha}t^{1+\frac{\alpha}{2}}\right)&\quad\text{if
}\ \alpha>-2;\\\\[14.22636pt]
Ct^{1-2B^{\prime}}=Ct^{-\sqrt{1+4B^{2}}}&\quad\text{if }\
\alpha=-2.\end{array}\right.$ (3.18)
Since the function $\widetilde{z}=g/w$ solves
$(w^{2}\widetilde{z}^{\prime})^{\prime}+(K+B^{2}t^{\alpha})w^{2}\widetilde{z}\leq
0\qquad\text{on }[1,+\infty),$
by $(4)$ of Theorem 4, $z$ (and hence $g$) oscillates provided
$\limsup_{t\rightarrow+\infty}\int_{t_{0}}^{t}\Big{(}\sqrt{K(\sigma)+B^{2}\sigma^{\alpha}}-\sqrt{\chi_{w^{2}}(\sigma)}\Big{)}\mathrm{d}\sigma=+\infty$
which, by Remark 4, is equivalent to
$\limsup_{t\rightarrow+\infty}\int_{t_{0}}^{t}\sqrt{K(\sigma)+B^{2}\sigma^{\alpha}}\mathrm{d}\sigma+\frac{1}{2}\log\int_{t}^{+\infty}\frac{\mathrm{d}\sigma}{w^{2}(\sigma)}=+\infty$
(3.19)
By (3.18), conditions (3.14) and (3.19) are equivalent, thus the conclusion.
$(2)$. The proof is the same. Indeed, it is enough to consider the following
positive solution $w$ of $w^{\prime\prime}+B^{2}t^{-2}w=0$:
$\begin{array}[]{ll}w(t)=\displaystyle t^{B^{\prime\prime}},\quad
B^{\prime\prime}=\frac{1+\sqrt{1-4B^{2}}}{2}&\quad\text{if }\
B\in[0,1/2);\\\\[14.22636pt] w(t)=\displaystyle\sqrt{t}\log t&\quad\text{if }\
B=1/2.\end{array}$ (3.20)
Again, in both cases $1/w^{2}\in L^{1}(+\infty)$. ∎
###### Remark 13.
_Note that, for $B=0$, we recover another proof of the original Calabi
oscillation criterion, which is different from that described in the previous
section._
Polynomial lower bounds for $K$ are clearly chosen for their simplicity.
Indeed, the statement in its full generality only requires a positive solution
$w$ of $w^{\prime\prime}+Ww\geq 0$, where the weight $W$ has only to satisfy
$K+W\geq 0$. In this way, arbitrary lower bounds for $K$ are allowed, and up
to finding a suitable positive $w$ the oscillatory conditions are explicit.
This improves on Moore oscillation criterion, where the existence of the limit
in (2.16) is essential for the proof of Theorem 6 to work. The same discussion
holds for Theorem 8, up to the further requirement that $w$ is sufficiently
well-behaved as $t\rightarrow 0^{+}$. From this perspective, Theorem 8
improves on Theorem 7, whose proof seems to us to be hardly generalizable when
the lower bound for $K$ is nonconstant.
The procedure described above, which loosely speaking allows to translate the
potential up to inserting a weight, can be iterated. In this way, we can
obtain finer and finer criteria in a very simple way. We now describe how to
proceed in this direction. The first example is the following
###### Theorem 10 (Positivity and nonoscillation criteria).
Let $K\in L^{\infty}_{\mathrm{loc}}(\mathbb{R}^{+}_{0})$.
* $(1)$
Suppose that
$K(t)\leq\frac{1}{4(1+t)^{2}}\left[1+\frac{1}{\log^{2}(1+t)}\right]\qquad\text{on
}\ \mathbb{R}^{+}.$ (3.21)
Then, every solution $g$ of
$\left\\{\begin{array}[]{l}g^{\prime\prime}+K(t)g\geq 0\\\\[5.69046pt]
g(0)=0,\quad g^{\prime}(0)=1\end{array}\right.$ (3.22)
is positive on $\mathbb{R}^{+}$ and satisfies $g(t)\geq C\sqrt{t\log
t}\log\log t$, for some $C>0$ and for $t>3$.
* $(2)$
Suppose that
$K(t)\leq\frac{1}{4t^{2}}\left[1+\frac{1}{\log^{2}t}\right]\qquad\text{on }\
[t_{0},+\infty),$ (3.23)
for some $t_{0}>0$. Then, every solution $g$ of $g^{\prime\prime}+Kg=0$ is
nonoscillatory.
###### Proof.
$(1)$. By Sturm argument, it is sufficient to prove the desired conclusion
under the additional assumptions that $g$ satisfies (3.22) with the equality
sign, and that
$K(t)\geq\frac{1}{4(1+t)^{2}}.$
Let $w(t)=\sqrt{1+t}\log(1+t)$ be the solution of (3.22) with the equality
sign and with $K(t)=[4(1+t)^{2}]^{-1}$. Then, $\widetilde{z}=g/w$ solves
$\left\\{\begin{array}[]{l}\displaystyle(w^{2}\widetilde{z}^{\prime})^{\prime}+\left[K(s)-\frac{1}{4(1+t)^{2}}\right]w^{2}\widetilde{z}=0\quad\text{on
}\mathbb{R}^{+}\\\\[11.38092pt]
\widetilde{z}(0)=1,\qquad\widetilde{z}^{\prime}(0)=0.\end{array}\right.$
(3.24)
Applying $(1)$ of Theorem 4, $\widetilde{z}$ is positive provided
$K(t)-\frac{1}{4(1+t)^{2}}\leq\chi_{w^{2}}(t)=\frac{1}{4(1+t)^{2}\log^{2}(1+t)},$
which is (3.21), and $\widetilde{z}$ satisfies
$\widetilde{z}(t)\geq-C\sqrt{\int_{t}^{+\infty}\frac{\mathrm{d}\sigma}{w^{2}(\sigma)}}\log\int_{t}^{+\infty}\frac{\mathrm{d}\sigma}{w^{2}(\sigma)}=C\frac{\log\log
t}{\sqrt{\log t}},$
for some $C>0$. The lower bound for $g$ follows at once by the definition of
$\widetilde{z}$.
To prove $(2)$, again by Sturm argument we can assume that the inequality
$K(t)\geq 1/[4t^{2}]$ holds. Proceeding along the same lines as for $(1)$ with
the choice $w=\sqrt{t}\log t$, and using $(2)$ of Theorem 4, we reach the
desired conclusion. ∎
The next prototype case illustrates the sharpness of our criteria. Let
$K(t)=\frac{1}{4t^{2}}+\frac{c^{2}}{4t^{2}\log^{2}t},\qquad\text{on
}[2,+\infty),$
where $c>0$ is a constant. Then, applying $(2)$ of Theorem 9, case $B=1/2$ we
deduce that $g$ oscillates whenever $c>1$. On the other hand, if $c\leq 1$, by
Theorem 10 $g$ is nonoscillatory. However, on $[2,+\infty)$
$\frac{1}{4}<k(t)=t\int_{t}^{+\infty}K(\sigma)\mathrm{d}\sigma\leq\frac{1}{4}+t\frac{c^{2}}{4t}\int_{t}^{+\infty}\frac{\mathrm{d}\sigma}{\sigma\log^{2}\sigma}=\frac{1}{4}+\frac{c^{2}}{4\log
t},$
hence the Hille-Nehari criterion cannot detect neither the oscillatory nor the
nonoscillatory behaviour of $g$ depending on $c$. Similarly, also Moore
criterion is not sharp enough. The proof of Theorem 10 suggests an iterative
improving procedure. In the general case, suppose that we are given an
ordinary differential equation of the type $(vz^{\prime})^{\prime}+Avz=0$,
with $v$ such that $\chi$ can be defined. By Sturm argument, there is no loss
of generality if we assume that $A\geq\chi$. An explicit solution $w$ of
$(vw^{\prime})^{\prime}+\chi vw=0$
is given by
$w(t)=-\sqrt{\int_{t}^{+\infty}\frac{\mathrm{d}s}{v(s)}}\log\int_{t}^{+\infty}\frac{\mathrm{d}s}{v(s)},$
and it is positive on some intervall $[r_{0},+\infty)$. Then,
$\widetilde{z}=z/w$ solves
$(\bar{v}\widetilde{z}^{\prime})^{\prime}+(A-\chi)\bar{v}\widetilde{z}=0\qquad\text{on
}[r_{0},+\infty),$
where $\bar{v}=vw^{2}$, which implies that $\widetilde{z}$, and therefore $z$,
are nonoscillatory if $(vw^{2})^{-1}\in L^{1}(+\infty)$ and
$A(r)-\chi(r)\leq\chi_{vw^{2}}(r),$
and oscillatory if $(vw^{2})^{-1}\in L^{1}(+\infty)$ and
$\limsup_{t\rightarrow+\infty}\int_{t_{0}}^{t}\Big{(}\sqrt{A(s)-\chi(s)}-\sqrt{\chi_{vw^{2}}(s)}\Big{)}\mathrm{d}s=+\infty.$
Now, the procedure can be pushed a step further by considering
$\widetilde{z}$. This enables us to construct finer and finer critical curves.
As an example, we refine Theorem 10. Suppose that
$K(t)\geq\frac{1}{4t^{2}}+\frac{1}{4t^{2}\log^{2}t}$
on, say, $[2,+\infty)$. Then, as in the proof of Theorem 10, define
$w=\sqrt{t}\log t$ and $v=w^{2}=t\log^{2}t$. Since $w$ is a positive solution
of $w^{\prime\prime}+(4t^{2})^{-1}w=0$ on some $[r_{1},+\infty)$, $z=g/w$ is
well defined and solves $(vz^{\prime})^{\prime}+Avz=0$ on $[r_{1},+\infty)$,
where
$A(t)=K(t)-\frac{1}{4t^{2}}\geq\frac{1}{4t^{2}\log^{2}t}=\chi_{w^{2}}(t)=\chi(t).$
Now, the function
$w_{2}(t)=-\sqrt{\int_{t}^{+\infty}\frac{\mathrm{d}s}{v(s)}}\log\int_{t}^{+\infty}\frac{\mathrm{d}s}{v(s)}=\frac{\log\log
t}{\sqrt{\log t}}$
is a solution of $(vw_{2}^{\prime})^{\prime}+\chi vw_{2}=0$, positive after
some $r_{2}\geq r_{1}$. Setting
$v_{2}(t)=v(t)w_{2}(t)^{2}=t\log t\log^{2}\log t,$
then
$\frac{1}{v_{2}(t)}\in L^{1}(+\infty),$
and the function $z_{2}=z/w_{2}$ is a solution of
$(v_{2}z_{2}^{\prime})^{\prime}+A_{2}v_{2}z_{2}=0$ on $[r_{2},+\infty)$, where
$A_{2}(t)=A(t)-\chi(t)=K(t)-\frac{1}{4t^{2}}-\frac{1}{4t^{2}\log^{2}t}\geq 0.$
Thus $z_{2}$, and hence $z$ and $g$, is nonoscillatory provided
$A_{2}(t)\leq\chi_{v_{2}}(t),\qquad\text{that is,}\qquad
K(t)\leq\frac{1}{4t^{2}}+\frac{1}{4t^{2}\log^{2}t}+\frac{1}{4t^{2}\log^{2}t\log^{2}\log
t},$
and, by (2.11), it is oscillatory if
$\limsup_{t\rightarrow+\infty}\left(\int_{t_{2}}^{t}\sqrt{K(\sigma)-\frac{1}{4\sigma^{2}}-\frac{1}{4\sigma^{2}\log^{2}\sigma}}\mathrm{d}\sigma-\frac{1}{2}\log\log\log
t\right)=+\infty.$
###### Remark 14.
_We mention that, with the aid of the change of variables ( 2.13) and (2.14),
Theorems 8, 9 and 10 can be applied to get sharp extensions of index estimates
for stationary Schrödinger operators on $\mathbb{R}^{m}$, $m\geq 3$, that
highly improve on classical results of M. Reed and B. Simon [17], and W.
Kirsch and B. Simon [11]. The interested reader can consult [2], Theorem
6.50._
## References
* [1] W. Ambrose, _A theorem of Myers_ , Duke Math. J. 24 (1957), 345–348.
* [2] B. Bianchini, L. Mari, and M. Rigoli, _On some aspects of oscillation theory and geometry_ , preprint.
* [3] , _Spectral radius, index estimates for Schrödinger operators and geometric applications_ , J. Funct. An. 256 (2009), 1769–1820.
* [4] E. Calabi, _On Ricci curvature and geodesics_ , Duke Math. J. 34 (1967), 667–676.
* [5] J. Cheeger, M. Gromov, and M. Taylor, _Finite propagation speed, Kernel estimates for functions of the Laplace operator and the geometry of complete Riemannian manifolds_ , J. Diff. Geom. 17 (1982), 15–53.
* [6] K.D. Elworthy and S. Rosenberg, _Manifolds with wells of negative curvature_ , Invent. Math. 103 (1991), no. 3, 471–495, With an appendix by Daniel Ruberman.
* [7] W.B. Fite, _Concerning the zeros of the solutions of certain differential equations_ , Trans. Amer. Math. Soc. 19 (1918), 341–352.
* [8] G.J. Galloway, _Compactness criteria for Riemannian manifolds_ , Proc. Amer. Math. Soc. 84 (1982), 106–110.
* [9] F.F. Guimar$\tilde{\mathrm{a}}$es, _The integral of the scalar curvature of complete manifolds without conjugate points_ , J. Diff. Geom. 36 (1992), no. 3, 651–662.
* [10] E. Hille, _Non-oscillation theorems_ , Trans. Amer. Math. Soc. 64 (1948), 234–252.
* [11] W. Kirsch and B. Simon, _Corrections to the classical behavior of the number of bound states of Schrödinger operators_ , Ann. Phys. 183 (1988), 122–130.
* [12] N. N. Lebedev, _Special Functions and Their Applications_ , Dover N.Y., 1972.
* [13] P. Mastrolia, M. Rimoldi, and G. Veronelli, _Myers’ type theorems and some related oscillation results_ , Available at arXiv:1002.2076.
* [14] R.A. Moore, _The behavior of solutions of a linear differential equation of second order_ , Pacific J. Math. 5 (1955), 125–145.
* [15] Z. Nehari, _Oscillation criteria for second-order linear differential equations_ , Trans. Amer. Math. Soc. 85 (1957), 428–445.
* [16] S. Pigola, M. Rigoli, and A.G. Setti, _Vanishing and finiteness results in Geometric Analisis. A generalization of the Böchner technique_ , Progress in Math., vol. 266, Birkäuser, 2008.
* [17] M. Reed and B. Simon, _Methods of Modern Mathematical Physics. IV. Analysis of Operators_ , Academic Press, New York-London, 1978.
* [18] S. Rosenberg and D. Yang, _Bounds on the fundamental group of a manifold with almost nonnegative Ricci curvature_ , J. Math. Soc. Japan 46 (1994), no. 2, 267–287.
* [19] C.A. Swanson, _Comparison and Oscillation Theory for Linear differential operators_ , Academic press, New York and London, 1968.
* [20] A. Wintner, _A criterion of oscillatory stability_ , Quart. Appl. Math. 7 (1949), 115–117.
* [21] J.Y. Wu, _Complete manifolds with a little negative curvature_ , Amer. J. Math. 113 (1991), no. 4, 567–572.
|
arxiv-papers
| 2011-12-16T03:22:42 |
2024-09-04T02:49:25.371838
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bruno Bianchini and Luciano Mari and Marco Rigoli",
"submitter": "Luciano Mari",
"url": "https://arxiv.org/abs/1112.3703"
}
|
1112.3707
|
# Time domain calculation of the electromagnetic self-force on eccentric
geodesics in Schwarzschild spacetime
Roland Haas Theoretical Astrophysics, California Institute of Technology,
Pasadena, CA 91125 Center for Relativistic Astrophysics, Georgia Institute of
Technology, Atlanta, GA 30332
###### Abstract
I calculate the self-force acting on a particle with electric charge $q$
moving on a generic geodesic around a Schwarzschild black hole. Using methods
similar to those developed for the scalar field case discussed in Haas (2007),
I investigate the relative sizes of the conservative (half-advanced plus half-
retarded) and dissipative (half-advanced minus half-retarded) pieces of the
self-force. I also display the regularization parameters used in the mode-sum
regularization scheme.
###### pacs:
04.25.-g, 04.40.-b, 41.60.-m, 45.50.-j, 02.60.Cb
## I Introduction
This is the second paper of a series of papers studying the self-force on a
point particle in generic geodesic orbit around a Schwarzschild black hole. I
extend the previous calculation of the scalar self-force Haas and Poisson
(2006) to electromagnetism, studying in particular the effects of the
conservative part of the self-force.
A test particle in orbit around a black hole will follow a geodesic. Going
beyond the test mass limit, this is no longer true and the particle’s path
will deviate from a geodesic of the background spacetime. As seen from the
background spacetime, the particle is said to experiences a self-force due to
its interaction with its own field. In order to accurately model the motion of
the body, including its inspiral toward the black hole, I seek to evaluate the
self-force and calculate its effect on the motion. Several methods to achieve
this have been proposed in the literature Barack and Ori (2000); Vega and
Detweiler (2008); Barack et al. (2007). I elect to use the mode-sum
regularization scheme introduced by Barack and Ori Barack and Ori (2000),
which been proven to be highly accurate.
In this paper, rather than dealing with the gravitational problem, I focus on
the technically simpler problem of a point particle endowed with an electric
charge $q$ orbiting a Schwarzschild black hole of mass $M$. In this context I
use a numerical simulation to check the analytically calculated regularization
parameters used in the mode-sum regularization scheme, which I calculate in a
manner analogous to Haas and Poisson (2006). This calculation also makes it
possible to investigate the behaviour of the conservative (half-advanced plus
half-retarded) part of the self-force in the strong-field limit, extending
previous work by Pound and Poisson Pound and Poisson (2008). Different from
the scalar case calculation, where the conservative self-force is suppressed,
the conservative electromagnetic self-force appears at the same post Newtonian
order as the gravitational conservative self-force. Agreement, even if only
qualitative, between the results for the electromagnetic problem, where our
physical intuition allows us to understand the mechanisms at work, and those
for a point mass recently explored by Barack and Sago (2011); Warburton et al.
(2011) can thus help provide a clearer understanding of the mechanisms at work
in the gravitational case as well.
Throughout the paper I use geometrized units in which $G=c=1$ and the sign
conventions of Misner et al. (1973).
### I.1 The problem
Since my approach is essentially identical to that described in Haas and
Poisson (2006) and Haas (2007) (paper I and paper II from now on), I will only
briefly introduce the required notation.
The first order self-force is calculated on a geodesic of Schwarzschild
spacetime, whose metric is written in Schwarzschild coordinates as
$\mathrm{d}s^{2}=-f\mathrm{d}t^{2}+f^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega\text{,}$
(1)
where $f=\left(1-\frac{2M}{r}\right)$,
$\mathrm{d}\Omega=\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}\right)$
is the metric on a two-sphere, and $t$, $r$, $\theta$ and $\phi$ are the usual
Schwarzschild coordinates. I numerically solve the Maxwell equations
$\displaystyle g^{\beta\gamma}\nabla_{\gamma}F_{\alpha\beta}(x)=4\pi
j_{\alpha}(x)\text{,}$ (2)
$\displaystyle\nabla_{[\gamma}F_{\alpha\beta]}(x)=0\text{,}$ (3)
$\displaystyle
j_{\alpha}(x)=q\int_{\gamma}u_{\alpha}(\tau)\delta_{4}\bm{(}x,z(\tau)\bm{)}\mathrm{d}\tau\text{,}$
(4)
where $\nabla_{\alpha}$ is the covariant derivative compatible with the metric
$g_{\alpha\beta}$, $F_{\alpha\beta}$ is the Faraday field tensor sourced by a
charge $q$ which moves along a world line $\gamma:\tau\mapsto z(\tau)$
parametrized by proper time $\tau$. The current density $j_{\alpha}(x)$
appearing on the right-hand side is written in terms of a scalarized four-
dimensional Dirac $\delta$-function
$\delta_{4}(x,x^{\prime})\equiv\delta(x_{0}-x^{\prime}_{0})\delta(x_{1}-x^{\prime}_{1})\delta(x_{2}-x^{\prime}_{2})\delta(x_{3}-x^{\prime}_{3})/\sqrt{-\det(g_{\alpha\beta})}$.
After having obtained the Faraday tensor I regularize it using the mode-sum
regularization scheme introduced by Barack and Ori Barack and Ori (2000)
$F^{R}_{(\mu)(\nu)}=F^{\text{ret}}_{(\mu)(\nu)}-q\sum_{\ell}\left[A_{(\mu)(\nu)}\Bigl{(}\ell+\frac{1}{2}\Bigr{)}+B_{(\mu)(\nu)}\right.\\\
\mbox{}\left.+\frac{C_{(\mu)(\nu)}}{\ell+\frac{1}{2}}+\frac{D_{(\mu)(\nu)}}{(\ell-\frac{1}{2})(\ell+\frac{3}{2})}+\cdots\right]\text{,}$
(5)
where indices in parenthesis $(\mu)$ signify components with respect to an
orthonormal tetrad $e^{\alpha}_{\ (\mu)}$ and the coefficients
$A_{(\mu)(\nu)}$, $B_{(\mu)(\nu)}$, $C_{(\mu)(\nu)}$, and $D_{(\mu)(\nu)}$ are
independent of $\ell$; they are listed in Appendix B. Finally I compute the
regularized self-force
$F^{\text{self}}_{\alpha}\equiv qF^{R}_{\alpha\beta}u^{\beta}$ (6)
from the regularized Faraday tensor and the four velocity of the particle.
### I.2 Organization of this paper
In Sec. II I introduce the ideas behind the discretization scheme used in the
numerical simulation. Sec. III describes the choices I make in order to handle
the problems of specifying initial data and proper boundary conditions. In
Sec. VII I describe the tests I performed in order to validate my
implementation of the numerical method. Sec. VIII contains sample results for
a small number of representative simulations. Finally in Sec. IX I calculate
the conservative self-force for the same set of simulations. The appendices
contain technical details and an alternative calculation using the vector
potential instead of the Faraday tensor.
## II Numerical method
In this section I describe the algorithm used to integrate the Maxwell
equations numerically. I use the second-order algorithm introduced by Lousto
and Price Lousto and Price (1997) suitably extended to handle a coupled system
of equations.
### II.1 Wave equations for the Faraday tensor
I introduce a vector potential $A_{\alpha}$ in terms of which the Faraday
tensor is given by
$\displaystyle F_{\alpha\beta}$
$\displaystyle=A_{\beta,\alpha}-A_{\alpha,\beta}\text{,}$ (7)
where a comma denotes an ordinary derivative. I use vector spherical harmonics
$Z^{\ell m}_{A}=\partial_{A}Y^{\ell m}$ and $X^{\ell
m}_{A}={\epsilon_{A}}^{B}\partial_{B}Y^{\ell m}$, where $\epsilon_{AB}$ is the
Levi-Civita tensor associated with the metric $\Omega_{AB}$ on the two-sphere
($\epsilon_{\theta\phi}=\sin\theta$), as introduced in Regge (1957); Martel
and Poisson (2005). I decompose the vector potential and the current density
into
$\displaystyle A_{a}(t,r,\theta,\phi)$ $\displaystyle=A^{\ell
m}_{a}(t,r)Y_{\ell m}(\theta,\phi)\text{,}$ (8a) $\displaystyle
j_{a}(t,r,\theta,\phi)$ $\displaystyle=j^{\ell m}_{a}(t,r)Y_{\ell
m}(\theta,\phi)$ for $a=t,r$, (8b) $\displaystyle A_{A}(t,r,\theta,\phi)$
$\displaystyle=v_{\ell m}(t,r)Z_{A}^{\ell m}(\theta,\phi)$
$\displaystyle\quad+\tilde{v}_{\ell m}(t,r)X_{A}^{\ell
m}(\theta,\phi)\text{,}$ (8c) $\displaystyle j_{A}(t,r,\theta,\phi)$
$\displaystyle=j^{\text{even}}_{\ell m}(t,r)Z_{A}^{\ell m}(\theta,\phi)$
$\displaystyle\quad+j^{\text{odd}}_{\ell m}(t,r)X_{A}^{\ell m}(\theta,\phi)$
for $A=\theta,\phi$, (8d)
where a summation over $\ell$ and $m$ is implied. Substituting these into Eq.
(2) I arrive at two sets of coupled equations for the even ($A^{\ell m}_{a}$,
$v_{\ell m}$) and odd ($\tilde{v}_{\ell m}$) modes
$\displaystyle-f\frac{\partial^{2}A_{t}^{\ell
m}}{\partial{r}^{2}}+f\frac{\partial^{2}A_{r}^{\ell m}}{\partial t\partial
r}-\frac{2f}{r}\frac{\partial A_{t}^{\ell m}}{\partial
r}+\frac{2f}{r}\frac{\partial A_{r}^{\ell m}}{\partial t}$
$\displaystyle\mbox{}-\frac{\ell(\ell+1)}{r^{2}}\frac{\partial v_{\ell
m}}{\partial t}+\frac{\ell(\ell+1)}{r^{2}}A_{t}^{\ell m}=4\pi j_{t}^{\ell
m}\text{,}$ (9a) $\displaystyle f^{-1}\frac{\partial^{2}A_{r}^{\ell
m}}{\partial{t}^{2}}-f^{-1}\frac{\partial^{2}A_{t}^{\ell m}}{\partial
t\partial r}-\frac{\ell(\ell+1)}{r^{2}}\frac{\partial v_{\ell m}}{\partial r}$
$\displaystyle\mbox{}+\frac{\ell(\ell+1)}{r^{2}}A_{r}^{\ell m}=4\pi
j_{r}^{\ell m}\text{,}$ (9b) $\displaystyle f^{-1}\frac{\partial^{2}v_{\ell
m}}{\partial{t}^{2}}-f\frac{\partial^{2}v_{\ell
m}}{\partial{r}^{2}}-\frac{2M}{r^{2}}\frac{\partial v_{\ell m}}{\partial
r}+f\frac{\partial A_{r}^{\ell m}}{\partial r}$
$\displaystyle\mbox{}-f^{-1}\frac{\partial A_{t}^{\ell m}}{\partial
t}+\frac{2M}{r^{2}}A_{r}^{\ell m}=4\pi j^{\text{even}}_{\ell m}\text{,}$ (9c)
$\displaystyle f^{-1}\frac{\partial^{2}\tilde{v}_{\ell
m}}{\partial{t}^{2}}-f\frac{\partial^{2}\tilde{v}_{\ell
m}}{\partial{r}^{2}}-\frac{2M}{r^{2}}\frac{\partial\tilde{v}_{\ell
m}}{\partial r}$
$\displaystyle\mbox{}+\frac{\ell(\ell+1)}{r^{2}}\tilde{v}_{\ell m}=4\pi
j^{\text{odd}}_{\ell m}\text{,}$ (9d)
where
$\displaystyle j^{\ell m}_{t}=-\frac{qf}{r_{0}^{2}}\bar{Y}^{\ell
m}(\frac{\pi}{2},\varphi_{0})\delta(r-r_{0})\text{,}$ (10a) $\displaystyle
j^{\ell m}_{r}=\frac{q\dot{r}_{0}}{Er_{0}^{2}}\bar{Y}^{\ell
m}(\frac{\pi}{2},\varphi_{0})\delta(r-r_{0})\text{,}$ (10b) $\displaystyle
j^{\text{even}}_{\ell m}=-\frac{imqfJ}{\ell(\ell+1)Er_{0}^{2}}\bar{Y}^{\ell
m}(\frac{\pi}{2},\varphi_{0})\delta(r-r_{0})\text{,}$ (10c) $\displaystyle
j^{\text{odd}}_{\ell
m}=-\frac{qfJ}{\ell(\ell+1)Er_{0}^{2}}\partial_{\theta}\bar{Y}^{\ell
m}(\frac{\pi}{2},\varphi_{0})\delta(r-r_{0})\text{.}$ (10d)
In the equation above an overbar denotes complex conjugation, an overdot
denotes differentiation with respect to $\tau$, $E=-u_{t}$ is the particle’s
conserved energy per unit mass, $J=u_{\phi}$ its conserved angular momentum
per unit mass, and $u^{\alpha}=\frac{\mathrm{d}z^{\alpha}}{\mathrm{d}\tau}$ is
its four velocity. Quantities bearing a subscript “$0$” are evaluated at the
particle’s position; they are functions of $\tau$ that are obtained by solving
the geodesic equation
$u^{\beta}\nabla_{\beta}u^{\alpha}=0$ (11)
in the background spacetime. Without loss of generality, I have confined the
motion of the particle to the equatorial plane $\theta=\frac{\pi}{2}$.
The three even mode equations Eq. (9a) – Eq. (9c) are not yet amenable to a
numerical treatment, as they are highly coupled. In order to obtain a more
convenient set of equation I define the auxiliary fields
$\displaystyle\psi^{\ell m}\equiv-r^{2}\left(\frac{\partial A_{t}^{\ell
m}}{\partial r}-\frac{\partial A_{r}^{\ell m}}{\partial t}\right)\text{,}$
(12) $\displaystyle\chi^{\ell m}\equiv f\,\left(A_{r}^{\ell m}-\frac{\partial
v^{\ell m}}{\partial r}\right)\text{,}$ (13) $\displaystyle\xi^{\ell m}\equiv
A_{t}^{\ell m}-\frac{\partial v^{\ell m}}{\partial t}\text{,}$ (14)
which, up to scaling factors, are just the even multipole moments of the $tr$,
$r\phi$ and $t\phi$ components of the Faraday tensor
$\displaystyle F_{tr}$ $\displaystyle=\sum_{\ell,m}\frac{\psi^{\ell
m}}{r^{2}}\,Y^{\ell m}\text{,}$ (15) $\displaystyle F_{tA}$
$\displaystyle=\sum_{\ell,m}(-\xi^{\ell m}\,Z^{\ell m}_{A}+\tilde{v}^{\ell
m}_{,t}\,X^{\ell m}_{A})\text{,}$ (16) $\displaystyle F_{rA}$
$\displaystyle=\sum_{\ell,m}(\frac{\chi^{\ell m}}{f}\,Z^{\ell
m}_{A}+\tilde{v}^{\ell m}_{,r}\,X^{\ell m}_{A})\text{,}$ (17) $\displaystyle
F_{\theta\varphi}$ $\displaystyle=\sum_{\ell,m}\tilde{v}_{\ell m}\,(X^{\ell
m}_{\phi,\theta}-X^{\ell m}_{\theta,\phi})$
$\displaystyle=-\sum_{\ell,m}\ell(\ell+1)\tilde{v}_{\ell
m}\,\sin(\theta)Y^{\ell m}\text{.}$ (18)
I note that the three fields $\psi^{\ell m}$, $\chi^{\ell m}$ and $\xi^{\ell
m}$ are not independent of each other, in fact knowledge of $\psi^{\ell m}$ is
sufficient to reconstruct $\chi^{\ell m}$ and $\xi^{\ell m}$. Eq. (9a) can be
rearranged to yield
$\displaystyle\xi^{\ell m}=-\frac{f}{\ell(\ell+1)}\frac{\partial\psi^{\ell
m}}{\partial r}-\frac{4\pi}{\ell(\ell+1)}j_{t}^{\ell m}\text{,}$ (19) and
similarly from Eq. (9b) $\displaystyle\chi^{\ell
m}=-\frac{1}{\ell(\ell+1)}\frac{\partial\psi^{\ell m}}{\partial t}-\frac{4\pi
f}{\ell(\ell+1)}j_{r}^{\ell m}\text{,}$ (20)
showing that knowledge of $\psi^{\ell m}$ is sufficient to reconstruct the
even multipole components of the Faraday tensor. In this work however I choose
to solve for $\chi^{\ell m}$ and $\xi^{\ell m}$ directly, rather than to
numerically differentiate $\psi^{\ell m}$ to obtain them. The gain in speed
from reducing the number of equations does not seem to offset the additional
time required to calculate $\psi^{\ell m}$ accurately enough to obtain good
approximations for its derivatives at the location of the particle. In this
approach Eqs. (19) and (20) are treated as constraints that the dynamical
variables have to satisfy.
Dropping the superscripts $\ell$, $m$ for notational convenience and following
Cunningham and Price (1979) I form linear combinations of derivatives of
Eqs.(9a) – (9c). I use
$[\partial_{r}(r^{2}\,\text{\eqref{eqn:mw-r}})-\partial_{t}(r^{2}\,\text{\eqref{eqn:mw-t}})]$
for $\psi$ and find
$\displaystyle\frac{\partial^{2}\psi}{\partial{r^{*}}^{2}}-\frac{\partial^{2}\psi}{\partial{t}^{2}}-V\psi=S_{\psi}\text{,}$
(21a) $\displaystyle S_{\psi}=4\pi f\left[\frac{\partial(r^{2}j_{t}^{\ell
m})}{\partial r}-\frac{\partial(r^{2}j_{r}^{\ell m})}{\partial
t}\right]\text{,}$ (21b) where $V=\ell(\ell+1)\frac{r-2M}{r^{3}}$ and
$r^{*}=r+2M\ln(\frac{r}{2M}-1)$ is the Regge-Wheeler tortoise coordinate.
Similarly I use
$[f\,\text{\eqref{eqn:mw-r}}-\partial_{r}(f\,\text{\eqref{eqn:mw-v}})]$ for
$\chi$ and $[\text{\eqref{eqn:mw-t}}-\partial_{t}\text{\eqref{eqn:mw-v}}]$ for
$\xi$. I find
$\displaystyle\frac{\partial^{2}\chi}{\partial{r^{*}}^{2}}-\frac{\partial^{2}\chi}{\partial{t}^{2}}-V\chi$
$\displaystyle=S_{\chi}\text{,}$ (21c) $\displaystyle S_{\chi}=4\pi
f\Biggl{[}\frac{\partial(fj^{\text{even}}_{\ell m})}{\partial r}$
$\displaystyle-fj_{r}^{\ell m}\Biggr{]}\text{,}$ (21d)
$\displaystyle\frac{\partial^{2}\xi}{\partial{r^{*}}^{2}}-\frac{\partial^{2}\xi}{\partial{t}^{2}}-V\xi-
V_{\xi}\psi$ $\displaystyle=S_{\xi}\text{,}$ (21e) $\displaystyle S_{\xi}=4\pi
f\Biggl{[}\frac{\partial(fj^{\text{even}}_{\ell m})}{\partial t}$
$\displaystyle-fj_{t}^{\ell m}\Biggr{]}\text{,}$ (21f)
where $V_{\xi}=\frac{2(r-3M)(r-2M)}{r^{5}}$. While still partially coupled
Eqs. (21b) – (21f) are much easier to deal with than the original set Eqs.
(9a) – (9c). The coupling is in the form of a staggering, which allows me to
first solve for $\psi$ and use this result in the calculation of $\xi$. On the
other hand, the source terms appearing on the right-hand side contain
derivatives of Dirac’s $\delta$-function resulting in fields that are
discontinuous at the location of the particle. Lousto’s scheme is designed to
cope with precisely this situation.
I derive explicit expressions for the source terms $S_{\alpha}$ on the right
hand sides
$\displaystyle
S_{\alpha}=G_{\alpha}(t)f_{0}\delta(r-r_{0})+F_{\alpha}(t)f\delta^{\prime}(r-r_{0})\text{,}$
(22a) $\displaystyle G_{\psi}(t)=-\frac{4\pi
q}{E^{2}}f_{0}\,\left(\ddot{r}_{0}-\frac{im\dot{r}_{0}J}{r_{0}^{2}}\right)\bar{Y}_{\ell
m}(\frac{\pi}{2},\varphi_{0})\text{,}$ (22b) $\displaystyle F_{\psi}(t)=4\pi
qf_{0}\left(\frac{\dot{r}_{0}^{2}}{E^{2}}-1\right)\bar{Y}_{\ell
m}(\frac{\pi}{2},\varphi_{0})\text{,}$ (22c) $\displaystyle
G_{\chi}(t)=-\frac{4\pi q\dot{r}_{0}}{Er_{0}^{2}}f_{0}\bar{Y}_{\ell
m}(\frac{\pi}{2},\varphi_{0})\text{,}$ (22d) $\displaystyle
F_{\chi}(t)=-\frac{4\pi qJim}{E\ell(\ell+1)r_{0}^{2}}f_{0}^{2}\bar{Y}_{\ell
m}(\frac{\pi}{2},\varphi_{0})\text{,}$ (22e)
$\displaystyle\begin{split}G_{\xi}(t)&=-4\pi
q\biggl{\\{}\frac{Jim}{E^{2}\ell(\ell+1)r_{0}^{2}}\biggl{[}\biggl{(}\frac{2M}{r_{0}^{2}}-\frac{2f_{0}}{r_{0}}\biggr{)}\dot{r}_{0}\\\
&\qquad-\frac{imJ}{r_{0}^{2}}\biggr{]}-\frac{1}{r_{0}^{2}}\biggr{\\}}f_{0}\bar{Y}_{\ell
m}(\frac{\pi}{2},\varphi_{0})\text{,}\end{split}$ $\displaystyle
F_{\xi}(t)=\frac{4\pi
qJim\dot{r}_{0}}{E^{2}\ell(\ell+1)r_{0}^{2}}f_{0}^{2}\bar{Y}_{\ell
m}(\frac{\pi}{2},\varphi_{0})\text{.}$ (22f)
My functions $G_{\alpha}$ and $F_{\alpha}$ correspond to $G/f_{0}$ and $F/f$
in Lousto and Price (1997), respectively, they are independent of $r$ (but do
contain terms in $r_{0}(t)$). I prefer this form of the source terms over the
form given in Lousto and Price (1997) since it simplifies the integral over
the source term Eq. (3.6) of Lousto and Price (1997)
$\iint\mathrm{d}AS=2\int_{t_{1}}^{t_{2}}\biggl{[}\frac{G\bm{(}r_{0}(t),t\bm{)}}{1-2M/r_{0}(t)}\\\
-\frac{\partial}{\partial
r}\biggl{(}\frac{F(r,t)}{1-2M/r}\biggr{)}\bigg{|}_{r=r_{0}(t)}\biggr{]}\mathrm{d}t\\\
\pm
2\frac{F\bm{(}r_{0}(t_{1}),t_{1}\bm{)}}{[1-2M/r_{0}(t_{1})]^{2}}[1\mp\dot{r}^{*}_{0}(t_{1})]^{-1}\\\
\pm
2\frac{F\bm{(}r_{0}(t_{2}),t_{2}\bm{)}}{[1-2M/r_{0}(t_{2})]^{2}}[1\pm\dot{r}^{*}_{0}(t_{2})]^{-1}\text{.}$
(23)
Since $G^{\text{Lousto}}=f_{0}G_{\alpha}(t)$ and
$G^{\text{Lousto}}=fF_{\alpha}(t)$, the first term in square brackets inside
the integral simplifies, while the second term vanishes completely.
$F_{\alpha}$ only appears in the boundary terms.
### II.2 Constraint equations
The full set of Maxwell equations consists of the inhomogeneous equations Eq.
(2) as well as the homogeneous constraints Eq. (3) which have to be satisfied
by a solution to Eq. (2). In the usual approach introducing a vector potential
$A_{\alpha}$ implies that the constraints are identically satisfied since they
reduce to the Bianchi identities for the second derivatives of $A_{\alpha}$.
When solving for the components of the Faraday tensor directly there is no a
priory guarantee that a solution to Eq. (21b) – (21f), and (9d) satisfies Eq.
(3). It turns out, however, that a decomposition into spherical harmonics is
sufficient to show that all but one of the constraints are identically
satisfied. The one that is not identically true is the $tr\varphi$ (or
$tr\theta$) equation, which in terms of $\psi$, $\chi$ and $\xi$ reads
$\frac{\psi}{r^{2}}-\frac{\chi_{,t}}{f}+\xi_{,r}=0\text{.}$ (24)
If the fields satisfy the sourced Maxwell equations Eqs. (9a), (9b), then Eq.
(24) is just the evolution equation for $\psi$. Thus Eq. (24) is valid
whenever $\psi$ satisfies the consistency relations Eq. (19) and (20).
Analytically then, the situation is clear. Given a set of compatible initial
conditions for $\psi$, $\chi$ and $\xi$ which initially satisfy the constraint
equations, a solution to the system of Eq. (21b) – (21f), (9d) satisfies the
full set of Maxwell equations at all later times, too.
Numerically I monitor but do not enforce Eq. (19) and (20). I generally find
that violations of the constraints are at least three orders of magnitude
smaller than the field quantities themselves. Figures 1 and 2 compare $\chi$
obtained from its evolution equations to that obtained from Eq. (20).
Figure 1: Violations of the constraint
$Z_{\chi}=\chi+\frac{1}{\ell(\ell+1)}\frac{\partial\psi}{\partial t}=0$ in the
vacuum region away from the location of particle. I plot the $\chi$ and
$\log_{10}\left\lvert{Z_{\chi}}\right\rvert$ as obtained on a spatial slice at
time $t=600\,M$. For this slightly eccentric orbit ($p=7.0$, $e=0.3$) using a
stepsize $h=1/512M$ the errors in the $\ell=2$, $m=2$ mode are at least three
orders of magnitude smaller than the field values. The exponentially growing
signal between $300M\lesssim r^{*}\lesssim 500$ is a remnant of the initial
data pulse travelling outward. Figure 2: Violations of the constraint
$Z_{\chi}=\chi+\frac{1}{\ell(\ell+1)}\frac{\partial\psi}{\partial t}=0$ at the
location of the particle as a function of time. I display $\chi$ and
$\log_{10}\left\lvert{Z_{\chi}}\right\rvert$ for the $\ell=5$, $m=3$ mode of a
particle on an eccentric orbit with $p=7.8001$, $e=0.9$ with stepsize
$h=1/256\,M$. During the time $400\,M\lesssim t\lesssim 800\,M$ the particle
is in the whirl phase. The exponentially decaying signal before $t\approx
250M$ is the initial data pulse.
### II.3 Monopole mode
For the electromagnetic field, the monopole mode $\ell=0$ is non-radiative.
The vector harmonics $Z^{\ell m}_{A}$ and $X^{\ell m}_{A}$ cannot be defined
in this case and the only surviving multipole mode is $\psi$. For the monopole
case Eq. (21b) reduces to a wave equation in flat space
$\frac{\partial^{2}\psi}{\partial{r^{*}}^{2}}-\frac{\partial^{2}\psi}{\partial{t}^{2}}=4\pi
f\left[\frac{\partial(r^{2}j_{t}^{0,0})}{\partial
r}-\frac{\partial(r^{2}j_{r}^{0,0})}{\partial t}\right]\text{,}$ (25)
which is simple enough so that I can solve it analytically. A straightforward
calculation shows that
$\psi(t,r^{*})=-\sqrt{4\pi}q\theta(r^{*}-r^{*}_{0}(t))$ (26)
satisfies Eq. (25) and corresponds to no outgoing radiation
$(\partial_{t}-\partial_{r^{*}})\psi=0$ at the event horizon and no ingoing
radiation $(\partial_{t}+\partial_{r^{*}})\psi=0$ at spatial infinity.
### II.4 Discretization—even sector
Lousto’s method is directly applicable to terms of the form
$-\frac{\partial^{2}\psi}{\partial{t}^{2}}+\frac{\partial^{2}\psi}{\partial{r^{*}}^{2}}$,
$V(r)\psi$ (ie. the wave operator and potential terms) on the left-hand side
of the equation and the source terms $S_{\alpha}(t)$ on the right hand side.
Here $\psi$ is used as a placeholder for any one of $\psi$, $\chi$ or $\xi$;
$V(r)$ is an expression depending only on $r$. I discretize these as
$\displaystyle\iint_{\text{\hbox
to0.0pt{cell\hss}}}\,\mathrm{d}u\,\mathrm{d}v\,\left(-\frac{\partial^{2}\psi}{\partial{t}^{2}}+\frac{\partial^{2}\psi}{\partial{r^{*}}^{2}}\right)=-4\left[\psi_{3}+\psi_{2}-\psi_{1}-\psi_{4}\right]\text{,}$
(27) $\displaystyle\iint_{\text{\hbox
to0.0pt{cell\hss}}}\,\mathrm{d}u\,\mathrm{d}v\,V(r)\psi=\begin{cases}h^{2}V_{0}\,\sum_{i}\psi_{i}+O(h^{4})&\text{vacuum
cells}\\\ V_{0}\,\sum_{i}A_{i}\psi_{i}+O(h^{3})&\text{sourced
cells,}\end{cases}$ (28)
and
$\displaystyle\iint_{\text{\hbox
to0.0pt{cell\hss}}}\,\mathrm{d}u\,\mathrm{d}v\,S_{\alpha}(t)$
$\displaystyle=2\int_{t_{1}}^{t_{2}}G_{\alpha}\bm{(}t,r_{0}(t)\bm{)}\,\mathrm{d}t$
$\displaystyle\qquad\pm\frac{2F_{\alpha}\bm{(}t_{1},r_{0}(t_{1})\bm{)}}{1-2M/r(t_{1})}[1\mp\dot{r}_{0}(t_{1})/E]^{-1}$
$\displaystyle\qquad\pm\frac{2F_{\alpha}\bm{(}t_{2},r_{0}(t_{2})\bm{)}}{1-2M/r(t_{2})}[1\pm\dot{r}_{0}(t_{2})/E]^{-1}\text{,}$
(29)
where $u=t-r^{*}$, $v=t+r^{*}$ are null coordinates, $\psi_{1}$,…,$\psi_{4}$
refer to values of the field at the points labelled $1$,…,$4$ in Fig. 3,
$h=\Delta_{t}=\Delta_{r^{*}}/2$ is the step size, $V_{0}$ is the value of the
potential at the centre of the cell, $A_{1}$,…,$A_{4}$ are the areas indicated
in Fig. 3 and $t_{1}$ and $t_{2}$ are the times at which the particle enters
and leaves the cell, respectively.
Figure 3: Points used to calculate the integral over the potential terms. Grid
points are indicated by blue circles.
Spelled out explicitly the evolution equations for vacuum cells are
$\displaystyle\psi_{3}$
$\displaystyle=-\psi_{2}+(1-\frac{h^{2}}{2}V_{0})(\psi_{1}+\psi_{4})\text{,}$
(30a) $\displaystyle\chi_{3}$
$\displaystyle=-\chi_{2}+(1-\frac{h^{2}}{2}V_{0})(\chi_{1}+\chi_{4})\text{,}$
(30b) $\displaystyle\xi_{3}$
$\displaystyle=-\xi_{2}+(1-\frac{h^{2}}{2}V_{0})(\xi_{1}+\xi_{4})$
$\displaystyle\quad-\frac{h^{2}}{4}V_{\xi,0}(\psi_{1}+\psi_{2}+\psi_{3}+\psi_{4})\text{,}$
(30c)
and for sourced cells
$\displaystyle\psi_{3}$
$\displaystyle=-[1+\frac{V_{0}}{4}(A_{2}-A_{3})]\psi_{2}+[1-\frac{V_{0}}{4}(A_{4}+A_{3})]\psi_{4}$
$\displaystyle\quad+[1-\frac{V_{0}}{4}(A_{1}+A_{3})]\psi_{1}$
$\displaystyle\quad-\frac{1}{4}(1-\frac{V_{0}}{4}A_{3})\iint\mathrm{d}u\,\mathrm{d}v\,S_{\psi}(t)\text{,}$
(31a) $\displaystyle\chi_{3}$
$\displaystyle=-[1+\frac{V_{0}}{4}(A_{2}-A_{3})]\chi_{2}+[1-\frac{V_{0}}{4}(A_{4}+A_{3})]\chi_{4}$
$\displaystyle\quad+[1-\frac{V_{0}}{4}(A_{1}+A_{3})]\chi_{1}$
$\displaystyle\quad-\frac{1}{4}(1-\frac{V_{0}}{4}A_{3})\iint\mathrm{d}u\,\mathrm{d}v\,S_{\chi}(t)\text{,}$
(31b) $\displaystyle\xi_{3}$
$\displaystyle=-[1+\frac{V_{0}}{4}(A_{2}-A_{3})]\xi_{2}+[1-\frac{V_{0}}{4}(A_{4}+A_{3})]\xi_{4}$
$\displaystyle\quad+[1-\frac{V_{0}}{4}(A_{1}+A_{3})]\xi_{1}$
$\displaystyle\quad-\frac{1}{4}V_{\xi,0}(A_{1}\psi_{1}+A_{2}\psi_{2}+A_{3}\psi_{3}+A_{4}\psi_{4})$
$\displaystyle\quad-\frac{1}{4}(1-\frac{V_{0}}{4}A_{3})\iint\mathrm{d}u\,\mathrm{d}v\,S_{\xi}(t)\text{.}$
(31c)
### II.5 Discretization–odd sector
When written in terms of $r^{*}$, Eq. (9d), which governs the odd modes
$\tilde{v}^{\ell m}$, is
$\displaystyle\frac{\partial^{2}\tilde{v}_{\ell
m}}{\partial{r^{*}}^{2}}-\frac{\partial^{2}\tilde{v}_{\ell
m}}{\partial{t}^{2}}-\frac{\ell(\ell+1)(r-2M)}{r^{3}}\tilde{v}_{\ell m}=-4\pi
fj^{\text{odd}}_{\ell m}\text{,}$ (32) $\displaystyle j^{\text{odd}}_{\ell
m}=-\frac{qJ}{\ell(\ell+1)Er_{0}^{2}}\partial_{\theta}\bar{Y}^{\ell
m}(\frac{\pi}{2},\varphi_{0})\delta(r^{*}-r^{*}_{0})\text{.}$ (33)
Eq. (32) is of the form of the scalar wave equation discussed in paper II. I
re-use the fourth order numerical code described there with
$V=\frac{\ell(\ell+1)(r-2M)}{r^{3}}$,
$S=4\pi\frac{qfJ}{\ell(\ell+1)Er_{0}^{2}}\partial_{\theta}\bar{Y}^{\ell
m}(\frac{\pi}{2},\varphi_{0})$. This yields accurate results for $\tilde{v}$
and its derivatives.
## III Initial values and boundary conditions
I follow the approach detailed in paper II for the scalar self-force and do
not specify physical initial data or an outer boundary condition. I
arbitrarily choose the fields to vanish on the characteristic slices
$u=u_{0}=t_{0}-r^{*}_{0}$ and $v=v_{0}=t_{0}+r^{*}_{0}$
$\psi(u=u_{0})=\psi(v=v_{0})=0\text{,}$ (34)
thereby adding a certain amount of spurious waves to the solution which show
up as an initial burst.
I implement ingoing wave boundary conditions near the event horizon,
sufficiently close that _numerically_ $r\approx 2M$, so that the potential
terms in Eqs. (21b) – (21f) vanish. This happens at $r^{*}\approx-73\,M$ and I
implement the ingoing waves condition $\partial_{u}\psi=0$ there. Near the
outer boundary this is not possible, since the potential decays slowly.
Instead I choose to evolve the full domain of dependence of the initial data
surface, hiding the effects of the boundary.
## IV Particle motion
I use the same approach as described in paper II to evolve the particle’s
motion, i.e. I introduce the semi-latus rectum $p$, the eccentricity $e$ and a
fictitious angle $\chi$, not to be confused with the Faraday tensor component
$\chi$ defined in Eq. (14), such that
$r(\tau)=\frac{pM}{1+e\cos\chi(\tau)}\text{.}$ (35)
The evolution is then governed by
$\displaystyle\begin{split}\frac{\mathrm{d}\chi}{\mathrm{d}t}=\frac{(p-2-2e\cos\chi)(1+e\cos\chi)^{2}}{(Mp^{2})}\\\
\mbox{}\times\sqrt{\frac{p-6-2e\cos\chi}{(p-2-2e)(p-2+2e)}}\text{,}\end{split}$
(36)
$\displaystyle\frac{\mathrm{d}\varphi}{\mathrm{d}t}=\frac{(p-2-2e\cos\chi)(1+e\cos\chi)^{2}}{p^{3/2}M\sqrt{(p-2-2e)(p-2+2e)}}\text{.}$
(37)
I use the embedded Runge-Kutta-Fehlberg (4, 5) algorithm provided by the GNU
Scientific Library routine gsl_odeiv_step_rkf45 and an adaptive step-size
control to evolve the position of the particle forward in time.
## V Extraction of field data at the particle
I use a straightforward one-sided extrapolation of field values to the right
of the particle’s position to extract values for $\psi$ and
$\partial_{r^{*}}\psi$. Specifically I fit a fourth order polynomial
$p(x)=\sum_{n=0}^{4}\frac{c_{i}}{n!}x^{n}\text{,}$ (38)
where $x=r^{*}-r^{*}_{0}$ to the five points to the right of the particle’s
current position and extract $\psi$ and $\partial_{r^{*}}\psi$ as $c_{0}$ and
$c_{1}$, respectively. In order to calculate
$\frac{\partial\psi(t_{0},r^{*}_{0})}{\partial t}$ I follow Sago and
calculate $\frac{\mathrm{d}\psi\bm{(}t,r^{*}(t)\bm{)}}{\mathrm{d}t}$ on the
world line of the particle. Since this can be calculated using either the
field values on the world line
$\frac{\mathrm{d}\psi\bm{(}t,r^{*}(t)\bm{)}}{\mathrm{d}t}=\\\
\mbox{}\frac{\psi\bm{(}t+h,r^{*}(t+h)\bm{)}-\psi\bm{(}t-h,r^{*}(t-h)\bm{)}}{2h}+O(h^{2})\text{,}$
(39)
or as
$\frac{\mathrm{d}\psi\bm{(}t,r^{*}(t)\bm{)}}{\mathrm{d}t}=\frac{\partial\psi}{\partial
t}+\frac{\partial\psi}{\partial
r^{*}}\frac{\mathrm{d}r^{*}_{0}}{\mathrm{d}t}\text{,}$ (40)
where both $\frac{\partial\psi}{\partial r^{*}}$ and
$\frac{\mathrm{d}r^{*}_{0}}{\mathrm{d}t}=\dot{r}_{0}/E$ are known, this allows
me to find
$\frac{\partial\psi}{\partial
t}=\frac{\mathrm{d}\psi\bm{(}t,r^{*}(t)\bm{)}}{\mathrm{d}t}-\frac{\partial\psi}{\partial
r^{*}}\frac{\mathrm{d}r^{*}_{0}}{\mathrm{d}t}\text{.}$ (41)
I repeat this procedure to the left of the particle. As a check for the
extraction procedure, I compare the difference between the right hand and left
hand values $\left[\psi\right]=\psi_{\text{right}}-\psi_{\text{left}}$ with
the analytically calculated jump conditions of appendix D.1. Similarly I check
whether the numerical solutions obtained for $\chi$ and $\xi$ directly are
consistent with Eqs. (20) and (19), which give them in terms of derivatives of
$\psi$.
## VI Regularization of the mode sum
The regularization procedure operates on scalar spherical harmonic modes of
the multipole coefficients $F^{\ell m}_{(\mu)(\nu)}$ of the Faraday tensor. As
a first step I use the auxiliary fields $\psi$, $\chi$ and $\xi$ to
reconstruct
$\displaystyle
A^{\ell^{\prime}m^{\prime}}_{r,t}-A^{\ell^{\prime}m^{\prime}}_{t,r}=\frac{\psi}{r^{2}}\text{,}$
(42a)
$\displaystyle\partial_{t}v^{\ell^{\prime}m^{\prime}}-A^{\ell^{\prime}m^{\prime}}_{t}=-\xi\text{,}$
(42b) and
$\displaystyle\partial_{r}v^{\ell^{\prime}m^{\prime}}-A^{\ell^{\prime}m^{\prime}}_{r}=-\frac{\chi}{f}\text{,}$
(42c)
the combinations of the vector potential modes needed to obtain the even
sector of a tensor spherical harmonic decomposition of the Faraday tensor. The
auxiliary field $\tilde{v}$ and its derivatives provide the odd sector of the
decomposition.
Using the comples pseudo-Cartesian tetrad $e^{\alpha}_{\ (0)}$, $e^{\alpha}_{\
(\pm)}$ and $e^{\alpha}_{\ (3)}$ introduced in paper I, I define tetrad
components
$F^{\text{ret}}_{(\mu)(\nu)}\equiv F^{\text{ret}}_{\alpha\beta}e^{\alpha}_{\
(\mu)}e^{\beta}_{\ (\nu)}$ (43)
of the Faraday tensor.
I construct the spherical harmonic modes of $F^{\text{ret}}_{(\mu)(\nu)}$
using the coupling coefficients displayed in Eq. (77).
$F^{\ell
m,\text{ret}}_{(\mu)(\nu)}=\sum_{\ell^{\prime},m^{\prime}}\left[C^{ab}_{(\mu)(\nu)}(\ell^{\prime}m^{\prime}|\ell
m)\left(A^{\ell^{\prime}m^{\prime}}_{b,a}-A^{\ell^{\prime}m^{\prime}}_{a,b}\right)\right.\\\
\left.+D^{a}_{(\mu)(\nu)}(\ell^{\prime}m^{\prime}|\ell
m)\left(\partial_{a}v^{\ell^{\prime}m^{\prime}}-A^{\ell^{\prime}m^{\prime}}_{a}\right)\right.\\\
\left.+E^{a}_{(\mu)(\nu)}(\ell^{\prime}m^{\prime}|\ell
m)\partial_{a}\tilde{v}^{\ell^{\prime}m^{\prime}}\right.\\\
\left.+E_{(\mu)(\nu)}(\ell^{\prime}m^{\prime}|\ell
m)\tilde{v}^{\ell^{\prime}m^{\prime}}\right]$ (44)
I calculate the multipole coefficients of $F^{\ell,\text{ret}}_{(\mu)(\nu)}$
as
$F^{\ell,\text{ret}}_{(\mu)(\nu)}=\sum_{m}F^{\ell
m,\text{ret}}_{(\mu)(\nu)}(t,r_{0})Y_{\ell
m}(\frac{\pi}{2},\varphi_{0})\text{,}$ (45)
and regularize them as in Eq. (5).
$F^{R}_{(\mu)(\nu)}=\sum_{\ell}\biggl{\\{}F^{\ell,\text{ret}}_{(\mu)(\nu)}-q\Bigl{[}A_{(\mu)(\nu)}\Bigl{(}\ell+\frac{1}{2}\Bigr{)}+B_{(\mu)(\nu)}+\\\
\frac{C_{(\mu)(\nu)}}{\ell+\frac{1}{2}}+\frac{D_{(\mu)(\nu)}}{(\ell-\frac{1}{2})(\ell+\frac{3}{2})}\Bigr{]}\biggr{\\}}$
(46)
I calculate the regularized self-force using
$F^{R}_{(\mu)}=qF^{R}_{(\mu)(\nu)}u^{(\nu)}$. Finally I reconstruct the vector
components of the self-force by from the tetrad components
$\displaystyle F^{R}_{t}=\sqrt{f_{0}}F^{R}_{(0)}\text{,}$ (47a) $\displaystyle
F^{R}_{r}=\frac{1}{\sqrt{f_{0}}}\operatorname{Re}\left(F^{R}_{(+)}e^{-i\varphi_{0}}\right)\text{,}$
(47b) $\displaystyle
F^{R}_{\phi}=r_{0}\operatorname{Im}\left(F^{R}_{(+)}e^{-i\varphi_{0}}\right)\text{.}$
(47c)
## VII Numerical tests
In this section I present the tests I performed to validate my numerical
evolution code. I performed the same set of tests as described in paper II.
First, in order to check the second-order convergence rate of the code, I
performed regression runs with increasing resolution. As a second test, I
computed the regularized self-force for several different combinations of
orbital elements $p$ and $e$ and checked that the multipole coefficients decay
with $\ell$ as expected. This provided a very sensitive check on the overall
implementation of the numerical scheme as well as the analytical calculations
that lead to the regularization parameters.
### VII.1 Convergence tests
Convergence tests are a straightforward way to test the implementation of a
numerical scheme. I performed regression runs for my second-order convergent
code using a non-zero charge $q$ and an eccentric orbit. I extract the field
at the position of the particle, and thus also test the implementation of the
extraction algorithm described in section V. I choose the $\ell=6$, $m=4$ mode
of the field generated by a particle on a mildly eccentric geodesic orbit with
$p=7$, $e=0.3$. As shown in Fig. 4 the convergence is approximately of second
order.
Figure 4: Convergence test of the numerical algorithm in the sourced case. I
show differences between simulations using different step sizes of 16, 32 and
64 cells per $M$. Displayed are the rescaled differences
$\delta_{32-16}=\xi(h=1/32M)-\xi(h=1/16M)$ etc. of the field values at the
position of the particle for a simulation with $\ell=6$, $m=4$ and $p=7$,
$e=0.3$. I see that the convergence is approximately second-order. The curves
are rescaled in such a way as to provide an estimate for the error of the
highest resolution run compared to the real ($h\equiv 0$) solution.
In the region $150\,M\lesssim t\lesssim 400\,M$ the two curves lie on top of
each other, as expected for a second-order convergent algorithm. In the region
from $400\,M$ to $450\,M$ there is some difference between the two lines,
caused by cell crossing effects similar to those discussed in paper II.
### VII.2 Discontinuity across the world line
The singular source term on the right hand side of Eqs. (21b) – (21f) implies
that the fields $\psi$, $\chi$ and $\xi$ are discontinuous across the world
line. Since the jump conditions can be calculated analytically as done in
appendix D.1, I can check whether the numerical results faithfully reproduce
the expected behaviour. Using the methods described in section V I obtain one-
sided extrapolation for the field values and their spatial derivatives. For
the highest resolution run used in the regression analysis in section VII.1 I
find that the numerical results for $\xi$ agree with the analytical
calculation of the jump condition up to terms of the order of $10^{-8}$; two
orders of magnitude smaller than the estimated numerical error of $10^{-6}$.
For $\partial_{r^{*}}\xi$ the situation is reversed, with the numerical error
in the jump condition being about an order of magnitude larger than the
numerical error in the field derivative itself. The accuracy of the numerical
derivatives is therefore limited by the accuracy of the extraction scheme,
resulting in about three significant figures for the set of parameters
displayed in Fig. 4. However the regularization calculation is constructed in
such a way that no derivatives of the fields need be obtained in order to
calculate the self-force. I therefore feel that I can accept the reduced
accuracy provided by the simple extraction scheme.
### VII.3 High-$\ell$ behaviour of the multipole coefficients
Inspection of Eq. (5) reveals that a plot of $F^{\ell}_{(\mu)(\nu)}$ as a
function of $\ell$ (for a fixed value of $t$) should display a linear growth
in $\ell$ for large $\ell$. Removing the $A_{(\mu)(\nu)}$ term should produce
a constant curve, removing the $B_{(\mu)(\nu)}$ term (given that
$C_{(\mu)(\nu)}=0$) should produce a curve that decays as $\ell^{-2}$, and
finally, removing the $D_{(\mu)(\nu)}$ term should produce a curve that decays
as $\ell^{-4}$. It is a powerful test of the overall implementation to check
whether the numerical data behaves as expected. Fig. 5 plots the remainders as
obtained from my numerical simulation, demonstrating the expected behaviour.
Figure 5: Multipole coefficients of the dimensionless Faraday tensor component
$\frac{M^{2}}{q}\operatorname{Im}F^{R}_{(+)(-)}$ for a particle on an
eccentric orbit ($p=7.2$, $e=0.5$). The coefficients are extracted at
$t=500\,M$ along the trajectory shown in Fig. 6. The plots show several stages
of the regularization procedure, with a closer description of the curves to be
found in the text. A uniform stepsize of $h=1/512\,M$ was used.
It displays, on a logarithmic scale, the absolute value of
$\operatorname{Im}F^{\ell,R}_{(+)(-)}$, the imaginary part of the
$F^{R}_{(+)(-)}$ tetrad component of the Faraday tensor. The orbit is
eccentric ($p=7.2$, $e=0.5$), and all components of the self-force require
regularization. The first curve (in triangles) shows the unregularized
multipole coefficients that increase linearly in $\ell$, as confirmed by
fitting a straight line to the data. The second curve (in squares) shows
partially regularized coefficients, obtained after the removal of
$(\ell+1/2)A_{(\mu)(\nu)}$; this clearly approaches a constant for large
values of $\ell$. The curve made up of diamonds shows the behaviour after
removal of $B_{(\mu)(\nu)}$; because $C_{(\mu)(\nu)}=0$, it decays as
$\ell^{-2}$, a behaviour that is confirmed by a fit to the $\ell\geq 5$ part
of the curve. Finally, after removal of
$D_{(\mu)(\nu)}/[(\ell-\frac{1}{2})\,(\ell+\frac{3}{2})]$ the terms of the sum
decrease in magnitude approximately as $\ell^{-4}$ when fitting to the data
points $\ell\geq 7$. This result depends slightly on the range of points used
for the fit. I expect this to be due to the fact that I stop at $\ell=15$,
which seems to be not large enough to show the asymptotic behaviour. Extending
the range to very high values of $\ell$ proved to be very difficult, since the
numerical code is only second order convergent, so that the numerical errors
become dominant by the time the asymptotic behaviour begins to show.
Each one of the last two curves would result in a converging sum, but the
convergence is faster after subtracting the $D_{(\mu)(\nu)}$ terms. I thereby
gain about one order of magnitude in the accuracy of the estimated sum.
Figure 5 provides a sensitive test of the implementation of both the numerical
and analytical parts of the calculation. Small mistakes in either one will
cause the difference in Eq. (5) to have a vastly different behaviour.
### VII.4 Accuracy of the numerical method
In this work I are less demanding with the numerical accuracy then I were in
paper II, where I describe a very high accuracy numerical code. Implementing
suach a code is very tedious even for the scalar case, and much more so for
the electromagnetic case treated here. Therefore I implement a simpler method
that allows me to access the physics of the problem without being hindered by
technical problems due to a complicated numerical method.
An estimate for the truncation error arising from cutting short the summation
in Eq. (5) at some $\ell_{\text{max}}$ can be calculated by considering the
behaviour of the remaining terms for large $\ell$. Detweiler et. al. Detweiler
et al. (2003) showed that the remaining terms scale as $\ell^{-4}$ for large
$\ell$. They find the functional form of the terms to be
$\frac{E\mathcal{P}_{3/2}}{(2\ell-3)(2\ell-1)(2\ell+3)(2\ell+5)}\text{,}$ (48)
where $\mathcal{P}_{3/2}=36\sqrt{2}$. I fit a function of this form to the
tail end of a plot of the multipole coefficients to find the coefficient $E$
in Eq. (48). Extrapolating to $\ell\rightarrow\infty$ I find that the
truncation error is
$\displaystyle\varepsilon$
$\displaystyle=\sum_{\ell=\ell_{\text{max}}}^{\infty}[\text{Eq.~{}\eqref{eqn:Eterm}}]$
(49)
$\displaystyle=\frac{12\sqrt{2}E\ell_{\text{max}}}{(2\ell_{\text{max}}+3)(2\ell_{\text{max}}+1)(2\ell_{\text{max}}-1)(2\ell_{\text{max}}-3)}\text{,}$
(50)
where $\ell_{\text{max}}$ is the value at which I cut the summation short.
A second source of error lies in the numerical calculation of the retarded
solution to the wave equation. This error depends on the step size $h$ used to
evolve the field forward in time. For a numerical scheme of a given
convergence order, I can estimate this discretization error by extrapolating
from simulations using different step sizes down to $h=0$. This is what was
done in the graphs shown in Sec. VII.1.
I display results for the mildly eccentric orbit shown in Fig. 6 with data
extracted at $t=500\,M$, that is at the instant shown in Fig. 5. At this
moment, the multipole coefficients of $\operatorname{Re}(F^{R}_{(+)})$ decay
as expected, but e.g. the $\operatorname{Im}(F^{R}_{(+)})$ component decays
faster with $\ell$ for the range of modes $0\leq\ell\leq 13$ modes that were
calculated. I choose an orbit of low eccentricity as high eccentricity causes
the field values to be plagued by high frequency noise, as discussed in paper
II. This makes it impossible to reliably estimate the discretization error for
these orbits.
Table 1 lists typical values for the errors discussed above.
error estimation | mildly eccentric orbit
---|---
relative truncation error in $\frac{M^{2}}{q^{2}}\operatorname{Re}(F^{R}_{(+)})$ | $2\times 10^{-4}$
relative discretization error in $\frac{M^{2}}{q}\psi$ | $\approx 10^{-7}$
Table 1: Estimated values for the various errors in the components of the
self-force as described in the text. I show the truncation and discretization
errors for the mildly eccentric orbit ($p=7.2$, $e=0.5$). The truncation error
is calculated using a plot similar to the one shown in Fig. 5. The
discretization error is estimated using a plot similar to that in Fig. 4 for
the $\ell=2$, $m=2$ mode.
## VIII Sample results
In this section I describe some results obtained from my numerical
calculation.
### VIII.1 Mildly eccentric orbit
I choose a particle on an eccentric orbit with $p=7.2$, $e=0.5$ which starts
at $r=pM/(1-e^{2})$, halfway between periastron and apastron. The field is
evolved for $600\,M$ with a uniform resolution of 512 grid points per $M$,
both in the $t$ and $r^{*}$ directions, for all values of $\ell$. Multipole
coefficients for $1\leq\ell\leq 15$ are calculated and used to reconstruct the
regularized self-force $F_{\alpha}$ along the geodesic. Figure 7 shows the
result of the calculation.
Figure 6: Trajectory of a particle with $p=7.2$, $e=0.5$. The cross-hair
indicates the point where the data for Fig. 5 was extracted. Figure 7:
Regularized dimensionless self-force $\frac{M^{2}}{q^{2}}F_{t}$,
$\frac{M^{2}}{q^{2}}F_{r}$ and $\frac{M}{q^{2}}F_{\phi}$ on a particle on an
eccentric orbit with $p=7.2$, $e=0.5$.
For the choice of parameters used to calculate the force shown in Fig. 7, the
error bars corresponding to the truncation error Eq. (49) (which are already
much larger than than the discretization error) would be of the order of the
line thickness and have not been drawn.
Already for this small eccentricity, I see that the self-force is most
important when the particle is closest to the black hole (ie. for
$200\,M\lesssim t\lesssim 400\,M$). The self-force acting on the particle is
very small once the particle has moved away to $r\approx 15\,M$.
### VIII.2 Zoom-whirl orbit
Particles on highly eccentric orbits are of most interest as sources of
gravitational radiation. For nearly parabolic orbits with $e\lesssim 1$ and
$p\gtrsim 6+2e$, a particle revolves around the black hole a number of times,
moving on a nearly circular trajectory close to the event horizon (“whirl
phase”), before moving away from the black hole (“zoom phase”). During the
whirl phase the particle is in the strong field region of the spacetime,
emitting copious amounts of radiation. Figures 8 and 9 show the trajectory of
a particle and the force on such an orbit with $p=7.8001$, $e=0.9$ calculated
using a uniform step size of $h=1/256\,$ throughout the range $1\leq\ell\leq
15$.
Figure 8: Trajectory of a particle on a zoom-whirl orbit with $p=7.8001$,
$e=0.9$. The cross-hairs indicate the positions where the data shown in Fig.
10 and 11 was extracted. Figure 9: Self-force acting on a particle. Shown is
the dimensionless self-force $\frac{M^{2}}{q^{2}}F_{t}$,
$\frac{M^{2}}{q^{2}}F_{r}$ and $\frac{M}{q^{2}}F_{\phi}$ on a zoom-whirl orbit
with $p=7.8001$, $e=0.9$. No error bars showing an estimate error are shown,
as the errors shown are to small to show up on the graph. Notice that the
self-force is essentially zero during the zoom phase $900\,M\lesssim t\lesssim
1200\,M$ and reaches a constant value very quickly after the particle enters
into the whirl phase.
Even more so than for the mildly eccentric orbit discussed in Sec. VIII.1, the
self-force (and thus the amount of radiation produced) is much larger while
the particle is close to the black hole than when it zooms out. The force
graph is very similar to that obtained for the scalar self-force in paper II,
however the overshooting behaviour at the onset and near the end of the whirl
phase is not as pronounced.
Since the rates of change in energy $E$ and angular momentum $J$ of the
trajectory are directly related to the self-force
$\dot{E}=-a_{t}\text{,}\qquad\dot{J}=a_{\phi}\text{,}$ (51)
it is easy to see that the self-force shown in Fig. 9 confirms the expectation
that the self-force decreases both the energy and angular momentum of the
particle while radiation is emitted.
In Fig. 10 and Fig. 11 I show plots of $F^{\ell}_{(0)}$ constructed from
$F^{\ell}_{(\mu)(\nu)}$ after the removal of the $A_{(\mu)(\nu)}$,
$B_{(\mu)(\nu)}$, and $D_{(\mu)(\nu)}$ terms.
Figure 10: Multipole coefficients of
$\frac{M^{2}}{q}\operatorname{Re}F^{R}_{(0)}$ for a particle on a zoom-whirl
orbit ($p=7.8001$, $e=0.9$). The coefficients are extracted at $t=525\,M$ when
the particle is deep within the whirl phase. Here $\dot{r}\approx 0$ and the
behaviour of $F^{R}_{(\mu),\ell}$ is very close to that for a circular orbit,
requiring very little regularization. Red triangles are used for the
unregularized multipole coefficients $F_{(0),\ell}$, squares, diamonds and
disks are used for the partly regularized coefficients after the removal of
the $A_{(0)}$, $B_{(0)}$ and $D_{(0)}$ terms respectively. Figure 11:
Multipole coefficients of $\frac{M^{2}}{q}\operatorname{Re}F^{R}_{(0)}$ for a
particle on a zoom-whirl orbit ($p=7.8001$, $e=0.9$). The coefficients are
extracted at $t=1100\,M$ when the particle is far away from the black hole. As
$\dot{r}$ is non-zero, all components of the self-force require regularization
and I see that the dependence of the multipole coefficients on $\ell$ is as
predicted by Eq. 5. After the removal of the regularization parameters
$A_{(\mu)(\nu)}$, $B_{(\mu)(\nu)}$, and $D_{(\mu)(\nu)}$ the remainder is
proportional to $\ell^{0}$, $\ell^{-2}$ and $\ell^{-4}$ respectively.
## IX Effects of the conservative self-force
In this section only, I will use the subscript “0” to denote quantities
evaluated on the unperturbed geodesic, and no subscript to denote quantities
evaluated on the perturbed world-line.
I follow the literature (see e.g. Pound and Poisson (2008)) and define the
dissipative part to be the half retarded minus half advanced force and the
conservative part to be the half retarded plus half advanced force
$\displaystyle
F^{\text{diss}}_{\alpha}\equiv\frac{1}{2}\left(F^{\text{ret}}_{\alpha}-F^{\text{adv}}_{\alpha}\right)\text{,}$
(52) $\displaystyle
F^{\text{cons}}_{\alpha}\equiv\frac{1}{2}\left(F^{\text{ret}}_{\alpha}+F^{\text{adv}}_{\alpha}\right)\text{.}$
(53)
The conservative force is the time reversal invariant part of the self-force.
It does not affect the radiated energy or angular momentum fluxes $\dot{E}$
and $\dot{J}$; it shifts the values of $E$ and $J$ away from their geodesic
values, affecting the orbital motion and the phase of the emitted waves.
To obtain expressions for $E$ and $J$ under the influence of the self-force, I
employ the procedure described in Diaz-Rivera et al. (2004). I begin by
writing down the normalization condition for the four velocity
$-1=u^{\alpha}u_{\alpha}=-\frac{E^{2}}{f}+\frac{J^{2}}{r^{2}}\text{,}$ (54)
as well as the $r$-component of the geodesic equation
$\frac{F^{r}}{m}=\ddot{r}-\frac{M}{(r-2M)r}\dot{r}^{2}-\frac{(r-2M)J^{2}}{r^{4}}+\frac{ME^{2}}{(r-2M)r}\text{,}$
(55)
where $F^{r}=qF^{r}_{\ \mu}u^{\mu}$ is the radial component of the self-force.
Solving Eq. (54) and (55) I find
$\displaystyle E^{2}=E_{0}^{2}-\frac{(r-2M)r}{r-3M}\frac{F^{r}}{m}\text{,}$
(56) $\displaystyle J^{2}=J_{0}^{2}-\frac{r^{4}}{r-3M}\frac{F^{r}}{m}\text{,}$
(57)
where
$\displaystyle E_{0}^{2}$
$\displaystyle=\dot{r}^{2}+\frac{(r-2M)r\ddot{r}}{r-3M}+\frac{(r-2M)^{2}}{(r-3M)r}\text{,}$
(58) $\displaystyle J_{0}^{2}$
$\displaystyle=\frac{r^{4}\ddot{r}}{r-3M}+\frac{Mr^{2}}{r-3M}\text{.}$ (59)
I stress that $E_{0}$ and $J_{0}$ are not the geodesic _values_ for energy and
angular momentum. They are of the correct form but are evaluated using the
_accelerated_ values for $r$, $\dot{r}$ and $\ddot{r}$ (instead of the
geodesic values $r_{0}$, $\dot{r}_{0}$, etc.).
For small perturbing force of order $\varepsilon$ I expand Eqs. (56) and (57)
in terms of the perturbation strength and find
$\displaystyle E=E_{0}+\Delta E\approx
E_{0}-\varepsilon\frac{(r-2M)r}{2(r-3M)E_{0}}\frac{F^{r}}{m}+O(\varepsilon^{2})\text{,}$
(60) $\displaystyle J=J_{0}+\Delta J\approx
J_{0}-\varepsilon\frac{r^{4}}{2(r-3M)J_{0}}\frac{F^{r}}{m}+O(\varepsilon^{2})\text{,}$
(61)
where $F^{r}$ is evaluated with the help of the unperturbed four velocity
$u_{0}^{\alpha}=[E_{0}/f,\dot{r}_{0},0,J_{0}/r_{0}^{2}]$. The fractional
changes $\Delta E/E_{0}$ and $\Delta J/J_{0}$ are given by
$\displaystyle\Delta
E/E_{0}=-\varepsilon\frac{(r-2M)r}{2(r-3M)E_{0}^{2}}\frac{F^{r}}{m}+O(\varepsilon^{2})\text{,}$
(62) $\displaystyle\Delta
J/J_{0}=-\varepsilon\frac{r^{4}}{2(r-3M)J_{0}^{2}}\frac{F^{r}}{m}+O(\varepsilon^{2})\text{.}$
(63)
Once the perturbations in $E$ and $J$ are known, I calculate the change in the
angular frequency
$\Omega\equiv\frac{\mathrm{d}\varphi}{\mathrm{d}t}=\frac{r-2M}{r^{3}}\frac{J}{E}\text{.}$
(64)
For small perturbing force I expand in powers of the perturbation strength
$\Omega=\frac{r_{0}-2M}{r_{0}^{3}}\frac{J_{0}}{E_{0}}\biggl{[}1-\varepsilon\biggl{(}\frac{r^{4}}{2(r-3M)J_{0}^{2}}\\\
-\frac{(r-2M)r}{2(r-3M)E_{0}^{2}}\biggr{)}\frac{F^{r}}{m}\biggr{]}+O(\varepsilon^{2})\text{.}$
(65)
The relative change $\Delta\Omega/\Omega_{0}$ is given by
$\Delta\Omega/\Omega_{0}=-\varepsilon\biggl{(}\frac{r^{4}}{2(r-3M)J_{0}^{2}}-\frac{(r-2M)r}{2(r-3M)E_{0}^{2}}\biggr{)}\frac{F^{r}}{m}\\\
+O(\varepsilon^{2})\text{.}$ (66)
### IX.1 Circular orbits
The effect of the conservative self-force is most clearly observed for
circular orbits, where the unperturbed angular frequency $\Omega$ as well as
the shift due to the perturbation are constant in time.
For a particle in circular motion the self-force is constant in time and it
turns out that the radial component is entirely conservative whereas the $t$
and $\phi$ components are entirely dissipative. For circular orbits, the
unperturbed values of $E$ and $J$ are given by
$\displaystyle E_{0}$
$\displaystyle=\frac{r_{0}-2M}{\sqrt{r_{0}(r_{0}-3M)}}\text{,}$ (67)
$\displaystyle J_{0}$ $\displaystyle=r_{0}\sqrt{\frac{M}{r_{0}-3M}}\text{,}$
(68)
and substituting these into Eq. (65) I find
$\Omega=\sqrt{\frac{M}{r_{0}^{3}}}-\frac{(r_{0}-3M)}{2mM}\sqrt{\frac{M}{r_{0}}}F_{r}+O(\varepsilon^{2})\text{,}$
(69)
where the first term is just the angular frequency for an unperturbed geodesic
at radius $r_{0}$. The fractional change $\Delta\Omega/\Omega_{0}$ is then
$\frac{\Delta\Omega}{\Omega_{0}}=-\frac{(r_{0}-3M)r_{0}}{2mM}F_{r}+O(\varepsilon^{2})\text{.}$
(70)
Similarly the fractional changes in $E$ and $J$: $\Delta E/E_{0}$ and $\Delta
J/J_{0}$ are given by
$\displaystyle\Delta E/E_{0}$ $\displaystyle=-\frac{r_{0}}{2m}F_{r}\text{,}$
(71) $\displaystyle\Delta J/J_{0}$
$\displaystyle=-\frac{(r_{0}-2M)r_{0}}{2mM}F_{r}\text{.}$ (72)
Figure 12: Fractional change $\Delta\Omega/\Omega_{0}$ induced by the presence
of the conservative self-force. The effect of the self-force is to move the
radius of the orbit outward, decreasing its angular frequency.
Figure 12 shows the fractional change in $\Omega_{0}$, $E$ and $J$ as a
function of the orbit’s radius $r_{0}$.
### IX.2 Eccentric orbits
For eccentric orbits the self-force is no longer constant in time and I have
to numerically calculate both the retarded and the advanced self-force in
order to construct the conservative self-force. I find the advanced force by
running the simulation backwards in time. That is I start the evolution on the
very last time slice and evolve backwards in time until I reach the slice
corresponding to $t=0$. I reverse the boundary condition at the event horizon
to be be outgoing radiation only $(\partial_{t}+\partial_{r^{*}})\psi=0$ and
adjust the outer boundary so as to simulate only the backwards domain of
dependence of the initial slice. I do not change the trajectory of the
particle. I do not change the regularization parameters, since they depend
only on the local behaviour of the field and are insensitive to the boundary
conditions far away.
#### IX.2.1 Conservative force on zoom-whirl orbits
I calculate the conservative self-force on a zoom-whirl orbit with $p=7.8001$,
$e=0.9$. Figs. 13 and 14
Figure 13: $r$ component of the dimensionless self-force acting on a particle
on a zoom-whirl orbit ($p=7.8001$, $e=0.9$) around a Schwarzschild black hole.
Shown are the retarded (solid, red), advanced (dashed, green), conservative
(dotted, blue) and dissipative (finely dotted, pink) force acting on the
particle. Figure 14: $\varphi$ component of the dimensionless self-force
acting on a particle on a zoom-whirl orbit ($p=7.8001$, $e=0.9$) around a
Schwarzschild black hole. Shown are the retarded (solid, red), advanced
(dashed, green), conservative (dotted, blue) and dissipative (finely dotted,
pink) force acting on the particle.
display the breakdown of the self-force into retarded and advanced, and
conservative and dissipative parts for a particle on a zoom-whirl orbit. In
both plots the force is very weak when the particle is in the zoom phase
$t\lesssim 400\,M$ or $t\gtrsim 800\,M$ and nearly constant while the particle
is in the whirl phase $400\,M\lesssim t\lesssim 800\,M$. Inspection of the
behaviour of the $r$ component reveals that it is almost exclusively
conservative, with only a tiny dissipative effect when the particle enters or
leaves the whirl phase. This result is consistent with the observation that
the particle moves on a nearly circular trajectory while in the whirl phase,
for which the radial component is precisely conservative. Similarly the $\phi$
component is almost entirely dissipative, with only a small conservative
contribution when the particle enters or leaves the whirl phase, its maximum
coinciding with that of $\ddot{r}$ (not shown on the graph).
I calculate the relative changes in $E$, $J$ and $\Omega$ under the influence
of the self-force using Eqs. (62), (63), (66). Fig. 15
Figure 15: Relative change in $\Omega$, $E$, $J$ for a particle on a zoom-
whirl orbit due to the conservative electromagnetic self-force.
displays the relative changes $\Delta E/E_{0}$, $\Delta J/J_{0}$ and
$\Delta\Omega/\Omega_{0}$ for a particle on a zoom whirl orbit $p=7.8001$,
$e=0.9$. The change in $E$, $J$ and $\Omega$ is strongest in the whirl phase
when $r\approx 4.1M$. It is consistent with the shift experienced by a
particle on a circular orbit at $4.1M$.
### IX.3 Effects on the innermost stable orbit
In the gravitational case, considerable work has been done to identify gauge
invariant effects of the self-force Detweiler (2008a); Sago et al. (2008). The
electromagnetic self-force is not subject to the same ambiguity thus it can
help shed light on the gravitational case as well by providing a clear
distinction between kinetic and dynamic effects. In this section I calculate
the effect of the conservative self-foce on the location of the innermost
stable circular orbit around a Schwarzschild black hole. Such a calculation
was first performed for the scalar self-force by Diaz-Rivera et al. (2004),
where a highly accurate frequency domain numerical scheme was used. Recently
Barack and Sago (2009, 2011) have extended this calculation to gravity, using
their time domain code to perform the intergration of the wave equation. Since
the code presented in this paper is in the time domain as well, it is closest
in spirit to Barack and Sago (2009).
## X Retardation of the self-force
For scalar perturbation in a weak gravitational field Poisson Pfenning and
Poisson (2002) showed that the self-force is delayed with respect to the
particle motion by the light travel time from the particle to the central body
and back to the particle again. In a spacetime where the central body is
compact the treatment of Pfenning and Poisson (2002) is no longer directly
applicable, but I still expect some retardation in the self-force when
compared to the particle’s motion. To study this effect, I calculate the self-
force on an eccentric orbit with $p=78$, $e=0.9$; ten times larger than the
zoom-whirl orbit discussed earlier. The large orbit was chosen so as to be
able to clearly see any possible retardation which might not be visible if the
particle’s orbit is deep within the strong field region close to the black
hole.
Figure 16: $r$ component of the dimensionless self-force acting on a particle
on an orbit with $p=78$, $e=0.9$. Shown are the retarded and advanced forces
as well as $\dot{r}$. The vertical line at $t\approx 2383\,M$ marks the time
of closest approach to the black hole. Figure 17: $\phi$ component of the
dimensionless self-force acting on a particle on an orbit with $p=78$,
$e=0.9$. Shown are the retarded and advanced forces as well as $\dot{r}$. The
vertical line at $t\approx 2383\,M$ marks the time of closest approach to the
black hole.
Figures 16 and 17 display plots of the $r$ and $\phi$ components of the self-
force acting on the particle close to periastron. Shown are the retarded and
advanced forces as well as the particle’s radial velocity $\dot{r}$. Without
considering retardation I expect the self-force to be strongest when the
particle is closest to the black hole, when $\dot{r}=0$, as evident in Fig.
13. Clearly for the $r$ component displayed in Fig. 16 the retarded and
advanced forces both peak at a time very close to the zero crossing of
$\dot{r}$, suggesting very little time delay in the $r$ component of the self-
force. In Fig. 17 on the other hand the retarded and advanced $\phi$-component
of the self-force peaks away from the time of closest approach
$t_{\text{min}}$. Inspection of the graph shows that the delay (advance)
between the time of closest approach and the peak in the retarded (advanced)
force is compatible with a delay of $\Delta t_{\text{min}}\approx
2(r_{\text{min}}-3.0\,M)\approx 74\,M$. Using a delay of $\Delta t\approx
2[r_{0}(t)-3.0\,M]$ and plotting $F^{\text{ret}}_{\varphi}(t+\Delta t)$ and
$-F^{\text{adv}}_{\varphi}(t-\Delta t)$ versus $t$ both curves visually lie on
top of each other and the maximum is located at $t_{\text{min}}$ as shown in
Fig. 18 below. This suggests that the self-force is in large parts due to
radiation that travels into the strong field region close to the black hole
and is scattered back to the particle. The time delay can then be loosely
interpreted as the time it takes the signal to travel to the light ring around
the black hole and back to the particle. This interpretation is loose for two
reasons: First $r^{*}$ and not $r$ is associated with the light travel time.
Using $r^{*}$, however, does not lead to a better overlap of the curves once a
suitable constant offset chosen. Second, for the zoom-whirl orbit shown in
Fig. 9 the (shallow) maximum in the self-force is offset by only $\Delta
t\approx 2[r_{0}(t)-1.0\,M]$ which leads to a reasonable overlap of the two
curves. Interestingly using $r^{*}$ instead of $r$ yields a worse overlap. For
very large orbits $p=780$, $e=0.9$ it is impossible to read off the small
constant offset to the dominant $2r_{0}(t)$ contribution.
## XI Weak field limit
As a last application I use my code to compare the numerical self-force in the
weak field region to the self-force calculated using the weak field expression
$\bm{f}_{\text{self}}=\lambda_{c}\frac{q^{2}}{m}\frac{M}{r^{3}}\hat{\bm{r}}+\lambda_{rr}\frac{2}{3}\frac{q^{2}}{m}\frac{\mathrm{d}\bm{g}}{\mathrm{d}t}\text{,}\qquad\bm{g}=-\frac{M}{r^{2}}\hat{\bm{r}}\text{,}$
(73)
of Pound and Poisson (2008); DeWitt-Morette and Dewitt (1964). I calculate the
self-force for a particle on an eccentric orbits with $e=0.9$ and $p=78$ or
$p=780$. Fig. 18
Figure 18: $\phi$ component of the retarded (solid red line) and (dashed green
line) negative advanced self-forces acting on a particle with $p=78$, $e=0.9$.
The forces have been shifted by $\Delta t\approx 2[r_{0}(t)-3.0\,M]$. Also
shown is the self-force calculated using the weak field expression Eq. 73
(blue dotted line).
shows the retarded and (negative) advanced forces shifted by $\Delta t\approx
2[r_{0}(t)-3.0\,M]$ as well as the analytic force calculated using Eq. (73).
At this distance there are still some differences between the (shifted)
retarded field and the weak field expression. One reason for this lies in the
choice of a suitable $r$ coordinate to correspond to the $r$ coordinate in the
weak field expression. In this work I use the areal Schwarzschild $r$, but the
isotropic coordinate $\bar{r}=\frac{r-M}{2}+\frac{\sqrt{r\,(r-2*M)}}{2}$ or
even the tortoise $r^{*}$ could be used as well. Neither one yields a good
agreement between the two curves.
For $p=780$ using a shift of $\Delta t=2r_{0}(t)$ the agreement between
numerical data and analytic expression is excellent as is evident in Fig. 19.
At this distance $r$, $\bar{r}$ and $r^{*}$ are indistinguishable.
Figure 19: $\phi$ component of the retarded self-force acting on a particle on
an orbit with $p=780$, $e=0.9$ close to periastron. Shown are the numerical
(solid, red) and shifted analytical (dashed, green) forces. The agreement
between numerical and analytical calculation is excellent, the discrepancy for
$t\lesssim 7500\,M$ is due to initial data contamination.
## XII Conclusions
I calculated the self-force acting a on an electromagnetic point charge in
orbit around a Schwarzschild black hole. To do so I calculated the
regularization parameters $A$,$B$, and $D$ in section B and implemented a
second order accurate numerical scheme in section II.
I find the behaviour of the electromagnetic self-force to be similar but not
identical to that of the scalar self-force. In both cases the self-force is
strongest when the particle is closest to the black hole. Further, during the
whirl phase of a zoom-whirl orbit with its nearly constant radius, the self-
force is very close to that of a particle in circular orbit at this radius. On
the other hand, the overshooting effect upon entering the whirl phase which
was observed in the scalar case is much weaker in the electromagnetic case.
I calculated the effects of the conservative self-force on circular orbits,
where it reduces the angular frequency and thus affects the phasing of the
observed waves. I find this effect to be much stronger in the electromagnetic
case than in the scalar case discussed in Diaz-Rivera et al. (2004). In
particular during the nearly circular whirl phase of a zoom-whirl orbit I find
that $\Omega$ decreases by $\approx 0.06\frac{q^{2}}{Mm}$. Due to the
smallness of the ratio $\frac{q^{2}}{Mm}$ this change is tiny for one orbit,
however since it accumulates over the inspiral, its effect on the total phase
shift during the full inspiral can be of order unity. This statement is not
directly transferable to the gravitational case since the radius $r_{0}$ of
the orbit is not a gauge invariant quantity. Therefore I cannot distinguish
between changes in $\Omega$ due to effects of the self-force and due to gauge
choices. To obtain a meaningful measure of the effect of the gravitational
self-force I need to compare two gauge invariant quantities, e.g. $\Omega$ and
the gauge invariant $u_{t}$ of Detweiler (2008b).
I investigated the retardation of the self-force with respect to the motion of
the particle. I found that the retardation is very weak for the $r$ component
of the force and strong in the $t$ and $\varphi$ components, which are linked
to radiated energy and angular momentum. In the later cases the retardation is
compatible with a delay of $\Delta t\approx 2(r_{0}(t)-R_{\text{delay}})$,
where $R_{\text{delay}}$ is a constant depending on the particle’s orbit.
###### Acknowledgements.
I thank Eric Poisson and Steve Detweiler for useful discussions and
suggestions. I gratefully acknowledge support by the Natural Sciences and
Engineering Council of Canada. This work was supported in part by NSF grants
PHY-0903973 and PHY-0904015. This work was made possible by the facilities of
the Shared Hierarchical Academic Research Computing Network
(SHARCNET:www.sharcnet.ca) as well as the e FoRCE cluster at Georgia Tech.
## Appendix A Translation tables
I require coupling coefficients to translate between the tensor harmonic modes
of the Faraday tensor and the scalar harmonic modes of the tetrad components
of the Faraday tensor.
As a first step, I reconstruct the Faraday tensor modes from the numerical
variables. For the even mode auxiliary fields $\psi$, $\chi$ and $\xi$ this
reconstruction can be done algebraically while the odd sector requires a
numerical differentiation of the numerical variable $\tilde{v}$. The
reconstruction relations were already displayed in Eqs. (15) – (18), which
involves both the even and odd modes.
In terms of the vector potential the Faraday tensor modes are reconstructed
using the defining equation Eq. (7). In this case, the reconstruction of the
Faraday tensor reads
$\displaystyle F_{tr}$ $\displaystyle=\sum_{\ell,m}(A^{\ell m}_{r,t}-A^{\ell
m}_{t,r})\,Y^{\ell m}\text{,}$ (74a) $\displaystyle F_{tA}$
$\displaystyle=\sum_{\ell,m}[(v^{\ell m}_{,t}-A^{\ell m}_{t})\,Z^{\ell
m}_{A}+\tilde{v}^{\ell m}_{,t}\,X^{\ell m}_{A}]\text{,}$ (74b) $\displaystyle
F_{rA}$ $\displaystyle=\sum_{\ell,m}[(v^{\ell m}_{,r}-A^{\ell m}_{r})\,Z^{\ell
m}_{A}+\tilde{v}^{\ell m}_{,r}\,X^{\ell m}_{A}]\text{,}$ (74c) $\displaystyle
F_{\theta\varphi}$ $\displaystyle=\sum_{\ell,m}\tilde{v}_{\ell m}\,(X^{\ell
m}_{\varphi,\theta}-X^{\ell m}_{\theta,\varphi})\text{.}$ (74d)
Clearly both the expansion Eqs. (15) – (18) and the one in Eqs. (74a) – (74d)
are of the same form and it is only necessary to obtain one set of translation
coefficients to handle both the calculation using $\psi$, $\chi$ and $\xi$ in
the main text and the one using the vector potential that will be presented in
appendix C.
The tetrad components $F_{(\mu)(\nu)}$ are decomposed in terms of scalar
spherical harmonics
$F_{(\mu)(\nu)}=\sum_{\ell,m}F^{\ell m}_{(\mu)(\nu)}Y_{\ell m}\text{,}$ (75)
where each mode is given by
$F^{\ell m}_{(\mu)(\nu)}=\int F_{(\mu)(\nu)}\bar{Y}_{\ell
m}\,\mathrm{d}\Omega\text{.}$ (76)
To obtain expressions for the coupling coefficients I substitute
$F_{(\mu)(\nu)}=F_{\alpha\beta}e^{\alpha}_{\ (\mu)}e^{\beta}_{\ (\nu)}$ into
Eq. (76)
$\displaystyle F^{\ell m}_{(\mu)(\nu)}$
$\displaystyle=\int\mathrm{d}\Omega\,F_{(\mu)(\nu)}\bar{Y}^{\ell m}$
$\displaystyle=\int\mathrm{d}\Omega\,\left(A_{\beta,\alpha}-A_{\alpha,\beta}\right)e^{\alpha}_{\
(\mu)}e^{\beta}_{\ (\nu)}\bar{Y}^{\ell m}$
$\displaystyle=\int\sum_{\ell^{\prime},m^{\prime}}\Bigl{[}\left(A_{b,a}-A_{a,b}\right)e^{a}_{\
(\mu)}e^{b}_{\ (\nu)}\bar{Y}^{\ell m}$
$\displaystyle\quad+\left(A_{b,A}-A_{A,b}\right)e^{A}_{\ (\mu)}e^{b}_{\
(\nu)}\bar{Y}^{\ell m}$
$\displaystyle\quad+\left(A_{B,a}-A_{a,B}\right)e^{a}_{\ (\mu)}e^{B}_{\
(\nu)}\bar{Y}^{\ell m}$
$\displaystyle\quad+\left(A_{B,A}-A_{A,B}\right)e^{A}_{\ (\mu)}e^{B}_{\
(\nu)}\bar{Y}^{\ell m}\Bigr{]}\mathrm{d}\Omega$
$\displaystyle\equiv\sum_{\ell^{\prime},m^{\prime}}\left[C^{ab}_{(\mu)(\nu)}(\ell^{\prime}m^{\prime}|\ell
m)\left(A^{\ell^{\prime}m^{\prime}}_{b,a}-A^{\ell^{\prime}m^{\prime}}_{a,b}\right)\right.$
$\displaystyle\quad\left.+D^{a}_{(\mu)(\nu)}(\ell^{\prime}m^{\prime}|\ell
m)\left(\partial_{a}v^{\ell^{\prime}m^{\prime}}-A^{\ell^{\prime}m^{\prime}}_{a}\right)\right.$
$\displaystyle\quad\left.+E^{a}_{(\mu)(\nu)}(\ell^{\prime}m^{\prime}|\ell
m)\partial_{a}\tilde{v}^{\ell^{\prime}m^{\prime}}\right.$
$\displaystyle\quad\left.+E_{(\mu)(\nu)}(\ell^{\prime}m^{\prime}|\ell
m)\tilde{v}^{\ell^{\prime}m^{\prime}}\right]\text{,}$ (77)
which defines the coupling coefficients. It is often possible to express these
coupling coefficients in terms of linear combinations of the coupling
coefficients derived in paper I for the scalar field.
To simplify the notation of the coupling coefficients I use
$\displaystyle\gamma^{\ell m}$
$\displaystyle=\sqrt{\frac{(\ell+m)(\ell+m+1)}{(2\ell+1)(2\ell+3)}}\text{,}$
(78) $\displaystyle\epsilon^{\ell m}$
$\displaystyle=\sqrt{\frac{(\ell+m+1)(\ell-m+1)}{(2\ell+1)(2\ell+3)}}\text{,}$
(79)
as shorthands for recurring combinations of terms. With these the reusable
scalar coupling coefficients are written as
$\displaystyle C^{r}_{(+)}(\ell^{\prime}m^{\prime}|\ell m)$
$\displaystyle=-\gamma^{\ell-1,m}\sqrt{f}\delta_{\ell^{\prime}\ell-1}\delta_{m^{\prime}m-1}$
$\displaystyle\quad+\gamma^{\ell,-m+1}\sqrt{f}\delta_{\ell^{\prime}\ell+1}\delta_{m^{\prime}m-1}$
(80) $\displaystyle C_{(+)}(\ell^{\prime}m^{\prime}|\ell m)$
$\displaystyle=\gamma^{\ell-1,m}\frac{\ell-1}{r}\delta_{\ell^{\prime}\ell-1}\delta_{m^{\prime}m-1}$
$\displaystyle\quad+\gamma^{\ell,-m+1}\frac{\ell+2}{r}\delta_{\ell^{\prime}\ell+1}\delta_{m^{\prime}m-1}\text{,}$
(81) $\displaystyle C_{(-)}(\ell^{\prime}m^{\prime}|\ell m)$
$\displaystyle=-\gamma^{\ell-1,-m}\frac{\ell-1}{r}\delta_{\ell^{\prime}\ell-1}\delta_{m^{\prime}m+1}$
$\displaystyle\quad-\gamma^{\ell,m+1}\frac{\ell+2}{r}\delta_{\ell^{\prime}\ell+1}\delta_{m^{\prime}m+1}\text{.}$
(82)
Similarly it proves useful to define lower order coupling coefficients for the
odd sector, which is absent in the scalar case.
$\displaystyle E_{(+)}(\ell^{\prime}m^{\prime}|\ell
m)=-\frac{i}{r}\sqrt{(\ell-m+1)(\ell+m)}\delta_{\ell^{\prime}\ell}\delta_{m^{\prime}m-1}\text{,}$
(83) $\displaystyle E_{(-)}(\ell^{\prime}m^{\prime}|\ell
m)=-\frac{i}{r}\sqrt{(\ell+m+1)(\ell-m)}\delta_{\ell^{\prime}\ell}\delta_{m^{\prime}m+1}\text{.}$
(84)
In terms of the scalar coupling coefficients, the first coefficient for the
expansion of $F_{(0)(+)}$ is given by
$\displaystyle C^{tr}_{(0)(+)}(\ell^{\prime}m^{\prime}|\ell m)$
$\displaystyle=\int
Y^{\ell^{\prime}m^{\prime}}e^{t}_{(0)}e^{r}_{(+)}\bar{Y}^{\ell
m}\,\mathrm{d}\Omega$ $\displaystyle=\frac{1}{\sqrt{f}}\int
Y^{\ell^{\prime}m^{\prime}}e^{r}_{(+)}\bar{Y}^{\ell m}\,\mathrm{d}\Omega$ (85)
$\displaystyle=C^{r}_{(+)}(\ell^{\prime}m^{\prime}|\ell m)/\sqrt{f}\text{,}$
and all other combinations of $a$, $b$ and $(\mu)$, $(\nu)$ lead to vanishing
$C^{ab}_{(\mu)(\nu)}$. Similarly for the remaining non-vanishing coefficients
for the $F_{(0)(+)}$ component
$\displaystyle D^{t}_{(0)(+)}(\ell^{\prime}m^{\prime}|\ell m)$
$\displaystyle=C_{(+)}(\ell^{\prime}m^{\prime}|\ell m)/\sqrt{f}\text{,}$ (86)
$\displaystyle E^{t}_{(0)(+)}(\ell^{\prime}m^{\prime}|\ell m)$
$\displaystyle=E_{(+)}(\ell^{\prime}m^{\prime}|\ell m)/\sqrt{f}\text{.}$ (87)
The coupling coefficients for $F_{(+)(-)}$ contain both even and odd modes.
The first non-vanishing one is $D^{r}_{(+)(-)}(\ell^{\prime}m^{\prime}|\ell
m)$, which is given by
$\displaystyle D^{r}_{(+)(-)}(\ell^{\prime}m^{\prime}|\ell m)$
$\displaystyle=\sqrt{f}\gamma^{\ell,-m+1}C_{(-)}(\ell^{\prime}m^{\prime}|\ell+1,m-1)$
$\displaystyle\quad-\sqrt{f}\gamma^{\ell-1,m}C_{(-)}(\ell^{\prime}m^{\prime}|\ell-1,m-1)$
$\displaystyle\quad+\sqrt{f}\gamma^{\ell,m+1}C_{(+)}(\ell^{\prime}m^{\prime}|\ell+1,m+1)$
$\displaystyle-\sqrt{f}\gamma^{\ell-1,-m}C_{(+)}(\ell^{\prime}m^{\prime}|\ell-1,m+1)\text{,}$
(88)
while the coefficients coupling to odd modes are
$\displaystyle E^{r}_{(+)(-)}(\ell^{\prime}m^{\prime}|\ell m)$
$\displaystyle=\sqrt{f}\gamma^{\ell,-m+1}E_{(-)}(\ell^{\prime}m^{\prime}|\ell+1,m-1)$
$\displaystyle\quad-\sqrt{f}\gamma^{\ell-1,m}E_{(-)}(\ell^{\prime}m^{\prime}|\ell-1,m-1)$
$\displaystyle\quad+\sqrt{f}\gamma^{\ell,m+1}E_{(+)}(\ell^{\prime}m^{\prime}|\ell+1,m+1)$
$\displaystyle-\sqrt{f}\gamma^{\ell-1,-m}E_{(+)}(\ell^{\prime}m^{\prime}|\ell-1,m+1)\text{,}$
(89) $\displaystyle E_{(+)(-)}(\ell^{\prime}m^{\prime}|\ell m)$
$\displaystyle=-2i/r^{2}(\ell+1)(\ell+2)\epsilon^{\ell
m}\delta_{\ell^{\prime}\ell+1}\delta_{m^{\prime}m}$
$\displaystyle\quad-2i/r^{2}(\ell-1)\ell\epsilon^{\ell-1,m}\delta_{\ell^{\prime}\ell-1}\delta_{m^{\prime}m}\text{.}$
(90)
## Appendix B Regularization parameters
I mimic the treatment in paper I and start from the covariant expression for
the singular vector potential tensor in Eq. (464) of Poisson (2004)
$\nabla_{\beta}A^{S}_{\alpha}(x)=-\frac{q}{2r^{2}}U_{\alpha\beta^{\prime}}u^{\beta^{\prime}}\nabla_{\beta}r-\frac{q}{2r_{\text{adv}}^{2}}U_{\alpha\beta^{\prime\prime}}u^{\beta^{\prime\prime}}\nabla_{\beta}r_{\text{adv}}\\\
\mbox{}+\frac{q}{2r}U_{\alpha\beta^{\prime};\beta}u^{\beta^{\prime}}+\frac{q}{2r}U_{\alpha\beta^{\prime};\gamma^{\prime}}u^{\beta^{\prime}}u^{\gamma^{\prime}}\nabla_{\beta}u+\frac{q}{2r_{\text{adv}}}U_{\alpha\beta^{\prime\prime};\beta}u^{\beta^{\prime\prime}}\\\
\mbox{}+\frac{q}{2r_{\text{adv}}}U_{\alpha\beta^{\prime\prime};\gamma^{\prime\prime}}u^{\beta^{\prime\prime}}u^{\gamma^{\prime\prime}}\nabla_{\beta}v+\frac{1}{2}qV_{\alpha\beta^{\prime}}u^{\beta^{\prime}}\nabla_{\beta}u\\\
\mbox{}-\frac{1}{2}qV_{\alpha\beta^{\prime\prime}}u^{\beta^{\prime\prime}}\nabla_{\beta}v-\frac{1}{2}q\int_{u}^{v}\nabla_{\beta}V_{\alpha\mu}\bm{(}x,z(\tau)\bm{)}u^{\beta}(\tau)\,\mathrm{d}\tau\text{.}$
(91)
I have introduced a large number of symbols. $x$ is the point where the field
is evaluated, $x^{\prime}$ and $x^{\prime\prime}$ are the retarded and
advanced points of $x$ on the world line $z(\tau)$. They are connected to $x$
with unique future-directed and past-directed null geodesics, respectively.
$u(x)$ and $v(x)$ are the retarded and advanced time functions such that
$x^{\prime}=z(\tau=u)$, $x^{\prime\prime}=z(\tau=v)$. $u^{\alpha^{\prime}}$
and $u^{\alpha^{\prime\prime}}$ are the four velocity at $x^{\prime}$ and
$x^{\prime\prime}$ respectively. Further I define Synge’s world function
$\sigma(x,\bar{x})$ which is numerically equal to half the squared geodesic
distance between two points $x$ and $\bar{x}$. Using its gradient
$\sigma_{\alpha}=\nabla_{\alpha}\sigma(x,\bar{x})$, I define
$r=u^{\alpha^{\prime}}\sigma_{\alpha^{\prime}}(x,x^{\prime})$ and
$r_{\text{adv}}=-u^{\alpha^{\prime\prime}}\sigma_{\alpha^{\prime\prime}}(x,x^{\prime\prime})$,
the affine parameter distances of $x$ away from the world line along its
future/past light cone. The potentials $U$ and $V$ appearing in Eq. (91) are
the direct and tail parts of the retarded Green function
$G_{\alpha\bar{\beta}}(x,\bar{x})$ associated with the wave operator.
From the definition of $r$, $r_{\text{adv}}$, $u$, and $v$ it follows that
(see Section 3.3.3 of Poisson (2004))
$\displaystyle\nabla_{\alpha}u=-\sigma_{\alpha}(x,x^{\prime})/r\text{,}$ (92)
$\displaystyle\nabla_{\alpha}v=\sigma_{\alpha}(x,x^{\prime\prime})/r_{\text{adv}}\text{,}$
(93)
$\displaystyle\nabla_{\alpha}r=-\sigma_{\alpha^{\prime}\beta^{\prime}}u^{\alpha^{\prime}}u^{\beta^{\prime}}\nabla_{\alpha}u+\sigma_{\alpha^{\prime}\alpha}u^{\alpha^{\prime}}\text{,}$
(94)
$\displaystyle\nabla_{\alpha}r_{\text{adv}}=-\sigma_{\alpha^{\prime\prime}\beta^{\prime\prime}}u^{\alpha^{\prime\prime}}u^{\beta^{\prime\prime}}\nabla_{\alpha}u-\sigma_{\alpha^{\prime\prime}\alpha}u^{\alpha^{\prime\prime}}\text{,}$
(95)
which are valid for geodesic motion.
The potentials $U_{\alpha\beta^{\prime}}$, $U_{\alpha\beta^{\prime\prime}}$
are determined by Eq. (322) of Poisson (2004)
$\displaystyle{U_{\alpha}}^{\beta^{\prime}}=g^{\beta^{\prime}}_{\phantom{\beta^{\prime}}\alpha}\Delta^{1/2}(x,x^{\prime})\text{,}$
(96)
$\displaystyle{U_{\alpha}}^{\beta^{\prime\prime}}=g^{\beta^{\prime\prime}}_{\phantom{\beta^{\prime\prime}}\alpha}\Delta^{1/2}(x,x^{\prime\prime})\text{,}$
(97)
where $g^{\bar{\mu}}_{\phantom{\bar{\mu}}\nu}$ is the parallel propagator from
$x^{\nu}$ to $\bar{x}^{\mu}$ and
$\Delta\equiv\det\bm{(}-g^{\alpha^{\prime}}_{\phantom{\alpha^{\prime}}\alpha}\sigma^{\alpha}_{;\beta^{\prime}}\bm{)}$
is the van Vleck determinant. Of the potentials $V_{\alpha\beta^{\prime}}$ and
$V_{\alpha\beta^{\prime\prime}}$ appearing in Eq. (91) I only need to know the
scaling behaviour following from Eq. (320) of Poisson (2004):
$\displaystyle V_{\alpha\beta^{\prime}}=O(\varepsilon^{2})\text{,}$ (98)
$\displaystyle V_{\alpha\beta^{\prime\prime}}=O(\varepsilon^{2})\text{,}$ (99)
$\displaystyle\nabla_{\beta}V_{\alpha\mu}=O(\varepsilon)\text{.}$ (100)
These expressions are valid in vacuum spacetimes where the Ricci tensor
vanishes.
Again mirroring the calculation in paper I, I introduce the arbitrary point
$\bar{x}\equiv z(\bar{\tau})$ on the world line and expand the quantities in
Eq. (91) in terms of a Taylor expansion around $\bar{x}$. I introduce the
convenient quantities
$\displaystyle\bar{r}\equiv\sigma_{\bar{\alpha}}(x,\bar{x})u^{\bar{\alpha}}\text{,}$
(101) $\displaystyle
s^{2}\equiv(g^{\bar{\alpha}\bar{\beta}}+u^{\bar{\alpha}}u^{\bar{\beta}})\sigma_{\bar{\alpha}}(x,\bar{x})\sigma_{\bar{\beta}}(x,\bar{x})\text{,}$
(102)
together with the time differences
$\Delta_{+}\equiv v-\bar{\tau}\text{,}\qquad\Delta_{-}\equiv u-\bar{\tau}$
(103)
from the advanced (retarded) point to the reference point $\bar{x}$.
I also use the expansion of the derivatives of the parallel propagator around
the point $\bar{x}$
$\displaystyle
g^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\beta;\bar{\gamma}}=-g^{\bar{\beta}}_{\phantom{\bar{\beta}}\beta}\left(\frac{1}{2}R^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\bar{\beta}\bar{\gamma}\bar{\delta}}\sigma^{\bar{\delta}}-\frac{1}{6}R^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\bar{\beta}\bar{\gamma}\bar{\delta};\bar{\varepsilon}}\sigma^{\bar{\delta}}\sigma^{\bar{\varepsilon}}\right)+O(\varepsilon^{3})\text{,}$
(104)
$\displaystyle\begin{split}g^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\beta;\gamma}&=-g^{\bar{\beta}}_{\phantom{\bar{\beta}}\beta}g^{\bar{\gamma}}_{\phantom{\bar{\gamma}}\gamma}\left(\frac{1}{2}R^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\bar{\beta}\bar{\gamma}\bar{\delta}}\sigma^{\bar{\delta}}-\frac{1}{3}R^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\bar{\beta}\bar{\gamma}\bar{\delta};\bar{\varepsilon}}\sigma^{\bar{\delta}}\sigma^{\bar{\varepsilon}}\right)\\\
&\quad+O(\varepsilon^{3})\text{,}\end{split}$ (105)
as well as an expansion for the second derivative of Synge’s world function
$\sigma_{\bar{\alpha}\bar{\beta}}=g_{\bar{\alpha}\bar{\beta}}-\frac{1}{3}R_{\bar{\alpha}\bar{\gamma}\bar{\beta}\bar{\delta}}\sigma^{\bar{\gamma}}\sigma^{\bar{\delta}}\\\
+\frac{1}{12}R_{\bar{\alpha}\bar{\gamma}\bar{\beta}\bar{\delta};\bar{\varepsilon}}\sigma^{\bar{\gamma}}\sigma^{\bar{\delta}}\sigma^{\bar{\varepsilon}}+O(\varepsilon^{4})\text{,}$
(106)
and the van Vleck determinant
$\Delta^{1/2}=1+O(\epsilon^{4})\text{,}$ (107)
which I calculate using the methods described in Sec. (2.4.2) of Poisson
(2004).
I make use of the fact that the bi-tensors
$\displaystyle U_{\alpha}(\tau)\equiv U_{\alpha\mu}u^{\mu}\text{,}$ (108)
$\displaystyle U_{\alpha\beta}(\tau)\equiv U_{\alpha\mu;\beta}u^{\mu}\text{,
and}$ (109) $\displaystyle\dot{U}_{\alpha}(\tau)\equiv
U_{\alpha\mu;\nu}u^{\mu}u^{\nu}$ (110)
appearing in Eq. (91) do not bear a free index on the world line, making them
scalars on the world line. With $w$ being either $u$ or $v$ and $\Delta\equiv
w-\bar{\tau}=\Delta_{\mp}$ I expand these as
$\displaystyle
U_{\alpha}(w)=U_{\alpha}+\dot{U}_{\alpha}\Delta+\frac{1}{2}\ddot{U}_{\alpha}\Delta^{2}+\frac{1}{6}U_{\alpha}^{(3)}\Delta^{3}+O(\varepsilon^{4})\text{,}$
(111) $\displaystyle
U_{\alpha\beta}(w)=U_{\alpha\beta}+\dot{U}_{\alpha\beta}\Delta+\frac{1}{2}\ddot{U}_{\alpha\beta}\Delta^{2}+O(\varepsilon^{3})\text{,
and}$ (112)
$\displaystyle\dot{U}_{\alpha}(w)=\dot{U}_{\alpha}+\ddot{U}_{\alpha}\Delta+\frac{1}{2}U_{\alpha}^{(3)}\Delta^{2}+O(\varepsilon^{3})\text{,}$
(113)
where it is understood that the coefficient functions are evaluated at
$\tau=\bar{\tau}$.
Repeatedly taking derivatives of Eq. (97) and contracting with $u^{\bar{\mu}}$
I find for the first set of coefficients
$\displaystyle
U_{\alpha}=g^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\alpha}u_{\bar{\alpha}}+O(\varepsilon^{4})\text{,}$
(114)
$\displaystyle\dot{U}_{\alpha}=g^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\alpha}\left(\frac{1}{2}R_{\bar{\alpha}uu\sigma}-\frac{1}{6}R_{\bar{\alpha}uu\sigma|\sigma}\right)+O(\varepsilon^{3})\text{,}$
(115)
$\displaystyle\ddot{U}_{\alpha}=\frac{1}{3}g^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\alpha}R_{\bar{\alpha}uu\sigma|u}+O(\varepsilon^{2})\text{,}$
(116) $\displaystyle U^{(3)}_{\alpha}=0+O(\varepsilon)\text{,}$ (117)
where I have introduced the notation $R_{\bar{\alpha}uu\sigma}\equiv
R_{\bar{\alpha}\bar{\beta}\bar{\gamma}\bar{\delta}}u^{\bar{\beta}}u^{\bar{\gamma}}\sigma^{\bar{\delta}}$
and $R_{\bar{\alpha}uu\sigma|\sigma}\equiv
R_{\bar{\alpha}\bar{\beta}\bar{\gamma}\bar{\delta};\bar{\varepsilon}}u^{\bar{\beta}}u^{\bar{\gamma}}\sigma^{\bar{\delta}}\sigma^{\bar{\varepsilon}}$;
I will use this notation and its natural extension to higher derivatives and
different combinations of $u^{\bar{\mu}}$ and $\sigma^{\bar{\mu}}$ frequently
below.
Similarly I find for the second set
$\displaystyle
U_{\alpha\beta}=g^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\alpha}g^{\bar{\beta}}_{\phantom{\bar{\beta}}\beta}\left(\frac{1}{2}R_{\bar{\alpha}u\bar{\beta}\sigma}-\frac{1}{3}R_{\bar{\alpha}u\bar{\beta}\sigma|\sigma}\right)+O(\varepsilon^{3})\text{,}$
(118)
$\displaystyle\begin{split}\dot{U}_{\alpha\beta}&=g^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\alpha}g^{\bar{\beta}}_{\phantom{\bar{\beta}}\beta}\left(\frac{1}{2}R_{\bar{\alpha}u\bar{\beta}u}+\frac{1}{6}R_{\bar{\alpha}u\bar{\beta}\sigma|u}-\frac{1}{3}R_{\bar{\alpha}u\bar{\beta}u|\sigma}\right)\\\
&\quad+O(\varepsilon^{2})\text{,}\end{split}$ (119)
$\displaystyle\ddot{U}_{\alpha\beta}=\frac{1}{3}g^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\alpha}g^{\bar{\beta}}_{\phantom{\bar{\beta}}\beta}R_{\bar{\alpha}u\bar{\beta}u|u}+O(\varepsilon)\text{.}$
(120)
Note that the third set does not involve new coefficients, but only those
already calculated for $U_{\alpha}$.
Finally I copy expressions for $\Delta_{\pm}$, $r$, $r_{\text{adv}}$, $u$, $v$
and their gradients from paper I
$\displaystyle\Delta_{\pm}=(\bar{r}\pm s)\mp\frac{(\bar{r}\pm
s)^{2}}{6s}R_{u\sigma u\sigma}\mp\frac{(\bar{r}\pm
s)^{2}}{24s}\left[(\bar{r}\pm s)R_{u\sigma u\sigma|u}-R_{u\sigma
u\sigma|\sigma}\right]+O(\varepsilon^{5})\text{,}$ (121) $\displaystyle
r=s-\frac{\bar{r}^{2}-s^{2}}{6s}R_{u\sigma
u\sigma}-\frac{\bar{r}-s}{24s}\left[(\bar{r}-s)(\bar{r}+2s)R_{u\sigma
u\sigma|u}-(\bar{r}+s)R_{u\sigma
u\sigma|\sigma}\right]+O(\varepsilon^{5})\text{,}$ (122) $\displaystyle
r_{\text{adv}}=s-\frac{\bar{r}^{2}-s^{2}}{6s}R_{u\sigma
u\sigma}-\frac{\bar{r}+s}{24s}\left[(\bar{r}+s)(\bar{r}-2s)R_{u\sigma
u\sigma|u}-(\bar{r}-s)R_{u\sigma
u\sigma|\sigma}\right]+O(\varepsilon^{5})\text{,}$ (123)
$\displaystyle\begin{split}\nabla_{\alpha}u&=\frac{1}{s}g^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\alpha}\biggl{\\{}\left[\sigma_{\bar{\alpha}}+(\bar{r}-s)u_{\bar{\alpha}}\right]\\\
&\quad+\left[\frac{1}{6}(\bar{r}-s)R_{\bar{\alpha}\sigma
u\sigma}-\frac{1}{3}(\bar{r}-s)^{2}R_{\bar{\alpha}u\sigma
u}+\frac{\bar{r}^{2}-s^{2}}{6s^{2}}R_{u\sigma
u\sigma}\sigma_{\bar{\alpha}}+\frac{(\bar{r}-s)^{2}(\bar{r}+2s)}{6s^{2}}R_{\sigma
u\sigma u|u}u_{\bar{\alpha}}\right]\\\
&\quad+\biggl{[}-\frac{1}{12}(\bar{r}-s)R_{\bar{\alpha}u\sigma
u|\sigma}+\frac{1}{8}(\bar{r}-s)^{2}R_{\bar{\alpha}u\sigma
u|\sigma}+\frac{1}{24}(\bar{r}-s)^{2}R_{\bar{\alpha}\sigma
u\sigma|u}-\frac{1}{12}(\bar{r}-s)^{3}R_{\bar{\alpha}u\sigma u|u}\\\
&\quad+\frac{\bar{r}-s}{24s^{2}}\left((\bar{r}-s)(\bar{r}+2s)R_{u\sigma
u\sigma|u}-(\bar{r}+s)R_{u\sigma
u\sigma|\sigma}\right)\sigma_{\bar{\alpha}}\\\
&\quad+\frac{(\bar{r}-s)^{2}}{24s^{2}}\left((\bar{r}-s)(\bar{r}+3s)R_{u\sigma
u\sigma|\sigma}-(\bar{r}+2s)R_{u\sigma
u\sigma|\sigma}\right)u_{\bar{\alpha}}\biggr{]}+O(\varepsilon^{5})\biggr{\\}}\text{,}\end{split}$
(124)
$\displaystyle\begin{split}\nabla_{\alpha}v&=-\frac{1}{s}g^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\alpha}\biggl{\\{}\left[\sigma_{\bar{\alpha}}+(\bar{r}+s)u_{\bar{\alpha}}\right]\\\
&\quad+\left[\frac{1}{6}(\bar{r}+s)R_{\bar{\alpha}\sigma
u\sigma}-\frac{1}{3}(\bar{r}+s)^{2}R_{\bar{\alpha}u\sigma
u}+\frac{\bar{r}^{2}-s^{2}}{6s^{2}}R_{u\sigma
u\sigma}\sigma_{\bar{\alpha}}+\frac{(\bar{r}+s)^{2}(\bar{r}-2s)}{6s^{2}}R_{\sigma
u\sigma u|u}u_{\bar{\alpha}}\right]\\\
&\quad+\biggl{[}-\frac{1}{12}(\bar{r}+s)R_{\bar{\alpha}u\sigma
u|\sigma}+\frac{1}{8}(\bar{r}+s)^{2}R_{\bar{\alpha}u\sigma
u|\sigma}+\frac{1}{24}(\bar{r}+s)^{2}R_{\bar{\alpha}\sigma
u\sigma|u}-\frac{1}{12}(\bar{r}+s)^{3}R_{\bar{\alpha}u\sigma u|u}\\\
&\quad+\frac{\bar{r}+s}{24s^{2}}\left((\bar{r}+s)(\bar{r}-2s)R_{u\sigma
u\sigma|u}-(\bar{r}-s)R_{u\sigma
u\sigma|\sigma}\right)\sigma_{\bar{\alpha}}\\\
&\quad+\frac{(\bar{r}+s)^{2}}{24s^{2}}\left((\bar{r}+s)(\bar{r}-3s)R_{u\sigma
u\sigma|\sigma}-(\bar{r}-2s)R_{u\sigma
u\sigma|\sigma}\right)u_{\bar{\alpha}}\biggr{]}+O(\varepsilon^{5})\biggr{\\}}\text{,}\end{split}$
(125)
$\displaystyle\begin{split}\nabla_{\alpha}r&=-\frac{1}{s}g^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\alpha}\left\\{\left[\sigma_{\bar{\alpha}}+\bar{r}u_{\bar{\alpha}}\right]+\left[\frac{1}{6}\bar{r}R_{\bar{\alpha}\sigma
u\sigma}-\frac{1}{3}(\bar{r}^{2}-s^{2})R_{\bar{\alpha}u\sigma
u}+\frac{\bar{r}^{2}+s^{2}}{6s^{2}}R_{u\sigma
u\sigma}\sigma_{\bar{\alpha}}+\frac{\bar{r}(\bar{r}^{2}-s^{2})}{6s^{2}}R_{u\sigma
u\sigma}u_{\bar{\alpha}}\right]\right.\\\
&\quad\left.+\left[-\frac{1}{12}\bar{r}R_{\bar{\alpha}\sigma
u\sigma|\sigma}+\frac{1}{8}(\bar{r}^{2}-s^{2})R_{\bar{\alpha}u\sigma
u|\sigma}+\frac{1}{24}(\bar{r}^{2}-s^{2})R_{\bar{\alpha}\sigma
u\sigma|u}\right.\right.\\\
&\quad\left.\left.-\frac{1}{12}(\bar{r}-s)^{2}(\bar{r}+2s)R_{\bar{\alpha}u\sigma
u|u}+\frac{1}{24s^{2}}\left((\bar{r}-s)(\bar{r}^{2}+\bar{r}s+4s^{2})R_{u\sigma
u\sigma|u}-(\bar{r}^{2}+s^{2})R_{u\sigma
u\sigma|\sigma}\right)\sigma_{\bar{\alpha}}\right.\right.\\\
&\quad\left.\left.+\frac{\bar{r}-s}{24s^{2}}\left((\bar{r}-s)(\bar{r}^{2}+2\bar{r}s+3s^{2})R_{u\sigma
u\sigma|u}-\bar{r}(\bar{r}+s)R_{u\sigma
u\sigma|\sigma}\right)u_{\bar{\alpha}}\right]+O(\epsilon^{5})\right\\}\text{,}\end{split}$
(126)
$\displaystyle\begin{split}\nabla_{\alpha}r_{\text{adv}}&=-\frac{1}{s}g^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\alpha}\left\\{\left[\sigma_{\bar{\alpha}}+\bar{r}u_{\bar{\alpha}}\right]+\left[\frac{1}{6}\bar{r}R_{\bar{\alpha}\sigma
u\sigma}-\frac{1}{3}(\bar{r}^{2}-s^{2})R_{\bar{\alpha}u\sigma
u}+\frac{\bar{r}^{2}+s^{2}}{6s^{2}}R_{u\sigma
u\sigma}\sigma_{\bar{\alpha}}+\frac{\bar{r}(\bar{r}^{2}-s^{2})}{6s^{2}}R_{u\sigma
u\sigma}u_{\bar{\alpha}}\right]\right.\\\
&\quad\left.+\left[-\frac{1}{12}\bar{r}R_{\bar{\alpha}\sigma
u\sigma|\sigma}+\frac{1}{8}(\bar{r}^{2}-s^{2})R_{\bar{\alpha}u\sigma
u|\sigma}+\frac{1}{24}(\bar{r}^{2}-s^{2})R_{\bar{\alpha}\sigma
u\sigma|u}\right.\right.\\\
&\quad\left.\left.-\frac{1}{12}(\bar{r}+s)^{2}(\bar{r}-2s)R_{\bar{\alpha}u\sigma
u|u}+\frac{1}{24s^{2}}\left((\bar{r}+s)(\bar{r}^{2}-\bar{r}s+4s^{2})R_{u\sigma
u\sigma|u}-(\bar{r}^{2}+s^{2})R_{u\sigma
u\sigma|\sigma}\right)\sigma_{\bar{\alpha}}\right.\right.\\\
&\quad\left.\left.+\frac{\bar{r}+s}{24s^{2}}\left((\bar{r}+s)(\bar{r}^{2}-2\bar{r}s+3s^{2})R_{u\sigma
u\sigma|u}-\bar{r}(\bar{r}-s)R_{u\sigma
u\sigma|\sigma}\right)u_{\bar{\alpha}}\right]+O(\epsilon^{5})\right\\}\text{.}\end{split}$
(127)
After substituting Eqs. (92) – (127) into Eq. (91) (all of them) and sorting
out the orders I find the final expression for the covariant expansion of
$A^{S}_{\alpha;\beta}$
$A^{S}_{\alpha;\beta}=qg^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\alpha}g^{\bar{\beta}}_{\phantom{\bar{\beta}}\beta}\left\\{\left[\frac{1}{s^{3}}u_{\bar{\alpha}}\sigma_{\bar{\beta}}+\frac{\bar{r}}{s^{3}}u_{\bar{\alpha}}u_{\bar{\beta}}\right]+\left[\frac{\bar{r}}{6s^{3}}u_{\bar{\alpha}}R_{\bar{\beta}\sigma
u\sigma}+\left(\frac{\bar{r}}{2s^{3}}\sigma_{\bar{\beta}}+\frac{\bar{r}^{2}-s^{2}}{2s^{3}}u_{\bar{\beta}}\right)R_{\bar{\alpha}uu\sigma}\right.\right.\\\
\mbox{}\left.\left.+\frac{\bar{r}^{2}-s^{2}}{3s^{2}}u_{\bar{\alpha}}R_{\bar{\beta}uu\sigma}+\frac{1}{2s}R_{\bar{\alpha}u\bar{\beta}\sigma}+\frac{3\bar{r}^{2}-s^{2}}{6s^{5}}R_{u\sigma
u\sigma}u_{\bar{\alpha}}\sigma_{\bar{\beta}}+\frac{\bar{r}(\bar{r}^{2}-s^{2})}{2s^{5}}R_{u\sigma
u\sigma}u_{\bar{\alpha}}u_{\bar{\beta}}+\frac{\bar{r}}{2s}R_{\bar{\alpha}u\bar{\beta}u}\right]\right.\\\
\mbox{}\left.+\left[-\frac{\bar{r}}{12s^{3}}u_{\bar{\alpha}}R_{\bar{\beta}\sigma
u\sigma|\sigma}-\frac{\bar{r}^{2}-s^{2}}{24s^{3}}u_{\bar{\alpha}}R_{\bar{\beta}\sigma\sigma
u|u}-\left(\frac{\bar{r}}{6s^{3}}\sigma_{\bar{\beta}}+\frac{\bar{r}^{2}-s^{2}}{6s^{3}}u_{\bar{\beta}}\right)R_{\bar{\alpha}uu\sigma|\sigma}-\frac{\bar{r}^{2}-s^{2}}{8s^{3}}u_{\bar{\alpha}}R_{\bar{\beta}uu\sigma|\sigma}\right.\right.\\\
\mbox{}\left.\left.-\frac{1}{3s^{2}}R_{\bar{\alpha}u\bar{\beta}\sigma|\sigma}+\left(\frac{\bar{r}^{2}-s^{2}}{6s^{3}}\sigma_{\bar{\beta}}+\frac{\bar{r}(\bar{r}^{2}-3s^{2})}{6s^{3}}u_{\bar{\beta}}\right)R_{\bar{\alpha}uu\sigma|u}+\frac{\bar{r}(\bar{r}^{2}-3s^{2})}{12s^{3}}u_{\bar{\alpha}}R_{\bar{\beta}uu\sigma|u}-\frac{\bar{r}}{3s}R_{\bar{\alpha}u\bar{\beta}u|\sigma}\right.\right.\\\
\mbox{}\left.\left.+\frac{\bar{r}}{6s}R_{\bar{a}u\bar{b}\sigma|u}+\frac{\bar{r}^{2}+s^{2}}{6s^{3}}R_{\bar{a}u\bar{b}u|u}+\left(-\frac{3\bar{r}^{2}-s^{2}}{24s^{5}}R_{u\sigma
u\sigma|\sigma}+\frac{\bar{r}(\bar{r}^{2}-s^{2})}{8s^{5}}R_{u\sigma
u\sigma|u}\right)u_{\bar{\alpha}}\sigma_{\bar{\beta}}\right.\right.\\\
\mbox{}\left.\left.+\left(-\frac{\bar{r}(\bar{r}^{2}-s^{2})}{8s^{5}}R_{u\sigma
u\sigma|\sigma}+\frac{(\bar{r}^{2}-s^{2})^{2}}{8s^{5}}R_{u\sigma
u\sigma|u}\right)u_{\bar{\alpha}}u_{\bar{\beta}}\right]\right\\}+O(\varepsilon^{2})\text{,}$
(128)
where terms in square brackets are of the same power in $\varepsilon$.
I copy the results for the coordinate expansion of
$\sigma_{\bar{\alpha}}(x,\bar{x})$ and
$g^{\bar{\alpha}}_{\phantom{\bar{\alpha}}\beta}(x,\bar{x})$ from Eqs. (3.16) –
(3.19) and Eqs. (3.30) – (3.33) of paper I. I use
$\displaystyle\begin{split}-\sigma_{\bar{\alpha}}(x,\bar{x})&=g_{\alpha\beta}w^{\beta}+A_{\alpha\beta\gamma}w^{\beta}w^{\gamma}+A_{\alpha\beta\gamma\delta}w^{\beta}w^{\gamma}w^{\delta}\\\
&\mbox{}+A_{\alpha\beta\gamma\delta\varepsilon}w^{\beta}w^{\gamma}w^{\delta}w^{\varepsilon}+O(\varepsilon^{5})\text{,}\end{split}$
(129) $\displaystyle
A^{\alpha}_{\phantom{\alpha}\beta\gamma}\equiv\frac{1}{2}\Gamma^{\alpha}_{\phantom{\alpha}\beta\gamma}\text{,}$
(130) $\displaystyle
A^{\alpha}_{\phantom{\alpha}\beta\gamma\delta}\equiv\frac{1}{6}\left(\Gamma^{\alpha}_{\phantom{\alpha}\beta\gamma,\delta}+\Gamma^{\alpha}_{\phantom{\alpha}\beta\mu}\Gamma^{\mu}_{\phantom{\mu}\gamma\delta}\right)\text{,}$
(131)
$\displaystyle\begin{split}A^{\alpha}_{\phantom{\alpha}\beta\gamma\delta\epsilon}&\equiv\frac{1}{24}\left(\Gamma^{\alpha}_{\phantom{\alpha}\beta\gamma,\delta\epsilon}+\Gamma^{\alpha}_{\phantom{\alpha}\beta\gamma,\mu}\Gamma^{\mu}_{\phantom{\mu}\delta\epsilon}\right.\\\
&\mbox{}\left.+2\Gamma^{\alpha}_{\phantom{\alpha}\beta\mu}\Gamma^{\mu}_{\phantom{\mu}\gamma\delta,\epsilon}+\Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}\Gamma^{\mu}_{\phantom{\mu}\beta\gamma}\Gamma^{\nu}_{\phantom{\nu}\delta\epsilon}\right)\text{,}\end{split}$
(132)
as well as
$\displaystyle\begin{split}g^{\bar{\mu}}_{\phantom{\bar{\mu}}\alpha}(x,\bar{x})&={\delta^{\mu}}_{\alpha}+B^{\mu}_{\phantom{\mu}\alpha\beta}w^{\beta}+B^{\mu}_{\phantom{\mu}\alpha\beta\gamma}w^{\beta}w^{\gamma}\\\
&\mbox{}+B^{\mu}_{\phantom{\mu}\alpha\beta\gamma\delta}w^{\beta}w^{\gamma}w^{\delta}+O(\varepsilon^{4})\text{,}\text{,}\end{split}$
(133) $\displaystyle
B^{\mu}_{\phantom{\mu}\alpha\beta}\equiv\Gamma^{\mu}_{\phantom{\mu}\alpha\beta}\text{,}$
(134) $\displaystyle
B^{\mu}_{\phantom{\mu}\alpha\beta\gamma}\equiv\frac{1}{2}\left(\Gamma^{\mu}_{\phantom{\mu}\alpha\beta,\gamma}+\Gamma^{\mu}_{\phantom{\mu}\beta\nu}\Gamma^{\nu}_{\phantom{\nu}\alpha\gamma}\right)\text{,}$
(135)
$\displaystyle\begin{split}B^{\mu}_{\phantom{\mu}\alpha\beta\gamma\delta}&\equiv\frac{1}{12}\left(2\Gamma^{\mu}_{\phantom{\mu}\alpha\beta,\gamma\delta}+2\Gamma^{\nu}_{\phantom{\nu}\alpha\beta}\Gamma^{\mu}_{\phantom{\mu}\nu\gamma,\delta}\right.\\\
&\mbox{}\left.-\Gamma^{\nu}_{\phantom{\nu}\beta\gamma}\Gamma^{\mu}_{\phantom{\mu}\alpha\nu,\delta}+4\Gamma^{\mu}_{\phantom{\mu}\beta\nu}\Gamma^{\nu}_{\phantom{\nu}\alpha\gamma,\delta}\right.\\\
&\mbox{}\left.+\Gamma^{\nu}_{\phantom{\nu}\beta\gamma}\Gamma^{\mu}_{\phantom{\mu}\alpha\delta,\nu}-\Gamma^{\mu}_{\phantom{\mu}\beta\nu}\Gamma^{\nu}_{\phantom{\nu}\alpha\lambda}\Gamma^{\lambda}_{\phantom{\lambda}\gamma\delta}\right.\\\
&\mbox{}\left.+\Gamma^{\mu}_{\phantom{\mu}\nu\lambda}\Gamma^{\nu}_{\phantom{\nu}\alpha\beta}\Gamma^{\lambda}_{\phantom{\lambda}\gamma\delta}+2\Gamma^{\mu}_{\phantom{\mu}\beta\nu}\Gamma^{\nu}_{\phantom{\nu}\gamma\lambda}\Gamma^{\lambda}_{\phantom{\lambda}\alpha\delta}\right)\text{,}\end{split}$
(136)
where $w^{\alpha}\equiv x^{\alpha}-x^{\bar{\alpha}}$ is the coordinate
distance between $x$ and $\bar{x}$. Together with Eq. (128) these equations
form an expansion in $w^{\alpha}$ of the singular part of the gradient of the
vector potential around a point $x$ near the world line of the particle. I
finally calculate the tetrad components of the singular Faraday tensor as
$F^{S}_{(\mu)(\nu)}=(A^{S}_{\beta,\alpha}-A^{S}_{\alpha,\beta})e^{\alpha}_{\
(\mu)}e^{\beta}_{\ (\nu)}\text{.}$ (137)
From this point on I proceed exactly as described in Section V of paper I
using Maple and GRTensorII to perform the calculations. I find, after an
extremely tedious calculation,
$\displaystyle
A_{(0)(+)}=\operatorname{sign}(\Delta)\Bigl{[}\frac{i\dot{r}_{0}J}{r_{0}\mathfrak{f}a^{2}}-\frac{1}{r_{0}^{2}}\Bigr{]}e^{i\varphi_{0}}\text{,}$
(138) $\displaystyle
B_{(0)(+)}=\biggl{\\{}\Bigl{[}-\frac{iE(J^{2}-r_{0}^{2})\dot{r}_{0}}{a^{3}\pi\mathfrak{f}J}+\frac{E(2-\mathfrak{f})}{\pi\mathfrak{f}ar_{0}}\Bigr{]}\mathcal{E}-\frac{ir_{0}^{2}E\dot{r}_{0}}{a^{3}J\mathfrak{f}\pi}\mathcal{K}\biggr{\\}}e^{i\varphi_{0}}\text{,}$
(139)
$\displaystyle\begin{split}D&{}_{(0)(+)}=\Bigg{\\{}\biggl{[}\frac{iEr_{0}^{2}(-14r_{0}^{2}J^{2}+J^{4}+r_{0}^{4})\dot{r}_{0}^{3}}{8\pi
J\mathfrak{f}a^{7}}-\frac{(-r_{0}\mathfrak{f}J^{2}+2r_{0}J^{2}+7r_{0}^{3}\mathfrak{f}-14r_{0}^{3})E\dot{r}_{0}^{2}}{8a^{5}\mathfrak{f}\pi}\\\
&\mbox{}+i\frac{\bigl{(}8MJ^{8}-14MJ^{6}r_{0}^{2}-3r_{0}^{5}J^{4}-80MJ^{4}r_{0}^{4}+4J^{4}r_{0}^{5}\mathfrak{f}-7r_{0}^{7}J^{2}-68Mr_{0}^{6}J^{2}+4r_{0}^{9}-26Mr_{0}^{8}-4r_{0}^{9}\mathfrak{f}\bigr{)}E\dot{r}_{0}}{8r_{0}^{5}a^{5}\mathfrak{f}J\pi}\\\
&\mbox{}-\frac{(8Mr_{0}\mathfrak{f}J^{6}-8r_{0}^{3}MJ^{4}+38J^{4}r_{0}^{3}\mathfrak{f}-2r_{0}^{6}J^{2}-16Mr_{0}^{5}J^{2}+3J^{2}r_{0}^{6}\mathfrak{f}+54J^{2}r_{0}^{5}\mathfrak{f}+20r_{0}^{7}\mathfrak{f}+5r_{0}^{8}\mathfrak{f}-6r_{0}^{8})E}{8r_{0}^{7}a^{3}\mathfrak{f}\pi}\biggr{]}\mathcal{E}\\\
&\mbox{}+\biggl{[}\frac{iEr_{0}^{4}(7J^{2}-r_{0}^{2})\dot{r}_{0}^{3}}{8\pi
J\mathfrak{f}a^{7}}-\frac{(2-\mathfrak{f})r_{0}^{3}E\dot{r}_{0}^{2}}{2a^{5}\mathfrak{f}\pi}\\\
&\mbox{}+\frac{\bigl{(}4Mr_{0}\mathfrak{f}J^{4}+20J^{2}r_{0}^{3}\mathfrak{f}+8Mr_{0}^{3}J^{2}+14r_{0}^{5}M\mathfrak{f}-2r_{0}^{6}+12Mr_{0}^{5}+r_{0}^{6}\mathfrak{f}\bigr{)}E}{8r_{0}^{5}a^{3}\mathfrak{f}\pi}\\\
&\mbox{}-\frac{i(2MJ^{6}-9Mr_{0}^{2}J^{4}-2J^{2}r_{0}^{5}\mathfrak{f}-20Mr_{0}^{4}J^{2}-2r_{0}^{7}\mathfrak{f}+2r_{0}^{7}-13Mr_{0}^{6})E\dot{r}_{0}}{4r_{0}^{3}a^{5}\mathfrak{f}J\pi}\biggr{]}\mathcal{K}\Bigg{\\}}e^{i\varphi_{0}}\text{,}\end{split}$
(140) $\displaystyle
A_{(+)(-)}=\operatorname{sign}(\Delta)\frac{2iEJ}{a^{2}r_{0}\mathfrak{f}}e^{i\varphi_{0}}\text{,}$
(141)
$\displaystyle\begin{split}B_{(+)(-)}&=-2i\Biggl{\\{}\biggl{[}-\frac{(r_{0}^{2}-J^{2})\dot{r}_{0}^{2}}{a^{3}\pi
J\mathfrak{f}}+\frac{-J^{2}r_{0}\mathfrak{f}+2r_{0}J^{2}+2r_{0}^{3}-2r_{0}^{3}\mathfrak{f}}{r_{0}^{3}aJ\pi}\biggr{]}\mathcal{E}+\biggl{[}\frac{r_{0}^{2}\dot{r}_{0}^{2}}{a^{3}\mathfrak{f}J\pi}-\frac{2(1-\mathfrak{f})}{aJ\pi}\biggr{]}\mathcal{K}\Biggr{\\}}e^{i\varphi_{0}}\text{,}\end{split}$
(142)
$\displaystyle\begin{split}D_{(+)(-)}&=-2i\Bigg{\\{}\biggl{[}-\frac{r_{0}^{2}(-14r_{0}^{2}J^{2}+J^{4}+r_{0}^{4})\dot{r}_{0}^{4}}{8\mathfrak{f}\pi
Ja^{7}}-\Bigl{(}4M\mathfrak{f}J^{8}-7Mr_{0}^{2}\mathfrak{f}J^{6}+2J^{4}r_{0}^{5}\mathfrak{f}+2J^{4}r_{0}^{4}M-J^{4}r_{0}^{5}\\\
&\mbox{}-43J^{4}r_{0}^{4}\mathfrak{f}M-7J^{2}r_{0}^{7}\mathfrak{f}-27J^{2}Mr_{0}^{6}\mathfrak{f}-11Mr_{0}^{8}\mathfrak{f}-r_{0}^{9}\mathfrak{f}+r_{0}^{9}-2r_{0}^{8}M\Bigr{)}r_{0}^{1/2}\dot{r}_{0}^{2}\Big{/}\Bigl{(}4r_{0}^{5}a^{5}J\mathfrak{f}\pi\Bigr{)}\\\
&\mbox{}-\Bigl{(}8M\mathfrak{f}J^{8}-8J^{6}Mr_{0}^{2}+30Mr_{0}^{2}\mathfrak{f}J^{6}-2J^{4}r_{0}^{5}+10J^{4}r_{0}^{4}\mathfrak{f}M-24J^{4}r_{0}^{4}M+3J^{4}r_{0}^{5}\mathfrak{f}\\\
&\mbox{}-28J^{2}Mr_{0}^{6}+J^{2}r_{0}^{7}\mathfrak{f}-28J^{2}Mr_{0}^{6}\mathfrak{f}-20Mr_{0}^{8}\mathfrak{f}-12r_{0}^{8}M\Bigr{)}\Big{/}\Bigl{(}8r_{0}^{7}a^{3}J\pi\Bigr{)}\biggr{]}\mathcal{E}\\\
&\mbox{}+\biggl{[}-\frac{r_{0}^{4}(7J^{2}-r_{0}^{2})\dot{r}_{0}^{4}}{8\mathfrak{f}\pi
Ja^{7}}\\\
&\mbox{}+\frac{4M\mathfrak{f}J^{6}-16J^{4}Mr_{0}^{2}+4r_{0}^{2}\mathfrak{f}MJ^{4}-18J^{2}r_{0}^{4}\mathfrak{f}M-28J^{2}r_{0}^{4}M-J^{2}\mathfrak{f}r_{0}^{5}-12r_{0}^{6}M-20r_{0}^{6}\mathfrak{f}M}{8r_{0}^{5}a^{3}J\pi}\\\
&\mbox{}+\Bigl{(}2M\mathfrak{f}J^{6}-9r_{0}^{2}\mathfrak{f}MJ^{4}+J^{2}r_{0}^{5}-2J^{2}r_{0}^{4}M-5J^{2}\mathfrak{f}r_{0}^{5}-14J^{2}r_{0}^{4}\mathfrak{f}M-2r_{0}^{6}M+r_{0}^{7}\\\
&\mbox{}-11r_{0}^{6}\mathfrak{f}M-\mathfrak{f}r_{0}^{7}\Bigr{)}\dot{r}_{0}^{2}\Big{/}\Bigl{(}4r_{0}^{5/2}a^{5}J\mathfrak{f}\pi\Bigr{)}\biggr{]}\mathcal{K}\Bigg{\\}}e^{i\varphi_{0}}\text{,}\end{split}$
(143)
where $\mathfrak{f}=\sqrt{\frac{r_{0}-2M}{r_{0}}}$, $a^{2}=r_{0}^{2}+J^{2}$.
Here, the rescaled elliptic integrals $\mathcal{E}$ and $\mathcal{K}$ are
defined by
$\mathcal{E}\equiv\frac{2}{\pi}\int_{0}^{\pi/2}(1-k\sin^{2}\psi)^{1/2}\,\mathrm{d}\psi=F\left(-{\frac{1}{2}},{\frac{1}{2}};1;k\right)\text{,}$
(144)
and
$\mathcal{K}\equiv\frac{2}{\pi}\int_{0}^{\pi/2}(1-k\sin^{2}\psi)^{-1/2}\,\mathrm{d}\psi=F\left({\frac{1}{2}},{\frac{1}{2}};1;k\right)\text{,}$
(145)
in which $F(a,b;c;x)$ are the hypergeometric functions and $k\equiv
J^{2}/(r_{0}^{2}+J^{2})$.
## Appendix C Vector potential calculation
In this section I describe a variant of the numerical calculation discussed in
the main part of the paper that uses the vector potential instead of the
Faraday tensor. To this end I decompose the vector potential and the sources
in terms of vectorial spherical harmonics
$\displaystyle A_{a}(t,r^{*},\theta,\phi)$
$\displaystyle={\textstyle\frac{1}{r}}A^{\ell m}_{a}(t,r^{*})Y_{\ell
m}(\theta,\phi)\text{,}$ (146a) $\displaystyle j_{a}(t,r^{*},\theta,\phi)$
$\displaystyle=j^{\ell m}_{a}(t,r^{*})Y_{\ell m}(\theta,\phi)$ for
$a=t,r^{*}$, (146b) $\displaystyle A_{A}(t,r^{*},\theta,\phi)$
$\displaystyle=v_{\ell m}(t,r^{*})Z_{A}^{\ell m}(\theta,\phi)$
$\displaystyle\quad+\tilde{v}_{\ell m}(t,r^{*})X_{A}^{\ell
m}(\theta,\phi)\text{,}$ (146c) $\displaystyle j_{A}(t,r^{*},\theta,\phi)$
$\displaystyle=j^{\text{even}}_{\ell m}(t,r^{*})Z_{A}^{\ell m}(\theta,\phi)$
$\displaystyle\quad+j^{\text{odd}}_{\ell m}(t,r^{*})X_{A}^{\ell
m}(\theta,\phi)$ for $A=\theta,\phi$, (146d)
and substitute this into the Maxwell equations for the vector potential in the
Lorenz gauge $g^{\alpha\beta}A_{\alpha;\beta}=0$:
$g^{\mu\nu}A_{\alpha;\mu\nu}-{R^{\beta}}_{\alpha}A^{\beta}=-4\pi
j_{\alpha}\text{,}$ (147)
where $R_{\alpha\beta}$ is the spacetime’s Ricci tensor, which vanishes in
Schwarzschild spacetime. Substituting Eq. (146) into Eq. (147) I arrive at two
decoupled sets of equations for the even ($A^{\ell m}_{a}$, $v_{\ell m}$) and
odd ($\tilde{v}_{\ell m}$) modes
$\displaystyle-\frac{\partial^{2}A^{\ell
m}_{t}}{\partial{t}^{2}}+\frac{\partial^{2}A^{\ell
m}_{t}}{\partial{r^{*}}^{2}}+\frac{2M}{r^{2}}\left(\frac{\partial A^{\ell
m}_{r^{*}}}{\partial t}-\frac{\partial A^{\ell m}_{t}}{\partial
r^{*}}\right)-VA^{\ell m}_{t}=-4\pi rfj^{\ell m}_{t}\text{,}$ (148)
$\displaystyle\begin{split}-\frac{\partial^{2}A^{\ell
m}_{r^{*}}}{\partial{t}^{2}}+\frac{\partial^{2}A^{\ell
m}_{r^{*}}}{\partial{r^{*}}^{2}}+\frac{2M}{r^{2}}\left(\frac{\partial A^{\ell
m}_{t}}{\partial t}-\frac{\partial A^{\ell m}_{r^{*}}}{\partial
r^{*}}\right)-\left(V+2\frac{f^{2}}{r^{2}}\right)A^{\ell m}_{r^{*}}+fVv_{\ell
m}=-4\pi rfj^{\ell m}_{r^{*}}\text{,}\end{split}$ (149)
$\displaystyle-\frac{\partial^{2}v_{\ell
m}}{\partial{t}^{2}}+\frac{\partial^{2}v_{\ell
m}}{\partial{r^{*}}^{2}}-Vv_{\ell m}+2\frac{f}{r^{2}}A^{\ell m}_{r^{*}}=-4\pi
fj^{\text{even}}_{\ell m}\text{,}$ (150)
$\displaystyle-\frac{\partial^{2}\tilde{v}_{\ell
m}}{\partial{t}^{2}}+\frac{\partial^{2}\tilde{v}_{\ell
m}}{\partial{r^{*}}^{2}}-V\tilde{v}_{\ell m}=-4\pi fj^{\text{odd}}_{\ell
m}\text{,}$ (151)
where $V$ and $j^{\ell m}_{\alpha}$ is defined as in Eqs. (21b) and (10) in
the main text.
### C.1 Numerical method
I discretize the set of reduced equations Eqs. (148) – (151) using Lousto’s
method as described in section II of the main text. Since the source terms on
the right hand side are less singular for the vector potential than they are
for the Faraday tensor, I do not have to distinguish between sourced and
vacuum cells in the integral over the potential terms.
Terms containing first derivatives $\frac{\partial\psi}{\partial t}$,
$\frac{\partial\psi}{\partial r^{*}}$, where now and in the remainder of the
appendix $\psi$ stands for any of $A^{\ell m}_{t}$, $A^{\ell m}_{r^{*}}$,
$v^{\ell m}$ or $\tilde{v}^{\ell m}$, were not treated in Lousto and Price
(1997), but, for generic vacuum cells, can be handled in a straightforward
manner
$\displaystyle\iint_{\text{\hbox
to0.0pt{cell\hss}}}\,\mathrm{d}u\,\mathrm{d}v\,V(r)\frac{\partial\psi}{\partial
t}=2h(\psi_{3}-\psi_{2})V_{0}+O(h^{4})\text{,}$ (152)
$\displaystyle\iint_{\text{\hbox
to0.0pt{cell\hss}}}\,\mathrm{d}u\,\mathrm{d}v\,V(r)\frac{\partial\psi}{\partial
r^{*}}=2h(\psi_{4}-\psi_{1})V_{0}+O(h^{4})\text{.}$ (153)
This fails for cells traversed by the particle, since the field is only
continuous across the world line but not differentiable. For these cells I
take recourse to Lousto’s original algorithm, which has to deal with a similar
issue, and use
$\displaystyle\iint_{\text{\hbox
to0.0pt{cell\hss}}}\,\mathrm{d}u\,\mathrm{d}v\,V(r)\frac{\partial\psi}{\partial
t}=V_{0}\,\sum_{i}A_{i}\partial_{t}\psi_{i}+O(h^{3})\text{,}$ (154)
$\displaystyle\iint_{\text{\hbox
to0.0pt{cell\hss}}}\,\mathrm{d}u\,\mathrm{d}v\,V(r)\frac{\partial\psi}{\partial
r^{*}}=V_{0}\,\sum_{i}A_{i}\partial_{r^{*}}\psi_{i}+O(h^{3})\text{,}$ (155)
where $A_{1}$,…,$A_{4}$ are the subareas indicated in Fig. 3 and
$\partial_{t}\psi_{1}$, …, $\partial_{t}\psi_{4}$, $\partial_{r^{*}}\psi_{1}$,
…, $\partial_{r^{*}}\psi_{4}$ are zeroth order accurate approximations to the
derivatives in the subareas. I calculate these using grid points outside of
the cell on the same side of the world line as the corresponding subarea, e.g.
$\partial_{r^{*}}\psi_{1}=\frac{\psi(t,r^{*}-h)-\psi(t,r^{*}-3h)}{2h}+O(h)\text{.}$
(156)
### C.2 Gauge condition
In contrast to the scalar field, the electromagnetic vector potential has to
satisfy a gauge condition
$Z\equiv g^{\alpha\beta}A_{\alpha;\beta}=0\text{.}$ (157)
Analytically the gauge condition is preserved by the evolution equations, so
that it is sufficient to impose it on the initial data. Numerically, however,
small violations of the gauge condition due to the numerical approximation can
be amplified exponentially and come to dominate the numerical data. To handle
this situation I introduce a gauge damping scheme as described in Gundlach et
al. (2005); Barack and Lousto (2005). That is I add a term of the form
$\frac{4M}{r^{2}}Z=\frac{4M}{r^{2}}\left(-\frac{1}{r-2M}\frac{\partial
A_{t}}{\partial t}+\frac{1}{r-2M}\frac{\partial A_{r^{*}}}{\partial
r^{*}}\right.\\\
\mbox{}\left.+\frac{1}{r^{2}}A_{r^{*}}-\frac{\ell(\ell+1)}{r^{2}}v\right)$
(158)
to the $t$ components of the evolution equations Eqs. (147), which dampens out
violations of the gauge condition. This choice proved to be numerically stable
for the radiative ($\ell>0$) modes but unstable for the monopole ($\ell=0$)
mode.
### C.3 Monopole mode
The monopole moment of an electromagnetic field is non-radiative. This makes
its behaviour sufficiently different from that of the radiative ($\ell>0$)
modes that the approach outlined earlier fails for $\ell=0$. In this case Eq.
(147) reduces to a set of coupled equations for $A_{a}^{0,0}$ only. Rather
than solving the system of equations directly for $A_{t}^{0,0}$ and
$A_{r^{*}}^{0,0}$ I use the analytical result for the $F_{tr}$ component of
the Faraday tensor derived in section II.3 in the main part of the paper. This
proves to be sufficient to reconstruct the combination
$A^{0,0}_{r,t}-A^{0,0}_{t,r}$ appearing in Eq. (77).
### C.4 Initial values and boundary conditions
I handle the problem of initial data and boundary conditions the same way as
in the main text, that is I arbitrarily choose the fields to vanish on the
characteristic slices $u=u_{0}$ and $v=v_{0}$
$A_{\alpha}(u=u_{0})=A_{\alpha}(v=v_{0})=0\text{,}$ (159)
thereby adding a certain amount of spurious waves to the solution which show
up as an initial burst. Gauge violations in this initial data are damped out
along with those arising during the evolution.
I implement ingoing wave boundary conditions near the event horizon and choose
a numerical domain that covers the full domain of dependence of the initial
data near the outer boundary.
### C.5 Extraction of the field data at the particle
In order to extract the value of the fields and their first derivatives at the
position of the particle, I use a variant of the extraction scheme described
in paper II. I introduce a piecewise polynomial
$p(x)=\begin{cases}c_{0}+c_{1}x+\frac{c_{3}}{2}x^{2}&\text{if $x<0$}\\\
c^{\prime}_{0}+c^{\prime}_{1}x+\frac{c^{\prime}_{3}}{2}x^{2}&\text{if
$x>0$}\end{cases}$ (160)
in $x\equiv r^{*}-r_{0}^{*}$ on the current slice. Its coefficients to the
left and right of the world line are linked by jump conditions
$c_{n}=c^{\prime}_{n}+\left[\partial^{n}_{r^{*}}\psi\right]$ listed in
Appendix D.2. Fitting this polynomial to the three grid points closest to the
particle, I extract approximations for $\psi(t_{0},r^{*}_{0})$ and
$\frac{\partial\psi(t_{0},r^{*}_{0})}{\partial r^{*}}$ which are just the
coefficients $c_{0}$, $c_{1}$ respectively. Once I have obtained these, I
proceed as in section V of the main part of the paper following Sago to
obtain values for $\frac{\partial\psi(t_{0},r^{*}_{0})}{\partial t}$.
### C.6 Results
Using the vector potential code described above I can reproduce the results
obtained from the Faraday tensor method discussed in the main paper. The
differences are small, typically of the order of $10^{-3}\%$ of the field
values as shown in Fig. 20.
Figure 20: Differences between $F_{tr}^{\ell m}$ calculated using the vector
potential and calculated using the Faraday tensor method for $\ell=2$, $m=2$
mode of field for the zoom-whirl orbit shown in Fig. 8. Displayed are the
difference and the actual field. The stepsizes were $h=1.041\bar{6}\times
10^{-2}\,M$ and $h=1/512\,M$ for the vector potential calculation and the
Faraday tensor calculation respectively.
I expect the Faraday tensor code to yield more accurate results since the
costly numerical differentiation that is necessary in the vector potential
calculation is absent. Nevertheless I can reproduce e.g. the correct decay
behaviour of the multipole coefficients for a zoom-whirl orbit as shown in
Fig. 21.
Figure 21: Multipole coefficients of
$\frac{M^{2}}{q}\operatorname{Re}F^{R}_{(0)}$ for a particle on a zoom-whirl
orbit ($p=7.8001$, $e=0.9$), calculated using a stepsize of $h=0.125M$ for the
$\ell=1$ modes and increasing the resolution linearly with $\ell$ for
$\ell>1$. The coefficients are extracted at $t=1100\,M$ when the particle is
deep within the zoom phase. Red triangles are used for the unregularized
multipole coefficients $F_{(0),\ell}$, squares, diamonds and disks are used
for the partly regularized coefficients after the removal of the $A_{(0)}$,
$B_{(0)}$ and $D_{(0)}$ terms respectively.
## Appendix D Jump conditions
### D.1 Faraday tensor calculation
Since the source term in Eqs. (21b) – (21f) contains a term proportional to
$\delta^{\prime}(r^{*}-r^{*}_{0})$, the field is discontinuous across the
world line of the particle. I use
$\left[\partial_{t}^{n}\partial_{r^{*}}^{m}\psi\right]=\lim_{\varepsilon\rightarrow
0^{+}}[\partial_{t}^{n}\partial_{r^{*}}^{m}\psi(t_{0},r_{0}^{*}+\varepsilon)\\\
-\partial_{t}^{n}\partial_{r^{*}}^{m}\psi(t_{0},r_{0}^{*}-\varepsilon)]$ (161)
to denote the jump in $\partial^{n}_{t}\partial^{m}_{r^{*}}\psi$ across the
world line. I only calculate jump conditions in the $r^{*}$ direction up to
$\left[\partial_{r^{*}}\psi\right]$, which I find by substituting the ansatz
$\displaystyle\psi$ $\displaystyle=\psi_{<}(t,r^{*})\theta(r^{*}_{0}-r^{*})$
$\displaystyle\quad+\psi_{>}(t,r^{*})\theta(r^{*}-r^{*}_{0})$ (162)
into Eqs. (21b) – (21f) and its $t$ and $r^{*}$ derivatives. Demanding in each
step that the singularity structure on the left hand side matches that of the
sources (and their derivatives) on the right hand side yields the jump
conditions
$\displaystyle\left[\psi\right]$
$\displaystyle=\frac{F_{\psi}}{f_{0}[(\partial_{t}r^{*}_{0})^{2}-1]}\text{,}$
(163) and $\displaystyle\left[\partial_{r^{*}}\psi\right]$
$\displaystyle=-\frac{G_{\psi}}{(\partial_{t}r^{*}_{0})^{2}-1}$
$\displaystyle\quad-\frac{\partial_{t}^{2}r^{*}_{0}\left[3\,(\partial_{t}r^{*}_{0})^{2}+1\right]F_{\psi}}{f_{0}\,[(\partial_{t}r^{*}_{0})^{2}-1]^{3}}$
$\displaystyle\quad+2\frac{\partial_{t}r^{*}_{0}\,\partial_{t}\left(F_{\psi}/f_{0}\right)}{[(\partial_{t}r^{*}_{0})^{2}-1]^{2}}\text{,}$
(164)
where $\psi$ stands for either one of $\psi$, $\chi$, or $\xi$.
### D.2 Vector potential calculation
Since the source term in Eq. (147) is singular, the field is only continuous
across the world line of the particle, but not smooth. I use
$\left[\partial_{t}^{n}\partial_{r^{*}}^{m}\psi\right]=\lim_{\varepsilon\rightarrow
0^{+}}[\partial_{t}^{n}\partial_{r^{*}}^{m}\psi(t_{0},r_{0}^{*}+\varepsilon)\\\
-\partial_{t}^{n}\partial_{r^{*}}^{m}\psi(t_{0},r_{0}^{*}-\varepsilon)]$ (165)
to denote the jump in $\partial^{n}_{t}\partial^{m}_{r^{*}}\psi$ across the
world line. For my purposes I only need the jump conditions in the $r^{*}$
direction up to $\left[\partial^{2}_{r^{*}}\psi\right]$, which I find by
substituting the ansatz
$\displaystyle A^{\ell m}_{a}(t,r^{*})$ $\displaystyle=A^{\ell
m}_{a,<}(t,r^{*})\theta(r^{*}_{0}-r^{*})$ $\displaystyle\quad+A^{\ell
m}_{a,>}(t,r^{*})\theta(r^{*}-r^{*}_{0})\text{,}$ (166) $\displaystyle v^{\ell
m}(t,r^{*})$ $\displaystyle=v^{\ell m}_{<}(t,r^{*})\theta(r^{*}_{0}-r^{*})$
$\displaystyle\quad+v^{\ell m}_{>}(t,r^{*})\theta(r^{*}-r^{*}_{0})\text{,}$
(167) $\displaystyle\tilde{v}^{\ell m}(t,r^{*})$
$\displaystyle=\tilde{v}^{\ell m}_{<}(t,r^{*})\theta(r^{*}_{0}-r^{*})$
$\displaystyle\quad+\tilde{v}^{\ell m}_{>}(t,r^{*})\theta(r^{*}-r^{*}_{0})$
(168)
into Eqs. (148) – (151) and its $t$ and $r^{*}$ derivatives. Demanding in each
step that the singularity structure on the left hand side matches that of the
sources (and their derivatives) on the right hand side yields the jump
conditions
$\displaystyle\left[A^{\ell m}_{a}\right]=\left[w^{\ell m}\right]=0\text{,}$
(169) $\displaystyle\left[\partial_{r^{*}}A^{\ell
m}_{a}\right]=\frac{E^{2}}{E^{2}-\dot{r}_{0}^{2}}S_{a}\text{,}$ (170)
$\displaystyle\left[\partial_{r^{*}}w^{\ell
m}\right]=\frac{E^{2}}{E^{2}-\dot{r}_{0}^{2}}S_{\text{even/odd}}\text{,}$
(171) $\displaystyle\begin{split}\left[\partial^{2}_{r^{*}}A^{\ell
m}_{a}\right]=\left(\frac{2ME^{4}}{r_{0}^{2}(E^{2}-\dot{r}_{0}^{2})^{2}}-f_{0}\frac{(3\dot{r}_{0}^{2}+E^{2})E^{2}\ddot{r}_{0}}{(E^{2}-\dot{r}_{0}^{2})^{3}}\right)S_{a}\\\
\mbox{}+\frac{2ME^{3}\dot{r}_{0}}{r_{0}^{2}(E^{2}-\dot{r}_{0}^{2})^{2}}S_{b}-f_{0}\frac{2E^{2}\dot{r}_{0}}{(E^{2}-\dot{r}_{0}^{2})^{2}}\dot{S}_{a}\\\
\text{,}\end{split}$ (172)
$\displaystyle\begin{split}\left[\partial^{2}_{r^{*}}w^{\ell
m}\right]=-f_{0}\frac{(3\dot{r}_{0}^{2}+E^{2})E^{2}\ddot{r}_{0}}{(E^{2}-\dot{r}_{0}^{2})^{3}}S_{\text{even/odd}}\\\
\mbox{}-f_{0}\frac{2E^{2}\dot{r}_{0}}{(E^{2}-\dot{r}_{0}^{2})^{2}}\dot{S}_{\text{even/odd}}\text{,}\end{split}$
(173)
where $a,b\in\\{t,r^{*}\\},a\neq b$, $w\in\\{v,\tilde{v}\\}$.
## References
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* Warburton et al. (2011) N. Warburton, S. Akcay, L. Barack, J. R. Gair, and N. Sago (2011), * Temporary entry *, eprint 1111.6908.
* Misner et al. (1973) C. W. Misner, K. S. Thorne, and J. A. Wheeler, _Gravitation_ (W. H. Freeman, San Francisco, 1973), ISBN 0716703343.
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* Regge (1957) T. Regge, Phys. Rev. 108, 1063 (1957).
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* Cunningham and Price (1979) C. T. Cunningham and R. H. Price, The Astrophysical Journal 230, 870 (1979).
* (14) N. Sago, talk in 10th Capra meeting at UAH (2007).
* Detweiler et al. (2003) S. Detweiler, E. Messaritaki, and B. F. Whiting, Phys. Rev. D 67, 104016 (2003), eprint gr-qc/0205079.
* Diaz-Rivera et al. (2004) L. M. Diaz-Rivera, E. Messaritaki, B. F. Whiting, and S. Detweiler, Physical Review D (Particles, Fields, Gravitation, and Cosmology) 70, 124018 (pages 14) (2004), eprint gr-qc/0410011, URL http://link.aps.org/abstract/PRD/v70/e124018.
* Detweiler (2008a) S. L. Detweiler, Phys. Rev. D77, 124026 (2008a), eprint 0804.3529.
* Sago et al. (2008) N. Sago, L. Barack, and S. L. Detweiler, Phys. Rev. D78, 124024 (2008), eprint 0810.2530.
* Barack and Sago (2009) L. Barack and N. Sago, Phys. Rev. Lett. 102, 191101 (2009), eprint 0902.0573.
* Pfenning and Poisson (2002) M. J. Pfenning and E. Poisson, Phys. Rev. D 65, 084001 (2002), eprint gr-qc/0012057.
* DeWitt-Morette and Dewitt (1964) C. DeWitt-Morette and B. S. Dewitt, _Relativite, groups et topologie = Relativity, groups and topology : lectures delivered at les Houches during the 1963 session of the Summer School of_ (Gordon and Breach, New York, 1964).
* Detweiler (2008b) S. Detweiler, Phys. Rev. D 77, 124026 (2008b), eprint 0804.3529.
* Poisson (2004) E. Poisson, Living Reviews in Relativity 7 (2004), URL http://www.livingreviews.org/lrr-2004-6.
* Gundlach et al. (2005) C. Gundlach, J. M. Martin-Garcia, G. Calabrese, and I. Hinder, Class. Quant. Grav. 22, 3767 (2005), eprint gr-qc/0504114.
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|
arxiv-papers
| 2011-12-16T04:20:35 |
2024-09-04T02:49:25.382661
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Roland Haas",
"submitter": "Roland Haas",
"url": "https://arxiv.org/abs/1112.3707"
}
|
1112.3760
|
# The reduction of plankton biomass induced by mesoscale stirring: a modeling
study in the Benguela upwelling.
Ismael Hernández-Carrasco Vincent Rossi Cristóbal López Emilio Hernández-
García IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos
(CSIC-UIB), 07122 Palma de Mallorca, Spain Veronique Garçon Laboratoire
d’Études en Géophysique et Océanographie Spatiale, CNRS, Observatoire Midi-
Pyrénées, 14 avenue Edouard Belin, Toulouse, 31401 Cedex 9, France
###### Abstract
Recent studies, both based on remote sensed data and coupled models, showed a
reduction of biological productivity due to vigorous horizontal stirring in
upwelling areas. In order to better understand this phenomenon, we consider a
system of oceanic flow from the Benguela area coupled with a simple
biogeochemical model of Nutrient-Phyto-Zooplankton (NPZ) type. For the flow
three different surface velocity fields are considered: one derived from
satellite altimetry data, and the other two from a regional numerical model at
two different spatial resolutions. We compute horizontal particle dispersion
in terms of Lyapunov Exponents, and analyzed their correlations with
phytoplankton concentrations. Our modelling approach confirms that in the
south Benguela there is a reduction of biological activity when stirring is
increased. Two-dimensional offshore advection and latitudinal difference in
Primary Production, also mediated by the flow, seem to be the dominant
processes involved. We estimate that mesoscale processes are responsible for
30 to 50% of the offshore fluxes of biological tracers. In the northern area,
other factors not taken into account in our simulation are influencing the
ecosystem. We suggest explanations for these results in the context of studies
performed in other eastern boundary upwelling areas.
## I Introduction
Marine ecosystems of the Eastern Boundary Upwelling zones are well known for
their major contribution to the world ocean productivity. They are
characterized by wind-driven upwelling of cold nutrient-rich waters along the
coast that supports elevated plankton and pelagic fish production (Mackas et
al., 2006). Variability is introduced by strong advection along the shore,
physical forcings by local and large scales winds, and high submeso- and meso-
scale activities over the continental shelf and beyond, linking the coastal
domain with the open ocean.
The Benguela Upwelling System (BUS) is one of the four major Eastern Boundary
Upwelling Systems (EBUS) of the world. The coastal area of the Benguela
ecosystem extends from southern Angola (around 17∘S) along the west coast of
Namibia and South Africa (36∘S). It is surrounded by two boundary currents,
the warm Angola Current in the north, and the temperate Agulhas Current in the
south. The BUS can itself be subdivided into two subdomains by the powerful
Luderitz upwelling cell (Hutchings et al., 2009). Most of the biogeochemical
activity occurs within the upwelling front and the coast, although it can be
extended further offshore toward the open ocean by the numerous filamental
structures developing offshore (Monteiro, 2009). In the BUS, as in the other
major upwelling areas, high mesoscale activity due to eddies and filaments
impacts strongly marine planktonic ecosystem over the continental shelf and
beyond (Brink and Cowles, 1991; Martin, 2003; Sandulescu et al., 2008; Rossi
et al., 2009).
The purpose of this study is to analyze the impact of horizontal stirring on
phytoplankton dynamics in the BUS within an idealized two dimensional
modelling framework. Based on satellite data of the ocean surface, Rossi et
al. (2008, 2009) recently suggested that mesoscale activity has a negative
effect on chlorophyll standing stocks in the four EBUS. This was obtained by
correlating remote sensed chlorophyll data with a Lagrangian measurement of
lateral stirring in the surface ocean (see Methods section). This result was
unexpected since mesoscale physical structures, particularly mesoscale eddies,
have been related to higher planktonic production and stocks in the open ocean
(McGillicuddy et al., 2007) as well as off a major EBUS (Correa-Ramirez et
al., 2007). A more recent and thorough study performed by Gruber et al. (2011)
in the California and the Canary current systems extended the initial results
from Rossi et al. (2008, 2009). Based on satellite derived estimates of net
Primary Production, of upwelling strength and of Eddy Kinetic Energy (EKE) as
a measure the intensity of mesoscale activity, they confirmed the suppressive
effect of mesoscale structures on biological production in upwelling areas.
Investigating the mechanism behind this observation by means of on 3D eddy-
resolving coupled models, Gruber et al. (2011) showed that mesoscale eddies
tend to export offshore and downward a certain pool of nutrients not being
effectively used by the biology in the coastal areas. This process they called
”nutrients leakage” is also having a negative feedback by diminishing the pool
of deep nutrients available in the surface waters being re-upwelled
continuously.
In our work, we focused on the Benguela area, being the most contrasting area
of all EBUS in terms of stirring intensity (Rossi et al., 2009). Although the
mechanisms studied by Gruber et al. (2011) seem to involve 3D dynamics, the
initial observation of this suppressive effect was essentially based on two-
dimensional (2D) datasets (Rossi et al., 2008). In this work we use 2D
numerical analysis in a semi-realistic framework to better understand the
effects of a 2D turbulent flow on biological dynamics, apart from the complex
3D bio-physical processes. The choice of this simple horizontal numerical
approach is indeed supported by other theoretical 2D studies that also
displayed a negative correlation between stirring and biomass (Tél et al.,
2005; MacKiver and Neufeld, 2009; Neufeld and Hernández-García, 2009).
Meanwhile, since biological productivity in upwelling areas rely on the (wind-
driven) vertical uplift of nutrients, we introduced in our model a nutrient
source term with an intensity and spatial distribution corresponding to the
upwelling characteristics. Instead of the commonly used EKE, which is an
Eulerian diagnostic tool, we used here a Lagrangian measurement of mesoscale
stirring that has been demonstrated as a powerful tool to study patchy
chlorophyll distributions influenced by dynamical structures at mesoscale,
such as upwelling filaments (Calil and Richards, 2010). The Lagrangian
perspective provides a complementary insight to transport phenomena in the
ocean with respect to the Eulerian one. In particular, the concept of
Lagrangian Coherent Structure may give a global idea of transport in a given
area, separating regions with different dynamical behavior, and signaling
avenues and barriers to transport, which are of great relevance for the marine
biological dynamics. While the Eulerian approach describes the characteristics
of the velocity field, the Lagrangian one addresses the effects of this field
on transported substances, which is clearly more directly related to the
biological dynamics. For example the work by Hernández-Carrasco et al. (2012)
describes currents in the world ocean having the same level of Eddy Kinetic
Energy but having two different stirring characteristics, as quantified by
Lagrangian tools. Further discussions comparing Lagrangian and Eulerian
diagnostics can be found, for example, in d’Ovidio et al. (2009) and the above
cited Hernández-Carrasco et al. (2012). To consider velocity fields with
different characteristics and to test the effect of the spatial resolution,
different flow fields are used, one derived from satellite and two produced by
numerical simulations at two different spatial resolutions. Our modelled
chlorophyll-a concentrations are compared with observed distributions of
chlorophyll-a (a metric for phytoplankton) obtained from the SeaWiFS satellite
sensor.
This paper is organized as follows. Sec. II is a brief description of the
different datasets used in this study. Sec. III depicts the methodology, which
includes the computation of the finite-size Lyapunov exponents, and the
numerical plankton-flow 2D coupled model. Then, our results are analyzed and
discussed in Sec. IV. Finally in Sec. V, we summed-up our main findings.
## II Satellite and simulated data
We used three different 2D surface velocity fields of the Benguela area. Two
are obtained from the numerical model Regional Ocean Model System (ROMS), and
the other one from a combined satellite product.
### II.1 Surface velocity fields derived from regional simulations.
ROMS is a free surface, hydrostatic, primitive equation model, and we used
here an eddy-resolving climatologically forced run provided by (Gutknecht et
al., 2013). At each grid point, linear horizontal resolution is the same in
both the longitudinal, $\phi$, and latitudinal, $\theta$, directions, which
leads to angular resolutions $\Delta\phi=\Delta_{0}$ and
$\Delta\theta=\Delta\phi\cos{\theta}$. The numerical model was run onto 2
different grids: a coarse one at spatial resolution of
$\Delta_{0}=1/4^{\circ}$, and a finer one at $\Delta_{0}=1/12^{\circ}$ of
spatial resolution. In the following we label the dataset from the coarser
resolution run as ROMS1/4, and the finer one as ROMS1/12. For both runs,
vertical resolution is variable with $30$ layers in total, while only data
from the surface upper layer are used in the following. Since the flows are
obtained from climatological forcings, they would represent a mean annual
cycle of the typical surface currents of the Benguela region.
### II.2 Surface velocity field derived from satellite
A velocity field derived from satellite observations is compared to the
simulated fields described previously. It consists of surface currents
computed from a combination of wind-driven Ekman currents, at 15 m depth,
derived from Quickscat wind estimates, and geostrophic currents calculated
using time-variable Sea Surface Heights (SSH) obtained from satellite (Sudre
and Morrow, 2008). These SSH were calculated from mapped altimetric sea level
anomalies combined with a mean dynamic topography. This velocity field,
labeled as Satellite1/4, covers a period from June 2002 to June 2005 with a
spatial resolution of $\Delta_{0}=1/4^{\circ}$ in both longitudinal and
latitudinal directions.
### II.3 Ocean color as a proxy for phytoplankton biomass
To validate simulated plankton concentrations, we use a three-year-long time
series, from January 2002 to January 2005, of ocean color data. Phytoplankton
pigment concentration (chlorophyll-a) is obtained from monthly Sea viewing
Wide Field-of-view Sensor (SeaWiFS) products, generated by the NASA Goddard
Earth Science (GES)/Distributed Active Archive Center (DAAC). Gridded global
data were used with a resolution of approximately 9 by 9 km.
## III Methodology
### III.1 Finite-Size Lyapunov Exponents (FSLEs)
FSLEs (Artale et al., 1997; Aurell et al., 1997; Boffetta et al., 2001)
provides a measure of dispersion, and thus of stirring and mixing, as a
function of the spatial resolution. This Lagrangian tool allows isolating the
different regimes corresponding to different length scales of the oceanic
flows, as well as identifying Lagrangian Coherent Structures (LCSs) present in
the data (Tew Kai et al., 2009). FSLE are computed from $\tau$, the time
required for two particles of fluid (one of them placed at x) to separate from
an initial distance of $\delta_{0}$ (at time $t$) to a final distance of
$\delta_{f}$, as
$\lambda(\textbf{x},t,\delta_{0},\delta_{f})=\frac{1}{\tau}\log{\frac{\delta_{f}}{\delta_{0}}}.$
(1)
It is natural to choose the initial points x on the nodes of a grid with
lattice spacing coinciding with the initial separation of fluid particles
$\delta_{0}$. Then, values of $\lambda$ are obtained in a grid with lattice
separation $\delta_{0}$. In most of this work the resolution of the FSLE
field, $\delta_{0}$, is chosen equal to the resolution of the velocity field,
$\Delta_{0}$. Other choices of parameter are possible and $\delta_{0}$ can
take any value, even much smaller than the resolution of the velocity field
(Hernández-Carrasco et al., 2011). This opens many possibilities that will not
be fully explored in this work (see also Fig. 3 and A.1) . Using similar
parameters for the FSLEs’ computation, We also investigate the response of the
coupled biophysical system to variable resolution of the velocity field, (see
Hernández-Carrasco et al. (2011) for further details about the sensitivity and
robustness of the FSLEs).
The field of FSLEs thus depends on the choice of two length scales: the
initial, $\delta_{0}$ and the final $\delta_{f}$ separations. As in previous
works (d’Ovidio et al., 2004, 2009; Hernández-Carrasco et al., 2011) we focus
on transport processes at mesoscale, so that $\delta_{f}$ is taken as about
110 $km$, or 1∘, which is the order of the size of mesoscale eddies at mid
latitudes. To compute $\lambda$ we need to know the trajectories of the
particles, which gives the Lagrangian character to this quantity. The
equations of motion that describe the horizontal evolution of particle
trajectories in longitudinal and latitudinal spherical coordinates,
$\textbf{x}=(\phi,\theta)$, are:
$\displaystyle\frac{d\phi}{dt}$ $\displaystyle=$
$\displaystyle\frac{u(\phi,\theta,t)}{R\cos{\theta}},$ (2)
$\displaystyle\frac{d\theta}{dt}$ $\displaystyle=$
$\displaystyle\frac{v(\phi,\theta,t)}{R},$ (3)
where $u$ and $v$ represent the eastwards and northwards components of the
surface velocity field, and $R$ is the radius of the Earth (6371 km).
The ridges of the FSLE field can be used to define the Lagrangian Coherent
Structures (LCSs) (Haller and Yuan, 2000; d’Ovidio et al., 2004, 2009; Tew Kai
et al., 2009; Hernández-Carrasco et al., 2011), which are useful to
characterize the flow from the Lagrangian point of view (Joseph and Legras,
2002; Koh and Legras, 2002). Since we are only interested in the ridges of
large FSLE values, the ones which significantly affect stirring, LCSs can be
computed by the high values of FSLE which have a line-like shape. We compute
FSLEs by integrating backwards-in-time the particle trajectories since
attracting LCSs (and its associated unstable manifolds) have a direct physical
interpretation (Joseph and Legras, 2002; d’Ovidio et al., 2004, 2009).
Tracers, such as temperature and chlorophyll-a, spread along the attracting
LCSs, thus creating their typical filamental structure (Lehan et al., 2007;
Calil and Richards, 2010).
### III.2 The Biological model
The plankton model is similar to the one used in previous studies by Oschlies
and Garçon (1998, 1999) and Sandulescu et al. (2007, 2008). It describes the
interaction of a three-level trophic chain in the mixed layer of the ocean,
including phytoplankton $P$, zoo-plankton $Z$ and dissolved inorganic nutrient
$N$, whose concentrations evolve in time according to the following equations:
$\displaystyle\dfrac{dN}{dt}=F_{N}=\Phi_{N}-\beta\dfrac{N}{\kappa_{N}+N}P+\mu_{N}\left((1-\gamma)\dfrac{\alpha\eta
P^{2}}{\alpha+\eta P^{2}}Z+\mu_{P}P+\mu_{z}Z^{2}\right),$ (4)
$\displaystyle\dfrac{dP}{dt}=F_{P}=\beta\dfrac{N}{\kappa_{N}+N}P-\dfrac{\alpha\eta
P^{2}}{\alpha+\eta
P^{2}}Z-\mu_{P}P,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(5) $\displaystyle\dfrac{dZ}{dt}=F_{Z}=\gamma\dfrac{\alpha\eta
P^{2}}{\alpha+\eta
P^{2}}Z-\mu_{Z}Z^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(6)
where the dynamics of the nutrients, Eq. (4), is determined by nutrient supply
due to the vertical transport $\Phi_{N}$, its uptake by phytoplankton (2nd
term) and its recycling by bacteria from sinking particles (remineralization)
(3rd term). Vertical mixing which brings subsurface nutrients into the mixed
surface layer of the ocean is parameterized in our coupled model (see below),
since the hydrodynamical part considers only horizontal 2D transport. The
terms in Eq. (5) represent the phytoplankton growth by consumption of $N$
(i.e. primary production $PP=\dfrac{N}{\kappa_{N}+N}P$), the grazing by
zooplankton ($G_{z}=\dfrac{\alpha\eta P^{2}}{\alpha+\eta P^{2}}Z$), and
natural mortality of phytoplankton. The last equation, Eq. (6), represents
zooplankton growth by consuming phytoplankton minus zooplankton quadratic
mortality.
An important term of our model is the parameterization of the vertical
transport of nutrients by coastal upwelling. Assuming constant nutrient
concentration $N_{b}$ below the mixed layer, this term can be expressed as:
$\Phi_{N}(\textbf{x},t)=S(\textbf{x},t)(N_{b}-N(\textbf{x},t)),$ (7)
where the function $S$, which depends on time and space (on the two
dimensional location x), determines the amplitude and the spatial distribution
of vertical mixing in the model, thus specifying the strength of the coastal
upwelling. Thus, the function $S$ represents the vertical transport due to
coastal upwelling in our 2D model. Upwelling intensity along the coast is
characterized by a number of coastal cells of enhanced vertical Ekman driven
transport that are associated with similar fluctuations of the alongshore wind
(Demarcq et al., 2003; Veitch et al., 2009). Following these results, we
defined our function $S$ as being null over the whole domain except in a 0.5∘
wide coastal strip, varying in intensity depending on the latitude concerned
(see Fig. 1). Six separate upwelling cells, peaking at approximately 33∘S,
31∘S, 27.5∘S, 24.5∘S, 21.5∘S, 17.5∘S,can be discerned. They are named Cape
Peninsula, Columbine/Namaqua, Luderitz, Walvis Bay, Namibia and Cunene,
respectively, Luderitz being the strongest. For the temporal dependence, $S$
switches between a summer and a winter parameterization displayed in Fig. 1.
When $\Phi_{N}$ is fixed to either its summer or its winter shape described in
Fig. 1, the dynamical system given by Eqs. (4,5,6) evolves towards an
equilibrium distribution for $N$, $P$ and $Z$. The transient time to reach
equilibrium is typically $60$ days with the initial concentrations used (see
Sec. III.3). The parameters are set following a study by Pasquero et al.
(2004) and are listed in Table 1.
Figure 1: Shape and values of the strength ($S$) of the upwelling cells used in the simulations for winter and summer seasons (following Veitch et al. (2009)). Parameter | Symbol | Value
---|---|---
Phytoplankton growth rate | $\beta$ | 0.66 day-1
Prey capture rate | $\eta$ | 1.0 (mmol N m-3)-2 day-1
Assimilation efficiency of Zooplankton | $\gamma$ | 0.75
Maximum grazing rate | $a$ | 2.0 day-1
Half-saturation constant for N uptake | $k_{N}$ | 0.5 mmol N m-3
Inefficiency of remineralization | $\mu_{N}$ | 0.2
Specific mortality rate | $\mu_{P}$ | 0.03 day-1
(Quadratic) mortality | $\mu_{Z}$ | 0.2 (mmol N m-3)-1 day-1
Nutrient concentration bellow mixed layer | $N_{b}$ | 8.0 mmol N m-3
Table 1: List of parameters used in the biological model.
### III.3 Coupling hydrodynamical and biological models in Benguela.
We used the velocity fields provided by (Sudre and Morrow, 2008) and
(Gutknecht et al., 2013) to do offline coupling with the NPZ model. The
evolution of simulated concentrations advected within a flow is determined by
the coupling between the hydrodynamical and biological models, as described by
an advection-reaction-diffusion system. The complete model is given by the
following system of partial differential equations:
$\displaystyle\dfrac{\partial N}{\partial t}+\textbf{v}\cdot\nabla
N=F_{N}+D\nabla^{2}N,$ (8) $\displaystyle\dfrac{\partial P}{\partial
t}+\textbf{v}\cdot\nabla P=F_{P}+D\nabla^{2}P,$ (9)
$\displaystyle\dfrac{\partial Z}{\partial t}+\textbf{v}\cdot\nabla
Z=F_{Z}+D\nabla^{2}Z.$ (10)
The biological model is the one described previously by the functions $F_{N}$,
$F_{P}$ and $F_{Z}$. Horizontal advection is the 2D velocity field v, which is
obtained from satellite data or from the ROMS model. We add also an eddy
diffusion term, via the $\nabla^{2}$ operator, acting on $N$, $P$, and $Z$ to
incorporate the unresolved small-scales which are not explicitly taken into
account by the velocity fields used.
The eddy diffusion coefficient, $D$, is given by Okubo’s formula (Okubo,
1971), $D(l)=2.055*10^{-4}~{}l^{1.15}$, where $l$ is the value of the
resolution, in meters, corresponding to the angular resolution $l=\Delta_{0}$.
The formula gives the values $D$=26.73 $m^{2}/s$ for Satellite1/4 and ROMS1/4,
and $D$=7.4$~{}m^{2}/s$ for ROMS1/12.
The coupled system Eqs. (8),(9) and (10) is solved numerically by the semi-
Lagrangian algorithm described in Sandulescu et al. (2007), combining Eulerian
and Lagrangian schemes. The initial concentrations of the tracers were taken
from Koné et al. (2005) and they are $N_{0}=1\ mmolNm^{-3}$ , $P_{0}=0.1\
mmolNm^{-3}$, and $Z_{0}=0.06\ mmolNm^{-3}$. The inflow conditions at the
boundaries are specified in the following way: at the eastern corner, and at
the western and southern edges of the computational domain fluid parcels enter
with very low concentrations ($N_{L}=0.01N_{0}\ mmolNm^{-3}$,
$P_{L}=0.01P_{0}\ mmolNm^{-3}$, and $Z_{L}=0.01Z_{0}\ mmolNm^{-3}$). Across
the northern boundary, fluid parcels enter with higher concentrations
($N_{H}=5\ mmolNm^{-3}$, $P_{H}=0.1\ mmolNm^{-3}$, and $Z_{H}=0.06\
mmolNm^{-3}$). Nitrate concentrations are derived from CARS climatology
(Condie and Dunn, 2006), while P and Z concentrations are taken from Koné et
al. (2005). The integration time step is $dt=6$ hours.
To convert the modeled $P$ values, originally in $mmolN.m^{-}3$, into
$mg~{}m^{-3}$ of chlorophyll, we used a standard ratio of
$Chloro/Nitrogen=1.59$ as prescribed by Hurtt and Armstrong (1996) and Doney
(1996). In the following we refer to as “simulated chlorophyll” for the
concentrations derived from the simulated phytoplankton P, after applying the
conversion ratio (see above); and we use “observed chlorophyll” for the
chlorophyll-a measured by SeaWIFS.
## IV Results and discussion
### IV.1 Validation of our simple 2D idealized setting using satellite data
#### IV.1.1 Horizontal stirring
We compute the FSLE with an initial separation of particles equal to the
spatial resolution of each velocity field ($\delta_{0}$= 1/4∘ for Satellite1/4
and ROMS1/4, and $\delta_{0}$= 1/12∘ for ROMS1/12), an a final distance of
$\delta_{f}$= 1∘ to focus on transport processes by mesoscale structures at
mid latitudes. The areas of more intense horizontal stirring due to mesoscale
activity can be identified by large values of temporal averages of backward
FSLEs (see Figure 2). While there are visible differences between the results
from the different velocity fields, especially in the small-scale patterns,
the spatial pattern are quantitatively well reproduced. For instance, spatial
correlation coefficient $R^{2}$ between FSLEs map from Satellite1/4 and from
ROMS1/4 is $0.81$. Correlation coefficients between Satellite1/4 and ROMS1/12
on one hand, and between ROMS1/4 and ROMS1/12 on the other hand, are lower
($0.61$ and $0.77$ respectively) since the FSLE were computed on a different
resolution. More details on the effect on the grid resolution when computing
FSLEs can be found in Hernández-Carrasco et al. (2011). For all datasets, high
stirring values are observed in the southern region, while the northern area
displays significantly lower values, in line with Rossi et al. (2009). Note
that the separation is well marked for Satellite1/4 where high and low values
of FSLE occur below and above a line at $27^{\circ}$ approximately. In the
case of ROMS flow fields, the stirring activity is more homogeneously
distributed, although the north-south gradient is still present. We associate
this latitudinal gradient with the injection of energetic Agulhas rings, the
intense jet/bathymetry interactions and with other source of flow
instabilities in the southern Benguela. Following Gruber et al. (2011) we
compute the EKE, another proxy of the intensity of mesoscale activity. There
are regions with distinct dynamical characteristics as the southern subsystem
is characterized by larger EKE values than the northern area, in good
agreement with the analysis arising from FSLEs (Fig. 2). Spatial correlations
(not shown) indicate that EKE and FSLE patterns are well correlated using a
non-linear fitting (power law). For instance, EKE and FSLE computed on the
velocity field from Satellite1/4 exhibit a $R^{2}$ of $0.86$ for the non-
linear fitting: $FSLE=0.009\cdot EKE^{0.49}$. This is in agreement with the
initial results from Waugh et al. (2006); Waugh and Abraham (2008), for a
related dispersion measurement, and confirmed for the Benguela region by the
thorough investigations of EKE/FSLE relationship by Hernández-Carrasco et al.
(2012).
Figure 2: Spatial distribution of time average of weekly FSLE maps in the
Benguela region. a) Three years average using data set Satellite1/4; b) one
year average using ROMS1/4; c) one year average using ROMS1/12. The units of
the colorbar are $1/days$. The black lines are contours of annual EKE. The
separation between contour levels is 100$cm^{2}/s^{2}$.
To analyze the variability of horizontal mixing with latitude, we compute
longitudinal averages of the plots in Fig. 2 for two different coastally-
oriented strips extended: a) from the coast to $3^{\circ}$ offshore, and b)
from $3^{\circ}$ to $6^{\circ}$ offshore (see Fig. 3). It allows analyzing
separately subareas characterized by distinct bio-physical characteristics
(see also Rossi et al. (2009)), the coastal upwelling ($3^{\circ}$ strip) with
high plankton biomasses and moderated mesoscale activity, and the open ocean
(from $3^{\circ}$ to $6^{\circ}$ offshore) with moderated plankton biomasses
and high mesocale activity. It is clear that horizontal stirring decreases
with decreasing latitude. In Fig. 3 (a) we see that, for Satellite1/4, the
values of FSLEs decay from $0.18\ days^{-1}$ in the southern to $0.03\
days^{-1}$ in the northern area, with similar significant decays for ROMS1/4
and ROMS1/12. Specifically the North-South difference for Satellite1/4,
ROMS1/4 and ROMS1/12 are of the order of $0.15\ days^{-1}$ , $0.15\ days^{-1}$
and $0.08\ days^{-1}$, respectively, confirming a lower latitudinal gradient
for the case of ROMS1/12.
Note that there are differences in the stirring values (FSLEs) depending on
the type of data, their resolution, the averaging strip, and the grid size of
FSLE computation. In general, considering velocities with the same resolution,
the lower values correspond to Satellite1/4 as compared to ROMS1/4. On
average, values of stirring from ROMS1/4 are larger than those from ROMS1/12,
whereas we would expect the opposite considering the higher resolution of the
latter simulation favoring small scales processes. However, this comparison is
hampered by the fact that spatial means of FSLE values are reduced when
computing them on grids of higher resolution, because the largest values
become increasingly concentrated in thinner lines, a consequence of their
multifractal character (Hernández-Carrasco et al., 2011). Indeed, one can not
compare consistently two FSLEs field computed on a different resolution,
whatever the intrinsic resolution of the velocity field is. The FSLEs computed
on a 1/4∘ grid (black and red lines on Fig. 3) cannot be directly compared to
FSLE fields computed on a 1/12∘ grid (green line Fig. 3) (see Hernández-
Carrasco et al. (2011)). Note however that when FSLEs are computed using the
ROMS1/12 and ROMS1/4 flows but on the same FSLE grid with a fixed resolution
of 1/12∘, one finds smaller values of FSLEs for the coarser velocity field
(ROMS1/4) (see green and blue lines in Fig. 3). The effect of reducing the
velocity spatial resolution on the FSLE calculations is considered more
systematically in A.1. FSLE values obtained from the same FSLE-grid increase
as the resolution of the velocity-grid becomes finer (Fig. 12) A general
observation consistent between all datasets is that horizontal mixing is
slightly less intense and more variable in the region of coastal upwelling
(from the coast to 3∘ offshore) than within the transitional area with the
open ocean (3-6∘ offshore). Note also that a low-stirring region is observed
within the 3∘ width coastal strip from $28^{\circ}$ to $30^{\circ}$S on all
calculations. These observations confirm that the ROMS model is representing
well the latitudinal variability of the stirring as measured from FSLE based
on satellite data. These preliminary results indicate that Lyapunov exponents
and methods could be used as a diagnostic to validate the representation of
mesoscale activity in eddy-resolving oceanic models, as suggested recently by
Titaud et al. (2011). Overall, the variability of stirring activity in the
Benguela derived from the simulated flow fields is in good agreement with the
satellite observations.
Figure 3: Zonal average on coastal bands of the FSLE time averages from Fig. 2
as a function of latitude. a) from the coast to 3 degrees offshore; b) between
3 and 6 degrees offshore.
#### IV.1.2 Simulated phytoplankton concentrations
Evolution of $N$, $P$ and $Z$ over space and time is obtained by integrating
the systems described by Eqs. (8), (9) and (10). The biological model is
coupled to the velocity field after the spin-up time needed to reach stability
($60$ days). Analysing the temporal average of simulated chlorophyll (Fig. 4),
we found that coastal regions with high $P$ extend approximately, depending on
latitude, between half a degree and two degrees offshore. It is comparable
with the pattern obtained from the satellite-derived chlorophyll data (Fig.4
d)). The spatial correlation between averaged simulated and satellite
chlorophyll is as follows: $R^{2}=0.85$ for Satellite1/4 versus SeaWIFS;
$R^{2}=0.89$ for ROMS1/4 versus SeaWIFS and $R^{2}=0.85$ for ROMS1/12 versus
SeaWIFS. Despite the very simple setting of our model, including the
parameterization of the coastal upwelling, the distribution of phytoplankton
biomass is relatively well simulated in the Benguela area. Note however that
our simulated chlorophyll values are about $\simeq$ 3-4 times lower than
satellite data. Many biological and physical factors not taken into account in
this simple setting could be invoked to explain this offset. Another possible
explanation is the low reliability of ocean color data in the optically
complex coastal waters (Mélin et al., 2007).
Figure 4: Spatial distribution of: a) Three years average of simulated
chlorophyll using Satellite1/4, b) One year average of simulated chlorophyll
using ROMS1/4, c) Same than b) but using ROMS1/12, d) Three years average of
observed chlorophyll derived from monthly SeaWIFS data. The units of the
colorbar are $mg/m^{3}$. Logarithmic scale is used to improve the
visualization of gradients in nearshore area.
We now examine the latitudinal distribution of $P$ comparing the outputs of
the numerical simulations versus the satellite chlorophyll-a over different
coastally oriented strips (Fig.5). Simulated $P$ concentrations are higher in
the northern than in the southern area of Benguela, in good agreement with the
chlorophyll-a data derived from satellite. A common feature is the minimum
located just below the Luderitz upwelling cell (28∘S), which may be related to
the presence of a physical boundary, already studied and named the LUCORC
barrier by Shannon et al. (2006) and Lett et al. (2007). The decrease of $P$
concentration is clearly visible in the open ocean region of the Satellite1/4
case (Fig. 5 b)). Correlations of zonal averages between simulated and
satellite chlorophyll-a are poor when considering the whole area ($R^{2}$
ranging from 0.1 to 0.5). However, when considering each subsystem (northern
and southern) independently, high correlation coefficients are found for the
south Benguela ($R^{2}$ around 0.75), but not for the north. This indicates
that our simple modelling approach is able to simulate the spatial patterns of
chlorophyll in the south Benguela, but not properly in the northern part. In
the north, other factors not considered here (such as the 3D flow, the varying
shelf width, the external inputs of nutrients, realistic non-climatologic
forcings, complex biogeochemical processes, etc…) seem to play an important
role in determining the surface chlorophyll-a observed from space.
Figure 5: Zonal mean of simulated chlorophyll on a coastally oriented strip
from the coast to 3 degrees (a) and from 3 degrees to 6 degrees offshore (b),
plotted as a function of latitude. Zonal average of observed chlorophyll
(SeaWIFS) over a coastal band from the coast to 3 degrees (c) and from 3
degrees to 6 degrees offshore (d).
### IV.2 Relationship between phytoplankton and horizontal stirring.
In Fig. 6 we show six selected snapshots of chlorophyll concentrations every
$8$ days during a $32$ days period for ROMS1/12. Since both ROMS simulations
were climatologically forced runs, the dates do not correspond to a specific
year. The most relevant feature is the larger value of concentrations near the
coast due to the injection of nutrients. Obviously the spatial distribution of
$P$ is strongly influenced by the submeso- and meso-scale structures such as
filaments and eddies, especially in the southern subsystem. Differences are
however observed between the three data sets. In particular, it seems that for
Satellite1/4 and ROMS1/12 the concentrations extend further offshore than for
ROMS1/4 (not shown). In A.1 we provide additional analysis of the effect of
the velocity spatial resolution on phytoplankton evolution. We found that
velocity data with different resolution produces similar phytoplankton
patterns but larger absolute values of concentrations as the spatial
resolution of the velocity field is refined (see Mahadevan and Archer (2000);
Levy et al. (2001)), supporting the need to compare different spatial
resolutions.
Several studies (Lehan et al., 2007; d’Ovidio et al., 2009; Calil and
Richards, 2010) have shown that transport of chlorophyll distributions in the
marine surface is linked to the motion of local maxima or ridges of the FSLEs.
This is also observed in our numerical setting when superimposing contours of
high values of FSLE (locating the LCSs) on top of phytoplankton concentrations
for ROMS1/12 (see Fig. 6). In some regions $P$ concentrations are constrained
and stirred by lines of FSLE. For instance, the elliptic eddy-like structure
at $13\ ^{\circ}$E, $32\ ^{\circ}$S is characterized by high phytoplankton
concentrations at its edge, but relatively low in its core. This reflects the
fact that tracers, even active such as chlorophyll, still disperse along the
LCSs.
Figure 6: Snapshots every $8$ days of large (top $30\%$) values of FSLE
superimposed on simulated chlorophyll concentrations calculated from ROMS1/12
in $mg/m^{3}$. Logarithmic scale for chlorophyll concentrations is used to
improve the visualization of the structures
From Fig. 5 it is clear that phytoplankton biomass has a general tendency to
decrease with latitude, an opposite tendency to the one exhibited by stirring
(as inferred from the FSLEs and EKE distributions in Figs. 2 and 3) for the
three data sets. Moreover, note that the minimum of phytoplankton located just
below the LUCORC barrier at $28^{\circ}$S (Fig. 5) coincides with a local
maximum of stirring that might be responsible for this barrier (Fig. 3 a).
Spatial mean and latitudinal variations of FSLE and chlorophyll-a analyzed
together suggest an inverse relationship between those two variables. The 2D
vigorous stirring in the south and its associated offshore export seem
sufficient to simulate reasonably well the latitudinal patterns of $P$. The
numerous eddies released from the Agulhas system and generally travelling
north-westward, associated with the elevated mesoscale activity in the south
Benguela, might inhibit the development of $P$ and export unused nutrients
toward the open ocean. Although Gruber et al. (2011) invoked the offshore
subduction of unused nutrients (3D effect), our results suggest that 2D
offshore advection and intense horizontal mixing could by themselves affect
negatively the phytoplankton growth in the southern Benguela.
To study quantatively the negative effect of horizontal stirring on
phytoplankton concentration, we examine the correlation between the spatial
averages – over each subregion (North and South) and the whole area of study –
of every weekly map of FSLE and the spatial average of the corresponding
weekly map of simulated $P$, considering each of the three velocity fields
(Fig.7). For all cases, a negative correlation between FSLEs and chlorophyll
emerges: the higher the surface stirring/mixing, the lower the biomass
concentration. The correlation coefficient taking into account the whole area
is quite high for all the plots, $R^{2}$=0.77 for Satellite1/4, 0.70 for
ROMS1/4 and 0.84 for ROMS1/12 , and the slopes (blue lines in Fig.7 have the
following values: -1.8 for Satellite1/4, -0.8 for ROMS1/4 and -2.3 for
ROMS1/12. The strongest negative correlation is found for the setting with
ROMS1/12. Note that, similarly to the results of Rossi et al. (2008, 2009) and
Gruber et al. (2011), the negative slope is larger but less robust when
considering the whole area rather than within every subregion. Moreover, if we
average over the coastal strip (from coast to 3∘ offshore) and only in the
south region (Fig.7 d),e),f) ) we find high values of the correlation
coefficient for the Satellite1/4, and ROMS1/12 cases. The suppressive effect
of stirring might be dominant only when stirring is intense, as in the south
Benguela. Gruber et al. (2011) stated that the reduction of biomass due to
eddies may extend beyond the regions of the most intense mesoscale activity,
including the offshore areas that we do not simulate in this work.
Figure 7: Weekly values of spatial averages of simulated chlorophyll versus
weekly values of spatial averages of FSLE, where the average are over the
whole area (6 ∘ from the coast) and in North and South subareas of Benguela.
a) Satellite1/4, b) ROMS1/4 and c) ROMS1/12. Right column plots the average
over 3∘ offshore in the south region: d) Satellite1/4, e) ROMS1/4 and f)
ROMS1/12
In the following we analyse the bio-physical mechanisms behind this negative
relationship.
### IV.3 What causes the variable biological responses within regions of
distinct dynamical properties?
In the following, our analysis is focused on the setting using ROMS1/12 as the
previous results revealed that the negative correlation is more robust.
Similar results and conclusions can be obtained from the simulations using the
two other velocity fields (not shown), attesting of the reliability of our
approach (see correlation coefficients and slopes in Fig. 7).
To understand why simulated chlorophyll-a concentrations differs in both
subsystems, as is the case in satellite observations, we compute annual
budgets of $N,P,Z$ and biological rates (Primary Production $PP$, Grazing and
Remineralization) in the case of the biological module alone (Table 2) and
when coupled with a realistic flow (Table 3). Considering the biological
module alone, we found that $PP$ in the north subsystem is slightly higher
than in the southern one (4$\%$, see also Table 2 ), essentially due to the
differential nutrient inputs $\Phi_{N}$. However, when considering the full
coupled system (hydrodynamic and biology), the latitudinal difference in $PP$
increases significantly (32$\%$, see also Table 3). This latitudinal
difference is in agreement with the patterns of $PP$ derived from remote-
sensed data by Carr (2002). These results indicate that the flow is the main
responsible of the difference in PP. Additional computations (see A.2) also
confirm the minor effect of the biological module ($\Phi_{N}$), as compared
with the flow, on the observed latitudinal differences in $PP$.
Annual budgets only biological system
---
| South | North | North-South difference ($\%$)
Nutrients ($mmolNm^{-3}$) | 821 | 1305 | 37
Phytoplankton ($mmolNm^{-3}$) | 57.0 | 57.7 | 1
Zooplankton ($mmolNm^{-3}$) | 113 | 115 | 2
Primary Production ($mmolNm^{-3}yr^{-1}$) | 35 | 36 | 4
Grazing ($mmolNm^{-3}yr^{-1}$) | 33 | 35 | 4
$\Phi_{N}$ ($mmolNm^{-3}yr^{-1}$) | 28 | 29 | 3
Remineralization ($mmolNm^{-3}yr^{-1}$) | 7.0 | 7.4 | 4
Table 2: Budgets of N,P,Z and biological rates (Primary Production, Grazing,
$\Phi_{N}$, and remineralization) for the biological model. Annual budgets
hydrodynamics-biology coupled system
---
| South | North | North-South difference ($\%$)
Nutrients ($mmolNm^{-3}$) | 849 | 1937 | 56
Phytoplankton ($mmolNm^{-3}$) | 147 | 198 | 26
Zooplankton ($mmolNm^{-3}$) | 231 | 347 | 33
Primary Production ($mmolNm^{-3}yr^{-1}$) | 63 | 98 | 32
Grazing ($mmolNm^{-3}yr^{-1}$) | 56 | 87 | 35
$\Phi_{N}$ ($mmolNm^{-3}yr^{-1}$) | 81 | 91 | 10
Remineralization ($mmolNm^{-3}yr^{-1}$) | 11 | 18 | 4
Table 3: Budgets of N,P,Z and biological rates (Primary Production, Grazing,
$\Phi_{N}$, and remineralization) for the bio-flow coupled model.
Gruber et al. (2011)) suggested that the offshore advection of plankton
biomass enhanced by mesoscale structures might be responsible for the
suppressive effect of stirring in upwelling areas. To test this mechanism, we
next analyze the net horizontal transport of biological tracers by the flow.
In particular, we have computed the zonal, $JC_{\phi}$, and meridional,
$JC_{\theta}$, advective fluxes of $N,P,Z$ (the diffusive fluxes being much
smaller):
$\displaystyle JC_{\phi}(\textbf{x},t)$ $\displaystyle=$ $\displaystyle
u(\textbf{x},t)C(\textbf{x},t),$ (11) $\displaystyle
JC_{\theta}(\textbf{x},t)$ $\displaystyle=$ $\displaystyle
v(\textbf{x},t)C(\textbf{x},t),$ (12)
where $u$ and $v$ are the zonal and meridional components of the velocity
field respectively, and with $C$ we denote the N, P and Z concentrations, all
of them given at a specific point in the 2D-space and time $(\textbf{x},t)$.
$JC$ is the flux of the concentration, $C$, i.e., $JN_{\phi}$ is the zonal
flux of nutrients (eastward positive), $JP_{\theta}$ is the meridional flux
(northward positive) of phytoplankton, and so on. Annual averages of daily
fluxes were computed, and then a zonal average as a function of the latitude
was calculated for the different coastal bands considered all along this
paper. Fig. 8 shows these calculations for the velocity field from ROMS1/12,
while similar results were found for the other data sets (not shown). Similar
behavior is observed for the fluxes of $N$, $P$ and $Z$: zonal fluxes are
almost always negative, so that westward transport dominates, and meridional
fluxes are predominantly positive so that they are directed to the north.
Comparing North and South in the 3∘ coastal band, it is observed that at high
latitudes the zonal flux has larger negative values than at low latitudes, and
the meridional flux presents larger positive values at higher latitudes. In
other words, the northwestward transport of biological material is more
intense in the southern than in the northern regions, suggesting a higher
’flushing rate’. It also suggests that unused nutrients from the southern
Benguela might be advected toward the northern areas, possibly promoting even
further the local ecosystem.
To estimate the transport of recently upwelled nutrients by LCSs and other
mesoscale structures, apart from the mean flow, we compute the zonal and
meridional fluxes of biological tracers using the smoothed ROMS1/12 velocity
field at the spatial resolution equivalent to 1/2∘ (see A.1 for more details).
The results, plotted in Fig. 8 (red lines), show that in general the fluxes
are less intense in the coarser than in the finer velocity, indicating that
there is a contribution to net transport due to the submeso- and meso-scale
activity. To estimate the quantitative contribution of mesoscale processes, we
compute the difference of the fluxes of the different biological tracers $C$ =
$N,P,Z$, $Q_{JC}$, in the coarser velocity field with respect to the original
velocity field. The values of $Q_{JC}$ range from 30 to 50$\%$, indicating
that the contribution of the mesocale to the net transport of the biological
concentrations is important. Moreover, the values of $Q_{JC}$ are larger in
the south than in the north confirming that the mesoscale-induced transport is
more intense in the south.
Lachkar and Gruber (2011) showed that mesoscale processes reduce the
efficiency of nutrients utilization by phytoplankton due to their influence on
residence times. The longer residence times (i.e. the less mesoscale activity)
seem to favor the accumulation of biomass. To test this effect in our
simulations, we compute the residence times (RT), defined as the the time
interval that a particle remains in the coastal trip of 5∘ wide. The spatial
distribution (not shown) of the annual average of RT indicates that the
longest RT are found in the north region. In fact, zonal analysis reveals that
RT has a tendency to increase as the latitude decreases, with a mean value in
the North equals to 249$days$, and 146$days$ in the South. This suggests that
regions with weak fluxes are associated with long residence times and high
growth rate of phytoplankton. On the other hand, high mesoscale activity is
favoring the northwestward advection which decreases the residence times,
associated to lower growth rate of plankton.
Figure 8: Zonal mean of zonal and meridional fluxes of $N,P,Z$ concentrations
for the ROMS1/12 case, averaged from the coast to 3∘ offshore.
This effect and the role of horizontal advection is confirmed by performing
numerical simulations where no biological dynamics is considered. This amounts
to solving Eq. (4) with $P=Z=0$ considering solely lateral transport, so that
$N$ is a passive scalar with sources. In Fig. 9 we see the results (for the
$ROMS1/12$ case, similar for the other datasets). There is a very small tracer
concentration in the southern domain, and the differences north-south are more
pronounced than the case including the plankton dynamics (see Fig. 5). This
supports further the fact that the main actor on the spatial distribution of
biomasses is the horizontal transport.
Figure 9: a) Spatial distribution of time average of the passive scalar
concentration (see details at the end of subsection IV.3). b) Comparison of
latitudinal profile of time averages of the passive scalar, as a function of
latitude, for zonal average over different coastal bands.
## V Conclusions
We have studied the biological dynamics in the Benguela area by considering a
simple biological NPZ model coupled with different velocity fields (satellite
and model). Although in a simple framework, a reduction of phytoplankton
concentrations in the coastal upwelling for increasing mesoscale activity has
been successfully simulated. Horizontal stirring was estimated by computing
the FSLEs and was correlated negatively with chlorophyll stocks. Similar
correlations are found, though not presented in this manuscript, for the
primary production. Some recent observational and modelling studies proposed
the “nutrient leakage” as a mechanism to explain this negative correlation.
Here we argue that Lagrangian Coherent Structures, mainly mesoscale eddies and
filaments, transport a significant fraction (30-50%) of the recently upwelled
nutrients nearshore toward the open ocean before being efficiently used by the
pelagic food web. The fluxes of nutrients and organic matter, due to the mean
flow and its mesoscale structures, reflect that transport is predominantly
westward and northward. Biomass is transported towards open ocean or to the
northern area. In addition to the direct effect of transport, primary
production is also negatively affected by high levels of turbulence,
especially in the south Benguela. Although some studies dealt with 3D effects,
we have shown that 2D advection processes seems to play an important role in
this suppressive effect. Our analysis suggests that the inhibiting effect of
the mesoscale activity on the plankton occurs when the stirring reaches high
levels, as in the south Benguela. However, this effect is not dominant under
certain levels of turbulence. It might indicate that planktonic ecosystems in
oceanic regions with vigorous mesoscale dynamics can be, as a first
approximation, easily modeled just by including a realistic flow field. The
small residence times of waters in the productive area will smooth out all the
other neglected biological factors in interaction.
Our findings confirm the unexpected role that mesoscale activity has on
biogeochemical dynamics in the productive coastal upwelling. Strong vertical
velocities are known to be associated with these physical structures and they
might have another direct effect by transporting downward rich nutrient waters
below the euphotic zone. Further studies are needed such as 3D realistic
modelling that take into account the strong vertical dynamics in upwelling
regions to test the complete mechanisms involved.
## Acknowledgments
I.H-C was supported by a FPI grant from MINECO to visit LEGOS. We acknowledge
support from MINECO and FEDER through projects FISICOS (FIS2007-60327) and
ESCOLA (CTM2012-39025-C02-01). V. G. thanks CNES funding through Hiresubcolor
project. We are also grateful to J. Sudre for providing us velocity data sets
both from ROMS and from the combined satellite product. Ocean color data were
produced by the SeaWiFS project at GES and were obtained from DAAC.
## Appendix A Sensitivity analysis
A number of numerical experiments were done to investigate the sensitivity of
the coupled bio-physical model with respect to different variables.
### A.1 Sensitivity with respect to different spatial resolution of the
velocity field
In this experiment we used a velocity field from ROMS1/12 smoothed out towards
a resolution 1/4∘, and to be compared with $ROMS1/4$ and $ROMS1/12$ at their
original spatial resolution. We coarse-grained the velocity field with a
convolution kernel weighted with a local normalization factor, and keeping the
original resolution for the data so that land points are equally well
described as in the original data. The coarsening kernel with scale factor
$s$, $k_{s}$, is defined as:
$k_{s}(x,y)=e^{-\frac{(x^{2}+y^{2})}{2s^{2}}}.$ (13)
To avoid spurious energy dump at land points we have introduced a local
normalization weight given by the convolution: $k_{s}(x,y)*M(x,y)$, where
$M(x,y)$ is the sea mask. For points far from the land the weight is just the
normalization of $k_{s}$, and for points surrounded by land the weight takes
the contribution from sea points only. Thus $v_{s}$, the velocity field
coarsened by a scale factor $s$, is obtained from the original velocity field
$v$ as:
$v_{s}=\frac{k_{s}*v}{k_{s}*M}.$ (14)
In Fig. 10 we compare two ROMS1/12 smoothed velocity fields at scales $s$=3
and $s$=6 (with an equivalent spatial resolution 1/4∘ and 1/2∘, respectively)
with the original velocity field from ROMS1/12. It is clear that the
circulation pattern is smoothed as $s$ is increased. The FSLE computations
using these smoothed velocity fields are shown in Fig 11. When the spatial
resolution is reduced to $1/4^{\circ}$ the FSLEs and small-scale contributions
decrease, but the main global features remain, as indicated in the study by
Hernández-Carrasco et al. (2011). Further coarsening to $1/2^{\circ}$ smoothes
most of the structures except the most intense ones.
Figure 10: Vectors of a velocity field from $ROMS1/12$: a) at original
resolution. b) smoothed by a scale factor of $s$=3, obtaining and equivalent
spatial resolution of 1/4∘, c) smoothed by a scale factor of s=6, obtaining
and equivalent spatial resolution of 1/2∘. The snapshots correspond to day 437
of the simulation. Figure 11: Snapshots of spatial distributions of FSLEs
backward 437 days in time starting from day 437 of $ROMS1/12$ at the same FSLE
grid resolution of 1/12∘, and using the velocity fields at different
resolutions: a) at original resolution 1/12∘. b) smoothed velocity field at
equivalent 1/4∘ and c) smoothed velocity field at equivalent 1/2∘.
The latitudinal variations of the zonal averages performed on the time
averages of the FSLE maps plotted in Fig. 11 are compared in Fig.12. The mean
FSLEs values strongly diminish when the velocity resolution is sufficiently
smoothed out. This is due to the progressive elimination of mesoscale
structures that are the main contributors to stirring processes. Also the
latitudinal variability of stirring diminishes for the very smoothed velocity
field (blue line in Fig. 12 ). Thus, latitudinal differences of stirring in
the Benguela system are likely to be related to mesoscale structures (eddies,
filaments, fronts, etc.) contained in the velocity fields.
Figure 12: Latitudinal profile of the zonal mean values of annual averaged
backward FSLEs (51 snapshots weekly separated) at the same FSLE grid
resolution of 1/12∘, and using different smoothed velocity fields.
We have also computed the phytoplankton using these smoothed velocity fields.
Some instantaneous spatial distributions can be seen in Fig 13. The filaments
of phytoplankton disappear in the very smoothed velocity field (1/2∘). The
spatial distribution of the annual average of phytoplankton concentrations for
the different velocity field shows, however, quite similar patterns (not
shown).
In the time series of $N$, $P$ and $Z$ budgets for the coarser velocities one
observes the same behavior as for the original velocity field (not shown).
Figure 13: Snapshots of simulated chlorophyll field using different velocity
fields: a) $ROMS1/12$ at original resolution 1/12∘, b) smoothed $ROMS1/12$
velocity field at equivalent 1/4∘, c) smoothed $ROMS1/12$ velocity field at
equivalent 1/2∘, and d) $ROMS1/4$ at original resolution 1/4∘. The units of
the colorbar are $mg/m^{3}$.
### A.2 Sensitivity with respect to different parameterization of the coastal
upwelling of nutrients.
In section III.2 we mimicked coastal upwelling of nutrient via a source term
in the nutrients equation which is determined by the function $S$, and was
considered spatiotemporally variable. Here we explore the plankton dynamics
using spatially and temporally homogeneous upwelling along the coast. $S$ is
fixed to an average value $S=0.1\ day^{-1}$ along the coast at any time. In
Fig. 14 we show the annual average of $P$ for the $ROMS1/12$ (top panel), and
the comparisons with the inhomogeneous case for the zonal mean (bottom panel).
Therefore, this test suggests that the way we simulate vertical mixing along
the coast has not a large effect on the 2D biological dynamics, which will be
mainly determined by the interplay with horizontal advection.
Figure 14: Comparison between zonal average on different coastal bands of
annual time average of simulated chlorophyll, using homogeneous upwelling and
the non-homogeneous upwelling cells described in Fig. 1.
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|
arxiv-papers
| 2011-12-16T11:03:15 |
2024-09-04T02:49:25.399664
|
{
"license": "Public Domain",
"authors": "Ismael Hern\\'andez-Carrasco, Vincent Rossi, Emilio\n Hern\\'andez-Garc\\'ia, Veronique Gar\\c{c}on and Crist\\'obal L\\'opez",
"submitter": "Ismael Hernandez-Carrasco",
"url": "https://arxiv.org/abs/1112.3760"
}
|
1112.3862
|
# Five Exponential Diophantine Equations and Mayhem Problem M429
Konstantine Zelator
Department of Mathematics, Computer Science and Statistics
212 Ben Franklin Hall
Bloomsburg University of Pennsylvania
400 East Second Street
Bloomsburg, PA 17815
USA
and
P.O. Box 4280
Pittsburgh, PA 15203
e-mails: konstantine zelator@yahoo.com
and kzelator@bloomu.edu
## 1 Introduction
In the March 2010 issue of the journal Crux Mathematicorum with Mathematical
Mayhem, mayhem problem M429 was proposed (see reference [1]):
Determine all positive integers $a,b,c$ that satisfy,
$\begin{array}[]{rcll}a^{(b^{c})}&=&(a^{b})^{c};&{\rm or\ equivalently}\\\ \\\
a^{b^{c}}&=&a^{bc}.\end{array}$
A solution, by this author, was published in the December 2010 issue of Crux
Mathematicorum with Mathematical Mayhem (see [2]). According to this solution,
the following ordered triples of positive integers $a,b,c$ are precisely those
that satisfy the above exponentialequation:
The triples of the form $(1,b,c)$, with $b,c$ being any positive integers;
the triples of the form $(a,b,1)$, with $a,b$ positive integers and with
$a\geq 2$;
and the triples of the form $(a,2,2)$ with $a\in{\mathbb{Z}}^{+}$, and $a\geq
2$.
In the language of diophantine equations, we are dealing with the three-
variable diophantine equation
$x^{(y^{z})}=x^{yz}.$ (1)
Accordingly, the above results can be expressed in Theorem 1 as follows.
###### Theorem 1.
Consider the three-variable diophantine equation, $x^{(y^{z})}=x^{yz}$, over
the set of positive integers ${\mathbb{Z}}^{+}$. If $S$ is the solution set of
the above diophantine equation, then $S=S_{1}\bigcup S_{2}\bigcup S_{3}$,
where $S_{1},S_{2},S_{3}$ are the pairwise disjoint sets,
$\begin{array}[]{rcl}S_{1}&=&\left\\{\left.(1,b,c)\right|b,c\in{\mathbb{Z}}^{+}\right\\};\\\
\\\ S_{2}&=&\left\\{\left.(a,b,1)\right|a\geq
2,a,b\in{\mathbb{Z}}^{+}\right\\};\\\ \\\
S_{3}&=&\left\\{\left.(a,2,2)\right|a\geq 2\ {\rm and}\
a\in{\mathbb{Z}}^{+}\right\\}.\end{array}$
Motivated by mayhem problem M429, in this work we tackle another four
exponential, three-variable diophantine equations. These are:
$x^{(y^{z})}=x^{(z^{y})},$ (2)
$x^{(y^{z})}=y^{xz},$ (3)
$x^{yz}=y^{xz},$ (4)
and
$x^{(y^{z})}=z^{xy}$ (5)
In Section 2, we state Theorems 2, 3, 4, and 5. In Theorems 2, 3 and 4, the
solutions sets of the diophantine equations (2), (3), and (4) are stated.
These three solution sets are determined with the aid of the two-variable
exponential diophantine equation found in Section 3, whose solution set is
given in Result 2.
The proofs of Theorems 2,3, and 4, are given in Section 4. The proof of
Theorem 5 is presented in Section 5. In Theorem 5, some solutions to equation
(5) are given.
## 2 The four theorems
###### Theorem 2.
Consider the three-variable diophantine equation (over ${\mathbb{Z}}^{+}$),
$x^{(y^{z})}=x^{z^{y}}.$
Let $S$ be the solution set of this equation.
Then, $S=S_{1}\bigcup S_{2}\bigcup S_{3}\bigcup S_{4}\bigcup S_{5}$, where
$\begin{array}[]{rcll}S_{1}&=&\left\\{\left.(1,b,c)\right|b,c\in{\mathbb{Z}}^{+}\right\\}&{\rm
where}\\\ \\\ S_{2}&=&\left\\{\left.(a,1,1)\right|a\geq
2,a\in{\mathbb{Z}}^{+}\right\\}&\\\ \\\
S_{3}&=&\left\\{\left.(a,b,b)\right|a\geq 2,b\geq
2,a,b\in{\mathbb{Z}}^{+}\right\\}&\\\ \\\
S_{4}&=&\left\\{\left.(a,4,2)\right|a\geq 2,\
a\in{\mathbb{Z}}^{+}\right\\}&\\\ \\\
S_{5}&=&\left\\{\left.(a,2,4)\right|a\geq
2,a\in{\mathbb{Z}}^{+}\right\\}&\end{array}$
###### Theorem 3.
Consider the three-variable diophantine equation (over ${\mathbb{Z}}^{+}$),
$x^{(y^{z})}=y^{xz}$
Let $S$ be the solution set of this equation. Then, $S=S_{1}\bigcup
S_{2}\bigcup S_{3}\bigcup S_{4}\bigcup S_{5}$ where
$\begin{array}[]{lrcl}&S_{1}&=&\left\\{\left.(1,1,c)\right|c\in{\mathbb{Z}}^{+}\right\\},\\\
\\\ &S_{2}&=&\left\\{\left.(a,a,1)\right|a\geq
2,a\in{\mathbb{Z}}^{+}\right\\}\\\ \\\ {\rm(singleton\
set)}&S_{3}&=&\left\\{(4,2,1)\right\\},\\\ \\\ {\rm(singleton\
set)}&S_{4}&=&\left\\{(2,4,1)\right\\}\\\ \\\
&S_{5}&=&\left\\{\left.(b^{c},b,c)\right|b\geq 2,c\geq
2,b,c\in{\mathbb{Z}}^{+}\right\\}\end{array}$
###### Theorem 4.
Consider the three-variable diophantine equation (over ${\mathbb{Z}^{+}}$
$x^{yz}=y^{xz}.$
Let $S$ be its solution set. Then,
$S=S_{1}\bigcup S_{2}\bigcup S_{3}\bigcup S_{4}\bigcup S_{t}\bigcup
S_{6}\bigcup S_{7},$
where
$\begin{array}[]{lrcl}&S_{1}&=&\left\\{\left.(1,1,c)\right|c\in{\mathbb{Z}}^{+}\right\\}\\\
\\\ &S_{2}&=&\left\\{\left.(a,a,1)\right|a\geq
2,a\in{\mathbb{Z}}^{+}\right\\}\\\ \\\ {\rm(singleton\
set)}&S_{3}&=&\left\\{(4,2,1)\right\\},\\\ \\\ {\rm(singleton\
set)}&S_{4}&=&\left\\{(2,4,1)\right\\}\\\ \\\
&S_{5}&=&\left\\{\left.(a,a,c)\right|a\geq 2,c\geq
2,a,c\in{\mathbb{Z}}^{+}\right\\}\\\ \\\
&S_{6}&=&\left\\{\left.(4,2,c)\right|c\geq 2,\
c\in{\mathbb{Z}}^{+}\right\\}\\\ \\\
&S_{7}&=&\left\\{\left.(2,4,c)\right|c\geq,\
c\in{\mathbb{Z}}^{+}\right\\}\end{array}$
###### Theorem 5.
Consider the three-variable equation (over ${\mathbb{Z}}^{+}$)
$x^{(y^{z})}=z^{xy}$
1. (i)
Let $S$ be the set of those solutions, $(x,y,z)$ such that at least one of
$x,y$, or $z$ is equal to $1$. Then
$S=\left\\{\left.(1,b,1)\right|b\in{\mathbb{Z}}^{+}\right\\}$
2. (ii)
The only solution $(x,y,z)$ to the above equation, such that $x\geq 2,\ y\geq
2,\ z\geq 2$, and with $x=z$, is the triple $(2,2,2)$
3. (iii)
Let $F$ be the family of solutions $(x,y,z)$ such that $x\geq 2,\ y\geq 2,\
z\geq 2$ and with $y=z\neq x$. Then
$F=\left\\{\left.(b^{b},b,b)\right|b\geq 2,\ b\in{\mathbb{Z}}^{+}\right\\}$
## 3 A key exponential diophantine equation
The diophantine equation, $x^{y}=y^{x}$, over the positive integers, is
instrumental in determining the solution sets of the diophantine equations
(2), (3), and (4). The following, Result 1, can be found in W. Sierpinski’s
book, “Elementary Theory of Numbers”, (see reference [3]). The proof is about
half a page long.
###### Result 1.
Consider the two-variable equation, $x^{y}=y^{x}$, over the set of positive
rational numbers, ${\mathbb{Q}}^{+}$. Then all the solutions to this equation,
with $x$ and $y$ being positive rationals, and with $y>x$, are given by
$x=\left(1+\dfrac{1}{n}\right)^{n},\ \ \ y=\left(1+\dfrac{1}{n}\right)^{n+1},$
where $n$ is a positive integer: $n=1,2,3,\ldots$ .
A simple or cursory examination of the formulas in Result 1 easily leads to
Result 2. Observe that these formulas can be written in the form,
$x=\left(\dfrac{n+1}{n}\right)^{n},\ \ \ y=\left(\dfrac{n+1}{n}\right)^{n+1}.$
For $n=1$, we obtain the integer solution $x=2$ and $y=4$. However, for $n\geq
2$, the number $\dfrac{n+1}{n}$ is a proper rational, i.e., a rational which
is not an integer.
This is clear since $n$ and $n+1$ are relatively prime, and $n\geq 2$. Thus,
since for $n\geq 2$, $\dfrac{n+1}{n}$ is a proper rational, so must be any
positive integer power of $\dfrac{n+1}{n}$. This observation takes us
immediately to Result 2 below.
###### Result 2.
Consider the two-variable diophantine equation (over ${\mathbb{Z}}^{+}$)
$x^{y}=y^{x}.$
Let $S$ be its solution set. Then, $S=S_{1}\bigcup S_{2}\bigcup S_{3}$. Where
$\begin{array}[]{lrcl}&S_{1}&=&\left\\{\left.(a,a)\right|a\in{\mathbb{Z}}^{+}\right\\},\\\
\\\ {\rm(singleton\ set)}&S_{2}&=&\left\\{(4,2)\right\\},\\\ \\\ {\rm and\
(singleton\ set)}&S_{3}&=&\left\\{(2,4)\right\\}\end{array}$
Result 2 is used in the proofs of Theorems 2, 3, and 4 below.
## 4 Proofs of Theorems 2, 3, and 4
1. (1)
###### Proof.
Theorem 2 Suppose that $(a,b,c)$ is a solution to equation (2). We have
$a^{(b^{c})}=a^{(c^{b})}$ (6)
If $a=1$, then $b$ and $c$ can be arbitrary positive integers; and (6) is
satisfied.
If $b=1$ and $a\geq 2$, then by (6) we get
$a=a^{c}$. (6a)
Since $a\geq 2$, by inspection, we see that (6a) is satisfied only when $c=1$.
So, we obtain the solutions of the form $(a,1,1)$ with $a\geq 2$. If $a\geq
2,\ b\geq 2$, and $c=1$, equation (6) yields
$a^{b}=a,$
which is impossible with $a\geq 2$ and $b\geq 2$.
Finally, assume that $a\geq 2,\ b\geq 2$, and $c\geq 2$ in (6). Then (6)
$\Leftrightarrow$ (since $a\geq 2$) $b^{c}=c^{b}$; and by Result 2, it follows
that either $b=4$ and $c=2$; or $b=2$ and $c=4$; or $b=c$. We have shown that
if $(a,b,c)$ is a positive integer solution of equation (2), then $(a,b,c)$
must belong to one of the sets $S_{1},S_{2},S_{3},S_{4}$, or $S_{5}$.
Conversely, a routine calculation shows that any member of these five sets is
a solution to (2). ∎
2. (2)
###### Proof.
Theorem 3. Let $(a,b,c)$ be a solution to equation (3). We then have,
$a^{(b^{c})}=b^{ac}$ (7)
If $a=1$, then by (7), $1=b^{c}$, which in turn implies $b=1$; and $c$ an
arbitrary positive integer.
If $a\geq 2$ and $b=1$, (7) becomes impossible for any value of $c$. If $a\geq
2,\ b\geq 2$, and $c=1$, (7) yields $a^{b}=b^{a}$; and by Result 2 we must
have either $a=4$ and $b=2$, or $a=2$ and $b=4$; or $a=b$. If $a\geq 2,\ b\geq
2,\ c\geq 2$. Then by (7),
$a^{(b^{c})}=(b^{c})^{a}$ (7a)
Combining (7a) with Result 2 implies that either $a=4$ and $b^{c}=2$, which is
impossible since $b\geq 2$ and $c\geq 2$, or that $a=2$ and $b^{c}=4$, which
gives $a=2=b=c$. Or, the third possibility, $a=b^{c}$. We have shown that if
$(a,b,c)$ is a positive integer solution of equation (3), then it must belong
to one of the sets $S_{1},\ S_{2},\ S_{3},\ S_{4}$ or $S_{5}$. Conversely, a
routine calculation shows that any member of these five sets is a solution to
(3). ∎
3. (3)
###### Proof.
Theorem 4. Let $(a,b,c)$ be a positive integer solution to equation (4)
$a^{bc}=b^{ac}$ (8)
If $a=1$, we obtain $1=b^{c}$; and so $b=1$, with $c$ being an arbitrary
positive integer.
If $a\geq 2$ and $b=1$, (8) gives $a^{c}=1$, which is impossible since $a\geq
2$.
If $a\geq 2,\ b\geq 2$, and $c=1$, we obtain from (8)
$a^{b}=b^{a}$ (8a)
Equation (8a), combined with Result 2, implies that either $a=4$ and $b=2$; or
$a=2$ and $b=4$; or $a=b$. If $a\geq 2,\ b\geq 2$, and $c\geq 2$, we have from
(8)
$a^{bc}=b^{ac}\Leftrightarrow(a^{b})^{c}=(b^{a})^{c}$ (8b)
Equation (8b) demonstrates that the $c$th powers of the positive integers
$a^{b}$ and $b^{a}$ are equal. Since these two integers are greater than $1$,
equation (8b) implies
$a^{b}=b^{a}$
which once more, when combined with Result 2, implies either $a=4$ and $b=2$
or $a=2$ and $b=4$; or $a=b$.
We have shown that if $(a,b,c)$ is a positive integer solution of equation
(4), it must belong to one of the sets $S_{1},\ S_{2},\ S_{3},\ S_{4},\
S_{5},\ S_{6},$ or $S_{7}$. Conversely, a routine calculation establishes that
any member of these seven sets is a solution to (4). ∎
4. (5)
Proof of Theorem 5 The following lemma can be easily proved by using
mathematical induction. We omit the details. We will use the lemma in the
proof of Theorem 5.
###### Lemma 1.
1. (i)
If $b\geq 3$, then $b^{n-1}>n$ for all positive integers $n\geq 2$.
2. (ii))
$2^{n-1}>n$, for all positive integers $n\geq 3$.
3. (iii)
If $c\geq 2$, then $c^{n}>n$, for all positive integers $n$.
###### Proof.
Theorem 5
1. (i)
Let $(a,b,c)$ be a solution to equation (5) with at least one of $a,b,c$ being
equal to $1$.
If $a=1$, (5) implies $1=c^{b}$, and so $c=1$ as well; and $b$ is an arbitrary
positive integer.
If $b=1$ and $a\geq 2$ we get $a=c^{a}$ which is impossible if $c\geq 2$, by
Lemma 1(iii); and clearly, $c\neq 1$ since $a\geq 2$.
Also, the case $b\geq 2,\ a\geq 2$, and $c=1$ is ruled out by inspection. We
conclude that if $(a,b,c)$ is a solution to (5), with one of $a,b,c$ being
$1$, then it must be of the form $(1,b,1)$. Conversely, a straightforward
calculation established that $(1,b,1)$ is a solution of (5) for every positive
integer $b$.
2. (ii)
Let $(a,b,c)$ be a solution to (5) with $a\geq 2,\ b\geq 2,\ c\geq 2$, and
$a=c$. We have, by (5), $a^{(b^{a})}=a^{ab}\Leftrightarrow$ (since $a\geq 2$)
$b^{a}=ab$, or equivalently, $b^{a-1}=a$, which, when combined with Lemma 1,
parts (i) and (ii), implies that either $b\geq 3$ and $a=1$; which is ruled
out since $a\geq 2$; or alternatively, $b=2$ and $a\leq 2$ which gives $a=2$.
We obtain $a=b=c=2$. Conversely, $(2,2,2)$ is a solution to equation (5),
$2^{4}=2^{4}$.
3. (iii)
Let $(a,b,c)$ be a solution to (5) with $a\geq 2,\ b\geq 2,\ c\geq 2$, and
with $b=c\neq a$. We have,
$a^{(b^{b})}=b^{ab};$
or equivalently,
$a^{(b^{b})}=(b^{b})^{a}$ (9)
Equation (9) combined with Result 2 implies that, either $a=4$ and $b^{b}=2$
or $a=2$ and $b^{b}=4$; or $a=b^{b}$. The first possibility is ruled out since
$b^{b}\geq 2^{2}>2$, because $b\geq 2$ . The second possibility yields
$b^{b}=4,\ b=2$, but then also have $a=2$ and so $a=b=c=2$, contrary to
$b=c\neq a$. The third possibility establishes $(a,b,c)=(b^{b},b,b)$.
Conversely, $(b^{b},b,b)$ is a solution to (5) for any positive integer $b\geq
2$. Both sides of (5) are equal to $b^{(b^{b+1})}$.
∎
## References
* [1] Crux Mathematicorum with Mathematical Mayhem, 36, No.2, March, 2010. Mayhem problem, M429, p. 73. Proposed by Samuel Gómez Moreno.
* [2] Crux Mathematicorum with Mathematical Mayhem, 36, No. 8, December, 2010. Solution to mayhem problem M429, p. 492, by Konstantine Zelator.
* [3] W. Sierpinski, Elementary Theory of Numbers, Warsaw, 1964\. Printed by ProQuest, UMI Books on Demand.
ISBN: 0-598-52758-3, pp. 106-107.
|
arxiv-papers
| 2011-12-13T19:00:22 |
2024-09-04T02:49:25.412107
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Konstantine Zelator",
"submitter": "Konstantine Zelator",
"url": "https://arxiv.org/abs/1112.3862"
}
|
1112.3988
|
Information Flow in Interaction Networks
Aleksandar Stojmirović and Yi-Kuo Yu***to whom correspondence should be
addressed
National Center for Biotechnology Information
National Library of Medicine
National Institutes of Health
Bethesda, MD 20894
United States
Interaction networks, consisting of agents linked by their interactions, are
ubiquitous accross many disciplines of modern science. Many methods of
analysis of interaction networks have been proposed, mainly concentrating on
node degree distribution or aiming to discover clusters of agents that are
very strongly connected between themselves. These methods are principally
based on graph-theory or machine learning.
We present a mathematically simple formalism for modelling context-specific
information propagation in interaction networks based on random walks. The
context is provided by selection of sources and destinations of information
and by use of potential functions that direct the flow towards the
destinations. We also use the concept of dissipation to model the aging of
information as it diffuses from its source.
Using examples from yeast protein-protein interaction networks and some of the
histone acetyltransferases involved in control of transcription, we
demonstrate the utility of the concepts and the mathematical constructs
introduced in this paper.
## 1 Introduction
Interaction networks are abundant and have recently gained significant
publicity in many diverse modern disciplines such as electronics (Cancho _et
al._ , 2001), sociology (Wasserman and Faust, 1994; Newman, 2004) and
epidemiology (Barthelemy _et al._ , 2005). In its simplest form, an
interaction network consists of a collection of entities (or agents), where
two agents are linked if they interact in some way. For example, in an
acquaintance network (Amaral _et al._ , 2000), the agents represent persons
and two persons are linked together if they know each other while the Woldwide
Web network consists of web pages with links between pages (Broder _et al._ ,
2000). Mathematically, networks correspond exactly to graphs (or multigraphs),
with agents as vertices and links as edges, which can be weighted and/or
directed depending on the exact application being modeled. The key to analysis
of interaction networks is the assumption of information transitivity:
information can flow through or can be exchanged via paths of interactions.
Biology in post-genomic era also contains numerous examples of molecular
networks (Galitski, 2004). Metabolic networks have been modeled by
representing metabolites as nodes and chemical reactions as links: two
metabolites are linked if they participate in the same reaction (Ma and Zeng,
2003). Genetic networks have genes as nodes with two genes being linked if
they interact through directed transcriptional regulation (Guelzim _et al._ ,
2002). Protein-protein interaction networks have proteins as nodes, with the
links representing physical interactions (binding) between proteins
(Pellegrini _et al._ , 2004). Large scale high-throughput studies in model
organisms such as Saccharomyces cerevisiae (baker’s yeast) (Ito _et al._ ,
2001; Uetz _et al._ , 2000), Drosophilla melanogaster (fruit-fly) (Giot _et
al._ , 2003), Caenorhabditis elegans (roundworm) (Li _et al._ , 2004) and
humans (Stelzl _et al._ , 2005; Rual _et al._ , 2005), provided extensive
datasets of protein-protein interactions, stored in publicly-available
databases such as the Database of Interacting Proteins (DIP) (Xenarios _et
al._ , 2002; Salwinski _et al._ , 2004). Unfortunately, there is very little
consistency between the protein-protein interaction data coming from different
high-throughput experiments (Sprinzak _et al._ , 2003) and significant effort
has been expended in devising ways to discover false positives and false
negatives (Suthram _et al._ , 2006). This problem is not restricted to
protein-protein interactions: microarray data also contains non-negligible
inconsistencies (Miklos and Maleszka, 2004) .
Numerous approaches have been proposed for analysis of biological and, in
particular, protein-protein interaction networks (Aittokallio and Schwikowski,
2006). However, due to space restrictions, we will refer to just a few. Most
algorithms aim to discover ‘functional modules’ (Hartwell _et al._ , 1999),
representing well connected clusters of nodes with the same or similar
function, by using clustering techniques from graph theory and/or machine
learning (Steffen _et al._ , 2002; Spirin and Mirny, 2003; Rives and Galitski,
2003; Pereira-Leal _et al._ , 2004; Nabieva _et al._ , 2005; Xiong _et al._ ,
2005; Chua _et al._ , 2006; Chen and Yuan, 2006; Hwang _et al._ , 2006). Very
frequently, these techniques make use of additional experimental data which is
not present in the network structure itself. For example, methods for
discovery of complexes from protein-protein interaction networks often refer
to the data from dataset from different species (Kelley _et al._ , 2003;
Sharan _et al._ , 2005a, b), microarray expression studies (Steffen _et al._ ,
2002; Chen and Yuan, 2006), or human-curated functional classifications
(Nabieva _et al._ , 2005; Chua _et al._ , 2006).
Our approach to analyzing interaction networks is very different, relying
solely on the network structure. We model diffusion of information through the
network by discrete-time random walks moving from the nodes representing the
sources of information to their destinations. The choice of sources and
destinations provides the _context of analysis_ with the nodes most affected
by information flow being called _Information Transduction Modules_. We use
two modes of diffusion, dual to each other, which we call absorbing and
emitting, with our absorbing mode directly corresponding to deeply
investigated absorbing Markov chains (Kemeny and Snell, 1976). Random walks
and corresponding Markov chains are one of the subjects of spectral graph
theory (Chung, 1997) but we do not use eigenspace decomposition in our work,
instead relying on a basic matrix algebra approach similar to that of Kemeny
and Snell (1976).
The algorithm Functional Flow by Nabieva _et al._ (2005), also modeling
diffusion of information from sources, is closest to our emitting model.
However, to delineate a certain biological context, we additionally direct the
flow from sources to selected destinations using potential functions and allow
the information content to dissipate (evaporate) from the network at each time
step, thus modeling natural ‘aging’ of information.
Our models allow investigation of several types of biological questions from
protein-protein interaction networks. Many proteins perform their function in
cooperation with other proteins through, often large, protein complexes. Thus,
to elucidate the function of a given protein, it is useful to know the most
likely members of complexes it may belong to and their relations to each
other. Additionally, if two proteins are known to have similar function, what,
if any, are the proteins they share in their respective complexes? To help
answer such questions, we employ our absorbing diffusion mode.
The answers to the above questions can provide the general interaction
environment of one or more proteins. It is also very instructive to identify
specific modules mediating interactions between distant (in network terms)
proteins. Our emitting diffusion mode can be used to find possible candidates
for members of such modules. Furthermore, analysis of interaction modules
obtained from considering different proteins in the same biological context
may lead to discovery of fundamental units of information transduction. To
achieve this we developed the concept of information interference. More
concrete definitions will be presented in the body of the text.
This paper is organized as follows. Section 2 outlines the theory behind our
models of information diffusion in networks. For better readability, all the
theorems and proofs, using mainly the basic concepts and results from the
matrix algebra are given in Appendix (the reader may wish to consult the
standard linear algebra textbooks such as (Hoffman and Kunze, 1971) or (Bapat
and Raghavan, 1997) for background). Section 3 introduces the methods of
analysis of results obtained using the concepts of Section 2, while Section 4
presents concrete examples centered around yeast histone acetyltransferases.
We finish with discussion and conclusion in Section 5.
## 2 Theory
### 2.1 Preliminaries
We represent an interaction network as a weighted directed graph
$\Gamma=(V,E,w)$ where $V$ is a finite set of vertices of size $n$,
$E\subseteq V\times V$ is a set of edges and $w$ is a non-negative real-valued
function on $V\times V$ that is positive on $E$, giving the weight of each
edge (the weight of non-existing edge is defined to be $0$). Assuming an
ordering of vertices in $V$, we represent a real-valued function on $V$ as a
state (column) vector $\mathbf{\boldsymbol{\varphi}}\in\mathbb{R}^{n}$ and the
connectivity of $\Gamma$ by the _weight_ matrix $\mathbf{W}$ where
$W_{ij}=w(i,j)$ (the weight of an edge from $i$ to $j$). If $\Gamma$ is an
unweighted undirected graph, $\mathbf{W}$ is the adjacency matrix of $\Gamma$
where
$W_{ij}=\begin{cases}2&\text{if $i=j$ and $(i,i)\in E$},\\\ 1&\text{if $i\neq
j$ and $(i,j)\in E$},\\\ 0&\text{if $(i,j)\not\in E$.}\end{cases}$ (1)
Throughout this paper, we will not make distinction between a vertex $v\in V$
and its corresponding state given by a particular ordering of vertices.
Let $\mathbf{P}$ denote the $n\times n$ _transition_ matrix of $\Gamma$ where
$P_{ij}=\frac{W_{ij}}{\sum_{k}W_{ik}},$ (2)
that is, $\mathbf{P}$ is the weight matrix of $\Gamma$ normalized by row. The
matrix $\mathbf{P}$ can be used to model random walks on $\Gamma$: for any
pair of vertices $i$ and $j$, $P_{ij}$ gives the probability of the random
walk moving from vertex $i$ to vertex $j$ in one time step, which is
proportional to the weight $W_{ij}$. Since the matrix $\mathbf{P}$ is
stochastic (all rows sum to unity), it can also be interpreted as the
transition matrix for Markov chain on the set $V$. In the following sections
we will model information diffusion as a random walk on $\Gamma$ with
particular starting and terminating points.
### 2.2 Constrained diffusion
In this section we select certain vertices as sources or sinks of information
and solve for the number of times a vertex is visited. Let $S$ denote the set
of selected vertices, let $T=V\setminus S$ and let $m=\left|T\right|$.
Assuming that the first $n-m$ states correspond to vertices in $S$, we write
the matrix $\mathbf{P}$ in the canonical form:
$\mathbf{P}=\left[\begin{array}[]{cc}\mathbf{P}_{SS}&\mathbf{P}_{ST}\\\
\mathbf{P}_{TS}&\mathbf{P}_{TT}\end{array}\right].$ (3)
Here $\mathbf{P}_{AB}$ denotes a matrix giving probabilities of moving from
$A$ to $B$ where $A,B$ stand for either $S$ or $T$. The states (vertices)
belonging to the set $T$ are called _transient_.
#### 2.2.1 Absorption in sinks
Suppose now that the set $S$ represents the set of _sinks_ of information: any
information reaching a sink vertex is absorbed and cannot not leave it. Let
$\mathbf{F}(t)$ denote an $m\times(n-m)$ matrix such that $F_{ij}(t)$ is the
probability that the information originating at $i\in T$ is absorbed at $j\in
S$ in $t$ or fewer steps. Since information can only be absorbed once in any
state $s\in S$, it follows that the information reaching $j$ avoided all other
sinks. For the same reason, $F_{ij}(t)$ can be interpreted as the expected
number of visits to the state $j$ of a random walk starting at $i$ for all
times up to $t$.
Absorption at $j$ after not more than $t$ steps can be achieved in two ways:
either the content reached vertex $j$ in the first step, with probability
$P_{ij}$ or it moved to some transient vertex $k$ in the first step and was
absorbed by $j$ from there in at most $t-1$ steps, with probability
$P_{ik}F_{kj}(t-1)$. Therefore, we have for all $t=1,2,\ldots$,
$F_{ij}(t+1)=P_{ij}+\sum_{k\in T}P_{ik}F_{kj}(t),$ (4)
or in the matrix form
$\mathbf{F}(t+1)=\mathbf{P}_{TS}+\mathbf{P}_{TT}\mathbf{F}(t).$ (5)
We solve for the long-term or equilibrium state, where
$\mathbf{F}(t+1)=\mathbf{F}(t)=\mathbf{F}$. In this case, Equation (5) becomes
$\mathbf{F}=\mathbf{P}_{TS}+\mathbf{P}_{TT}\mathbf{F},$ (6)
or
$(\mathbb{I}-\mathbf{P}_{TT})\mathbf{F}=\mathbf{P}_{TS},$ (7)
where $\mathbb{I}$ denotes the identity matrix. If
$\mathbb{I}-\mathbf{P}_{TT}$ is invertible, let
$\mathbf{G}=(\mathbb{I}-\mathbf{P}_{TT})^{-1}$. Equation (7) then has a unique
solution
$\mathbf{F}=\mathbf{G}\mathbf{P}_{TS}.$ (8)
#### 2.2.2 Diffusion from sources
Now consider the dual problem where $S$ is a set of sources of information.
Each source emits a unit of information at each time step and no information
can enter any source: we assume any information entering a source vanishes.
Let $\mathbf{H}(t)$ denote an $(n-m)\times m$ matrix such that $H_{ij}(t)$ is
the total expected number of times the transient vertex $j$ is visited by a
random walk emitted from source $i$ for the time up to $t$.
The information emitted from $i$ can arrive at $j$ at time $t$ in two
different ways: either the content was emitted from $i$ at time $t$ and
reached $j$ directly, or it was emitted at an earlier time step, was located
at some transient vertex at time $t-1$ and moved from there to $j$ at time
$t$. The former option contributes $P_{ij}$ while the latter contributes
$H_{ik}(t-1)P_{kj}$ for all $k\in T$ towards $H_{ij}$. Therefore, we have for
all $t=1,2,\ldots$,
$H_{ij}(t+1)=P_{ij}+\sum_{k\in T}H_{ik}(t)P_{kj},$ (9)
or in the matrix form
$\mathbf{H}(t+1)=\mathbf{P}_{ST}+\mathbf{H}(t)\mathbf{P}_{TT}.$ (10)
Similarly to the previous case, we are interested in the steady state,
representing the total expected number of visits, where
$\mathbf{H}(t+1)=\mathbf{H}(t)=\mathbf{H}$. In this case, Equation (10)
becomes
$\mathbf{H}=\mathbf{P}_{ST}+\mathbf{H}\mathbf{P}_{TT},$ (11)
or
$\mathbf{H}(\mathbb{I}-\mathbf{P}_{TT})=\mathbf{P}_{ST}.$ (12)
If $\mathbb{I}-\mathbf{P}_{TT}$ is invertible, Equation (12) has a unique
solution
$\mathbf{H}=\mathbf{P}_{ST}\mathbf{G}.$ (13)
#### 2.2.3 Existence and interpretation of solutions
It can immediately be observed that existence of solutions to Equation (12)
and Equation (7) are equivalent: they both depend on the existence of the
inverse of $\mathbb{I}-\mathbf{P}_{TT}$. Specifically, they are special cases
of the discrete Laplace equation on $T$ with the Dirichlet boundary condition
on $S$ (Chung, 1997; Chung and Yau, 2000).
Given a square matrix $\mathbf{M}$, the matrix $\mathbb{I}-\mathbf{M}$ is
often called the _discrete Laplace operator_ of $\mathbf{M}$. Let
$\Delta=\mathbb{I}-\mathbf{P}_{TT}$ ($\Delta$ is the discrete Laplace operator
of $\mathbf{P}$ restricted to $T$). Equation (7) can then be written as
$\Delta\mathbf{F}=\mathbf{P}_{TS}.$ (14)
Denote by $\mathbf{e}_{k}$ the $k$-th standard basis (column) vector of length
$n-m$ where $(\mathbf{e}_{k})_{j}=\delta_{kj}$ ($\delta$ here is the
Kronecker’s delta). Let $\mathbf{f}_{k}=\mathbf{F}\mathbf{e}_{k}$ denote the
$k$-th column of $\mathbf{F}$ and let
$\mathbf{p}_{k}=\mathbf{P}_{TS}\mathbf{e}_{k}$. Then, solving Equation (14) is
equivalent to solving the discrete Laplace equation
$\Delta\mathbf{f}_{k}=\mathbf{p}_{k}$ (15)
for all $k\in S$. The standard basis vectors $\mathbf{e}_{k}$ provide exactly
the _Dirichlet boundary conditions_ on the set $S$ (the set $S$ can be assumed
to be a boundary of $T$).
It is also easy to see that Equation (12) can be written as
$\mathbf{H}\Delta=\mathbf{P}_{ST}.$ (16)
Hence, the solution to (16) is obtained by solving the discrete Laplace
equation in terms of the discrete Laplace operator of the transpose of
$\mathbf{P}$.
The _Green’s function_ is defined to be the inverse of the Laplacian. In our
case the inverse of $\Delta$ is exactly the matrix
$\mathbf{G}=(\mathbb{I}-\mathbf{P}_{TT})^{-1}$ and hence the existence of
solutions to Equations (12) and (7) is equivalent to existence of the Green’s
functions to the corresponding Laplacian. In the absorbing Markov chain theory
(Kemeny and Snell, 1976), the matrix $\mathbf{G}$ is known as the _Fundamental
matrix_ of the corresponding absorbing Markov chain. The entry $G_{ij}$
represents the mean number of times the random walk reaches vertex $j\in T$
having started in state $i\in T$.
We now present some elementary sufficient conditions for existence of the
Green’s functions of the discrete Laplacians of the graphs. The full proofs
are given in Appendix A. For the development of the discrete Green’s functions
(for undirected graphs) in terms of the eigenvalues and eigenfunctions of the
Laplacian, we refer the reader to the paper by Chung and Yau (2000).
###### Proposition 2.1.
Suppose that $\Gamma$ is a weighted directed graph such that for every $p\in
T$ there exists $s\in S$ such that there exists a directed path from $p$ to
$s$. Then, the matrix $\mathbb{I}-\mathbf{P}_{TT}$ is invertible and
$(\mathbb{I}-\mathbf{P}_{TT})^{-1}=\sum_{k=0}^{\infty}(\mathbf{P}_{TT})^{k}.$
(17)
Proposition 2.1 thus guarantees existence of the Green’s functions if every
transient vertex can be connected to a source or sink via a directed path. If
the underlying graph is undirected, this condition can be rephrased as
follows: every connected component of $V$ contains at least one vertex from
$S$.
In the context of information diffusion, the connectivity condition implies
that all information entering the transient set at any specific time must
eventually leave it, either by absorption into $S$ when $S$ is a set sinks, or
by dissipation when $S$ represents the set of sources. We will further discuss
the concept of dissipation in 2.3.
Assuming the Green’s function exists, the entries of the matrices $\mathbf{F}$
and $\mathbf{H}$ can be interpreted in several different ways. Fundamentally,
both $F_{ij}$ and $H_{ij}$ represent the total expected number of times the
vertex $j$ is visited by the information originating at the vertex $i$ while
avoiding all members of the boundary set $S$ (the proofs are given in Appendix
B.1). It is also clear, by Equation (17), that $\mathbf{F}$ and $\mathbf{H}$
are both non-negative matrices and that
$\mathbf{F}=\lim_{t\to\infty}\mathbf{F}(t)$ and
$\mathbf{H}=\lim_{t\to\infty}\mathbf{H}(t)$. In addition, the rows of
$\mathbf{F}$ all sum to $1$ (Lemma B.3 in Appendix B.2) and thus $F_{ij}$ is
the overall probability an information originating from transient vertex $i$
is absorbed at the sink $j$ while avoiding all other sinks.
If we assume that a random walk deposits a fixed amount of information content
each time it visits a node, we can interpret $H_{ij}$ is the overall amount of
information content originating from the source $i$ deposited at the transient
vertex $j$. If $\Gamma$ is an undirected graph with symmetric weight matrix
$\mathbf{W}$ and $S$ contains a single source, the value of $H_{ij}$ is
directly proportional to the degree of the transient vertex $j$ (Appendix
B.2). Hence, in this case, the total average number of times of visits for
each transient node is proportional to its degree. This is no longer true if
$\mathbf{W}$ is not symmetric.
Furthermore, we can interpret $F_{ij}$ as the sum of probabilities of paths
originating at the vertex $i\in T$ and terminating at the vertex $j\in S$ that
avoid all other nodes in the set $S$, and $H_{ij}$ as the sum of probabilities
of paths originating at the vertex $i\in S$ and terminating at the vertex
$j\in T$, also avoiding all other nodes in the set $S$. Each such path has a
finite but unbounded length. However, unlike $F_{ij}$, $H_{ij}$ does not
represent a probability because the events of the information being located at
$j$ at the times $t$ and $t^{\prime}$ are not mutually exclusive (a random
walk can be at $j$ at time $t$ and revisit it at time $t^{\prime}$). For
$F_{ij}$, the absorbing events at different times are mutually exclusive.
### 2.3 Information dissipation
It was mentioned previously that the requirement that every transient node is
connected to a node in the set $S$ is effectively equivalent to the property
that all information content entering the transient set leaves it at the nodes
in $S$. In the present section we extend our model to allow the information to
dissipate not only at those nodes but also at the transient nodes.
Let $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ be vectors of length $n$
such that for all $i\in V$, $\alpha_{i}>0$ and $\beta_{i}>0$. We form the
matrix $\mathbf{\tilde{P}}$ with entries
$\tilde{P}_{ij}=\alpha_{i}\beta_{j}P_{ij},$ (18)
and use the new matrix to compute the matrices $\mathbf{\tilde{F}}$ and
$\mathbf{\tilde{H}}$ by replacing the matrix $\mathbf{P}$ in the previous
section with $\tilde{P}$ so that.
$\mathbf{\tilde{F}}=\mathbf{\tilde{G}}\mathbf{\tilde{P}}_{TS}.$ (19)
and
$\mathbf{\tilde{H}}=\mathbf{\tilde{P}}_{ST}\mathbf{\tilde{G}}.$ (20)
where $\mathbf{\tilde{G}}=(\mathbb{I}-\mathbf{\tilde{P}}_{TT})^{-1}$, provided
$\mathbb{I}-\mathbf{\tilde{P}}_{TT}$ is invertible.
The entry $\alpha_{i}$ gives the proportion of the signal leaving the vertex
$i$ that is retained (we call the value of $1-\alpha_{i}$ the _outgoing
dissipation coefficient_ of the node $i$) while the entry $\beta_{j}$ gives
the proportion of the signal entering the vertex $j$ that is retained (the
value $1-\beta_{j}$ is called the _incoming dissipation coefficient_ of the
node $j$). The case where $\alpha_{i}=\beta_{i}=1$ for all $i\in V$ gives back
the original matrix $\mathbf{P}$. Note that our definition allows entries of
$\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ that are greater than $1$,
corresponding to negative dissipation coefficients. Such coefficients lead to
amplification of the signal. However, in order for the Green’s function
$\mathbf{\tilde{G}}$ to exist, any amplification should be balanced by
dissipation.
We now establish a sufficient condition for existence of $\mathbf{\tilde{G}}$.
The proof, as well as a discussion of its generalization, is given in Appendix
A.1.
###### Proposition 2.2.
Let $\alpha_{*}=\max\\{\alpha_{i}:i\in V\\}$ and
$\beta_{*}=\max\\{\beta_{i}:i\in V\\}$ and suppose $\alpha_{*}\beta_{*}<1$.
Then, the matrix $\mathbb{I}-\mathbf{\tilde{P}}_{TT}$ is invertible and
$(\mathbb{I}-\mathbf{\tilde{P}}_{TT})^{-1}=\sum_{k=0}^{\infty}(\mathbf{\tilde{P}}_{TT})^{k}.$
(21)
Proposition 2.2 makes no assumptions on the connectivity of the graph: the
equilibrium solutions exist regardless of the graph topology. The reason for
the removal of the connectivity conditions is that a unit of information
originating anywhere in the network has a nonzero probability of being
dissipated at each time step and therefore will disappear in the long term,
with a portion possibly reaching a sink in the absorbing model. The vectors of
coefficients $\alpha$ and $\beta$ provide us with the ability to consider
different rates of dissipation at different vertices. We demonstrate the
utility of the extended model in examples involving protein-protein
interaction networks (Section 4), where we use vertex specific dissipation to
construct ‘evaporating nodes’ that dissipate most of the information coming in
but allow unrestricted outward flow.
A possible further generalization of this model is for the entries of the
vectors $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ to be functions of the
state variable of the dynamical system instead of constants. The dynamical
system in this case would become non-linear, allowing us to model
amplification or dissipation of the information depending on the time specific
state of the system.
### 2.4 Potentials
Our models so far, including the dissipation modifications described above,
model ‘free diffusion’ of information through the network: the likelihood for
the signal to move from vertex $i$ to vertex $j$ is proportional to the
relative weight of the edge $(i,j)$ among all edges emanating from $i$
(dissipation only affects the total amount transmitted). In order to direct
the flow of information towards or away from selected nodes, we adjust the
weights of edges of our network graph $\Gamma$ using _potentials_ , real-
valued monotone functions defined on the nodes that depend on the distances
from selected points.
Let $\rho$ denote the path-metric on the weighted directed connected graph
$\Gamma=(V,E,w)$, where for all $i,j\in V$, $\rho(i,j)$ denotes the sum of the
reciprocals of the weights of the edges forming the shortest directed path
from $i$ to $j$. Suppose $R$ is a subset of $T$ such that for each $k\in R$
there exists a monotone potential function
$\theta_{k}:\mathbb{R}\to\mathbb{R}$. For each vertex $j\in V$ define the
_total potential_ at $j$, denoted $\Theta(j)$ by
$\Theta(j)=\sum_{k\in R}\theta_{k}(\rho(j,k)).$ (22)
Let $\hat{\Gamma}$ denote the new weighted directed graph $(V,E,\hat{w})$
where
$\hat{W}_{ij}=W_{ij}\exp\left(-\Theta(j)\right).$ (23)
The form of Equation (23) ensures that the signal preferentially diffuses from
each vertex towards the vertices adjacent to it that have lower potential
relative to other adjacent vertices.
A vertex $i\in V$ is called a _destination_ if $\Theta$ has a minimum at $i$.
There can be multiple destinations in a network. The natural candidates for
destinations are the members of the set $S$ since all information entering
them does not leave them. Some transient states, with the weights of their
outgoing edges adjusted to partially accumulate the signal, are also good
candidates for destinations.
Let $K$ be a subset of $T$ and let $0\leq\gamma\leq 1$. From the already
modified graph $\hat{\Gamma}$, we form the graph $\Gamma^{\prime}$ represented
by the weight matrix $\mathbf{W}^{\prime}$ where
$W^{\prime}_{ij}=\begin{cases}\hat{W}_{ij}&\text{if $i\not\in K$,}\\\
\gamma\hat{W}_{ij}&\text{if $i\in K$ and $i\neq j$,}\\\
\hat{W}_{ij}+(1-\gamma)\sum_{k\neq i}\hat{W}_{ik}&\text{if $j\in K$ and
$i=j$}.\end{cases}$ (24)
The effect of this modification is to turn each vertex $i\in K$, called a
_pseudosink_ , into a partial sink: some proportion of the weights of edges
emanating out of $i$ is transferred to the edge pointing back to $i$. The
parameter $\gamma$, representing the proportion of information allowed to
leave each pseudosink while the remainder is accumulated, is called the
_pseudosink leakage coefficient_. The value $\gamma=1$ implies no change in
edge weights.
The value $\gamma=0$ is a special case because no directed path exists between
pseudosinks and source nodes in the resulting graph $\Gamma^{\prime}$ and
Proposition 2.1 does not apply. In this case, there are two possibilities
leading to the existence of the Green’s function: either set the outgoing
dissipation coefficient of the pseudosinks to something less than $1$, or
treat the pseudosinks as parts of the boundary set $S$, as a ‘non-emitting
source’ defined in 3.2 below.
Note that, while dissipation is applied to the transition matrix $\mathbf{P}$,
potentials and pseudosinks are applied to the weight matrix $\mathbf{W}$ prior
to normalization. Since applications of potentials and pseudosinks do not
commute, potentials are applied before pseudosinks, although pseudosinks can
be potential centers (members of the set $R$).
## 3 Theoretical Methods for Analysis
In the previous section we introduced the basic concepts related to our models
of diffusion of information through networks as well as some modifications to
the underlying graph and the transition matrix that lead to biologically
realistic models. After all modifications are applied, we obtain the matrices
$\mathbf{\tilde{F}}$ and $\mathbf{\tilde{H}}$, the Green’s functions arising
where $S$ represents sinks and sources, respectively. Here we turn to the
practical interpretation of these results, which depend on the boundary
conditions imposed on the vertices in $S$.
### 3.1 Absorbing model
In the case where $S$ represents sinks of information (the _absorbing model_),
the entries of the matrix $\mathbf{\tilde{F}}$ have a clear probabilistic
interpretation: $\tilde{F}_{ij}$ is the probability that information starting
at transient vertex $i$ reaches the sink $j$ while avoiding all other sinks,
taking into account the dissipation as well as the new weights induced by the
potentials. Generally, each sink $j$ exerts a ‘region of influence’, including
the transient points with large $\tilde{F}_{ij}$. Depending on the
distributions of sinks within the network, some transient node may have a
$\tilde{F}_{ij}$ small for all $j$: information emerging from these points is
more likely to dissipate than to reach any of the sinks.
If $S^{\prime}\subset S$ is a selection of sink nodes, then $\sum_{j\in
S^{\prime}}F_{ij}$ gives the total probability of information reaching the set
$S^{\prime}$ from the vertex $i$, avoiding all other nodes in $S$. In this
context, we call the nodes in $S^{\prime}$ _explicit sinks_ (since we
investigate the probabilities of reaching them) and the remaining nodes in $S$
_implicit sinks_ , the points that serve as sinks of information but are not
considered. Furthermore, if the sinks are treated as general boundary points,
with boundary values not restricted to $0$ and $1$, the entries of
$\mathbf{\tilde{F}}$ can be interpreted as temperatures (Zhang _et al._ ,
2007).
### 3.2 Emitting model
Where $S$ represents sources (the _emitting model_), the entries of
$\mathbf{\tilde{H}}$ can be interpreted as visiting times or as information
contents: $\tilde{H}_{ij}$ is the total information content emitted from the
source $i$ deposited at the transient vertex $j$. Information is dissipated at
all sources and the value of $\tilde{H}_{ij}$ is dependent on transient
dissipation coefficients $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ and
the potentials. For biological applications, we will consider the case where
at least one pseudosink is present in addition to one or several sources, with
the potential directing the flow towards the pseudosinks. The distribution of
entries of the $i$-th row of $\mathbf{\tilde{H}}$ will then describe the
_information transduction module_ (ITM) involved in transfer of information
from $i$ to the pseudosinks, with the nodes with largest entries being most
significant.
Let $\boldsymbol{\xi}$ denote the vector of length $\left|S\right|$ such that
for all $i\in S$, $\xi_{i}\geq 0$. We call $\xi_{i}$ the _source strength_ of
the source $i$, representing the amount of information emitted from $i$ at
each time step. In this context, we call $i\in S$ an _emitting source_ if
$\xi_{i}>0$ and a _non-emitting source_ if $\xi_{i}=0$. Non-emitting sources
are essentially information ‘black holes’, dissipating any information coming
in and not emitting any.
#### 3.2.1 Total content
For any $i\in S$, let $\boldsymbol{\epsilon}_{i}$ denote the standard $i$-th
row basis vector of length $n-m$, where
$(\boldsymbol{\epsilon}_{i})_{j}=\delta_{ij}$. For $x>0$ define the vector
$\boldsymbol{\phi}_{i}$ by
$\boldsymbol{\phi}_{i}=\xi_{i}\boldsymbol{\epsilon}\mathbf{\tilde{H}},$ (25)
that is, $\boldsymbol{\phi}_{i}$ denotes the $i$-th row of
$\mathbf{\tilde{H}}$ multiplied by $\xi_{i}$. Its entries give the amount of
information content originating from the source $i$ of strength $\xi_{i}$
deposited at transient vertices. The value of
$\left\|\boldsymbol{\phi}_{i}\right\|_{1}$ is then the total amount of content
originating at source $i$ deposited at the transient states. In our examples
in the following sections we choose the source strengths $\boldsymbol{\xi}$ so
that $\left\|\boldsymbol{\phi}_{i}\right\|_{1}$ is the same for all $i\in S$
(we call the resulting vectors $\phi_{i}$ normalized content vectors). The
_joint information content_ vector, denoted $\boldsymbol{\tau}$, is defined by
$\boldsymbol{\tau}=\sum_{i\in S}\boldsymbol{\phi}_{i}.$ (26)
The vector $\boldsymbol{\tau}$ implicitly depends on the matrix
$\mathbf{\tilde{H}}$ and the source strength vector $\boldsymbol{\xi}$: we
have $\boldsymbol{\tau}=\boldsymbol{\xi}\mathbf{\tilde{H}}$.
#### 3.2.2 Participation ratio
Let $\mathbf{x}\in\mathbb{R}^{n}$ be any vector and recall that for any $0\leq
p<\infty$, the $\ell_{p}$-norm of $\mathbf{x}$, denoted
$\left\|\mathbf{x}\right\|_{p}$, is given by
$\left\|\mathbf{x}\right\|_{p}=\left(\sum_{k}\left|x_{k}\right|^{p}\right)^{1/p}$.
Define the _participation ratio_ of $\mathbf{x}$, denoted $\pi(\mathbf{x})$ by
$\pi(\mathbf{x})=\frac{\left\|\mathbf{x}\right\|_{1}^{2}}{\left\|\mathbf{x}\right\|_{2}^{2}}=\frac{\left(\sum_{k}\left|x_{k}\right|\right)^{2}}{\sum_{k}x_{k}^{2}}.$
(27)
Participation ratio is well known under a slightly different definition in the
physics literature (Thouless, 1974). It gives the number of components of
$\mathbf{x}$ whose magnitude is ‘significant’. Clearly, $\pi$ is independent
of the scale of $\mathbf{x}$: we have for any $\lambda>0$,
$\pi(\lambda\mathbf{x})=\pi(\mathbf{x})$. We illustrate the usage by examples.
###### Example 3.1.
Let $\mathbf{x}=[1,1,1,1,1]$. Then, $\pi(\mathbf{x})=\frac{5^{2}}{5}=5$. All
components are equally significant and this is reflected in the participation
ratio.
###### Example 3.2.
Now consider $\mathbf{x}=[1,1,0,0,0]$. We have,
$\pi(\mathbf{x})=\frac{2^{2}}{2}=2$. Only the first two components are non-
zero and are of equal magnitude.
###### Example 3.3.
Finally, let
$\mathbf{x}=\left[1,\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16}\right]$.
We obtain $\pi(\mathbf{x})\approx 2.8181$. Here all five components are non-
zero but their magnitudes differ significantly. The participation ratio here
implies that the first two components and to a large extent the third are
significant while the remaining two are much smaller.
In our biological examples, we use $\pi(\boldsymbol{\tau})$ to choose the
number of the transient vertices with largest total mass to display as a
‘significant’ subgraph, together with all sources and pseudosinks.
#### 3.2.3 Interference
Given the vector of source strengths $\boldsymbol{\xi}$, the entry of
$\tau_{j}$ can be interpreted as providing the total amount of information
deposited at the vertex $j$. It is also possible to investigate the
interaction of the signals from different sources using the concept of
destructive interference.
For any vector $\mathbf{x}\in\mathbb{R}^{n}$, let $\mu$ denote an
_interference function_ such that
$0\leq\mu(\mathbf{x})\leq\left\|\mathbf{x}\right\|_{1}$. When applied to a
vector containing information content from different sources, interference
function is interpreted as removing some of the information present due to the
interaction of the various information types and returning the remaining
information content. Interference functions can take various forms depending
on the nature of the types of information in each application.
###### Example 3.4.
Suppose $\mathbf{x}$ consists of two components representing information types
that are assumed to completely cancel out each other. In this case, the
interference function takes the form
$\mu(\mathbf{x})=\left|x_{1}-x_{2}\right|$.
###### Example 3.5.
When $\mathbf{x}$ has more than two components, there are may possible ways to
generalize the above example. We distinguish two general modes of
interference: exclusive and partial. Exclusive interference mode represents
the case where simultaneous presence of all types of information is necessary
for destructive interference. For example, if each information type carries
the same weight, the interference function is:
$\mu(\mathbf{x})=\sum_{k}\left(x_{k}-\nu\right),$ (28)
where $\displaystyle\nu=\min_{k}x_{k}$.
###### Example 3.6.
We call the partial interference the case where presence of all types of
information is not necessary. It can be modeled in many ways depending on the
desired interpretation. For example, if there are three sources, we can use
complex numbers to set $\mu$ so that
$\mu(\mathbf{x})=\left|\sum_{k=1}^{3}x_{k}\exp\left(\frac{\iota
k\pi}{3}\right)\right|,$ (29)
where $\iota$ denotes the imaginary unit. In this case, some content is lost
when any two types of signal are present but all three must be present for
complete annihilation.
Given the interference function $\mu$, define the _interference strength
function_ $\psi:\mathbb{R}^{n}\to\mathbb{R}\cup\\{\infty\\}$ by
$\psi(\mathbf{x})=\begin{cases}\left\|\mathbf{x}\right\|_{1}\log\left(\frac{\left\|\mathbf{x}\right\|_{1}}{\mu(\mathbf{x})}\right)&\text{if
$\left\|\mathbf{x}\right\|_{1}>0$,}\\\ 0&\text{if
$\left\|\mathbf{x}\right\|_{1}=0$.}\end{cases}$ (30)
By the definition of $\mu$
Since $0\leq\mu(\mathbf{x})\leq\left\|\mathbf{x}\right\|_{1}$, it follows that
$\psi$ takes non-negative values (including $+\infty$). The value of $\psi$ is
infinite if $\mu(\mathbf{x})=0$ (perfect interference) and finite otherwise.
For an $m\times n$ matrix $\mathbf{X}$ define the vector
$\boldsymbol{\sigma}(\mathbf{X})$ of length $n$ having the components
$\sigma_{i}(\mathbf{X})=\psi(\mathbf{X}\mathbf{e}_{i})$ (31)
(recall that $\mathbf{e}_{i}$ is the standard column basis vector and hence
$\mathbf{X}\mathbf{e}_{i}$ represents the $i$-th column of $\mathbf{X}$). We
will call $\boldsymbol{\sigma}$ the _interference strength vector_.
For our applications, the entries of the matrix $\mathbf{X}$ above are
interpreted as information contents over some graph: $X_{ij}$ is the the
content of type $i$ at the vertex $j$. For each node $j$, the $\ell_{1}$-norm
in Equation (30) can be interpreted in this context as the total information
content at $j$ and the value of $\mu$ applied to the $j$-th column of
$\mathbf{X}$ as the information content remaining after interference. Hence,
interference strength of each node measures how much information content was
lost by interference, adjusted by the node’s joint information content.
The matrix $\mathbf{\tilde{H}}$ is therefore a natural input to $\psi$ and
$\boldsymbol{\sigma}$, however other derived matrices can be used such as
$\mathbf{\tilde{H}}$ adjusted for source strength by multiplying each row by
its corresponding source strength $\xi_{i}$. Furthermore, rows of $\mathbf{X}$
can come from different $\mathbf{\tilde{H}}$ matrices, using different
potentials or dissipation coefficients, as long as the underlying vertex set
is the same. The general purpose of interference strength is to measure the
amount of interaction or overlap between different ITMs.
## 4 Biological Examples
The theory and methods outlined in previous sections can be applied to any
interaction network. This section will present some examples using biological
networks, more specifically, yeast protein-protein interaction networks. Since
the interaction data obtained using many high-throughput methods is generally
inconsistent (Sprinzak _et al._ , 2003), we use the core yeast dataset from
DIP, version ScereCR20060402, consisting of 2554 proteins and 5952
interactions for all our examples. The core dataset, obtained using the
methods of Deane _et al._ (2002), contains only the most reliable interactions
from the DIP dataset of all yeast protein-protein interactions.
Our examples are restricted to investigation of information transduction
modules related to yeast histone acetyltransferases (HATs). Histones are
nuclear proteins that are major components of eukaryotic chromatin (Wolffe,
1992): eukaryotic DNA is organized as a repeating array of nucleosomes
consisting of 146 bp of DNA wound around a histone octamer consisting of two
of each of histone proteins H2A (Hta1, Hta2 in yeast), H2B (Htb1, Htb2 in
yeast), H3 (Hht1, Hht2 in yeast) and H4 (Hhf1, Hhf2 in yeast). It has been
repeatedly demonstrated that transcription is strongly influenced by the
chromatin structure and DNA-histone interactions in particular. The regions of
DNA that interact with histones are generally unavailable for transcription
and transcriptional activation and deactivation are connected with chromatin
alterations (Wolffe, 2001).
Histone acetyltransferases are enzymes that acetylate histones, leading to
weakening of the nucleosome structure and making the DNA involved accessible
to transcription factors (Struhl, 1998; Workman and Kingston, 1998).
Saccharomyces cerevisiae contains several HATs from two major classes with a
variety of biological functions and substrate specificities (Sterner and
Berger, 2000). The proteins Hat1, Gcn5, Elp3, Spt10 and Hpa2 belong to the
GNAT superfamily (Neuwald and Landsman, 1997), while Esa1, Sas2 and Sas3
belong to the MYST family (Borrow _et al._ , 1996; Smith _et al._ , 1998). The
proteins TAF1 (TATA-binding protein associated factor), a subunit of the TFIID
complex, and Nut1 (Med5), a subunit of the mediator complex (Biddick and
Young, 2005), have also been associated with histone acetyltransferase
activity (Mizzen _et al._ , 1996; Lorch _et al._ , 2000).
Unfortunately, the core dataset does not contain the relevant data for all
known HATs. The HATs Hpa2 and Spt10 are not present in the core while HAT1 has
interactions only with Hat2 and its substrate Hhf2. We chose to primarily
concentrate on HATs Gcn5, Esa1 and Elp3 because they are well researched and
the interaction data is abundant. They are all involved in transcriptional
activation, unlike Sas2, which promotes silencing (Osada _et al._ , 2001).
Gcn5 is the best characterized of all HATs, preferentially acetylating histone
H3 (Sternglanz and Schindelin, 1999). It forms the catalytic subunit of the
ADA and SAGA transcriptional activation complexes (Grant _et al._ , 1997). In
addition to Gcn5, the SAGA complex also contains the proteins Tra1, TAF5,
TAF6, TAF9, TAF10, TAF12, Hfi1 (Ada1), Ada2, Ngg1 (Ada3), Spt3, Spt7, Spt8 and
Spt20 (Ada5) (Timmers and Tora, 2005). The ADA complex contains a subset of
proteins from the SAGA complex, namely Gcn5, Hfi1, Ada2, Ngg1 and Spt20, plus
the adaptor protein Ahc1 (Eberharter _et al._ , 1999). The TAF proteins in
SAGA also belong to the TFIID complex, which overall consists of 15 subunits
including a TATA-binding protein and 14 TAFs (Sanders and Weil, 2000).
Esa1 is the catalytic subunit of the NuA4 histone acetyltransferase complex
essential for growth in yeast (Smith _et al._ , 1998; Allard _et al._ , 1999)
that catalyses acetlyaltion of the histone H4. It has been established that
the NuA4 complex, containing, in addition to Esa1, the proteins Tra1, Epl1
Yng2, Eaf1, Eaf2, Eaf3, Eaf5, Eaf6, Act1, Arp4 and Yaf9, is recruited by a
variety of transcriptional complexes as a transcriptional coactivator and is
involved in DNA repair (Doyon and Cote, 2004).
Elp3 is a part of the six component elongator complex , which is associated
with RNA polymerase II during transcript elongation (Wittschieben _et al._ ,
1999). The elongator complex also includes the proteins Iki3 (Elp1), Elp2–4,
Iki1 (Elp5) and Elp6 (Krogan and Greenblatt, 2001).
This section contains four examples of the application of our models, depicted
in Figures 1–5. Subsection 4.2 describes possible complexes associated with
the HATs Gcn5, Esa1 and Elp3, taken individually and in competition, that can
be inferred from the protein-protein interaction network using the absorbing
model. Subsection 4.3 investigates possible physical interaction interfaces
between the MADS box protein Mcm1 (Shore and Sharrocks, 1995) and the HATs
Esa1 and Gcn5. In this case, the emitting model is employed to discover the
pathways through which Mcm1 can recruit the above HATs and whether they are
recruited through the same interface. Before presenting our results we
describe the model parameters and computational techniques used.
### 4.1 Parameters and computation
#### 4.1.1 Dissipation
For all our examples, we set $\alpha_{i}=1$ for every node $i$ in our
interaction network so that the outgoing flow from any node is not dissipated.
Modeling the incoming dissipation the coefficients $\beta_{i}$ can take two
values: one for ‘ordinary’ and one for _evaporating_ vertices. In our examples
that use the absorbing model (4.2), $\beta_{i}$ is set to $0.70$ for ordinary
nodes and $0.01$ for evaporating nodes while the examples using the emitting
model (4.3) set $0.87$ for ordinary nodes and $0.01$ for evaporating nodes.
The evaporating nodes consisted of cytoskeleton proteins Act1, Myo1, Myo2,
Myo3, Myo4, Myo5, Smy1, Smy2, Sla1, Arc40, Arp2, Rvs167, Tpm1, Tpm2, Aip1 and
Las17 and histones (Hta1, Hta2, Htb1, Htb2, Hht1, Hhf2, Htz1, Hho1).
The coefficients for the ordinary nodes were chosen using the following
reasoning. For the emitting model we considered the dissipation rate that
would allow the random walk emitted from the source to reach an ‘average’ node
along the shortest path to it with the probability slightly less than $0.5$,
say $0.49$. We found that the average length of the shortest path between two
points in the yeast core dataset is $5.23$ and hence our coefficient is
$0.49^{(1/5.23)}=0.872$, which is rounded to $0.87$. A different coefficient
was needed for the absorbing examples because we were interested in only the
immediate complexes containing our selected HATs: the coefficient
$\beta_{i}=0.87$ would lead to most of the members of the RNA polymerase II
holoenzyme to be retrieved as members of the resulting ITM. We chose to
consider the shortest paths of length $2$, rather than of the average length
$5.23$. Using the same calculation as above, we obtain $0.49^{(1/2)}=0.7$.
The reason for having evaporating nodes with larger dissipation rate is that
both the cytoskeleton proteins and the histones form extended structures in
the cell and the nucleus, respectively. In our physical interaction network,
we assume that information can flow from one protein to another through an
intermediate node if all three nodes are brought close together in space and
time. Information is not likely to flow through proteins that are parts of
extended structures because proteins with completely different biological
function may bind them at different locations and at different times.
Therefore, allowing significant information flow through such nodes would
yield biologically implausible results.
However, depending on the exact context of the investigation, such nodes may
have an important role to play and removing them completely from the
interaction networks or assigning them to the boundary set $S$ would not be
appropriate. Hence, we set a very high incoming dissipation rate at
evaporating nodes while allowing the information to originate from them. In
terms of our models, this approach means that the evaporating nodes will have
very small visiting times in the emitting models and hence will not be
components of any ITM. On the other hand, depending on the exact network
topology, they may be part of ITMs obtained by the emitting model. Note that
other proteins that bind their interacting partners in a non space and time
specific manner can be chosen as additional evaporating nodes; we chose
histones and cytoskeleton proteins due to their direct relevance to our
selected examples.
#### 4.1.2 Potentials
All our examples use attracting potentials centered at each pseudosink or
sink. The potential function, heuristic in nature, is the same in every
example has the the form
$\theta_{k}(x)=\begin{cases}a_{1}x&\text{if $0<x\leq b$,}\\\
a_{1}x+a_{2}(x-b)^{2}&\text{if $x>b$,}\end{cases}$ (32)
where $a_{1}=0.8181$, $a_{2}=0.05$, $b=2$ and $k$ is any pseudosink or a sink.
The potential function shown above is long-range, affecting the whole graph,
with a linear portion for short ranges $0\leq x\leq 2$ and quadratic for
distances larger than $2$. We do not expect to see qualitative changes in the
results if the form of the potential function is modified as long as it has
the effect of attracting information towards the destination.
The sources (in the case of emitting models) and evaporating points were
excluded from the graph prior to calculating distances (their distances from
the centers were set to an arbitrary large number) in order to exclude the
paths passing through them from consideration. The reason for excluding the
paths passing through sources was that, by construction, the information never
enters a source from a transient vertex, while the evaporating points were
excluded because most of the signal entering them is dissipated.
#### 4.1.3 Numerical implementation
The code for computation of the results was implemented in the Python
programming language, using the NumPy and SciPy packages (Jones _et al._ ,
2001–). In particular, the computation of the matrices $\mathbf{\tilde{F}}$
and $\mathbf{\tilde{G}}$ (Equations (19–20)) was performed by the embedded
FORTRAN code from the UMFPACK (Davis, 2004) solver of sparse systems of linear
equations, using the Automatically Tuned Linear Algebra Software (ATLAS)
(Whaley and Petitet, 2005) implementation of Basic Linear Algebra Subprograms
(BLAS). The graphical representations of the subgraphs of interest were
produced by the neato program from the Graphviz graph visualization suite
(Gansner and North, 2000).
### 4.2 HAT complexes: absorbing examples
(a) |
---|---
|
(b) |
|
(c) |
|
Figure 1: ITMs obtained by running the absorbing model with Esa1(a), Gcn5(b)
and Elp3(c) as a sink. The shades of grey at the nodes represent the
probability of the information originating at the corresponding protein being
absorbed at the sink, the darker nodes indicating higher probability.
Figure 1 shows the three subgraphs of the yeast core interaction graph
consisting of the top scoring nodes according to the absorbing model with
Esa1, Gcn5 and Elp3 as single sinks, respectively. The information orginating
at the proteins shown has more than $0.07$ probability of being absorbed by
the sink (under the influence of the potential centered at the sink) as
opposed to being dissipated. Hence, the subgraphs show the proteins that are
likely to be in the same complex with the HATs chosen as sinks.
Figure 1(a), with Esa1 as the sink, shows all the proteins from the NuA4
complex that are available in the core dataset as highly significant. Some of
the proteins from ADA and SAGA complexes can also be seen because Tra1 belongs
to these complexes as well as to NuA4. The four types of histones forming the
histone octamer can also be seen interacting with Arp4. The proteins Vps51–54
on the right of Figure 1(a) belong to the Vps Fifty-three thethering (VFT)
complex, involved in vesicle assembly (Reggiori _et al._ , 2003). The proteins
Tlg1 and Ypt6 are interacting partners of the VFT complex (Reggiori _et al._ ,
2003). The relation between VFT and NuA4 is not established as these two
complexes are localized in different cellular compartments: NuA4 in the
nucleus and VFT in golgi-vacuole transport vesicles. The relationship observed
in Figure 1(a) results exclusively from the Yng2–Vps51 interaction, which was
orginally observed in a yeast-two-hybrid screen by Ito _et al._ (2000, 2001).
Based on the above information, it appears that VFT and NuA4 complexes do not
interact _in vivo_. Note that the histones as well as actin, although selected
as evaporating points, can be seen in the figure because the outgoing flow
from evaporating nodes is allowed.
In a similar fashion, Figure 1(b), with Gcn5 as the sink, shows the members of
SAGA, ADA and TFIID transcriptional activator complexes as well as many other
transcription factors, mostly members of subcomplexes of the RNA polymerase II
holoenzyme. Also worth mentioning is Cti6, which bridges the Cyc8-Tup1
corepressor and the SAGA coactivator to overcome repression of the GAL1 gene
(Papamichos-Chronakis _et al._ , 2002). The Cyc8 protein is also shown while
Tup1 is not, most likely because it is involved in many other interactions
away from Gcn5, bringing down its relative significance. Figure 1(c), with
Elp3 as the sink, clearly outlines the elongator complex, as well as some
members of the core RNA polymerase II complex (Rbp2–5, Rbp7, Rpc10, Rpo26)
(Myer and Young, 1998).
Figure 2 shows the top scoring nodes according to the absorbing model with
Esa1, Gcn5 and Elp3 as simultaneous sinks with attracting potentials. In this
case, the information originating at the depicted nodes has more than $0.05$
total probability of being absorbed by any of the sinks as opposed to being
dissipated.
Fewer nodes can be seen in this figure as compared to Figure 1 because the
three attracting potentials are now involved that may cancel each other out.
It can be seen that the elongator complex centered around Elp3 is not
connected to the subgraph around Esa1 and Gcn5. Although all of the NuA4,
SAGA, ADA and elongator complexes belong to the RNA polymerase II holoenzyme,
they do so at different times. The NuA4, ADA and SAGA complexes have a role in
initiation of transcription while the elongator complex is involved in
transcript elongation (Martinez, 2002). The green (mixture of cyan and yellow)
color of Tra1 is indicative of the fact that it is a subunit of both
Esa1-containing NuA4 complex and the Gcn5-containing SAGA complex.
Figure 2: ITM obtained by running the absorbing model with Esa1, Gcn5 and Elp3
as simultaneous sinks. The strength of each of cyan, yellow and magenta color
component of the node shows the square root of the probability of absorption
at Esa1, Gcn5 and Elp3, respectively.
### 4.3 Transcription factor interaction interfaces; emitting examples
Mcm1 is a yeast transcription factor essential for cell viability. It controls
many cellular functions including cell cycle transition (Althoefer _et al._ ,
1995), mating (Mead _et al._ , 2002) and arginine metabolism (Messenguy and
Dubois, 1993), through interactions with different cofactors. It has been
determined that Mcm1 acts both as an activator and a repressor of
transcription (Bruhn _et al._ , 1992; Messenguy and Dubois, 1993) and here we
explore the possible ways it can interact with the NuA4 and SAGA HAT
complexes.
(a) |
---|---
|
(b) |
|
Figure 3: ITMs resulting from the emitting model with Mcm1 as a source and
Esa1 as a pseudosink using the original yeast core dataset (a) and the
modified dataset additionally including the edges Tra1–Gal4 and Tra1–Gcn4 (b).
The proteins containing the largest amounts of deposited information are
shown, with the information content indicated by shading (darkest nodes
contain the most information).
Figure 3(a) shows the subgraph consisting of the $22$ proteins with the
largest deposited information content obtained by running our emitting model
with Mcm1 as a source and Esa1 as a pseudosink. The number of proteins to
display ($20$ plus the source and the pseudosink) was chosen because the
participation ratio for the information content vector (excluding the source
and the pseudosink) was $20.33$.
The ITM shown in Figure 3(a) gives the likely pathways of physical interaction
from Mcm1 to Esa1, according to the yeast core interaction dataset. It can be
immediately observed that Esa1 is reached solely through Tra1, which is known
to be the general interaction domain of both NuA4 and SAGA HAT complexes
(Allard _et al._ , 1999; Grant _et al._ , 1998). Directly associated with Mcm1
are the proteins Arg80–Arg82, belonging to the ArgR complex involved in
regulation of arginine metabolism (Dubois and Messenguy, 1991). The majority
of the ITM is dominated by the members of the SRB mediator subcomplex of the
RNA polymerase II holoenzyme (Srb2, Srb4, Srb7) (Biddick and Young, 2005) and
the TFIID, SAGA and ADA complexes. Also prominent are transcriptional
activators Gal4 and Gcn4 (Hinnebusch, 2005; Traven _et al._ , 2006).
The subgraph image suggests two possible interaction pathways: the main (based
on the intensities of deposited information) through Srb4 and members of
SAGA/ADA complex and the alternative through Ume6–TAF10–Spt7. Ume6 is a DNA
binding protein that acts as a transcriptional repressor by recruiting histone
deacetylases, which have the catalytic activity opposite to the HATs (Kassir
_et al._ , 2003). While simultaneous existence of activating and repressing
pathways is biologically plausible, we do not anticipate both pathways to be
in action at the same time. On the other hand, interaction of Mcm1 with the
NuA4 through any of the above pathways _in vivo_ is doubtful because both
pathways lead through the interacting partners of Tra1 in the SAGA complex
that are not associated with it in the NuA4 complex (Doyon and Cote, 2004;
Timmers and Tora, 2005). Note that the direct physical interaction of the
ArgR/Mcm1 complex and the SAGA complex was hypothesized by Ricci _et al._
(2002) in relation to regulation of arginine metabolism.
Nevertheless, it is likely that the yeast core dataset does not contain all
the interactions of Tra1 and that the interactions not in the dataset may
provide us with the plausible explanation. Brown _et al._ (2001) have
indicated that HAT complexes are recruited through Tra1 by Gal4 and Gcn4
transcriptional activators. To investigate if adding the implied edges would
significantly change the resulting ITM we added the Gcn4–Tra1 and Gal4–Tra1
links to the core dataset and rerun the emitting model with all other
parameters unchanged. The resulting ITM, with participation ratio of $21.66$,
is shown in Figure 3(b). We observe few changes: the proteins Ssn3, Srb5, Srb6
and Gal11, belonging to the mediator complex, replaced Cti6 and Srb7, thus
placing more emphasis to the mediator complex.
In this example, our emitting model appears to be quite robust to changes in
the pseudosink leakage parameter $\gamma$. Using the original core dataset, in
addition to the original run with $\gamma=0.3$, we ran our model with
$\gamma=0$, $\gamma=0.5$ and $\gamma=1$, obtaining participation ratios of
$19.43$, $20.34$ and $20.75$ and very little change in constitution of the
ITMs. For example, when $\gamma=1$, the new ITM contains the NuA4 proteins
Arp4 and Yng2 in the place of Cti6 and Srb7. Hence, larger pseudosink leakage
coefficient allows exploration of the nodes surrounding the pseudosinks
without affecting the remainder of the ITM in a major way. Such exploration is
very desirable for protein-protein interaction networks because it reveals
more of the complexes around pseudosinks, thus giving some of the
characteristics of the absorbing model to the emitting model. Note that many
of the interacting partners of the sources are found in the ITM solely due to
proximity of the source.
(a) |
---|---
|
(b) |
|
Figure 4: ITM resulting from the emitting model with Esa1 and Gcn5 as sources
and Mcm1 as a pseudosink: (a) information content, (b) interference strength.
To explore the extend the HATs Esa1 and Gcn5 share their interaction interface
with Mcm1 we set Esa1 and Gcn5 as sources and Mcm1 as a pseudosink
destination. Figure 4 shows the ITM based on the total information content
(participation ratio $24.62$, 28 nodes shown), with the nodes shaded according
to total content and interference strength. The proteins shown as nodes in
Figure 4 have appeared in one of the previous figures, mostly forming parts of
NuA4, SAGA/ADA, TFIID and mediator complexes. The nodes with the largest total
content are Tra1, Ada2, Ngg1 and Srb4 and the latter three are also the nodes
with by far the largest interference strength. This fact does not surprise us
because although Tra1 is a member of both NuA4 and SAGA complexes, information
flowing from Gcn5 to Mcm1 largely avoids it.
The paths used by the information emitted from Esa1 and Gcn5 separately can
best be seen in a color figure (Figure 5(a)) where the information content
from Esa1 and Gcn5 is shown as cyan and yellow, respectively. The nodes
colored strongly cyan contain mostly information from Esa1 while those colored
yellow contain mostly the information from Gcn5. The nodes colored green
contain information from both sources. In this way it can be observed that
members of NuA4 contain the information solely from Esa1, some SAGA proteins
contain the information solely from Gcn5, while Ada2, Ngg1 and Srb4 contain a
significant amount of information from both sources.
(a) |
---|---
|
(b) |
|
Figure 5: Information content of members of the ITM arising from the emitting
model with Esa1 and Gcn5 as sources and Mcm1 as a pseudosink: (a) using the
yeast core dataset; (b) using the modified dataset additionally including the
edges Tra1–Gal4 and Tra1–Gcn4. The strength of the cyan and yellow color
component of the node corresponds to the information content originating from
Esa1 and Gcn5, respectively.
Using additional links based on Brown _et al._ (Figure 5(b)) produces effects
similar to Figure 3(b): the common interface through the mediator complex is
emphasized at the expense of the paths through the SAGA complex. For example,
note the difference in color of Spt7, Gcn4 and Gal4 between Figure 5(a) and
Figure 5(b). The common interface through the mediator complex appears
biologically more plausible than directly through members of the SAGA complex
but we are as yet unable to find direct evidence in the literature confirming
either possibility.
## 5 Discussion and conclusion
The proposed information diffusion models appear to capture some of the
essential features of the yeast protein-protein interaction network in our
examples. Our absorbing model performed well in identifying complexes related
to sinks while the emitting model with pseudosinks is able to illuminate the
possible interaction interfaces between sources and pseudosinks. Application
of the concept of destructive interference in this context provides a way to
assess the degree of overlap of different ITMs.
The salient feature of our models is a novel use of attraction potentials and
dissipation. While the entries of the Green’s function can be interpreted in
graph-theoretic terms as sums of weights of paths from a source to a transient
vertex (for the emitting model) or from a transient vertex to a sink (for the
absorbing model), the potentials, together with the choice of boundary,
provide a unique context for information diffusion in the network. The weights
of the edges and hence the nature of the underlying graphs are changed every
time a different potential is applied, thus bringing forward different aspects
of the network. The potential function used for our examples was heuristic in
nature and we hope that our work would generate interest in developing
theoretical foundations for directed information propagation through networks.
Dissipation coefficients provide a natural and extremely flexible way of
controlling the spread of information content through the network. While
Girvan and Newman (2002) proposed a similar formulation for penalizing longer
paths connecting two nodes in a network, they did so in the context of
hierarchical clustering and using a single dissipation rate. Node specific
dissipation rates are important because they allow construction of
‘evaporating nodes’ and possible integration of additional information to our
model. Having the dissipation rates dependent on the environment of the node
may lead to a more sophisticated model of information transduction.
When modelling physical cellular protein networks, the main limitation of our
approach is that the the publicly available representations of protein-protein
interaction networks contain a limited amount of information. Each interaction
is shown as either occurring or not occurring, without reference to the
dynamics, time-scale, or specificity of binding. Furthermore, the spatial
location of the interactions on the protein molecules is not available, so
that it cannot be determined if a protein known to belong to two separate
complexes, such as Tra1 in our examples, can belong to both at the same time
and therefore transmit information between them. Therefore, our model of
protein cellular networks is only metaphorical at this stage. However, our
diffusion paradigm can be adapted to account for additional information about
proteins, such as their concentrations, cellular compartment localizations,
post-translational modifications or rate constants for binding interactions,
as it becomes available. One way to do that is to associate each protein to a
vector instead of a scalar value and to construct an evolution operator that
reflects the nature of the additional information. In such circusmstances, the
dynamics of information flow could be as revealing as the steady state we use
at this stage.
The quality of the interaction dataset also has a strong influence to the
outcomes of our models. Addition or deletion of edges may make the results
more realistic, as in our emitting examples, but also may completely alter the
ITM produced, if a particular edge provides a shortcut towards the
destination. Hence, in order to obtain the results useful in field of
application, it is imperative to use datasets of interactions that precisely
reflect the network being investigated. In the case of yeast protein-protein
interactions, Collins _et al._ (2007) were recently able to derive a
significantly more reliable collection of interactions, primarily based on two
large-scale studies of protein complexes by tandem affinity purification of
complexes followed by mass spectroscopic identification of individual proteins
(Gavin _et al._ , 2006; Krogan _et al._ , 2006). It is interesting that the
same transcriptional complexes encountered in our examples are prominent in
the unified physical interactome map presented by Collins _et al._ (2007).
The problem of ‘shortcuts’ through the network was also observed by Steffen
_et al._ (2002), who completely eliminated certain nodes in their effort to
model signal transduction pathways using the yeast protein-protein
interactions. Our evaporating nodes, with a very large incoming dissipation
rate, have a similar role with an added advantage that they can be visible as
parts of complexes observed using the absorbing model. The list of evaporating
nodes used by us is not exhaustive and it would be necessary to add further
classes of proteins to it for large-scale investigations of the yeast protein
interactome using our methods.
In this paper, we introduced a flexible mathematical framework for analysis of
interaction networks and indicated its utility by examples. We believe that
the ability to select a particular context for information propagation by
setting various model parameters will be extremely useful for addressing
questions involving interaction networks in biology and many other
disciplines.
## 6 Acknowledgments
We thank Drs John Wootton and David Landsman for encouragement and comments.
This work was supported by the Intramural Research Program of the National
Library of Medicine at National Institutes of Health.
## Appendix A Existence of Green’s Functions
In this appendix we provide the elementary proofs of the results about
existence of the Green’s functions stated in the main text. As before,
$\Gamma=(V,E,w)$ denotes a weighted directed graph with $N$ vertices, with the
weight matrix $\mathbf{W}$ and transition matrix $\mathbf{P}$. We also have
$T\subset V$ and $S=V\setminus T$.
Recall that for every matrix $\mathbf{M}$, the induced $\ell_{\infty}$ norm of
$\mathbf{M}$, written $\left\|\mathbf{M}\right\|_{\infty}$, is defined by
$\left\|\mathbf{M}\right\|_{\infty}=\sup_{\mathbf{x}\in\mathbb{R}^{n}}\frac{\left\|\mathbf{M}\mathbf{x}\right\|_{\infty}}{\left\|\mathbf{x}\right\|_{\infty}},$
(33)
where $\left\|\mathbf{x}\right\|_{\infty}=\max_{i}\left|x_{i}\right|$. One can
easily show that
$\left\|\mathbf{M}\right\|_{\infty}=\max_{i}\sum_{j}\left|M_{ij}\right|.$ (34)
Also recall that the spectral radius of a square matrix $\mathbf{M}$ is
defined to be the largest absolute value of its eigenvalues. It is well known
that that for every eigenvalue $\lambda$ of $\mathbf{M}$ and any
$k=1,2,\ldots$,
$\left|\lambda\right|\leq\left\|\mathbf{M}^{k}\right\|_{\infty}^{1/k}.$ (35)
###### Lemma A.1.
Let $\mathbf{M}$ be a square matrix with the spectral radius strictly less
than $1$. Then,
1. (i)
$\mathbf{M}^{k}\to\mathbf{0}$ as $k\to\infty$,
2. (ii)
The matrix $\mathbb{I}-\mathbf{M}$ is invertible and
$(\mathbb{I}-\mathbf{M})^{-1}=\sum_{k=0}^{\infty}\mathbf{M}^{k}$.
###### Proof.
By the Jordan matrix decomposition, we can write
$\mathbf{M}=\mathbf{V}\boldsymbol{\Lambda}\mathbf{V}^{-1}$ for some matrix
$\mathbf{V}$, where $\boldsymbol{\Lambda}$ is a block-diagonal matrix of the
form
$\boldsymbol{\Lambda}=\left[\begin{array}[]{cccc}\mathbf{B}_{1}&\mathbf{0}&\cdots&\mathbf{0}\\\
\mathbf{0}&\mathbf{B}_{2}&\cdots&\mathbf{0}\\\ \vdots&\vdots&\ddots&\vdots\\\
\mathbf{0}&\mathbf{0}&\cdots&\mathbf{B}_{N}\end{array}\right],$
with each of the sub-blocks $\mathbf{B}_{j}$, $1\leq j\leq N$, is of the form
$\mathbf{B}_{j}=\lambda_{j}\mathbb{I}+\mathbf{C}_{j}$ where
$\mathbf{C}_{j}=\left[\begin{array}[]{ccccc}0&1&0&\cdots&0\\\
0&0&1&\cdots&0\\\ \vdots&\vdots&\ddots&\ddots&\vdots\\\ 0&0&0&\cdots&1\\\
0&0&0&\cdots&0\end{array}\right]$
and $\lambda_{1},\ldots\lambda_{N}$ are eigenvalues of $\mathbf{M}$. Hence,
$\mathbf{M}^{k}=\mathbf{V}\boldsymbol{\Lambda}^{k}\mathbf{V}^{-1}$ and
$\boldsymbol{\Lambda}^{k}=\left[\begin{array}[]{cccc}\mathbf{B}_{1}^{k}&\mathbf{0}&\cdots&\mathbf{0}\\\
\mathbf{0}&\mathbf{B}_{2}^{k}&\cdots&\mathbf{0}\\\
\vdots&\vdots&\ddots&\vdots\\\
\mathbf{0}&\mathbf{0}&\cdots&\mathbf{B}_{N}^{k}\end{array}\right].$
For each eigenvalue $\lambda_{j}$ and each block $\mathbf{B}_{j}$, we can
write
$\mathbf{B}_{j}^{k}=(\lambda_{j}\mathbb{I}+\mathbf{C}_{j})^{k}=\sum_{p=0}^{k}\binom{k}{p}\lambda_{j}^{k-p}\mathbf{C}_{j}^{p}.$
It can easily be shown that for each $j$, $\mathbf{C}_{j}$ is a nilpotent
matrix, that is, if $\mathbf{C}_{j}$ is an $m\times m$ matrix, then
$\mathbf{C}^{m}=\mathbf{0}$. Therefore, for $k\geq m-1$,
$\mathbf{B}_{j}^{k}=\lambda_{j}^{k-m+1}\left(\sum_{p=0}^{m-1}\binom{k}{p}\lambda_{j}^{m-p-1}\mathbf{C}_{j}^{p}\right).$
Observe that the above expression in parenthesis gives an (upper triangular)
matrix whose entries are $m-1$-th degree polynomials in $k$ and hence, that
the whole expression for $\mathbf{B}_{j}^{k}$ is dominated by
$\lambda_{j}^{k-m+1}$. Since, by the spectral radius assumption,
$\left|\lambda_{j}\right|<1$ for each $i$, it follows that for each $j$,
$\mathbf{B}_{j}^{k}\to\mathbf{0}$ as $k\to\infty$ and hence
$\boldsymbol{\Lambda}^{k}\to\mathbf{0}$ as $k\to\infty$ by the block
structure. This proves the first statement.
For the second statement suppose that $\mathbb{I}-\mathbf{M}$ is singular.
Then $\mathbb{I}-\mathbf{M}$ has $0$ as an eigenvalue and hence $\lambda=1$ is
an eigenvalue of $\mathbf{M}$, contradicting our assumption about the spectral
radius of $\mathbf{M}$. Therefore, $\mathbb{I}-\mathbf{M}$ is invertible.
Furthermore, it can easily be obtained using the block diagonal structure of
$\boldsymbol{\Lambda}$ and the ratio test that the sum
$\sum_{k=0}^{\infty}\mathbf{M}^{k}$ converges, Hence,
$(\mathbb{I}-\mathbf{M})\sum_{k=0}^{\infty}\mathbf{M}^{k}=\sum_{k=0}^{\infty}\mathbf{M}^{k}-\sum_{k=0}^{\infty}\mathbf{M}^{k+1}=\mathbb{I}+\sum_{k=1}^{\infty}\mathbf{M}^{k}-\sum_{k=1}^{\infty}\mathbf{M}^{k}=\mathbb{I}.$
∎
Since the matrix $\mathbf{P}$ is stochastic, we have
$\left\|\mathbf{P}\right\|_{\infty}=1$ and hence the spectral radius of
$\mathbf{P}$ is bounded by $1$. Since $\mathbf{P}_{TT}$ is a submatrix of
$\mathbf{P}$, we have $\left\|\mathbf{P}_{TT}\right\|_{\infty}\leq 1$ and its
spectral radius is also bounded by $1$. To prove Proposition 2.1 (denoted
Proposition A.5 below) we will show that the spectral radius of
$\mathbf{P}_{TT}$ is strictly smaller than $1$ if there is some vertex in $S$
that can be reached from any transient node via a directed path. Before
presenting the main proof, we require several lemmas.
###### Lemma A.2.
Let $\mathbf{B}$ and $\mathbf{C}$ be $n\times n$ matrices with non-negative
entries such that $\left\|\mathbf{B}\right\|_{\infty}\leq 1$ and
$\left\|\mathbf{C}\right\|_{\infty}\leq 1$ and let
$\mathbf{D}=\mathbf{C}\mathbf{B}$. Suppose there exists $1\leq p\leq n$ such
that $0<\sum_{j}B_{pj}<1$. Then, for every $1\leq i\leq n$ such that
$C_{ip}>0$,
$\sum_{j}D_{ij}<1.$
###### Proof.
Let $K=\\{k:C_{ik}>0\\}$. Then $p\in K$ and
$\displaystyle\sum_{j}D_{ij}$ $\displaystyle=\sum_{j}\sum_{k}C_{ik}B_{kj}$
$\displaystyle=\sum_{k\in K}C_{ik}\sum_{j}B_{kj}$ $\displaystyle\leq\sum_{k\in
K\setminus\\{p\\}}C_{ik}\left\|\mathbf{B}\right\|_{\infty}+C_{ip}\sum_{j}B_{pj}$
$\displaystyle<\sum_{k\in K\setminus\\{p\\}}C_{ik}+C_{ip}$ $\displaystyle\leq
1.$
∎
###### Lemma A.3.
Let $\Gamma$ be a weighted directed graph with weight matrix $\mathbf{W}$. Let
$i$ and $j$ be distinct nodes of $\Gamma$ connected by a directed path from
$i$ to $j$ of length $n\geq 1$. Then $W^{n}_{ij}>0$.
###### Proof.
We use induction. If $i$ and $j$ are connected with a path of length $1$, then
there exists an edge $(i,j)\in E$ and hence $W_{ij}>0$. Assume that
$W^{m}_{ij}>0$ if $i$ and $j$ are connected by a directed path from $i$ to $j$
of length $m$. Suppose $i$ and $j$ are connected by a path of length $m+1$.
Then there exists a vertex $k$ such that $i$ and $k$ are connected by a
directed path from $i$ to $k$ of length $m$ and there exists a directed edge
$(k,j)$. Hence, by our assumption $W^{m}_{ik}>0$ and $W_{kj}>0$. Therefore,
$W^{m+1}_{ij}=\sum_{k^{\prime}\in V}W^{m}_{ik^{\prime}}W_{k^{\prime}j}\geq
W^{m}_{ik}W_{kj}>0.$
∎
###### Lemma A.4.
Let $\mathbf{M}=\mathbf{P}_{TT}$, let $i\in T$ and suppose there exists $s\in
S$ such that there exists a directed path from $i$ to $s$ of length $m$. Then
for all $n\geq m$,
$\sum_{k\in T}M^{n}_{ik}<1.$ (36)
###### Proof.
Let $i\in T$ and let $s\in S$ be a vertex such that there exists a directed
path from $i$ to $s$ of length $m$. Let $J$ be the set of vertices in $T$
directly adjacent to a vertex in $S$. Then, by our assumption, for every $i\in
T$ there exists $j\in J$ such that there exists a directed path from $i$ to
$j$ of length $m-1$. Since the matrix $\mathbf{P}_{TT}$ can be treated as the
weight matrix for the subgraph of $\Gamma$ restricted to vertices in $T$, it
follows by Lemma A.3 that $M^{m-1}_{ij}>0$.
Since every point in $J$ is adjacent to a point in $S$, it also follows that
$\sum_{k\in T}M_{jk}<1$. Clearly, $\left\|\mathbf{M}\right\|_{\infty}\leq 1$
and hence $\left\|\mathbf{M}^{m-1}\right\|_{\infty}\leq 1$. Applying Lemma A.2
to the matrices $\mathbf{M}$ and $\mathbf{M}^{m-1}$ we obtain that for every
$i\in T$, $\sum_{k\in T}M^{m}_{ik}<1$.
Let $t\geq m$ and assume $\sum_{k\in T}M^{t}_{ik}<1$. We have
$\sum_{k\in T}M^{t+1}_{ik}=\sum_{k\in T}\sum_{k^{\prime}\in
T}M^{t}_{ik^{\prime}}M_{k^{\prime}k}=\sum_{k^{\prime}\in
T}M^{t}_{ik^{\prime}}\sum_{k\in T}M_{k^{\prime}k}\leq\sum_{k^{\prime}\in
T}M^{t}_{ik^{\prime}}<1$
and our result follows by induction. ∎
###### Proposition A.5.
Suppose that for every $p\in T$ there exists $s\in S$ such that there exists a
directed path from $p$ to $s$. Then, the matrix $\mathbb{I}-\mathbf{P}_{TT}$
is invertible and
$(\mathbb{I}-\mathbf{P}_{TT})^{-1}=\sum_{k=0}^{\infty}(\mathbf{P}_{TT})^{k}.$
(37)
###### Proof.
Let $\mathbf{M}=\mathbf{P}_{TT}$. Observe that our assumption implies that for
every $i\in T$ there exists $s\in S$ such that there exists a directed path
from $i$ to $s$ of length at most $N$. By Lemma A.4, we have for every $i\in
T$, $\sum_{k\in T}M^{N}_{ik}<1$. Hence,
$\left\|\mathbf{M}^{N}\right\|_{\infty}<1$ and therefore the spectral radius
of $\mathbf{M}=\mathbf{P}_{TT}$ is strictly smaller than $1$. Our result
follows by Lemma A.1. ∎
### A.1 Information dissipation
###### Proposition A.6.
Let $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ be vectors of length $N$
such that for all $i\in V$, $\alpha_{i}>0$ and $\beta_{i}>0$. Define the
$N\times N$ matrix $\mathbf{\tilde{P}}$ with entries
$\tilde{P}_{ij}=\alpha_{i}\beta_{j}P_{ij},$
Let $\alpha_{*}=\max\\{\alpha_{i}:i\in V\\}$ and
$\beta_{*}=\max\\{\beta_{i}:i\in V\\}$ and suppose $\alpha_{*}\beta_{*}<1$.
Then, the matrix $\mathbb{I}-\mathbf{\tilde{P}}_{TT}$ is invertible and
$(\mathbb{I}-\mathbf{\tilde{P}}_{TT})^{-1}=\sum_{k=0}^{\infty}(\mathbf{\tilde{P}}_{TT})^{k}.$
(38)
###### Proof.
Let $\mathbf{M}=\mathbf{\tilde{P}}_{TT}$ and let $i\in T$. Then,
$\displaystyle\sum_{j\in T}M_{ij}$ $\displaystyle=\sum_{j\in
T}\alpha_{i}\beta_{j}P_{ij}\leq\alpha_{*}\beta_{*}\sum_{j\in T}P_{ij}<1.$
Hence, $\left\|\mathbf{M}\right\|_{\infty}<1$ and thus the spectral radius of
$\mathbf{\tilde{P}}_{TT}$ is strictly smaller than $1$. Our result then
follows by Lemma A.1. ∎
More generally, it is possible to interpret dissipation in the light of
Proposition A.5 by constructing a new graph $\tilde{\Gamma}$ with the vertex
set $\tilde{V}=V\cup\\{v\\}$, where $v$ denotes an additional vertex. The
weight matrix of $\tilde{\Gamma}$, denoted $\mathbf{\tilde{W}}$, has entries
$\tilde{W}_{ij}=\begin{cases}\alpha_{i}\beta_{j}P_{ij}&\text{if $i\in V$ and
$j\in V$,}\\\ 1-\sum_{k\in V}\alpha_{i}\beta_{k}P_{ik}&\text{if $i\in V$ and
$j=v$,}\\\ 0&\text{if $i=v$.}\end{cases}$ (39)
Clearly, a random walk on $\tilde{\Gamma}$ is equivalent to a random walk on
$\Gamma$ with dissipation: the dissipated information is directed towards the
additional vertex $v$ and then disappears. If we place $v$ in the boundary set
$\tilde{S}$, by Proposition A.5, the necessary condition for existence of the
Green’s function $(\mathbb{I}-\mathbf{\tilde{P}}_{TT})^{-1}$ is that from
every transient node $i$ there exists a directed path to either a node $s\in
S$ or a node $j\in T$ such that $\sum_{k\in V}\alpha_{j}\beta_{k}P_{jk}<1$
(such node $j$ is adjacent to $v$ in the graph $\tilde{\Gamma}$. Proposition
A.6 then just represents the special case where every transient vertex is
adjacent to $v$ in $\tilde{\Gamma}$.
## Appendix B Interpretations of the matrices $\mathbf{F}$ and $\mathbf{H}$
### B.1 $\mathbf{F}$ and $\mathbf{H}$ as matrices of expected visiting times
We will show that both $F_{ij}$ and $H_{ij}$ can be interpreted as the
expected number of times a random walk originating at the vertex $i$ visits
the vertex $j$, while avoiding all vertices in the boundary set $S$. Note that
in the case of the matrix $\mathbf{F}$, we have $i\in T$ and $j\in S$ while
for the matrix $\mathbf{H}$, $i\in S$ and $j\in T$. We will use $\mathbb{E}$
to denote the expectation operator.
###### Lemma B.1.
Suppose the boundary set $S$ represents sinks and let $Z_{ij}$ be a random
variable denoting the total number of times a random walk starting at $i\in T$
is absorbed at $j\in S$. Then,
$\mathbb{E}(Z_{ij})=F_{ij}.$ (40)
###### Proof.
Let $Y_{ij}(t)$ be the random variable taking the value $1$ if the random walk
originating at $i\in T$ is absorbed at $j\in S$ at time $t$, with probability
$\sum_{k\in T}P_{ik}^{t-1}P_{kj}$, and taking the value $0$ otherwise. We have
$Z_{ij}=\sum_{t=1}^{\infty}Y_{ij}(t)$ and $\mathbb{E}(Y_{ij}(t))=\sum_{k\in
T}P_{ik}^{t-1}P_{kj}$. Thus,
$\displaystyle\mathbb{E}(Z_{ij})$
$\displaystyle=\mathbb{E}\left(\sum_{t=1}^{\infty}Y_{ij}(t)\right)$
$\displaystyle=\sum_{t=1}^{\infty}\mathbb{E}(Y_{ij}(t))$
$\displaystyle=\sum_{t=1}^{\infty}\sum_{k\in T}P_{ik}^{t-1}P_{kj}$
$\displaystyle=\sum_{k\in T}\sum_{t=0}^{\infty}P_{ik}^{t}P_{kj}$
$\displaystyle=\sum_{k\in T}G_{ik}P_{kj}$ $\displaystyle=F_{ij}.\qed$
###### Lemma B.2.
Suppose the boundary set $S$ represents sources and let $Z_{ij}$ be a random
variable denoting the total number of times a random walk starting at $i\in S$
visits the node $j\in T$. Then,
$\mathbb{E}(Z_{ij})=H_{ij}.$ (41)
###### Proof.
In the same fashion as above, let $Y_{ij}(t)$ be the random variable taking
the value $1$ if the random walk originating at $i\in S$ is at $j\in T$ at
time $t$, with probability $\sum_{k\in T}P_{ik}P^{t-1}_{kj}$, and taking the
value $0$ otherwise. We have $Z_{ij}=\sum_{t=1}^{\infty}Y_{ij}(t)$ and
$\mathbb{E}(Y_{ij}(t))=\sum_{k\in T}P_{ik}P_{kj}^{t-1}$. Thus,
$\displaystyle\mathbb{E}(Z_{ij})$
$\displaystyle=\mathbb{E}\left(\sum_{t=1}^{\infty}Y_{ij}(t)\right)$
$\displaystyle=\sum_{t=1}^{\infty}\mathbb{E}(Y_{ij}(t))$
$\displaystyle=\sum_{t=1}^{\infty}\sum_{k\in T}P_{ik}P^{t-1}_{kj}$
$\displaystyle=\sum_{k\in T}\sum_{t=0}^{\infty}P_{ik}P^{t}_{kj}$
$\displaystyle=\sum_{k\in T}P_{ik}G_{kj}$ $\displaystyle=H_{ij}.\qed$
### B.2 Invariants of $\mathbf{F}$ and $\mathbf{H}$
Let $\mathbf{1}\in\mathbb{R}^{n}$ denote the vector whose entries are all
$1$’s. Since all rows of $\mathbf{P}$ sum to unity, it follows that
$\mathbf{P}\mathbf{1}=\mathbf{1}$ and hence $\mathbf{1}$ is a right
eigenvector of $\mathbf{P}$ for the eigenvalue $\lambda=1$. Define
$\mathbf{d}$ as a vector of length $n$ having entries $d_{i}=\sum_{j}W_{ij}$.
If $\Gamma$ is unweighted graph, $d_{i}$ gives the degree of the node $i$.
Assuming $\mathbf{W}$ is symmetric,
$\sum_{k}P_{kj}d_{k}=\sum_{k}W_{kj}=\sum_{k}W_{jk}=d_{j}$
and therefore $\mathbf{d}$ is a left eigenvector of $\mathbf{P}$ corresponding
to the eigenvalue $\lambda=1$. This leads to the following result.
###### Lemma B.3.
Suppose that the matrix $\mathbb{I}-\mathbf{P}_{TT}$ is invertible. Let
$\mathbf{u}$ and $\mathbf{v}$ be the left and right eigenvector of the matrix
$\mathbf{P}$ corresponding to the eigenvalue $\lambda=1$, respectively. Write
$\mathbf{u}=[\mathbf{u}_{S}\ \mathbf{u}_{T}]$ and
$\mathbf{v}=\left[\begin{array}[]{c}\mathbf{v}_{S}\\\
\mathbf{v}_{T}\end{array}\right]$. Then,
$\mathbf{u}_{T}=\mathbf{u}_{S}\mathbf{H},$ (42)
and
$\mathbf{v}_{T}=\mathbf{F}\mathbf{v}_{S}.$ (43)
###### Proof.
Using the canonical form of the matrix $\mathbf{P}$ (Equation (3)) and the
fact that $\mathbf{u}$ and $\mathbf{v}$ are left and right eigenvectors of
$\mathbf{P}$ respectively, we obtain
$\mathbf{u}_{T}=\mathbf{u}_{S}\mathbf{P}_{ST}+\mathbf{u}_{T}\mathbf{P}_{TT},$
(44)
and
$\mathbf{v}_{T}=\mathbf{P}_{TS}\mathbf{v}_{S}+\mathbf{P}_{TT}\mathbf{v}_{T}.$
(45)
Rearranging Equations (44) and (45) leads to
$\mathbf{u}_{T}(\mathbb{I}-\mathbf{P}_{TT})=\mathbf{u}_{S}\mathbf{P}_{ST},$
(46)
and
$(\mathbb{I}-\mathbf{P}_{TT})\mathbf{v}_{T}=\mathbf{P}_{TS}\mathbf{v}_{S}.$
(47)
Our result then follows as the consequence of invertibility of
$\mathbb{I}-\mathbf{P}_{TT}$. ∎
Since $\mathbf{1}$ is a right eigenvector of $\mathbf{P}$, it follows from
(43) that for all $i$, $\sum_{j\in S}F_{ij}=1$. Furthemore, recall that if
$\Gamma$ is an undirected graph, $\mathbf{W}$ is symmetric and $\mathbf{d}$ is
a left eigenvector of $\mathbf{P}$ for $\lambda=1$. Assuming the matrix
$\mathbf{H}$ exists, we obtain from Lemma B.3 that, if $S$ contains a single
point, the matrix $\mathbf{H}$ is a row vector, which is a multiple of
$\mathbf{d}_{T}$.
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|
arxiv-papers
| 2011-12-16T22:33:10 |
2024-09-04T02:49:25.421327
|
{
"license": "Public Domain",
"authors": "Aleksandar Stojmirovi\\'c and Yi-Kuo Yu",
"submitter": "Aleksandar Stojmirovi\\'c",
"url": "https://arxiv.org/abs/1112.3988"
}
|
1112.4042
|
# Volume growth, number of ends and the topology of a complete submanifold
Vicent Gimeno Departament de Matemàtiques- Institut of New Imaging
Technologies, Universitat Jaume I, Castellon, Spain. gimenov@uji.es and
Vicente Palmer Departament de Matemàtiques- Institut of New Imaging
Technologies, Universitat Jaume I, Castellon, Spain. palmer@mat.uji.es
###### Abstract.
Given a complete isometric immersion $\varphi:P^{m}\longrightarrow N^{n}$ in
an ambient Riemannian manifold $N^{n}$ with a pole and with radial sectional
curvatures bounded from above by the corresponding radial sectional curvatures
of a radially symmetric space $M^{n}_{w}$, we determine a set of conditions on
the extrinsic curvatures of $P$ that guarantees that the immersion is proper
and that $P$ has finite topology in the line of the results in [24] and [25].
When the ambient manifold is a radially symmetric space, it is shown an
inequality between the (extrinsic) volume growth of a complete and minimal
submanifold and its number of ends which generalizes the classical inequality
stated in [1] for complete and minimal submanifolds in $\mathbb{R}^{n}$. We
obtain as a corollary the corresponding inequality between the (extrinsic)
volume growth and the number of ends of a complete and minimal submanifold in
the Hyperbolic space together with Bernstein type results for such
submanifolds in Euclidean and Hyperbolic spaces, in the vein of the work [12].
###### Key words and phrases:
volume growth, minimal submanifold, end, Hessian-Index comparison theory,
extrinsic distance, total extrinsic curvature, second fundamental form, gap
theorem, Bernstein-type theorem.
###### 2000 Mathematics Subject Classification:
Primary 53A20 53C40; Secondary 53C42
* Work partially supported by the Caixa Castelló Foundation, and DGI grant MTM2010-21206-C02-02.
## 1\. Introduction
A natural question in Riemannian geometry is to explore the influence of the
curvature conduct of a complete Riemannan manifold on its geometric and
topological properties. Classical results concernig this are the gap theorems
showed by Greene and Wu in [7], (see too [8]), and, when it is considered a
minimal submanifold (properly) immersed in the Euclidean space
$\mathbb{R}^{n}$, the Berstein-type theorems showed by Anderson in [1] and by
Schoen in [32]. Greene and Wu’s results states, roughly speaking, that a
Riemannian manifold with a pole and with faster than quadratic decay of its
sectional curvatures is isometric to the Euclidean space. On the other hand,
Anderson proved, as a corollary of a generalization of the Chern-Osserman
theorem on complete and minimal submanifolds of $\mathbb{R}^{n}$ with finite
total (extrinsic) curvature, that any of such submanifolds having one end is
an affine $n$-plane. More examples concerning submanifolds immersed in an
ambient Riemannian manifold and the analysis of its (intrinsic and extrinsic)
curvature behavior are the gap results, (of Bernstein-type), given by Kasue
and Sugahara in [12] (see Theorems A and B), where an accurate (extrinsic)
curvature decay forces to minimal, (or not) submanifolds with one end of the
Euclidean and Hyperbolic spaces to be totally geodesic, and the gap results
for minimal submanifolds in the Euclidean space with controlled scalar
curvature given by Kasue in [13].
The estimation of the number of ends of these submanifolds plays a fundamental
rôle in all the Bernstein-type results above mentioned. In this way, it is
proved in [1] (see Theorems 4.1 and 5.1 in that paper) that given a complete
and minimal submanifold $\varphi:P^{m}\longrightarrow\mathbb{R}^{n}$, ($m>2$)
having finite total curvature $\int_{P}\|B^{P}\|^{m}d\sigma<\infty$, its
(extrinsic) volume growth, defined as the quotient
$\frac{\operatorname{Vol}(\varphi(P)\cap B^{0,n}_{t})}{\omega_{n}t^{n}}$ is
bounded from above by the number of ends of $P$, $\mathcal{E}(P)$, namely
(1.1) $\lim_{t\to\infty}\frac{\operatorname{Vol}(\varphi(P)\cap
B^{0,n}_{t})}{\omega_{n}t^{n}}\leq\mathcal{E}(P)$
where $B^{b,n}_{t}$ denotes the metric $t-$ ball in the real space form of
constant curvature $b$, $I\\!\\!K^{n}(b)$, and $\|B^{P}\|$ denotes the
Hilbert-Schmidt norm of the second fundamental form of $P$ in
$\mathbb{R}^{n}$. If moreover $\mathcal{E}(P)=1$, it is concluded (using
inequality (1.1)) the Bernstein-type result above alluded, namely, that
$P^{m}$ is an affine plane, i.e. totally geodesic in $\mathbb{R}^{n}$, (see
Theorem 5.2 in [1]).
In the paper [3] it was proved that inequality (1.1) is in fact an equality
when the minimal submanifold in $\mathbb{R}^{n}$ exhibits an accurate decay of
its extrinsic curvature $\|B^{P}\|$ and in the paper [12] it was proved that,
if the submanifold $P$ has only one end and the decay of its extrinsic
curvature $\|B^{P}\|$ is faster than linear, (when the ambient space is
$\mathbb{R}^{n}$) or than exponential, (when the ambient space is
$\mathbb{H}^{n}(b)$), then it is is totally geodesic.
Within this study of the behavior at infinity of complete and minimal
submanifolds with finite total curvature immersed in the Euclidean space, it
was proved also in [1] and in [22] that the immersion of a complete and
minimal submanifold $P$ in $\mathbb{R}^{n}$ or $\mathbb{H}^{n}(b)$ satisfying
$\int_{P}\|B^{P}\|^{m}d\sigma<\infty$ is proper and that $P$ is of finite
topological type.
We should mention here the results in [24] and in [25], where has been stated
new conditions on the decay of the extrinsic curvature for a completely
immersed submanifold $P$ in the Euclidean space ([24]) and in a Cartan-
Hadamard manifold ([25]) which guarantees the properness of the submanifold
and the finiteness of its topology.
In view of these results, it seems natural to consider the following three
issues:
1. (1)
Can the properness/finiteness results in [24] and [25] be extended to
submanifolds immersed in spaces which have not necessarily non-positive
curvature?,
2. (2)
Do we have an analogous to inequality (1.1) between the extrinsic volume
growth and the number of ends when we consider a minimal submanifold
(properly) immersed in Hyperbolic space which exhibit an accurate extrinsic
curvature decay?.
3. (3)
Moreover, is it possible to deduce from this inequality a Bernstein-type
result in the line of [1] and [12]?.
We provide in this paper a (partial) answer to these questions, besides other
lower bounds for the number of ends for (non-minimal) submanifolds in the
Euclidean and Hyperbolic spaces and other gap results related with these
estimates. As a preliminary view of our results, we have the following
theorems, Theorem 1.1 and Theorem 1.2, which follows directly from our Theorem
3.5. In Theorem 1.1 we have the answer to the two last questions, namely,
setting equation (1.1), but in the Hyperbolic case, and a Bernstein-type
result for minimal submanifolds in the Hyperbolic space, in the line studied
by Kasue and Sugahara in [12], (see assertion (A-iv) of Theorem A). On the
other hand, Theorem 1.2 encompasses a slightly less general version of
assertion (A-i) of Theorem A in [12].
###### Theorem 1.1.
Let $\varphi:P^{m}\longrightarrow\mathbb{H}^{n}(b)$ be a complete, proper and
minimal immersion with $m>2$. Let us suppose that for sufficiently large
$R_{0}$ and for all points $x\in P$ such that $r(x)>R_{0}$, (i.e. outside a
compact),
$\|B^{P}_{x}\|\leq\frac{\delta(r(x))}{e^{2\sqrt{-b}\,r(x)}}$
where $r(x)=d_{\mathbb{H}^{n}(b)}(o,\varphi(x))$ is the (extrinsic) distance
in $\mathbb{H}^{n}(b)$ of the points in $\varphi(P)$ to a fixed pole
$o\in\mathbb{H}^{n}(b)$ such that $\varphi^{-1}(o)\neq\emptyset$ and
$\delta(r)$ is a smooth function such that $\delta(r)\to 0$ when $r\to\infty$.
Then:
1. (1)
The finite number of ends $\mathcal{E}(P)$ is related with the volume growth
by
$\operatorname{Sup}_{t>0}\frac{D_{t}(o)}{\operatorname{Vol}(B_{t}^{b,m})}\leq\mathcal{E}(P)$
where $D_{t}(o)=\\{x\in P:r(x)<t\\}=\\{x\in P:\varphi(x)\in B^{b,n}_{t}(o)\\}$
is the extrinsic ball of radius $t$ in $P$, (see Definition 2.1).
2. (2)
If $P$ has only one end, $P$ is totally geodesic in $\mathbb{H}^{n}(b)$
When the ambient manifold is $\mathbb{R}^{n}$, we have the following
Bernstein-type result as in [12]:
###### Theorem 1.2.
Let $\varphi:P^{m}\longrightarrow\mathbb{R}^{n}$ be a complete non-compact,
minimal and proper immersion with $m>2$. Let us suppose that for sufficiently
large $R_{0}$ and for all points $x\in P$ such that $r(x)>R_{0}$, (i.e.
outside the compact extrinsic ball $D_{R_{0}}(o)$ with
$\varphi^{-1}(o)\neq\emptyset$),
$\|B^{P}_{x}\|\leq\frac{\epsilon(r(x))}{r(x)}$
where $\epsilon(r)$ is a smooth function such that $\epsilon(r)\to 0$ when
$r\to\infty$. Then:
1. (1)
The finite number of ends $\mathcal{E}(P)$ is related with the volume growth
by
$\operatorname{Sup}_{t>0}\frac{\operatorname{Vol}(D_{t})}{\operatorname{Vol}(B_{t}^{0,m})}\leq\mathcal{E}(P)$
2. (2)
If $P$ has only one end, $P$ is totally geodesic in $\mathbb{R}^{n}$.
These results, that we shall prove in Section 8, (together the corollaries of
Section 4), follows from two main theorems, stablished in Section 3. In the
first (Theorem 3.1) we show that a complete isometric immersion
$\varphi:P^{m}\longrightarrow N^{n}$, ($m>2$), with controlled second
fundamental form in a complete Riemannian manifold which possess a pole and
has controlled radial sectional curvatures is proper and has finite topology.
In the second (Theorem 3.4) it is proved that a complete and proper isometric
immersion $\varphi:P^{m}\longrightarrow M^{n}_{w}$, ($m>2$), with controlled
second fundamental form in a radially symmetric space $M^{n}_{w}$ with
sectional curvatures bouded from below by a radial function has its volume
growth bounded from above by a quantity which involve its (finite) number of
ends.
The proof of both theorems follows basically the argumental lines of the
proofs given in [24] and [25] and some ideas in [3]. An important difference
to these results is that, on our side, we allow to the ambient manifold to
have positive sectional curvatures, bounding from above only the sectional
curvatures of the planes containing radial directions. However, to show the
properness of the immersion in [25], the ambient manifold must have non-
positive sectional curvatures, and to assure the finiteness of the topology of
the immersion $P$, this ambient manifold must be, in addition, simply
connected, (i.e. a Cartan-Hadamard manifold). This difference is based in
following considerations.
To obtain the finiteness of the topology in Theorem 3.1, we show that the
restricted, (to the submanifold) extrinsic distance to a fixed pole (in the
ambient manifold) has no critical points outside a compact and then, we apply
classical Morse theory. To show that the extrinsic distance function has no
critical points we compute its Hessian as we can find it in [16] and [27].
These results are, in its turn, based in the Jacobi-Index analysis for the
Hessian of the distance function given in [6], in particular, its Theorem A,
(see Subsection 2.3). This comparison theorem is different of the Hessian
comparison Theorem 1.2 used in [25]: while in this last theorem, the space
used as a model to compare is the real space form with constant sectional
curvature equal to the bound on the sectional curvatures of the given
Riemannian manifold, in our adaptation of Theorem A in [6], (see Theorem
2.10), only the sectional curvatures of the planes containing radial
directions from the pole are bounded by the corresponding radial sectional
curvatures in a radially symmetric space used as a model.
We also note at this point that although we use the definition of pole given
by Greene and Wu in [6], (namely, the exponential must be a diffeomorphism at
a pole), in fact, the comparison of the Hessians in Theorem A holds along
radial geodesics from the poles defined as those points which have not
conjugate points, as in [25].
### 1.1. Outline
The outline of the paper is the following. In Section §.2 we present the
definiton of extrinsic ball, together the basic facts about the Hessian
comparison theory of restricted distance function we are going to use and an
isoperimetric inequality for the extrinsic balls which plays an important rôle
in the proof of Theorem 3.4 . Section §.3 is devoted to the statement of the
main results (Theorem 3.1, Theorem 3.4 and Theorem 3.5). We shall present in
Section 4 two lists of results based in Theorems 3.1, 3.4 and 3.5: the first
set of consequences is devoted to bound from above the volume growth of a
submanifold by the number of its ends, in several contexts, obtaining moreover
some Bernstein-type results. In the second set of corollaries are stated some
compactification theorems for submanifolds in $\mathbb{R}^{n}$, in
$\mathbb{H}^{n}$ and in $\mathbb{H}^{n}\times\mathbb{R}^{l}$. Sections §.5,
§.6, §.7 are devoted to the proof of Theorems 3.1, 3.4, and 3.5, respectively.
Theorem 1.1, Theorem 1.2 and the corollaries stated in Section §.4 are proved
in Section §.8.
## 2\. Preliminaires
### 2.1. The extrinsic distance
We assume throughout the paper that $\varphi:P^{m}\longrightarrow N^{n}$ is an
isometric immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$
into a complete Riemannian manifold $N^{n}$ with a pole $o\in N$, (this is the
precise meaning we shall give to the word submanifold along the text) . Recall
that a pole is a point $o$ such that the exponential map
$\exp_{o}\colon T_{o}N^{n}\to N^{n}$
is a diffeomorphism. For every $x\in N^{n}-\\{o\\}$ we define
$r(x)=r_{o}(x)=\operatorname{dist}_{N}(o,x)$, and this distance is realized by
the length of a unique geodesic from $o$ to $x$, which is the radial geodesic
from $o$. We also denote by $r|_{P}$ or by $r$ the composition
$r\circ\varphi:P\to\mathbb{R}_{+}\cup\\{0\\}$. This composition is called the
extrinsic distance function from $o$ in $P^{m}$. The gradients of $r$ in $N$
and $r|_{P}$ in $P$ are denoted by $\nabla^{N}r$ and $\nabla^{P}r$,
respectively. Then we have the following basic relation, by virtue of the
identification, given any point $x\in P$, between the tangent vector fields
$X\in T_{x}P$ and $\varphi_{*_{x}}(X)\in T_{\varphi(x)}N$
(2.1) $\nabla^{N}r=\nabla^{P}r+(\nabla^{N}r)^{\bot},$
where $(\nabla^{N}r)^{\bot}(\varphi(x))=\nabla^{\bot}r(\varphi(x))$ is
perpendicular to $T_{x}P$ for all $x\in P$.
###### Definition 2.1.
Given $\varphi:P^{m}\longrightarrow N^{n}$ an isometric immersion of a
complete and connected Riemannian $m$-manifold $P^{m}$ into a complete
Riemannian manifold $N^{n}$ with a pole $o\in N$, we denote the extrinsic
metric balls of radius $t>0$ and center $o\in N$ by $D_{t}(o)$. They are
defined as the subset of $P$:
$D_{t}(o)=\\{x\in P:r(\varphi(x))<t\\}=\\{x\in P:\varphi(x)\in
B^{N}_{t}(o)\\}$
where $B^{N}_{t}(o)$ denotes the open geodesic ball of radius $t$ centered at
the pole $o$ in $N^{n}$. Note that the set $\varphi^{-1}(o)$ can be the empty
set.
###### Remark 2.2.
When the imersion $\varphi$ is proper, the extrinsic domains $D_{t}(o)$ are
precompact sets, with smooth boundary $\partial D_{t}(o)$. The assumption on
the smoothness of $\partial D_{t}(o)$ makes no restriction. Indeed, the
distance function $r$ is smooth in $N-\\{o\\}$ since $N$ is assumed to possess
a pole $o\in N$. Hence the composition $r|_{P}$ is smooth in $P$ and
consequently the radii $t$ that produce smooth boundaries $\partial D_{t}(o)$
are dense in $\mathbb{R}$ by Sard’s theorem and the Regular Level Set Theorem.
We now present the curvature restrictions which constitute the geometric
framework of our study.
###### Definition 2.3.
Let $o$ be a point in a Riemannian manifold $N$ and let $x\in N-\\{o\\}$. The
sectional curvature $K_{N}(\sigma_{x})$ of the two-plane $\sigma_{x}\in
T_{x}N$ is then called a $o$-radial sectional curvature of $N$ at $x$ if
$\sigma_{x}$ contains the tangent vector to a minimal geodesic from $o$ to
$x$. We denote these curvatures by $K_{o,N}(\sigma_{x})$.
### 2.2. Model spaces
Throughout this paper we shall assume that the ambient manifold $N^{n}$ has
its $o$-radial sectional curvatures $K_{o,N}(x)$ bounded from above by the
expression $K_{w}(r(x))=-w^{\prime\prime}(r(x))/w(r(x))$, which are precisely
the radial sectional curvatures of the $w$-model space $\,M^{m}_{w}\,$ we are
going to define.
###### Definition 2.4 (See [23], [10] and [6]).
A $w-$model $M_{w}^{m}$ is a smooth warped product with base
$B^{1}=[0,\Lambda[\,\subset\mathbb{R}$ (where $0<\Lambda\leq\infty$), fiber
$F^{m-1}=\mathbb{S}^{m-1}_{1}$ (i.e. the unit $(m-1)$-sphere with standard
metric), and warping function
$w\colon[0,\Lambda[\to\mathbb{R}_{+}\cup\\{0\\}$, with $w(0)=0$,
$w^{\prime}(0)=1$, and $w(r)>0$ for all $r>0$. The point $o_{w}=\pi^{-1}(0)$,
where $\pi$ denotes the projection onto $B^{1}$, is called the center point of
the model space. If $\Lambda=\infty$, then $o_{w}$ is a pole of $M_{w}^{m}$.
###### Proposition 2.5.
The simply connected space forms $\mathbb{K}^{m}(b)$ of constant curvature $b$
are $w-$models with warping functions
$w_{b}(r)=\begin{cases}\frac{1}{\sqrt{b}}\sin(\sqrt{b}\,r)&\text{if $b>0$}\\\
\phantom{\frac{1}{\sqrt{b}}}r&\text{if $b=0$}\\\
\frac{1}{\sqrt{-b}}\sinh(\sqrt{-b}\,r)&\text{if $b<0$}.\end{cases}$
Note that for $b>0$ the function $Q_{b}(r)$ admits a smooth extension to
$r=\pi/\sqrt{b}$.
###### Proposition 2.6 (See Proposition 42 in Chapter 7 of [23]. See also [6]
and [10]).
Let $M_{w}^{m}$ be a $w-$model with warping function $w(r)$ and center
$o_{w}$. The distance sphere $S^{w}_{r}$ of radius $r$ and center $o_{w}$ in
$M_{w}^{m}$ is the fiber $\pi^{-1}(r)$. This distance sphere has the constant
mean curvature $\eta_{w}(r)=\frac{w^{\prime}(r)}{w(r)}$.
On the other hand, the $o_{w}$-radial sectional curvatures of $M_{w}^{m}$ at
every $x\in\pi^{-1}(r)$ (for $r>0$) are all identical and determined by
$K_{o_{w},M_{w}}(\sigma_{x})=-\frac{w^{\prime\prime}(r)}{w(r)}.$
and the sectional curvatures of $M_{w}^{m}$ at every $x\in\pi^{-1}(r)$ (for
$r>0$) of the tangent planes to the fiber $S^{w}_{r}$ are also all identical
and determined by
$K(r)=K_{M_{w}}(\Pi_{S^{w}_{r}})=\frac{1-(w^{\prime}(r))^{2}}{w^{2}(r)}.$
###### Remark 2.7.
The $w-$model spaces are completely determined via $w$ by the mean curvatures
of the spherical fibers $S^{w}_{r}$:
$\,\eta_{w}(r)=w^{\prime}(r)/w(r)\,\quad,$
by the volume of the fiber
$\,\operatorname{Vol}(S^{w}_{r})\,=V_{0}\,w^{m-1}(r)\,\quad,$
and by the volume of the corresponding ball, for which the fiber is the
boundary
$\,\operatorname{Vol}(B^{w}_{r})\,=\,V_{0}\,\int_{0}^{r}\,w^{m-1}(t)\,dt\,\quad.$
Here $V_{0}$ denotes the volume of the unit sphere $S^{0,m-1}_{1}$, (we denote
in general as $S^{b,m-1}_{r}$ the sphere of radius $r$ in the real space form
$I\\!\\!K^{m}(b)$) . The latter two functions define the isoperimetric
quotient function as follows
$\,q_{w}(r)\,=\,\operatorname{Vol}(B^{w}_{r})/\operatorname{Vol}(S^{w}_{r})\quad.$
Besides the rôle of comparison controllers for the radial sectional curvatures
of $N^{n}$, we shall need two further purely intrinsic conditions on the model
spaces:
###### Definition 2.8.
A given $w-$model space $\,M^{m}_{w}\,$ is called balanced from below and
balanced from above, respectively, if the following weighted isoperimetric
conditions are satisfied:
$\displaystyle\text{Balance from below:}\quad q_{w}(r)\,\eta_{w}(r)$
$\displaystyle\geq 1/m\quad\text{for all}\quad r\geq 0\quad;$
$\displaystyle\text{Balance from above:}\quad q_{w}(r)\,\eta_{w}(r)$
$\displaystyle\leq 1/(m-1)\quad\text{for all}\quad r\geq 0\quad.$
A model space is called totally balanced if it is balanced both from below and
from above.
###### Remark 2.9.
If $\,K_{w}(r)\geq-\eta_{w}^{2}(r)\,$ then $\,M_{w}^{m}\,$ is balanced from
above. If $\,K_{w}(r)\leq 0\,$ then $\,M_{w}^{m}\,$ is balanced from below,
see the paper [16] for a detailed list of examples.
### 2.3. Hessian comparison analysis
The 2.nd order analysis of the restricted distance function $r_{|_{P}}$
defined on manifolds with a pole is governed by the Hessian comparison Theorem
A in [6].
This comparison theorem can be stated as follows, when one of the spaces is a
model space $M^{m}_{w}$, (see [27]):
###### Theorem 2.10 (See [6], Theorem A).
Let $N=N^{n}$ be a manifold with a pole $o$, let $M=M_{w}^{m}$ denote a
$w-$model with center $o_{w}$. Suppose that every $o$-radial sectional
curvature at $x\in N\setminus\\{o\\}$ is bounded from above by the
$o_{w}$-radial sectional curvatures in $M_{w}^{m}$ as follows:
$K_{o,N}(\sigma_{x})\,\leq\,-\frac{w^{\prime\prime}(r)}{w(r)}$
for every radial two-plane $\sigma_{x}\in T_{x}N$ at distance
$r=r(x)=\operatorname{dist}_{N}(o,x)$ from $o$ in $N$. Then the Hessian of the
distance function in $N$ satisfies
(2.2) $\displaystyle{{\rm Hess}\,}^{N}(r(x))(X,X)$ $\displaystyle\,\geq\,{{\rm
Hess}\,}^{M}(r(y))(Y,Y)$
$\displaystyle=\eta_{w}(r)\left(\|X\|^{2}-\langle\nabla^{M}r(y),Y\rangle_{M}^{2}\right)$
$\displaystyle=\eta_{w}(r)\left(\|X\|^{2}-\langle\nabla^{N}r(x),X\rangle_{N}^{2}\right)$
for every vector $X$ in $T_{x}N$ and for every vector $Y$ in $T_{y}M$ with
$\,r(y)=r(x)=r\,$ and
$\,\langle\nabla^{M}r(y),Y\rangle_{M}=\langle\nabla^{N}r(x),X\rangle_{N}\,$.
###### Remark 2.11.
As we mentioned in the Introduction, inequality (2.2) is true along the
geodesics emanating from $o$ and $o_{w}$ which are free of conjugate points of
$o$ and $o_{w}$, (see Remark 2.3 in [6]). Other relevant observation is that
the bound given in inequality (2.2) does not depend on the dimension of the
model space, (see Remark 3.7 in [27]).
We present now a technical result concerning the Hessian of a radial function,
namely, a function which only depends on the distance function $r$. For the
proof of this result, and the rest of the results in this subsection, we refer
to the paper [27].
###### Proposition 2.12.
Let $N=N^{n}$ be a manifold with a pole $o$. Let
$r=r(x)=\operatorname{dist}_{N}(o,x)$ be the distance from $o$ to $x$ in $N$.
Let $F:\mathbb{R}\longrightarrow\mathbb{R}$ a smooth function. Then, given
$q\in N$ and $X,Y\in T_{q}N$,
(2.3) $\begin{split}{{\rm Hess}\,}^{N}F\circ
r|_{q}(X,Y)&=F^{\prime\prime}(r)(\nabla^{N}r\otimes\nabla^{N}r)(X,Y)\\\
&+F^{\prime}(r){{\rm Hess}\,}^{N}r|_{q}(X,Y)\end{split}$
Now, let us consider a complete isometric immersion
$\varphi:P^{m}\longrightarrow N$ in a Riemannian ambient manifold $N^{n}$ with
pole $o$, and with distance function to the pole $r$. We are going to see how
the Hessians (in $P$ and in $N$), of a radial function defined in the
submanifold are related via the second fundamental form $B^{P}$ of the
submanifold $P$ in $N$. As before, we identify, given any $q\in P$, the
tangent vectors $X\in T_{q}P$ with $\varphi_{*_{q}}X\in T\varphi(q)N$ along
the next results.
###### Proposition 2.13.
Let $N^{n}$ be a manifold with a pole $o$, and let us consider an isometric
immersion $\varphi:P^{m}\longrightarrow N$. If $r|_{P}$ is the extrinsic
distance function, then, given $q\in P$ and $X,Y\in T_{q}P$,
(2.4) ${{\rm Hess}\,}^{P}r|_{q}(X,Y)={{\rm
Hess}\,}^{N}r|_{\varphi(q)}(X,Y)+\langle
B^{P}_{q}(X,Y),\nabla^{N}r|_{q}\rangle$
where $B^{P}_{q}$ is the second fundamental form of $P$ in $N$ at the point
$q\in P$.
Now, we apply Proposition 2.12 to $F\circ r|_{P}=F\circ r\circ\varphi$,
(considering $P$ as the Riemannian manifold where the function is defined), to
obtain an expression for ${{\rm Hess}\,}^{P}F\circ r|_{P}(X,Y)$ . Then, let us
apply Proposition above to ${{\rm Hess}\,}^{P}r|_{P}(X,Y)$, and we finally
get:
###### Proposition 2.14.
Let $N=N^{n}$ be a manifold with a pole $o$, and let $P^{m}$ denote an
immersed submanifold in $N$. Let $r|_{P}$ be the extrinsic distance function.
Let $F:\mathbb{R}\longrightarrow\mathbb{R}$ be a smooth function. Then, given
$q\in P$ and $X,Y\in T_{q}P$,
(2.5) $\begin{split}{{\rm Hess}\,}^{P}F\circ
r|_{q}(X,Y)&=F^{\prime\prime}(r(q))\langle\,\nabla^{N}r|_{q},X\,\rangle\langle\,\nabla^{N}r|_{q},Y\,\rangle\\\
&+F^{\prime}(r(q))\\{{{\rm Hess}\,}^{N}r|_{q}(X,Y)\\\
&+\langle\nabla^{N}r|_{q},B^{P}_{q}(X,Y)\,\rangle\,\\}\end{split}$
### 2.4. Comparison constellations and Isoperimetric inequalities
The isoperimetric inequalities satisfied by the extrinsic balls in minimal
submanifolds are on the basis of the monotonicity of the volume growth
function $f(r)=\frac{Vol(D_{r})}{Vol(B_{r}^{w})}$, a key result to prove
Theorem 1.1. We have the following theorem.
###### Theorem 2.15 (See [16], [17], [18], [19] and [26]).
Let $\varphi:P^{m}\longrightarrow N^{n}$ be a complete, proper and minimal
immersion in an ambient Riemannian manifold $N^{n}$ which possess at least one
pole $o\in N$. Let us suppose that the $o-$radial sectional curvatures of $N$
are bounded from above by the $o_{w}-$radial sectional curvatures of the
$w-$model space $M_{w}^{m}$:
$K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r(x))}{w(r(x))}\,\,\,\forall
x\in N$
and assume that $M^{m}_{w}$ is balanced from below. Let $D_{r}$ be an
extrinsic $r$-ball in $P^{m}$, with center at a pole $o\in N$ in the ambient
space $N$. Then:
(2.6) $\frac{\operatorname{Vol}(\partial
D_{r})}{\operatorname{Vol}(D_{r})}\geq\frac{\operatorname{Vol}(S^{w}_{r})}{\operatorname{Vol}(B^{w}_{r})}\,\,\,\,\,\textrm{for
all}\,\,\,r>0\quad.$
Furthermore, if $\varphi^{-1}(o)\neq\emptyset$,
(2.7)
$\operatorname{Vol}(D_{r})\geq\operatorname{Vol}(B^{w}_{r})\,\,\,\,\textrm{for
all}\,\,\,r>0\quad.$
Moreover, if equality in inequalities (2.6) or (2.7) holds for some fixed
radius $R$ and if the balance of $M^{m}_{w}$ from below is sharp
$q_{w}(r)\,\eta_{w}(r)\,>\,1/m\,$ for all $r$, then $D_{R}$ is a minimal cone
in the ambient space $N^{n}$, so if $N^{n}$ is the hyperbolic space
$\,\mathbb{H}^{n}(b)\,$, $\,b<0\,$, then $P^{m}\,$ is totally geodesic in
$\mathbb{H}^{n}(b)$.
If, on the other hand, the ambient space is $\mathbb{R}^{n}$ and equality in
inequalities (2.6) or (2.7) holds for all radius $r>0$ then $P^{m}$ is totally
geodesic in $\mathbb{R}^{n}$.
On the other hand, and also as a consequence of inequality (2.6), the volume
growth function $f(r)=\frac{Vol(D_{r})}{Vol(B_{r}^{w})}$ is a non-decreasing
function of $r$.
## 3\. Main Results
We prove in this section our main results, stablishing a set of conditions
that assures that our submanifolds are properly immersed and have finite
topology and bounding from below, under certain conditions, the number of its
ends.
###### Theorem 3.1.
Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric immersion of a
complete non-compact Riemannian $m$-manifold $P^{m}$ into a complete
Riemannian manifold $N^{n}$ with a pole $o\in N$ and satisfying
$\varphi^{-1}(o)\neq\emptyset$. Let us suppose that:
1. (1)
The $o-$radial sectional curvatures of $N$ are bounded from above by the
$o_{w}-$radial sectional curvatures of the $w-$model space $M_{w}^{m}$:
$K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r(x))}{w(r(x))}\,\,\,\forall
x\in N.$
2. (2)
The second fundamental form $B^{P}_{x}$ in $x\in P$ satisfies that, for
sufficiently large radius $R_{0}$, and for some constant $c\in]0,1[$:
$\|B^{P}_{x}\|\leq c\,\eta_{w}(\rho^{P}(x))\,\,\,\forall x\in
P-B^{P}_{R_{0}}(x_{o})$
where $\rho^{P}(x)$ denotes the intrinsic distance in $P$ from some fixed
$x_{o}\in\varphi^{-1}(o)$ to $x$.
3. (3)
For any $r>0$, $w^{\prime}(r)\geq d>0$ and $(\eta_{w}(r))^{\prime}\leq 0$.
Then $P$ is properly immersed in $N$ and it is $C^{\infty}$\- diffeomorphic to
the interior of a compact smooth manifold $\overline{P}$ with boundary.
###### Remark 3.2.
To show that $\varphi$ is proper, we shall use Theorem 2.10. Hence, it is
enough to assume that $o$ is a pole in the sense that there are not conjugate
points along any geodesic emanating from $o$, (see [5] and [30]). Therefore
our statement about the properness of the immersion includes ambient manifolds
$N$ that admit non-negative sectional curvatures, unlike the ambient manifold
in Theorem 1.2 in [25]. On the other hand, to prove the finiteness of the
topology of $P$ we need to assume that the ambient manifold $N$ posses a pole
as it is defined in [6], namely, a point $p\in N$ where $exp_{p}$ is a
$C^{\infty}$ diffeomorphism. However, although our ambient manifold must be
diffeomorphic to $\mathbb{R}^{n}$ in this case, (as in Theorem 1.2 in [25],
where the ambient space must be a Cartan-Hadamard manifold), also admits non-
negative sectional curvatures.
To complete the benchmarking with the hypotheses in [24] and [25], we are
going to compare the assumptions (2) and (3) in Theorem 3.1 with the notion of
“submanifold with tamed second fundamental form” introduced in [24]. It is
straightforward to check that if $\varphi:P^{m}\longrightarrow N^{n}$ is an
immersion of a complete Riemannian $m$\- manifold $P^{m}$ into a complete
Riemannian manifold $N^{n}$ with sectional curvatures $K_{N}\leq b\leq 0$, and
$P$ has tamed second fundamental form, in the sense of Definition 1.1 in [25],
then there exists $R_{0}>0$ such that for all $r\geq R_{0}$, the quantity
$a_{r}:=\operatorname{Sup}\\{\frac{w_{b}}{w_{b}^{\prime}}(\rho^{P}(x))\|B^{P}_{x}\|:x\in
P-B^{P}_{r}\\}$
satisfies $a_{r}<1$.
Hence, taking $r=R_{0}$, we have that for all $x\in P-B^{P}_{R_{0}}$, and some
$c\in(0,1)$,
$\|B^{P}_{x}\|\leq c\eta_{w_{b}}(\rho^{P}(x))\,.$
On the other hand, when $b\leq 0$, then $w_{b}^{\prime}(r)\geq 1>0\,\,\forall
r>0$ and $(\eta_{w_{b}}(r))^{\prime}\leq 0\,\,\forall r>0$.
All these observations make us consider our Theorem 3.1 as a natural and
slight generalization of assertions (b) and (c) of Theorem 1.2 in [25].
Observe that if we assume the properness of the immersion we obtain the
following version of Theorem 3.1, where we can remove the hypothesis about the
decrease of the function $\eta_{w}(r)$ because the norm of the second
fundamental form $\|B^{P}_{x}\|$ is bounded by the value of $\eta_{w}$ at
$r(x)$ instead of $\rho^{P}(x)$ :
###### Theorem 3.3.
Let $\varphi:P^{m}\longrightarrow N^{n}$ be an isometric and proper immersion
of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a complete
Riemannian manifold $N^{n}$ with a pole $o\in N$ and satisfying
$\varphi^{-1}(o)\neq\emptyset$. Let us suppose that, as in Theorem 3.1, the
$o-$radial sectional curvatures of $N$ are bounded from above as
$K_{o,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r(x))}{w(r(x))}\,\,\,\forall
x\in N\,,$
and for any $r>0$, $w^{\prime}(r)\geq d>0$. Let us assume moreover that the
second fundamental form $B^{P}_{x}$ in $x\in P$ satisfies that, for
sufficiently large radius $R_{0}$:
$\|B^{P}_{x}\|\leq c\,\eta_{w}(r(x))\,\,\,\forall x\in P-D_{R_{0}}(o)$
where $c$ a positive constant such that $c<1$ .
Then $P$ is $C^{\infty}$\- diffeomorphic to the interior of a compact smooth
manifold $\overline{P}$ with boundary.
We are going to see how to estimate the area growth function of $P$, defined
as $g(r)=\frac{Vol(\partial D_{r})}{Vol(S_{r}^{w})}$ by the number of ends of
the immersion $P$, $\mathcal{E}(P)$, when the ambient space $N$ is a radially
symmetric space.
###### Theorem 3.4.
Let $\varphi:P^{m}\longrightarrow M^{n}_{w}$ be an isometric and proper
immersion of a complete non-compact Riemannian $m$-manifold $P^{m}$ into a
model space $M^{n}_{w}$ with pole $o_{w}$. Suppose that
$\varphi^{-1}(o_{w})\neq\emptyset$, $m>2$ and moreover:
1. (1)
The norm of second fundamental form $B^{P}_{x}$ in $x\in P$ is bounded from
above outside a (compact) extrinsic ball $D_{R_{0}}(o)\subseteq P$ with
sufficiently large radius $R_{0}$ by:
$\|B^{P}_{x}\|\,\leq\,\frac{\epsilon(r(x))}{(w^{\prime}(r(x)))^{2}}\eta_{w}(r(x))\,\,\,\forall
x\in P-D_{R_{0}}$
where $\epsilon$ is a positive function such that $\epsilon(r)\to 0$ when
$r\to\infty$.
2. (2)
For $r$ sufficiently large, $w^{\prime}(r)\geq d>0$.
Then, for sufficiently large $r$, we have:
(3.1) $\frac{Vol(\partial
D_{r})}{Vol(S_{r}^{w})}\leq\frac{\mathcal{E}(P)}{\left(1-4\epsilon(r)\right)^{\frac{(m-1)}{2}}}$
where $\mathcal{E}(P)$ is the (finite) number of ends of $P$.
When we consider minimal immersions in the model spaces, we have the following
result, which is an inmediate corollary from the above theorem, and Theorem
2.15 in Section 2.
###### Theorem 3.5.
Let $\varphi:P^{m}\longrightarrow M^{n}_{w}$ be a complete non-compact, proper
and minimal immersion into a ballanced from below model space $M^{n}_{w}$ with
pole $o_{w}$. Suppose that $\varphi^{-1}(o_{w})\neq\emptyset$ and $m>2$. Let
us assume moreover the hypotheses (1) and (2) in Theorem 3.4.
Then
1. (1)
The (finite) number of ends $\mathcal{E}(P)$ is related with the (finite)
volume growth by
(3.2)
$1\leq\lim_{r\to\infty}\frac{Vol(D_{r})}{Vol(B_{r}^{w})}\leq\mathcal{E}(P)$
2. (2)
If $P$ has only one end, P is a minimal cone in $M_{w}^{n}$.
## 4\. Corollaries
As we have said in the Introduction, we have divided the list of results based
in Theorem 3.1 and in Theorem 3.4 in two series of corollaries. The first set
of consequences follows the line of Theorem 1.1 and Theorem 1.2, (which are in
fact the main representatives of these results) presenting upper bounds for
the volume and area growth of a complete and proper immersion in the real
space form $I\\!\\!K^{n}(b)$, ($b\leq 0$), in terms of the number of its ends.
In the second set of corollaries, are stated compactification theorems for
complete and proper immersions in $\mathbb{R}^{n}$, $\mathbb{H}^{n}(b)$ and
$\mathbb{H}^{n}(b)\times\mathbb{R}^{l}$.
The first of these corollaries constitutes a non-minimal version of Theorem
1.1:
###### Corollary 4.1.
Let $\varphi:P^{m}\longrightarrow\mathbb{H}^{n}(b)$ be a complete non-compact
and proper immersion with $m>2$. Let us suppose that for sufficiently large
$R_{0}$ and for all points $x\in P$ such that $r(x)>R_{0}$, (i.e. outside the
compact extrinsic ball $D_{R_{0}}(o)$ with $\varphi^{-1}(o)\neq\emptyset$),
$\|B^{P}_{x}\|\leq\frac{\delta(r(x))}{e^{2\sqrt{-b}\,r(x)}}$
where $r(x)=d_{\mathbb{H}^{n}(b)}(o,\varphi(x))$ is the (extrinsic) distance
in $\mathbb{H}^{n}(b)$ of the points in $\varphi(P)$ to a fixed pole
$o\in\mathbb{H}^{n}(b)$ and $\delta(r)$ is a smooth function such that
$\delta(r)\to 0$ when $r\to\infty$.Let $\\{t_{i}\\}_{i=1}^{\infty}$ be any
non-decreasing sequence such that $t_{i}\to\infty$ when $i\to\infty$. Then the
finite number of ends $\mathcal{E}(P)$ is related with the area growth of $P$
by:
$\liminf_{i\to\infty}\frac{\operatorname{Vol}(\partial
D_{t_{i}})}{\operatorname{Vol}(S_{t_{i}}^{b,m-1})}\leq\mathcal{E}(P)$
The corresponding non-minimal statement of Theorem 1.2 is:
###### Corollary 4.2.
Let $\varphi:P^{m}\longrightarrow\mathbb{R}^{n}$ be a complete non-compact and
proper immersion with $m>2$. Let us suppose that for sufficiently large
$R_{0}$ and for all points $x\in P$ such that $r(x)>R_{0}$, (i.e. outside the
compact extrinsic ball $D_{R_{0}}(o)$ with $\varphi^{-1}(o)\neq\emptyset$),
$\|B^{P}_{x}\|\leq\frac{\epsilon(r(x))}{r(x)}$
where $r(x)=d_{\mathbb{R}^{n}}(o,\varphi(x))$ is the (extrinsic) distance in
$\mathbb{R}^{n}$ of the points in $\varphi(P)$ to a fixed pole
$o\in\mathbb{R}^{n}$ and $\epsilon(r)$ is a smooth function such that
$\epsilon(r)\to 0$ when $r\to\infty$. Let $\\{t_{i}\\}_{i=1}^{\infty}$ be any
non-decreasing sequence such that $t_{i}\to\infty$ when $i\to\infty$. Then the
finite number of ends $\mathcal{E}(P)$ is related with the area growth by:
$\liminf_{i\to\infty}\frac{\operatorname{Vol}(\partial
D_{t_{i}})}{\operatorname{Vol}(S_{t_{i}}^{0,m-1})}\leq\mathcal{E}(P)$
Concerning the compactification results we have the following result given by
Bessa, Jorge and Montenegro in [24] and by Bessa and Costa in [25]:
###### Corollary 4.3.
Let $\varphi:P^{m}\longrightarrow I\\!\\!K^{n}(b)$ be a complete non-compact
immersion in the real space form $I\\!\\!K^{n}(b)$, ($b\leq 0$). Let us
suppose that for all points $x\in P\setminus B^{P}_{R_{0}}(o)$ (for
sufficientlty large $R_{0}$, where $o$ is a pole in $I\\!\\!K^{n}(b)$ such
that $\varphi^{-1}(o)\neq\emptyset$) :
$\|B^{P}_{x}\|\leq c\,h_{b}(\rho^{P}(x))$
where $\rho^{P}(x)$ is the (intrinsic) distance to a fixed
$x_{o}\in\varphi^{-1}(o)$ and $c$ is a positive constant such that $c<1$ and
$h_{b}(r)=\eta_{w_{b}}(r)=\begin{cases}\phantom{\sqrt{b}}1/r&\text{if
$b=0$}\\\ \sqrt{-b}\coth(\sqrt{-b}\,r)&\text{if $b<0$}\quad.\end{cases}$
is the mean curvature of the geodesic spheres in $I\\!\\!K^{n}(b)$. Then $P$
is properly immersed in $I\\!\\!K^{n}(b)$ and it is diffeomorphic to the
interior of a compact smooth manifold $\overline{P}$ with boundary.
Our last result concerns isometric immersions in
$\mathbb{H}^{n}(b)\times\mathbb{R}^{l}$:
###### Corollary 4.4.
Let $\varphi:P^{m}\longrightarrow\mathbb{H}^{n}(b)\times\mathbb{R}^{l}$ be a
complete non-compact immersion. Let us consider a pole
$o\in\mathbb{H}^{n}(b)\times\mathbb{R}^{l}$ such that
$\varphi^{-1}(o)\neq\emptyset$. Let us suppose that for all points $x\in
P\setminus B^{P}_{R_{0}}(x_{o})$, where $x_{o}\in\varphi^{-1}(o)$ and for
$R_{0}$ sufficiently large:
$\|B_{x}\|\leq\frac{c}{\rho^{P}(x)}\,\,.$
Here $\rho^{P}(x)$ denotes the intrinsic distance in $P$ from the fixed
$x_{o}\in\varphi^{-1}(o)$ to $x$ and $c$ is a positive constant such that
$c<1$. Then $P$ is properly immersed in
$\mathbb{H}^{n}(b)\times\mathbb{R}^{l}$ and it is diffeomorphic to the
interior of a compact smooth manifold $\overline{P}$ with boundary.
## 5\. Proof of Theorem 3.1
### 5.1. $P$ is properly immersed
Let us define the following function:
(5.1) $F(r):=\int_{0}^{r}w(t)dt$
Observe that $F$ is injective, because $F^{\prime}(r)=w(r)>0\,\,\,\forall
r>0$, and $F(r)\to\infty$ when $r\to\infty$. Applying Theorem 2.10 and
Proposition 2.14, we obtain, for all $x\in P$, and given $X\in T_{x}P$,
(5.2) $\displaystyle{{\rm Hess}\,}^{P}_{x}F(r)(X,X)$ $\displaystyle\geq
w^{\prime}(r(x))\|X\|^{2}+w(r(x))\langle B^{P}_{x}(X,X),\nabla^{N}r\rangle$
$\displaystyle\geq
w^{\prime}(r(x))\|X\|^{2}-w(r(x))\|B^{P}_{x}\|\,\,\|X\|^{2}$
By hypotesis there exist a geodesic ball $B^{P}_{r_{1}}(x_{0})$ in $P$, with
$r_{1}\geq R_{0}$, such that for any $x\in P\setminus B^{P}_{r_{1}}(x_{0})$,
$\|B^{P}_{x}\|\ \leq c\eta_{w}(\rho^{P}(x))$. On the other hand, as
$\eta_{w}(r)$ is non-increasing and $r(x)\leq\rho^{P}(x)$ because $\varphi$ is
isometric, we have $c\eta_{w}(\rho^{P}(x))\leq c\eta_{w}(r(x))$, so if $x\in
P\setminus B^{P}_{r_{1}}$ :
(5.3) $\displaystyle{{\rm Hess}\,}^{P}_{x}F(r)(X,X)$ $\displaystyle\geq
w^{\prime}(r(x))\|X\|^{2}-w(r)c\eta_{w}(\rho^{P}(x))\,\|X\|^{2}$
$\displaystyle\geq w^{\prime}(r(x))\|X\|^{2}\left(1-c\right)\geq
d\left(1-c\right)>0$
The above result implies that there exists $r_{1}\geq R_{0}$ such that $F\circ
r$ is a strictly convex function outside the geodesic ball in $P$ centered at
$x_{0}$, $B^{P}_{r_{1}}(x_{0})$. And hence, as $r(x)\leq\rho^{P}(x)$ for all
$x\in P$, (and therefore $B^{P}_{r_{1}}(x_{0})\subseteq D_{r_{1}}$), $F\circ
r$ is a strictly convex function outside the extrinsic disc $D_{r_{1}}$.
Let $\sigma:[0,\rho^{P}(x)]\to P^{m}$ be a minimizying geodesic from $x_{0}$
to $x$.
If we denote as $f=F\circ r$, let us define $h:\mathbb{R}\to\mathbb{R}$ as
$h(s)=F(r(\sigma(s)))=f(\sigma(s))$
Then,
(5.4)
$(f\circ\sigma)^{\prime}(s)=h^{\prime}(s)=\sigma^{\prime}(s)(f)=\langle\nabla^{P}f(\sigma(s)),\sigma^{\prime}(s)\rangle$
and hence,
(5.5) $\displaystyle(f\circ\sigma)^{\prime\prime}(s)$
$\displaystyle=h^{\prime\prime}(s)=\sigma^{\prime}(s)(\langle\nabla^{P}f(\sigma(s)),\sigma^{\prime}(s)\rangle)=\langle\nabla^{P}_{\sigma^{\prime}(s)}\nabla^{P}f(\sigma(s)),\sigma^{\prime}(s)\rangle$
$\displaystyle+\langle\nabla^{P}f(\sigma(s)),\nabla^{P}_{\sigma^{\prime}(s)}\sigma^{\prime}(s)\rangle=Hess^{P}_{\sigma(s)}f(\sigma(s))(\sigma^{\prime}(s),\sigma^{\prime}(s))$
We have from (5.3) that $(f\circ\sigma)^{\prime\prime}(\tau)={{\rm
Hess}\,}^{P}f(\sigma(\tau))(\sigma^{\prime},\sigma^{\prime})\geq d(1-c)$ for
all $\tau\geq r_{1}$ . And for $\tau<r_{1}$,
$(f\circ\sigma)^{\prime\prime}(\tau))\geq a=\inf_{x\in B^{P}_{r_{1}}}\\{{{\rm
Hess}\,}^{P}f(x)(\nu,\nu),\,|\nu|=1\\}$. Then
$\displaystyle(f\circ\sigma)^{\prime}(s)$ $\displaystyle=$
$\displaystyle(f\circ\sigma)^{\prime}(0)+\int_{0}^{s}(f\circ\sigma)^{\prime\prime}(\tau)d\tau$
$\displaystyle\geq$
$\displaystyle(f\circ\sigma)^{\prime}(0)+\int_{0}^{r_{1}}a\,d\tau+d\,\int_{r_{1}}^{s}(1-c)d\tau$
$\displaystyle\geq$
$\displaystyle(f\circ\sigma)^{\prime}(0)+a\,r_{1}+d\,(1-c)(s-r_{1})$
On the other hand, as
(5.7)
$\nabla^{P}f(\sigma(s))=\nabla^{P}F(r(\sigma(s)))=F^{\prime}(r(\sigma(s)))\nabla^{P}r|_{\sigma(s)}=w(r(\sigma(s)))\nabla^{P}r|_{\sigma(s)}$
then
$\nabla^{P}f(\sigma(0))=w(r(\sigma(0)))\nabla^{P}r|_{\sigma(0)}=w(0)\nabla^{P}r|_{\sigma(0)}=0$
so we have that
(5.8)
$(f\circ\sigma)^{\prime}(0)=\langle\nabla^{P}f(\sigma(0)),\sigma^{\prime}(0)\rangle=0$
We also have that $(f\circ\sigma)(0)=F(r(\sigma(0)))=F(0)=0$. Hence, applying
inequality (5.1),
(5.9)
$f(\sigma(s))=(f\circ\sigma)(0)+\int_{0}^{s}(f\circ\sigma)^{\prime}(\tau)d\tau\geq
ar_{1}s+d(1-c)\\{\frac{1}{2}s^{2}-r_{1}s\\}$
Therefore,
$\displaystyle F(r(x))$ $\displaystyle=$ $\displaystyle
f(x)=f(\sigma(\rho^{P}(x)))=\int_{0}^{\rho^{P}(x)}(f\circ\sigma)^{\prime}(s)\,ds$
$\displaystyle\geq$
$\displaystyle\int_{0}^{\rho^{P}(x)}a\,r_{1}+d\,(1-c)(s-r_{1})\,ds$
$\displaystyle=$ $\displaystyle a\,r_{1}\rho^{P}(x)\
+d\,(1-c)\left(\frac{\rho^{P}(x)^{2}}{2}-r_{1}\,\rho^{P}(x)\right)$
Hence, if $\rho^{P}\to\infty$ then $F(r(x))\to\infty$ and then, as $F$ is
strictly increasing, $r\to\infty$ so the immersion is proper.
### 5.2. $P$ has finite topology
We are going to see that $\nabla^{P}r$ never vanishes on $P\setminus
D_{r_{1}}$. To show this, we consider, as in the previous subsection, any
geodesic in $P$ emanating from the pole $o$, $\sigma(s)$. We have, using
inequality (5.1), that
(5.11)
$\langle\nabla^{P}f(\sigma(s)),\sigma^{\prime}(s)\rangle=(f\circ\sigma)^{\prime}(s)\geq
a\,r_{1}+d\,(1-c)(s-r_{1})>0\,\,\forall s>r_{1}$
Hence, as $\|\sigma^{\prime}(s)\|=1\,\,\forall s$, then
$\|\nabla^{P}f(\sigma(s))\|>0$ for all $s>r_{1}$. But we have computed
$\nabla^{P}f(\sigma(s))=w(r(\sigma(s)))\nabla^{P}r|_{\sigma(s)}$, so, as
$w(r)>0\,\,\forall r>0$, then $\|\nabla^{P}r|_{\sigma(s)}\|>0\,\,\forall
s>r_{1}$ and hence, $\nabla^{P}r|_{\sigma(s)}\neq 0\,\,\forall s>r_{1}$. We
have proved that $\nabla^{P}r$ never vanishes on $P\setminus B^{P}_{r_{1}}$,
so we have too that $\nabla^{P}r$ never vanishes on $P\setminus D_{r_{1}}$.
Let
$\phi:\partial D_{r_{1}}\times[r_{1},+\infty)\to P\setminus D_{r_{1}}$
be the integral flow of a vector field
$\frac{\nabla^{P}r}{\|\nabla^{P}r\|^{2}}$ with
$\phi(p,r_{1})=p\in\partial D_{r_{1}}$
It is obvious that $r(\phi(p,t))=t$ and
$\phi(\cdot,t):\partial D_{r_{1}}\to\partial D_{t}$
is a diffeomorphism. So $P$ has finitely many ends, and each of its ends is of
finite topological type.
In fact, applying Theorem 3.1 in [20], we conclude that, as the extrinsic
annuli $A_{r_{1},R}(o)=D_{R}(o)\setminus D_{r_{1}}(o)$ contain no critical
points of the extrinsic distance function $r:P\longrightarrow\mathbb{R}^{+}$,
then $D_{R}(o)$ is diffeomorphic to $D_{r_{1}}(o)$ for all $R\geq r_{1}$ and
hence the annuli $A_{r_{1},R}(o)$ are diffeomorphic to $\partial
D_{r_{1}}\times[r_{1},R]$.
###### Remark 5.1.
To show Theorem 3.3, we argue as in the beginning of the proof of Theorem 3.1:
with the same function $F(r)$ we obtain inequality (5.2). But now we have as
hypothesis that $\|B^{P}_{x}\|\leq c\,\eta_{w}(r(x))$, so we don’t need that
$\eta_{w}^{\prime}(r)\leq 0$ to get inequality (5.3).
## 6\. Proof of Theorem 3.4
We are going to see first that $P$ has finite topology. As $P$ is properly
immersed, we shall apply Theorem 3.3 and for that, it must be checked that
hypotheses in that theorem are acomplished. First, we have hypothesis (1) in
Theorem 3.3 because $N=M^{n}_{w}$. On the other hand, as $w^{\prime}(r)\geq
d>0\forall r>0$ and, for some $R_{0}$, we have that
$\|B^{P}_{x}\|\leq\frac{\epsilon(r(x))}{(w^{\prime}(r(x)))^{2}}\eta_{w}(r(x))\,\,\,\forall
x\in P-D_{R_{0}}$ where $\epsilon$ is a positive function such that
$\epsilon(r)\to 0$ when $r\to\infty$, hence
$0\leq\lim_{r\to\infty}\frac{\epsilon(r)}{(w^{\prime}(r))^{2}}\leq\lim_{r\to\infty}\frac{\epsilon(r)}{d^{2}}=0$.
Therefore, for some constant $c<1$, there exist $R_{0}$ such that
$\|B^{P}_{x}\|\leq c\eta_{w}(r(x))\,\,\,\forall x\in P-D_{R_{0}}$. Therefore,
as $\varphi:P\longrightarrow M^{n}_{w}$ is a proper immersion, we have by
Theorem 3.3 that $P$ has finite topological type and thus $P$ has finitely
many ends, each of finite topological type. Hence we have, in an analogous way
than in [1], and for $r_{1}\geq R_{0}$ as in Section 5:
(6.1) $P-D_{r_{1}}=\cup_{k=1}^{\mathcal{E}(P)}V_{k}$
where $V_{k}$ are disjoint, smooth domains in $P$. Along the rest of the
proof, we will work on each end $V_{k}$ separately. Let $V$ denote one element
of the family $\\{V_{k}\\}_{k=1}^{\mathcal{E}(P)}$, and, given a fixed radius
$t>r_{1}$, let $\partial V(t)$ denote the set $\partial V(t)=V\cap\partial
D_{t}=V\cap S^{w}_{t}$, where $S^{w}_{t}$ is the geodesic $t$-sphere in
$M^{n}_{w}$. This set is a hypersurface in $P^{m}$, with normal vector
$\frac{\nabla^{P}r}{\|\nabla^{P}r\|}$, and we are going to estimate its
sectional curvatures when $t\to\infty$.
Suppose that $e_{i},e_{j}$ are two orthonormal vectors of $T_{p}\partial V(t)$
on the point $p\in\partial V(t)$. Then the sectional curvature of the plane
expanded by $e_{i},e_{j}$ is, using Gauss formula:
(6.2) $\displaystyle K_{\partial V(t)}$
$\displaystyle(e_{i},e_{j})=K_{P}(e_{i},e_{j})+\langle B^{\partial
V-P}(e_{i},e_{i}),B^{\partial V-P}(e_{j},e_{j})\rangle$
$\displaystyle-\|B^{\partial
V-P}(e_{i},e_{j})\|^{2}=K_{N}(e_{i},e_{j})+\langle B^{\partial
V-P}(e_{i},e_{i}),B^{\partial V-P}(e_{j},e_{j})\rangle$
$\displaystyle-\|B^{\partial V-P}(e_{i},e_{j})\|^{2}+\langle
B^{P}(e_{i},e_{i}),B^{P}(e_{j},e_{j})\rangle-\|B^{P}(e_{i},e_{j})\|^{2}$
$\displaystyle\geq K_{N}(e_{i},e_{j})+\langle B^{\partial
V-P}(e_{i},e_{i}),B^{\partial V-P}(e_{j},e_{j})\rangle$
$\displaystyle-\|B^{\partial V-P}(e_{i},e_{j})\|^{2}-2\|B^{P}\|^{2}$
where $B^{\partial V-P}$ is the second fundamental form of $\partial V(t)$ in
$P$. But this second fundamental form is for two vector fields $X,Y$ in
$T\partial V(t)$:
(6.3) $\displaystyle B^{\partial V-P}(X,Y)$
$\displaystyle=\langle\nabla_{X}^{P}Y,\frac{\nabla^{P}r}{||\nabla^{P}r||}\rangle\frac{\nabla^{P}r}{||\nabla^{P}r||}=\langle\nabla_{X}^{P}Y,\nabla^{P}r\rangle\frac{\nabla^{P}r}{||\nabla^{P}r||^{2}}$
$\displaystyle=X(\langle
Y,\nabla^{P}r\rangle)\frac{\nabla^{P}r}{||\nabla^{P}r||^{2}}-\langle
Y,\nabla^{P}_{X}\nabla^{P}r\rangle\frac{\nabla^{P}r}{||\nabla^{P}r||^{2}}$
$\displaystyle=-{{\rm
Hess}\,}^{P}r(X,Y)\frac{\nabla^{P}r}{||\nabla^{P}r||^{2}}$
Then, since, for all $X,Y\in T_{p}M^{n}_{w}$
(6.4) ${{\rm Hess}\,}^{M^{n}_{w}}r(X,Y)=\eta_{w}(r)\langle X,Y\rangle-\langle
X,\nabla^{M^{n}_{w}}r\rangle\langle Y,\nabla^{M^{n}_{w}}r\rangle$
we have, (using the fact that $e_{i}$ are tangent to the fiber $S^{w}_{t}$,
and Proposition 2.6), that
(6.5) $K_{M^{n}_{w}}(e_{i},e_{j})=K(t)=\frac{1}{w^{2}(t)}-\eta^{2}_{w}(t)$
so for any $p\in\partial V(t)$ such that $t=r(p)$ is sufficiently large:
(6.6) $\displaystyle K_{\partial V(t)}(e_{i},e_{j})\geq$ $\displaystyle
K_{M^{n}_{w}}(e_{i},e_{j})+\frac{{{\rm Hess}\,}^{P}_{p}r(e_{i},e_{i}){{\rm
Hess}\,}^{P}_{p}r(e_{j},e_{j})}{||\nabla^{P}r||^{2}}$
$\displaystyle-\frac{{{\rm
Hess}\,}^{P}_{p}r(e_{i},e_{j})^{2}}{||\nabla^{P}r||^{2}}-2\|B^{P}\|^{2}$
$\displaystyle\geq
K(t)+\frac{\left(\eta_{w}(t)-\|B^{P}\|\right)^{2}-\|B^{P}\|^{2}}{||\nabla^{P}r||^{2}}-2\|B^{P}\|^{2}$
$\displaystyle\geq\eta^{2}_{w}(t)\left(1-2\frac{\|B^{P}\|}{\eta_{w}(t)}-2\left(\frac{\|B^{P}\|}{\eta_{w}(t)}\right)^{2}+\frac{K(t)}{\eta^{2}_{w}(t)}\right)$
$\displaystyle\geq\eta^{2}_{w}(t)\left(1-4\frac{\|B^{P}\|}{\eta_{w}(t)}+\frac{K(t)}{\eta^{2}_{w}(t)}\right)$
$\displaystyle=\eta^{2}_{w}(t)\left(1+\frac{K(t)}{\eta^{2}_{w}(t)}\right)\left(1-4\frac{\frac{\|B^{P}\|}{\eta_{w}(t)}}{1+\frac{K(t)}{\eta^{2}_{w}(t)}}\right)$
$\displaystyle\geq\frac{1}{w^{2}(t)}\left(1-4\|B^{P}\|w^{\prime}(t)w(t)\right)\geq\frac{1}{w^{2}(t)}\left(1-4\epsilon(t)\right)$
where we recall that, by hypothesis,
$\|B^{P}\|\leq\frac{\epsilon(t)}{(w^{\prime}(t))^{2}}\eta_{w}(t)$ for all
$t=r(x)>R_{0}$, and $\epsilon$ is a positive function such that
$\epsilon(r)\to 0$ when $r\to\infty$.
If we denote as $\delta(t)=\frac{1}{w^{2}(t)}\left(1-4\epsilon(t)\right)$ we
have for each $t$ sufficiently large that $K_{\partial
V(t)}(e_{i},e_{j})\geq\delta(t)$ holds everywhere on $\partial V(t)$ and
$\delta(t)$ is a positive constant. Then, the Ricci curvature of $\partial
V(t)$ is bounded from below, for these sufficiently large radius $t$ as
$Ricc_{\partial V(t)}(\xi,\xi)\geq\delta(t)(m-1)\|\xi\|^{2}>0\,\,\forall\xi\in
T\partial V(t)$
so, applying Myers’ Theorem $\partial V(t)$ is compact and has diameter
$d(\partial V(t))\leq\frac{\pi}{\sqrt{\delta(t)}}$ (see [30]). Applying on the
other hand Bishop’s Theorem, (see Theorem 6 in [2]), we obtain:
(6.7) $\displaystyle\operatorname{Vol}(\partial
V(t))\leq\frac{\operatorname{Vol}(S^{0,m-1}(1))}{\sqrt{\delta(t)^{m-1}}}$
and hence
(6.8) $\displaystyle\frac{\operatorname{Vol}(\partial
V(t))}{\operatorname{Vol}(S_{t}^{w})}\leq$
$\displaystyle\frac{1}{w(t)^{m-1}\sqrt{\delta(t)^{m-1}}}$
$\displaystyle=\frac{1}{\left(1-4\epsilon(t)\right)^{(m-1)/2}}$
Therefore, since for $t$ large enough $Vol(\partial
D_{t}(o))\leq\sum_{i=1}^{\mathcal{E}(P)}Vol(\partial V_{i}(t))$ where $V_{i}$
denotes each end of $P$ then:
(6.9) $\displaystyle\frac{\operatorname{Vol}(\partial
D_{t}(o))}{\operatorname{Vol}(S_{t}^{w})}\leq\frac{\mathcal{E}(P)}{\left(1-4\epsilon(t)\right)^{(m-1)/2}}$
## 7\. Proof of Theorem 3.5
To show assertion (1) we apply Theorem 2.15 and inequality (3.1) in Theorem
3.4 to obtain, for $r$ sufficiently large, (we suppose that
$\varphi^{-1}(o_{w})\neq\emptyset$, and take $o\in\varphi^{-1}(o_{w})$ in
order to have that
$\operatorname{Vol}(D_{r}(o))\geq\operatorname{Vol}(B^{w}_{r})$ for all $r>0$)
:
(7.1) $\displaystyle 1\leq$
$\displaystyle\frac{\operatorname{Vol}(D_{r}(o))}{\operatorname{Vol}(B^{w}_{r})}\leq\frac{\operatorname{Vol}(\partial
D_{r}(o))}{\operatorname{Vol}(S_{r}^{w})}$
$\displaystyle\leq\frac{\mathcal{E}(P)}{\left(1-4\epsilon(r)\right)^{(m-1)/2}}$
Moreover, we know (again using Theorem 2.15) that the volume growth function
is non-decreasing.
Therefore, taking limits in (7.1) when $r$ goes to $\infty$, we obtain:
(7.2)
$1\leq\lim_{r\to\infty}\frac{\operatorname{Vol}(D_{r}(o))}{\operatorname{Vol}(B^{w}_{r})}=\operatorname{Sup}_{r>0}\frac{\operatorname{Vol}(D_{r}(o))}{\operatorname{Vol}(B^{w}_{r})}\leq\mathcal{E}(P)$
Now, to prove assertion (2), we have, if $P$ has one end, that
(7.3)
$1\leq\operatorname{Sup}_{r>0}\frac{\operatorname{Vol}(D_{r}(o))}{\operatorname{Vol}(B^{w}_{r})}\leq
1$
Hence, as
$f(r)=\frac{\operatorname{Vol}(D_{r}(o))}{\operatorname{Vol}(B^{w}_{r})}$ is
non-decreasing, then $f(r)=1\,\,\forall r>0$, so we have equality in
inequality (2.6) for all $r>0$, and $P$ is a minimal cone, (see [17] for
details).
## 8\. Proof of Theorems 1.1 and 1.2 and the Corollaries
### 8.1. Proof of Theorem 1.1
We are going to apply Theorem 3.5. To do that, we must to check hypotheses (1)
and (2) in Theorem 3.4.
We have, in this case, that the ambient manifold is the hyperbolic space
$\mathbb{H}^{n}(b)$. Therefore all of its points are poles, so there exist at
least $o\in\mathbb{H}^{n}(b)$ such that $\varphi^{-1}(o)\neq\emptyset$. As it
is known, Hyperbolic space $\mathbb{H}^{n}(b)$ is a model space with
$w(r)=w_{b}(r)=\frac{1}{\sqrt{-b}}\sinh\sqrt{-b}r$ so
$w_{b}^{\prime}(r)=\cosh\sqrt{-b}r\geq 1\,\,\forall r>0$.
Therefore, hypothesis (2) in Theorem 3.4 is fulfilled in this context.
Concerning hypothesis (1), it is straightforward that
(8.1) $\displaystyle\|B^{P}_{x}\|$
$\displaystyle\leq\frac{\delta(r(x))}{e^{2\sqrt{-b}\,r(x)}}\leq\frac{\epsilon(r)\sqrt{-b}}{\sinh\sqrt{-b}r\cosh\sqrt{-b}r}$
$\displaystyle=\frac{\epsilon(r)}{\cosh^{2}\sqrt{-b}r}\sqrt{-b}\coth\sqrt{-b}r=\frac{\epsilon(r)}{(w_{b}^{\prime}(r))^{2}}\eta_{w_{b}}(r)$
where $\epsilon(r)=\frac{\delta(r(x))}{4\sqrt{-b}}$ goes to $0$ when $r$ goes
to $\infty$.
Hence, also hypothesis (1) in Theorem 3.4 is fulfilled so, applying inequality
(3.2) in Theorem 3.5, (because $P$ is minimal)
(8.2)
$1\leq\lim_{r\to\infty}\frac{Vol(D_{r})}{Vol(B_{r}^{w_{b}})}\leq\mathcal{E}(P)$
Finally, when $P$ has one end, then
$\lim_{r\to\infty}\frac{Vol(D_{r})}{Vol(B_{r}^{w_{b}})}=1$. Since $P$ is
minimal, by Theorem 2.15, $f(r)=\frac{Vol(D_{r})}{Vol(B_{r}^{w_{b}})}$ is a
monotone non-decreasing function, and, on the other hand, $f(r)\geq
1\,\,\forall r>0$ because inequality (2.7). Hence $f(r)=1\,\,\forall r>0$, so
$f^{\prime}(r)=0\,\,\forall r>0$. This last equality implies the equality in
inequality (2.6) for all $r>0$, (see [17] or [18] for details), and we apply
equality assertion in Theorem 2.15 to conclude that $P$ is totally geodesic in
$\mathbb{H}^{n}(b)$.
### 8.2. Proof of Theorem 1.2
In this case, we apply Theorem 3.5, being $M^{n}_{w}=\mathbb{R}^{n}$, i.e.,
being $w(r)=w_{0}(r)=r$, ($b=0$). Hence, $w_{0}^{\prime}(r)=1>0\,\,\forall
r>0$ and $\eta_{0}(r)=\frac{1}{r}$ and hypotheses (1) and (2) in this theorem
are trivially satisfied.
When $P$ has only one end we conclude as before that the volume growth
function is constant so we conclude equality in (2.6) for all radius $r>0$.
Hence $P$ is totally geodesic in $\mathbb{R}^{n}$ applying the corresponding
equality assertion in Theorem 2.15.
### 8.3. Proof of Corollary 4.1
We are considering now a complete and proper immersion in $\mathbb{H}^{n}(b)$,
as in Theorem 1.1, but $P$ is not necessarily minimal. In this setting
hypotheses (1) and (2) in Theorem 3.4 are fulfilled (as we have checked in the
proof above, without using minimality). Hence taking limits in (3.1) when we
consider an increasing sequence $\\{t_{i}\\}_{i=1}^{\infty}$ such that
$t_{i}\to\infty$ when $i\to\infty$, we have:
$\liminf_{i\to\infty}\frac{\operatorname{Vol}(\partial
D_{t_{i}})}{\operatorname{Vol}(S_{t_{i}}^{b,m-1})}\leq\mathcal{E}(P)$
### 8.4. Proof of Corollary 4.2
Hypotheses (1) and (2) in Theorem 3.4 are trivially satisfied and we argue as
in the proof of Corollary 4.1 to obtain the result.
### 8.5. Proof of Corollary 4.3
We apply Theorem 3.1. Our ambient manifold is $I\\!\\!K^{n}(b)$, ($b\leq 0$),
so hypothesis (1) about the bounds for the radial sectional curvature holds,
and as $w(r)=w_{b}(r)$ hence $w_{b}^{\prime}(r)\geq 1>0\,\,\forall r>0$ and
$\eta_{w_{b}}^{\prime}(r)\leq 0\,\,\forall r>0$. This means that hypothesis
(3) is fulfilled. Hypothesis (2) in Theorem 3.1 holds because
$\|B^{P}_{x}\|\leq c\,h_{b}(\rho^{P}(x))$
where $\rho^{P}(x)$ is the (intrinsic) distance to a fixed
$x_{o}\in\varphi^{-1}(o)$ and $c$ is a positive constant such that $c<1$.
### 8.6. Proof of Corollary 4.4
We apply again Theorem 3.1, having into account that the ambient space is the
Cartan-Hadamard manifold $\mathbb{H}^{n}(b)\times\mathbb{R}^{l}$ and the model
space used to compare is $\mathbb{R}^{m}$, with $w(r)=w_{0}(r)=r$.
## References
* [1] M. Anderson The compactification of a minimal submanifold by the Gauss Map., Preprint IEHS (1984).
* [2] I. Chavel Eigenvalues in Riemannian geometry, vol. 115 of Pure and Applied Mathematics, Academic Press Inc., Orlando, FL, 1984. Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk.
* [3] Q. Chen On the volume growth and the topology of complete minimal submanifolds of a Euclidean space J. Math. Sci. Univ. Tokyo 2 (1995), 657-669.
* [4] S. S. Chern & R. Osserman Complete minimal surfaces in euclidean $n$-space., J. d’Analyse Math. vol. 19, 15-34, (1967).
* [5] M. P. do Carmo Riemannian Geometry, Birkhauser, Boston Inc., 1992.
* [6] R. Greene and H. Wu Function theory on manifolds which possess a pole., Lecture Notes in Math., vol. 699, Springer-Verlag, Berlin and New York, 1979\.
* [7] R. Greene and H. Wu Gap theorems for noncompact Riemannian manifolds, Duke Math. J. 49 (1982), 731-756.
* [8] R. Greene and H. Wu, On a new gap phenomenon in riemannian geometry, Proc. Nac. Acad. Sci. USA. 79 (1982), 714-715.
* [9] R. E. Greene, P. Petersen & S. Zhu Riemannian Manifolds of Faster-Than-Quadratic Curvature Decay, Int. Math. Research Notices, 1994, No. 9.
* [10] A. Grigor’yan Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999), 135–249.
* [11] Luquésio P. Jorge & W. Meeks III The topology of complete minimal surfaces of finite total Gaussian curvature., Topology, vol. 22 (2), 203-221, (1983).
* [12] A. Kasue, & K. Sugahara Gap theorems for certain submanifolds of Euclidean spaces and hyperbolic space forms., Osaka J. Math. 24 (1987), 679-704.
* [13] A. Kasue Gap theorems for minimal submanifolds of Euclidean space., J. Math. Soc. Japan 38 (3) (1986), 473-492.
* [14] H. B. Lawson Lectures on minimal submanifolds., Monografias de Matemática, IMPA, Rio de Janeiro, Brazil.
* [15] S. Markvorsen On the mean exit time from a minimal submanifold., J. Diff. Geom. 29 (1989), 1–8
* [16] S. Markvorsen & V. Palmer Torsional rigidity of minimal submanifolds., Proc. London Math Soc.(3) 93 (2006) 253-272.
* [17] S. Markvorsen and V. Palmer The relative volume growth of minimal submanifolds., Archiv der Mathematik, 79 (2002), 507–514.
* [18] S. Markvorsen and V. Palmer Generalized isoperimetric inequalities for extrinsic balls in minimal submanifolds., Journal reine angew. Math., 551 (2002), 101–121.
* [19] S. Markvorsen and V. Palmer On the isoperimetric rigidity of extrinsic minimal balls., Differential Geometry and its Applications , 18 (2003), 47–54.
* [20] J. Milnor Morse theory, Lecture Notes in Math., 699, (1979), Springer Verlag, Berlin.
* [21] S. Muller & V. Sverak On surfaces of finite total curvature., J. Diff. Geometry, 42, 229-258 (1995).
* [22] G. De Oliveira Compactification of minimal submanifolds of hyperbolic space., Comm. Analysis and Geometry, 1 (1993), 1-29.
* [23] B. O’Neill Semi-Riemannian Geometry; With Applications to Relativity., Academic Press (1983).
* [24] G. Pacelli Bessa & L. Jorge & J. Fabio Montenegro Complete submanifolds of $\mathbb{R}^{n}$ with finite topology., Communications in Analysis and Geometry, ISSN 1019-8385, Vol. 15, 4, 2007 , 725-732
* [25] G. Pacelli Bessa & M. Silvana Costa .On Submanifolds With Tamed Second Fundamental Form., Glasgow Mathematical Journal, 51, 2009, 669-680
* [26] V. Palmer Isoperimetric inequalities for extrinsic balls in minimal submanifolds and their applications., J. London Math. Soc. (2) 60, 2 (1999), 607–616.
* [27] V. Palmer On deciding whether a submanifold is parabolic of hyperbolic using its mean curvature , Simon Stevin Transactions on Geometry, vol 1. 131-159, Simon Stevin Institute for Geometry, Tilburg, The Netherlands, 2010.
* [28] R. Osserman Global properties of minimal surfaces in $\mathbb{E}^{\,3}$ and $\mathbb{E}^{\,n}$., Ann. of Math. vol. 80, 340-364, (1964).
* [29] Q.H. Ruan New gap theorem on complete Riemannian manifolds., (2006). Retrieved from http://arxiv.org/abs/math/0605360
* [30] T. Sakai Riemannian Geometry, Translations of Mathematical Monographs, vol. 149, A.M.S.1996Addison-Wesley, Reading, MA 1990.
* [31] K. Shiohama Total curvature and minimal areas of complete open surfaces., Proc. Amer. Math. Soc, 94 num 2 (1985), 310-316.
* [32] R. Schoen Uniqueness, symmetry and embededness of minimal surfaces., J. Diff. Geometry, 18 (1983), 791-809.
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|
arxiv-papers
| 2011-12-17T09:17:29 |
2024-09-04T02:49:25.437393
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vicent Gimeno and Vicente Palmer",
"submitter": "Vicent Gimeno",
"url": "https://arxiv.org/abs/1112.4042"
}
|
1112.4155
|
# Mixing and CP-violation studies in charm decays at LHCb
On behalf of the LHCb collaboration
The University of Oxford
E-mail
###### Abstract:
Studies of charm physics with the 2010 LHCb data sample are presented. Time-
integrated searches for CP violation in $D^{+}\to K^{-}K^{+}\pi^{+}$ and
$D^{0}\to K^{-}K^{+},~{}\pi^{-}\pi^{+}$ are discussed.
## 1 Introduction
The charm sector is a promising place to probe for new physics effects. Mixing
is now well-established [1] at a level which is consistent with but at the
upper end of Standard Model (SM) expectations [2]. Three types of $C\\!P$
violation (CPV) are possible: in the decay amplitudes, in the mixing between
$D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$, and in the
interference between mixing and decay. The first is referred to as direct CPV,
and the second and third as indirect CPV. Only direct CPV is possible in
$D^{+}$ decays, due to the absence of mixing. In the SM indirect $C\\!P$
violation is expected to be small and direct $C\\!P$ violation in singly-
Cabibbo-suppressed modes such as those discussed below is naively expected to
be $\mathcal{O}(10^{-3})$ or less [3], though larger values cannot be excluded
from first principles [4]. In the presence of new physics the rate of $C\\!P$
violation could plausibly be enhanced to $\mathcal{O}(10^{-2})$. At the time
of the conference no evidence for CPV in charm had yet been found, though
first indications have since emerged in the 2011 LHCb data [5].
## 2 Search for CPV in $D^{+}\to K^{-}K^{+}\pi^{+}$
Direct $C\\!P$ violation arises when two different amplitudes with non-zero
relative weak and strong phases contribute to decays to the same final state.
In two-body decays this must imply contributions from different Feynmann
diagrams, such as from tree and penguin processes. In multi-body decays the
same mechanism exists, but in addition a rich variety of intermediate resonant
states can contribute to the decay, each naturally producing a different
strong phase with well-defined variation across the Dalitz plane. Thus, the
interference between these amplitudes can give rise to observable asymmetries
which change across the Dalitz plane.
We search for such asymmetries at LHCb [6] by comparing the Dalitz plot
distributions of $D^{+}\to K^{-}K^{+}\pi^{+}$ and its conjugate process
$D^{-}\to K^{+}K^{-}\pi^{-}$ (Fig. 1), applying a model-independent technique
of comparing the binned, normalized distributions. Normalizing the two Dalitz
plots to the same total number of events cancels any production asymmetry and
suppresses many systematic effects that are mainly expressed as an overall
efficiency asymmetry. The statistical technique used to test for consistency
between the $D^{+}$ and $D^{-}$ Dalitz plots, and to localize the asymmetry if
one is found, is based on the Miranda approach (see Ref. [7] and also Ref.
[8]). A variety of different binnings are used in order to test for different
manifestations of $CP$ violation.
Figure 1: Mass spectra and Dalitz plot. The mass spectra after selection for
(a) $K^{-}\pi^{+}\pi^{+}$ and (b) $K^{-}K^{+}\pi^{+}$ are shown, with the
signal and sideband mass windows indicated. For those candidates in the
$D^{+}\to K^{-}K^{+}\pi^{+}$ signal window, the Dalitz plot is shown on the
right.
Control modes are analysed to validate the method. The main tool is the
Cabibbo-favoured $D_{s}^{+}\to K^{-}K^{+}\pi^{+}$ control mode, which has the
same final state as the signal as well as similar kinematics and Dalitz plot
structure. As expected, no evidence of any asymmetry is found in this mode
(e.g. p-value of 34% for 25-bin adaptive binning), nor in the sidebands around
the $D^{+}$ mass window. In addition, the analysis is repeated for the
Cabibbo-favoured $D^{+}\to K^{-}\pi^{+}\pi^{+}$ mode. This is more sensitive
to systematic effects, since (a) the yield is ten times larger than that of
the signal mode, and (b) the kaon imbalance can induce momentum-dependent
detector efficiency asymmetries which would not be present in the signal mode.
Nonetheless, only weak indications of asymmetries are seen (e.g. p-value of
12% for 25-bin adaptive binning). Thus, systematic effects in the more robust
$D^{+}\to K^{-}K^{+}\pi^{+}$ signal mode are negligible. The final, unblinded
results are shown in Table 1: no evidence of $CP$ violation is found in the
2010 data. For further details, see Ref. [9, 5].
Binning | Bins | $\chi^{2}/{\rm ndf}$ | $p$-value (%)
---|---|---|---
Adaptive I | 25 | 32.0/24 | 12.7
Adaptive II | 106 | 123.4/105 | 10.6
Uniform I | 199 | 191.3/198 | 82.1
Uniform II | 530 | 519.5/529 | 60.5
Table 1: $\chi^{2}/{\rm ndf}$ and $p$-values for consistency with no CPV for
the $D^{+}\to K^{-}K^{+}\pi^{+}$ decay mode with four different binnings.
## 3 Search for CPV in $D^{0}\to K^{-}K^{+},~{}\pi^{-}\pi^{+}$
As discussed in Section 1, both direct and indirect CPV can contributed to the
time-integrated $CP$ asymmetry in these singly Cabibbo suppressed decays to
$CP$-even final states. The indirect $CP$ asymmetry is universal to a very
good approximation [10], although the measured value is affected by the
$D^{0}$ decay time acceptance of the experiment [11]. However, the direct $CP$
asymmetry in general varies between final states, and in the limit of U-spin
symmetry is equal and opposite between $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ [3].
Thus, the difference in time-integrated asymmetry between the two final
states, $\Delta A_{CP}$, is sensitive to direct CPV but has limited
sensitivity to indirect CPV:
$\Delta
A_{CP}=a^{\mathrm{dir}}_{C\\!P}(K^{-}K^{+})-a^{\mathrm{dir}}_{C\\!P}(\pi^{-}\pi^{+})+\frac{\Delta\langle
t\rangle}{\tau}\,a^{\mathrm{ind}}_{C\\!P},$
where $\frac{\Delta\langle t\rangle}{\tau}=0.10\pm 0.01$ is the difference in
normalized time acceptance for the two final states at LHCb,
$a^{\mathrm{dir}}_{C\\!P}(f)$ is the direct $CP$ asymmetry for final state
$f$, and $a^{\mathrm{ind}}_{C\\!P}$ is the indirect $CP$ asymmetry.
The observable $\Delta A_{CP}$ also has the advantage of being highly robust
against systematic effects. The measured (raw) asymmetry between $D^{0}\to f$
and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\to\bar{f}$, where the
initial flavour of the $D$ is established with a $D^{*+}\to
D^{0}\pi_{\mathrm{s}}$ tag, can be written at first order as:
$A_{\mathrm{RAW}}(f)\approx
A_{C\\!P}(f)\,+\,A_{\mathrm{D}}(f)\,+\,A_{\mathrm{D}}(\pi_{\mathrm{s}})\,+\,A_{\mathrm{P}}(D^{*+}),$
(1)
where $A_{CP}$, $A_{\mathrm{D}}$, and $A_{\mathrm{P}}$ are the relevant
physics, detector efficiency, and production asymmetries, respectively. Within
a local kinematic region, $A_{\mathrm{D}}(\pi_{\mathrm{s}})$ and
$A_{\mathrm{P}}(D^{*+})$ are independent of the $D^{0}$ decay mode and thus
cancel in the difference $\Delta A_{CP}$. Further,
$A_{\mathrm{D}}(K^{-}K^{+})$ and $A_{\mathrm{D}}(\pi^{-}\pi^{+})$ are zero by
construction, since the final state is spinless and self-conjugate. Thus, all
detector and production effects cancel in $\Delta A_{CP}$ at first order. To
ensure good behaviour at second order, the data are divided into 12 disjoint
kinematic bins, as well as being partitioned according to trigger conditions
and magnetic field polarity. Taking the weighted average of the individual
measurements, we obtain $\Delta A_{CP}=(-0.28\pm 0.70\pm 0.25)\%$, where the
first uncertainty is statistical and the second is systematic (taking into
account modeling of the lineshapes [0.06%], the $D^{0}$ mass window [0.20%],
multiple candidates [0.13%], and the kinematic binning [0.01%]). For further
details, see Ref. [12].
## 4 Conclusions and prospects
LHCb’s charm physics programme is off to a strong start. Several proof-of-
concept measurements have been made on the 2010 data sample of 38
$\mbox{\,pb}^{-1}$, and the first results on the much larger 2011 and 2012
data sets are now forthcoming.
## References
* [1] Heavy Flavor Averaging Group, D. Asner et al., Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton Properties, arXiv:1010.1589.
* [2] A. F. Falk, Y. Grossman, Z. Ligeti, Y. Nir and A. A. Petrov, The $D^{0}$-$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mass difference from a dispersion relation, Phys. Rev. D 69 (2004) 114021, [hep-ph/0402204].
* [3] See e.g. S. Bianco, F. L. Fabbri, D. Benson and I. Bigi, A Cicerone for the physics of charm, Riv. Nuovo Cim. 26N7 (2003) 1 [hep-ex/0309021]; M. Bobrowski, A. Lenz, J. Riedl and J. Rohrwild, How large can the SM contribution to $CP$ violation in $D^{0}$-$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing be?, JHEP 1003 (2010) 009 [arXiv:1002.4794]; Y. Grossman, A. L. Kagan and Y. Nir, New physics and $CP$ violation in singly Cabibbo suppressed $D$ decays, Phys. Rev. D 75 (2007) 036008 [hep-ph/0609178].
* [4] J. Brod, A. L. Kagan and J. Zupan, On the size of direct $CP$ violation in singly Cabibbo-suppressed $D$ decays, arXiv:1111.5000.
* [5] LHCb Collaboration, R. Aaij et al., Evidence for $CP$ violation in time-integrated $D^{0}\to h^{-}h^{+}$ decay rates, arXiv:1112.0938 (submitted to Phys. Rev. Lett.).
* [6] LHCb Collaboration, A. A. Alves, Jr. et al., The LHCb Detector at the LHC, JINST 3 (2008) S08005.
* [7] I. Bediaga, I. I. Bigi, A. Gomes, G. Guerrer, J. Miranda and A. C. d. Reis, On a $CP$ anisotropy measurement in the Dalitz plot, Phys. Rev. D 80 (2009) 096006 [arXiv:0905.4233].
* [8] BABAR Collaboration, B. Aubert et al., Search for $CP$ Violation in Neutral $D$ Meson Cabibbo-suppressed Three-body Decays, Phys. Rev. D 78 (2008) 051102 [arXiv:0802.4035].
* [9] LHCb Collaboration, R. Aaij et al., Search for CP violation in $D^{+}\to K^{-}K^{+}\pi^{+}$ decays, arXiv:1110.3970 (accepted by Phys. Rev. D).
* [10] A. L. Kagan and M. D. Sokoloff, On Indirect $CP$ Violation and Implications for $D^{0}$-$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ and $B^{0}_{s}$-$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mixing, Phys. Rev. D 80 (2009) 076008 [arXiv:0907.3917].
* [11] CDF Collaboration, T. Aaltonen et al., Measurement of $CP$-violating asymmetries in $D^{0}\to\pi^{+}\pi^{-}$ and $D^{0}\to K^{+}K^{-}$ decays at CDF, arXiv:1111.5023.
* [12] LHCb Collaboration, A search for time-integrated $CP$ violation in $D^{0}\rightarrow h^{+}h^{-}$ decays and a measurement of the $D^{0}$ production asymmetry, LHCb-CONF-2011-023.
|
arxiv-papers
| 2011-12-18T13:31:07 |
2024-09-04T02:49:25.449378
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Matthew Charles (for the LHCb Collaboration)",
"submitter": "Matthew Charles",
"url": "https://arxiv.org/abs/1112.4155"
}
|
1112.4318
|
# Geometric measure of quantum discord and the geometry of a class of two-
qubit states
Wei Song wsong1@mail.ustc.edu.cn(Corresponding~Author) School of Electronic
and Information Engineering, Hefei Normal University, Hefei 230061, China
Long-Bao Yu School of Electronic and Information Engineering, Hefei Normal
University, Hefei 230061, China Ping Dong School of Electronic and
Information Engineering, Hefei Normal University, Hefei 230061, China Da-
Chuang Li School of Electronic and Information Engineering, Hefei Normal
University, Hefei 230061, China Hefei National Laboratory for Physical
Sciences at the Microscale and Department of Modern Physics, University of
Science and Technology of China, Hefei 230026, China Ming Yang School of
Physics and Material Science, Anhui University, Hefei 230039, China Zhuo-
Liang Cao School of Electronic and Information Engineering, Hefei Normal
University, Hefei 230061, China
###### Abstract
We investigate the geometric picture of the level surfaces of quantum
entanglement and geometric measure of quantum discord(GMQD) of a class of
X-states, respectively. This pictorial approach provides us a direct
understanding of the structure of entanglement and GMQD. The dynamic evolution
of GMQD under two typical kinds of quantum decoherence channels is also
investigated. It is shown that there exists a class of initial states for
which the GMQD is not destroyed by decoherence in a finite time interval.
Furthermore, we establish a factorization law between the initial and final
GMQD, which allows us to infer the evolution of entanglement under the
influences of the environment.
###### pacs:
03.67.Mn, 03.65.Ud, 03.65.Yz
## I Introduction
Entanglement is regarded as an ingredient resource for performing almost all
quantum information processing tasksNeilsen:2000 . The situation started to
change until a computational model was presented named deterministic quantum
computation with one qubit(DQC1)Datta:2008 . Quantum discord was considered to
be the figure of merit for this model of quantum computation. Ever since,
quantum discord has attracted much attentionOllivier:2001 ; Henderson:2001 ;
Luo:2008 ; Modi:2010 ; Dakic:2010 ; Chen:2011 ; Shi:2011a ; Shi:2011b ;
Cornelio:2011 ; Ferraro:2010 ; Streltsov:2011 ; Piani:2011 ; Al-Qasimi:2011 ;
Bennett:2011 ; Galve:2011 ; Luo:2010 ; Hassan:2010 ; Yu:2011 ; Bylicka:2010 ;
Huang:2011 ; Fanchini:2011 ; Zhang:2011a ; Zhang:2011b ; Shabani:2009 ;
Bradler:2010 . Quantum discord is a measure of nonclassical correlations
between two subsystems of a quantum system. It quantifies how much a system
can be disturbed when people observe it to gather information. Such quantum
correlations may be present in separable states and have a non vanishing value
for almost all quantum statesFerraro:2010 . On the other hand, the quantum
discord can be used to indicate the quantum phase transitions better than
entanglement in certain physical systems at a finite temperature
Dillenschneider:2008 ; Werlang:2010 . In particular, two operational
interpretations of quantum discord have been proposed, one in thermodynamics
Zurek:2003 and the other from the information theoretic perspective through
the state merging protocol Cavalcanti:2011 ; Madhok:2011 . These results
establish the status of quantum discord as another important resource for
quantum informational processing tasks besides entanglement.
Originally, the first definition of quantum discord was given by Ollivier and
Zurek Ollivier:2001 and, independently, by Henderson and Vedral Henderson:2001
. The quantum discord of a composite system AB is defined by
$D_{A}\equiv\mathop{\min}\limits_{\left\\{{E_{i}^{A}}\right\\}}\sum\limits_{i}{p_{i}}S\left({\rho_{B|i}}\right)+S\left({\rho_{A}}\right)-S\left({\rho_{AB}}\right)$,
where $S\left({\rho_{AB}}\right)=Tr\left({\rho_{AB}\log_{2}\rho_{AB}}\right)$
is the von Neumann entropy and the minimum is taken over all positive operator
valued measures(POVMs) $\left\\{{E_{i}^{A}}\right\\}$ on the subsystem $A$
with $p_{i}=Tr\left({E_{i}^{A}\rho_{AB}}\right)$ being the probability of the
i-th outcome and
$\rho_{B|i}={{Tr_{A}\left({E_{i}^{A}\rho_{AB}}\right)}\mathord{\left/{\vphantom{{Tr_{A}\left({E_{i}^{A}\rho_{AB}}\right)}{p_{i}}}}\right.\kern-1.2pt}{p_{i}}}$
being the conditional state of subsystem $B$. In a more restrict sense, the
minimum is often taken over the von Neumann measurements. However, it is
notoriously difficult to compute because of the minimization taken over all
possible POVM, or von Neumann measurements. At present, there are only a few
analytical results including the Bell-diagonal states Luo:2008 , rank-2 states
Shi:2011b ; Cen:2011 and Gaussian states Giorda:2010 ; Adesso:2010 . In
addition, a simple algorithm to evaluate the quantum discord for two-qubit
X-states is proposed by Ali _et al._ Ali:2010 with minimization taken over
only a few cases. Unfortunately, their algorithm is valid only for a family of
X-statesChen:2011 ; Lu:2011a . Recently Shi _et al._ Shi:2011c present an
efficient method to solve this problem. For the general two-qubit states, the
evaluation of quantum discord remains a nontrivial task and only some lower
and upper bounds are available Yu:2011c . In order to avoid the difficulties
in minimization procedures a geometric view of quantum discord was introduced.
Generally, there are two versions of geometric measure of quantum
discord(GMQD). In the first version the concept of relative entropy is used as
a distance measure of correlations Modi:2010 . The second version is defined
by the Hilbert-Schmidt norm measure Dakic:2010 . The relative-entropy-based
discords have the drawback that their analytical expressions are known only
for certain limited classes of states. Below we only consider the second
version of GMQD. Especially, Dakic _et al_ Dakic:2012 show that the GMQD is
related to the fidelity of remote state preparation which provides an
operational meaning to GMQD. Formerly, this geometric measure of quantum
discord is defined by
$D_{A}^{g}=\mathop{\min}\limits_{\chi\in\Omega_{0}}\left\|{\rho-\chi}\right\|^{2}$,
where $\Omega_{0}$ denotes the set of zero-discord states and
$\left\|{X-Y}\right\|^{2}=Tr\left({X-Y}\right)^{2}$ is the square norm in the
Hilbert-Schmidt space. The subscript $A$ denotes that the measurement is taken
on the system $A$. An arbitrary two-qubit state can be written in Bloch
representation:
$\displaystyle\rho=\frac{1}{4}\left[{I\otimes
I+\sum\limits_{i}^{3}{\left({x_{i}\sigma_{i}\otimes
I+y_{i}I\otimes\sigma_{i}}\right)+\sum\limits_{i,j=1}^{3}{R_{ij}\sigma_{i}\otimes\sigma_{j}}}}\right]$
(1)
where $x_{i}=Tr\rho\left({\sigma_{i}\otimes
I}\right),y_{i}=Tr\rho\left({I\otimes\sigma_{i}}\right)$ are components of the
local Bloch vectors, $\sigma_{i},i\in\left\\{{1,2,3}\right\\}$ are the three
Pauli matrices, and $R_{ij}=Tr\rho\left({\sigma_{i}\otimes\sigma_{j}}\right)$
are components of the correlation tensor. For the two-qubit case, the zero-
discord state is of the form
$\chi=p_{1}\left|{\psi_{1}}\right\rangle\left\langle{\psi_{1}}\right|\otimes\rho_{1}+p_{2}\left|{\psi_{2}}\right\rangle\left\langle{\psi_{2}}\right|\otimes\rho_{2}$,
where
$\left\\{{\left|{\psi_{1}}\right\rangle,\left|{\psi_{2}}\right\rangle}\right\\}$
is a single-qubit orthonormal basis. Then an analytic expression of the GMQD
is given by Dakic:2010 :
$\displaystyle
D_{A}^{g}\left(\rho\right)=\frac{1}{4}\left({\left\|x\right\|^{2}+\left\|R\right\|^{2}-k_{\max}}\right)$
(2)
where $x=\left({x_{1},x_{2},x_{3}}\right)^{T}$ and $k_{\max}$ is the largest
eigenvalue of matrix $K=xx^{T}+RR^{T}$.
Figure 1: (Color online). The geometry of the set of valid states with
different $r$ and $s$, repectively.
(a)$r$=$s$=0.3,(b)$r$=$s$=0.5,(c)$r$=0.4,$s$=0.1
In this paper we shall investigate the level surfaces of entanglement and GMQD
for a class of two-qubit states using the geometric picture presented in
Lang:2010 . It is well known that the set of Bell-diagonal states for two
qubits can be depicted as a tetrahedron in three dimensions in Bloch
representation Horodecki:2009 . Analogous to entanglement, Lang and Caves have
depicted the level surfaces of quantum discord for Bell-diagonal states. More
recently, Girolami and Adesso Girolami:2011 and independently Batle _et al._
Batle:2011 provided numerical evidence, from which one can infer that discord
and GMQD may be different. It is thus worth investigating the GMQD from the
geometric picture. In this sense, our research provides a direct understanding
of the structure of GMQD. We consider a class of X-states that the Bloch
vectors are $z$ directional, which include Bell-diagonal states as a special
case. We show that the level surface of GMQD is very different from the
quantum discord. On the other hand, the dynamics of quantum discord has
attracted much attention due to the inevitable interaction with environment.
The dynamic behavior of GMQD is also investigated under two typical kinds of
decoherence channels. We find a class of states for which the GMQD is not
destroyed by decoherence in a finite time interval. Interestingly, we also
obtain a factorization law for GMQD which allows us to infer the evolution of
entanglement under the influences of the environment, _e.g._ the phase damping
channel(PDC), and the depolarizing channel(DPC).
The paper is organized as follows. In Sec.II, we present the geometric picture
of the level surfaces of quantum entanglement and GMQD of a class of X-states
with $z$ directional Bloch vectors, respectively. It is shown that the surface
of constant GMQD varies with the local Bloch vectors. In Sec.III, we
investigate the dynamic evolution of GMQD under quantum decoherence channels,
and obtain analytic results for two typical kinds of quantum decoherence
channels. A summary is given in Sec.IV.
## II geometrical picture of entanglement and GMQD
For analytical simplicity, we consider the following two-qubit X states:
$\displaystyle\rho=\frac{1}{4}\left[{I\otimes I+\textbf{r}\cdot\sigma\otimes
I+I\otimes\textbf{s}\cdot\sigma+\sum\limits_{i=1}^{3}{c_{i}\sigma_{i}\otimes\sigma_{i}}}\right]$
(3)
where we choose the Bloch vectors as $z$ directional with
$\textbf{r}=\left({0,0,r}\right)$,$\textbf{s}=\left({0,0,s}\right)$. The GMQD
can be calculated explicitly for this state, thus allowing us to get analytic
results. If $\textbf{r}=\textbf{s}=0$, $\rho$ is reduced to the two-qubit
Bell-diagonal states. Horodecki have shown that Bell-diagonal states belongs
to a tetrahedron with vertices $(1,-1,1),(-1,1,1),(1,1,-1)$,and $(-1,-1,-1)$
in the Bloch representation. From the positivity of the eigenvalues of $\rho$
in Eq.(3), we have
$\displaystyle
0\leqslant\frac{{\text{1}}}{{\text{4}}}\left({1-\sqrt{r^{2}-2rs+s^{2}+c_{1}^{2}+2c_{1}c_{2}+c_{2}^{2}}-c_{3}}\right)\leqslant
1,$ $\displaystyle
0\leqslant\frac{{\text{1}}}{{\text{4}}}\left({1+\sqrt{r^{2}-2rs+s^{2}+c_{1}^{2}+2c_{1}c_{2}+c_{2}^{2}}-c_{3}}\right)\leqslant
1,$ $\displaystyle
0\leqslant\frac{{\text{1}}}{{\text{4}}}\left({1-\sqrt{r^{2}+2rs+s^{2}+c_{1}^{2}-2c_{1}c_{2}+c_{2}^{2}}+c_{3}}\right)\leqslant
1,$ $\displaystyle
0\leqslant\frac{{\text{1}}}{{\text{4}}}\left({1+\sqrt{r^{2}+2rs+s^{2}+c_{1}^{2}-2c_{1}c_{2}+c_{2}^{2}}+c_{3}}\right)\leqslant
1$
For fixed parameters $r$ and $s$, the above inequalities become a three-
parameter set, whose geometry can be depicted in the three dimensional
correlation state space. In Fig.1 we plot the physical region with different
$r$ and $s$, respectively. Fig.1 shows that physical regions of the state
$\rho$ shrink with larger $r$ and $s$. We plot in Fig.2 the level surfaces of
constant concurrence with fixed $r$ and $s$ for three cases. Here, we choose
concurrence to measure entanglement which is defined as
$C=\max\left\\{{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}}\right\\}$,
where the $\lambda_{i}$ are, in decreasing order, the square roots of the
eigenvalues of the matrix
$\rho\sigma_{y}\otimes\sigma_{y}\rho^{*}\sigma_{y}\otimes\sigma_{y}$ where
$\rho^{*}$ is the complex conjugate of $\rho$. As shown in Fig.2, the level
surfaces of constant concurrence for the state $\rho$ defined in Eq.(3)
consist of four discrete pieces, and the areas decrease when the concurrence
increases.An extremal case is the four vertices $(1,-1,1),(-1,1,1),(1,1,-1)$,
and $(-1,-1,-1)$ of the tetrahedron corresponding to the four Bell states with
maximal concurrence. Finally, we investigate the GMQD from the geometric
picture. For the state $\rho$, the GMQD can be calculated in the method
presented in Ref. Lu:2010c By introducing a matric $\mathcal{R}$ defined by
$\mathcal{R}=\left({{\begin{array}[]{*{20}c}1&{y^{T}\hfill}\\\ x&R\hfill\\\
\end{array}}}\right)$ (5)
and $3\times 4$ matric $\mathcal{{R^{\prime}}}$ through deleting the first row
of $\mathcal{R}$, then the GMQD is given by
$\displaystyle
D_{A}^{g}\left(\rho\right)=\frac{1}{4}\left[{\left({\sum\limits_{k}{\lambda_{{}_{k}}^{2}}}\right)-\mathop{\max}\limits_{k}\lambda_{{}_{k}}^{2}}\right]$
(6)
where ${\lambda_{k}}$ is the singular values of $\mathcal{R^{\prime}}$. For
the two-qubit state $\rho$ in Eq.(3), we obtain
$\displaystyle
D_{A}^{g}\left(\rho\right)=\frac{1}{4}\left({c_{1}^{2}+c_{2}^{2}+c_{3}^{2}+r^{2}-{\text{Max}}\left({c_{1}^{2},c_{2}^{2},c_{3}^{2}+r^{2}}\right)}\right)$
(7)
The geometric picture is depicted in terms of the constant GMQD in Fig.3. From
these plots one can see that the shape of the constant GMQD is quite different
from quantum discord. It also shows different shapes for different local Bloch
vectors. The constant surfaces are cut off by the physical region of state
$\rho$. For small discord the surface is continuous, and it becomes discrete
pieces for larger discord. At the four vertices
$(1,-1,1),(-1,1,1),(1,1,-1)$,and $(-1,-1,-1)$ of the tetrahedron the GMQD
reaches its maximal value. Furthermore, one can see that GMQD is neither
concave nor convex as shown in Fig.3.
Figure 2: (Color online). Surfaces of constant concurrence. (a)$r$=$s$
=0.3,$C(\rho)$=0.03,(b)$r$=$s$=0.5,$C(\rho)$=0.35,(c)$r$=0.4,$s$=0.1,$C(\rho)$=0.03
Figure 3: (Color online). Surfaces of constant GMQD. (a)$r$=$s$
=0.3,$D(\rho)$=0.03,(b)$r$=$s$=0.5,$D(\rho)$=0.35,(c)$r$=0.4,$s$=0.1,$D(\rho)$=0.08
## III Dynamics of GMQD under local decoherence channels
In this section we consider the state affected by the action of two
independent channels and calculate the GMQD analytically. The dynamics of
quantum discord has been investigated in both Markovian and non-Markovian
environments and has been demonstrated experimentally Werlang:2009b ;
Maziero:2009 ; Wang:2010 ; Mazzola:2010 ; Li:2011 ; Fanchini:2010 ;
Auccaise:2011 ; Xu:2010a ; Xu:2010b . It has been shown that the behaviors of
quantum discord and GMQD may be different. It is thus desirable to consider
the evolution of GMQD under different decoherence channels. Here, we consider
two typical kinds of decoherence channels: the phase damping channel(PDC), and
the depolarizing channel(DPC). To calculate the dynamics of GMQD, we turn to
the Heisenberg picture to describe the quantum channels. In order to obtain
the analytic expressions of GMQD of the state subject to local decoherence
channels, we need to calculate the expection matrix $\mathcal{R}$. In the
Heisenberg picture Lu:2010c ; Wang:2010d , the expectation matrix
$\mathcal{R}$ is given by
$\displaystyle\mathcal{R}_{ij}=\left({M_{A}\mathcal{R}_{0}M_{B}^{T}}\right)_{ij}$
(8)
where $\mathcal{R}_{0}=Tr\left({\sigma_{i}\otimes\sigma_{j}\rho_{0}}\right)$,
$i\in\left\\{{0,1,2,3}\right\\}$, $\rho_{\text{0}}$ is the initial state, and
$M_{A\left(B\right)}$ is the transmission matrix of each local channel. For
simplicity, we choose the local channels to be identical. In this case, the
transmission matrices can be written as
$M_{PDC}=\left({{\begin{array}[]{*{20}c}1&0&0&0\hfill\\\ 0&1-p&0&0\hfill\\\
0&0&1-p&0\hfill\\\ 0&0&0&1\hfill\\\ \end{array}}}\right),$ (9)
$M_{DPC}=\left({{\begin{array}[]{*{20}c}1&0&0&0\hfill\\\ 0&1-p&0&0\hfill\\\
0&0&1-p&0\hfill\\\ 0&0&0&1-p\hfill\\\ \end{array}}}\right),$ (10)
For state (3),
$\mathcal{R}_{0}=\left\\{{\left\\{{1,0,0,s}\right\\},\left\\{{0,c_{1},0,0}\right\\},\left\\{{0,0,c_{2},0}\right\\}}\right.$
$\left.{\left\\{{r,0,0,c_{3}}\right\\}}\right\\}$. According to the above
formula, we have
$\displaystyle
D_{PDC}^{g}=\frac{1}{4}\left[{\left({1-p}\right)^{4}c_{1}^{2}+\left({1-p}\right)^{4}c_{2}^{2}+r^{2}+c_{3}^{2}}\right.$
$\displaystyle\left.{-\max\left\\{{\left({1-p}\right)^{4}c_{1}^{2},\left({1-p}\right)^{4}c_{2}^{2},r^{2}+c_{3}^{2}}\right\\}}\right]$
(11)
$\displaystyle
D_{DPC}^{g}=\frac{1}{4}\left[{\left({1-p}\right)}\right.^{4}c_{1}^{2}+\left({1-p}\right)^{4}c_{2}^{2}$
$\displaystyle+\left({1-p}\right)^{2}r^{2}+\left({1-p}\right)^{4}c_{3}^{2}$
$\displaystyle-\max\left\\{{\left({1-p}\right)^{4}}\right.c_{1}^{2},\left({1-p}\right)^{4}c_{2}^{2}$
$\displaystyle\left.{\left.{\left({1-p}\right)^{2}r^{2}+\left({1-p}\right)^{4}c_{3}^{2}}\right\\}}\right]$
(12)
For some Bell-diagonal states, it has been shown that quantum discord is not
destroyed by decoherence for some finite time interval Mazzola:2010 . A
natural question arises: Whether such a phenomena exists for GMQD? We consider
the state in Eq.(3) undergoes two identical PDCs. In this case,
$p=1-\exp(-\gamma t)$, where $\gamma$ is the phase damping rate. For
$c_{1}=0,c_{2}^{2}>r^{2}+c_{3}^{2}$, suppose ${p_{1}}$ satisfies the equation
$\left({1-p_{1}}\right)^{4}c_{2}^{2}=r^{2}+c_{3}^{2}$. If ${p<p_{1}}$, from
Eq.(11) we have $D_{PDC}^{g}=\frac{1}{4}\left({r^{2}+c_{3}^{2}}\right)$ which
is independent of time. Therefore, we conclude that for a finite time interval
the GMQD does not decay despite the presence of local phase damping noises. It
is directly seen that such a phenomena also exists for the case
$c_{2}=0,c_{1}^{2}>r^{2}+c_{3}^{2}$. These results show that GMQD remains
intact under the action of some special kinds of quantum channels. It should
be noted that similar phenomenon have also been noticed by Karpat _et al_
Karpat:2011 for the qubit-qutrit systems. In the geometric picture, this
behavior corresponds to the state evolving along a straight line in the
constant GMQD tube until it enconters another constant GMQD tube.
Hitherto, we have only considered the time evolution of GMQD under PDC or DPC
described by Eq.(11) and Eq.(12). Next, we want to derive a more general
result on GMQD relating the initial and final state of GMQD. Inspired by the
famous factorization law for entanglement decay derived by Konrad _et al._
Konrad:2008 , we find an analogous factorization law between the initial and
final GMQD of the class of two-qubit states defined in Eq.(3) subject to two
different local decoherence channels. We state our result as the following
theorem.
Theorem. Consider the class of X-states defined in Eq.(3), with each qubit
being subject to the local decoherence channels, _i.e._ the phase damping
channel(PDC) or the depolarizing channel(DPC). The time evolution of GMQD
satisfies
$\displaystyle
D^{g}\left[{\left({\$_{1}\otimes\$_{2}}\right)\rho(t)}\right]\geqslant
2D^{g}\left[{\left({\$_{1}\otimes\$_{2}}\right)\left|{\beta_{i}}\right\rangle\left\langle{\beta_{i}}\right|}\right]D^{g}\left[{\rho}(0)\right].$
(13)
where the local decoherence channels are represented by ${\$_{1}}$ and
${\$_{2}}$, $\rho_{0}$ is the initial state and
${\left|{\beta_{i}}\right\rangle},i\in\left\\{{1,2,3,4}\right\\}$ denotes one
of the four Bell states.
Proof. First we consider the state $\rho(0)$ is effected by the action of two
identical local PDCs, the time evolution of GMQD is given by Eq.(11). For
convenience we divide the proof into three cases.
Case 1. $r^{2}+c_{3}^{2}\geqslant\left\\{{c_{1}^{2},c_{2}^{2}}\right\\}$. In
this case, it is easy to show that the inequality becomes equality.
Case 2. $c_{1}^{2}\geqslant c_{2}^{2}\geqslant r^{2}+c_{3}^{2}$. First, using
Eq.(7) we have
$D^{g}\left({\rho(0)}\right)=\frac{1}{4}\left({c_{2}^{2}+r^{2}+c_{3}^{2}}\right)$.
Suppose ${p_{0}}$ satisfies the equation
$\left({1-p_{0}}\right)^{4}c_{1}^{2}=r^{2}+c_{3}^{2}$. If $p\leqslant p_{0}$,
the GMQD of the state $\rho$ is given by
$\displaystyle D_{{}^{PDC}}^{g}\left(\rho(0)\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[{\left({1-p}\right)^{4}c_{2}^{2}+r^{2}+c_{3}^{2}}\right]$
$\displaystyle\geqslant$
$\displaystyle\frac{1}{4}\left[{\left({1-p}\right)^{4}c_{2}^{2}+\left({1-p}\right)^{4}\left({r^{2}+c_{3}^{2}}\right)}\right]$
$\displaystyle=$
$\displaystyle\left({1-p}\right)^{4}D^{g}\left({\rho(0)}\right)$
$\displaystyle=$ $\displaystyle
2D^{g}\left[{\left({\$_{PDC}\otimes\$_{PDC}}\right)\left|{\beta_{i}}\right\rangle\left\langle{\beta_{i}}\right|}\right]D^{g}\left({\rho(0)}\right)$
If $p>p_{0}$, then
$\displaystyle D_{{}^{PDC}}^{g}\left(\rho(t)\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[{\left({1-p}\right)^{4}c_{1}^{2}+\left({1-p}\right)^{4}c_{2}^{2}}\right]$
$\displaystyle\geqslant$
$\displaystyle\frac{1}{4}\left[{\left({1-p}\right)^{4}c_{2}^{2}+\left({1-p}\right)^{4}\left({r^{2}+c_{3}^{2}}\right)}\right]$
$\displaystyle=$
$\displaystyle\left({1-p}\right)^{4}D^{g}\left({\rho(0)}\right)$
$\displaystyle=$ $\displaystyle
2D^{g}\left[{\left({\$_{PDC}\otimes\$_{PDC}}\right)\left|{\beta_{i}}\right\rangle\left\langle{\beta_{i}}\right|}\right]D^{g}\left({\rho(0)}\right)$
Case 3. $c_{1}^{2}\geqslant r^{2}+c_{3}^{2}\geqslant c_{2}^{2}$. For
$p\leqslant p_{0}$ or $p>p_{0}$, the proof is similar to case 2.
By simply exchanging $c_{1}$ and $c_{2}$ we can verify the above relations for
the cases $c_{2}^{2}\geqslant c_{1}^{2}\geqslant r^{2}+c_{3}^{2}$ and
$c_{2}^{2}\geqslant r^{2}+c_{3}^{2}\geqslant c_{1}^{2}$. For the case of two
identical local DPCs, one can prove the above results in the same way as for
PDCs. In the following, we consider the first qubit is subject to the PDC and
the second qubit is subject to the DPC. The expectation matrix $\mathcal{R}$
can be calculated according to the formula
$\mathcal{R}=M_{A}\mathcal{R}_{0}M_{B}^{T}$, where $M_{A\left(B\right)}$ is
the transformation matrix of PDC(DPC). Thus, the GMQD is given by
$\displaystyle
D^{g}\left({\rho\left(t\right)}\right)=\frac{1}{4}\left[{\left({1-p}\right)}\right.^{4}c_{1}^{2}+\left({1-p}\right)^{4}c_{2}^{2}+r^{2}+\left({1-p}\right)^{2}c_{3}^{2}$
$\displaystyle\left.{{-\text{
max}}\left\\{{\left({1-p}\right)^{4}c_{1}^{2},\left({1-p}\right)^{4}c_{2}^{2},r^{2}+\left({1-p}\right)^{2}c_{3}^{2}}\right\\}}\right]$
where we have assumed that the parameter $p$ is the same in the two
decoherence channels. Then it suffices to consider three separate cases.
Case 1. $r^{2}+c_{3}^{2}\geqslant\left\\{{c_{1}^{2},c_{2}^{2}}\right\\}$. Then
$\displaystyle
D^{g}\left({\rho\left(t\right)}\right)=\frac{1}{4}\left[{\left({1-p}\right)^{4}c_{1}^{2}+\left({1-p}\right)^{4}c_{2}^{2}}\right]$
$\displaystyle=2D^{g}\left[({\$_{PDC}\otimes\$_{DPC})\left({\left|{\beta_{i}}\right\rangle\left\langle{\beta_{i}}\right|}\right)}\right]D^{g}\left({\rho\left(0\right)}\right)$
(17)
Case 2. ${c_{1}^{2}\geqslant c_{2}^{2}\geqslant r^{2}+c_{3}^{2}}$. Suppose
$p^{\prime}_{0}$ satisfies
$\left({1-p^{\prime}_{0}}\right)^{4}c_{1}^{2}=r^{2}+\left({1-p^{\prime}_{0}}\right)^{2}c_{3}^{2}$.
If $p\leqslant p^{\prime}_{0}$, we have
$\displaystyle
D^{g}\left({\rho\left(t\right)}\right)=\frac{1}{4}\left[{\left({1-p}\right)^{4}c_{2}^{2}+r^{2}+\left({1-p}\right)^{2}c_{3}^{2}}\right]$
$\displaystyle\geqslant\frac{1}{4}\left[{\left({1-p}\right)^{4}c_{2}^{2}+\left({1-p}\right)^{4}r^{2}+\left({1-p}\right)^{4}c_{3}^{2}}\right]$
$\displaystyle=2D^{g}\left[({\$_{PDC}\otimes\$_{DPC})\left({\left|{\beta_{i}}\right\rangle\left\langle{\beta_{i}}\right|}\right)}\right]D^{g}\left({\rho\left(0\right)}\right)$
(18)
If $p>p^{\prime}_{0}$, we obtain
$\displaystyle
D^{g}\left({\rho\left(t\right)}\right)=\frac{1}{4}\left[{\left({1-p}\right)^{4}c_{1}^{2}+\left({1-p}\right)^{4}c_{2}^{2}}\right]$
$\displaystyle\geqslant\frac{1}{4}\left({1-p}\right)^{4}\left[{c_{2}^{2}+r^{2}+c_{3}^{2}}\right]$
$\displaystyle=2D^{g}\left[({\$_{PDC}\otimes\$_{DPC})\left({\left|{\beta_{i}}\right\rangle\left\langle{\beta_{i}}\right|}\right)}\right]D^{g}\left({\rho\left(0\right)}\right)$
(19)
Case 3. ${c_{1}^{2}\geqslant r^{2}+c_{3}^{2}\geqslant c_{2}^{2}}$. In this
case, the proof is similar to case 2.
One can also directly verify that the above relation holds for the other
cases.$\hfill\blacksquare$
The theorem above provides us a method to compute the lower bound of the time
evolution of a class of X-states under two typical kinds of decoherence
channels, without resorting to the time evolution of the underlying quantum
state itself. This inequality also holds for the one-sided PDC or DPC, which
is summarized as the following Corollary:
Corollary 1. For the class of X-states defined in Eq.(3), with one qubit being
subject to PDC or DPC, we have
$\displaystyle D^{g}\left({\rho\left(t\right)}\right)\geqslant
2D^{g}\left[({\$_{i}\otimes
I)\left({\left|{\beta_{i}}\right\rangle\left\langle{\beta_{i}}\right|}\right)}\right]D^{g}\left({\rho\left(0\right)}\right)$
(20)
Moreover, for $r=s=0$, _i.e._ Bell diagonal states, using Eq.(12), one can
directly calculate the above inequality becomes equality which we summarize as
follows:
Corollary 2. For arbitrary Bell-diagonal states subject to two DPCs, the
evolution of GMQD is given by
$\displaystyle
D_{DPC}^{g}\left(\rho(t)\right)=2D^{g}\left[{\left({\$_{DPC}\otimes\$_{DPC}}\right)\left|{\beta_{i}}\right\rangle\left\langle{\beta_{i}}\right|}\right]D^{g}\left(\rho(0)\right)$
where the two DPCs may be different. By far we have considered the time
evolution of GMQD in the presence of two typical kinds of local decoherence
noise. Another important decoherence noise is the amplitude damping
channel(ADC). Evidence can show that the above relation also holds for two
identical ADCs, i.e.
$\displaystyle D^{g}\left({\rho\left(t\right)}\right)\geqslant
2D^{g}\left[({\$_{ADC}\otimes\$_{ADC})\left({\left|{\beta_{i}}\right\rangle\left\langle{\beta_{i}}\right|}\right)}\right]D^{g}\left({\rho\left(0\right)}\right)$
In Fig.4, we plot the evolution of GMQD under two identical ADCs and its lower
bound in Eq.(22) when (a)$c_{1}=0.1,c_{2}=0.1,c_{3}=0.2,r=s=0.3$;
(b)$c_{1}=0.2,c_{2}=0.05,c_{3}=0.3,r=0.4,s=0.1$.
Figure 4: (Color online).Plots of the dynamics of GMQD under two identical
ADCs(blue line) and its lower bound defined in Eq.(22)(red
line).(a)$c_{1}=0.1,c_{2}=0.1,c_{3}=0.2,r=s=0.3$;
(b)$c_{1}=0.2,c_{2}=0.05,c_{3}=0.3,r=0.4,s=0.1$.
## IV DISCUSSIONS AND CONCLUSIONS
In this work, we investigated the level surfaces of GMQD for a class of two-
qubit X-states from the geometric picture. First, we plot the physical region
for a class of two-qubit X-states with fixed local Bloch vectors. It is shown
that physical regions of the state have different geometry with the Bell-
diagonal states and shrink with larger Bloch vectors. Second, the geometric
picture is depicted in terms of the constant concurrence and GMQD,
respectively. We find that the shape of the surfaces has close relationship
with the value of GMQD and local Bloch vectors. Finally, we also investigate
the dynamics of GMQD under two typical kinds of decoherence channels and
obtain analytic results of the evolution of GMQD. It is shown that there
exists a class of initial states for which the GMQD is not destroyed by
decoherence in a finite time interval. Moreover, a direct factorization
relationship between the initial and final GMQD subject to two typical kinds
of decoherence channels is derived. This factorization law allows us to infer
the evolution of entanglement under the influences of the environment without
resorting to the time evolution of the initial quantum state itself. An open
question is whether this law holds under general local decoherence channels.
Our results imply that further study on the dynamics of GMQD is required.
## V ACKNOWLEDGMENTS
We are grateful to the referee for valuable suggestions. This work was
supported by the National Natural Science Foundation of China under Grant
No.10905024,No.11005029,No.11104057,No.11204061, the Key Project of Chinese
Ministry of Education under Grant No.211080, and the Key Program of the
Education Department of Anhui Province under Grant No.KJ2011A243,
No.KJ2012A244, No.KJ2012A245, the Anhui Provincial Natural Science Foundation
under Grant No.11040606M16, No.10040606Q51, the Doctoral Startup Foundation of
Hefei Normal University under Grant No.2011rcjj03.
Note added. After completing this manuscript, we became aware of an
interesting related works by Yao Yao _et al_Yao:2011 recently.
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|
arxiv-papers
| 2011-12-19T12:37:06 |
2024-09-04T02:49:25.464270
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wei Song, Long-Bao Yu, Ping Dong, Da-Chuang Li, Ming Yang, Zhuo-Liang\n Cao",
"submitter": "Song Wei",
"url": "https://arxiv.org/abs/1112.4318"
}
|
1112.4394
|
# Additive Gaussian Processes
David Duvenaud
Department of Engineering
Cambridge University
dkd23@cam.ac.uk
&Hannes Nickisch
MPI for Intelligent Systems
Tübingen, Germany
hn@tue.mpg.de
&Carl Edward Rasmussen
Department of Engineering
Cambridge University
cer54@cam.ac.uk
###### Abstract
We introduce a Gaussian process model of functions which are
$\textit{a}dditive$. An additive function is one which decomposes into a sum
of low-dimensional functions, each depending on only a subset of the input
variables. Additive GPs generalize both Generalized Additive Models, and the
standard GP models which use squared-exponential kernels. Hyperparameter
learning in this model can be seen as Bayesian Hierarchical Kernel Learning
(HKL). We introduce an expressive but tractable parameterization of the kernel
function, which allows efficient evaluation of all input interaction terms,
whose number is exponential in the input dimension. The additional structure
discoverable by this model results in increased interpretability, as well as
state-of-the-art predictive power in regression tasks.
## 1 Introduction
Most statistical regression models in use today are of the form:
$g(y)=f(x_{1})+f(x_{2})+\dots+f(x_{D})$. Popular examples include logistic
regression, linear regression, and Generalized Linear Models[1]. This family
of functions, known as Generalized Additive Models (GAM)[2], are typically
easy to fit and interpret. Some extensions of this family, such as smoothing-
splines ANOVA [3], add terms depending on more than one variable. However,
such models generally become intractable and difficult to fit as the number of
terms increases.
At the other end of the spectrum are kernel-based models, which typically
allow the response to depend on all input variables simultaneously. These have
the form: $y=f(x_{1},x_{2},\dots,x_{D})$. A popular example would be a
Gaussian process model using a squared-exponential (or Gaussian) kernel. We
denote this model as SE-GP. This model is much more flexible than the GAM, but
its flexibility makes it difficult to generalize to new combinations of input
variables.
In this paper, we introduce a Gaussian process model that generalizes both
GAMs and the SE-GP. This is achieved through a kernel which allow additive
interactions of all orders, ranging from first order interactions (as in a
GAM) all the way to $D$th-order interactions (as in a SE-GP). Although this
kernel amounts to a sum over an exponential number of terms, we show how to
compute this kernel efficiently, and introduce a parameterization which limits
the number of hyperparameters to $O(D)$. A Gaussian process with this kernel
function (an additive GP) constitutes a powerful model that allows one to
automatically determine which orders of interaction are important. We show
that this model can significantly improve modeling efficacy, and has major
advantages for model interpretability. This model is also extremely simple to
implement, and we provide example code.
We note that a similar breakthrough has recently been made, called
Hierarchical Kernel Learning (HKL)[4]. HKL explores a similar class of models,
and sidesteps the possibly exponential number of interaction terms by cleverly
selecting only a tractable subset. However, this method suffers considerably
from the fact that cross-validation must be used to set hyperparameters. In
addition, the machinery necessary to train these models is immense. Finally,
on real datasets, HKL is outperformed by the standard SE-GP [4].
## 2 Gaussian Process Models
Gaussian processes are a flexible and tractable prior over functions, useful
for solving regression and classification tasks[5]. The kind of structure
which can be captured by a GP model is mainly determined by its _kernel_ : the
covariance function. One of the main difficulties in specifying a Gaussian
process model is in choosing a kernel which can represent the structure
present in the data. For small to medium-sized datasets, the kernel has a
large impact on modeling efficacy.
| + | | = | |
---|---|---|---|---|---
$k_{1}(x_{1},x_{1}^{\prime})$ | | $k_{2}(x_{2},x_{2}^{\prime})$ | | $k_{1}(x_{1},x_{1}^{\prime})+k_{2}(x_{2},x_{2}^{\prime})$ | $k_{1}(x_{1},x_{1}^{\prime})k_{2}(x_{2},x_{2}^{\prime})$
1D kernel | | 1D kernel | | 1st order kernel | 2nd order kernel
$\downarrow$ | | $\downarrow$ | | $\downarrow$ | $\downarrow$
| + | | = | |
$f_{1}(x_{1})$ | | $f_{2}(x_{2})$ | | $f_{1}(x_{1})+f_{2}(x_{2})$ | $f(x_{1},x_{2})$
draw from | | draw from | | draw from | draw from
1D GP prior | | 1D GP prior | | 1st order GP prior | 2nd order GP prior
Figure 1: A first-order additive kernel, and a product kernel. Left: a draw
from a first-order additive kernel corresponds to a sum of draws from one-
dimensional kernels. Right: functions drawn from a product kernel prior have
weaker long-range dependencies, and less long-range structure.
Figure 1 compares, for two-dimensional functions, a first-order additive
kernel with a second-order kernel. We can see that a GP with a first-order
additive kernel is an example of a GAM: Each function drawn from this model is
a sum of orthogonal one-dimensional functions. Compared to functions drawn
from the higher-order GP, draws from the first-order GP have more long-range
structure.
We can expect many natural functions to depend only on sums of low-order
interactions. For example, the price of a house or car will presumably be well
approximated by a sum of prices of individual features, such as a sun-roof.
Other parts of the price may depend jointly on a small set of features, such
as the size and building materials of a house. Capturing these regularities
will mean that a model can confidently extrapolate to unseen combinations of
features.
## 3 Additive Kernels
We now give a precise definition of additive kernels. We first assign each
dimension $i\in\\{1\dots D\\}$ a one-dimensional _base kernel_
$k_{i}(x_{i},x^{\prime}_{i})$. We then define the first order, second order
and $n$th order additive kernel as:
$\displaystyle k_{add_{1}}({\bf x,x^{\prime}})$ $\displaystyle=$
$\displaystyle\sigma_{1}^{2}\sum_{i=1}^{D}k_{i}(x_{i},x_{i}^{\prime})$ (1)
$\displaystyle k_{add_{2}}({\bf x,x^{\prime}})$ $\displaystyle=$
$\displaystyle\sigma_{2}^{2}\sum_{i=1}^{D}\sum_{j=i+1}^{D}k_{i}(x_{i},x_{i}^{\prime})k_{j}(x_{j},x_{j}^{\prime})$
(2) $\displaystyle k_{add_{n}}({\bf x,x^{\prime}})$ $\displaystyle=$
$\displaystyle\sigma_{n}^{2}\sum_{1\leq i_{1}<i_{2}<...<i_{n}\leq
D}\left[\prod_{d=1}^{n}k_{i_{d}}(x_{i_{d}},x_{i_{d}}^{\prime})\right]$ (3)
where $D$ is the dimension of our input space, and $\sigma_{n}^{2}$ is the
variance assigned to all $n$th order interactions. The $n$th covariance
function is a sum of ${D\choose n}$ terms. In particular, the $D$th order
additive covariance function has ${D\choose D}=1$ term, a product of each
dimension’s covariance function:
$k_{add_{D}}({\bf
x,x^{\prime}})=\sigma_{D}^{2}\prod_{d=1}^{D}k_{d}(x_{d},x_{d}^{\prime})$ (4)
In the case where each base kernel is a one-dimensional squared-exponential
kernel, the $D$th-order term corresponds to the multivariate squared-
exponential kernel:
$k_{add_{D}}({\bf
x,x^{\prime}})=\sigma_{D}^{2}\prod_{d=1}^{D}k_{d}(x_{d},x_{d}^{\prime})=\sigma_{D}^{2}\prod_{d=1}^{D}\exp\Big{(}-\frac{(x_{d}-x_{d}^{\prime})^{2}}{2l^{2}_{d}}\Big{)}=\sigma_{D}^{2}\exp\Big{(}-\sum_{d=1}^{D}\frac{(x_{d}-x_{d}^{\prime})^{2}}{2l^{2}_{d}}\Big{)}$
(5)
also commonly known as the Gaussian kernel. The full additive kernel is a sum
of the additive kernels of all orders.
### 3.1 Parameterization
The only design choice necessary in specifying an additive kernel is the
selection of a one-dimensional base kernel for each input dimension. Any
parameters (such as length-scales) of the base kernels can be learned as usual
by maximizing the marginal likelihood of the training data.
In addition to the hyperparameters of each dimension-wise kernel, additive
kernels are equipped with a set of $D$ hyperparameters
$\sigma_{1}^{2}\dots\sigma_{D}^{2}$ controlling how much variance we assign to
each order of interaction. These “order variance” hyperparameters have a
useful interpretation: The $d$th order variance hyperparameter controls how
much of the target function’s variance comes from interactions of the $d$th
order. Table 1 shows examples of normalized order variance hyperparameters
learned on real datasets.
Table 1: Relative variance contribution of each order in the additive model, on different datasets. Here, the maximum order of interaction is set to 10, or smaller if the input dimension less than 10. Values are normalized to sum to 100. Order of interaction | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---
pima | $0.1$ | $0.1$ | $0.1$ | $0.3$ | $1.5$ | ${\bf 96.4}$ | $1.4$ | $0.0$ | |
liver | $0.0$ | $0.2$ | ${\bf 99.7}$ | $0.1$ | $0.0$ | $0.0$ | | | |
heart | ${\bf 77.6}$ | $0.0$ | $0.0$ | $0.0$ | $0.1$ | $0.1$ | $0.1$ | $0.1$ | $0.1$ | $22.0$
concrete | ${\bf 70.6}$ | $13.3$ | $13.8$ | $2.3$ | $0.0$ | $0.0$ | $0.0$ | $0.0$ | |
pumadyn-8nh | $0.0$ | $0.1$ | $0.1$ | $0.1$ | $0.1$ | $0.1$ | $0.1$ | ${\bf 99.5}$ | |
servo | ${\bf 58.7}$ | $27.4$ | $0.0$ | $13.9$ | | | | | |
housing | $0.1$ | $0.6$ | ${\bf 80.6}$ | $1.4$ | $1.8$ | $0.8$ | $0.7$ | $0.8$ | $0.6$ | $12.7$
On different datasets, the dominant order of interaction estimated by the
additive model varies widely. An additive GP with all of its variance coming
from the 1st order is equivalent to a GAM; an additive GP with all its
variance coming from the $D$th order is equivalent to a SE-GP.
Because the hyperparameters can specify which degrees of interaction are
important, the additive GP is an extremely general model. If the function we
are modeling is decomposable into a sum of low-dimensional functions, our
model can discover this fact and exploit it (see Figure 5) . If this is not
the case, the hyperparameters can specify a suitably flexible model.
### 3.2 Interpretability
As noted by Plate[6], one of the chief advantages of additive models such as
GAM is their interpretability. Plate also notes that by allowing high-order
interactions as well as low-order interactions, one can trade off
interpretability with predictive accuracy. In the case where the
hyperparameters indicate that most of the variance in a function can be
explained by low-order interactions, it is useful and easy to plot the
corresponding low-order functions, as in Figure 2.
| |
---|---|---
Figure 2: Low-order functions on the concrete dataset. Left, Centre: By
considering only first-order terms of the additive kernel, we recover a form
of Generalized Additive Model, and can plot the corresponding 1-dimensional
functions. Green points indicate the original data, blue points are data after
the mean contribution from the other dimensions’ first-order terms has been
subtracted. The black line is the posterior mean of a GP with only one term in
its kernel. Right: The posterior mean of a GP with only one second-order term
in its kernel.
### 3.3 Efficient Evaluation of Additive Kernels
An additive kernel over $D$ inputs with interactions up to order $n$ has
$O(2^{n})$ terms. Naïvely summing over these terms quickly becomes
intractable. In this section, we show how one can evaluate the sum over all
terms in $O(D^{2})$.
The $n$th order additive kernel corresponds to the $n$th elementary symmetric
polynomial[7] [8], which we denote $e_{n}$. For example: if $\bf x$ has 4
input dimensions ($D=4$), and if we let $z_{i}=k_{i}(x_{i},x_{i}^{\prime})$,
then
$\displaystyle k_{add_{1}}({\bf x,x^{\prime}})$
$\displaystyle=e_{1}(z_{1},z_{2},z_{3},z_{4})=z_{1}+z_{2}+z_{3}+z_{4}$
$\displaystyle k_{add_{2}}({\bf x,x^{\prime}})$
$\displaystyle=e_{2}(z_{1},z_{2},z_{3},z_{4})=z_{1}z_{2}+z_{1}z_{3}+z_{1}z_{4}+z_{2}z_{3}+z_{2}z_{4}+z_{3}z_{4}$
$\displaystyle k_{add_{3}}({\bf x,x^{\prime}})$
$\displaystyle=e_{3}(z_{1},z_{2},z_{3},z_{4})=z_{1}z_{2}z_{3}+z_{1}z_{2}z_{4}+z_{1}z_{3}z_{4}+z_{2}z_{3}z_{4}$
$\displaystyle k_{add_{4}}({\bf x,x^{\prime}})$
$\displaystyle=e_{4}(z_{1},z_{2},z_{3},z_{4})=z_{1}z_{2}z_{3}z_{4}$
The Newton-Girard formulae give an efficient recursive form for computing
these polynomials. If we define $s_{k}$ to be the $k$th power sum:
$s_{k}(z_{1},z_{2},\dots,z_{D})=\sum_{i=1}^{D}z_{i}^{k}$, then
$k_{add_{n}}({\bf
x,x^{\prime}})=e_{n}(z_{1},\dots,z_{D})=\frac{1}{n}\sum_{k=1}^{n}(-1)^{(k-1)}e_{n-k}(z_{1},\dots,z_{D})s_{k}(z_{1},\dots,z_{D})$
(6)
Where $e_{0}\triangleq 1$. The Newton-Girard formulae have time complexity
$O(D^{2})$, while computing a sum over an exponential number of terms.
Conveniently, we can use the same trick to efficiently compute all of the
necessary derivatives of the additive kernel with respect to the base kernels.
We merely need to remove the kernel of interest from each term of the
polynomials:
$\displaystyle\frac{\partial k_{add_{n}}}{\partial z_{j}}$
$\displaystyle=e_{n-1}(z_{1},\dots,z_{j-1},z_{j+1},\dots z_{D})$ (7)
This trick allows us to optimize the base kernel hyperparameters with respect
to the marginal likelihood.
### 3.4 Computation
The computational cost of evaluating the Gram matrix of a product kernel (such
as the SE kernel) is $O(N^{2}D)$, while the cost of evaluating the Gram matrix
of the additive kernel is $O(N^{2}DR)$, where R is the maximum degree of
interaction allowed (up to D). In higher dimensions, this can be a significant
cost, even relative to the fixed $O(N^{3})$ cost of inverting the Gram matrix.
However, as our experiments show, typically only the first few orders of
interaction are important for modeling a given function; hence if one is
computationally limited, one can simply limit the maximum degree of
interaction without losing much accuracy.
Additive Gaussian processes are particularly appealing in practice because
their use requires only the specification of the base kernel. All other
aspects of GP inference remain the same. All of the experiments in this paper
were performed using the standard GPML toolbox111Available at
http://www.gaussianprocess.org/gpml/code/; code to perform all experiments is
available at the author’s website.222Example code available at:
http://mlg.eng.cam.ac.uk/duvenaud/
## 4 Related Work
Plate[6] constructs a form of additive GP, but using only the first-order and
$D$th order terms. This model is motivated by the desire to trade off the
interpretability of first-order models, with the flexibility of full-order
models. Our experiments show that often, the intermediate degrees of
interaction contribute most of the variance.
A related functional ANOVA GP model[9] decomposes the _mean_ function into a
weighted sum of GPs. However, the effect of a particular degree of interaction
cannot be quantified by that approach. Also, computationally, the Gibbs
sampling approach used in [9] is disadvantageous.
Christoudias et al.[10] previously showed how mixtures of kernels can be
learnt by gradient descent in the Gaussian process framework. They call this
_Bayesian localized multiple kernel learning_. However, their approach learns
a mixture over a small, fixed set of kernels, while our method learns a
mixture over all possible products of those kernels.
### 4.1 Hierarchical Kernel Learning
| | |
---|---|---|---
HKL kernel | GP-GAM kernel | Squared-exp GP | Additive GP kernel
| | kernel |
Figure 3: A comparison of different models. Nodes represent different
interaction terms, ranging from first-order to fourth-order interactions. Far
left: HKL can select a hull of interaction terms, but must use a pre-
determined weighting over those terms. Far right: the additive GP model can
weight each order of interaction seperately. Neither the HKL nor the additive
model dominate one another in terms of flexibility, however the GP-GAM and the
SE-GP are special cases of additive GPs.
Bach[4] uses a regularized optimization framework to learn a weighted sum over
an exponential number of kernels which can be computed in polynomial time. The
subsets of kernels considered by this method are restricted to be a hull of
kernels.333In the setting we are considering in this paper, a hull can be
defined as a subset of all terms such that if term $\prod_{j\in J}k_{j}(\bf
x,x^{\prime})$ is included in the subset, then so are all terms $\prod_{j\in
J/i}k_{j}(\bf x,x^{\prime})$, for all $i\in J$. For details, see [4]. Given
each dimension’s kernel, and a pre-defined weighting over all terms, HKL
performs model selection by searching over hulls of interaction terms. In [4],
Bach also fixes the relative weighting between orders of interaction with a
single term $\alpha$, computing the sum over all orders by:
$k_{a}({\bf x,x^{\prime}})=v_{D}^{2}\prod_{d=1}^{D}\left(1+\alpha
k_{d}(x_{d},x_{d}^{\prime})\right)$ (8)
which has computational complexity $O(D)$. However, this formulation forces
the weight of all $n$th order terms to be weighted by $\alpha^{n}$.
Figure 3 contrasts the HKL hull-selection method with the Additive GP
hyperparameter-learning method. Neither method dominates the other in
flexibility. The main difficulty with the approach of [4] is that
hyperparameters are hard to set other than by cross-validation. In contrast,
our method optimizes the hyperparameters of each dimension’s base kernel, as
well as the relative weighting of each order of interaction.
### 4.2 ANOVA Procedures
Vapnik [11] introduces the support vector ANOVA decomposition, which has the
same form as our additive kernel. However, they recommend approximating the
sum over all $D$ orders with only one term “of appropriate order”, presumably
because of the difficulty of setting the hyperparameters of an SVM. Stitson et
al.[12] performed experiments which favourably compared the support vector
ANOVA decomposition to polynomial and spline kernels. They too allowed only
one order to be active, and set hyperparameters by cross-validation.
A closely related procedure from the statistics literature is smoothing-
splines ANOVA (SS-ANOVA)[3]. An SS-ANOVA model is estimated as a weighted sum
of splines along each dimension, plus a sum of splines over all pairs of
dimensions, all triplets, etc, with each individual interaction term having a
separate weighting parameter. Because the number of terms to consider grows
exponentially in the order, in practice, only terms of first and second order
are usually considered. Learning in SS-ANOVA is usually done via penalized-
maximum likelihood with a fixed sparsity hyperparameter.
In contrast to these procedures, our method can easily include all $D$ orders
of interaction, each weighted by a separate hyperparameter. As well, we can
learn kernel hyperparameters individually per input dimension, allowing
automatic relevance determination to operate.
### 4.3 Non-local Interactions
By far the most popular kernels for regression and classification tasks on
continuous data are the squared exponential (Gaussian) kernel, and the Matérn
kernels. These kernels depend only on the scaled Euclidean distance between
two points, both having the form: $k({\bf
x,x^{\prime}})=f(\sum_{d=1}^{D}\left(x_{d}-x_{d}^{\prime}\right)^{2}/l_{d}^{2})$.
Bengio et al.[13] argue that models based on squared-exponential kernels are
particularily susceptible to the curse of dimensionality. They emphasize that
the locality of the kernels means that these models cannot capture non-local
structure. They argue that many functions that we care about have such
structure. Methods based solely on local kernels will require training
examples at all combinations of relevant inputs.
| | |
---|---|---|---
1st order interactions | 2nd order interactions | 3rd order interactions | All interactions
$k_{1}+k_{2}+k_{3}$ | $k_{1}k_{2}+k_{2}k_{3}+k_{1}k_{3}$ | $k_{1}k_{2}k_{3}$ |
| | (Squared-exp kernel) | (Additive kernel)
Figure 4: Isocontours of additive kernels in 3 dimensions. The third-order
kernel only considers nearby points relevant, while the lower-order kernels
allow the output to depend on distant points, as long as they share one or
more input value.
Additive kernels have a much more complex structure, and allow extrapolation
based on distant parts of the input space, without spreading the mass of the
kernel over the whole space. For example, additive kernels of the second order
allow strong non-local interactions between any points which are similar in
any two input dimensions. Figure 4 provides a geometric comparison between
squared-exponential kernels and additive kernels in 3 dimensions.
## 5 Experiments
### 5.1 Synthetic Data
Because additive kernels can discover non-local structure in data, they are
exceptionally well-suited to problems where local interpolation fails.
| | |
---|---|---|---
True Function | Squared-exp GP | Additive GP | Additive GP
& data locations | posterior mean | posterior mean | 1st-order functions
Figure 5: Long-range inference in functions with additive structure.
Figure 5 shows a dataset which demonstrates this feature of additive GPs,
consisting of data drawn from a sum of two axis-aligned sine functions. The
training set is restricted to a small, L-shaped area; the test set contains a
peak far from the training set locations. The additive GP recovered both of
the original sine functions (shown in green), and inferred correctly that most
of the variance in the function comes from first-order interactions. The
ability of additive GPs to discover long-range structure suggests that this
model may be well-suited to deal with covariate-shift problems.
### 5.2 Experimental Setup
On a diverse collection of datasets, we compared five different models. In the
results tables below, GP Additive refers to a GP using the additive kernel
with squared-exp base kernels. For speed, we limited the maximum order of
interaction in the additive kernels to 10. GP-GAM denotes an additive GP model
with only first-order interactions. GP Squared-Exp is a GP model with a
squared-exponential ARD kernel. HKL444Code for HKL available at
http://www.di.ens.fr/~fbach/hkl/ was run using the all-subsets kernel, which
corresponds to the same set of kernels as considered by the additive GP with a
squared-exp base kernel.
For all GP models, we fit hyperparameters by the standard method of maximizing
training-set marginal likelihood, using L-BFGS [14] for 500 iterations,
allowing five random restarts. In addition to learning kernel hyperparameters,
we fit a constant mean function to the data. In the classification
experiments, GP inference was done using Expectation Propagation [15].
### 5.3 Results
Tables 2, 3, 4 and 5 show mean performance across 10 train-test splits.
Because HKL does not specify a noise model, it could not be included in the
likelihood comparisons.
Table 2: Regression Mean Squared Error Method | bach | concrete | pumadyn-8nh | servo | housing
---|---|---|---|---|---
Linear Regression | $1.031$ | $0.404$ | $0.641$ | $0.523$ | $0.289$
GP GAM | $1.259$ | $0.149$ | $0.598$ | $0.281$ | $0.161$
HKL | $\mathbf{0.199}$ | $0.147$ | $0.346$ | $0.199$ | $0.151$
GP Squared-exp | $\mathbf{0.045}$ | $0.157$ | $\mathbf{0.317}$ | $\mathbf{0.126}$ | $\mathbf{0.092}$
GP Additive | $\mathbf{0.045}$ | $\mathbf{0.089}$ | $\mathbf{0.316}$ | $\mathbf{0.110}$ | $\mathbf{0.102}$
Table 3: Regression Negative Log Likelihood Method | bach | concrete | pumadyn-8nh | servo | housing
---|---|---|---|---|---
Linear Regression | $2.430$ | $1.403$ | $1.881$ | $1.678$ | $1.052$
GP GAM | $1.708$ | $0.467$ | $1.195$ | $0.800$ | $0.457$
GP Squared-exp | $\mathbf{-0.131}$ | $0.398$ | $\mathbf{0.843}$ | $0.429$ | $\mathbf{0.207}$
GP Additive | $\mathbf{-0.131}$ | $\mathbf{0.114}$ | $\mathbf{0.841}$ | $\mathbf{0.309}$ | $\mathbf{0.194}$
Table 4: Classification Percent Error Method | breast | pima | sonar | ionosphere | liver | heart
---|---|---|---|---|---|---
Logistic Regression | $7.611$ | $24.392$ | $26.786$ | $16.810$ | $45.060$ | $\mathbf{16.082}$
GP GAM | $\mathbf{5.189}$ | $\mathbf{22.419}$ | $\mathbf{15.786}$ | $\mathbf{8.524}$ | $\mathbf{29.842}$ | $\mathbf{16.839}$
HKL | $\mathbf{5.377}$ | $24.261$ | $\mathbf{21.000}$ | $9.119$ | $\mathbf{27.270}$ | $\mathbf{18.975}$
GP Squared-exp | $\mathbf{4.734}$ | $\mathbf{23.722}$ | $\mathbf{16.357}$ | $\mathbf{6.833}$ | $\mathbf{31.237}$ | $\mathbf{20.642}$
GP Additive | $\mathbf{5.566}$ | $\mathbf{23.076}$ | $\mathbf{15.714}$ | $\mathbf{7.976}$ | $\mathbf{30.060}$ | $\mathbf{18.496}$
Table 5: Classification Negative Log Likelihood Method | breast | pima | sonar | ionosphere | liver | heart
---|---|---|---|---|---|---
Logistic Regression | $0.247$ | $0.560$ | $4.609$ | $0.878$ | $0.864$ | $0.575$
GP GAM | $\mathbf{0.163}$ | $\mathbf{0.461}$ | $\mathbf{0.377}$ | $\mathbf{0.312}$ | $\mathbf{0.569}$ | $\mathbf{0.393}$
GP Squared-exp | $\mathbf{0.146}$ | $0.478$ | $\mathbf{0.425}$ | $\mathbf{0.236}$ | $\mathbf{0.601}$ | $0.480$
GP Additive | $\mathbf{0.150}$ | $\mathbf{0.466}$ | $\mathbf{0.409}$ | $\mathbf{0.295}$ | $\mathbf{0.588}$ | $\mathbf{0.415}$
The model with best performance on each dataset is in bold, along with all
other models that were not significantly different under a paired t-test. The
additive model never performs significantly worse than any other model, and
sometimes performs significantly better than all other models. The difference
between all methods is larger in the case of regression experiments. The
performance of HKL is consistent with the results in [4], performing
competitively but slightly worse than SE-GP.
The additive GP performed best on datasets well-explained by low orders of
interaction, and approximately as well as the SE-GP model on datasets which
were well explained by high orders of interaction (see table 1). Because the
additive GP is a superset of both the GP-GAM model and the SE-GP model,
instances where the additive GP performs slightly worse are presumably due to
over-fitting, or due to the hyperparameter optimization becoming stuck in a
local maximum. Additive GP performance can be expected to benefit from
integrating out the kernel hyperparameters.
## 6 Conclusion
We present additive Gaussian processes: a simple family of models which
generalizes two widely-used classes of models. Additive GPs also introduce a
tractable new type of structure into the GP framework. Our experiments
indicate that such additive structure is present in real datasets, allowing
our model to perform better than standard GP models. In the case where no such
structure exists, our model can recover arbitrarily flexible models, as well.
In addition to improving modeling efficacy, the additive GP also improves
model interpretability: the order variance hyperparameters indicate which
sorts of structure are present in our model.
Compared to HKL, which is the only other tractable procedure able to capture
the same types of structure, our method benefits from being able to learn
individual kernel hyperparameters, as well as the weightings of different
orders of interaction. Our experiments show that additive GPs are a state-of-
the-art regression model.
#### Acknowledgments
The authors would like to thank John J. Chew and Guillaume Obozonksi for their
helpful comments.
## References
* [1] J.A. Nelder and R.W.M. Wedderburn. Generalized linear models. Journal of the Royal Statistical Society. Series A (General), 135(3):370–384, 1972.
* [2] T.J. Hastie and R.J. Tibshirani. Generalized additive models. Chapman & Hall/CRC, 1990.
* [3] G. Wahba. Spline models for observational data. Society for Industrial Mathematics, 1990.
* [4] Francis Bach. High-dimensional non-linear variable selection through hierarchical kernel learning. CoRR, abs/0909.0844, 2009.
* [5] C.E. Rasmussen and CKI Williams. Gaussian Processes for Machine Learning. The MIT Press, Cambridge, MA, USA, 2006.
* [6] T.A. Plate. Accuracy versus interpretability in flexible modeling: Implementing a tradeoff using Gaussian process models. Behaviormetrika, 26:29–50, 1999.
* [7] I.G. Macdonald. Symmetric functions and Hall polynomials. Oxford University Press, USA, 1998.
* [8] R.P. Stanley. Enumerative combinatorics. Cambridge University Press, 2001.
* [9] C.G. Kaufman and S.R. Sain. Bayesian functional anova modeling using Gaussian process prior distributions. Bayesian Analysis, 5(1):123–150, 2010.
* [10] M. Christoudias, R. Urtasun, and T. Darrell. Bayesian localized multiple kernel learning. Technical report, 2009.
* [11] V.N. Vapnik. Statistical learning theory, volume 2. Wiley New York, 1998.
* [12] M. Stitson, A. Gammerman, V. Vapnik, V. Vovk, C. Watkins, and J. Weston. Support vector regression with ANOVA decomposition kernels. Advances in kernel methods: Support vector learning, pages 285–292, 1999.
* [13] Y. Bengio, O. Delalleau, and N. Le Roux. The curse of highly variable functions for local kernel machines. Advances in neural information processing systems, 18, 2006.
* [14] J. Nocedal. Updating quasi-newton matrices with limited storage. Mathematics of computation, 35(151):773–782, 1980.
* [15] T.P. Minka. Expectation propagation for approximate Bayesian inference. In Uncertainty in Artificial Intelligence, volume 17, pages 362–369, 2001.
|
arxiv-papers
| 2011-12-19T16:22:09 |
2024-09-04T02:49:25.473436
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "David Duvenaud, Hannes Nickisch, Carl Edward Rasmussen",
"submitter": "David Duvenaud",
"url": "https://arxiv.org/abs/1112.4394"
}
|
1112.4466
|
# Star Formation Rates in Molecular Clouds and the Nature of the Extragalactic
Scaling Relations
Charles J. Lada Harvard-Smithsonian Center for Astrophysics, 60 Garden Street
Cambridge, MA 02138, USA clada@cfa.harvard.edu Jan Forbrich Harvard-
Smithsonian Center for Astrophysics, 60 Garden Street Cambridge, MA 02138, USA
jforbrich@cfa.harvard.edu Marco Lombardi University of Milan, Department of
Physics, via Celoria 16, 220133 Milan, Italy marco.lombardi@gmail.com João F.
Alves Institute for Astronomy, University of Vienna, Türkenschanzstrasse 17,
1180 Vienna, Austria joao.alves@univie.ac.at
###### Abstract
In this paper we investigate scaling relations between star formation rates
and molecular gas masses for both local Galactic clouds and a sample of
external galaxies. We specifically consider relations between the star
formation rates and measurements of dense, as well as total, molecular gas
masses. We argue that there is a fundamental empirical scaling relation that
directly connects the local star formation process with that operating
globally within galaxies. Specifically, the total star formation rate in a
molecular cloud or galaxy is linearly proportional to the mass of dense gas
within the cloud or galaxy. This simple relation, first documented in previous
studies, holds over a span of mass covering nearly nine orders of magnitude
and indicates that the rate of star formation is directly controlled by the
amount of dense molecular gas that can be assembled within a star formation
complex. We further show that the star formation rates and total molecular
masses, characterizing both local clouds and galaxies, are correlated over
similarly large scales of mass and can be described by a family of linear star
formation scaling laws, parameterized by $f_{DG}$, the fraction of dense gas
contained within the clouds or galaxies. That is, the underlying star
formation scaling law is always linear for clouds and galaxies with the same
dense gas fraction. These considerations provide a single unified framework
for understanding the relation between the standard (non-linear) extragalactic
Schmidt-Kennicutt scaling law, that is typically derived from CO observations
of the gas, and the linear star formation scaling law derived from HCN
observations of the dense gas.
Stars: formation, Galaxies: star formation
††slugcomment: Submitted to ApJ 10/20/2011, Accepted 12/16/2011
## 1 Introduction
Knowledge of the physical factors that control the conversion of interstellar
gas into stars is of fundamental importance for both developing a predictive
physical theory of star formation and understanding the evolution of galaxies
from the earliest epochs of cosmic history to the present time. An essential
first step to obtaining such knowledge is to establish empirically the
underlying relation or relationships that most directly connect the rate of
star formation in a galaxy to some general physical property of the
interstellar gas from which stars form. A little more than a half-century ago,
Schmidt (1959) conjectured that this might take the form of a scaling relation
between the rate of star formation and some power, n, of the surface density
of atomic (HI) gas. From evaluation of the distributions of local HI gas and
stars orthogonal to the Galactic plane, he suggested that n$\approx$ 2\.
Subsequent studies comparing the surface densities of OB stars and HII regions
with those of atomic gas within nearby external galaxies produced scaling laws
with similar, super-linear, power-law indices (e.g., Sanduleak 1969; Hamajima
& Tosa 1975).
By the 1980s it became clear that molecular, not atomic, clouds were the sites
of star formation in galaxies. The ability to make sensitive CO molecular-line
observations enabled, for the first time, the measurement of total gas surface
densities ($\Sigma_{HI+H_{2}}$) in external galaxies while advancements in
infrared and ultraviolet observations led to significant improvements in the
measurements of star formation rates. Significant effort was then expended by
a number of researchers to systematically measure star formation rates and
total gas surface densities in increasingly large samples of galaxies (e.g.,
Kennicutt 1989 and references therein). These efforts culminated in the study
of Kennicutt (1998a) who compiled galaxy averaged measurements of star
formation rates and gas surface densities for a large sample of galaxies
including normal spirals and starbursts. He derived a scaling relation between
the star formation rate surface density ($\Sigma_{SFR}$) and total gas surface
density ($\Sigma_{HI+H_{2}}$) that was characterized by a power-law index of n
$\approx$ 1.4. This value was shallower than that Schmidt and others found for
individual galaxies using only atomic gas but still super-linear. Wong and
Blitz ( 2002), employing spatially resolved observations of seven nearby,
molecular rich, spiral galaxies, showed that the star formation rate was
better correlated with the molecular hydrogen surface density,
$\Sigma_{H_{2}}$, than with the atomic surface density, but still obtained n
$\approx$ 1.4. More recently, Bigiel et al. (2008) analyzed spatially resolved
observations of 18 nearby galaxies containing both atomic rich and molecular
rich objects and confirmed that $\Sigma_{SFR}$ was better correlated with
$\Sigma_{H_{2}}$ than $\Sigma_{HI}$, but they determined that n $=$ 1.0 ($\pm$
0.2) for the $\Sigma_{SFR}$ – $\Sigma_{H_{2}}$ relation. However, recent
observations of M 101 and M 81 have suggested that the index of the scaling
law can vary within a galaxy with values of n ranging between 1 and 2 (Suzuki
et al. 2010).
Among the more interesting investigations of the extragalactic scaling laws
for star formation was that of Gao and Solomon (2004) who used molecular-line
emission from HCN, rather than CO, to trace the molecular gas. They found a
linear (n $=$ 1) correlation between the total far-infrared luminosities and
the HCN molecular-line luminosities of a large sample of star forming galaxies
including normal spirals and luminous and ultra-luminous infrared galaxies.
Since the total infrared luminosity is a good proxy for the total star
formation rate (SFR) and the HCN luminosity a good proxy for the total amount
of dense (i.e., n(H2) $\geq$ 3 $\times$ 104 cm-3) gas in a galaxy, this also
implied a linear correlation between the SFR and the mass of dense molecular
gas.
The various determinations of differing power-law indices for the
extragalactic star formation scaling relations present a somewhat confused and
problematic picture. Particularly since the difference between a linear and
non-linear scaling relation can have significant consequences for the
theoretical understanding of the star formation process in galaxies. Therefore
it is important to understand the nature of such differences. Are the
different scaling relations consistent with each other? Are the differences
due to such effects as the choice of the samples studied (e.g., normal spirals
vs starbursts, CO rich vs. HI rich galaxies, distant vs. nearby systems, etc.)
or the different quantities actually measured (e.g., SFR vs. $\Sigma_{SFR}$,
$\Sigma_{HI+H_{2}}$ vs. $\Sigma_{H_{2}}$, or CO vs. HCN, etc.), or the
systematic uncertainties in the quantities measured (e.g., observational
tracers or IMFs adopted for SFR determinations, conversion factor for
transforming CO measurements into H2 masses, etc.), or some linear combination
of all these effects? Do any of these scaling relations represent the
fundamental underlying physical relationship that most directly connects star
formation activity with interstellar gas?
Schmidt’s original scaling law was determined from observations of the local
region of the Galaxy. Since our knowledge of the local Milky Way has improved
profoundly over the last half century, it would seem that important insights
into the relation between star formation and interstellar gas could and should
be derived from observations of local star formation activity. In a previous
paper (Lada et al. 2010; hereafter Paper I) we presented a study of the star
formation activity in a sample of local (d $<$ 0.5 kpc) molecular clouds with
total masses between 103 and 105 M⊙. We employed infrared extinction
measurements derived from wide-field surveys to determine accurate cloud
masses and mass surface densities, and compiled from the literature both
ground and space-based infrared surveys of young stellar objects to construct
complete inventories of star formation within the clouds of our local sample.
We found the specific star formation rates (i.e., the star formation rates per
unit cloud mass) in these clouds to vary by an order of magnitude, independent
of total cloud mass. However, we also found the dispersion in the specific
star formation rate, to be minimized (and reduced by a factor of 2-3) if one
considers only the mass of molecular gas characterized by high extinction in
calculating the specific star formation rates. As a result we showed that the
(total) star formation rate in local clouds is linearly proportional to the
cloud mass contained above an extinction threshold of AK $\geq$ 0.8
magnitudes, corresponding to a gas surface density threshold of
$\Sigma_{H_{2}}$ $\approx$ 116 M⊙pc-2. Similar surface density thresholds for
star formation in local clouds have been suggested in other recent studies
(e.g., Goldsmith et al 2008; Heiderman et al. 2010). Given the density
stratification of molecular clouds, we argued that such surface density
thresholds also correspond to volume density thresholds of n(H2) $\approx$ 104
cm-3. These findings are consistent with and reinforce those of Wu et al.
(2005) who had already demonstrated a linear correlation between far-infrared
luminosity and HCN luminosity (i.e., between SFR and dense gas mass) for more
massive and distant star formation regions in the Milky Way.
The correspondence between these results and those obtained by Gao and Solomon
(2004) for external galaxies is intriguing and especially striking because the
scalings of the Galactic and extragalactic power-law relations, that together
span more than nine orders of magnitude in cloud mass, agree to within a
factor of 2-3. This suggested to us that the close relationship between the
star formation rates and the dense gas masses of molecular clouds could be the
underlying physical relation that connects star formation activity with
interstellar gas over vast spatial scales from the immediate vicinity of the
sun to the most distant galaxies.
However, if this is so, how does one understand these observations in the
context of the classical Schmidt-Kennicutt scaling relations based on CO
observations? These classical relations are often super linear and moreover,
as Heiderman et al. (2010) point out, they under predict the $\Sigma_{SFR}$ in
local regions by factors of 17 - 50 (see also Evans et al. 2009). In this
paper we attempt to address this issue by re-examining the extinction
observations of local clouds to include low extinction material and re-
examining the CO observations of the clouds studied by Gao and Solomon. We
show that all the observations can be understood within a self-consistent
framework in which the differences are primarily due to the dense gas
fractions that characterize the molecular gas being observed, supporting a
hypothesis originally put forward by Gao and Solomon (2004).
## 2 The SFR-Molecular Mass Diagram
### 2.1 The Local Clouds
In Figure 1 we plot the relation between the (total) star formation rate, SFR,
and gas mass for the 11 clouds in the Paper I sample. The SFRs are from Table
2 of Paper I and are the averaged rates over a timescale of 2 Myrs. However,
here we plot for each cloud two different masses derived from the infrared
extinction measurements. The filled circles represent cloud masses measured
above an infrared (K-band) extinction threshold of 0.8 magnitudes and
correspond to the dense gas masses ($M_{DG}$) of the clouds. The open circles
represent cloud masses measured above a lower infrared extinction threshold of
0.1 magnitudes and correspond to the total gaseous masses ($M_{TG}$) of the
clouds. These latter masses should also approximately correspond to those that
would be traced by CO emission, while the former masses approximately
correspond to those that would be traced by HCN emission. The parallel dashed
lines represent a series of linear relations between SFR and mass. The top
line is the best fit linear relation for the high extinction (dense gas)
masses (following Paper I). The two lower lines are the same relation only
shifted or scaled in the horizontal direction by one and two orders of
magnitude in mass, respectively. We can now express the star formation scaling
law for these clouds as:
$SFR\equiv\dot{M}_{*}=4.6\times 10^{-8}f_{DG}M_{G}(M_{\odot})\ \ \ M_{\odot}\
yr^{-1}\ \ \ $ (1)
where $M_{G}$ is molecular mass measured at a particular extinction threshold
and corrected for the presence of Helium and $f_{DG}$ is the fraction of dense
gas, i.e., $M_{DG}=f_{DG}M_{G}$. The three parallel lines correspond to
$f_{DG}$ $=$ 1.0, 0.1 and 0.01 from left to right, respectively. These
scalings essentially represent the fraction of the measured mass that is above
the 0.8 magnitude extinction threshold or equivalently above a volume density
threshold of roughly n(H2) = 104 cm-3(Paper I). These lines also correspond to
lines of constant gas depletion times of 20 Myr, 200 Myr and 2 Gyr,
respectively. For the open symbols on the plot, $M_{G}$ = $M_{TG}$, the total
mass of the molecular cloud.
The interesting aspect of this plot is that the low extinction (total) masses
also appear to follow a linear scaling law, similar to that of the high
extinction (high density) masses. Indeed, a formal least-squares fit to the
former data produces a slightly sub-linear index value of 0.81 $\pm$ 0.19. The
total cloud masses, $M_{TG}$, appear to follow and scatter around the relation
given by Equation 1, if $f_{DG}$ $=$ 0.1. However the magnitude of the scatter
around this linear relation is significantly higher than that for the high
extinction masses around the best-fit line given by Equation 1 (i.e., $f_{DG}$
$=$ 1 and $M_{G}=M_{DG}$). Star formation occurs almost exclusively in gas
characterized by high densities (n($H_{2}$) $>$ 104 cm-3; Lada 1992) and the
origin of the large scatter in the star formation scaling law for the total
cloud masses is a direct result of the large variations in the dense gas (high
extinction) fractions that are observed for these clouds (Paper I). In
contrast to classical Schmidt-Kennicutt extragalactic scaling laws, there is
no evidence for a super-linear scaling for the star formation law for local
clouds, even when the total masses of the clouds are considered.
### 2.2 Galaxies
In order to compare galaxies with the galactic clouds on the SFR-Molecular
Mass diagram we use the sample of galaxies observed by Gao and Solomon (2004;
hereafter GS04). Their sample consists of normal spirals and starburst
galaxies, including luminous and ultra-luminous infrared galaxies (i.e., LIRGs
& ULIRGs). We selected this sample for comparison with our local cloud sample
because it is the only sample of galaxies with systematically measured
molecular masses using both a tracer of high density gas, HCN, and a tracer of
total cloud mass, CO. In addition, the SFRs of the galaxies in the sample are
all derived in the same manner from a homogeneous set of infrared
observations.
It is a priori unclear whether the star formation rates and/or gas masses
reported for the GS04 galaxy sample are directly comparable to those reported
in Paper I for the local cloud sample. The SFR for the local clouds was
determined by direct counting of nearly complete inventories of Young Stellar
Objects in each cloud and assuming a star formation timescale of 2 Myrs, while
the SFRs for the GS04 galaxies are galaxy-wide averages that were derived from
conversion of a FIR flux into a mass growth rate using stellar population
synthesis models and assuming, among other parameters, a simple Salpeter IMF
and a timescale of 10-100 Myrs (Kennicutt 1998b). Gao and Solomon use the most
simple form of the virial theorem to convert HCN luminosity to a galaxy
averaged dense gas mass, while in Paper I masses are calculated from direct
integration of resolved extinction measurements of individual clouds and the
assumption of a standard gas-to-dust ratio. Moreover, even if both mass
calculations are accurate, it is not obvious that the AK $=$ 0.8 mag contour
encompasses exactly the same mass as would be detected in HCN emission
averaged over an entire cloud or galaxy. We therefore would not necessarily
expect the Galactic clouds and the Gao-Solomon galaxies to fall onto the exact
same line in the SFR-Molecular Mass diagram (for dense gas masses) and, they
do not. Although previous studies (GS04, Paper I) independently found the
relation between SFR and dense gas mass to be linear for both local clouds and
galaxies, the respective coefficients (intercepts) differed by a factor of
2.7, with the galactic relation predicting higher SFRs for a given amount of
dense gas. However, given the fact that these two linear relations together
span nine orders of magnitude in mass, and their coefficients are consistent
within the quoted errors (Paper I), it seems reasonable to conclude that they
represent one and the same relation.
Indeed, in a study of massive, but relatively distant, Galactic molecular
clouds, Wu et al (2005) demonstrated a linear correlation between FIR and LHCN
for those clouds that was nearly identical (with similar coefficients) to that
found by Gao and Solomon (2004). This finding thus extended the correlation
between these two quantities over a range of more than 7-8 orders of magnitude
and indicated that both the GS04 galaxies and Galactic clouds should lie on
the same SFR-Mass relation for the dense gas component traced by HCN
observations. Because the SFRs and masses calculated for the local sample are
likely more robust than those determined for the Wu et al. clouds and the GS04
galaxies, we decided to adjust the coefficient of the GS04 relation to match
that of the local sample for the dense molecular gas. In principle, this could
be accomplished by either, a) adjusting the star formation rates upward, b)
adjusting the HCN masses downward, or c) simultaneously adjusting some
appropriate linear combination of both these quantities. It is not obvious
which of these alternatives would be most appropriate, and given the
complexities and uncertainties in calculating both the star formation rates
and dense gas masses for these galaxies, choice c) might be the best option.
However, for simplicity we elected to match the coefficients by adjusting the
GS04 star formation rates upward (by log($\Delta$SFR) = 0.43) so that they
match those of Paper I when linearly extrapolated down to local cloud masses.
We emphasize here that the primary results and conclusions of this paper (see
§3) are independent of the details (i.e., a, b or c) of how we choose to
adjust the coefficient of the GS04 relation to match the relation for local
clouds.
In Figure 2 we extend the SFR-Molecular Mass plot to scales that can include
measurements of entire galaxies and we plot the galaxies in the GS04 sample.
As with the local sample we plot two sets of masses for the GS04 galaxies.
Again, the filled symbols correspond to dense gas masses, as measured using
HCN emission. The open symbols correspond to total cloud masses measured from
CO emission. The dense gas masses of the galaxies are those determined by
GS04. Since GS04 did not report total gas (CO) masses for the galaxies in
their sample, we made use of the CO(1-0) luminosities reported by GS04 to
derive the total gas masses. We applied a conversion of $M_{\rm gas}$/$L_{\rm
CO}$ = 1.36 $\times$ $\alpha_{\rm G}$ with a Galactic value of $\alpha_{\rm
G}=3.2$ $M_{\odot}$ (K km s-1 pc2)-1 (e.g., Genzel et al. 2010). The star
formation rates for these galaxies are those determined by GS04 after the
upward adjustment described described above.
Adjusting the GS04 SFRs upward implicitly assumes that the SFRs determined
from LFIR underestimate the true star formation rates, at least when
extrapolated to local clouds. In an attempt to assess this possibility we
investigated the relation between LFIR and SFR in the local cloud sample. In
the local cloud sample of Paper I, the SFR is dominated by the Orion A and B
molecular clouds which account for 67% of the total SFR for all the clouds in
the sample. Following the prescription of GS04 we used IRAS observations to
determine the FIR luminosity of a 100 pc diameter region encompassing both the
Orion A and B clouds. We calculated the FIR luminosity to be 5.4 $\times$ 105
L⊙. Using the relation $\dot{M}_{\rm SFR}\approx 2\times 10^{-10}(L_{\rm
IR}/L_{\odot})\ M_{\odot}yr^{-1}$, (following GS04 and Kennicutt 1998b), this
corresponds to SFR $=$ 1.1 $\times$ 10-4 M⊙yr-1, a value which is a factor of
8 lower than the combined SFR (8.7 $\times$ 10-4 M⊙yr-1) determined for the
Orion A and B clouds in Paper I. We note that much of this deficit is likely
due to the fact that the extragalactic FIR prescription for SFRs is
appropriate for star formation timescales of 10-100 Myrs and a well sampled
IMF at high stellar masses while the SFRs for the local cloud sample are
derived for a 2 Myr timescale and for a young stellar population that does not
as completely sample the high mass end of the IMF. Nonetheless, these
considerations suggest that at least some upward adjustment of the GS04 SFRs
may be necessary for comparison with local clouds.
Another consequence of the upward adjustment of the star formation rates is
that of a corresponding decrease in the estimated total molecular gas
depletion times for the GS04 galaxies. This decrease would amount to a factor
of 2.7 for the adjustment factor we adopted and have potentially important
consequences for our understanding of galaxy evolution. These decreased gas
depletion times for the GS04 galaxies are consistent those that describe the
local galactic clouds (e.g., Figure 1). However, we hesitate in drawing too
firm a conclusion regarding this particular issue since it does depend
somewhat on our choice of adjustment options (i.,e., a, b, or c). For example,
if we selected option (b) above, only the depletion time for the dense gas
component of the galaxies would be lowered. It is also interesting to note in
this context that the depletion times for the dense star-forming gas are
typically an order of magnitude lower than those estimated for the total
molecular gas component in both galaxies and local clouds, and this remains
true independent of any adjustments to the galaxy data.
As discussed earlier, instead of adjusting the star formation rates, we could
have adjusted the GS04 galaxy masses (downward) by the same constant offset in
log(M). By not correcting the mass estimates we are assuming that the
molecular-line derived masses and the extinction derived masses accurately
reflect the same cloud material, that is, $M_{DG}$ $=$ $M_{HCN}$ and $M_{TG}$
$=$ $M_{CO}$. To assess this possibility for the case of the total cloud
masses, $M_{TG}$, we compared the extinction measurements with CO observations
of a subset of the local cloud sample. We obtained CO data for five of the
clouds from the archive of the CfA 1.2 m Millimeter-wave Telescope (Dame et
al. 2001). The 12CO observations were averaged over the individual clouds and
the integrated CO intensities were measured for each cloud. Applying the
standard CO-to-H2 conversion factor of 2 x 1020 cm-2 (K km s-1)-1 (Dame et al.
2001) to convert the integrated intensities to H2 column densities, we
determined the mass of each cloud. We found these CO derived masses to all
agree with the corresponding extinction (AK $\geq$ 0.1 mag) derived masses to
better than 12%, indicating that the extinction (AK $>$ 0.1 mag) and CO
derived total masses both trace the same cloud material for local clouds. This
suggests that total masses derived from CO can be a good proxy for extinction
derived total masses and thus that the masses derived from CO observations of
galaxies can be compared directly with those of the local cloud sample,
provided that the galaxy measurements trace the summed CO emission from a
population of GMCs. If there is any diffuse CO emission from inert, non star-
forming, molecular gas contributing to the galaxy-averaged CO measurements,
then the CO masses derived for galaxies overestimate the masses in star
forming GMCs. In such a case the CO derived masses for the galaxies would have
to be adjusted downwards to compare to the local observations.
A similar comparison of extinction and HCN derived masses is not possible for
the local clouds since the corresponding HCN observations of these clouds do
not exist. This is unfortunate because the HCN masses derived by GS04 are
likely upper limits to the true masses (Gao and Solomon 2004b). For example,
if the clouds are bound but not virialized then the derived masses could be
somewhat underestimated. Thus, although it appears that the extragalactic CO
derived masses can be directly placed on the SFR-Molecular Cloud Mass diagram
without any systematic adjustment, the situation is somewhat less certain for
the HCN masses derived by GS04. However, we note that the average ratio of
dense gas (i.e., AV $\geq$ 0.8 mag) to total cloud mass (i.e., AV $\geq$ 0.1
mag.) calculated from the extinction data is $<f_{DG}>$ $=$ 0.10 $\pm$ 0.06
for the sample of local clouds. For the GS04 sample of galaxies we find
$<f_{DG}>$ = 0.16 $\pm$ 0.14 comparing the HCN and CO derived masses. The
relatively close correspondence of $f_{DG}$ for these two samples is
consistent with the idea that the high extinction and HCN observations trace
roughly similar fractions of the total cloud masses and thus similar dense
material in clouds and galaxies, (i.e., $M_{DG}=M_{HCN}$). This suggests that
the extragalactic HCN and CO observations of Gao and Solomon likely trace
similar material as observed in the extinction observations of Galactic clouds
by Lada et al.(2010) and thus both sets of masses can be directly placed on
the SFR-Molecular Mass diagram without systematic adjustment.
We note here that instead of plotting galaxies on the SFR–Molecular Mass
diagram many authors traditionally prefer to plot them on the
$\Sigma_{\rm{SFR}}$ – $\Sigma_{gas}$ diagram, arguing that these two latter
quantities are not affected by the (correlated) errors induced by inaccuracies
in the galaxy distance measurements. However, we prefer to plot the total
formation rate, SFR, as function of the gas mass, $M_{G}$, to better compare
the local sample with the extragalactic one. In doing so, we acknowledge the
fact that the distance-squared factor entering the evaluation of the total
mass and total star-formation rate could induce a potentially strong
correlation between these two variables. This correlation, in turn, might hide
the real power-law index of the underlying relation, making it appear closer
to unity than it is in reality (this is a consequence of the fact that the
distance enters with the same exponent, two, in both quantities). On the other
hand, simple statistical arguments and numerical checks show that the measured
slope of the relation is significantly biased only in the limit where the
relative error on the square of the distance is of the same order of
magnitude, or larger, than the range spanned by the data. In our case, the
extragalactic data set spans approximately 4 orders of magnitude, and
distances errors are on the order of $30\%$ or less, and therefore we are
affected by a negligible bias in the measurement of the slope of the
underlying relation using the SFR-Molecular Mass diagram.
## 3 Discussion and Conclusions
The SFR-Molecular Mass diagram of Figure 2 provides a physical context for
understanding the star formation scaling laws over spatial scales ranging from
those of local molecular clouds to those of entire galaxies. The close
correlation of the star formation rate with the mass of dense gas over these
immense scales has been established in previous studies (Wu et al. 2005, Paper
I). Here we find that a close relation also appears to hold between the SFR
and the total molecular mass over a similarly large range, 8-9 orders of
magnitude in both quantities. Both the local clouds and galaxies appear to
scatter around the linear relation given by Equation 1 for $f_{DG}$ $=$ 0.1
and $M_{G}=M_{TG}$. From extrapolation of the results for local clouds we
suggest that this particular line corresponds to the case where 10% of the
measured gas mass is in the form of dense, star forming material for the
galaxies as well as for the local clouds. The smaller scatter of the galaxies
around this relation compared to that of the local clouds is likely the result
of the fact that the galaxy measurements are averages over entire systems.
These results indicate that, similar to the situation for dense gas, the star
formation scaling law for total (H2 \+ He) gas mass is likely linear across
all scales for molecular clouds with similar dense gas fractions. This notion
is reinforced by the recent observations of Daddi et al. (2010) who studied
infrared-selected BzK galaxies at $z\sim 1.5$ and found evidence for unusually
high gas fractions and extended molecular reservoirs in these distant systems.
Using the star formation rates and CO gas masses provided by Daddi et al.
(2010), we plot these six galaxies (open triangles) on Figure 2 and find that
the BzK galaxies occupy an area in the SFR-Molecular Mass plot that is close
to the linear relation described by Equation 1, consistent with the locations
of Gao and Solomon galaxies and the extrapolation of the local Galactic cloud
sample.
These results lead us to the conclusion that there is a basic and universal
physical process of star formation that presently operates in our Milky Way
galaxy and is also responsible for the bulk of star forming activity occurring
in external galaxies both in the present epoch (z $\approx$ 0; GS04) and
perhaps at much earlier (z $\approx 1-2$; Daddi et al. 2010) cosmic times. It
is a process in which the rate of star formation is simply and directly
controlled by the amount of dense molecular gas that can be assembled within a
star forming complex. In most situations massive molecular clouds appear to be
able to convert only about 10% or less of their total mass into a sufficiently
dense (n(H2) $\geq$ 104 cm-3) form to actively produce stars. This may be
considered as the normal process of star formation in GMCs.
Closer inspection of Figure 2 suggests that for starburst galaxies,
particularly the ULIRGS, this standard scenario may be modified. As the SFRs
for starbursts (i.e., LIRGs and ULIRGs in Figure 2) increase with gas mass,
the open symbols (CO derived gas masses) appear to approach and then merge
with the filled symbols (HCN derived gas masses), almost overlapping at the
highest SFRs. As originally hypothesized by Gao and Solomon (2004), we
interpret this to indicate that these galaxies are characterized by an
increasingly high dense gas fraction and consequently, the CO observations
begin to trace nearly the same material as the HCN observations. Nonetheless,
the star formation rate is still dictated by the amount of dense gas within
the galaxies. This interpretation is also favored by Heiderman et al. (2010)
who suggested that the maximal starburst activity occurs when $f_{DG}=\rm 1$
which they posit to happen when the mass surface density exceeds values $\sim$
104 M⊙ pc-2. ULIRGS (e.g., Arp 200) are believed to be experiencing major
mergers and we suggest that this extreme process likely produces conditions
(e.g., high pressures) that could increase the dense gas fractions of the
molecular clouds within these systems (e.g., Blitz & Rosolowsky 2006). In
contrast the BzK galaxies studied by Daddi et al. (2010) have similarly high
SFRs but lower dense gas fractions. Their high star formation rates appear to
result from high global molecular gas mass fractions (i.e., M${}_{H_{2}}$/M∗),
as might be expected for very young galaxies.
We note that a linear relation in the SFR-Mass plane should transform to a
linear relation in the $\Sigma_{SFR}$-$\Sigma_{g}$ plane (provided the surface
densities for the galaxies are global averages) and we can express our star
formation scaling law in this latter plane as:
$\Sigma_{SFR}\propto f_{DG}\Sigma_{g}$ (2)
where $\Sigma_{g}$ refers to the H2 gas mass. Moreover, the Spitzer study of
Galactic clouds by Heiderman et al. (2010) suggested a linear star formation
law in the $\Sigma_{SFR}$-$\Sigma_{g}$ plane that holds for gas above a
threshold surface density of $\sim$ 130 M⊙ pc-2 (i.e., AK $>$ 0.9 mag) and
extrapolates smoothly to the GS04 galaxies.
Our result is apparently not consistent with the standard Schmidt-Kennicutt,
super-linear, scaling law (Kennicutt 1998a). Both are based on valid empirical
relations. However, here we argue that the underlying scaling law for star
formation is linear over all scales for all clouds and galaxies, provided they
are characterized by the same dense gas fraction. Kennicutt (1998a) uses total
(HI $+$ H2) gas mass surface densities with CO derived molecular masses and
combines results for both normal star-forming disk galaxies and starburst
galaxies to derive his star formation scaling law. Note that for these latter
galaxies the total gas surface densities are dominated by the molecular
component. The starbursts dominate his relation for $\Sigma_{gas}$ $>$ 100
M⊙pc-2. It is possible that the fit of a single relation to the combined
sample with CO determined masses is inappropriate and skewed by the starbursts
because $f_{DG}$ for starbursts is higher than that for normal star forming
spirals. Indeed, Gao and Solomon (2004) showed that using the masses
calculated from the CO observations produced a super-linear (n $\approx$ 1.7)
scaling law (in the SFR vs MG plane) for a sample that included their galaxies
and an additional number of luminous starbursts drawn form the literature.
Using gas masses derived solely from HCN observations, however, produces a
linear star formation law connecting both normal star forming galaxies and
starbursts. The standard Schmidt-Kennicutt relation may also be skewed at low
mass surface densities. For galaxies in this portion of the diagram, the HI
surface density is a large fraction of the total gas surface density and thus
the measured total gas surface density, $\Sigma_{HI+H_{2}}$, contains a large
component of inert, non-star forming, (HI) gas; this dilutes and lowers the
SFR corresponding to a fixed mass surface density, resulting in a steepening
of the slope of the $\Sigma_{SFR}$ vs $\Sigma_{gas}$ relation.
These two effects, the increasing dense gas fraction for the starbursts and
the dilution of the dense gas fraction by HI gas at low gas surface densities,
which act together to steepen the slope of the Schmidt-Kennicutt relation, can
also explain the finding of Heiderman et al. (2010) and Evans et al. (2009)
that the extrapolation of the extragalactic scaling relations to local scales
(i.e., low mass surface densities) lies below the data for Galactic clouds.
It can also be shown that our scaling law (Equation 1) is consistent with a
volumetric scaling law, $\dot{\rho_{*}}\propto\rho_{G}^{n}$ if and only if
$n=1$ and $\rho_{G}\geq\rho_{thres}$, where $\rho_{thres}/\mu$ corresponds to
the threshold volumetric number density for star formation for a mean particle
mass given by $\mu$ (i.e., n(H2) $\geq$ 104 cm-3).
As discussed earlier, taking the empirical, linear star-forming scaling
relations at face value leads to a simple interpretation of the observations
in Figures 1 & 2\. Namely, that the total rate of star formation,
$\dot{M}_{*}$, is directly proportional to the mass of dense molecular gas
above a threshold density, $M_{DG}=\int_{\rho_{thres}}^{\infty}M(\rho)d\rho$.
Moreover, once the gas has reached this threshold density, the SFR does not
depend on the exact value of the density but only on the total mass of gas
whose density has exceeded the threshold. This interpretation of the
observations differs from those that explain the observed non-linear index of
the Schmidt-Kennicutt law as resulting from star formation timescales dictated
by the free-fall time, e.g., SFR $\sim$ M/$\tau_{ff}$ $\sim$
$\rho/\rho^{-0.5}$ $\sim$ $\rho^{1.5}$ since $\tau_{ff}\sim\rho^{-0.5}$ (e.g.,
Elmegreen 1994; Krumholz & Thompson 2007; Narayanan et al. 2008). A recent
variant of such a model has been studied by Krumholz et al. (2011). They
propose that the underlying physical law governing the relation between star
formation rates and cloud properties is given by
$\dot{\rho_{*}}\propto\rho_{G}/\tau_{ff}$. They find that the standard
Schmidt-Kennicutt law can be linearized if the data are plotted in the
$\Sigma_{SFR}-\Sigma_{G}/\tau_{ff}$ plane as long as the free-fall times are
measured using the typically higher densities of the star forming gas rather
than those derived from the mean densities averaged over kpc scales. Their
interpretation differs from the one in this paper in that Krumholz et al.
(2011) posit that the positions of galaxies in the standard
$\Sigma_{SFR}$-$\Sigma_{G}$ plane are a consequence of both their gas surface
densities and their local free-fall times, while here we posit that the
locations of these galaxies instead depend on their gas surface densities and
their dense gas fractions. Although both interpretations are consistent with
the observations, they appear not to be consistent with each other. However,
we point out that Figure 1 empirically demonstrates that the locations of
Galactic clouds in the SFR-Mass diagram are in fact a result of their dense
gas fractions. Therefore it seems reasonable to infer that the locations of
galaxies in the diagram are due to the same cause.
Finally, we reiterate our point that the linear scaling law of Equation 1
implies that the process of star formation across entire galaxies as well as
individual local clouds is governed by a very similar and simple physical
principle: the rate at which molecular gas is turned into stars depends on the
mass of dense gas within a molecular cloud or cloud population. The underlying
star formation scaling law is linear over all scales for all clouds and
galaxies characterized by the same dense gas fraction. The star formation rate
appears therefore to be controlled by local processes and not by global,
galactic scale mechanisms, except to the extent that such mechanisms can alter
the dense gas fractions in the molecular gas. If this interpretation is
correct, then the key problem that needs to be addressed in future studies is
that of the origin of the dense gas component of molecular clouds.
We thank Leo Blitz, Tom Dame, Daniela Calzetti, Bruce Elmegreen, Debbie
Elmegreen, Neal Evans, Reinhard Genzel, and Mark Krumholz for informative
discussions and comments and Tom Dame for providing us with CO data.
## References
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Figure 1: The SFR-Molecular Mass diagram for local molecular clouds. The solid
symbols indicate cloud masses above an extinction threshold of AK $=$ 0.8
magnitudes (dense gas masses) while open circles correspond to cloud masses
above AK $=$ 0.1 magnitudes (total cloud masses). The parallel dashed lines
are linear relations that indicate constant fractions of dense (i.e., AK
$\geq$ 0.8 magnitudes; n(H2) $\geq$ 104 cm-3) gas. The top line is the best
linear fit to the solid symbols and represents the case where all the gas
measured is dense star forming material. (see text). Figure 2: The SFR-
Molecular Mass diagram for local molecular clouds and galaxies from the Gao
and Solomon (2004) sample. The solid symbols correspond to measurements of
dense cloud masses either from extinction observations of the galactic clouds
or HCN observations of the galaxies. The open symbols correspond to
measurements of total cloud masses of the same clouds and galaxies, either
from extinction measurements for the galactic clouds or CO observations for
the galaxies. For the galaxies, pentagons represent the locations of normal
spirals, while the positions of starburst galaxies are represented by squares
(LIRGS) and inverted triangles (ULIRGS). Triangles represent high-z BzK
galaxies. The star formation rates for the Gao and Solomon galaxies have been
adjusted upward by a factor of 2.7 to match those of galactic clouds when
extrapolated to local cloud masses. (see text).
|
arxiv-papers
| 2011-12-19T20:54:17 |
2024-09-04T02:49:25.482128
|
{
"license": "Public Domain",
"authors": "Charles J. Lada, Jan Forbrich, Marco Lombardi, and Joao F. Alves",
"submitter": "Charles J. Lada",
"url": "https://arxiv.org/abs/1112.4466"
}
|
1112.4556
|
# Effect of dynamical traps on chaotic transport in a meandering jet flow
M.Yu. Uleysky, M.V. Budyansky, and S.V. Prants
###### Abstract
We continue our study of chaotic mixing and transport of passive particles in
a simple model of a meandering jet flow [Prants, et al, Chaos 16, 033117
(2006)]. In the present paper we study and explain phenomenologically a
connection between dynamical, topological, and statistical properties of
chaotic mixing and transport in the model flow in terms of dynamical traps,
singular zones in the phase space where particles may spend arbitrary long but
finite time [Zaslavsky, Phys. D 168–169, 292 (2002)]. The transport of passive
particles is described in terms of lengths and durations of zonal flights
which are events between two successive changes of sign of zonal velocity.
Some peculiarities of the respective probability density functions for short
flights are proven to be caused by the so-called rotational-islands traps
connected with the boundaries of resonant islands (including the vortex cores)
filled with the particles moving in the same frame and the saddle traps
connected with periodic saddle trajectories. Whereas, the statistics of long
flights can be explained by the influence of the so-called ballistic-islands
traps filled with the particles moving from a frame to frame.
## 1 Introduction
Chaotic advection of water masses with their physical and biochemical
characteristics in quasi two-dimensional geophysical flows in the ocean and
atmosphere can be studied within the framework of Hamiltonian dynamics. In a
recent paper [1] we have studied chaotic mixing and transport of passive
particles in a simple kinematic model of a meandering jet flow motivated by
the problem of lateral mixing in the western boundary currents in the ocean.
We found all the possible bifurcations of advection equations, described the
structure of the phase space (which is the physical space for advected
particles), and computed some statistical characteristics of chaotic
transport. In the present paper we establish a phenomenological connection
between dynamical, topological, and statistical properties of chaotic
transport and mixing in the same flow. Specific singular zones in the phase
space where particles may spend arbitrary long but finite time (dynamical
traps in terminology by Zaslavsky [2]), are responsible for anomalous
statistical properties. The dynamical traps, connected with rotational islands
and saddle trajectories, are responsible, mainly, for anomalous mixing,
whereas those ones, connected with ballistic islands — for anomalous
transport. These dynamical traps may have strong impact on transport and
mixing in real geophysical jet flows.
Methods of the theory of dynamical systems are actively used to describe
advection of water (air) masses and their properties in the ocean and
atmosphere [1, 14, 13, 15, 16, 26, 3, 4, 5, 6, 7, 17, 18, 19, 20, 21, 32]. The
geophysical jet currents, like the Gulf Stream, the Kuroshio, the Antarctic
circumpolar current, and others in the ocean and the polar night Antarctic jet
in the atmosphere, are robust structures whose form typically changes in space
and time in a meander-like way. If advected particles rapidly adjust their own
velocity to that of background flow and do not affect the flow properties,
then the particles are called passive (scalars, tracers, or Lagrangian
particles) and their equation of motion is very simple
$\frac{d{\mathbf{r}}}{dt}={\mathbf{v}}({\mathbf{r}},t)$ (1)
where ${\mathbf{r}}=(x,y,z)$ and ${\mathbf{v}}=(u,v,w)$ are the position
vector and the particle velocity at a point $(x,y,z)$. If the corresponding
Eulerian velocity field is supposed to be regular, the vector Eq. (1) in
nontrivial cases is a set of three nonlinear deterministic differential
equations whose phase space is a physical space of advected particles. It is
well known from dynamical systems theory that solutions of this kind of
equations can be chaotic in the sense of exponential sensitivity to small
variations in initial conditions and/or control parameters. As to advection
equations, it was Arnold [8] who firstly suggested chaos in the field lines
(and, therefore, in trajectories) for a special class of three-dimensional
stationary flows (so-called ABC flows), and this suggestion has been confirmed
numerically by Hénon [9]. In the approximation of incompressible planar flows,
the velocity components can be expressed in terms of a stream function [10]:
$u=-\partial\Psi/\partial y$ and $v=\partial\Psi/\partial x$. The equations of
motion (1) in a two-dimensional incompressible flow have now a Hamiltonian
form with the streamfunction $\Psi$ playing the role of a Hamiltonian and the
coordinates $(x,y)$ of a particle playing the role of canonically conjugated
variables.
Advection of passive particles has been shown to be chaotic in a number of
theoretical [4, 5, 6, 7, 1, 14, 13, 15, 16, 17, 26, 32] and laboratory [18,
19, 20, 21] models of geophysical jet currents. In our recent paper [1] (Paper
I) we have studied mixing, transport, and chaotic advection in a simple
kinematic model of a meandering two-dimensional jet flow with a Bickley zonal
velocity profile $u\sim\operatorname{sech}^{2}y$ being motivated by the
problem of lateral mixing of water masses (together with salinity, heat,
nutrients, pollutants, and other passive scalars) in the western boundary
currents in the ocean. We derived advection equations in the frame moving with
the phase velocity of a running wave imposed on the Bickley jet, found the
stationary points, conditions of their stability, and all the possible
bifurcations of these equations which were shown to be autonomous in the co-
moving frame. Under a periodic perturbation of the wave amplitude, the phase
plane of the chosen model flow has been shown to consist of a central eastward
jet, peripheral westward currents, and chains of northern and southern
circulations (vortex cores) immersed in a chaotic sea which, in turn, contains
islands of regular motion. Statistical properties of chaotic transport of
advected particles have been characterized in terms of particle’s zonal
flights (any event between two successive changes of the sign of the
particle’s zonal velocity $u$). Probability density functions (PDFs) of
durations and lengths of flights, computed with a number of very long chaotic
trajectories, have been found to be complicated functions with local maxima
and fragments with exponential and power-law decays.
The aim of this paper is to study a phenomenological connection between
dynamical, topological, and statistical properties of chaotic mixing and
transport in the meandering jet flow considered in Paper I and to explain
transport properties by phenomenon of the so-called dynamical traps. Following
to Zaslavsky [2], the dynamical trap is a domain in the phase space of a
Hamiltonian system where a particle (or, its trajectory) can spend arbitrary
long finite time, performing almost regular motion, despite the fact that the
full trajectory is chaotic in any appropriate sense. In fact, it is the
definition of a quasi-trap. Absolute traps, where particles could spend an
infinite time, are possible in Hamiltonian systems only with a zero measure
set. The dynamical traps are due to a stickiness of trajectories to some
singular domains in the phase space, largely, to the boundaries of resonant
islands, saddle trajectories, and cantori. There are no classification and
description of the dynamical traps. Zaslavsky described two types of dynamical
traps in Hamiltonian systems: hierarchical-islands traps around chains of
resonant islands [27, 28, 12] and stochastic-layer traps which are stochastic
jets inside a stochastic sea where trajectories can spend a very long time
[29, 2, 12]. It is expected that classification and description of the most
typical dynamical traps would help us to construct kinetic equations which
will be able to describe transport properties of chaotic systems including
anomalous ones [2, 11, 12].
This paper is organized as follows. We start with advection equations derived
in Paper I in the frame of reference moving with the phase velocity of a
meander whose amplitude changes in time periodically. We compute in Sec. II
PDFs for lengths $x_{f}$ and durations $T_{f}$ of zonal flights for a number
of chaotic trajectories and show analytically that all the flights start and
finish only inside a strip confined by two curves whose form is defined by the
condition $u=0$. Some prominent peaks in statistics of short flights
($|x_{f}|<2\pi$) are proved to be caused by stickiness of trajectories to the
boundaries of rotational resonant islands filled with regular particles
rotating in the same frame (Sec. III). We call this type of dynamical traps as
a rotational-island trap (RIT). In Sec. IV we study dynamical traps connected
with periodic saddle trajectories, emerged from saddle points of the
unperturbed system (4) under the perturbation (5), and prove numerically that
the saddle traps (STs) contribute to the statistics of short flights as well.
Another type of islands, ballistic islands (filled with regular particles
moving from frame to frame), is proved to contribute to the statistics of long
flights ($|x_{f}|\gg 2\pi$) in Sec. V. Both the ballistic-islands trap (BITs)
and RITs belong to the class of hierarchical-island traps by the Zaslavsky’s
classification.
## 2 Description of chaotic transport in terms of flights
### 2.1 Basic features of the flow
We take the following specific stream function as a kinematic model of a
meandering jet flow in the laboratory frame of reference:
$\Psi^{\prime}(x^{\prime},y^{\prime},t^{\prime})=\\\
=-\Psi^{\prime}_{0}\tanh{\left(\frac{y^{\prime}-a\cos{k(x^{\prime}-ct^{\prime})}}{\lambda\sqrt{1+k^{2}a^{2}\sin^{2}{k(x^{\prime}-ct^{\prime})}}}\right)},$
(2)
where the width of the jet is $\lambda$. Meandering is provided by a running
wave with the amplitude $a$, the wave-number $k$, and the phase velocity $c$.
The normalized streamfunction in the frame moving with the phase velocity is
$\Psi=-\tanh{\left(\frac{y-A\cos x}{L\sqrt{1+A^{2}\sin^{2}x}}\right)}+Cy,$ (3)
where $x=k(x^{\prime}-ct^{\prime})$ and $y=ky^{\prime}$ are new scaled
coordinates, and $A=ak$, $L=\lambda k$, and $C=c/\Psi^{\prime}_{0}k$ are the
control parameters. Equations, governing advection of passive particles (1) in
the co-moving frame, are the following:
$\begin{gathered}\begin{aligned}
\dot{x}&=\frac{1}{L\sqrt{1+A^{2}\sin^{2}x}\cosh^{2}\theta}-C,\\\
\dot{y}&=-\frac{A\sin x(1+A^{2}-Ay\cos
x)}{L\left(1+A^{2}\sin^{2}x\right)^{3/2}\cosh^{2}\theta},\end{aligned}\\\
\theta=\frac{y-A\cos x}{L\sqrt{1+A^{2}\sin^{2}x}},\end{gathered}$ (4)
where dot denotes differentiation with respect to dimensionless time
$t=\Psi^{\prime}_{0}k^{2}t^{\prime}$.
In Paper I (for more details see [17]) we have found and analyzed all the
stationary points, their stability, and the bifurcations of the equations of
motion (4). Being motivated by the problem of mixing and transport of water
masses and their properties in oceanic western boundary currents like the Gulf
Stream and the Kuroshio, we chose the phase portrait shown in Fig. 1a among
all the possible flow regimes. Passive particles can move along stationary (in
the co-moving frame) streamlines in a different manner. They can move to the
east in the jet ($J$) and to the west in northern and southern (with respect
to the jet) peripheral currents ($P$). There are also particles rotating in
the northern and southern circulation cells (C) in a periodic way. The
northern separatrix connects the saddle points at $x_{s}^{(n)}=2\pi n$ and
$y_{s}^{(n)}=L\operatorname{Arcosh}\sqrt{1/LC}+A$ and the southern one
connects the saddle points at $x_{s}^{(s)}=(2n+1)\pi$ and
$y_{s}^{(s)}=-L\operatorname{Arcosh}\sqrt{1/LC}-A$, where $n=0,\pm 1,\dots$.
Figure 1: (a) Stationary streamfunction of a meandering jet in the co-moving
frame (3). The flow is divided into three different regimes: circulations
($C$), jet ($J$), and peripheral currents ($P$). (b) Poincaré section of the
perturbed meandering jet in the co-moving frame. The parameters of the steady
flow are: the jet’s width $L=0.628$, the meander’s amplitude $A_{0}=0.785$ and
its phase velocity $C=0.1168$. The perturbation amplitude and frequency are:
$\varepsilon=0.0785$ and $\omega=0.2536.$ (c) Turning points of a single
chaotic trajectory on the cylinder $0\leq x\leq 2\pi$ are in a strip confined
by two curves (6) with $A=A_{0}\pm\varepsilon$.
As a perturbation, we took in Paper I the simple periodic modulation of the
meander’s amplitude
$A=A_{0}+\varepsilon\cos(\omega t+\phi).$ (5)
Under the perturbation, there arise resonances between the perturbation
frequency $\omega$ and the frequencies $f$ of the particle’s rotation in the
circulations $C$. A frequency map $f(x_{0},y_{0})$, computed in Paper I (see
Fig. 2 in that paper), shows the values of $f$ for particles with initial
positions $(x_{0},y_{0})$ in the unperturbed flow. With a given value of the
perturbation frequency and fixed values of the other control parameters, the
vortex cores in the circulations survive, stochastic layers appear along the
unperturbed separatrix, and the central jet $J$ is a barrier to transport of
particles across the jet. In Paper I we fixed the scaled values of the
parameters of the unperturbed flow, the jet’s width $L=0.628$, the meander’s
amplitude $A_{0}=0.785$ and its phase velocity $C=0.1168$, that are in the
range of the realistic values for the Gulf Stream [22, 23], and took the
initial phase to be $\phi=\pi/2$. The perturbation frequency $\omega=0.2536$
chosen in Paper I is close to the values of the rotation frequency $f$ of the
particles circulating in the inner core of the regions $C$ (see Fig. 2 in
Paper I). In Fig. 1 b we show the Poincaré section (for a large number of
trajectories) of the meandering jet whose amplitude is modulated with the
frequency $w=0.2536$ and the strength $\varepsilon=0.0785$. The vortex cores
survive under this perturbation, the stochastic layers appear along the
unperturbed separatrix, and a central jet $J$ is a barrier to transport of
particles across the jet.
The equations of motion (4) with the perturbation (5) are symmetric under the
following transformations: (1) $t\to t$, $x\to\pi+x$, $y\to-y$ and (2)
$t\to-t$, $x\to\ -x$, $y\to y$. It implies that the meridional transport
(north-south and south-north) is symmetric but the zonal transport (west-east
and east-west) is symmetric under a time reversal. Due to these symmetries
motion can be considered on the cylinder with $0\leq x\leq 2\pi$ and $y\geq
0$. The part of the phase space with $2\pi n\leq x\leq 2\pi(n+1)$, $n=0,\pm
1,\dots$, is called a frame.
It is convenient to characterize chaotic mixing and transport in terms of
zonal flights. A zonal flight is a motion of a particle between two successive
changes of signs of its zonal velocity, i. e. the motion between two
successive events $\dot{x}=u=0$. Particles (and corresponding trajectories) in
chaotic jet flows can be classified in terms of the lengths of flights $x_{f}$
as follows. The trajectories with $|x_{f}|<2\pi$ correspond to the particles
moving in the same frame or in neighbor frames. In the global stochastic layer
there are particles moving chaotically forever in the same frame but they are
of a zero measure. Among the particles with inter-frame motion, there are
regular and chaotic ballistic ones. Regular ballistic trajectories can be
defined as those which cannot have two flights with $|x_{f}|>2\pi$ in
succession. They correspond to particles moving in regular regions of the
phase space persisting under the perturbation, (eastward motion in the jet and
western motion in the peripheral current) and those moving in the stochastic
layer (trajectories belonging to ballistic islands). Typical chaotic
trajectories have complicated distributions over the lengths and durations of
flights.
In the laboratory frame of reference, all the fluid particles move to the east
together with the jet flow and a flight is a motion between two successive
events when the particle’s zonal velocity $U$ is equal to the meander’s phase
velocity $c$. If $U<c$, the corresponding particle is left behind the meander
(it is a western flight in the co-moving frame), if $U>c$, it passes the
meander (an eastern flight in the co-moving frame). Short flights with
$|x_{f}|<2\pi$ (motion in the same spatial frame in the co-moving frame of
reference) correspond to the motion in the laboratory frame when two
successive events $U=c$ occur on the space interval less than the meander’s
spatial period $2\pi/k$. Ballistic flights between the spatial frames in the
co-moving frame with $|x_{f}|>2\pi$ correspond to the motion in the laboratory
frame when the particles move through more than one meander’s crest between
two successive events $U=c$.
### 2.2 Turning points
As in Paper I, we will characterize statistical properties of chaotic
transport by probability density functions (PDFs) of lengths of flights
$P(x_{f})$ and durations of flights $P(T_{f})$ for a number of very long
chaotic trajectory. Both regular and chaotic particles may change many times
the sign of their zonal velocity $\dot{x}=u$. From the condition $\dot{x}=0$
in Eq. (4), it is easy to find the equations for the curves which are loci of
turning points
$Y_{\pm}(x,A)=\pm L\sqrt{1+A^{2}\sin^{2}{x}}\times\\\
\times\operatorname{Arsech}{\sqrt{{LC\sqrt{1+A^{2}\sin^{2}{x}}}}}+A\cos x.$
(6)
We consider the northern curve, i. e., Eq. (6) with the positive sign. Taking
into account that the perturbation has the form (5), we realize that all the
northern turning points are inside a strip confined by two curves of the form
(6) with $A=A_{0}\pm\varepsilon$. Let us analyze the derivative over the
varying parameter A
$\frac{\partial Y}{\partial A}=\cos x+\\\
+\frac{ACL^{2}\sin^{2}x}{2D}\biggl{(}2\operatorname{Arsech}{\sqrt{D}}-\frac{1}{\sqrt{1-D}}\biggr{)},$
(7)
where $D=LC\sqrt{1+A^{2}\sin^{2}x}$. If the derivative at a fixed value of $x$
does not change its sign on the interval $A_{0}-\varepsilon\leq A\leq
A_{0}+\varepsilon$, then $Y$ varies from $Y(x,A_{0}-\varepsilon)$ to
$Y(x,A_{0}+\varepsilon)$, and for each value of $y$ we have a single value of
the perturbation parameter $A$. However, there may exist such values of $x$
for which the equation $\partial Y/\partial A=0$ has a solution on the
interval mentioned above. In this case one may have more than one values of
$A$ for a single value of $y$. Thus, the width of the strip, containing
turning points, is defined by the values of $Y$ at the extremum points and at
the end points of the interval of the values of $A$. In Fig. 1 c we show the
turning points of a single chaotic trajectory on the cylinder $0\leq x\leq
2\pi$ confined between two corresponding curves.
In the numerical simulation throughout the paper we use the Runge-Kutta
integration scheme of the fourth order with the constant time step $\Delta
t0.0247$. To study chaotic transport we have carried out numerical experiments
with tracers initially placed in the stochastic layer. It was found that
statistical properties of chaotic transport practically do not depend on the
number of tracers provided that the corresponding trajectories are
sufficiently long ($t\simeq 10^{8}$). The PDFs for the lengths $x_{f}$ and
durations $T_{f}$ of flights for five tracers with the computation time
$t=5\cdot 10^{8}$ for each tracer are shown in Figs. 2 a and b, respectively,
both for the eastward (e) and westward (w) motion. Both $P(x_{f})$ and
$P(T_{f})$ are complicated functions with local extrema decaying in a
different manner for different ranges of $x_{f}$ and $T_{f}$. The main aim of
our study of chaotic transport is to figure out the basic peculiarities of the
statistics and attribute them to specific zones in the phase space, namely, to
dynamical traps strongly influencing the transport.
Figure 2: Probability density functions of (a) lengths $x_{f}$ and (b)
durations $T_{f}$ of the westward (w) and eastward (e) flights. The PDFs
$P_{\rm w}(T_{f})$ and $P_{\rm e}(T_{f})$ are normalized to the number of
westward ($4.23\cdot 10^{7}$) and eastward ($4\cdot 10^{7}$) flights,
respectively. Statistics for five tracers with the computation time $t=5\cdot
10^{8}$ for each one.
## 3 Rotational-islands traps
It is well known, that in nonlinear Hamiltonian systems a complicated
structure of the phase space with islands, stochastic layers, and chains of
islands, immersed in a stochastic sea, arises under a perturbation due to a
variety of nonlinear resonances and their overlapping [24]. The motion is
quasiperiodic and stable in the islands. The boundaries of the islands are
absolute barriers to transport: particles can not go through them neither from
inside nor from outside. Invariant curves of the unperturbed system (see Fig.
1 a) are destroyed under the perturbation (5) (see Fig. 1 b). As the
perturbation strength $\varepsilon$ increases, a closed invariant curve with
frequency $f$ is destroyed at some critical value of $\varepsilon$. If the
$f/\omega$ is a rational number, the corresponding curve is replaces by an
island chain, while the curves with irrational frequencies are replaced by
cantori (for a review see [25]). There are uncountably many cantori forming a
complicated hierarchy. Numerical experiments with a variety of Hamiltonian
systems with different number of degrees of freedom provide an evidence for
the presence of strong partial barriers to transport around the island’s
boundaries (for review, see [12]) which manifest themselves on Poincaré
sections as domains with increased density of points.
In Paper I we have found that with chosen values of the control parameters
there exist in each frame a vortex core (which is an island of the primary
resonance $\omega=f$) immersed into a stochastic sea, where there are six
islands of a secondary resonance emerged from three islands of the primary
resonance $3f=2\omega$ (see Fig. 3 in Paper I). Chains of smaller size islands
are present around the vortex core and the secondary-resonance islands.
Particles belonging to all of these islands (including the vortex core) rotate
in the same frame performing short flights with the lengths $|x_{f}|<2\pi$. So
we will call them rotational islands and distinguish from the so-called
ballistic islands to be considered below.
Stickiness of particles to boundaries of the rotational islands has been
demonstrated in Paper I. It means that real fluid particles can be trapped for
a long time in a singular zone nearby the borders of the rotational islands
which we will call rotational-islands traps (RITs). To illustrate the effect
of the RITs we demonstrate in Figs. 3 and 4 the Poincaré sections of a chaotic
trajectory in the frame $0\leq x\leq 2\pi$ sticking to the vortex core and to
the secondary-resonance islands, respectively. The contour of the vortex core
is shown in Fig. 4 by the thick line. The small points are tracks of the
particle’s position at the moments of time $t_{n}=2\pi n/\omega$ (where
$n=1,2,\dots$) and the thin curves are fragments of the corresponding
trajectory on the phase plane. Increased density of points indicates the
presence of dynamical traps near the boundaries of the rotational islands.
Contribution of the vortex-core RIT (Fig. 3) to chaotic transport is expected
to be much more significant than the one of the RITs of the other islands
(Fig. 4).
Figure 3: The vortex-core trap. Poincaré section of a chaotic trajectory in
the frame $0\leq x\leq 2\pi$ with a fragment of a trajectory.
Figure 4: The secondary resonance islands trap. A fragment of a chaotic
trajectory sticking to the islands is shown.
Figure 5: The PDFs for the eastward (e) and westward (w) flights with the
length shorter than $2\pi$. The PDFs $P_{\rm w}(T_{f})$ and $P_{\rm e}(T_{f})$
are normalized to the number of westward ($4.19\cdot 10^{7}$) and eastward
($3.7\cdot 10^{7}$) flights, respectively. Statistics for five tracers with
the computation time $t=5\cdot 10^{8}$ for each one.
Figure 6: The vortex-core trap PDFs of durations $T_{f}$ of the eastward (e)
and westward (w) flights. (a) Regular quasiperiodic trajectory with the
duration $t=2\cdot 10^{5}$ inside the vortex core close to its boundary. Both
the PDFs are normalized to the number $8\cdot 10^{3}$ of corresponding
flights. (b) Chaotic trajectory with the duration $t=2\cdot 10^{5}$ sticking
to the boundary of the vortex core from the outside. Both the PDFs are
normalized to the number $4\cdot 10^{3}$ of corresponding flights.
It is reasonable to suppose that RITs contribute to the statistics of short
flights. By short flights we mean the flights with the length shorter than
$2\pi$. In Fig. 5 we show the part of the full PDF $P(T_{f})$ (Fig. 2 b) for
the eastward (e) and westward (w) short flights separately. There are a
comparatively small number of the eastward flights with $T_{f}<11$. Let us
note the prominent peak of the corresponding PDF at $T_{f}\simeq 11$ followed
by an exponential decay. As to the westward short flights, there are two small
local peaks around $T_{f}\simeq 17$ and $21$.
To estimate the contribution of the vortex-core RIT to the statistics of short
flights, we compute and compare the statistics of the durations of flights
$T_{f}$ for two trajectories: a regular quasiperiodic one with the initial
position close to the inner border of the vortex core (Fig. 6 a) and a chaotic
one with the initial position close to the vortex-core border from the outside
(Fig. 6 b). Each full rotation of a particle in a frame consists of two
flights, eastward and westward, with different values of $T_{f}$ because of
the zonal asymmetry of the flow. The statistics for the chaotic trajectory,
sticking to the vortex core (Fig. 6 b), may be considered as a distribution of
the durations of flights in the vortex-core RIT. The minimal flight duration
in this RIT is $T_{f}\simeq 11$ (the flights with smaller values of $T_{f}$
are rare and they occur outside the trap). Positions of the local maxima of
the PDF for the sticking trajectory in Fig. 6 b correlate approximately with
the corresponding local maxima of the PDF for the regular trajectory inside
the core in Fig. 6 a. The similar correlations have been found (but not shown
here) between the local maxima of the PDFs for the lengths of flights
$P(x_{f})$ for the interior regular and sticking chaotic trajectories. These
correlations and positions of the peaks prove numerically that short flights
with $|x_{f}|<2\pi$ and $11\lesssim T_{f}\lesssim 21$ may be caused by the
effect of vortex-core RIT. We conclude from Fig. 6 b that the vortex-core RIT
contributes to the statistics of the short flights in the range $11\lesssim
T_{f}\lesssim 20$ for the eastward flights with the prominent peak at
$T_{f}\simeq 11$ and in the range $15\lesssim T_{f}\lesssim 21$ for the
westward flights with small peaks at $T_{f}\simeq 17$ and $21$.
The effect of the RIT of the secondary-resonance islands is illustrated in
Fig. 4. To find the characteristic times of this RIT we compute two
trajectories: a regular quasiperiodic one with the initial position inside one
of these islands and a chaotic one with the initial position close to the
outer border of the island. The respective PDFs $P(T_{f})$, shown in Figs. 7 a
and b, demonstrate strong correlations between the corresponding peaks at
$T_{f}\simeq 12,23$, and $27$. Computed (but not shown here) PDFs $P(x_{f})$
for these trajectories confirm the effect of the islands RIT on the statistics
of short flights.
Figure 7: The secondary-resonance islands trap. The PDFs of durations $T_{f}$
of the eastward (e) and westward (w) flights. (a) Regular quasiperiodic
trajectory inside the islands with the duration $t=5\cdot 10^{5}$. Both the
PDFs are normalized to the number $1.5\cdot 10^{4}$ of corresponding flights.
(b) Chaotic trajectory sticking to the island’s boundary from the outside with
the duration $t=5\cdot 10^{5}$. $P_{\rm w}(T_{f})$ and $P_{\rm e}(T_{f})$ are
normalized to the number of westward ($1.1\cdot 10^{4}$) and eastward ($9\cdot
10^{3}$) flights, respectively.
## 4 Saddle traps
As a result of the periodic perturbation (5), the saddle points of the
unperturbed system (4) at $x_{s}^{(n)}=2\pi n$,
$y_{s}^{(n)}=L\operatorname{Arcosh}\sqrt{1/LC}+A$ and at
$x_{s}^{(s)}=(2n+1)\pi$, $y_{s}^{(s)}=-L\operatorname{Arcosh}\sqrt{1/LC}-A$
($n=0,\pm 1,\dots$) become periodic saddle trajectories. These hyperbolic
trajectories have their own stable and unstable manifolds and play a role of
specific dynamical traps which we call saddle traps (ST). In this section we
demonstrate that the STs influence strongly on chaotic mixing and transport of
passive particles and contribute, mainly, in the short-time statistics of
flights.
Tracers with initial positions close to a stable manifold of a saddle
trajectory are trapped for a while performing a large number of revolutions
along it. To illustrate the effect of the STs we show in Fig. 8 a and b
fragments of two chaotic trajectories sticking to the saddle trajectory and
performing about 20 full revolutions before escaping to the east (Fig. 8 a)
and to the west (Fig. 8 b). We have managed to detect and locate the
corresponding periodic unstable saddle trajectory which is situated in Figs. 8
a and b in the domain where a few fragments of the chaotic trajectory imposed
on each other. Because of the flow asymmetry, the duration of eastern flights
of a particle along the saddle trajectory $T_{\rm e}\simeq 11.9$ is shorter
than the duration of western flights $T_{\rm w}\simeq 12.9$. The black points
are the tracks of the particle’s positions on the flow plane at the moments of
time $t_{n}=2\pi n/\omega\simeq 24.8\,n$ (where $n=1,2,\dots$). They belong to
smooth curves which are fragments of the stable and unstable manifolds of the
saddle trajectory at the chosen initial phase $\phi=\pi/2$.
Figure 8: The saddle trap. Fragments of two chaotic trajectories sticking to
the periodic saddle trajectory one of which escapes to the east (a) and
another one to the west (b). (c) The number of the eastward $(N_{\rm e})$ and
westward $(N_{\rm w})$ short flights with duration $T_{f}$ for those two
trajectories. Statistics with two trajectories with the duration $t=10^{3}$
and the total number of western $N_{\rm w}=55$ and eastern $N_{\rm e}=51$
flights.
To estimate the contribution of the STs to the statistics of short flights
shown in Fig. 5, we compute and plot in Fig. 8 c the number of the eastward
$(N_{\rm e})$ and westward $(N_{\rm w})$ short flights with a given duration
$T_{f}$ for those two chaotic trajectories sticking to the saddle trajectory
arising from the saddle point with the position $x_{s}=0,y_{s}\simeq 2.02878$.
Each full rotation of the particles consists of an eastward flight with the
duration $T_{\rm e}\simeq 11.9$ and an westward flight with the duration
$T_{\rm w}\simeq 12.9$. The flights with $T_{\rm e}\simeq 11.9$ contribute to
the main peak in Fig. 5 and the flights with $T_{\rm w}\simeq 12.9$ to ‘‘the
wesward’’ plateau in that figure.
The mechanism of operation of the STs can be described as follows. Each saddle
trajectory $\gamma(t)$ possesses time-dependent stable $W_{s}(\gamma(t))$ and
unstable $W_{u}(\gamma(t))$ material manifolds composed of a continuous sets
of points through which pass at time $t$ trajectories of fluid particles that
are asymptotic to $\gamma(t)$ as $t\to\infty$ and $t\to-\infty$, respectively.
Under a periodic perturbation, the stable and unstable manifolds oscillate
with the period of the perturbation. It was firstly proved by Poincaré that
$W_{s}$ and $W_{u}$ may intersect each other transversally at an infinite
number of homoclinic points through which pass doubly asymptotic trajectories.
To give an image of a fragment of the stable manifold of the periodic saddle
trajectory, we distribute homogeneously $2.5\cdot 10^{5}$ particles in the
rectangular $[-0.4\leq x\leq 0.45;2\leq y\leq 2.1]$ and compute the time the
particles need to escape the rectangular. The color in Fig. 9 modulates the
time $T$ when particles with given initial positions $(x_{0},y_{0})$ reach the
western line at $x=-1$ or the eastern line at $x=1$. The particles with
initial positions marked by the black and white colors move close to the
stable manifold of the saddle trajectory and spend a maximal time near it
before escaping. The black and white diagonal curve in Fig. 9 is an image of a
fragment of the corresponding stable manifold. The particles with initial
positions to the north from the curve escape to the west along the unstable
manifold of the saddle trajectory whereas those with initial positions to the
south from the curve escape to the east along its another unstable manifold.
Figure 9: The saddle-trap map. Color modulates the time $T$ which $2.5\cdot
10^{5}$ particles with given initial positions ($x_{0},y_{0}$) need to reach
the lines at $x=-1$ or $x=1$ escaping to the west (w) and to the east (e),
respectively. The black and white diagonal curve is an image of a fragment of
the stable manifold of the saddle trajectory. The cross is a position of a
particle on that trajectory at the initial time moment. The integration time
is $t=500$.
We have found that particles quit the ST along the unstable manifolds in
accordance with specific laws. We distribute a large number of particles along
the segment with $x_{0}=0$ and $y_{0}=[2.02;2.06]$, crossing the stable
manifold $W_{s}$, and compute the time $T$ particles with given initial
latitude positions $y_{0}$ need to quit the ST. More precisely, $T(y_{0})$ is
a time moment when a particle with the initial position $y_{0}$ reaches the
lines with $x=-1$ or $x=1$. The ‘‘experimental’’ points in Fig. 10 a fit the
law $T_{\rm e}=(-85.81\pm 0.04)-(31.216\pm 0.007)\ln(y_{0s}-y_{0})$ for the
particles which quit the trap moving to the east and the law $T_{\rm
w}=(-60.61\pm 0.03)-(28.933\pm 0.006)\ln(y_{0}-y_{0s})$ for those particles
which move to the west when quitting the trap, where $y_{0s}=2.0405755472$ is
a crossing point of $W_{s}$ with the segment of initial positions.
Figure 10: (a) Time $T$ a particle with an initial latitude position $y_{0}$
needs to quit the saddle trap. (b) The number of short flights $n$ such a
particle performs before quitting the saddle trap. The ranges of $y_{0}$ from
which particles quit the trap moving to the west and east are denoted by ‘‘w’’
and ‘‘e’’, respectively.
The ST attracts particles and force them to rotate in its zone of influence
performing short flights, the number of which $n$ depends on particle’s
initial positions $y_{0}$. The $n(y_{0})$ is a steplike function (see Fig. 10
b) with the lengths of the steps decreasing in a geometric progression in the
direction to the singular point, $l_{j}=l_{0}\,q^{-j}$, where $l_{j}$ is the
length of the $j$-th step and $q\simeq 2.27$ for the western exits and
$q\simeq 2.20$ for the eastern ones. The seeming deviation from this law in
the range $y_{0}=[2.045;2.046]$ (see a small western segment between two
larger ones in Fig. 10 b) is explained by crossing the initial line
$y_{0}=[2.02;2.06]$ by the curve of zero zonal velocity $u$. To have the
correct law for the western exits, it is necessary to add the two segments of
that cut step. The asymmetry of the functions $T(y_{0})$ and $n(y_{0})$ is
caused by the asymmetry of the flow.
## 5 Ballistic-islands traps
Besides the rotational islands with particles moving around the corresponding
elliptic points in the same frame, we have found in Paper I ballistic islands
situated both in the stochastic layer and in the peripheral currents. Regular
ballistic modes [30] correspond to stable quasiperiodic inter-frame motion of
particles. Only the ballistic islands in the stochastic layer are important
for chaotic transport. Mapping positions of the regular ballistic trajectories
at the moments of time $t_{n}=2\pi n/\omega$ $(n=1,2,\dots)$ onto the first
frame, we obtain chains of ballistic islands both in the northern and southern
stochastic layers, i. e., between the borders of the northern (southern)
peripheral currents and of the corresponding vortex cores. A chain with three
large ballistic islands is situated in those stochastic layers. The particles,
belonging to these islands, move to the west, and their mean zonal velocity
can be easily calculated to be
$\left<u_{f}\right>=-2\pi/3T=-\omega/3\simeq-0.0845$. There are also chains of
smaller-size ballistic islands along the very border with the peripheral
currents.
We have demonstrated in Paper I a stickiness of chaotic trajectories to the
borders of those three large ballistic islands (see Figs. 6 and 7 in Paper I).
The Poincaré section with fragments of two chaotic trajectories in the
northern stochastic layer is shown in Fig. 11 a. One particle performs a long
flight sticking to the very border with the regular westward current, and
another one moves to the west sticking to the very boundaries of three large
ballistic islands. A magnification of a fragment of the border and tracks of a
sticking trajectory around a smaller-size ballistic island are demonstrated in
Fig. 11 b. Fig. 11 c demonstrates the effective size of the trap of the large
ballistic islands with tracks of a sticking trajectory around them.
Figure 11: (a) Poincaré section of the northern stochastic layer where
stickiness to the very border with the regular westward current and to three
large ballistic islands are shown. Increased density of points along the
border with the peripheral current is caused by the traps of the border
ballistic islands one of which is shown in (b). (c) The trap of the large
ballistic islands.
Figure 12: The distribution of a number of long westward flights with
$T_{f}\geq 10^{3}$ over their mean zonal velocities $\left<u_{f}\right>$. The
sharp peak corresponds to the trap connected with the very boundaries of the
large ballistic islands, the left wing — to a number of traps of families of
the border ballistic islands, and the right wing — to the trap situated around
the large ballistic islands. Statistics for five tracers with the total number
of long westward flights $N_{f}=5\cdot 10^{4}$ and the computation time
$t=5\cdot 10^{8}$ for each tracer.
It is reasonable to suppose that the ballistic-islands traps (BIT) contribute,
largely, to the statistics of long flights with $|x_{f}|\gg 2\pi$. All the
ballistic particles, moving both to the west and to the east, can finish a
flight and make a turn only in the strip shown in Fig. 1 c. The loci of the
corresponding turning points have a complicated fractal-like structure. We
consider further only long westward flights, taking place in the northern
stochastic layer, because it is much wider than the stochastic layer between
the regular central jet and the southern parts of the vortex cores where
eastward flights take place.
To distinguish between contributions of the traps of different ballistic
islands (and, maybe, other zones in the phase space) to the statistics of long
flights, we compute for five long chaotic trajectories (up to $t=5\cdot
10^{8}$) the distribution of a number of westward flights with $T_{f}\geq
10^{3}$ over the mean zonal velocities $\left<u_{f}\right>=x_{f}/T_{f}$ of the
particles performing such flights. The distribution in Fig. 12 has a prominent
peak centered at the mean zonal velocity $\left<u_{f}\right>\simeq-0.0845$
which corresponds to a large number of long flights of those particles (and
their trajectories) which stick to the very boundaries of the large ballistic
islands (see Fig. 11 a) moving with the mean velocity
$\left<u_{f}\right>\simeq-0.0845$. The flat left wing of the distribution
$N(\left<u_{f}\right>)$ corresponds to the traps of smaller-size ballistic
islands nearby the border with the peripheral current. There are different
families of these islands (see one of them in Fig. 11 b) with their own values
of the mean zonal velocity which are in the range
$-0.092\lesssim\left<u_{f}\right>\lesssim-0.0845$. Stickiness to the
boundaries of the border islands is weaker because they are smaller than the
large islands and their contribution to the statistics of long flights is
comparatively small.
The right wing of the distribution $N(\left<u_{f}\right>)$ with
$-0.084\lesssim\left<u_{f}\right>\lesssim-0.075$ deserves further
investigation. The value $\left<u_{f}\right>\simeq-0.075$ is a minimal value
of the zonal velocity for long westward flights possible in the northern
stochastic layer. Increasing the minimal duration of a flight from
$T_{f}=10^{3}$ to $T_{f}=(2\div 5)\cdot 10^{3}$, we have found splitting of
the broad distribution with $-0.084\lesssim\left<u_{f}\right>\lesssim-0.08$
into a number of small distinct peaks. Comparing trajectories with the values
of $\left<u_{f}\right>$ corresponding to these peaks, we have found that all
they move around the large ballistic islands. The particles with smaller
values of $\left<u_{f}\right>$ used to penetrate further to the south from the
islands more frequently than those with larger values of $\left<u_{f}\right>$
which prefer to spend more time in the northern part of the dynamical trap
connected with those islands. Thus, we attribute the right wing of the
distribution $N(\left<u_{f}\right>)$ to an effect of the trap situated around
the large ballistic islands.
To estimate the contribution of different BITs to the statistics of long
westward flights in Fig. 2 we have computed the PDFs $P(x_{f})$ and $P(T_{f})$
for particles performing westward flights with $x_{f}\geq 100$ and $T_{f}\geq
1000$ and with the mean zonal velocity $\left<u_{f}\right>$ to be chosen in
three different ranges shown in Fig. 12:
$-0.092\lesssim\left<u_{f}\right>\lesssim-0.085$ (particles sticking to the
border islands) $-0.085<\left<u_{f}\right>\lesssim-0.084$ (particles sticking
to the very boundary of three large islands), and
$-0.084<\left<u_{f}\right>\lesssim-0.075$ (the trap of the three large
islands). All the PDFs $P(x_{f})$ decay exponentially but with different
values of the exponents equal to $\nu\simeq-0.005$ and
$\nu\simeq-0.0018\div-0.0014$ for the traps of border and the large ballistic
islands, respectively. The tail of the PDF $P(x_{f})$ for westward flights,
shown in Fig. 2, decays exponentially with $\nu\simeq-0.0014$. Thus, the
contribution of the large island’s BIT to the statistics of long westward
flights is dominant. As to temporal PDFs $P(T_{f})$ for westward long flights,
they are neither exponential nor power-law like with strong oscillations at
the very tails. The slope for the border BITs is again smaller than for the
large ballistic islands trap.
## 6 Conclusion
A meandering jet is a fundamental structure in oceanic and atmospheric flows.
We described statistical properties of chaotic mixing and transport of passive
particles in a kinematic model of a meandering jet flow in terms of dynamical
traps in the phase (physical) space. The boundaries of rotational islands
(including the vortex cores) in circulation zones are dynamical traps (RITs)
contributing, mainly, to the statistics of short flights with $|x_{f}|<2\pi$.
Characteristic times and spatial scales of the RITs have been shown to
correlate with the PDFs for the lengths $x_{f}$ and durations $T_{f}$ of short
flights. The stable manifolds of periodic saddle trajectories play a role of
saddle traps (STs) with the specific values of the lengths and durations of
short flights of the particles sticking to the saddle trajectories. The
boundaries of ballistic islands in the stochastic layers (including those
situated along the border with the peripheral current) are dynamical traps
(BITs) contributing, mainly, to the statistics of very long flights with
$|x_{f}|\gg 2\pi$.
Dynamical traps are robust structures in the phase space of dynamical systems
in the sense that they present at practically all values of the corresponding
control parameters. We never know exact values of the parameters in real
flows, especially, in geophysical ones. We don’t know exactly the structure of
the corresponding phase space, however, we know that typical features, like
islands of regular motion, vortices, and jets, exist in real flows (see their
images in some laboratory flows in Ref. [31]). In this paper we chose specific
values of the control parameters for which specific PDFs have been computed
and explained by the effect of those dynamical traps that exist under the
chosen parameters. We have carried out computer experiments with different
values of the control parameters and found that the phase space structure has
been changed, of course, with changing the values of the parameters, but the
corresponding RITs, STs, and BITs with specific temporal and spatial
characteristics have been found to contribute to the corresponding statistics.
After finishing our work, we were acquainted with Ref. [32] where meridional
chaotic transport, associated with a similar kinematic model of a meandering
jet, has been studied by the method of lobe dynamics [33]. In difference from
our study of zonal chaotic transport, a geometric structure of cross jet
transport has been considered in Ref. [32] where values of the control
parameters have been chosen to be sufficiently large to break up the central
jet as a barrier to transport of particles across the jet. The mechanisms for
particles to cross the jet have been described in terms of lobe dynamics and
the mean time to cross the jet for particles entering the jet and the mean
residence time for particles in the jet have been estimated in Ref. [32]. We
have studied a more realistic situation (at least, for surface oceanic jet
currents) when the jet is an absolute barrier to cross jet transport and we
explained statistical properties of transport in terms of dynamical traps of
saddle trajectories, rotational and ballistic islands. The method of lobe
dynamics is hardly applicable for study zonal chaotic transport since it is
practically impossible to trace out lobe evolution for a large number of
frames.
## Acknowledgments
The work was supported by the Russian Foundation for Basic Research (Grant no.
06-05-96032), by the Program ‘‘Mathematical Methods in Nonlinear Dynamics’’ of
the Russian Academy of Sciences, and by the Program for Basic Research of the
Far Eastern Division of the Russian Academy of Sciences.
## References
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* [2] G.M. Zaslavsky, Phys. D. 168-169, 292 (2002).
* [3] H.A. Dijkstra, Nonlinear physical oceanography (Dordrecht, Kluwer, 2000).
* [4] S. Wiggins, Annu. Rev. Fluid Mech. 37, 295 (2005).
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* [18] J. Sommeria, S.D. Meyers, and H.L. Swinney, Nature (London) 337, 58 (1989).
* [19] R.P Behringer, S.D. Meyers, and H.L. Swinney, Phys. Fluids A 3, 1243 (1991).
* [20] T.H. Solomon, E.R. Weeks, and H.L. Swinney, Phys. Rev. Lett. 71, 3975 (1993).
* [21] T.H. Solomon, E.R. Weeks, and H.L. Swinney, Phys. D 76, 70 (1994).
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* [24] B.V. Chirikov, Phys. Rep. 52, 263 (1979).
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* [26] K.V. Koshel and S.V. Prants, Physics-Uspekhi 49, 1151 (2006) [Uspekhi Fizicheskikh Nauk 176, 1177 (2006)].
* [27] G.M. Zaslavsky, Chaos 5, 653 (1995).
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* [29] B.A. Petrovichev, A.V. Rogalsky, R.Z. Sagdeev, and G.M. Zaslavsky, Phys. Lett. A 150, 391 (1990).
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|
arxiv-papers
| 2011-12-20T03:42:57 |
2024-09-04T02:49:25.493570
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Yu. Uleysky, M. V. Budyansky, and S. V. Prants",
"submitter": "Michael Uleysky",
"url": "https://arxiv.org/abs/1112.4556"
}
|
1112.4581
|
# Limits on the Gas Disk Content of Two “Evolved” T Tauri Stars
Alan G. Aversa Steward Observatory, 933 N Cherry Ave., Tucson AZ 85721
###### Abstract
We derived upper limits of the circumstellar gas disk masses around the T
Tauri stars St 34 and RX J0432.8+1735 in order to place constraints on
theories of planet formation and to explore the evolution of the gas-to-dust
ratio during the epoch of disk dissipation around young sun-like stars. Since
sub-millimeter lines of 12CO trace of the cold, outer regions of circumstellar
disks, we observed 12CO $J=2-1$ emission with the 10 m Sub-Millimeter
Telescope (SMT) for two carefully chosen targets. St 34 is a rare classical T
Tauri star with an age of $8\pm 3$ Myr, and RX J0432.8+1735 is a rare weak-
emission T Tauri star with far-infrared excess. Both exhibit radial space
motion enabling us to distinguish disk emission from ambient cloud material.
Assuming a 12CO excitation temperature of 20 K, a 12CO line-width of 5 km s-1,
and optically-thin emission, we derive $3\sigma$ upper limits on the H2
circumstellar disk mass for St 34 and RX J0432.8+1735 to be $<4.20$ M⊕ for
both disks. Placing these results in the context of other studies, we discuss
their implications on planet formation models.
## 1 Introduction
Circumstellar disks of gas and dust, a natural result of the conservation of
angular momentum, are a common outcome of the star formation process. Kenyon &
Hartmann (1995) find that over half the low mass ($<$ 2 M⊙) pre–main-sequence
T Tauri stars in the Taurus-Auriga star formation region have more infrared
emission than expected from a normal stellar photosphere, indicating the
presence of a dusty circumstellar disk heated by the parent star as well as
active accretion. T Tauri stars fall into two categories: Weak-Line T Tauri
Stars (WTTSs), characterized by low H$\alpha$ equivalent widths, and Classical
T Tauri Stars (CTTSs), with higher H$\alpha$ equivalent widths indicative of
ongoing gas accretion. These circumstellar disks generally have the following
properties (Dutrey et al., 2007; Andrews et al., 2009): mass surface densities
$\Sigma(r)\propto r^{0\,\mathrm{to}\,-1.0}$, surface temperatures $T(r)\propto
r^{0\,\mathrm{to}\,-0.6}$(depending on disk flaring), and Keplerian rotational
velocities $V(r)\propto r^{0.5}$.
Gas giant planet formation depends upon the gas content of the circumstellar
disk from which they form. Primordial inner disks traced by hot dust disappear
after about 3 Myr with a range of $1-10$ Myr (Meyer et al., 2007). If gas
content and dust content in disks dissipate through similar mechanisms
(Damjanov et al., 2007), then we would expect gas to disappear on these same
timescales. To understand the timescales of planet formation, we must
understand how long a gas disk persists around its parent star. Accretion
rates (higher in CTTSs than WTTSs) also trace the time evolution of gas
content as gas must be present to accrete. Durisen et al. (2007) suggest that
gravitational instabilities in gas disks could account for their rapid ($<3$
Myr) dissipation, locking up mass in planets. In contrast, simulations of the
evolution of gas disk surface density for a 1 M☉ star due to photo-evaporation
indicate that gas disks disappear in about 6 Myr (Alexander et al., 2006;
Dullemond et al., 2007). Because there are various theoretical results for the
gas dispersal mechanisms and associated timescales, we might expect to observe
diverse properties for gas disks around T Tauri stars.
Ideally, sub-millimeter interferometric images of T Tauri stars would yield
the most information about circumstellar disks, including their orientation,
geometry, and gas content. Yet observations of this sort have only been
published for a few nearby stars (e.g., Sub-Millimeter Array (SMA)
observations of TW Hya; Qi et al., 2004). Infrared photometric campaigns with
Spitzer, such the Cores to Disks (Evans et al., 2003) and the Formation and
Evolution of Planetary Systems (FEPS; Meyer et al., 2006) Spitzer Legacy
projects, provide knowledge of the dust circumstellar disks based upon IR
excesses at many wavelengths including $24\micron$, which trace dust within a
few AU of the parent star, and at $70\micron$, which trace cooler dust at
larger radii. Silverstone et al. (2006) surveyed 74 stars with ages between
3-30 Myr finding no stars with mid–IR excess that were not gas rich accreting
T Tauri stars. Padgett et al. (2006) however identified a handful of WTTS
(lacking signatures of accretion) with evidence for mid– and far–IR excess
emission. Photometric IR observations do not, however, constrain the total gas
content in circumstellar disks because of a potentially highly variable gas-
to-dust ratio. Observing the rotational energy transitions of 12CO, a proxy
for molecular hydrogen (H2), and assuming the ISM abundance ratio
$\left[\mathrm{H}_{2}/\mathrm{CO}\right]\approx 10^{4}$ enables one to trace
the majority of cold gas in disks out to radii many times larger. Is the
timescale for gas dissipation similar to the timescale for dust dissipation
around T Tauri stars? This is the central question we address with our new
observations.
Expanding on previous works (e.g. Pascucci et al., 2006; van Kempen et al.,
2007), we search for 12CO $J=2-1$ emission for two “evolved” T Tauri stars in
Taurus to place constraints on their gas circumstellar disk masses. St 34 is
evolved in the sense that it is $8\pm 3$ Myr old, whereas RX J0432.8+1735 is
evolved in the sense that—although it is much younger ($1.0\pm 0.5$ Myr)—it
has already lost its inner accretion disk. In §2, we describe our selection of
sources and observations, then in §3 we derive upper limits on our sources’
gas disk masses. Lastly, in §4 we describe the implications of our results,
and in §5 we list our conclusions.
## 2 Observations
### 2.1 Selection of Sources
We observed the 12CO $J=2-1$ emission of the Taurus region objects St 34 and
RX J0432.8+1735, T Tauri stars with known 24 or $70\micron$ excesses (Kenyon &
Hartmann, 1995; Padgett et al., 2006, see Table 2.1). In order to detect gas
in their circumstellar disks, we must be able to distinguish emission from the
disk and surrounding molecular cloud. From available candidate “evolved” (in
age or shape of spectral energy distribution) T Tauri stars, we selected those
least likely to be contamined by the ambient CO emission from the parent
molecular cloud based on the Dame et al. (1987) CO survey. The radial
velocities of our sources differ from the systemic velocities of any CO
emission in the vicinity of our sources by $\sim 2-3$ km s-1.
Table 1: Candidate Source Summary
Object | $\alpha$ | $\delta$ | Heliocentric RV | Source $V_{\mathrm{LSR}}$aaWe corrected the heliocentric radial velocities (RVs) from the literature into local standard of rest ($V_{\mathrm{LSR}}$) radial velocities assuming the sun moves toward J2000 $(\alpha=18\fh 0,\delta=30\fdg 0)$ at 20 km s-1. | Cloud $V_{\mathrm{LSR}}$bbDetermined by the 12CO $J=1-0$ emission from the Dame et al. (1987) survey | $T_{\mathrm{eff}}$ | $\log{L^{\star}}$ | Mass | Age | General
---|---|---|---|---|---|---|---|---|---|---
| (J2000) | (J2000) | (km s-1) | (km s-1) | (km s-1) | (K) | ($\log{L_{\sun}}$) | ($M_{\sun}$) | (Myr) | Reference
RX J0432.8+1735 | 04 32 53.23 | 17 35 33.68 | $18.6$ | 7.3 | 10.2 | 3499ccWichmann et al. (2000) | $\sim-6.1\times 10^{-2}$ggEstimated by integrating the spectral energy distribution (SED) of RX J0432.8+1735 in Padgett et al. (2006) | $\cdots$ | $1\pm 0.5$ddD’Antona & Mazzitelli (1997) | a
St 34 | 04 54 23.70 | 17 09 54.00 | $17.9\pm 0.6$ | 6.6 | 8.37 | 3415 | $1.03\pm 0.06$ | $0.37\pm 0.08$eeFor both binary components | $8\pm 3$e,fe,ffootnotemark: | b
fffootnotetext: Isochronal age is given. The Li depletion age for both binary
components is $>25$ Myr.
References. — For RVs, see Wichmann et al. (2000); White & Hillenbrand (2005).
For general references, see (a) Padgett et al. (2006) and (b) White &
Hillenbrand (2005).
#### 2.1.1 St 34
St 34 (HBC 425) is a binary system of two CTTSs, separated by $\lesssim 0.78$
AU (White & Hillenbrand, 2005) based on the orbital solution for the system
(Downes & Keyes, 1988) in the Taurus-Auriga T association (Kenyon & Hartmann,
1995). Both components of the spectroscopic binary have roughly equal mass and
spectral types of M3 (White & Hillenbrand, 2005). White & Hillenbrand (2005)
observed St 34 in the optical with the HIRES spectrograph at Keck and derived
an isochronal age of $8\pm 3$ Myr for both components of the binary. Since
they did not detect any lithium (7Li) in the spectrum, St 34 must have
reached—assuming the stars are completely convective—an internal temperature
$>2\times 10^{6}$ K and since depleted all of its lithium. St 34 has a low
accretion rate of $2.5\times 10^{-10}$ M☉ yr-1, and the maximum radial
velocity difference between the two binary components of St 34 is 58.4 km s-1
(White & Hillenbrand, 2005). St 34, being one of the oldest known pre–main-
sequence (PMS) star still accreting from a proto-planetary disk, also has a
low dust mass of $\sim 2\times 10^{-10}$ M☉ for radii $\lesssim 0.7$ AU
(Hartmann et al., 2005).
#### 2.1.2 RX J0432.8+1735
RX J0432.8+1735 is a WTTS of spectral type M2 (Martín & Magazzù, 1999). Based
on the PMS tracks of D’Antona & Mazzitelli (1997) RX J0432.8+1735 is estimated
to be $1.0\pm 0.5$ Myr old. Padgett et al. (2006) observed RX J0432.8+1735
with Spitzer and noticed that its $24\micron$ flux is in excess of the
expected photospheric value by a factor of 3. Its lack of IR excess $\leq
12\micron$ suggests there may be a large inner hole in the disk. Based on
ROSAT observations, Carkner et al. (1996) discovered that RX J0432.8+1735 is
also an X-ray source. As RX J0432.8+1735 is classified as a WTTS star with no
estimates of its accretion rate, we assume it is not accreting.
### 2.2 Observing Procedure
On 26-27 November 2007, we observed the 12CO $J=2-1$ (230.53799 GHz) emission
line of our two T Tauri stars with the 10 m Heinrich Hertz Sub-Millimeter
Telescope (HHT) on Mt. Graham, Arizona. Observations were obtained with a 1 mm
dual polarization (Vpol, Hpol), sideband-separating, ALMA prototype receiver.
The upper sideband was tuned to 12CO $J=2-1$ while the lower sideband was
tuned to 13CO $J=2-1$. We used the Forbes Filter Bank (FFB) backend in 4 IF
mode, an upper and lower sideband each with 1 MHz and 250 kHz of spectral
resolution, respectively. The channel width, $\Delta\nu_{\mathrm{ch}}$, of our
spectrometer was $0.33$ km s-1. The 1 MHz resolution data were used to
determine main beam efficiencies $\eta_{\mathrm{mb}}$, and the $250$ kHz
resolution data were used to measure the 12CO line.
Table 2: Observation of Main Beam Efficiencies $\eta$ Planet | $\eta_{\mathrm{Vpol}}$ | $\eta_{\mathrm{Hpol}}$ | $\eta_{\mathrm{Vpol}}/\eta_{\mathrm{Hpol}}$
---|---|---|---
MarsaaAll Mars brightness temperature errors assumed to be 5% | $0.80\pm 0.05$ | $0.72\pm 0.04$ | $1.10\pm 0.09$
Mars | $0.80\pm 0.05$ | $0.71\pm 0.04$ | $1.12\pm 0.09$
Saturn | $0.93\pm 0.04$ | $0.70\pm 0.03$ | $1.33\pm 0.08$
Saturn | $0.94\pm 0.04$ | $0.69\pm 0.03$ | $1.36\pm 0.09$
Venus | $0.89\pm 0.03$ | $0.66\pm 0.03$ | $1.35\pm 0.08$
Venus | $0.88\pm 0.03$ | $0.66\pm 0.03$ | $1.34\pm 0.08$
Mars | $0.87\pm 0.05$ | $0.66\pm 0.04$ | $1.31\pm 0.10$
MarsbbEnd of first night | $0.87\pm 0.05$ | $0.66\pm 0.04$ | $1.33\pm 0.10$
MarsccBeginning of second night | $0.88\pm 0.05$ | $0.68\pm 0.04$ | $1.29\pm 0.10$
Mars | $0.89\pm 0.05$ | $0.68\pm 0.04$ | $1.31\pm 0.11$
Mars | $0.84\pm 0.04$ | $0.70\pm 0.04$ | $1.21\pm 0.10$
Mars | $0.88\pm 0.05$ | $0.68\pm 0.04$ | $1.29\pm 0.10$
Venus | $0.93\pm 0.03$ | $0.68\pm 0.03$ | $1.38\pm 0.09$
Using CLASS in the GILDAS data reduction package, we estimated the main beam
efficiencies by observing the planets shown in Table 2. Typical sideband
rejections, ignored in the calibration, were $>10$ dB. The main beam
efficiency $\eta$ was computed following Mangum (1993) and corrected for
single-sideband observations:
$\eta=\frac{T_{A}^{\star}(\mathrm{planet})}{J(\nu_{s},T_{\mathrm{planet}})-J(\nu_{s},T_{\mathrm{cmb}})}\times\left[1-\exp{\left(-\ln{(2)}\frac{\theta_{\mathrm{eq}}\theta_{\mathrm{pol}}}{\theta_{\mathrm{mb}}^{2}}\right)}\right]^{-1},$
(1)
where
$J(\nu,T_{b})=\frac{h\nu/k}{e^{\frac{h\nu}{kT_{\mathrm{mb}}}}-1}$ (2)
is the Planck function at brightness temperature $T_{b}$ and frequency $\nu$,
$T_{A}^{\star}$ is the single-sideband antenna temperature of the planet,
$T_{\mathrm{planet}}$ is the planet’s observed brightness temperature,
$T_{\mathrm{cmb}}=2.73$ K, $\theta_{\mathrm{eq}}$ and $\theta_{\mathrm{pol}}$
are respectively the planet’s equatorial and poloidal diameters in arcseconds,
and $\theta_{\mathrm{mb}}=33\arcsec$ at $\nu=230$ GHz. We adopted an average
Venus brightness temperature $T_{b}$ from Kuznetsov et al. (1982) of $287\pm
20$ K. For all other planets’ $T_{b}$, we used the JCMT online
database111http://www.jach.hawaii.edu/jac-bin/planetflux.pl. We derived a
ratio $\eta_{\mathrm{Vpol}}/\eta_{\mathrm{Hpol}}$ of the two IF’s mean main
beam efficiencies for both nights of $1.24\pm 0.04$. We used this ratio to
scale the Hpol polarization’s antenna temperature up to match the level of the
Vpol polarization’s antenna temperature. After fitting a baseline to each
spectrum, we averaged the sum of the scaled Hpol brightness temperatures and
the Vpol brightness temperatures: $\frac{1}{2}\langle
T_{A}^{\star}(\mathrm{Hpol,scaled})+T_{A}^{\star}(\mathrm{Vpol})\rangle=T_{A}^{\star}(\mathrm{sum})$.
Thus we computed the corrected main beam temperature as
$T_{\mathrm{mb}}=\frac{T_{A}^{\star}(\mathrm{sum})}{\eta_{\mathrm{Vpol}}}.$
(3)
The average beam efficiencies were $\langle\eta_{\mathrm{Hpol}}\rangle=0.68\pm
0.01$ and $\langle\eta_{\mathrm{Vpol}}\rangle=0.88\pm 0.01$ for both nights.
Since we had null detections for our two sources, we must assume a line-width
to calculate upper limits on the integrated intensity. We assumed typcial a
line-width of $\Delta\nu=10$ km s-1 ($=7.69$ MHz). If we assume the CO line is
well described by a Gaussian line shape, then the uncertainty in the
integrated intensity is given by
$\sigma_{I}=\sigma_{T_{\mathrm{mb}}}\sqrt{\frac{3\Delta\nu_{\mathrm{ch}}\Delta\nu}{\sqrt{\ln{2}}}},$
(4)
where $\sigma_{I}$ and $\Delta v$ are the CO line fluxes and the the full
width at half maximum (FWHM) and $\Delta v_{\mathrm{ch}}$ is the channel
spacing 0.33 km s-1; see Appendix I of Schlingman et al. (in prep.). The
observations are summarized in Table 3.
Table 3: Observational Summary
Source | Integration Time | $\sigma_{T_{\mathrm{mb}}}$ | $\sigma_{I}$ | $\log_{10}{(F_{\nu})}$ | $\overline{N}(T\mathrm{ex}=10,20,100\mathrm{~{}K})$ | $M_{\mathrm{H}_{2}}(T_{\mathrm{ex}}=10,20,100\mathrm{~{}K})$
---|---|---|---|---|---|---
| (sec) | (K) | (K km s-1) | ($\log_{10}{(\mathrm{W~{}cm}^{-2})}$) | (1013 cm-2) | (M⊕)
RX J0432.8+1735 | 3600 | 0.019 | 0.046 | $<-23.02$ | $<12.7,7.47,6.53$ | $<3.59,1.10,1.84$
St 34 | 6120 | 0.019 | 0.046 | $<-23.02$ | $<12.7,7.47,6.53$ | $<3.59,1.10,1.84$
Note. — $T_{\mathrm{mb}}$ is the main beam corrected brightness temperature
and $I$ is the corresponding intensity assuming a line width of 5 km s-1. $F$
is the $3\sigma$ line flux upper limit.
Note. — That $\sigma_{T_{\mathrm{mb}}}$ for both objects is the same is a
fluke.
## 3 Results & Analysis
The main-beam corrected spectra of our observations are shown in Figure 1.
Figure 1: Spectra of St 34 (left) and RX J0432.8+1735 (right). The vertical
line labeled “Cloud” represents the estimated background cloud velocity
determined by the Dame et al. (1987) CO $J=1-0$ survey, and the vertical line
labeled with the source’s name indicates that source’s $V_{\mathrm{LSR}}$. For
St 34, we show with an arrow the difference between the radial velocities of
its binary components with an arrow centered on the systemic velocity.
While 9$\arcmin$ is a rough spatial scale for comparison to the 33$\arcsec$
beam of the SMT, and since we did not detect a 12CO line in any of our
sources, the on-cloud results from SMT are consistent with Dame et al. (1987).
It is unlikely that high spatial frequency variations of 33$\arcsec$ scales
over 9$\arcmin$ regions have systemic velocity shifts of 2-3 km-1
Since we did not detect any 12CO line, we convert our $3\sigma$ noise into an
upper limit on the flux. A knowledge of the flux will enable us to estimate
upper limits on gas disk mass.
The observed flux is the double-integral of the observed intensity $I_{\nu}$
over frequency $\nu$ and solid angle $\Omega$:
$F=\int\int
I_{\nu}\,d\nu\,d\Omega=\int\int\frac{2k\nu^{3}T_{b}}{c^{3}}\,dv\,d\Omega.$ (5)
Assuming the brightness temperature does not vary substantially over the
telescope beam and that the line is a Gaussian with line-width $\Delta v$,
then the upper limit on the $3\sigma$ line flux $F$ is
$F<\frac{2k\nu^{3}(3\sigma_{T_{\mathrm{mb}}})}{c^{3}}\frac{\pi\theta^{2}}{4\ln{2}}\sqrt{\frac{4\ln{2}}{\pi}}\times\Delta
v=(5.05\times
10^{-15})\sigma_{T_{\mathrm{mb}}}\mathrm{erg~{}s}^{-1}\mathrm{~{}cm}^{-2}.$
(6)
The $3\sigma$ upper limits on $F$ are listed in Table 3.
Similarly, we can derive the column density in the optically thin limit to be
$\overline{N}(T_{\mathrm{ex}})=\frac{8\pi
k\nu^{2}}{hc^{3}g_{u}A_{ul}}\mathcal{F}(T_{\mathrm{ex}},E_{u},\nu)\int
T_{\mathrm{mb}}\,dv,$ (7)
where
$\mathcal{F}(T_{\mathrm{ex}},E_{u},\nu)\equiv\frac{J_{\nu}(T_{\mathrm{ex}})Q(T_{\mathrm{ex}})\exp{\left(\frac{E_{u}}{kT_{\mathrm{ex}}}\right)}}{J_{\nu}(T_{\mathrm{ex}})-J_{\nu}(T_{\mathrm{cmb}})},$
(8)
$A_{ul}$ is the Einstein A coefficient (spontaneous emission) and has units of
s-1.
Similar to the analysis of Pascucci et al. (2006), we assume an excitation
temperature $T_{\mathrm{ex}}\approx 20$ K. Then in our case for 12CO $J=2-1$,
$\mathcal{F}(T_{\mathrm{ex}},E_{u},\nu)\approx 28.00$. The Einstein
$A_{ul}=6.91\times 10^{-7}$ s-1 and partition function
$Q(20\mathrm{~{}K})=15.9$ (CDMS222http://www.ph1.uni-koeln.de/vorhersagen/).
We compute and tabulate in Table 3 the 12CO number densities and H2 gas masses
in the optically thin limit. Gas disk masses were derived from Scoville et al.
(1986),
$M_{\mathrm{H}_{2}}<\overline{N}(T_{\mathrm{ex}})\times\Bigg{\\{}\left[\frac{\mathrm{H}_{2}}{\mathrm{CO}}\right]\mu_{G}m_{\mathrm{H}_{2}}\frac{\pi\theta^{2}}{4}d^{2}\Bigg{\\}}\mathrm{~{}g}=110\sigma_{T_{\mathrm{mb}}}\mathrm{~{}M}_{\earth},$
(9)
where $\left[\mathrm{H}_{2}/\mathrm{CO}\right]\approx 10^{4}$, $\mu_{G}=1.36$
is the mean molecular weight, $m_{\mathrm{H}_{2}}$ is the mass of an H2
molecule, and $d\approx 140$ pc is the distance to Taurus.
## 4 Discussion
Theories of gaseous planet formation require knowledge of (1) how much gas
there is in circumstellar disks initially and (2) the rate at which gas is
depleted over time. To answer the first question, upper limits on the amount
of gas in very young protostellar disk systems puts limits directly on the
amount of mass available for gaseous planet formation in the systems observed.
Answers to the second question, requiring large samples over a wide range of
ages, are also necessary because of the competition between the timescale for
planet formation and gas disk dispersion timescales (for a recent review, see
Meyer, 2009).
There are several ways of constraining gas disk masses, each with its own
advantages and disadvantages. To understand gas disk timescales, one can also
analyze H$\alpha$ emission line profiles and determine gas accretion rates and
thus constrain the gas mass surface density at the inner edge of the disk
tracing perhaps global disk evolution (e.g. Fedele et al., 2009). Using UV
tracers of gas emission, Ingleby et al. (2009) describe HST observations
searching for evidence of hot gas in emission finding no evidence for H2
emission for WTTS in their sample. Ultraviolet absorption line from a
continuum source (e.g. Roberge et al., 2005) can help constrain cold mass in
disks, but it requires a continuum source that is bright in the far UV and an
edge-on geometry; therefore, it is observationally feasible only in special
circumstances. Near IR fluorescent H2 traces gas with excitation temperatures
$T_{\mathrm{ex}}>2000$ K (Bary et al., 2003) and mid-IR ro-vibrational lines
(e.g. Najita et al., 2007; Thi et al., 2001; Pascucci et al., 2006),
especially the $28.2~{}\mu$m and $17\mu$m Spitzer bands, trace gas up to
50-200 K. Our observations of rotational lines of CO trace cooler gas at
larger orbital radii. The disadvantage of such measurements is that CO can
freeze out at the coldest temperatures corresponding to the outer limits of
the disk $\gtrsim 30$-100 AU. Hughes et al. (2010) present evidence for
evolving gas to dust ratios in transitional disks which exhibit evidence for
optically-thick outer disks but possess inner holes and gaps.
St 34, being $8\pm 3$ Myr old, might have a lower gas surface density than
typical CTTSs. It is still accreting gas (White & Hillenbrand, 2005), albeit
at a low rate, but retains at least a low density inner disk. Our
non–detection in a search for cold gas implies that most of its outer disk
must have disappeared or frozen out onto grains. White & Hillenbrand (2005)
show that St 34 has an accretion rate $\langle\dot{M}\rangle=8.3\times
10^{-10}$ M☉ yr-1, so after 1 Myr much more gas than our upper limit of $4.20$
M⊕ would accrete ($\sim 276$ M⊕). Perhaps St 34 recently lost its outer disk
through photoevaporation (e.g. Gorti & Hollenbach, 2009) and we are witnessing
the “last gasp” of accretion onto the star. This is possible but not likely,
as it requires current observations to be taking place at a very special time.
Conversely, RX J0432.8+1735—being a much younger, $1.0\pm 0.5$ Myr WTTS—must
have either evaporated its disk or formed planets from its gas disk faster
than normal. Its being a WTTS is consistent with our null detection of its gas
content, although remnant amounts of gas less than a few $M_{\earth}$ could
still exist in its outer disk. That RX J0432.8+1735 is relatively young and
does not have detectable gas content could be significant considering that
there are older systems, such as Hen 3-600, a binary system at between $1-10$
Myr of age with apparent WTTS and CTTS components (Jayawardhana et al., 1999);
TW Hya, a CTTS at 8-10 Myr (Webb et al., 1999); and DM Tau at $\sim 8$ Myr
(Guilloteau & Dutrey, 1994). Thus something about the disk evaporation physics
is dramatically different in RX J0432.8+1735 than these oldest systems.
In Figure 2 we compare our sources’ gas disk masses and ages with the gas mass
upper limits of those sources $\leq 30$ Myr from Pascucci et al. (2006) and
with the gas mass determinations (solid circles) of BP Tau (13CO $J=2-1$;
Dutrey et al., 2003); DL Tau, DO Tau (12CO $J=2-1$; Koerner & Sargent, 1995);
and DM Tau, DR Tau, GG Tau a, GM Aur, GO Tau, LkCa 15, RY Tau (12CO $J=3-2$
and 13CO $J=3-2$; Thi et al., 2001). We note that this is not an exhaustive
compilation from the literature, but representative of recent results.
Assuming a 1:10 gas-to-dust ratio (D’Alessio et al., 2005), we would expect RX
J0432.8+1735 and St 34 to have at most 0.420 M⊕ and 0.420 M⊕ of dust,
respectively, for $T_{\mathrm{ex}}=20$ K. For St 34, Hartmann et al. (2005)
estimates a disk mass of 665 M⊕ located in a circumbinary disks between the
“wall” (the region defined to surround the two components of the St 34 binary;
$\sim 0.7$ AU) and $7$ AU. Assuming this mass is representative of a total
disk mass in St 34 out to $7$ AU, this would be consistent with a small ($<10$
AU), optically-thick CO disk with a gas-to-dust ratio of $\sim 100$; we do not
detect this due to beam dilation (cf. Pascucci et al., 2006).
St 34 has infrared excess for wavelengths longer than 3.6 $\mu$m (Hartmann et
al., 2005) and RX J0432.8+1735 has infrared excess for wavelengths longer than
24 $\mu$m (Padgett et al., 2006), yet St 34 is a binary CTTS with an accreting
inner disk and RX J0432.8+1735 is a WTTS. Binaries tend to disrupt inner gas
disks (Jensen, 1996) and may decrease disk lifetimes (Monin et al., 2007).
However, Armitage & Clarke (1996) have argued that close binaries affect
angular momentum exchange in the natural evolution of accretion disks
resulting in longer lived outer disks. Indeed Thébault et al. (2004) find that
planet formation around binaries might require a long-lived but massive disk.
Since circumbinary disks allow for long gas disk lifetimes, St 34 might have
had more time to form planets.
Figure 2: Gas circumstellar disk mass versus age of selected sources: our RX
J0432.8+1735 and St 34 upper limits (labeled); upper limits from the Pascucci
et al. (2006) sample (unlabeled upper limits) with ages $\leq 30$ Myr (the
Kelvin-Hemholtz contraction timescale for a 1 M☉ star); and exact mass
determinations (solid circles) of BP Tau (13CO $J=2-1$; Dutrey et al., 2003);
DL Tau, DO Tau (12CO $J=2-1$; Koerner & Sargent, 1995); and DM Tau, DR Tau, GG
Tau a, GM Aur, GO Tau, LkCa 15, RY Tau (12CO $J=3-2$ and 13CO $J=3-2$; Thi et
al., 2001). We assume errors in stellar ages to be 50%.
## 5 Conclusions
Assuming optically thin disks ($\tau\ll 1$), an excitation temperature
$T_{\mathrm{ex}}=20$ K, and a line-width $\Delta v=10$ km s-1, we do not
detect significant amounts of gas around three T Tauri stars: $<4.20$ M⊕ for
the PMS binary St 34 and $<4.20$ M⊕ for RX J0432.8+1735. St 34, a CTTS, is
still accreting gas although it is $8\pm 3$ Myr old, and the gas disk of RX
J0432.8+1735, a WTTS of $1.0\pm 0.5$ Myr, has disappeared quickly. Future
observations of larger samples will be required to understand the diversity of
disk lifetimes as a function of stellar properties.
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|
arxiv-papers
| 2011-12-20T06:34:36 |
2024-09-04T02:49:25.503141
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alan G. Aversa",
"submitter": "Alan Aversa",
"url": "https://arxiv.org/abs/1112.4581"
}
|
1112.4592
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
LHCb-PAPER-2011-011 CERN-PH-EP-2011-209 14 December 2011
Measurement of charged particle multiplicities in $pp$ collisions at
${\sqrt{s}=7\mathrm{\,Te\kern-2.07413ptV}}$ in the forward region
The LHCb Collaboration111Authors are listed on the following pages.
Abstract
The charged particle production in proton-proton collisions is studied with
the LHCb detector at a centre-of-mass energy of
${\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}}$ in different intervals of
pseudorapidity $\eta$. The charged particles are reconstructed close to the
interaction region in the vertex detector, which provides high reconstruction
efficiency in the $\eta$ ranges $-2.5<\eta<-2.0$ and $2.0<\eta<4.5$. The data
were taken with a minimum bias trigger, only requiring one or more
reconstructed tracks in the vertex detector. By selecting an event sample with
at least one track with a transverse momentum greater than
1${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ a hard QCD subsample is investigated.
Several event generators are compared with the data; none are able to describe
fully the multiplicity distributions or the charged particle density
distribution as a function of $\eta$. In general, the models underestimate the
charged particle production.
Keywords: minimum bias, underlying event, particle multiplicities, LHC, LHCb
LHCb Collaboration
R. Aaij23, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi42, C. Adrover6, A.
Affolder48, Z. Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J.
Anderson39, R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L.
Arrabito53, A. Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16,
R.J. Barlow50, C. Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10,
Th. Bauer23, A. Bay38, I. Bediaga1, S. Belogurov30, K. Belous34, I.
Belyaev30,37, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson46, J.
Benton42, R. Bernet39, M.-O. Bettler17, M. van Beuzekom23, A. Bien11, S.
Bifani12, T. Bird50, A. Bizzeti17,h, P.M. Bjørnstad50, T. Blake37, F. Blanc38,
C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V. Bocci22, A. Bondar33, N.
Bondar29, W. Bonivento15, S. Borghi47,50, A. Borgia52, T.J.V. Bowcock48, C.
Bozzi16, T. Brambach9, J. van den Brand24, J. Bressieux38, D. Brett50, M.
Britsch10, T. Britton52, N.H. Brook42, H. Brown48, A. Büchler-Germann39, I.
Burducea28, A. Bursche39, J. Buytaert37, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,37, A. Cardini15, L. Carson49, K. Carvalho
Akiba2, G. Casse48, M. Cattaneo37, Ch. Cauet9, M. Charles51, Ph.
Charpentier37, N. Chiapolini39, K. Ciba37, X. Cid Vidal36, G. Ciezarek49,
P.E.L. Clarke46,37, M. Clemencic37, H.V. Cliff43, J. Closier37, C. Coca28, V.
Coco23, J. Cogan6, P. Collins37, A. Comerma-Montells35, F. Constantin28, G.
Conti38, A. Contu51, A. Cook42, M. Coombes42, G. Corti37, G.A. Cowan38, R.
Currie46, B. D’Almagne7, C. D’Ambrosio37, P. David8, P.N.Y. David23, I. De
Bonis4, S. De Capua21,k, M. De Cian39, F. De Lorenzi12, J.M. De Miranda1, L.
De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi38,37, M.
Deissenroth11, L. Del Buono8, C. Deplano15, D. Derkach14,37, O. Deschamps5, F.
Dettori24, J. Dickens43, H. Dijkstra37, P. Diniz Batista1, F. Domingo
Bonal35,n, S. Donleavy48, F. Dordei11, A. Dosil Suárez36, D. Dossett44, A.
Dovbnya40, F. Dupertuis38, R. Dzhelyadin34, A. Dziurda25, S. Easo45, U.
Egede49, V. Egorychev30, S. Eidelman33, D. van Eijk23, F. Eisele11, S.
Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch. Elsasser39, D. Elsby55, D.
Esperante Pereira36, L. Estève43, A. Falabella16,14,e, E. Fanchini20,j, C.
Färber11, G. Fardell46, C. Farinelli23, S. Farry12, V. Fave38, V. Fernandez
Albor36, M. Ferro-Luzzi37, S. Filippov32, C. Fitzpatrick46, M. Fontana10, F.
Fontanelli19,i, R. Forty37, M. Frank37, C. Frei37, M. Frosini17,f,37, S.
Furcas20, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini51, Y.
Gao3, J-C. Garnier37, J. Garofoli52, J. Garra Tico43, L. Garrido35, D.
Gascon35, C. Gaspar37, N. Gauvin38, M. Gersabeck37, T. Gershon44,37, Ph.
Ghez4, V. Gibson43, V.V. Gligorov37, C. Göbel54, D. Golubkov30, A.
Golutvin49,30,37, A. Gomes2, H. Gordon51, M. Grabalosa Gándara35, R. Graciani
Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E.
Greening51, S. Gregson43, B. Gui52, E. Gushchin32, Yu. Guz34, T. Gys37, G.
Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42, S. Hansmann-Menzemer11, R.
Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44, J. He7, V. Heijne23, K.
Hennessy48, P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, E.
Hicks48, K. Holubyev11, P. Hopchev4, W. Hulsbergen23, P. Hunt51, T. Huse48,
R.S. Huston12, D. Hutchcroft48, D. Hynds47, V. Iakovenko41, P. Ilten12, J.
Imong42, R. Jacobsson37, A. Jaeger11, M. Jahjah Hussein5, E. Jans23, F.
Jansen23, P. Jaton38, B. Jean-Marie7, F. Jing3, M. John51, D. Johnson51, C.R.
Jones43, B. Jost37, M. Kaballo9, S. Kandybei40, M. Karacson37, T.M. Karbach9,
J. Keaveney12, I.R. Kenyon55, U. Kerzel37, T. Ketel24, A. Keune38, B. Khanji6,
Y.M. Kim46, M. Knecht38, P. Koppenburg23, A. Kozlinskiy23, L. Kravchuk32, K.
Kreplin11, M. Kreps44, G. Krocker11, P. Krokovny11, F. Kruse9, K.
Kruzelecki37, M. Kucharczyk20,25,37,j, T. Kvaratskheliya30,37, V.N. La Thi38,
D. Lacarrere37, G. Lafferty50, A. Lai15, D. Lambert46, R.W. Lambert24, E.
Lanciotti37, G. Lanfranchi18, C. Langenbruch11, T. Latham44, C. Lazzeroni55,
R. Le Gac6, J. van Leerdam23, J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J.
Lefrançois7, O. Leroy6, T. Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M.
Liles48, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, J.H. Lopes2, E. Lopez
Asamar35, N. Lopez-March38, H. Lu38,3, J. Luisier38, A. Mac Raighne47, F.
Machefert7, I.V. Machikhiliyan4,30, F. Maciuc10, O. Maev29,37, J. Magnin1, S.
Malde51, R.M.D. Mamunur37, G. Manca15,d, G. Mancinelli6, N. Mangiafave43, U.
Marconi14, R. Märki38, J. Marks11, G. Martellotti22, A. Martens8, L. Martin51,
A. Martín Sánchez7, D. Martinez Santos37, A. Massafferri1, Z. Mathe12, C.
Matteuzzi20, M. Matveev29, E. Maurice6, B. Maynard52, A. Mazurov16,32,37, G.
McGregor50, R. McNulty12, C. Mclean14, M. Meissner11, M. Merk23, J. Merkel9,
R. Messi21,k, S. Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, J. Molina
Rodriguez54, S. Monteil5, D. Moran12, P. Morawski25, R. Mountain52, I. Mous23,
F. Muheim46, K. Müller39, R. Muresan28,38, B. Muryn26, B. Muster38, M. Musy35,
J. Mylroie-Smith48, P. Naik42, T. Nakada38, R. Nandakumar45, I. Nasteva1, M.
Nedos9, M. Needham46, N. Neufeld37, C. Nguyen-Mau38,o, M. Nicol7, V. Niess5,
N. Nikitin31, A. Nomerotski51, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen49, K. Pal52, J. Palacios39, A.
Palano13,b, M. Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo47, C.
Parkes47,37, C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49, S.K.
Paterson49, G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos
Alvarez36, A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A.
Petrolini19,i, A. Phan52, E. Picatoste Olloqui35, B. Pie Valls35, B.
Pietrzyk4, T. Pilař44, D. Pinci22, R. Plackett47, S. Playfer46, M. Plo
Casasus36, G. Polok25, A. Poluektov44,33, E. Polycarpo2, D. Popov10, B.
Popovici28, C. Potterat35, A. Powell51, T. du Pree23, J. Prisciandaro38, V.
Pugatch41, A. Puig Navarro35, W. Qian52, J.H. Rademacker42, B.
Rakotomiaramanana38, M.S. Rangel2, I. Raniuk40, G. Raven24, S. Redford51, M.M.
Reid44, A.C. dos Reis1, S. Ricciardi45, K. Rinnert48, D.A. Roa Romero5, P.
Robbe7, E. Rodrigues47,50, F. Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43,
S. Roiser37, V. Romanovsky34, M. Rosello35,n, J. Rouvinet38, T. Ruf37, H.
Ruiz35, G. Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail47, B.
Saitta15,d, C. Salzmann39, M. Sannino19,i, R. Santacesaria22, C. Santamarina
Rios36, R. Santinelli37, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C.
Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina30, P. Schaack49, M.
Schiller24, S. Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt37, O.
Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A.
Sciubba18,l, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp49, N.
Serra39, J. Serrano6, P. Seyfert11, B. Shao3, M. Shapkin34, I. Shapoval40,37,
P. Shatalov30, Y. Shcheglov29, T. Shears48, L. Shekhtman33, O. Shevchenko40,
V. Shevchenko30, A. Shires49, R. Silva Coutinho44, T. Skwarnicki52, A.C.
Smith37, N.A. Smith48, E. Smith51,45, K. Sobczak5, F.J.P. Soler47, A.
Solomin42, F. Soomro18, B. Souza De Paula2, B. Spaan9, A. Sparkes46, P.
Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S.
Stone52,37, B. Storaci23, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, S.
Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26, S. T’Jampens4, E.
Teodorescu28, F. Teubert37, C. Thomas51, E. Thomas37, J. van Tilburg11, V.
Tisserand4, M. Tobin39, S. Topp-Joergensen51, N. Torr51, E. Tournefier4,49,
M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23, M. Ubeda Garcia37, A. Ukleja27,
P. Urquijo52, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez35, P.
Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis42, M. Veltri17,g, B. Viaud7, I.
Videau7, X. Vilasis-Cardona35,n, J. Visniakov36, A. Vollhardt39, D.
Volyanskyy10, D. Voong42, A. Vorobyev29, H. Voss10, S. Wandernoth11, J.
Wang52, D.R. Ward43, N.K. Watson55, A.D. Webber50, D. Websdale49, M.
Whitehead44, D. Wiedner11, L. Wiggers23, G. Wilkinson51, M.P. Williams44,45,
M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25, W. Witzeling37, S.A.
Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Xing52, Z. Yang3, R. Young46,
O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3, L. Zhang52, W.C. Zhang12, Y.
Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin37.
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
23Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraców, Poland
26AGH University of Science and Technology, Kraców, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55University of Birmingham, Birmingham, United Kingdom
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
The charged particle multiplicity is a basic observable that characterizes the
hadronic final state. The multiplicity distribution is sensitive to the
underlying QCD dynamics of the proton-proton collision. ALICE [1], ATLAS [2]
and CMS [3] have measured the charged multiplicity distributions mainly
covering the central region, while LHCb’s geometrical acceptance allows the
dynamics of the collision to be probed in the forward region. The forward
region is in particular sensitive to low Bjorken-$x$ QCD dynamics and multi-
parton interactions (MPI) [4].
In this analysis, the charged particles are reconstructed in the vertex
detector (VELO) surrounding the interaction region. The VELO was designed to
provide a uniform acceptance in the forward region with additional coverage of
the backward region. In the absence of almost any magnetic field in the VELO
region, the particle trajectories are straight lines and therefore no
acceptance corrections as a function of momentum are needed. Since the VELO is
close to the interaction region, the amount of material before the particle
detection is small, minimising the corrections for particle interactions with
detector material.
This paper is organized as follows. Section 2 gives a brief description of the
LHCb detector and the configuration used to record data in Spring 2010. The
Monte Carlo simulation and data selection are outlined in Sections 3 and 4
respectively, with Section 5 giving an overview of the analysis. The
systematic uncertainties are outlined in Section 6. The final results are
discussed in Section 7 and compared with different model expectations, before
concluding in Section 8.
## 2 LHCb detector
The LHCb detector is a single-arm magnetic dipole spectrometer with a polar
angular coverage with respect to the beam line of approximately 15 to 300 mrad
in the horizontal bending plane, and 15 to 250 mrad in the vertical non-
bending plane. The detector is described in detail elsewhere [5]. A right-
handed coordinate system is defined with its origin at the nominal proton-
proton interaction point, the $z$ axis along the beam line and pointing
towards the magnet, and the $y$ axis pointing upwards.
For the low luminosity running period of the LHC relevant for this analysis,
the probability of observing more than one collision in a proton-proton bunch
crossing (pile-up) is measured to be $(3.7\pm 0.4)\%$, dominated by a double
interaction. For the measurements presented in this paper the tracking
detectors are of particular importance. The LHCb tracking system consists of
the VELO surrounding the proton-proton interaction region, a tracking station
(TT) before the dipole magnet, and three tracking stations (T1–T3) after the
magnet. Particles traversing from the interaction region to the downstream
tracking stations experience an integrated bending-field of approximately 4
Tm.
The VELO consists of silicon microstrip modules, providing a measure of the
radial and azimuthal coordinates, $r$ and $\phi$, distributed in 23 stations
arranged along the beam direction. The first two stations at the most upstream
$z$ positions are instrumented to provide information on the number of visible
interactions in the detector at the first level of the trigger. The VELO is
constructed in two halves, movable in the $x$ and $y$ directions so that it
can be centered on the beam. During stable beam conditions the two halves are
located at their nominal closed position, with active silicon at only 8 mm
from the beams, providing full azimuthal coverage.
The TT station also uses silicon microstrip technology. The T1–T3 tracking
stations have silicon microstrips in the region close to the beam pipe,
whereas straw tubes are employed in the outer region.
Though the particle multiplicity is measured using only tracks reconstructed
with the VELO, momentum information is only available for “long” tracks. Long
tracks are formed from hits in the VELO (before the magnet) and in the T1–T3
stations (after the magnet). If available, measurements in the TT station are
added to the long track.
The LHCb trigger system consists of two levels. The first level is implemented
in hardware and is designed to reduce the event rate to a maximum of 1 MHz.
The complete detector is then read out and the data is sent to the second
level, a software trigger. For the early data taking period with low
luminosity used in this analysis a simplified trigger was used. The first
level trigger was operated in pass-through mode. A fast track reconstruction
was performed in the software trigger and events with at least one track
observed in the VELO were accepted.
## 3 Monte Carlo simulation
Monte Carlo event simulation is used to correct for acceptance, resolution
effects and for background characterisation. The detector simulation is based
on the Geant4 [6] package. Details of the detector simulation are given in
Ref. [5]. The simulated material of the components of the VELO was compared
with the masses measured at the time of production and agreement was found to
be within 15%. The Monte Carlo event samples are passed through reconstruction
and selection procedures identical to those for the data.
Elastic and inelastic proton-proton collisions are generated using the Pythia
6.4 event generator [7], with CTEQ6L parton density functions [8], which is
tuned to lower energy hadron collider data [9]. The inelastic processes
include both single and double diffractive components. The decay of the
generated particles is carried out by EvtGen [10], with final state radiation
handled by Photos [11]. Secondary particles produced in material interactions
are decayed through the Geant4 program.
## 4 Data selection
A sample of $3\times 10^{6}$ events, collected during May 2010, was used in
this analysis. In order to minimize the contribution of secondary particles
and misreconstructed (fake) tracks, only the tracks satisfying a set of
minimal quality criteria are accepted. To minimise fake tracks a cut on the
$\chi^{2}$ per degree of freedom of the reconstructed track, $\chi^{2}/{\rm
ndf}<5,$ is applied. To further reduce fake tracks, and reduce duplicate
tracks due to a split of the reconstructed trajectory, a cut of less than four
missing VELO hits compared to the expectation is applied. To ensure that
tracks originate from the primary interaction, the requirements
$d_{0}<2\,\rm\,mm$ and $z_{0}<3\sigma_{L}$ are applied, where $d_{0}$ is the
track’s closest distance to the beam line, $z_{0}$ is the distance along the
$z$ direction from the centre of the luminous region and $\sigma_{L}$ is the
width of the luminous region extracted from a Gaussian fit.
Tracks are considered for this analysis only if their pseudorapidity is in
either of the ranges $-2.5<\eta<-2.0$ and $2.0<\eta<4.5$. Pseudorapidity is
defined as $-\ln[\tan(\theta/2)]$ where $\theta$ is the polar angle of the
particle with respect to the $z$ direction. The forward range is divided in
five equal sub-intervals with $\Delta\eta=0.5$.
## 5 Analysis strategy
The reconstructed multiplicity distributions are corrected on an event by
event basis to account for the tracking and selection efficiencies and for the
background contributions. These corrected distributions are then used to
measure the charged particle multiplicities in each of the $\eta$ intervals
(bins) through an unfolding procedure. Only events with tracks in the $\eta$
bins are included in the distributions and subsequent normalisation. The
distributions are corrected for pile-up effects so they represent the charged
particle multiplicities, ${\rm n_{ch}}$, for single proton-proton
interactions. No unfolding procedure is required for the charged particles
pseudorapidity density distribution i.e. the mean number of charged particles
per single pp-collision and unit of pseudorapidity. Only the per track
corrections for background and tracking efficiency are needed. For this
distribution, at least one VELO track is required in the full forward $\eta$
range. Each of these elements of the analysis procedure are discussed in
subsequent subsections.
Hard interaction events are defined by requiring at least one long track with
$\mbox{$p_{\rm T}$}>1$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in the range
$2.5<\eta<4.5$ where the detector has high efficiency. The geometric
acceptance is no longer independent of momentum and therefore the
distributions require an additional correction.
In this analysis primary charged particles are defined as all particles for
which the sum of the ancestors’ mean lifetimes is smaller than $10$ ps;
according to this definition the decay products of beauty and charm are
primary particles.
### 5.1 Efficiency correction
The LHCb simulation is used to estimate the overall tracking and selection
efficiency as a function of pseudorapidity and azimuthal angle $\phi$. As the
VELO is outside the magnetic field region tracks are straight lines and no
study of acceptance as a function of momentum is necessary. It is found that
the efficiency (including acceptance) in the forward region is typically
greater than 90% while it is at least 85% in the backward region. Tracking
efficiency depends weakly on the event track multiplicity; this is taken into
account in the evaluation of the systematic error.
### 5.2 Background contributions
There are two main sources of background that can affect the measurement of
the multiplicity of charged particles: secondary particles misidentified as
primary and fake tracks. Other sources of background, such as beam-gas
interactions, are estimated to be negligible.
The correlation between the number of VELO hit clusters in an event and its
track multiplicity is in good agreement between the data and simulation,
indicating that the fraction of fake tracks is well understood. It is also
found that for each $\eta$ bin the multiplicity of fake tracks is linearly
dependent on the number of VELO clusters in the event. Therefore it is
possible to parameterise the fake contribution as a function of VELO clusters
using the Monte Carlo simulation.
The majority of secondary particles are produced in photon conversions in the
VELO material, and in the decay of long-lived strange particles such as
$K^{0}_{\rm\scriptscriptstyle S}$ and hyperons. While earlier LHCb
measurements show that the production of $K^{0}_{\rm\scriptscriptstyle S}$ is
reasonably described by the Monte Carlo generator [12], there are indications
that the production of $\mathchar 28931\relax$ particles is underestimated
[13]. This difference is accounted for in the systematic error associated with
the definition of primary particles.
The fraction of secondary particles is estimated as a function of both $\eta$
and $\phi$. In general, depending on the $\eta$ bin, the correction for non-
primary particles (from conversion and secondaries) changes the mean values of
the particle multiplicity distributions by $5-10\%$.
### 5.3 Correction and unfolding procedure
The procedure consists of three steps; a background subtraction is made,
followed by an efficiency correction and finally a correction for pile-up. The
procedure is applied to all measured track multiplicity distributions in each
of the different $\eta$ intervals.
In the first step, the distribution is corrected for fake tracks and non-
primary particles. A mean number of background tracks is estimated for each
event based on the parameterizations described in Section 5.2. A PDF
(probability density function) is built with this mean value assuming a
Poisson distribution for the number of background tracks. Hence, a PDF for the
number of prompt charged particles in a given event is then obtained. These
per event PDFs are summed up and normalized to obtain the reconstructed prompt
charged track multiplicity distribution i.e. the fraction of events with ${\rm
n_{tr}}$ tracks, ${\rm Prob(n_{tr})}$.
In the second step, the correction for the tracking efficiency is applied. For
each $\eta$ bin a mean efficiency, $\epsilon$, is calculated based on the per
track efficiency as function of $(\eta,\phi)$. As explained below, this is
used to unfold the background-subtracted track multiplicity distribution,
${\rm Prob(n_{tr})}$, to obtain the underlying charged particle multiplicity
distribution, ${\rm Prob(\tilde{n}_{ch})}$, where ${\rm\tilde{n}_{ch}}$ is the
number of primary produced particles of all proton-proton collisions in an
event.
For a given value of ${\rm\tilde{n}_{ch}}$, the probability to observe ${\rm
n_{tr}}$ reconstructed tracks given a reconstruction efficiency $\epsilon$ is
described by the binomial distribution
$p({\rm
n_{tr}},{\rm\tilde{n}_{ch}},\epsilon)=\left(\begin{array}[]{c}{\rm\tilde{n}_{ch}}\\\
{\rm n_{tr}}\end{array}\right)(1-\epsilon)^{{\rm\tilde{n}_{ch}}-{\rm
n_{tr}}}\epsilon^{{\rm n_{tr}}}.$ (1)
Hence, the observed track multiplicity distribution is given by
${\rm Prob(n_{tr})}=\sum_{{\rm\tilde{n}_{ch}}=0}^{\infty}{\rm
Prob(\tilde{n}_{ch})}\times p({\rm n_{tr}},{\rm\tilde{n}_{ch}},\epsilon).$ (2)
The values for ${\rm Prob(\tilde{n}_{ch})}$ are obtained by performing a fit
to ${\rm Prob(n_{tr})}$. The procedure has been verified using simulated data.
In the last step, the distributions are corrected for pile-up to obtain the
charged particle multiplicity distributions of single interaction events,
${\rm Prob(n_{ch})}$. This is done using an iterative procedure. For low
luminosity, ${\rm Prob(\tilde{n}_{ch})}$ has mainly two contributions: single
proton-proton interactions and a convolution of two single proton-proton
interactions. The starting assumption is that the observed distribution is the
single proton-proton interaction. From this, the convolution term is
calculated, and by subtracting it from the observed distribution, a first
order estimate for the single proton-proton distribution is obtained. This can
then be used to calculate again the convolution term and obtain a second order
estimate for the single proton-proton distribution. The procedure usually
converges after the second iteration. The pile-up correction typically changes
the mean value of the particle multiplicity distributions by $3-4\%$. It was
checked that the contribution from pile-up events with more than two proton-
proton collisions is negligible.
As mentioned before, no unfolding procedure is required for the charged
particles pseudorapidity density, only the per track corrections for
background tracks and tracking efficiency are applied. The distribution is
then normalized to the total number of proton-proton collisions including
pile-up collisions. In the case of hard interactions, the pseudorapidity
density distribution of the pile-up collisions without the $p_{\rm T}$ cut is
first subtracted. Finally, the distribution is normalized to the total number
of hard collisions.
## 6 Systematic uncertainty
### 6.1 Efficiency
Studies based on data and simulation show that the error on the tracking
efficiency for particles reaching the tracking stations T1-T3 is $<3\%$ [14].
The tracking efficiency reduces for low-momentum ($\mbox{$p_{\rm
T}$}<50$${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$) particles due to interactions
with the detector material and the residual magnetic field in the VELO region.
Since no momentum measurement exists for the reconstructed VELO tracks, the
estimate of a mean efficiency relies on the prediction of the LHCb Monte Carlo
model for the contribution of low-momentum particles to the total number of
particles. The simulation predicts that in the forward region the fraction of
particles below a transverse momentum of
50${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ is 2.4%. The corresponding average
single track efficiency in this $\eta$ range is measured to be 94%. In the two
extreme cases in which no particles with $p_{\rm T}$ below
50${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ were reconstructed or no such
particles were produced the average track efficiency would be reduced by 1.2%
or increased by 1.1% respectively. Assuming a 25% uncertainty on the number of
low momentum particles, as suggested by the comparison between the measured
particle multiplicity and Monte Carlo prediction, the additional contribution
to the track efficiency uncertainty is $<$ 1%. Adding this to the $3\%$ track
reconstruction uncertainty, gives an overall $4\%$ error on the track
efficiency used in the unfolding procedure. The systematic error contribution
is then estimated by unfolding the multiplicity distributions varying the
tracking efficiency by $\pm 4\%$.
### 6.2 Non-primary particles
The main systematic uncertainty on the contribution of non-primary particles
arises from the knowledge of the detector material (15%). Two thirds of non-
primary particles are due to conversions of photons from $\pi^{0}$ decays,
resulting in an 10% uncertainty. The multiplicity of $\pi^{0}$ scales with the
charged multiplicity, therefore no additional error is applied. Varying by
$\pm 40\%$ the production of $\Lambda$ results in an uncertainty of about 5%
on the non-primary contribution. A pessimistic assumption of a 25%
underestimation of the non-prompt contribution would change the mean and RMS
values of the particle multiplicity distributions by $-2\%$, which can be
neglected compared to the tracking efficiency uncertainty of $4\%$.
### 6.3 Pile-up
The pile-up corrections inherit a systematic uncertainty from the
determination of the mean number of visible interactions of $10\%$. This
correction to the pile-up fraction is small and is negligible compared to the
systematic uncertainty due to the track efficiency correction.
## 7 Results
Figure 1 shows the unfolded charged particle multiplicity distribution for
different bins in pseudorapidity, $\eta$. Figure 2 shows the multiplicity
distributions for the full forward range, $2.0<\eta<4.5$. There is a
requirement of at least one track in the relevant $\eta$ range. The
distributions are compared to several Monte Carlo event generators. Pythia
6.424 is compared with the data for a number of tunes including the LHCb tuned
settings [9]. In particular the Perugia0 and PerugiaNOCR tunings [15] are
shown. In addition, the Pythia 8.145 generator [16] was compared to the data
as well as Phojet 1-12.35 [17]. In general all generators underestimate the
multiplicity distributions, with the LHCb tune giving the best description of
the data; this tune does not use data from the LHC. The exclusion of the
Pythia diffractive processes in the Perugia tunes, Figs. 1b and 2b, also
improves the description of the data, particularly in the full forward region.
Figure 1: The multiplicity distribution in $\eta$ bins (shown as points with
statistical error bars) with predictions of different event generators. The
inner error bar represents the statistical uncertainty and the outer error bar
represents the systematic and statistical uncertainty on the measurements. The
data in both figures are identical with predictions from Pythia 6, Phojet and
Pythia 8 in (a) and predictions of the Pythia 6 Perugia tunes with and without
diffraction in (b).
Figure 2: The multiplicity distribution in the forward $\eta$ range (shown as
points with error bars) with predictions of different event generators. The
shaded bands represent the total uncertainty on the measurements. The data in
both figures are identical with predictions from Pythia 6, Phojet and Pythia 8
in (a) and predictions of the Pythia 6 Perugia tunes with and without
diffraction in (b).
The Koba-Nielsen-Olesen (KNO) scaling variable [18] has been used to compare
the data in the different $\eta$ bins. Figure 3 shows the KNO scaled
multiplicity distributions, $\Psi(u)=\left\langle{\rm
n_{ch}}\right\rangle\times{\rm Prob(n_{ch})}$ as a function of $u=\frac{\rm
n_{ch}}{\left\langle\rm n_{ch}\right\rangle}$. As the multiplicity
distributions measured are truncated the mean used was extracted by fitting a
negative binomial distribution. It clearly shows that the distributions in the
different $\eta$ bins are equivalent. In particular this illustrates that when
there is a requirement of at least one track in the $\eta$ bin the forward and
backward regions $(2.0<|\eta|<2.5)$ are identical.
Figure 3: The KNO distributions in different bins of $\eta$. Only the the
statistical uncertainties are shown.
The charged particle pseudorapidity density, $\rho,$ is shown as a function of
pseudorapidity in Fig. 4. The data have a marked asymmetry between the forward
and backward region; this is a consequence of the requirement of at least one
track in the full forward $\eta$ range. All models fail to describe the mean
charged particle multiplicity per unit of pseudorapidity. The models, to
varying degrees, also display the asymmetry but in none of the models is this
as large as in the data. The effect on the predictions of excluding
diffractive processes is shown in Fig. 4b using the Perugia tunes. There is a
better description of the $\eta$ distribution in the backward directions but
it still fails to describe the forward-backward asymmetry.
Figure 4: The charged particle densities as a function of $\eta$ (shown as
points with statistical error bars) and comparisons with predictions of event
generators, as indicated in the key. The shaded bands represent the total
uncertainty. The events are selected by requiring at least one charged
particle in the range $2.0<\eta<4.5$. The data in both figures are identical
with predictions from Pythia 6, Phojet and Pythia 8 in (a) and predictions of
the Pythia 6 Perugia tunes with and without diffraction in (b).
A sample of hard QCD events were studied by ensuring at least one track in the
pseudorapidity range $2.5<\eta<4.5$ has a transverse momentum $\mbox{$p_{\rm
T}$}>1$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In comparison to the data
without this $p_{\rm T}$ requirement, the multiplicity distributions have
larger high multiplicity tails, see Figs. 5 and 6. The data are again compared
to predictions of several event generators. In general the predictions are in
better agreement than for the minimum bias data but the pseudorapidity range
$4.0<\eta<4.5$ remains poorly described. As the $p_{\rm T}$ cut removes the
majority of diffractive events from Pythia 6 the comparisons with and without
diffraction are not shown.
Figure 5: The multiplicity distribution in $\eta$ bins (shown as points with
error bars) with predictions of different event generators. The inner error
bar represents the statistical uncertainty and the outer error bar represents
the systematic and statistical uncertainty on the measurements. The events
have at least one track with a $\mbox{$p_{\rm
T}$}>1.0$${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ in the pseudorapidity range
$2.5<\eta<4.5$. The data in both figures are identical with predictions from
Pythia 6, Phojet and Pythia 8 in (a) and predictions of the Pythia 6 Perugia
tunes in (b).
Figure 6: The multiplicity distribution in the forward $\eta$ range (shown as
points with statistical error bars) with predictions of different event
generators. The shaded bands represent the total uncertainty. The events have
at least one track with a $\mbox{$p_{\rm
T}$}>1.0$${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ in the pseudorapidity range
$2.5<\eta<4.5$. The data in both figures are identical with predictions from
Pythia 6, Phojet and Pythia 8 in (a) and predictions of the Pythia 6 Perugia
tunes in (b).
Figure 7: The data charged particle densities as a function of $\eta$ (shown
as points with statistical error bars) and comparisons with predictions of
event generators, as indicated in the key. The events have at least one track
with a $\mbox{$p_{\rm T}$}>1.0$${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ in the
pseudorapidity range $2.5<\eta<4.5$. The shaded bands represent the total
uncertainty.
The charged particle density as a function of pseudorapidity for the hard QCD
sample is shown in Fig. 7. The discontinuity observed in the data at
$\eta=2.5$ is an artefact of the event selection for the hard events. The
asymmetry between the forward and backward region is further amplified in this
sample. All models fail to describe the mean charged particle multiplicity per
unit of pseudorapidity. The models, to varying degrees, also display the
asymmetry but never give an effect as large as the data. The Perugia (NOCR)
tune gives the best description of the data in the backward direction but
fails to reproduce the size of the asymmetry.
## 8 Summary
The LHCb spectrometer acceptance, $2.0<\eta<4.5,$ allows the forward region to
be probed at the LHC. The charged multiplicity distributions at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ are measured with and without a
$p_{\rm T}$ event selection, making use of the high efficiency of the LHCb
VELO. Several event generators are compared to the data; none are fully able
to describe the multiplicity distributions or the charged density distribution
as a function of $\eta$ in the LHCb acceptance. In general, the models
underestimate the charged particle production, in agreement with the
measurements in the central region at the LHC.
## 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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## Appendix A Tables of charged particle multiplicities
Table 1: Charged particle multiplicity distribution in the pseudorapidity range $-2.5<\eta<-2.0$ for minimum bias events and for hard QCD events (see text). The first quoted uncertainty is statistical and the second is systematic. $n_{\rm ch}$ | Prob. in min. bias | Prob. in hard QCD
---|---|---
| events $\times 10^{3}$ | events $\times 10^{3}$
$1$ | $246.66\pm 0.40\pm 7.96$ | $155.54\pm 0.49\pm 6.47$
$2$ | $188.43\pm 0.41\pm 4.03$ | $146.92\pm 0.55\pm 5.26$
$3$ | $141.00\pm 0.41\pm 1.25$ | $132.46\pm 0.61\pm 3.20$
$4$ | $105.57\pm 0.42\pm 0.11$ | $114.15\pm 0.67\pm 1.75$
$5$ | $79.25\pm 0.43\pm 0.75$ | $96.44\pm 0.73\pm 0.24$
$6$ | $60.83\pm 0.45\pm 1.13$ | $79.84\pm 0.79\pm 0.48$
$7$ | $46.08\pm 0.48\pm 1.33$ | $63.40\pm 0.83\pm 1.33$
$8$ | $35.01\pm 0.50\pm 1.35$ | $51.30\pm 0.90\pm 1.63$
$9$ | $26.43\pm 0.52\pm 1.40$ | $40.66\pm 0.97\pm 1.81$
$10$ | $19.75\pm 0.55\pm 1.36$ | $31.50\pm 1.02\pm 1.86$
$11$ | $14.60\pm 0.57\pm 1.19$ | $24.16\pm 1.08\pm 1.83$
$12$ | $10.82\pm 0.59\pm 1.00$ | $18.03\pm 1.12\pm 1.64$
$13$ | $7.86\pm 0.61\pm 0.90$ | $13.96\pm 1.21\pm 1.61$
$14$ | $5.57\pm 0.63\pm 0.86$ | $9.56\pm 1.19\pm 1.28$
$15$ | $3.94\pm 0.65\pm 0.73$ | $7.14\pm 1.30\pm 1.09$
$16$ | $2.90\pm 0.67\pm 0.37$ | $5.10\pm 1.29\pm 1.11$
$17$ | $2.44\pm 0.68\pm 0.96$ | $4.48\pm 1.34\pm 1.28$
$18$ | $1.14\pm 0.70\pm 0.61$ | $2.13\pm 1.43\pm 2.03$
$19$ | $0.96\pm 0.71\pm 0.66$ | $1.78\pm 1.41\pm 0.19$
$20$ | $0.75\pm 0.72\pm 0.27$ | $1.46\pm 1.44\pm 0.60$
Table 2: Charged particle multiplicity distribution in the pseudorapidity range $2.0<\eta<2.5$ for minimum bias events and for hard QCD events (see text). The first quoted uncertainty is statistical and the second is systematic. $n_{\rm ch}$ | Prob. in min. bias | Prob. in hard QCD
---|---|---
| events $\times 10^{3}$ | events $\times 10^{3}$
$1$ | $244.35\pm 0.36\pm 7.66$ | $126.88\pm 0.38\pm 6.57$
$2$ | $191.00\pm 0.33\pm 4.02$ | $140.50\pm 0.43\pm 5.81$
$3$ | $142.72\pm 0.31\pm 1.44$ | $133.83\pm 0.44\pm 3.91$
$4$ | $106.75\pm 0.28\pm 0.10$ | $121.45\pm 0.44\pm 1.95$
$5$ | $80.27\pm 0.26\pm 0.73$ | $103.10\pm 0.43\pm 0.75$
$6$ | $61.09\pm 0.25\pm 1.22$ | $86.87\pm 0.42\pm 0.98$
$7$ | $46.22\pm 0.23\pm 1.42$ | $70.01\pm 0.41\pm 1.59$
$8$ | $34.57\pm 0.21\pm 1.45$ | $55.15\pm 0.39\pm 1.83$
$9$ | $26.09\pm 0.20\pm 1.38$ | $43.12\pm 0.36\pm 2.13$
$10$ | $19.30\pm 0.18\pm 1.34$ | $32.71\pm 0.34\pm 2.20$
$11$ | $14.08\pm 0.17\pm 1.17$ | $24.64\pm 0.32\pm 2.00$
$12$ | $10.17\pm 0.16\pm 1.07$ | $18.25\pm 0.29\pm 1.80$
$13$ | $7.23\pm 0.14\pm 0.98$ | $13.66\pm 0.28\pm 1.84$
$14$ | $5.43\pm 0.13\pm 0.82$ | $9.97\pm 0.25\pm 1.52$
$15$ | $3.55\pm 0.12\pm 0.60$ | $6.64\pm 0.22\pm 1.12$
$16$ | $2.60\pm 0.11\pm 0.40$ | $4.91\pm 0.21\pm 0.78$
$17$ | $1.78\pm 0.10\pm 0.65$ | $3.14\pm 0.18\pm 1.23$
$18$ | $1.35\pm 0.09\pm 0.28$ | $2.45\pm 0.17\pm 0.47$
$19$ | $0.82\pm 0.08\pm 0.22$ | $1.56\pm 0.15\pm 0.42$
$20$ | $0.62\pm 0.07\pm 0.19$ | $1.15\pm 0.13\pm 0.34$
Table 3: Charged particle multiplicity distribution in the pseudorapidity range $2.5<\eta<3.0$ for minimum bias events and for hard QCD events (see text). The first quoted uncertainty is statistical and the second is systematic. $n_{\rm ch}$ | Prob. in min. bias | Prob. in hard QCD
---|---|---
| events $\times 10^{3}$ | events $\times 10^{3}$
$1$ | $249.37\pm 0.35\pm 7.88$ | $121.02\pm 0.36\pm 6.72$
$2$ | $194.45\pm 0.33\pm 4.11$ | $140.71\pm 0.41\pm 6.20$
$3$ | $144.53\pm 0.29\pm 1.39$ | $138.90\pm 0.42\pm 4.26$
$4$ | $107.18\pm 0.27\pm 0.10$ | $125.71\pm 0.41\pm 2.10$
$5$ | $80.42\pm 0.24\pm 0.89$ | $108.13\pm 0.40\pm 0.34$
$6$ | $60.29\pm 0.22\pm 1.34$ | $87.75\pm 0.37\pm 1.24$
$7$ | $45.03\pm 0.20\pm 1.53$ | $70.69\pm 0.35\pm 1.85$
$8$ | $33.53\pm 0.18\pm 1.55$ | $55.79\pm 0.33\pm 2.31$
$9$ | $24.75\pm 0.16\pm 1.46$ | $42.12\pm 0.30\pm 2.40$
$10$ | $17.98\pm 0.15\pm 1.30$ | $31.82\pm 0.27\pm 2.23$
$11$ | $12.98\pm 0.13\pm 1.23$ | $23.37\pm 0.25\pm 2.10$
$12$ | $9.16\pm 0.12\pm 1.12$ | $16.64\pm 0.22\pm 1.95$
$13$ | $6.74\pm 0.11\pm 0.87$ | $12.07\pm 0.19\pm 1.52$
$14$ | $4.46\pm 0.09\pm 0.71$ | $8.43\pm 0.17\pm 1.27$
$15$ | $3.23\pm 0.08\pm 0.47$ | $5.97\pm 0.15\pm 0.88$
$16$ | $2.20\pm 0.07\pm 0.71$ | $4.07\pm 0.13\pm 1.31$
$17$ | $1.57\pm 0.06\pm 0.32$ | $2.78\pm 0.11\pm 0.52$
$18$ | $0.94\pm 0.05\pm 0.32$ | $1.86\pm 0.10\pm 0.51$
$19$ | $0.69\pm 0.05\pm 0.33$ | $1.26\pm 0.09\pm 0.56$
$20$ | $0.50\pm 0.04\pm 0.13$ | $0.92\pm 0.08\pm 0.20$
Table 4: Charged particle multiplicity distribution in the pseudorapidity range $3.0<\eta<3.5$ for minimum bias events and for hard QCD events (see text). The first quoted uncertainty is statistical and the second is systematic. $n_{\rm ch}$ | Prob. in min. bias | Prob. in hard QCD
---|---|---
| events $\times 10^{3}$ | events $\times 10^{3}$
$1$ | $257.54\pm 0.36\pm 8.38$ | $128.89\pm 0.38\pm 7.33$
$2$ | $199.12\pm 0.33\pm 4.08$ | $145.79\pm 0.41\pm 6.39$
$3$ | $147.50\pm 0.30\pm 1.23$ | $145.41\pm 0.43\pm 4.13$
$4$ | $108.21\pm 0.27\pm 0.31$ | $130.01\pm 0.42\pm 2.16$
$5$ | $79.83\pm 0.24\pm 1.10$ | $109.73\pm 0.41\pm 0.44$
$6$ | $58.83\pm 0.22\pm 1.50$ | $87.48\pm 0.38\pm 1.58$
$7$ | $43.25\pm 0.20\pm 1.67$ | $67.91\pm 0.35\pm 2.16$
$8$ | $31.48\pm 0.18\pm 1.64$ | $52.94\pm 0.32\pm 2.50$
$9$ | $22.72\pm 0.16\pm 1.48$ | $38.50\pm 0.29\pm 2.43$
$10$ | $16.12\pm 0.14\pm 1.28$ | $28.21\pm 0.26\pm 2.21$
$11$ | $11.37\pm 0.13\pm 1.19$ | $20.63\pm 0.24\pm 2.17$
$12$ | $7.89\pm 0.11\pm 1.07$ | $14.74\pm 0.21\pm 1.83$
$13$ | $5.63\pm 0.10\pm 0.81$ | $10.02\pm 0.18\pm 1.45$
$14$ | $3.54\pm 0.08\pm 0.67$ | $7.00\pm 0.16\pm 1.02$
$15$ | $2.53\pm 0.07\pm 0.71$ | $4.49\pm 0.13\pm 1.37$
$16$ | $1.79\pm 0.06\pm 0.38$ | $3.33\pm 0.12\pm 0.64$
$17$ | $1.07\pm 0.06\pm 0.29$ | $1.96\pm 0.10\pm 0.53$
$18$ | $0.75\pm 0.05\pm 0.17$ | $1.38\pm 0.09\pm 0.32$
$19$ | $0.49\pm 0.04\pm 0.22$ | $0.94\pm 0.08\pm 0.43$
$20$ | $0.35\pm 0.04\pm 0.10$ | $0.65\pm 0.07\pm 0.17$
Table 5: Charged particle multiplicity distribution in the pseudorapidity range $3.5<\eta<4.0$ for minimum bias events and for hard QCD events (see text). The first quoted uncertainty is statistical and the second is systematic. $n_{\rm ch}$ | Prob. in min. bias | Prob. in hard QCD
---|---|---
| events $\times 10^{3}$ | events $\times 10^{3}$
$1$ | $268.35\pm 0.37\pm 8.77$ | $139.99\pm 0.39\pm 7.61$
$2$ | $206.16\pm 0.34\pm 4.00$ | $158.42\pm 0.44\pm 6.72$
$3$ | $150.62\pm 0.31\pm 0.98$ | $151.42\pm 0.45\pm 4.01$
$4$ | $108.81\pm 0.28\pm 0.56$ | $133.07\pm 0.44\pm 1.67$
$5$ | $78.99\pm 0.25\pm 1.35$ | $110.17\pm 0.42\pm 0.92$
$6$ | $56.92\pm 0.22\pm 1.77$ | $84.74\pm 0.38\pm 1.91$
$7$ | $40.49\pm 0.20\pm 1.81$ | $65.65\pm 0.36\pm 2.61$
$8$ | $28.60\pm 0.18\pm 1.68$ | $48.06\pm 0.32\pm 2.71$
$9$ | $19.98\pm 0.16\pm 1.46$ | $34.60\pm 0.29\pm 2.49$
$10$ | $13.79\pm 0.14\pm 1.30$ | $24.49\pm 0.26\pm 2.26$
$11$ | $9.31\pm 0.12\pm 1.18$ | $16.62\pm 0.22\pm 2.05$
$12$ | $6.48\pm 0.11\pm 0.94$ | $11.50\pm 0.19\pm 1.51$
$13$ | $4.02\pm 0.09\pm 0.68$ | $7.40\pm 0.17\pm 1.18$
$14$ | $2.80\pm 0.08\pm 0.41$ | $5.09\pm 0.15\pm 0.75$
$15$ | $1.82\pm 0.07\pm 0.64$ | $3.48\pm 0.13\pm 1.27$
$16$ | $1.24\pm 0.06\pm 0.28$ | $2.23\pm 0.11\pm 0.45$
$17$ | $0.68\pm 0.05\pm 0.25$ | $1.35\pm 0.09\pm 0.43$
$18$ | $0.50\pm 0.04\pm 0.21$ | $0.85\pm 0.08\pm 0.47$
$19$ | $0.27\pm 0.04\pm 0.05$ | $0.55\pm 0.06\pm 0.14$
$20$ | $0.18\pm 0.03\pm 0.08$ | $0.31\pm 0.05\pm 0.18$
Table 6: Charged particle multiplicity distribution in the pseudorapidity range $4.0<\eta<4.5$ for minimum bias events and for hard QCD events (see text). The first quoted uncertainty is statistical and the second is systematic. $n_{\rm ch}$ | Prob. in min. bias | Prob. in hard QCD
---|---|---
| events $\times 10^{3}$ | events $\times 10^{3}$
$1$ | $284.08\pm 0.40\pm 9.11$ | $159.68\pm 0.01\pm 8.81$
$2$ | $215.09\pm 0.38\pm 4.25$ | $174.85\pm 0.01\pm 6.65$
$3$ | $155.18\pm 0.35\pm 0.72$ | $159.67\pm 0.01\pm 3.42$
$4$ | $109.77\pm 0.32\pm 1.07$ | $135.15\pm 0.01\pm 0.61$
$5$ | $76.74\pm 0.29\pm 1.76$ | $107.91\pm 0.01\pm 1.45$
$6$ | $53.34\pm 0.27\pm 1.97$ | $82.45\pm 0.01\pm 2.49$
$7$ | $36.49\pm 0.24\pm 1.93$ | $58.82\pm 0.01\pm 2.84$
$8$ | $24.57\pm 0.22\pm 1.75$ | $41.25\pm 0.01\pm 2.75$
$9$ | $16.30\pm 0.20\pm 1.50$ | $28.48\pm 0.01\pm 2.55$
$10$ | $10.63\pm 0.17\pm 1.25$ | $18.52\pm 0.01\pm 2.11$
$11$ | $6.76\pm 0.15\pm 1.00$ | $12.41\pm 0.01\pm 1.83$
$12$ | $4.20\pm 0.13\pm 0.70$ | $7.64\pm 0.01\pm 1.25$
$13$ | $2.92\pm 0.12\pm 0.57$ | $5.63\pm 0.01\pm 1.12$
$14$ | $1.48\pm 0.10\pm 0.86$ | $2.66\pm 0.01\pm 1.54$
$15$ | $1.15\pm 0.09\pm 0.33$ | $2.35\pm 0.01\pm 0.67$
$16$ | $0.55\pm 0.07\pm 0.21$ | $1.08\pm 0.01\pm 0.40$
$17$ | $0.35\pm 0.06\pm 0.28$ | $0.71\pm 0.01\pm 0.54$
$18$ | $0.24\pm 0.05\pm 0.12$ | $0.45\pm 0.01\pm 0.21$
$19$ | $0.09\pm 0.04\pm 0.13$ | $0.17\pm 0.01\pm 0.24$
$20$ | $0.07\pm 0.04\pm 0.02$ | $0.14\pm 0.01\pm 0.05$
Table 7: Charged particle multiplicity distribution in the pseudorapidity range $2.0<\eta<4.5$ for minimum bias events and for hard QCD events (see text). The first quoted uncertainty is statistical and the second is systematic. $n_{\rm ch}$ | Prob. in min. bias | Prob. in hard QCD
---|---|---
| events $\times 10^{3}$ | events $\times 10^{3}$
$1$ | $51.23\pm 0.16\pm 2.05$ | $5.38\pm 0.09\pm 0.45$
$2$ | $56.09\pm 0.18\pm 2.35$ | $10.02\pm 0.14\pm 1.10$
$3$ | $60.21\pm 0.20\pm 2.38$ | $14.69\pm 0.17\pm 2.04$
$4$ | $63.32\pm 0.21\pm 2.81$ | $21.62\pm 0.23\pm 2.16$
$5$ | $63.18\pm 0.23\pm 1.82$ | $26.22\pm 0.26\pm 1.88$
$6$ | $61.39\pm 0.24\pm 1.14$ | $31.38\pm 0.31\pm 1.94$
$7$ | $58.08\pm 0.25\pm 0.57$ | $35.13\pm 0.35\pm 1.87$
$8$ | $53.81\pm 0.26\pm 0.24$ | $37.72\pm 0.39\pm 1.67$
$9$ | $49.25\pm 0.27\pm 0.32$ | $39.37\pm 0.43\pm 2.27$
$10$ | $45.18\pm 0.28\pm 0.26$ | $42.69\pm 0.49\pm 2.31$
$11$ | $41.36\pm 0.29\pm 0.28$ | $43.07\pm 0.53\pm 1.37$
$12$ | $37.94\pm 0.31\pm 0.35$ | $43.97\pm 0.58\pm 1.39$
$13$ | $35.09\pm 0.32\pm 0.30$ | $43.52\pm 0.63\pm 1.71$
$14$ | $32.55\pm 0.34\pm 0.33$ | $45.25\pm 0.70\pm 2.01$
$15$ | $30.48\pm 0.36\pm 0.43$ | $43.98\pm 0.75\pm 0.86$
$16$ | $28.20\pm 0.38\pm 0.48$ | $43.48\pm 0.81\pm 0.90$
$17$ | $26.55\pm 0.40\pm 0.40$ | $43.85\pm 0.89\pm 0.74$
$18$ | $24.83\pm 0.43\pm 0.39$ | $42.96\pm 0.96\pm 0.34$
$19$ | $23.26\pm 0.45\pm 0.39$ | $41.47\pm 1.02\pm 0.24$
$20$ | $21.64\pm 0.48\pm 0.59$ | $40.21\pm 1.09\pm 0.29$
$21$ | $19.87\pm 0.19\pm 0.46$ | $37.97\pm 0.43\pm 0.51$
$23$ | $17.44\pm 0.20\pm 0.52$ | $35.08\pm 0.46\pm 0.67$
$25$ | $15.49\pm 0.21\pm 0.76$ | $32.39\pm 0.51\pm 0.87$
$27$ | $13.24\pm 0.22\pm 0.68$ | $30.02\pm 0.56\pm 1.42$
$29$ | $11.63\pm 0.23\pm 0.60$ | $26.14\pm 0.57\pm 1.54$
$31$ | $10.05\pm 0.24\pm 0.62$ | $23.18\pm 0.60\pm 1.38$
$33$ | $8.66\pm 0.25\pm 0.62$ | $20.40\pm 0.63\pm 1.45$
$35$ | $7.43\pm 0.26\pm 0.60$ | $17.59\pm 0.63\pm 1.52$
$37$ | $6.19\pm 0.26\pm 0.72$ | $15.85\pm 0.66\pm 1.88$
$39$ | $5.56\pm 0.26\pm 0.71$ | $13.11\pm 0.64\pm 1.45$
$41$ | $4.40\pm 0.25\pm 0.62$ | $11.22\pm 0.64\pm 1.32$
$43$ | $3.71\pm 0.25\pm 0.56$ | $9.55\pm 0.63\pm 1.24$
$45$ | $3.14\pm 0.24\pm 0.44$ | $7.74\pm 0.59\pm 1.27$
$47$ | $2.68\pm 0.23\pm 0.46$ | $6.21\pm 0.58\pm 1.40$
$49$ | $2.00\pm 0.22\pm 0.49$ | $5.38\pm 0.54\pm 1.09$
$51$ | $1.70\pm 0.12\pm 0.32$ | $4.18\pm 0.30\pm 1.09$
$54$ | $1.22\pm 0.11\pm 0.24$ | $3.04\pm 0.27\pm 0.69$
$57$ | $0.88\pm 0.09\pm 0.20$ | $2.26\pm 0.24\pm 0.49$
$60$ | $0.63\pm 0.08\pm 0.15$ | $1.58\pm 0.21\pm 0.45$
|
arxiv-papers
| 2011-12-20T07:48:16 |
2024-09-04T02:49:25.512332
|
{
"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 G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis, J.\n Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, D.\n S. Bailey, 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, F.\n Constantin, G. Conti, A. Contu, A. Cook, M. Coombes, G. Corti, G. A. Cowan,\n R. Currie, B. D'Almagne, C. D'Ambrosio, P. David, P. N. Y. David, I. De\n Bonis, S. De Capua, M. De Cian, F. De Lorenzi, J. M. De Miranda, L. De Paula,\n P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, M. Deissenroth, L. Del\n Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori, J. Dickens, H.\n 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, L. Est\\`eve, A. Falabella, E. Fanchini, 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, M. Frank,\n C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman,\n P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L. Garrido, D.\n Gascon, C. Gaspar, 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, B. Gui, E. Gushchin, Yu. Guz,\n T. Gys, G. Haefeli, C. Haen, S. C. Haines, T. Hampson, S. Hansmann-Menzemer,\n R. Harji, N. Harnew, J. Harrison, P. F. Harrison, J. He, V. Heijne, K.\n Hennessy, P. Henrard, J. A. Hernando Morata, E. van Herwijnen, E. Hicks, K.\n Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R. S. Huston, D.\n Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A.\n Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F.\n Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M. Kaballo, S. Kandybei, M.\n Karacson, T. M. Karbach, J. Keaveney, I. R. Kenyon, U. Kerzel, T. Ketel, A.\n Keune, B. Khanji, Y. M. Kim, M. Knecht, P. Koppenburg, A. Kozlinskiy, L.\n Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K.\n Kruzelecki, M. Kucharczyk, 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, L. Li\n Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. H. Lopes,\n 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, D. Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M.\n Matveev, E. Maurice, B. Maynard, A. Mazurov, G. McGregor, R. McNulty, C.\n Mclean, M. Meissner, M. Merk, J. Merkel, R. Messi, 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, M. Musy, J. Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I.\n Nasteva, M. Nedos, M. Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, V. Niess,\n N. Nikitin, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora\n Goicochea, P. Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A.\n 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. Petrella, A.\n 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, T. du\n Pree, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H.\n Rademacker, B. Rakotomiaramanana, M. S. Rangel, I. Raniuk, G. Raven, S.\n Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, K. Rinnert, D. A. Roa\n Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez, G. J.\n Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz,\n G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C.\n Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, S. Schleich, M. Schlupp, M. Schmelling, B.\n Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia,\n A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, B. Shao, 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, A. C. Smith, N. A. Smith, E. Smith, K.\n Sobczak, F. J. P. Soler, A. Solomin, F. Soomro, B. Souza De Paula, B. Spaan,\n A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S.\n Stone, B. Storaci, M. Straticiuc, U. Straumann, V. K. Subbiah, S. Swientek,\n M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F.\n Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S.\n Topp-Joergensen, N. Torr, E. Tournefier, M. T. Tran, A. Tsaregorodtsev, N.\n Tuning, M. Ubeda Garcia, A. Ukleja, P. Urquijo, U. Uwer, V. Vagnoni, G.\n Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M.\n Veltri, B. Viaud, I. Videau, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt,\n D. Volyanskyy, D. Voong, A. Vorobyev, H. Voss, S. Wandernoth, J. Wang, D. R.\n Ward, N. K. Watson, A. D. Webber, D. Websdale, M. Whitehead, D. Wiedner, L.\n Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi,\n M. Witek, W. Witzeling, S. A. Wotton, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z.\n Yang, R. Young, O. Yushchenko, M. Zavertyaev, F. Zhang, L. Zhang, W. C.\n Zhang, Y. Zhang, A. Zhelezov, L. Zhong, E. Zverev, A. Zvyagin",
"submitter": "Thomas Ruf",
"url": "https://arxiv.org/abs/1112.4592"
}
|
1112.4695
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2011-212 LHCb-PAPER-2011-026 December 20, 2011
Observation of $\overline{B}_{s}^{0}\rightarrow J/\psi f^{\prime}_{2}(1525)$
in $J/\psi K^{+}K^{-}$ final states
The LHCb Collaboration†††Authors are listed on the following pages.
The decay $\overline{B}_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ is investigated
using 0.16 fb-1 of data collected with the LHCb detector using 7 TeV $pp$
collisions. Although the $J/\psi\phi$ channel is well known, final states at
higher $K^{+}K^{-}$ masses have not previously been studied. In the
$K^{+}K^{-}$ mass spectrum we observe a significant signal in the
$f^{\prime}_{2}(1525)$ region as well as a non-resonant component. After
subtracting the non-resonant component, we find
${{\cal{B}}\left(\overline{B}_{s}^{0}\rightarrow J/\psi
f^{\prime}_{2}(1525)\right)}/{{\cal{B}}\left(\overline{B}_{s}^{0}\rightarrow
J/\psi\phi\right)}=(26.4\pm 2.7\pm 2.4)$%.
Keywords: LHC, $C\\!P$ violation, $B$ decays
PACS: 13.25.Hw, 14.40.Nd, 14.40.Be
Submitted to Physical Review Letters
The LHCb Collaboration
R. Aaij23, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi42, C. Adrover6, A.
Affolder48, Z. Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J.
Anderson39, R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L.
Arrabito53, A. Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16,
R.J. Barlow50, C. Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10,
Th. Bauer23, A. Bay38, I. Bediaga1, S. Belogurov30, K. Belous34, I.
Belyaev30,37, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson46, J.
Benton42, R. Bernet39, M.-O. Bettler17, M. van Beuzekom23, A. Bien11, S.
Bifani12, T. Bird50, A. Bizzeti17,h, P.M. Bjørnstad50, T. Blake37, F. Blanc38,
C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V. Bocci22, A. Bondar33, N.
Bondar29, W. Bonivento15, S. Borghi47,50, A. Borgia52, T.J.V. Bowcock48, C.
Bozzi16, T. Brambach9, J. van den Brand24, J. Bressieux38, D. Brett50, M.
Britsch10, T. Britton52, N.H. Brook42, H. Brown48, A. Büchler-Germann39, I.
Burducea28, A. Bursche39, J. Buytaert37, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,37, A. Cardini15, L. Carson49, K. Carvalho
Akiba2, G. Casse48, M. Cattaneo37, Ch. Cauet9, M. Charles51, Ph.
Charpentier37, N. Chiapolini39, K. Ciba37, X. Cid Vidal36, G. Ciezarek49,
P.E.L. Clarke46,37, M. Clemencic37, H.V. Cliff43, J. Closier37, C. Coca28, V.
Coco23, J. Cogan6, P. Collins37, A. Comerma-Montells35, F. Constantin28, A.
Contu51, A. Cook42, M. Coombes42, G. Corti37, G.A. Cowan38, R. Currie46, C.
D’Ambrosio37, P. David8, P.N.Y. David23, I. De Bonis4, S. De Capua21,k, M. De
Cian39, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi38,37, L. Del Buono8, C. Deplano15, D.
Derkach14,37, O. Deschamps5, F. Dettori24, J. Dickens43, H. Dijkstra37, P.
Diniz Batista1, F. Domingo Bonal35,n, S. Donleavy48, F. Dordei11, A. Dosil
Suárez36, D. Dossett44, A. Dovbnya40, F. Dupertuis38, R. Dzhelyadin34, A.
Dziurda25, S. Easo45, U. Egede49, V. Egorychev30, S. Eidelman33, D. van
Eijk23, F. Eisele11, S. Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch.
Elsasser39, D. Elsby55, D. Esperante Pereira36, L. Estève43, A.
Falabella16,14,e, E. Fanchini20,j, C. Färber11, G. Fardell46, C. Farinelli23,
S. Farry12, V. Fave38, V. Fernandez Albor36, M. Ferro-Luzzi37, S. Filippov32,
C. Fitzpatrick46, M. Fontana10, F. Fontanelli19,i, R. Forty37, M. Frank37, C.
Frei37, M. Frosini17,f,37, S. Furcas20, A. Gallas Torreira36, D. Galli14,c, M.
Gandelman2, P. Gandini51, Y. Gao3, J-C. Garnier37, J. Garofoli52, J. Garra
Tico43, L. Garrido35, D. Gascon35, C. Gaspar37, N. Gauvin38, M. Gersabeck37,
T. Gershon44,37, Ph. Ghez4, V. Gibson43, V.V. Gligorov37, C. Göbel54, D.
Golubkov30, A. Golutvin49,30,37, A. Gomes2, H. Gordon51, M. Grabalosa
Gándara35, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35, G.
Graziani17, A. Grecu28, E. Greening51, S. Gregson43, B. Gui52, E. Gushchin32,
Yu. Guz34, T. Gys37, G. Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42, S.
Hansmann-Menzemer11, R. Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44,
T. Hartmann56, J. He7, V. Heijne23, K. Hennessy48, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks48, K. Holubyev11, P. Hopchev4, W.
Hulsbergen23, P. Hunt51, T. Huse48, R.S. Huston12, D. Hutchcroft48, D.
Hynds47, V. Iakovenko41, P. Ilten12, J. Imong42, R. Jacobsson37, A. Jaeger11,
M. Jahjah Hussein5, E. Jans23, F. Jansen23, P. Jaton38, B. Jean-Marie7, F.
Jing3, M. John51, D. Johnson51, C.R. Jones43, B. Jost37, M. Kaballo9, S.
Kandybei40, M. Karacson37, T.M. Karbach9, J. Keaveney12, I.R. Kenyon55, U.
Kerzel37, T. Ketel24, A. Keune38, B. Khanji6, Y.M. Kim46, M. Knecht38, P.
Koppenburg23, A. Kozlinskiy23, L. Kravchuk32, K. Kreplin11, M. Kreps44, G.
Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki37, M. Kucharczyk20,25,37,j,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty50, A.
Lai15, D. Lambert46, R.W. Lambert24, E. Lanciotti37, G. Lanfranchi18, C.
Langenbruch11, T. Latham44, C. Lazzeroni55, R. Le Gac6, J. van Leerdam23,
J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J. Lefran$c$cois7, O. Leroy6, T.
Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles48, R. Lindner37, C. Linn11,
B. Liu3, G. Liu37, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-
March38, H. Lu38,3, J. Luisier38, A. Mac Raighne47, F. Machefert7, I.V.
Machikhiliyan4,30, F. Maciuc10, O. Maev29,37, J. Magnin1, S. Malde51, R.M.D.
Mamunur37, G. Manca15,d, G. Mancinelli6, N. Mangiafave43, U. Marconi14, R.
Märki38, J. Marks11, G. Martellotti22, A. Martens8, L. Martin51, A. Martín
Sánchez7, D. Martinez Santos37, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev29, E. Maurice6, B. Maynard52, A. Mazurov16,32,37, G. McGregor50, R.
McNulty12, M. Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S.
Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski25, R. Mountain52, I. Mous23, F. Muheim46, K.
Müller39, R. Muresan28,38, B. Muryn26, B. Muster38, M. Musy35, J. Mylroie-
Smith48, P. Naik42, T. Nakada38, R. Nandakumar45, I. Nasteva1, M. Nedos9, M.
Needham46, N. Neufeld37, C. Nguyen-Mau38,o, M. Nicol7, V. Niess5, N.
Nikitin31, A. Nomerotski51, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen49, B. Pal52, J. Palacios39, A.
Palano13,b, M. Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo47, C.
Parkes50,37, C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49, S.K.
Paterson49, G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos
Alvarez36, A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A.
Petrolini19,i, A. Phan52, E. Picatoste Olloqui35, B. Pie Valls35, B.
Pietrzyk4, T. Pilař44, D. Pinci22, R. Plackett47, S. Playfer46, M. Plo
Casasus36, G. Polok25, A. Poluektov44,33, E. Polycarpo2, D. Popov10, B.
Popovici28, C. Potterat35, A. Powell51, J. Prisciandaro38, V. Pugatch41, A.
Puig Navarro35, W. Qian52, J.H. Rademacker42, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk40, G. Raven24, S. Redford51, M.M. Reid44, A.C. dos Reis1,
S. Ricciardi45, K. Rinnert48, D.A. Roa Romero5, P. Robbe7, E. Rodrigues47,50,
F. Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43, S. Roiser37, V.
Romanovsky34, M. Rosello35,n, J. Rouvinet38, T. Ruf37, H. Ruiz35, G.
Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail47, B. Saitta15,d,
C. Salzmann39, M. Sannino19,i, R. Santacesaria22, C. Santamarina Rios36, R.
Santinelli37, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m,
A. Satta21, M. Savrie16,e, D. Savrina30, P. Schaack49, M. Schiller24, S.
Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt37, O. Schneider38, A.
Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba18,l, M.
Seco36, A. Semennikov30, K. Senderowska26, I. Sepp49, N. Serra39, J. Serrano6,
P. Seyfert11, M. Shapkin34, I. Shapoval40,37, P. Shatalov30, Y. Shcheglov29,
T. Shears48, L. Shekhtman33, O. Shevchenko40, V. Shevchenko30, A. Shires49, R.
Silva Coutinho44, T. Skwarnicki52, A.C. Smith37, N.A. Smith48, E. Smith51,45,
K. Sobczak5, F.J.P. Soler47, A. Solomin42, F. Soomro18, B. Souza De Paula2, B.
Spaan9, A. Sparkes46, P. Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39,
S. Stoica28, S. Stone52,37, B. Storaci23, M. Straticiuc28, U. Straumann39,
V.K. Subbiah37, S. Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26,
S. T’Jampens4, E. Teodorescu28, F. Teubert37, C. Thomas51, E. Thomas37, J. van
Tilburg11, V. Tisserand4, M. Tobin39, S. Topp-Joergensen51, N. Torr51, E.
Tournefier4,49, M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23, M. Ubeda
Garcia37, A. Ukleja27, P. Urquijo52, U. Uwer11, V. Vagnoni14, G. Valenti14, R.
Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis42, M.
Veltri17,g, B. Viaud7, I. Videau7, X. Vilasis-Cardona35,n, J. Visniakov36, A.
Vollhardt39, D. Volyanskyy10, D. Voong42, A. Vorobyev29, H. Voss10, S.
Wandernoth11, J. Wang52, D.R. Ward43, N.K. Watson55, A.D. Webber50, D.
Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G. Wilkinson51, M.P.
Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25, W.
Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Xing52, Z.
Yang3, R. Young46, O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3, L. Zhang52,
W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin37.
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
23Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraców, Poland
26AGH University of Science and Technology, Kraców, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55University of Birmingham, Birmingham, United Kingdom
56Physikalisches 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
The $\overline{B}_{s}^{0}\rightarrow J/\psi K^{+}K^{-}$ channel has previously
been studied only when the $K^{+}K^{-}$ are consistent with the decay of the
$\phi$ meson. This mode has been used to measure the $C\\!P$ violation in
$\overline{B}_{s}^{0}$ mixing, $\phi_{s}$, a key probe in the search for
physics beyond the Standard Model [1, 2, 3, 4].111Charge conjugate modes are
also considered throughout. In addition to the $\phi$ other resonant or non-
resonant final states may appear and affect the $C\\!P$ measurements,
including an S-wave contribution [5]. In this paper we study the entire
$K^{+}K^{-}$ mass spectrum, including a search for other final states that may
be useful in $C\\!P$ violation studies. These states may provide other ways of
measuring $\phi_{s}$, in decays with a different spin structure that may be
useful for revealing different aspects of $C\\!P$ violation.
We use a 0.16 fb-1 data sample collected with the LHCb detector [6] at a
center-of-mass energy of 7 TeV. The detector elements most important for this
analysis include a vertex locator, a silicon strip device that surrounds the
$pp$ interaction region in the LHC, and other downstream tracking devices
before and after a 4 Tm dipole magnet. Two ring-imaging Cherenkov detectors
are used to identify charged hadrons, while muons are identified using their
penetration through iron. This analysis is restricted to events accepted by a
di-muon trigger [7]. Subsequent selection criteria are applied that serve to
reject background, yet preserve high efficiencies, as determined by Monte
Carlo (MC) events generated using Pythia [8], and the LHCb detector simulation
based on Geant [9]. To be considered as a $J/\psi\rightarrow\mu^{+}\mu^{-}$
candidate, opposite sign tracks are required to have transverse momentum,
$p_{\rm T}$, greater than 500 MeV, be identified as muons, and give a good fit
to a common vertex.222We work in units where $c=1$. Di-muon candidates with
masses between $-48$ and +43 MeV of the $J/\psi$ peak are selected for further
analysis, where the r.m.s. resolution is 13.4 MeV. The invariant mass of the
$\mu^{+}\mu^{-}$ pair is constrained to the $J/\psi$ mass for further
analysis.
Kaon candidates are selected if their minimum distance from the closest
primary vertex is inconsistent with being produced at that vertex. They must
be positively identified based on the logarithm of the likelihood ratio
comparing two particle hypotheses (DLL). There are two criteria used; loose
corresponds to DLL($K-\pi)>0$, while tight has DLL($K-\pi)>10$ and
DLL($K-p)>-3$. We use the loose criterion for checking kaon identification
efficiencies, otherwise the tight criterion is used. In addition, the two
kaons must have the sum of the magnitudes of their $p_{\rm T}>900$ MeV.
To select $\overline{B}_{s}^{0}$ candidates we require that the $K^{+}K^{-}$
pair and the $J/\psi$ candidate give a good fit to a common secondary vertex
with a $\chi^{2}<5$ per degree of freedom. We also require that the
$\overline{B}_{s}^{0}$ candidate’s decay point must be $>$ 1.5 mm from the
primary vertex and that the negative of its momentum vector points back to the
primary.
The $\overline{B}_{s}^{0}$ candidate invariant mass is shown in Fig. 1. A
clear signal is seen, part of which comes from the previously known
$J/\psi\phi$ mode.
Figure 1: Invariant mass of $J/\psi K^{+}K^{-}$ combinations. The vertical
lines indicate the signal and sideband regions.
A check was made for possible resonant states decaying to $J/\psi K^{-}$ since
similar exotic states have been claimed [10], but no obvious structures are
visible in the invariant mass spectrum. Figure 2 shows the $K^{+}K^{-}$
invariant mass for both signal and sideband regions, where the signal region
extends $\pm$25 MeV around the $\overline{B}_{s}^{0}$ mass peak and the
sidebands extend from 35 MeV to 60 MeV on either side of the peak.
Figure 2: Invariant mass of $K^{+}K^{-}$ combinations. The histogram shows the
data in the signal region while the points (red) show the sidebands.
Apart from the large peak at the $\phi$ there is a structure near 1525 MeV. In
addition there is an excess of signal events over most of the mass range which
we refer to as non-resonant. We investigate the possibility of the peak to be
the $f^{\prime}_{2}(1525)$ resonance. The PDG quotes the mass of the
$f_{2}^{\prime}$ state as 1525$\pm$5 MeV and the width as 73${}^{+6}_{-5}$ MeV
[11]. Other states such as the $f_{2}(1270)$ and the $f_{0}(1500)$ have small
branching fractions into $K^{+}K^{-}$ of less than 5%, and are unlikely to
have large rates.
It is possible for the decay $\overline{B}^{0}\rightarrow J/\psi K^{-}\pi^{+}$
to fake our signal if the $\pi^{+}$ is misidentified as a $K^{+}$. A specific
example is given by $\overline{B}^{0}\rightarrow
J/\psi\overline{K}_{2}^{*}(1430)$ decays [12]. To examine if we are sensitive
to a reflection of this mode in the 1525 MeV di-kaon mass region, a simulation
was performed where the $\pi^{+}$ from the $\overline{K}_{2}^{*}(1430)$ was
interpreted as a $K^{+}$. Figure 3(a) shows that the reflection of this mode
does indeed peak in the di-kaon mass range around 1525 MeV. It also peaks in
the $\overline{B}_{s}^{0}$ signal region but is much wider than the
$\overline{B}_{s}^{0}$ mass peak. The region $25-200$ MeV above the
$\overline{B}_{s}^{0}$ mass peak provides a sample of misidentified
$\overline{B}^{0}\rightarrow J/\psi K^{-}\pi^{+}$ decays. By measuring the
number of $\overline{B}^{0}$ candidates in this region we can calculate the
number in the $\overline{B}_{s}^{0}$ signal region.
Figure 3: (a) The $m(K^{+}K^{-})$ distribution for simulated
$\overline{B}^{0}\rightarrow J/\psi\overline{K}_{2}^{*}(1430)$ decays where
the $\pi^{+}$ from the $\overline{K}_{2}^{*}(1430)$ decay is interpreted as a
$K^{+}$. The histogram shows $m(K^{+}K^{-})$ in the signal region of
$\overline{B}_{s}^{0}$ mass and the points in the sideband region. The
simulation corresponds to approximately 8 fb-1 of data. (b) The $m(J/\psi
K^{+}\pi^{-})$ distribution for $J/\psi K^{+}K^{-}$ data candidates $25-200$
MeV above the $\overline{B}_{s}^{0}$ mass, and with $m(K^{+}K^{-})$ within
$\pm$300 MeV of 1525 MeV, reinterpreted as $\overline{B}^{0}\rightarrow J/\psi
K^{-}\pi^{+}$ events. The fit is to a signal Gaussian whose mass and width are
allowed to vary as well as a quadratic background.
To determine the size of any $\overline{B}^{0}$ reflection in the
$f^{\prime}_{2}$ mass region we select events where the reconstructed $J/\psi
K^{+}K^{-}$ mass is in the range $25-200$ MeV above the $\overline{B}_{s}^{0}$
mass, reassign each of the two kaons in turn to the pion hypothesis, and plot
the $J/\psi K\pi$ mass. The resulting peak at the $\overline{B}^{0}$ mass has
36$\pm$10 events from the fit shown in Fig. 3(b). We calculate 37$\pm$10
events in the $\overline{B}_{s}^{0}$ signal region, using the shape from Monte
Carlo simulation, and use this number as a constraint in the fit described
below to determine the $f^{\prime}_{2}$ parameters and signal yields.
To test the $f^{\prime}_{2}$ hypothesis we perform a simultaneous fit to the
$\overline{B}_{s}^{0}$ candidate mass and the di-kaon mass. The
$f^{\prime}_{2}$ signal is parameterized by a spin-2 Breit-Wigner function
[13]. The width of the $f^{\prime}_{2}$ is fixed to the PDG value of 73 MeV
[11]. A contribution from non-resonant $K^{+}K^{-}$ is included as a linear
function in the di-kaon mass. The contribution from the $K^{-}\pi^{+}$
reflection is included using the di-kaon and $\overline{B}_{s}^{0}$ mass
shapes from the simulation, with the normalization fixed by the event yield in
Fig 3(b). The results of the fits are shown in Fig. 4. The $f_{2}^{\prime}$
mass from the fit is 1525$\pm$4 MeV and the yield is 269$\pm$26 events within
$\pm$125 MeV of the mass. If we allow the $f_{2}^{\prime}$ width to vary we
find a consistent value of 90${}^{+16}_{-14}$ MeV.
Figure 4: Projections of fits to (a) the $\overline{B}_{s}^{0}$ candidate mass
and (b) the di-kaon mass. The $f^{\prime}_{2}$ signal is parameterized by a
spin-2 Breit-Wigner function whose width is fixed to 73 MeV (dotted curve).
The combinatorial background is shown in the light shaded region, while the
darker shaded region shows the non-resonant $J/\psi K^{+}K^{-}$ component. The
long-dashed (red) line shows the misidentified $\overline{B}^{0}\rightarrow
J/\psi K^{-}\pi^{+}$ decays, and the (blue) line the total.
As we have not taken into account possible interferences between the
$f^{\prime}_{2}$ and other $J/\psi K^{+}K^{-}$ final states we do not provide
systematic uncertainties for these values. The decay angle of the $J/\psi$,
$\theta_{J/\psi}$, can test for pure spin-0, or the presence of a higher spin
state such as the spin-2 $f^{\prime}_{2}$ [11]. Here $\theta_{J/\psi}$ is
defined as the angle of the $\mu^{+}$ with respect to the
$\overline{B}_{s}^{0}$ direction in the $J/\psi$ rest frame. It is distributed
as
$f(\cos\theta_{J/\psi})=(1-p)\sin^{2}\theta_{J/\psi}+\frac{p}{2}\left(1+\cos^{2}\theta_{J/\psi}\right),$
(1)
where $1-p$ is the fraction of helicity zero and $p$ is the fraction of
helicity $\pm$1\. Shown in Fig. 5 is the background subtracted, acceptance
corrected $\cos\theta_{J/\psi}$ distribution for $K^{+}K^{-}$ masses in the
$f^{\prime}_{2}$ region. MC simulation is used to find the acceptance
correction. The points are extracted from the joint fit to the $m(J/\psi
K^{+}K^{-})$ and $m(K^{+}K^{-})$ distributions in the $K^{+}K^{-}$ mass region
within $1400-1650$ MeV for events in the peak above the non-resonant
$K^{+}K^{-}$. The fit result is $p=(0.57\pm 0.13)$, with $\chi^{2}$/number of
degrees of freedom (ndof) of 10/8 (27% probability). Fitting only with an
S-wave gives $\chi^{2}$/ndof of 27/9 (0.1% probability), showing that the data
are not likely to be pure spin-0, but are compatible with a higher spin state
consistent with an $f^{\prime}_{2}$ contribution.
Figure 5: Distribution of $\cos\theta_{J/\psi}$ for
$\overline{B}_{s}^{0}\rightarrow J/\psi f^{\prime}_{2}$ decays. The background
and non-resonant $K^{+}K^{-}$ components have been subtracted, and the data
have been corrected for acceptance. The fit to Eq. 1 is shown by the solid
line. Note that for pure S-wave the distribution would be
$\sin^{2}\theta_{J/\psi}$ ($p=0$), shown as the dotted curve, while for pure
helicity 1 ($p=1$) the data would be described by the dot-dashed curve.
The branching fraction of $\overline{B}_{s}^{0}\rightarrow J/\psi
f_{2}^{\prime}$ relative to $\overline{B}_{s}^{0}\rightarrow J/\psi\phi$ is
determined by assuming that the dominant background is S-wave and the signal
D-wave, so there is no interference between them.333Although there can be
interference as a function of the $K^{+}$ decay angle in the $f_{2}^{\prime}$
rest frame, integrating over this variable causes the net result to be zero.
The number of $J/\psi K^{+}K^{-}$ events is determined by a fit to the
$\overline{B}_{s}^{0}$ mass distribution, within $\pm$20 MeV of the $\phi$
mass. A small S-wave component in the $\phi$ mass region of (4.2$\pm$2.3)% is
subtracted [2]. Although there are the same final state particles in both
modes, the relative efficiency is (78$\pm$2)%, where the uncertainty arises
from simulation statistics. The efficiency ratio differs from unity due to the
different $p_{\rm T}$ distributions of the kaons in the final states. The kaon
identification efficiencies are corrected with respect to those given by the
MC simulation using a sample of $D^{*+}$ decays, where the kaon can be
selected without resorting to PID information. Typical corrections are on the
order of 5%.
To find the effective relative rate of $f^{\prime}_{2}$ decays we use the fit
where the width is allowed to vary. There are 320$\pm$33 $f^{\prime}_{2}$
events and 1774$\pm$42 $\phi$ events. Correcting for the relative efficiencies
and the explicit branching fractions
${\cal{B}}\left(f^{\prime}_{2}(1525)\rightarrow K^{+}K^{-}\right)=(44.4\pm
1.1)$%, and ${\cal{B}}\left(\phi\rightarrow K^{+}K^{-}\right)=(48.9\pm 0.5)$%
[11], we measure
$R\equiv\frac{{\cal{B}}\left(\overline{B}_{s}^{0}\rightarrow J/\psi
f^{\prime}_{2}(1525)\right)}{{\cal{B}}\left(\overline{B}_{s}^{0}\rightarrow
J/\psi\phi\right)}=(26.4\pm 2.7\pm 2.4)\%.$ (2)
The systematic uncertainty on $R$ has several contributions, as listed in
Table 1.
Table 1: Systematic uncertainties on $R$. Source | Change (%)
---|---
$f^{\prime}_{2}$ width | 6.3
Helicity | 4.0
Relative efficiency | 2.6
S-wave under $\phi$ | 2.3
$K^{+}K^{-}$ mass dependent efficiency | 2.3
Background shape | 1.3
$\overline{B}_{s}^{0}$ $p_{\rm T}$ distribution | 0.5
$\overline{B}_{s}^{0}$ mass resolution | 0.5
PID | 1.0
Signal shape | 1.0
${\cal{B}}\left(f^{\prime}_{2}(1525)\rightarrow K^{+}K^{-}\right)$ | 2.5
${\cal{B}}\left(\phi\rightarrow K^{+}K^{-}\right)$ | 1.0
Total | 9.2
The largest source of uncertainty is $f^{\prime}_{2}$ width. The error quoted
reflects changing the width by one standard deviation from the fitted value of
90 MeV. The helicity amplitudes of the $J/\psi f^{\prime}_{2}$ decay are
unknown, unlike the $J/\psi\phi$ amplitudes which are well measured [11]. The
difference between the values obtained using helicity zero and helicity one
$J/\psi$ MC samples is 4% compared to our central value. The S-wave
subtraction of the events in the $J/\psi\phi$ region causes a 2.3%
uncertainty. We include an uncertainty for the efficiency as a function of
$K^{+}K^{-}$ mass, as the tracking could be sensitive to the opening angle of
the kaon pair. Modifying the acceptance from a flat to linear function of mass
changes the yield by 2.3%. Varying the $\overline{B}_{s}^{0}$ $p_{\rm T}$
distribution within limits imposed by the data results in a small 0.5% change
in the rate. Changing the mass resolution by its error results in a 0.5%
change. A PID uncertainty of 1% is added to account for different momentum
distributions of the kaons in the two final states. As a check we note that
the ratio of the number of events in $J/\psi\phi$ with tight cuts to loose
cuts on the kaon identification is (61$\pm$2)% and the simulation gives a
consistent (60$\pm$1)%. Variation of the background and signal shapes makes
small differences.
In conclusion, we have made the first investigation of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow J/\psi K^{+}K^{-}$
final state over the entire range of $K^{+}K^{-}$ mass. There is a significant
non-resonant component that extends under the $\phi$ region which can affect
$C\\!P$ violation measurements [5]. We have also observed
$\overline{B}_{s}^{0}\rightarrow J/\psi f^{\prime}_{2}(1525)$ decays. The
branching fraction ratio relative to $J/\psi\phi$ is
$\frac{{\cal{B}}\left(\overline{B}_{s}^{0}\rightarrow J/\psi
f^{\prime}_{2}(1525)\right)}{{\cal{B}}\left(\overline{B}_{s}^{0}\rightarrow
J/\psi\phi\right)}=(26.4\pm 2.7\pm 2.4)\%,$ (3)
assuming that the background does not interfere with the signal amplitude.
This decay mode can also be used to measure $C\\!P$ violation in the
$\overline{B}_{s}^{0}$ system, although a different transversity analysis than
in $J/\psi\phi$ would be required as the final state is a combination of a
spin-1 $J/\psi$ and a spin-2 $f^{\prime}_{2}$ state. Some consideration has
been given to measuring $C\\!P$ violation in vector-tensor decays [14].
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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|
arxiv-papers
| 2011-12-20T14:16:47 |
2024-09-04T02:49:25.524899
|
{
"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 G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis, J.\n Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, D.\n S. Bailey, 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, 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, F.\n Constantin, A. Contu, A. Cook, M. Coombes, G. Corti, G. A. Cowan, R. Currie,\n C. D'Ambrosio, P. David, I. De Bonis, S. De Capua, M. De Cian, F. De Lorenzi,\n J. M. De Miranda, L. De Paula, P. De Simone, D. Decamp, M. Deckenhoff, H.\n Degaudenzi, L. Del Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori,\n J. Dickens, H. Dijkstra, P. Diniz Batista, F. Domingo Bonal, S. Donleavy, F.\n Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R.\n Dzhelyadin, A. Dziurda, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van\n Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch. Elsasser, D.\n Elsby, D. Esperante Pereira, L. Est\\`eve, A. Falabella, E. Fanchini, 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, 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, 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, G. Haefeli, C. Haen, S. C. Haines, T.\n Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P. F.\n Harrison, 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, P.\n Koppenburg, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P.\n Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, T. Kvaratskheliya, V. N. La\n 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. van 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, 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, R. Messi,\n S. Miglioranzi, D. A. Milanes, M.-N. Minard, S. Monteil, D. Moran, P.\n Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn,\n B. Muster, M. Musy, J. Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I.\n Nasteva, M. Nedos, M. Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, S. Nies,\n V. Niess, N. Nikitin, 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, K. Pal, J. Palacios, A. Palano, M. Palutan, J.\n 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. Petrella, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T.\n Pilar, 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, K. Rinnert, D. A. Roa Romero, P. Robbe, E.\n Rodrigues, F. Rodrigues, P. Rodriguez Perez, G. J. Rogers, S. Roiser, V.\n Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J.\n Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti, M. Sapunov,\n A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M.\n Schiller, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider,\n A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A.\n 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, A. C.\n Smith, N. A. Smith, E. Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F.\n Soomro, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V. K. Subbiah, S. Swientek, M. Szczekowski, P. Szczypka, T.\n Szumlak, S. T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Topp-Joergensen, N. Torr, E.\n Tournefier, M. T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A.\n Ukleja, P. Urquijo, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P.\n Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, B. Viaud, I. Videau,\n X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A.\n Vorobyev, H. Voss, K. Wacker, S. Wandernoth, J. Wang, D. R. Ward, N. K.\n Watson, A. D. Webber, D. Websdale, M. Whitehead, D. Wiedner, L. Wiggers, G.\n Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi, M. Witek,\n W. Witzeling, S. A. Wotton, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R.\n Young, O. Yushchenko, M. Zavertyaev, F. Zhang, L. Zhang, W. C. Zhang, Y.\n Zhang, A. Zhelezov, L. Zhong, E. Zverev, A. Zvyagin",
"submitter": "Sheldon Stone",
"url": "https://arxiv.org/abs/1112.4695"
}
|
1112.4698
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
LHCb-PAPER-2011-032 CERN-PH-EP-2011-206
Measurement of mixing and $C\\!P$ violation parameters in two-body charm
decays
The LHCb Collaboration 111Authors are listed on the following pages.
A study of mixing and indirect $C\\!P$ violation in $D^{0}$ mesons through the
determination of the parameters $y_{C\\!P}$ and $A_{\Gamma}$ is presented. The
parameter $y_{C\\!P}$ is the deviation from unity of the ratio of effective
lifetimes measured in $D^{0}$ decays to the $C\\!P$ eigenstate $K^{+}K^{-}$
with respect to decays to the Cabibbo favoured mode $K^{-}\pi^{+}$. The result
measured using data collected by LHCb in 2010, corresponding to an integrated
luminosity of $29\mbox{\,pb}^{-1}$, is
$y_{C\\!P}=(5.5\pm 6.3_{\rm stat}\pm 4.1_{\rm syst})\times 10^{-3}.$
The parameter $A_{\Gamma}$ is the asymmetry of effective lifetimes measured in
decays of $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
mesons to $K^{+}K^{-}$. The result is
$A_{\Gamma}=(-5.9\pm 5.9_{\rm stat}\pm 2.1_{\rm syst})\times 10^{-3}.$
A data-driven technique is used to correct for lifetime-biasing effects.
Submitted to JHEP
## The LHCb collaboration
R. Aaij23, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi42, C. Adrover6, A.
Affolder48, Z. Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J.
Anderson39, R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L.
Arrabito53, A. Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16,
R.J. Barlow50, C. Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10,
Th. Bauer23, A. Bay38, I. Bediaga1, S. Belogurov30, K. Belous34, I.
Belyaev30,37, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson46, J.
Benton42, R. Bernet39, M.-O. Bettler17, M. van Beuzekom23, A. Bien11, S.
Bifani12, T. Bird50, A. Bizzeti17,h, P.M. Bjørnstad50, T. Blake37, F. Blanc38,
C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V. Bocci22, A. Bondar33, N.
Bondar29, W. Bonivento15, S. Borghi47,50, A. Borgia52, T.J.V. Bowcock48, C.
Bozzi16, T. Brambach9, J. van den Brand24, J. Bressieux38, D. Brett50, M.
Britsch10, T. Britton52, N.H. Brook42, H. Brown48, A. Büchler-Germann39, I.
Burducea28, A. Bursche39, J. Buytaert37, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,37, A. Cardini15, L. Carson49, K. Carvalho
Akiba2, G. Casse48, M. Cattaneo37, Ch. Cauet9, M. Charles51, Ph.
Charpentier37, N. Chiapolini39, K. Ciba37, X. Cid Vidal36, G. Ciezarek49,
P.E.L. Clarke46,37, M. Clemencic37, H.V. Cliff43, J. Closier37, C. Coca28, V.
Coco23, J. Cogan6, P. Collins37, A. Comerma-Montells35, F. Constantin28, A.
Contu51, A. Cook42, M. Coombes42, G. Corti37, G.A. Cowan38, R. Currie46, C.
D’Ambrosio37, P. David8, P.N.Y. David23, I. De Bonis4, S. De Capua21,k, M. De
Cian39, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi38,37, L. Del Buono8, C. Deplano15, D.
Derkach14,37, O. Deschamps5, F. Dettori24, J. Dickens43, H. Dijkstra37, P.
Diniz Batista1, F. Domingo Bonal35,n, S. Donleavy48, F. Dordei11, A. Dosil
Suárez36, D. Dossett44, A. Dovbnya40, F. Dupertuis38, R. Dzhelyadin34, A.
Dziurda25, S. Easo45, U. Egede49, V. Egorychev30, S. Eidelman33, D. van
Eijk23, F. Eisele11, S. Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch.
Elsasser39, D. Elsby55, D. Esperante Pereira36, L. Estève43, A.
Falabella16,14,e, E. Fanchini20,j, C. Färber11, G. Fardell46, C. Farinelli23,
S. Farry12, V. Fave38, V. Fernandez Albor36, M. Ferro-Luzzi37, S. Filippov32,
C. Fitzpatrick46, M. Fontana10, F. Fontanelli19,i, R. Forty37, M. Frank37, C.
Frei37, M. Frosini17,f,37, S. Furcas20, A. Gallas Torreira36, D. Galli14,c, M.
Gandelman2, P. Gandini51, Y. Gao3, J-C. Garnier37, J. Garofoli52, J. Garra
Tico43, L. Garrido35, D. Gascon35, C. Gaspar37, N. Gauvin38, M. Gersabeck37,
T. Gershon44,37, Ph. Ghez4, V. Gibson43, V.V. Gligorov37, C. Göbel54, D.
Golubkov30, A. Golutvin49,30,37, A. Gomes2, H. Gordon51, M. Grabalosa
Gándara35, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35, G.
Graziani17, A. Grecu28, E. Greening51, S. Gregson43, B. Gui52, E. Gushchin32,
Yu. Guz34, T. Gys37, G. Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42, S.
Hansmann-Menzemer11, R. Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44,
T. Hartmann56, J. He7, V. Heijne23, K. Hennessy48, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks48, K. Holubyev11, P. Hopchev4, W.
Hulsbergen23, P. Hunt51, T. Huse48, R.S. Huston12, D. Hutchcroft48, D.
Hynds47, V. Iakovenko41, P. Ilten12, J. Imong42, R. Jacobsson37, A. Jaeger11,
M. Jahjah Hussein5, E. Jans23, F. Jansen23, P. Jaton38, B. Jean-Marie7, F.
Jing3, M. John51, D. Johnson51, C.R. Jones43, B. Jost37, M. Kaballo9, S.
Kandybei40, M. Karacson37, T.M. Karbach9, J. Keaveney12, I.R. Kenyon55, U.
Kerzel37, T. Ketel24, A. Keune38, B. Khanji6, Y.M. Kim46, M. Knecht38, P.
Koppenburg23, A. Kozlinskiy23, L. Kravchuk32, K. Kreplin11, M. Kreps44, G.
Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki37, M. Kucharczyk20,25,37,j,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty50, A.
Lai15, D. Lambert46, R.W. Lambert24, E. Lanciotti37, G. Lanfranchi18, C.
Langenbruch11, T. Latham44, C. Lazzeroni55, R. Le Gac6, J. van Leerdam23,
J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J. Lefrançois7, O. Leroy6, T.
Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles48, R. Lindner37, C. Linn11,
B. Liu3, G. Liu37, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-
March38, H. Lu38,3, J. Luisier38, A. Mac Raighne47, F. Machefert7, I.V.
Machikhiliyan4,30, F. Maciuc10, O. Maev29,37, J. Magnin1, S. Malde51, R.M.D.
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Märki38, J. Marks11, G. Martellotti22, A. Martens8, L. Martin51, A. Martín
Sánchez7, D. Martinez Santos37, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev29, E. Maurice6, B. Maynard52, A. Mazurov16,32,37, G. McGregor50, R.
McNulty12, M. Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S.
Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski25, R. Mountain52, I. Mous23, F. Muheim46, K.
Müller39, R. Muresan28,38, B. Muryn26, B. Muster38, M. Musy35, J. Mylroie-
Smith48, P. Naik42, T. Nakada38, R. Nandakumar45, I. Nasteva1, M. Nedos9, M.
Needham46, N. Neufeld37, C. Nguyen-Mau38,o, M. Nicol7, V. Niess5, N.
Nikitin31, A. Nomerotski51, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen49, K. Pal52, J. Palacios39, A.
Palano13,b, M. Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo47, C.
Parkes50,37, C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49, S.K.
Paterson49, G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos
Alvarez36, A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A.
Petrolini19,i, A. Phan52, E. Picatoste Olloqui35, B. Pie Valls35, B.
Pietrzyk4, T. Pilař44, D. Pinci22, R. Plackett47, S. Playfer46, M. Plo
Casasus36, G. Polok25, A. Poluektov44,33, E. Polycarpo2, D. Popov10, B.
Popovici28, C. Potterat35, A. Powell51, J. Prisciandaro38, V. Pugatch41, A.
Puig Navarro35, W. Qian52, J.H. Rademacker42, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk40, G. Raven24, S. Redford51, M.M. Reid44, A.C. dos Reis1,
S. Ricciardi45, K. Rinnert48, D.A. Roa Romero5, P. Robbe7, E. Rodrigues47,50,
F. Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43, S. Roiser37, V.
Romanovsky34, M. Rosello35,n, J. Rouvinet38, T. Ruf37, H. Ruiz35, G.
Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail47, B. Saitta15,d,
C. Salzmann39, M. Sannino19,i, R. Santacesaria22, C. Santamarina Rios36, R.
Santinelli37, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m,
A. Satta21, M. Savrie16,e, D. Savrina30, P. Schaack49, M. Schiller24, S.
Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt37, O. Schneider38, A.
Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba18,l, M.
Seco36, A. Semennikov30, K. Senderowska26, I. Sepp49, N. Serra39, J. Serrano6,
P. Seyfert11, M. Shapkin34, I. Shapoval40,37, P. Shatalov30, Y. Shcheglov29,
T. Shears48, L. Shekhtman33, O. Shevchenko40, V. Shevchenko30, A. Shires49, R.
Silva Coutinho44, T. Skwarnicki52, A.C. Smith37, N.A. Smith48, E. Smith51,45,
K. Sobczak5, F.J.P. Soler47, A. Solomin42, F. Soomro18, B. Souza De Paula2, B.
Spaan9, A. Sparkes46, P. Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39,
S. Stoica28, S. Stone52,37, B. Storaci23, M. Straticiuc28, U. Straumann39,
V.K. Subbiah37, S. Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26,
S. T’Jampens4, E. Teodorescu28, F. Teubert37, C. Thomas51, E. Thomas37, J. van
Tilburg11, V. Tisserand4, M. Tobin39, S. Topp-Joergensen51, N. Torr51, E.
Tournefier4,49, M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23, M. Ubeda
Garcia37, A. Ukleja27, P. Urquijo52, U. Uwer11, V. Vagnoni14, G. Valenti14, R.
Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis42, M.
Veltri17,g, B. Viaud7, I. Videau7, X. Vilasis-Cardona35,n, J. Visniakov36, A.
Vollhardt39, D. Volyanskyy10, D. Voong42, A. Vorobyev29, H. Voss10, S.
Wandernoth11, J. Wang52, D.R. Ward43, N.K. Watson55, A.D. Webber50, D.
Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G. Wilkinson51, M.P.
Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25, W.
Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Xing52, Z.
Yang3, R. Young46, O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3, L. Zhang52,
W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin37.
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
23Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraców, Poland
26AGH University of Science and Technology, Kraców, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55University of Birmingham, Birmingham, United Kingdom
56Physikalisches 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
Mixing of neutral $D^{0}$ mesons has only recently been established [1, 2, 3]
and first evidence for $C\\!P$ violation in the charm sector has just been
seen by LHCb [4]. In this work the mixing and $C\\!P$ violation parameters
$y_{C\\!P}$ and $A_{\Gamma}$ in the decays of neutral $D^{0}$ mesons into two
charged hadrons are studied. Both quantities are measured here for the first
time at a hadron collider. The observable $y_{C\\!P}$ is the deviation from
unity of the ratio of inverse effective lifetimes in the decay modes
$D^{0}\rightarrow K^{+}K^{-}$ and $D^{0}\rightarrow K^{-}\pi^{+}$
$y_{C\\!P}\equiv\frac{\hat{\Gamma}(D^{0}\rightarrow
K^{+}K^{-})}{\hat{\Gamma}(D^{0}\rightarrow K^{-}\pi^{+})}-1,$ (1)
where effective lifetime refers to the value measured using a single
exponential model. All decays implicitly include their charge conjugate modes,
unless explicitly stated otherwise. Similarly, $A_{\Gamma}$ is given by the
asymmetry of inverse effective lifetimes as
$A_{\Gamma}\equiv\frac{\hat{\Gamma}(D^{0}\rightarrow
K^{+}K^{-})-\hat{\Gamma}(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow
K^{+}K^{-})}{\hat{\Gamma}(D^{0}\rightarrow K^{+}K^{-})+\hat{\Gamma}(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}K^{-})}.$ (2)
The neutral $D^{0}$ mass eigenstates $|D_{1,2}\rangle$ with masses $m_{1,2}$
and widths $\Gamma_{1,2}$ can be expressed as linear combinations of the
flavour eigenstates as $|D_{1,2}\rangle=p|D^{0}\rangle\pm{}q|\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$ with complex coefficients
$p$ and $q$ satisfying $|p|^{2}+|q|^{2}=1$. The average mass and width are
defined as $m\equiv(m_{1}+m_{2})/2$ and
$\Gamma\equiv(\Gamma_{1}+\Gamma_{2})/2$; the mass and width difference are
used to define the mixing parameters $x\equiv(m_{2}-m_{1})/\Gamma$ and
$y\equiv(\Gamma_{2}-\Gamma_{1})/(2\Gamma$). The phase convention is chosen
such that $C\\!P|D^{0}\rangle=-|\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$ and $C\\!P|\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle=-|D^{0}\rangle$ which leads,
in the case of no $C\\!P$ violation ($p=q$), to $|D_{1}\rangle$ being the
$C\\!P$ odd and $|D_{2}\rangle$ the $C\\!P$ even eigenstate, respectively.
The parameter
$\lambda_{f}=\frac{q\bar{A}_{f}}{pA_{f}}=-\eta_{C\\!P}\left|\frac{q}{p}\right|\left|\frac{\bar{A}_{f}}{A_{f}}\right|e^{i\phi},$
(3)
contains the amplitude $A_{f}$ ($\bar{A}_{f}$) of $D^{0}$ ($\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$) decays to the $C\\!P$ eigenstate
$f$ with eigenvalue $\eta_{C\\!P}$. The mixing parameters $x$ and $y$ are
known to be at the level of $10^{-2}$ while both the phase and the deviation
of the magnitude from unity of $\lambda_{f}$ are experimentally only
constrained to about $0.2$ [5]. The direct $C\\!P$ violation, i.e. the
difference in the rates of $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays, is constrained to the
level of $10^{-2}$ and has recently been measured by LHCb [4]. Introducing
$|q/p|^{\pm 2}\approx 1\pm\mathrm{A_{\rm m}}$ and $|\bar{A}_{f}/A_{f}|^{\pm
2}\approx 1\pm\mathrm{A_{\rm d}}$, with the assumption that $\mathrm{A_{\rm
m}}$ and $\mathrm{A_{\rm d}}$ are small, and neglecting terms below $10^{-4}$
according to the experimental constraints, one obtains according to Ref. [6]
$y_{C\\!P}\approx\left(1+\frac{1}{8}\mathrm{A_{\rm
m}}^{2}\right)y\cos\phi-\frac{1}{2}\mathrm{A_{\rm m}}x\sin\phi.$ (4)
In the limit of no $C\\!P$ violation $y_{C\\!P}$ is equal to $y$ and hence
becomes a pure mixing parameter. However, once precise measurements of $y$ and
$y_{C\\!P}$ are made, any difference between $y$ and $y_{C\\!P}$ would be a
sign of $C\\!P$ violation.
Previous measurements of $y_{C\\!P}$ have been performed by BaBar and Belle.
The results are $y_{C\\!P}=(11.6\pm 2.2\pm 1.8)\times 10^{-3}$ [7] for BaBar
and $y_{C\\!P}=(13.1\pm 3.2\pm 2.5)\times 10^{-3}$ [2] for Belle. They are
consistent with the world average of $y=(7.5\pm 1.2)\times 10^{-3}$ [5].
The study of the lifetime asymmetry of $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mesons decaying into the singly
Cabibbo-suppressed final state $K^{+}K^{-}$ can reveal indirect $C\\!P$
violation in the charm sector. The measurement can be expressed in terms of
the quantity $A_{\Gamma}$. Using the same expansion as for $y_{C\\!P}$ leads
to
$\displaystyle A_{\Gamma}$ $\displaystyle\approx$
$\displaystyle\bigg{[}\frac{1}{2}(\mathrm{A_{\rm m}}+\mathrm{A_{\rm
d}})y\cos\phi-x\sin\phi\bigg{]}\frac{1}{1+y_{C\\!P}}$ (5)
$\displaystyle\approx$ $\displaystyle\frac{1}{2}(\mathrm{A_{\rm
m}}+\mathrm{A_{\rm d}})y\cos\phi-x\sin\phi.$
Despite this measurement being described in most literature as a determination
of indirect $C\\!P$ violation it is apparent that direct $C\\!P$ violation at
the level of $10^{-2}$ can have a contribution to $A_{\Gamma}$ at the level of
$10^{-4}$. Therefore precise measurements of both time-dependent and time-
integrated asymmetries are necessary to reveal the nature of $C\\!P$ violating
effects in the $D^{0}$ system.
The measurement of $A_{\Gamma}$ requires tagging the flavour of the $D^{0}$ at
production, which will be discussed in the following section. Previous
measurements of $A_{\Gamma}$ were performed by Belle and BaBar leading to
$A_{\Gamma}=(0.1\pm 3.0\pm 1.5)\times 10^{-3}$ [2] and $A_{\Gamma}=(2.6\pm
3.6\pm 0.8)\times 10^{-3}$ [8], respectively. They are consistent with zero,
hence showing no indication of $C\\!P$ violation.
## 2 Data selection
LHCb is a precision heavy flavour experiment which exploits the abundance of
charm particles produced in collisions at the Large Hadron Collider (LHC). The
LHCb detector [9] is a single arm spectrometer at the LHC with a
pseudorapidity acceptance of $2<\eta<5$ for charged particles. High precision
measurements of flight distances are provided by the Vertex Locator (VELO),
which consists of two halves with a series of semi-circular silicon microstrip
detectors. The VELO measurements, together with momentum information from
forward tracking stations and a $4~{}\mathrm{Tm}$ dipole magnet, lead to
decay-time resolutions of the order of one tenth of the $D^{0}$ lifetime. Two
Ring-Imaging Cherenkov (RICH) detectors using three different radiators
provide excellent pion-kaon separation over the full momentum range of
interest. The detector is completed by hadronic and electromagnetic
calorimeters and muon stations. The measurements presented here are based on a
data sample corresponding to an integrated luminosity of $29\mbox{\,pb}^{-1}$
of $pp$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ recorded
during the LHC run in 2010.
### 2.1 Trigger selection
The LHCb trigger consists of hardware and software (HLT) stages. The hardware
trigger is responsible for reducing the LHC $pp$ interaction rate from $\cal
O$(10) MHz to the rate at which the LHCb subdetectors can be read out,
nominally 1 MHz. It selects events based on the transverse momentum of track
segments in the muon stations, the transverse energy of clusters in the
calorimeters, and overall event multiplicity.
The HLT further reduced the event rate to about $2~{}{\rm\,kHz}$ in 2010, at
which the data was stored for offline processing. The HLT runs the same
software for the track reconstruction and event selection as is used offline
and has access to the full event information.
The first part of the HLT is based on the reconstruction of tracks and primary
interaction vertices in the VELO. Heavy flavour decays are identified by their
large lifetimes, which cause their daughter tracks to be displaced from the
primary interaction. The trigger first selects VELO tracks whose distance of
closest approach to any primary interaction, known as the impact parameter
(IP), exceeds $110~{}\,\upmu\rm m$. In addition the tracks are required to
have at least ten hits in the VELO to reduce further the accepted rate of
events. This cut limits the fiducial volume for $D^{0}$ decays and therefore
rejects events where the $D^{0}$ candidate has a large transverse component of
the distance of flight, causing an upper bound on the decay-time acceptance.
The term decay-time acceptance will be used throughout this paper to refer to
the selection efficiency as a function of the $D^{0}$ decay time. Selected
tracks are then used to define a region of interest in the tracking stations
after the dipole magnet, whose size is defined by an assumed minimum track
momentum of $8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$; hits inside these search
regions are used to form tracks traversing the full tracking system. Tracks
passing this selection are fitted, yielding a full covariance matrix, and a
final selection is made based on the track-fit quality and the track
$\chi^{2}({\rm IP})$. The $\chi^{2}({\rm IP})$ is a measure of the consistency
with the hypothesis that the IP is equal to zero. At least one good track is
required for the event to be accepted. The requirements on both the track IP
and on the $\chi^{2}({\rm IP})$ reduce the number of $D^{0}$ candidates with a
short decay time.
In the second part of the HLT, an exclusive selection of $D^{0}$ candidates is
performed by reconstructing two-track vertices. Further cuts are placed on the
$\chi^{2}({\rm IP})$ of the $D^{0}$ daughters and the displacement
significance of the $D^{0}$ vertex from the primary interaction, as well as a
requirement which limits the collinearity angle between the $D^{0}$ momentum
and the direction of flight, as defined by the primary and decay vertices.
These cuts all affect the distribution of the decay time of the $D^{0}$
candidates. Additional cuts are placed on track and vertex fit quality, and on
kinematic quantities such as the transverse momentum of the $D^{0}$ candidate,
which have no effect on the decay-time distribution.
### 2.2 Offline selection
Given the abundance of charm decays, the selection has been designed to
achieve high purity. It uses similar requirements to those made in the trigger
selection, though often with tighter thresholds. In addition it makes use of
the RICH information for separating kaons and pions to achieve a low
misidentification rate. A mass window of
$\pm{}16{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ (about $\pm 2\sigma$) is
applied to the invariant mass of the two $D^{0}$ daughter particles using the
appropriate mass hypotheses. After these criteria have been applied there is
negligible remaining cross-feed between the different two-body $D^{0}$ decay
modes.
Flavour tagging of the $D^{0}$ decays is done by reconstructing the
$D^{*+}\rightarrow D^{0}\pi_{\rm s}^{+}$ decay, where the charge of the slow
pion, $\pi_{\rm s}$, determines the flavour of the $D^{0}$ meson at
production. The selection applies loose requirements on the kinematics of the
bachelor pion and the quality of the $D^{*+}$ vertex fit. The most powerful
variable for selecting the $D^{*+}$ decay is the difference in the
reconstructed invariant masses of the $D^{*+}$ and the $D^{0}$ candidates,
$\Delta m$. Candidates are required to have $\Delta m$ in the range $|\Delta
m-145.4{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}|<2.0{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
Events with multiple signal candidates are excluded from the analysis. For
tagged $D^{0}$ decays this causes a reduction of the number of candidates of
about $15\%$ due to the high probability of assigning a random slow pion to
form a $D^{*+}$ candidate. The numbers of selected candidates are 286,155 for
$D^{0}\\!\rightarrow K^{-}\pi^{+}$ and 39,262 for $D^{0}\\!\rightarrow
K^{+}K^{-}$ decays.
## 3 Determination of proper-time acceptance effects
Since absolute lifetime measurements are used to extract $y_{C\\!P}$ and
$A_{\Gamma}$, it is essential to correct for lifetime-biasing effects. The
analysis uses a data-driven approach that calculates, for each candidate and
at every possible decay time, an acceptance value of zero or one which is
related to the trigger decision and offline selection. The final per-event
acceptance function is used in the normalisation of the decay-time probability
density function (PDF) as described in the following section.
The method used to determine decay-time acceptance effects is based on the so-
called “swimming” algorithm. This approach was first used at the NA11
spectrometer [10], further developed within DELPHI [11] and CDF [12, 13],
studied at LHCb [14, 15], and applied to the measurement of the
$B^{0}_{s}\rightarrow K^{+}K^{-}$ lifetime [16].
The key to this method is the ability to execute the LHCb trigger software,
including the reconstruction, in precisely the same configuration used during
data taking. This is made possible by the implementation of all lifetime-
biasing requirements of the trigger in software rather than in the hardware.
The acceptance as a function of decay time is evaluated per event by
artificially moving the position of the primary interaction vertices
reconstructed in the trigger along the direction of the $D^{0}$ momentum in
order to give the $D^{0}$ candidate a different decay time. In events
containing multiple primary vertices, all are moved coherently in the
direction of the $D^{0}$ momentum. An analogous procedure is used to obtain
the decay-time acceptance of the offline selection.
A decay-time acceptance function for any single event is in the simplest case
a step function, as shown in Fig. 1, since the kinematics and chosen decay
time of the $D^{0}$ decay fully determine whether the event is triggered by
this $D^{0}$ candidate or not. It is important to note that the acceptance
function for a given event does not depend on the measured decay time of that
event, $t_{\rm meas}$. Accepted (rejected) regions take an acceptance value of
$1$ ($0$). In this method decay-time independent selection efficiencies are
factorised out and hence do not affect the result. The presence of additional
interaction vertices can lead to regions of no acceptance and the VELO
geometry puts an upper limit on the accepted range. Thus, a general decay-time
acceptance function is given as a series of steps or top-hat functions. The
decay times at which the event enters or leaves one of these top hats are
called turning points. The acceptance functions of the trigger and offline
selections are combined to a single acceptance function by including only the
ranges which have been accepted by both selections.
(a)
(b)
(c)
Figure 1: Variation of the decay-time acceptance function for a two-body
$D^{0}$ decay when moving the primary vertex along the $D^{0}$ momentum
vector. The shaded, light blue regions show the bands for accepting a track
impact parameter. While the impact parameter of the negative track (IP2) is
too low in (a) it reaches the accepted range in (b). The actual measured decay
time, $t_{\rm meas}$, lies in the accepted region which continues to larger
decay times (c).
The idea of studying the decay-time dependence of the acceptance in principle
requires moving the hits produced by the $D^{0}$ decay products. The
implementation of moving the primary vertices instead leads to significant
technical simplifications. However, this procedure ignores the fact that
events are no longer accepted if the mother particle has such a long decay
time that one or both tracks can no longer be reconstructed inside the VELO.
This is a very small effect as a $D^{0}$ meson has to fly ten to a hundred
times its average distance of flight in order to escape detection in the VELO.
Nevertheless, this effect can be estimated based on the knowledge of the
position of the VELO modules and on the number of hits required to form a
track. Using the information on the position of the VELO sensors, the limit of
the acceptance is determined by swimming the tracks along the $D^{0}$ momentum
vector. The result is treated as another per event decay-time acceptance and
merged with the swimming results of the trigger and offline selections.
Finally, the track reconstruction efficiency in the trigger is reduced
compared to the offline reconstruction due to the requirements described in
Sect. 2. It has been verified, using a smaller sample acquired without a
lifetime biasing selection, that this relative reconstruction efficiency does
not depend on the decay time of the $D^{0}$ candidate with a precision of
$3\times 10^{-3}$, and therefore introduces no significant additional
acceptance effect.
## 4 Fitting method
The peak in $\Delta m$ from true $D^{*+}$ decays is parametrised as the sum of
three Gaussians; two of which have a common mean and a third which has a
slightly higher mean. The random $\pi_{\rm s}$ background PDF is given by
$f_{\pi_{\rm s}}(\Delta m)=\left(\frac{\Delta
m}{a}\right)^{2}\>\left(1-\exp(-\frac{\Delta
m-d}{c})\right)\>+\>b\left(\frac{\Delta m}{d}-1\right)\qquad\Delta m\geq d,$
(6)
where $a$ and $b$ define the slope at high values of $\Delta m$, $c$ defines
the curvature at low values of $\Delta m$ and $\Delta m=d$ defines the
threshold below which the function is equal to zero. Figure 2 shows the
$\Delta m$ vs $m_{D^{0}}$ distribution and Fig. 3 shows the fit to the mass
difference between the reconstructed invariant masses of $D^{*+}$ and $D^{0}$
candidates, $\Delta m$.
Figure 2: $\Delta m$ vs $m_{D^{0}}$ distribution for $D^{0}\rightarrow
K^{-}\pi^{+}$ candidates. The contribution of random slow pions extends around
the signal peak in the vertical direction while background is visible as a
horizontal band.
Figure 3: $\Delta m$ fit projections of (left) $D^{0}\rightarrow K^{-}\pi^{+}$
and (right) $D^{0}\rightarrow K^{+}K^{-}$ candidates. Shown are data (points),
the total fit (green, solid) and the background component (blue, dot-dashed).
The signal yield is extracted from fits to the reconstructed $D^{0}$ invariant
mass distribution after application of the cut in $\Delta m$. The fit model
for the signal peak has been chosen to be a double Gaussian and background is
modelled as a first-order polynomial. The background level is evaluated to be
about $1\%$ for $D^{0}\\!\rightarrow K^{-}\pi^{+}$ decays and about $3\%$ for
$D^{0}\\!\rightarrow K^{+}K^{-}$ decays. It consists of combinatorial
background and partially reconstructed or misidentified $D^{0}$ decays. If the
latter stem from a $D^{*+}$ decay they have a peaking distribution in $\Delta
m$ similar to signal candidates. The data in the mass sidebands are
insufficient to reliably describe the background shape in other variables, so
the background contribution is neglected in the time-dependent fit and a
systematic uncertainty is estimated accordingly.
Events inside the signal windows in $\Delta m$ and $m_{D^{0}}$ are used in the
lifetime fit, where $D^{0}$ mesons produced at the primary vertex (prompt)
have to be distinguished from those originating from $b$ hadron decays
(secondary). The combined PDF for this decay-time dependent fit is factorized
as
$f(\chi^{2}({\rm IP}_{D}),t,A)=\sum_{\begin{subarray}{c}{\rm class}\\\ ={\rm
prompt},\\\ {\rm secondary}\end{subarray}}f_{\text{IP}}(\chi^{2}({\rm
IP}_{D})|t,A,{\rm class})\>f_{t}(t|A,{\rm class})\>f_{\text{TP}}(A|{\rm
class})\>P({\rm class}).$ (7)
The four factors on the right-hand side of Eq. 7, which will be described in
detail below, are:
* •
the time-dependent PDFs for the $\ln\chi^{2}({\rm IP}_{D})$ values for prompt
and secondary $D^{0}$ mesons;
* •
the decay-time PDFs for prompt and secondary $D^{0}$ mesons;
* •
the PDF for the turning points which define the acceptance $A$;
* •
the fractions of prompt and secondary $D^{0}$ decays among the signal
candidates.
The separation of prompt and secondary $D^{0}$ mesons is done on a statistical
basis using the impact parameter of the $D^{0}$ candidate with respect to the
primary vertex, ${\rm IP}_{D}$. For prompt decays, this is zero up to
resolution effects, but can acquire larger values for secondary decays as the
$D^{0}$ candidate does not in general point back to the primary vertex. Given
an estimate of the vertex resolution is available on an event-by-event basis,
it is advantageous to use the $\chi^{2}$ of the ${\rm IP}_{D}$ instead of the
impact parameter value itself. The natural logarithm of this quantity,
$\ln(\chi^{2}({\rm IP}_{D}))$, allows for an easier parametrisation.
Empirically, the sum of two bifurcated Gaussians, i.e. Gaussians with
different widths on each side of the mean, and a third, symmetric Gaussian,
all sharing a common peak position, is found to be a suitable model to
describe the $\ln(\chi^{2}({\rm IP}_{D}))$ distribution for both prompt and
secondary $D^{0}$.
For the prompt $D^{0}$ class the $\ln(\chi^{2}({\rm IP}_{D}))$ distribution
does not change with $D^{0}$ decay time as the true value is zero at all times
and the resolution of ${\rm IP}_{D}$ can be assumed to be independent of the
measured decay time. For secondary $D^{0}$ decays the decay-time and
$\ln(\chi^{2}({\rm IP}_{D}))$ are correlated. The width of the
$\ln(\chi^{2}({\rm IP}_{D}))$ distribution is found to be approximately
constant in decay time for both prompt and secondary $D^{0}$ mesons. As Monte
Carlo simulation studies suggest that secondary decays have a larger width in
this variable, a scale factor between the widths for prompt and secondary
mesons is introduced. The mean value of $\ln(\chi^{2}({\rm IP}_{D}))$
increases with $D^{0}$ decay time, which reflects the fact that $D^{0}$ mesons
coming from other long-lived decays do not necessarily point back to the
primary vertex and that they may point further away the further their parent
particle flies. The functional form for this time dependence is based on
simulation and all parameters are determined in the fit to data.
The decay-time PDF, $f_{t}(t|A,{\rm class})$ is modelled as a single
exponential for the prompt $D^{0}$ class and as a convolution of two
exponentials for secondary decays. To account for resolution effects, these
are convolved with a single Gaussian resolution function. The parameters of
the resolution model are obtained from a fit to the decay time distribution of
prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ events. The resulting
dilution is equivalent to that of a single Gaussian with a width of
$50\rm\,fs$ [17]. The decay-time probability densities are properly normalized
by integrating their product with the acceptance function $A$, evaluated by
the swimming method, only over the decay-time intervals for which the event
would have been accepted. Hence, the acceptance turning points are used as
boundaries in the integration.
Finally, a PDF for the per-event acceptance function is needed. While the
first acceptance turning point, i.e. the one with the smallest decay time,
depends on the $D^{0}$ decay topology, the others are governed more by the
underlying event structure, e.g. the distribution of primary vertices. The
primary vertex distribution is independent of whether the $D^{0}$ candidate is
of prompt or secondary origin. Hence, the PDF can be approximated as
$f_{\text{TP}}(A|{\rm class})\approx f_{\text{TP}}(\text{TP}_{1}|{\rm
class})$, where $\text{TP}_{1}$ denotes the position of the first turning
point. The distribution for $f_{\text{TP}}(\text{TP}_{1}|{\rm prompt})$ is
obtained by applying a cut at $\ln\chi^{2}({\rm IP}_{D})<1$, thus selecting a
very pure sample of prompt decays. The distribution for
$f_{\text{TP}}(\text{TP}_{1}|{\rm secondary})$ is obtained from the
distribution of $\text{TP}_{1}$ weighted by the probability of each candidate
being of secondary decay origin.
An initial fit is performed using the full $\ln\chi^{2}({\rm IP}_{D})$
distribution and all parameters in the description of this term are then fixed
in the final fit. A cut is then applied requiring $\ln\chi^{2}({\rm
IP}_{D})<2$ in order to suppress the fractions of both background and
secondary candidates to less than a few percent. The final fit is performed on
this reduced sample. The effect of this procedure is estimated in the
systematic uncertainty evaluation.
## 5 Cross-checks and systematic uncertainties
The method for absolute lifetime measurements described in Sect. 4 comprises
three main parts whose accuracy and potential for biasing the measurement have
to be evaluated in detail:
* •
the determination of the event-by-event decay-time acceptance;
* •
the separation of prompt from secondary charm decays;
* •
the estimation of the decay time distribution of combinatorial background.
Since the contribution of combinatorial background is ignored in the fit, it
is important to evaluate the corresponding systematic uncertainty.
Furthermore, several other parameters are used in the fit whose systematic
effects have to be evaluated, e.g. the description of the decay-time
resolution. It is generally expected that the systematic uncertainties in
$y_{C\\!P}$ are similar to or larger than those in $A_{\Gamma}$ as in
$y_{C\\!P}$ two different final states contribute to the measurement.
Several consistency checks are performed by splitting the dataset into
subsets. The stability is tested as a function of run period, $D^{0}$ momentum
and transverse momentum, and primary vertex multiplicity. No significant trend
is observed and therefore no systematic uncertainty assigned.
The fitting procedure is verified using simplified Monte Carlo simulation
studies. No indication of a bias is observed and the statistical uncertainties
are estimated accurately.
As an additional check, a control measurement is performed using the lifetime
asymmetry of $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
decays to the Cabibbo favoured decay $D^{0}\\!\rightarrow K^{-}\pi^{+}$. The
result is in agreement with zero and the flavour-averaged $D^{0}$ lifetime is
found to be consistent with the world average. Detailed results are given in
Sect. 6. The fit results for $D^{0}\\!\rightarrow K^{+}K^{-}$ decays were not
looked at throughout the development of the method and the study of systematic
uncertainties for the analyses of $y_{C\\!P}$ and $A_{\Gamma}$.
### 5.1 Evaluation of systematic uncertainties
Particle decay times are measured from the distance between the primary vertex
and secondary decay vertex in the VELO. The systematic uncertainty from the
distance scale is determined by considering the potential error on the length
scale of the detector from the mechanical survey, thermal expansion and the
current alignment precision. A relative systematic uncertainty of $0.1\%$ is
assigned to the measurements of absolute lifetimes, translating into a
relative uncertainty of $0.1\%$ on $A_{\Gamma}$ and $y_{C\\!P}$.
The method to evaluate the turning points of the decay-time acceptance
functions described in Sect. 3 uses an iterative approach which estimates the
turning points to a precision of about $1\rm\,fs$. Two scenarios have been
tested: a common bias of all acceptance turning points and a common length
scaling of the turning points, which could originate from differences in the
length scale in the trigger and offline reconstructions. From a variation of
the bias and the scale, a systematic uncertainty of $0.1\times 10^{-3}$ on
$A_{\Gamma}$ and $y_{C\\!P}$ is determined.
The reconstruction acceptance is dominated by the VELO geometry, which is
accounted for by the method described in Sect. 3. This leads to a correction
of less than $1\rm\,fs$ on the absolute lifetime measurements, i.e. a relative
correction of about $0.24\%$. No further systematic uncertainty is assigned to
$A_{\Gamma}$ or $y_{C\\!P}$ as the size of this relative correction is
negligible. Additional studies of the reconstruction efficiency as a function
of variables governing the decay geometry did not provide any indication of
lifetime biasing effects.
The decay-time resolution is modelled by a single Gaussian. The width of the
resolution function is varied from its nominal value of $0.05{\rm\,ps}$
between $0.03{\rm\,ps}$ and $0.07{\rm\,ps}$. The range of variation was chosen
to cover possible alignment effects as well as effects from the different
final state used to evaluate the resolution. The result leads to a systematic
uncertainty of $0.1\times 10^{-3}$ for $A_{\Gamma}$ and $y_{C\\!P}$.
The fit range in decay time is restricted by lower and upper limits. The lower
limit is put in place to avoid instabilities in regions with extremely low
decay-time acceptances and very few events. The default cut value is
$0.25{\rm\,ps}$ which is close to the lower end of the observed range of
events. This cut is varied to both $0.2{\rm\,ps}$ and $0.3{\rm\,ps}$. The
result leads to a systematic uncertainty of $0.1\times 10^{-3}$ for
$A_{\Gamma}$ and $0.8\times 10^{-3}$ for $y_{C\\!P}$.
The upper limit of the fit range in decay time is put in place to minimise the
impact of long-lived background events. The default cut is put at $6{\rm\,ps}$
which corresponds to about $15$ $D^{0}$ lifetimes. This cut is varied to
$5{\rm\,ps}$ and $8{\rm\,ps}$. The result leads to a systematic uncertainty of
$0.2\times 10^{-3}$ for $A_{\Gamma}$ and $y_{C\\!P}$.
The description of the contribution from combinatorial background is studied
by varying its relative amount in the data sample and repeating the fit. This
is done by changing the $\Delta m$ window from the default of $\pm
2{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to $\pm
1{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $\pm
3{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The result leads to a systematic
uncertainty of $1.3\times 10^{-3}$ for $A_{\Gamma}$ and $0.8\times 10^{-3}$
for $y_{C\\!P}$.
Events that originate from secondary charm decays are the background with the
largest impact on the fit procedure as they have a very different decay-time
distribution compared to prompt charm decays, but they peak in the invariant
mass and $\Delta m$ distributions. Also a fraction of combinatorial background
events appear to be secondary-like in their $\ln\chi^{2}({\rm IP}_{D})$
distribution. The cut of $\ln\chi^{2}({\rm IP}_{D})<2$ removes a large
fraction of secondary-like events. However, it is important that the remainder
is properly modelled and does not bias the signal lifetime. Varying this cut
changes the relative number of secondary-like decays in the sample and
therefore tests the stability of the secondary description in the fit model.
The fraction of secondary-like combinatorial background events is also altered
with this test. The $\ln\chi^{2}({\rm IP}_{D})$ cut is varied from $1.5$ which
is just above the peak of the prompt distribution to $3.5$ where the
probability densities for prompt and secondary decays are about equal. The
result leads to a systematic uncertainty of $1.6\times 10^{-3}$ for
$A_{\Gamma}$ and $3.9\times 10^{-3}$ for $y_{C\\!P}$. The uncertainty is
significantly larger for $y_{C\\!P}$ than for $A_{\Gamma}$ as may be expected
from the difference in the background level in the channels involved in the
$y_{C\\!P}$ measurement.
Additional studies were performed to estimate the potential impact of
neglecting background events in the fit. A background component was added to a
simplified simulation. The background decay time distribution was generated
using extreme values of fits to the distribution observed in mass sidebands.
The average bias on the measurement of $y_{C\\!P}$ was about $2\times
10^{-3}$. Since this is consistent with the assigned systematic uncertainty,
we do not assign any additional uncertainty.
Furthermore, a background component was added to the $D^{0}$ decay-time PDF
with a fixed fraction and average lifetime. The fraction of this component,
which was assumed to be secondary-like, was varied. A change in the fit result
for $y_{C\\!P}$ of $0$ (all background secondary-like) to $4\times 10^{-3}$
(all background prompt-like) was observed. As it is known that a fraction of
the background events are secondary-like, this result is considered consistent
with the simplified simulation results.
### 5.2 Summary of systematic uncertainties
Table 1 summarises the systematic uncertainties evaluated as described above.
The main systematic uncertainties are due to neglecting the combinatorial
background and to the contribution of secondary-like decays. The total
systematic uncertainties for $A_{\Gamma}$ and $y_{C\\!P}$, obtained by
combining all sources in quadrature, are $2.1\times 10^{-3}$ and $4.1\times
10^{-3}$, respectively.
Table 1: Summary of systematic uncertainties. Effect | $A_{\Gamma}$ $(10^{-3})$ | $y_{C\\!P}$ $(10^{-3})$
---|---|---
Decay-time acceptance correction | $0.1$ | $0.1$
Decay-time resolution | $0.1$ | $0.1$
Minimum decay-time cut | $0.1$ | $0.8$
Maximum decay-time cut | $0.2$ | $0.2$
Combinatorial background | $1.3$ | $0.8$
Secondary-like background | $1.6$ | $3.9$
Total | $2.1$ | $4.1$
## 6 Results and conclusion
The measurement of $y_{C\\!P}$ is based on absolute lifetime measurements as
described in Sect. 4. It uses flavour-tagged events reconstructed in the decay
chain $D^{*+}\rightarrow D^{0}\pi^{+}$, with $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays fitted simultaneously per
decay mode. The $\ln\chi^{2}({\rm IP}_{D})$ projection of the final fit is
shown in Fig. 4.
Figure 4: $\ln\chi^{2}({\rm IP}_{D})$ fit projection of $D^{0}\rightarrow
K^{+}K^{-}$ candidates in logarithmic scale. Shown are data (points), the
total fit (green, solid), the prompt signal (blue, short-dashed), and the
secondary signal (purple, long-dashed).
The result for the lifetime measured in $D^{0}\\!\rightarrow K^{-}\pi^{+}$
decays is $\tau(D^{0})=410.2\pm 0.9\rm\,fs$ where the uncertainty is
statistical only. The result for the lifetime is found to be in agreement with
the current world average [18]. Combining with the $D^{0}\\!\rightarrow
K^{+}K^{-}$ lifetime measurement, $\tau(D^{0})=408.0\pm 2.4_{\rm
stat}\rm\,fs$, this leads to the final result for $y_{C\\!P}$ of
$y_{C\\!P}=(5.5\pm 6.3_{\rm stat}\pm 4.1_{\rm syst})\times 10^{-3}.$
The measurement of $A_{\Gamma}$ is performed based on the same dataset and
applying the same fitting method as used for the extraction of $y_{C\\!P}$. A
control measurement is performed using decays to the Cabibbo favoured mode
$D^{0}\\!\rightarrow K^{-}\pi^{+}$ by forming a lifetime asymmetry analogous
to Eq. 2. The measured flavour-tagged lifetimes are effective parameters since
the fitted distributions also include mistagged events. For the control
measurement using $D^{0}\\!\rightarrow K^{-}\pi^{+}$ decays this contamination
is ignored as it is negligible due to the Cabibbo suppression of the mistagged
decays. The result for the asymmetry is
$A_{\Gamma}^{K\pi,\mathrm{eff}}=(-0.9\pm 2.2_{\rm stat})\times 10^{-3}$ which
is consistent with zero, according to expectations.
For the extraction of $A_{\Gamma}$, the mistagged decays are taken into
account by expressing the measured effective lifetimes, $\tau^{\mathrm{eff}}$,
in terms of the flavour-tagged lifetimes, $\tau(D^{0})$ and $\tau(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})$, and the mistag rate,
$\epsilon_{\pm}$, where the sign is according to the sign of the tagging pion:
$\displaystyle\tau^{\mathrm{eff}}(D^{0})$ $\displaystyle\approx$
$\displaystyle(1-\epsilon_{+})\>\tau(D^{0})+\epsilon_{+}\>\tau(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})$ (8)
$\displaystyle\tau^{\mathrm{eff}}(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})$ $\displaystyle\approx$
$\displaystyle(1-\epsilon_{-})\>\tau(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})+\epsilon_{-}\>\tau(D^{0}).$ (9)
The mistag rates are assumed to be independent of the final state and are
extracted from the favoured $D^{0}\\!\rightarrow K^{-}\pi^{+}$ decays as half
the fraction of the random slow pion background in the signal region of the
$\Delta m$ distribution. They are found to be about $1.8\%$. The systematic
uncertainty due to this correction is negligible.
Figure 5: Proper-time fit projections of (left) $D^{0}\rightarrow K^{+}K^{-}$
and (right) $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow
K^{+}K^{-}$ candidates after application of the $\ln\chi^{2}({\rm IP}_{D})<2$
cut. Shown are data (points), the total fit (green, solid), the prompt signal
(blue, short-dashed), and the secondary signal (purple, long-dashed).
The projection of the decay-time fit to $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ candidates in $D^{0}\\!\rightarrow
K^{+}K^{-}$ decays is shown in Fig. 5. After applying the mistag correction,
the resulting value of $A_{\Gamma}$ is
$A_{\Gamma}=(-5.9\pm 5.9_{\rm stat}\pm 2.1_{\rm syst})\times 10^{-3}.$
Both results on $y_{C\\!P}$ and $A_{\Gamma}$ are compatible with zero and in
agreement with previous measurements [2, 7, 8]. Future updates are expected to
lead to significant improvements in the sensitivity. The systematic
uncertainty is expected to be reduced by an improved treatment of background
events which will be possible for the data taken in 2011.
## 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
| 2011-12-20T14:23:04 |
2024-09-04T02:49:25.533951
|
{
"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 G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis, J.\n Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, D.\n S. Bailey, 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, F.\n Constantin, A. Contu, A. Cook, M. Coombes, G. Corti, 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, F. De Lorenzi, J. M. De Miranda, L. De Paula, P. De Simone, D. Decamp,\n M. 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, L. Est\\`eve, A. Falabella, E.\n Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, 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, N. Gauvin, M.\n Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, G. Haefeli, C.\n Haen, S. 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, P. Koppenburg, A. Kozlinskiy, L. Kravchuk, K. Kreplin,\n M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, T.\n Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles,\n R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E. Lopez\n Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I.\n V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R. M. D. Mamunur,\n G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, 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, R. Messi, S. Miglioranzi, D. A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Nedos, M.\n Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora Goicochea, P.\n Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis,\n M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M.\n Patel, S. K. Paterson, G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A.\n Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.\n L. Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrella, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R. Plackett, S.\n Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B.\n Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig\n Navarro, W. Qian, J. H. Rademacker, B. Rakotomiaramanana, M. S. Rangel, I.\n Raniuk, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, K.\n Rinnert, D. A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez\n Perez, G. J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T.\n Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, C. Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R.\n Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M.\n Savrie, D. Savrina, P. Schaack, M. Schiller, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, A. C. Smith, N. A.\n Smith, E. Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F. Soomro, B. Souza\n 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. Topp-Joergensen, N. Torr, E. Tournefier, M. T.\n Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, P. Urquijo,\n U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S.\n Vecchi, J. J. Velthuis, M. Veltri, B. Viaud, I. Videau, X. Vilasis-Cardona,\n J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, H. Voss, 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. Zavertyaev, F.\n Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, E. Zverev, A.\n Zvyagin",
"submitter": "Marco Gersabeck",
"url": "https://arxiv.org/abs/1112.4698"
}
|
1112.4830
|
# Askey–Wilson Integral and its Generalizations
Paweł J. Szabłowski Department of Mathematics and Information Sciences,
Warsaw University of Technology
pl. Politechniki 1, 00-661 Warsaw, Poland pawel.szablowski@gmail.com
(Date: December 20, 2011)
###### Abstract.
We expand the Askey–Wilson (AW) density in a series of products of continuous
$q-$Hermite polynomials times the density that makes these polynomials
orthogonal. As a by-product we obtain the value of the AW integral as well as
the values of integrals of $q-$Hermite polynomial times the AW density
($q-$Hermite moments of AW density). Our approach uses nice, old formulae of
Carlitz and is general enough to venture a generalization. We prove that it is
possible and pave the way how to do it.
###### Key words and phrases:
Askey–Wilson integral, Askey–Wilson polynomials, q-Hermite polynomials,
expansion of ratio of densities, symmetric functions.
###### 1991 Mathematics Subject Classification:
33D45, 05A30, 05E05
## 1\. Introduction and Preliminaries
### 1.1. Introduction
We consider sequence of nonnegative, integrable functions:
$g_{n}:[-1,1]\longmapsto\mathbb{R}^{+}$ defined by the formula:
$g_{n}\left(x|\mathbf{a}^{\left(n\right)},q\right)\allowbreak=\allowbreak
f_{h}\left(x|q\right)\prod_{j=1}^{n}\varphi_{h}\left(x|a_{j},q\right),$
where
$\mathbf{a}^{\left(n\right)}\allowbreak=\allowbreak(a_{1},\ldots,a_{n}),$
functions $f_{h}$ and $\varphi_{h}$ defined by (1.16) and (1.14) denote in
fact respectively the density of measure that makes the so called continuous
$q-$Hermite polynomials orthogonal and the characteristic function of these
polynomials calculated at points $a_{j},$ $j=1,\ldots,n.$ Naturally functions
$g_{n}$ are symmetric with respect to vectors $\mathbf{a}^{\left(n\right)}.$
Our elementary but crucial for this paper observation is that examples of such
functions are the densities of measures that make orthogonal respectively the
so called continuous $q-$Hermite (q-Hermite, $n=0$), big $q-$Hermite (bqH,
$n=1),$ Al-Salam–Chihara (ASC, $n=2)$, continuous dual Hahn (C2H, $n=3),$
Askey–Wilson (AW, $n=4)$ polynomials. This observation makes functions $g_{n}$
important and what is more exciting allows possible generalization of both AW
integral as well as AW polynomials, i.e. go beyond $n\allowbreak=\allowbreak
4.$
Similar observations were in fact made in [10] when commenting on formula
10.11.19. Hence one can say that we are developing certain idea of [10].
On the other hand by the observation that these functions are symmetric in
variables $\mathbf{a}^{\left(n\right)}$ we enter the fascinating world of
symmetric functions.
The paper is organized as follows. Next Subsection 1.2 presents used notation
and basic families of orthogonal polynomials that will appear in the sequel.
We also present here important properties of these polynomials.
Section 2 is devoted to expanding functions $g_{n}$ in the series of the form:
$g_{n}\left(x|\mathbf{a}^{\left(n\right)},q\right)=A_{n}\left(\mathbf{a}^{\left(n\right)},q\right)f_{h}\left(x|q\right)\sum_{j\geq
0}\frac{T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)}{\left(q\right)_{j}}h_{j}\left(x|q\right),$
where $\left\\{h_{n}\right\\}$ denote q-Hermite polynomials,
$\left\\{T_{j}^{\left(n\right)}\right\\}$ are sequences of certain symmetric
functions and finally $\left\\{A_{n}\right\\}$ are the values of the integrals
$\int_{-1}^{1}g_{n}\left(x|\mathbf{a}^{\left(n\right)},q\right)dx,$
and symbol $\left(q\right)_{j}$ is explained at the beginning of next
Subsection.
We do this effectively for $n=0,\ldots 4,$ obtaining known results in a new
way. In Section 3 we show that defined above sequences do exist and present
the way how to obtain them recursively. We are unable however to present nice
compact forms of these sequences resembling those obtained for $n\leq 4,$ thus
posing several open questions (see Subsection 3.2) and leaving the field to
younger and more talented researchers.
The partially legible although not very compact form was obtained for
$\int_{-1}^{1}g_{5}\left(x|\mathbf{a}^{\left(5\right)},q\right)dx$ (see
(3.3)).
For $q\allowbreak=\allowbreak 0$, the case important for the rapidly
developing so called ’free probability’, we give simple, compact form for
$\int_{-1}^{1}g_{5}\left(x|\mathbf{a}^{\left(5\right)},0\right)dx$ (see
Theorem 2, ii)) paving the way to conjecture the compact form of (3.3).
Tedious, uninteresting proofs are shifted to Section 4.
### 1.2. Preliminaries
$q$ is a parameter. We will assume that $-1<q\leq 1$ unless otherwise stated.
Let us define $\left[0\right]_{q}\allowbreak=\allowbreak 0,$
$\left[n\right]_{q}\allowbreak=\allowbreak 1+q+\ldots+q^{n-1}\allowbreak,$
$\left[n\right]_{q}!\allowbreak=\allowbreak\prod_{j=1}^{n}\left[j\right]_{q},$
with $\left[0\right]_{q}!\allowbreak=1$ and
$\QATOPD[]{n}{k}_{q}\allowbreak=\allowbreak\left\\{\begin{array}[]{ccc}\frac{\left[n\right]_{q}!}{\left[n-k\right]_{q}!\left[k\right]_{q}!}&,&n\geq
k\geq 0\\\ 0&,&otherwise\end{array}\right..$
We will use the so called $q-$Pochhammer symbol for $n\geq 1:$
$\displaystyle\left(a;q\right)_{n}$ $\displaystyle=$
$\displaystyle\prod_{j=0}^{n-1}\left(1-aq^{j}\right),$
$\displaystyle\left(a_{1},a_{2},\ldots,a_{k};q\right)_{n}\allowbreak$
$\displaystyle=$
$\displaystyle\allowbreak\prod_{j=1}^{k}\left(a_{j};q\right)_{n}.$
with $\left(a;q\right)_{0}=1$.
Often $\left(a;q\right)_{n}$ as well as
$\left(a_{1},a_{2},\ldots,a_{k};q\right)_{n}$ will be abbreviated to
$\left(a\right)_{n}$ and $\left(a_{1},a_{2},\ldots,a_{k}\right)_{n},$ if it
will not cause misunderstanding.
It is easy to notice that
$\left(q\right)_{n}=\left(1-q\right)^{n}\left[n\right]_{q}!$ and that
(1.1)
$\QATOPD[]{n}{k}_{q}\allowbreak=\allowbreak\allowbreak\left\\{\begin{array}[]{ccc}\frac{\left(q\right)_{n}}{\left(q\right)_{n-k}\left(q\right)_{k}}&,&n\geq
k\geq 0\\\ 0&,&otherwise\end{array}\right..$
###### Remark 1.
Notice that $\left[n\right]_{1}\allowbreak=\allowbreak
n,\left[n\right]_{1}!\allowbreak=\allowbreak n!,$
$\QATOPD[]{n}{k}_{1}\allowbreak=\allowbreak\binom{n}{k},$
$\left(a;1\right)_{n}\allowbreak=\allowbreak\left(1-a\right)^{n}$ and
$\left[n\right]_{0}\allowbreak=\allowbreak\left\\{\begin{array}[]{ccc}1&if&n\geq
1\\\ 0&if&n=0\end{array}\right.,$ $\left[n\right]_{0}!\allowbreak=\allowbreak
1,$ $\QATOPD[]{n}{k}_{0}\allowbreak=\allowbreak 1,$ for $0\leq k\leq n,$
$\left(a;0\right)_{n}\allowbreak=\allowbreak\left\\{\begin{array}[]{ccc}1&if&n=0\\\
1-a&if&n\geq 1\end{array}\right..$
We will need the following sets of orthogonal polynomials
The Rogers–Szegö polynomials that are defined by the equality:
(1.2)
$w_{n}\left(x|q\right)\allowbreak=\allowbreak\sum_{k=0}^{n}\QATOPD[]{n}{k}_{q}x^{k},$
for $n\geq 0$ and $w_{-1}\left(x|q\right)\allowbreak=\allowbreak 0.$ They will
be playing an auxiliary rôle in the sequel.
In particular one shows (see e.g. [8]) that the polynomials defined by:
(1.3) $h_{n}\left(x|q\right)\allowbreak=\allowbreak
e^{in\theta}w_{n}\left(e^{-2i\theta}|q\right)$
where $x\allowbreak=\allowbreak\cos\theta,$ satisfy the following 3-term
recurrence:
(1.4) $h_{n+1}(x|q)=2xh_{n}(x|q)-(1-q^{n})h_{n-1}(x|q),$
with $h_{-1}\left(x|q\right)\allowbreak=\allowbreak 0,$
$h_{0}\left(x|q\right)\allowbreak=\allowbreak 1.$
These polynomials are called continuous $q-$Hermite polynomials. A lot is
known about their properties. For good reference see [8]. In particular we
know that
$\sup_{\left|x\right|\leq 1}\left|h_{n}\left(x|q\right)\right|\leq
w_{n}\left(1|q\right).$
###### Remark 2.
Notice that $h_{n}\left(x|0\right)\allowbreak\ $equals to $n-th$ Chebyshev
polynomial of the second kind. More about these polynomials one can find in
e.g. [10]. To analyze the case $q\allowbreak=\allowbreak 1$ let us consider
rescaled polynomials $h_{n}$ i.e.
$H_{n}\left(x|q\right)\allowbreak=\allowbreak
h_{n}\left(x\sqrt{1-q}/2|q\right)/\left(1-q\right)^{n/2}.$ Then equation (1.4)
takes a form:
$H_{n+1}\left(x|q\right)\allowbreak=\allowbreak
xH_{n}(x|q)-\left[n\right]_{q}H_{n-1}\left(x|q\right),$
which shows that $H_{n}\left(x|q\right)\allowbreak=\allowbreak H_{n}(x),$
where $\left\\{H_{n}\right\\}$ denote the so called ’probabilistic’ Hermite
polynomials i.e. polynomials orthogonal with respect to the measure with
density equal to $\exp\left(-x^{2}/2\right)/\sqrt{2\pi}.$ This observation
suggests that although the case $q\allowbreak=\allowbreak 1$ lies within our
interest it requires special approach. In fact it will be solved completely in
Section 3. For now we will assume that $\left|q\right|<1.$
In the sequel the following identities discovered by Carlitz (see Exercise
12.3(b) and 12.3(c) of [8]), true for $\left|q\right|,\left|t\right|<1$ :
(1.5)
$\sum_{k=0}^{\infty}\frac{w_{k}\left(1|q\right)t^{k}}{\left(q\right)_{k}}\allowbreak=\allowbreak\frac{1}{\left(t\right)_{\infty}^{2}},\sum_{k=0}^{\infty}\frac{w_{k}^{2}\left(1|q\right)t^{k}}{\left(q\right)_{k}}\allowbreak=\allowbreak\frac{\left(t^{2}\right)_{\infty}}{\left(t\right)_{\infty}^{4}},$
will enable to show convergence of many series considered in the sequel.
We have also the following so called ’linearization formula’ ([8], 13.1.25)
which can be dated back in fact to Rogers and Carlitz (see [10], 10.11.10 with
$\beta\allowbreak=\allowbreak 0$ or [16] for Rogers–Szegö polynomials):
(1.6)
$h_{n}\left(x|q\right)h_{m}\left(x|q\right)=\sum_{j=0}^{\min\left(n,m\right)}\QATOPD[]{m}{j}_{q}\QATOPD[]{n}{j}_{q}\left(q\right)_{j}h_{n+m-2k}\left(x|q\right),$
that will be our basic tool.
We will use the following two formulae of Carlitz presented in [6], that
concern properties of Rogers–Szegö polynomials. Let us define two sets of
functions
$\displaystyle\zeta_{n}\left(x|a,q\right)\allowbreak$ $\displaystyle=$
$\displaystyle\allowbreak\sum_{m\geq
0}\frac{a^{m}}{\left(q\right)_{m}}w_{n+m}\left(x|q\right),$
$\displaystyle\lambda_{n,m}\left(x,y|a,q\right)\allowbreak$ $\displaystyle=$
$\displaystyle\allowbreak\sum_{k\geq
0}\frac{a^{k}}{\left(q\right)_{k}}w_{n+k}\left(x|q\right)w_{m+k}\left(y|q\right),$
defined for $\left|x\right|,\left|y\right|\leq 1$,
$\left|a\right|<1\allowbreak$ and $n,m$ being nonnegative integers. Carlitz
proved ([6], (3.2), after correcting an obvious misprint) that
(1.7) $\displaystyle\zeta_{n}\left(x|a,q\right)\allowbreak$ $\displaystyle=$
$\displaystyle\allowbreak\zeta_{0}\left(x|a,q\right)\mu_{n}\left(x|a,q\right),$
(1.8) $\displaystyle\zeta_{0}\left(x|a,q\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\left(a,ax\right)_{\infty}},$
where functions $\mu_{n}$ are polynomials that are defined by:
(1.9)
$\mu_{n}\left(x|a,q\right)\allowbreak=\allowbreak\sum_{j=0}^{n}\QATOPD[]{n}{j}_{q}\left(a\right)_{j}x^{j},$
and that ([6], (1.4), case $m\allowbreak=\allowbreak 0$ also given in [8], Ex
12.3 (d))
(1.10)
$\frac{\lambda_{m,n}\left(x,y|a,q\right)}{\lambda_{0,0}\left(x,y|a,q\right)}=\sum_{j=0}^{m}\sum_{k=0}^{n}\QATOPD[]{n}{k}_{q}\QATOPD[]{m}{j}_{q}\frac{\left(ax\right)_{j}\left(ay\right)_{k}\left(xya\right)_{k+j}}{\left(xya^{2}\right)_{k+j}}x^{m-j}y^{n-k},$
with
(1.11)
$\lambda_{0,0}\left(x,y|a,q\right)=\frac{\left(xya^{2}\right)_{\infty}}{\left(a,ax,ay,axy\right)_{\infty}}.$
It is elementary to prove the following two properties of the polynomials
$\mu_{n},$ hence we present them without the proof.
###### Proposition 1.
(1.12) $\displaystyle x^{n}\mu_{n}\left(x^{-1}|a,q\right)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\QATOPD[]{n}{k}_{q}(-a)^{j}q^{\binom{j}{2}}w_{n-j}\left(x|q\right)$
(1.13) $\displaystyle w_{n}\left(x|q\right)\allowbreak$ $\displaystyle=$
$\displaystyle\allowbreak\sum_{k=0}^{n}\QATOPD[]{n}{k}_{q}a^{k}x^{n-k}\mu_{n-k}\left(x^{-1}|a,q\right).$
To perform our calculations we will need also the following two functions.
The generating function of the $q-$Hermite polynomials that is given by the
formula below (see [7], 3.26.11):
(1.14) $\varphi_{h}\left(x|t,q\right)\overset{df}{=}\sum_{j\geq
0}\frac{t^{j}}{\left(q\right)_{j}}h_{j}\left(x|q\right)\allowbreak=\allowbreak\frac{1}{\prod_{k=0}^{\infty}v\left(x|tq^{k}\right)},$
where $v\left(x|t\right)\allowbreak=\allowbreak 1-2tx+t^{2}.$ Notice that
$v\left(x|t\right)\geq 0$ for $\left|x\right|\leq 1$ and that from (1.5) it
follows that series in (1.3) converges for $\left|t\right|<1.$ Notice also
that from (1.5) it follows that:
(1.15) $\sup_{\left|x\right|\leq
1}\varphi_{h}\left(x|t,q\right)\allowbreak=\allowbreak
1/\left(\left|t\right|\right)_{\infty}^{2}.$
The density of the measure with respect to which polynomials $h_{n}$ are
orthogonal is given in e.g. [7], (3.26.2). Following it we have
$\int_{-1}^{1}h_{n}\left(x|q\right)h_{m}\left(x|q\right)f_{h}\left(x|q\right)dx=\left(q\right)_{n}\delta_{nm},$
where $\delta_{mn}$ denotes Kronecker’s delta, and
(1.16)
$f_{h}\left(x|q\right)\allowbreak=\allowbreak\frac{2\left(q\right)_{\infty}\sqrt{1-x^{2}}}{\pi}\prod_{k=1}^{\infty}l\left(x|q^{i}\right),$
where $l\left(x|a\right)\allowbreak=\allowbreak\left(1+a\right)^{2}-4ax^{2}.$
###### Remark 3.
We have
$f_{h}\left(x|0\right)=2\sqrt{1-x^{2}}/\pi,~{}~{}\varphi_{h}\left(x|a,0\right)=1/\left(1-2ax+a^{2}\right),$
for $\left|x\right|,\left|a\right|<1.$
After proper rescaling and normalization similar to the one performed in
Remark 2, the case $q\allowbreak=\allowbreak 1$ leads to:
$\exp\left(-x^{2}/2\right)/\sqrt{2\pi},~{}~{}\exp\left(ax-a^{2}/2\right),$
for $x,a\in\mathbb{R},$ as respectively the density of orthogonalizing measure
and the characteristic function. For details see [11] or [4].
## 2\. Main results
Since in our approach symmetric polynomials will appear let us introduce the
following set of symmetric polynomials of $k$ variables:
(2.1)
$S_{n}^{\left(k\right)}\left(a_{1},\ldots,a_{k}|q\right)=\sum_{\begin{subarray}{c}j_{1},\ldots,j_{k-1}\geq
0\\\ j_{1}+\ldots+j_{k-1}\leq
n\end{subarray}}\frac{\left(q\right)_{n}}{\prod_{m=0}^{k-1}\left(q\right)_{jm}\left(q\right)_{n-j_{1}-\ldots-
j_{k-1}}}a_{1}^{j_{1}}\ldots a_{k-1}^{j_{k-1}}a_{k}^{n-j_{1}-\ldots-j_{k-1}}.$
###### Remark 4.
Notice that
$S_{n}^{\left(k\right)}\left(a_{1},\ldots,a_{k}|1\right)\allowbreak=\allowbreak\left(\sum_{j=1}^{k}a_{j}\right)^{n}.$
###### Proof.
Obvious since
$\left.\frac{\left(q\right)_{n}}{\prod_{m=0}^{k-1}\left(q\right)_{jm}\left(q\right)_{n-j_{1}-\ldots-
j_{k-1}}}\right|_{q=1}\allowbreak=\allowbreak\frac{n!}{(n-\sum_{m=1}^{k-1}j_{m})!\prod_{m=1}^{k-1}j_{m}!}.$
###### Proposition 2.
Let $q\in\left(-1,1\right)$ then i)
(2.2) $\sum_{n\geq
0}\frac{t^{n}}{\left(q\right)_{n}}S_{n}^{\left(k\right)}\left(a_{1},\ldots,a_{k}|q\right)\allowbreak=\allowbreak\frac{1}{\prod_{j=1}^{k}\left(a_{i}t\right)_{\infty}},$
ii) for $\left|t\right|<1$ and $\forall j=1,\ldots,k$
(2.3)
$S_{n}^{\left(k\right)}\left(a_{1},\ldots,a_{k}|q\right)=\sum_{m=0}^{n}\QATOPD[]{n}{m}_{q}S_{m}^{\left(j\right)}(a_{1},\ldots,a_{j})S_{n-m}^{\left(k-j\right)}\left(a_{j+1},\ldots,a_{k}|q\right),$
If $q=1,$ then
$\sum_{n\geq
0}\frac{t^{n}}{n!}S_{n}^{\left(k\right)}\left(a_{1},\ldots,a_{k}|1\right)\allowbreak=\allowbreak\exp\left(t\sum_{j=0}^{k}a_{j}\right).$
iii)
$\left|S_{n}^{\left(k\right)}\left(a_{1},\ldots,a_{k}|q\right)\right|\leq\left(\max_{1\leq
j\leq
k}\left|a_{j}\right|\right)^{n}S_{n}^{\left(k\right)}\left(1,\ldots,1|q\right).$
###### Proof.
i) Notice that
$\sum_{n\geq
0}\frac{t^{n}}{\left(q\right)_{n}}S_{n}^{\left(k\right)}\left(a_{1},\ldots,a_{k}|q\right)\allowbreak=\allowbreak\sum_{n\geq
0}\sum_{\begin{subarray}{c}j_{1},\ldots,j_{k-1}\geq 0\\\
j_{1}+\ldots+j_{k-1}\leq
n\end{subarray}}\frac{(ta_{1})^{j_{1}}\ldots(ta_{k-1})^{j_{k-1}}(ta_{k})^{n-j_{1}-\ldots-
j_{k-1}}}{\prod_{m=0}^{k-1}\left(q\right)_{jm}\left(q\right)_{n-j_{1}-\ldots-
j_{k}}}.$
Secondly recall that
$\frac{1}{\left(a\right)_{\infty}}\allowbreak=\allowbreak\sum_{j\geq
0}\frac{a^{j}}{\left(q\right)_{j}}.$ Now the assertion is easy. ii) follows
either direct calculation or i) and the properties of characteristic
functions. iii) We use (2.1).
Recall (i.e. [8] or [7]) that there exist sets of orthogonal polynomials
forming a part of the so called ’AW scheme’ that are orthogonal with respect
to measures with densities mentioned below. Although our main interest is in
providing simple proof of the so called AW integral we will list related
densities for better exposition and for indicating the ways of possible
generalization of AW integrals and polynomials.
So let us mention first the so called big $q-$Hermite polynomials
$\left\\{h_{n}\left(x|a,q\right)\right\\}_{n\geq-1}$ whose orthogonalizing
measure has density for $\left|a\right|<1$. This density has a form (see [7]
(3.18.2)) which can be written with the help of functions $f_{h}$ and
$\varphi_{h}.$ Namely:
(2.4) $\displaystyle f_{bh}\left(x|a,q\right)\allowbreak$ $\displaystyle=$
$\displaystyle\allowbreak f_{h}\left(x|q\right)\varphi_{h}\left(x|a,q\right),$
(2.5)
$\displaystyle\int_{-1}^{1}h_{n}\left(x|a,q\right)h_{m}\left(x|a,q\right)f_{bh}\left(x|a,q\right)$
$\displaystyle=$ $\displaystyle\left(q\right)_{n}\delta_{mn}.$
The form of polynomials $h_{n}\left(x|a,q\right)$ and their relation to
$q-$Hermite polynomials is not important for our considerations. It can be
found e.g. in [7], (3.18.4) or in [13] , (2.11, 2.12). So for the sake of
completeness let us remark that from (2.4) it follows immediately that for
$\left|x\right|\leq 1,$ $\left|a\right|<1$
$f_{bh}\left(x|a,q\right)\allowbreak=\allowbreak
f_{h}\left(x|q\right)\sum_{n\geq
0}\frac{a^{n}}{\left(q\right)_{n}}h_{n}\left(x|q\right).$
Here and below, where we will present similar expansions convergence is almost
uniform since all these expansions are in fact the Fourier series and that the
Rademacher–Menshov theorem can be applied following (1.5).
Let us notice immediately that following (2.4) we have:
$\int_{-1}^{1}h_{n}\left(x|q\right)f_{bh}\left(x|a,q\right)dx=a^{n}.$
Secondly let us mention the so called Al-Salam–Chihara polynomials
$\left\\{Q_{n}\left(x|a,b,q\right)\right\\}_{n\geq-1}$ that are orthogonal
with respect to the measure that for $\left|a\right|,\left|b\right|<1$ has the
density of the form (compare [7], (3.8.2))
(2.6)
$f_{Q}\left(x|a,b,q\right)\allowbreak=\allowbreak\left(ab\right)_{\infty}f_{h}\left(x|q\right)\varphi_{h}\left(x|a,q\right)\varphi_{h}\left(x|b,q\right).$
We have the following Lemma that illustrates our method as well as to will
give a very simple proof of well known so called Poisson–Mehler formula as a
corollary.
###### Lemma 1.
For $\left|x\right|\leq 1,$ $\left|a\right|,\left|b\right|<1$ we have
$f_{Q}\left(x|a,b,q\right)\allowbreak=\allowbreak
f_{h}\left(x|q\right)\sum_{j=0}^{\infty}\frac{S_{j}^{(2)}\left(a,b\right)}{\left(q\right)_{j}}h_{j}\left(x|q\right).$
###### Proof.
Following (2.6) and (1.14) we have :
$f_{Q}\left(x|a,b,q\right)\allowbreak=\allowbreak\left(ab\right)_{\infty}f_{h}\left(x|q\right)\sum_{j,k\geq
0}\frac{a^{j}b^{k}}{\left(q\right)_{j}\left(q\right)_{k}}h_{j}\left(x|q\right)h_{k}\left(x|q\right).$
Now we use (1.6) and (1.1) and change the order of summation getting:
$\displaystyle f_{Q}\left(x|a,b,q\right)$ $\displaystyle=$
$\displaystyle\left(ab\right)_{\infty}f_{h}\left(x|q\right)\sum_{m\geq
0}\frac{\left(ab\right)^{m}}{\left(q\right)_{m}}\sum_{j,k\geq
m}\frac{a^{j-m}b^{k-m}}{\left(q\right)_{j-m}\left(q\right)_{k-m}}h_{j-k+m-k}\left(x|q\right)$
$\displaystyle=$
$\displaystyle\left(ab\right)_{\infty}f_{h}\left(x|q\right)\sum_{m\geq
0}\frac{\left(ab\right)^{m}}{\left(q\right)_{m}}\sum_{n,i\geq
0}\frac{a^{n}b^{i}}{\left(q\right)_{i}\left(q\right)_{n}}h_{n+i}\left(x|q\right)$
$\displaystyle=$ $\displaystyle f_{h}\left(x|q\right)\sum_{s\geq
0}\frac{h_{s}\left(x|q\right)}{\left(q\right)_{s}}\sum_{n=0}^{s}\QATOPD[]{s}{j}_{q}a^{n}b^{s-n}.$
As an immediate corollary of our result we have:
(2.7)
$\int_{-1}^{1}h_{n}\left(x|q\right)f_{Q}\left(x|a,b,q\right)dx=S_{n}^{\left(2\right)}\left(a,b|q\right).$
###### Remark 5.
Let $a\allowbreak=\allowbreak\rho e^{i\eta},$ $b\allowbreak=\allowbreak\rho
e^{-i\eta}$ and denote $y\allowbreak=\allowbreak\cos\eta.$ Then
i)
$S_{n}^{\left(2\right)}\left(a,b|q\right)\allowbreak=\allowbreak\rho^{n}h_{n}\left(y|q\right),$
ii)
$v\left(x|a\right)v\left(x|b\right)\allowbreak=\allowbreak\left(1-\rho^{2}\right)^{2}-4xy\rho\left(1+\rho^{2}\right)+4\rho^{2}\left(x^{2}+y^{2}\right)$
###### Proof.
i) is an immediate consequence of (1.3). ii) We have
$v\left(x|a\right)v\left(x|b\right)\allowbreak=\allowbreak(1-2\rho
xe^{i\eta}+\rho^{2}e^{2i\eta})(1-2\rho xe^{-i\eta}+\rho^{2}e^{-2i\eta})$
As a slightly more complicated corollary implied by Lemma 1 we have the
following famous Poisson–Mehler (PM) expansion formula:
###### Corollary 1.
For $\left|x\right|,\left|y\right|<1,$ $\left|\rho\right|<1$ we have
$\displaystyle\frac{\left(\rho^{2}\right)_{\infty}}{\prod_{k=0}^{\infty}\left(1-\rho^{2}q^{2k}\right)^{2}-4xy\rho
q^{k}\left(1+\rho^{2}q^{2k}\right)+4\rho^{2}q^{2k}\left(x^{2}+y^{2}\right)}$
$\displaystyle=$ $\displaystyle\sum_{j\geq
0}\frac{\rho^{j}}{\left(q\right)_{j}}h_{j}\left(x|q\right)h_{j}\left(y|q\right).$
###### Proof.
We take $a\allowbreak=\allowbreak\rho e^{i\eta},$
$b\allowbreak=\allowbreak\rho e^{-i\eta}$ and denote
$y\allowbreak=\allowbreak\cos\eta.$ Now we use (2.6) and Remark 5, ii) to get
left hand side multiplied by $f_{h}.$ Then we apply Lemma 1, and Remark 5, i)
to get right hand side of our PM formula also multiplied by $f_{h}$. Finally
we cancel out $f_{h}$ which is positive on $(-1,1).$
###### Remark 6.
The calculations we have performed while proving Lemma 1 are very much like
those performed in [8] while proving of Theorem 13.1.6 concerning Poisson
kernel (or Poisson–Mehler) formula. There exist may proofs of PM formula, see
e.g. [1] or recently obtained very short in [3]. In fact the formula (1) can
be dated back to Carlitz who in [17] formulated it for Rogers–Szegö
polynomials. The one presented above seems to be one of the shortest, was
obtained as a by-product and as already mentioned is almost the same as the
one presented in [8].
Notice that considering (2.7) with $a\allowbreak=\allowbreak\rho e^{i\eta},$
$b\allowbreak=\allowbreak\rho e^{-i\eta}$ and
$y\allowbreak=\allowbreak\cos\eta$ leads in view of Remark 5, i) to
$\int_{-1}^{1}h_{n}\left(x|q\right)f_{Q}\left(x|a,b,q\right)dx=\rho^{n}h_{n}\left(y|q\right),$
a nice symmetric formula that appeared in [2] in probabilistic context. Its
probabilistic interpretation was exploited further in [11].
Third in our sequence of families of polynomials that constitute AW scheme are
the so called continuous dual Hahn (C2H) polynomials. Again their relationship
to other sets of polynomials is not important. From [7], (3.3.2) it follows
that the density of measure that makes them orthogonal is given by the
following formula.
$f_{CH}\left(x|a,b,c,q\right)\allowbreak=\allowbreak\left(ab,ac,bc\right)f_{h}\left(x|q\right)\varphi_{h}\left(x|a,q\right)\varphi_{h}\left(x|b,q\right)\varphi_{h}\left(x|c,q\right).$
We have the following lemma.
###### Lemma 2.
$f_{CH}\left(x|a,b,c,q\right)=f_{h}\left(x|q\right)\sum_{n\geq
0}\frac{\sigma_{n}^{\left(3\right)}\left(a,b,c|q\right)}{\left(q\right)_{n}}h_{n}\left(x|q\right),$
where
(2.9)
$\sigma_{n}^{\left(3\right)}\left(a,b,c|q\right)=\sum_{j=0}^{n}\QATOPD[]{n}{j}_{q}q^{\binom{j}{2}}\left(-abc\right)^{j}S_{n-j}^{\left(3\right)}\left(a,b,c|q\right).$
###### Proof.
Is shifted to Section 4.
###### Remark 7.
Notice that for $\left|t\right|<1$
$\sum_{n\geq
0}\frac{t^{n}}{\left(q\right)_{n}}\sigma_{n}^{\left(3\right)}\left(a,c,b|q\right)\allowbreak=\allowbreak\frac{\left(abct\right)_{\infty}}{\left(at,bt,ct\right)_{\infty}}.$
###### Proof.
Using (2.9) we have:
$\displaystyle\sum_{n\geq
0}\frac{t^{n}}{\left(q\right)_{n}}\sigma_{n}^{\left(3\right)}\left(a,c,b|q\right)$
$\displaystyle=$ $\displaystyle\sum_{n\geq
0}\frac{t^{n}}{\left(q\right)_{n}}\sum_{j=0}^{n}\QATOPD[]{n}{j}_{q}q^{\binom{j}{2}}\left(-abc\right)^{j}S_{n-j}^{\left(3\right)}\left(a,b,c|q\right)$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}\frac{(-abct)^{j}}{\left(q\right)_{j}}q^{\binom{j}{2}}\sum_{n\geq
j}\frac{t^{n-j}}{\left(q\right)_{n-j}}S_{n-j}^{\left(3\right)}\left(a,b,c|q\right).$
Now it remains to change the index of summation in the second sum, use (2.2)
and use the fact that
$\sum_{j=0}^{\infty}\frac{(-abct)^{j}}{\left(q\right)_{j}}q^{\binom{j}{2}}\allowbreak=\allowbreak\left(abct\right)_{\infty}.$
###### Corollary 2.
For $\left|a|,|b\right|,\left|c\right|<1:$
$\int_{-1}^{1}h_{n}\left(x|q\right)f_{CH}\left(x|a,b,c,q\right)dx=\sigma_{n}^{\left(3\right)}\left(a,b,c|q\right).$
###### Proof.
Elementary.
Fourth family of polynomials that constitute AW scheme are the celebrated
Askey–Wilson polynomials. Again their form and relationship to other families
of polynomials of AW scheme is not important for our considerations. Recently
a relatively rich study of these relationships was done in [13] hence it may
serve as the reference. We need only the form of AW density. It is given e.g.
in [7], (3.1.2) and after necessary adaptation to our notation is presented
below:
$f_{AW}\left(x|a,b,c,d,q\right)=\frac{\left(ab,ac,ad,bc,bd,cd\right)_{\infty}}{\left(abcd\right)_{\infty}}f_{h}\left(x|a\right)\varphi_{h}\left(x|a,q\right)\varphi_{h}\left(x|b,q\right)\varphi_{h}\left(x|c,q\right)\varphi_{h}\left(x|d,q\right),$
for $\left|x\right|\leq 1,$
$\left|a\right|,\left|b\right|,\left|c\right|,\left|d\right|<1.$ Our main
result concerns this density and is the following:
###### Theorem 1.
For $\left|x\right|\leq 1,$
$\left|a\right|,\left|b\right|,\left|c\right|,\left|d\right|<1$
(2.10) $f_{AW}\left(x|a,b,c,d,q\right)=f_{h}\left(x|q\right)\sum_{n\geq
0}\frac{\sigma_{n}^{\left(4\right)}\left(a,b,c,d|q\right)}{\left(q\right)_{n}}h_{n}\left(x|q\right),$
where
(2.11)
$\sigma_{n}^{\left(4\right)}\left(a,b,c,d|q\right)=\sum_{j=0}^{n}\QATOPD[]{n}{j}_{q}\frac{\left(bd\right)_{j}}{\left(abcd\right)_{j}}S_{n-j}^{\left(2\right)}\left(b,d|q\right)\sum_{k=0}^{j}\QATOPD[]{j}{k}_{q}\left(cb\right)_{k}a^{k}\left(ad\right)_{j-k}c^{j-k},$
are symmetric functions of $a,$ $b,$ $c,$ $d.$
###### Proof.
is shifted to section 4.
As immediate corollaries we have the following fact.
###### Corollary 3.
For $\max(\left|a\right|,\left|b\right|,\left|c\right|,\left|d\right|)<1:$
(2.12)
$\int_{-1}^{1}h_{n}\left(x\right)f_{AW}\left(x|a,b,c,d,q\right)dx=\sigma_{n}^{\left(4\right)}\left(a,b,c,d|q\right).$
###### Proof.
Follows directly from (2.10).
###### Remark 8.
Notice that from (2.10) follows in fact the value of AW integral, since we see
that $\int_{-1}^{1}f_{AW}\left(x|a,b,c,d|q\right)\allowbreak=\allowbreak 1$
which means that the integral
(2.13)
$\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{1}{\sqrt{1-x^{2}}}\prod_{n\geq
0}\frac{l\left(x|q^{n}\right)}{v\left(x|aq^{n}\right)v\left(x|bq^{n}\right)v\left(x|cq^{n}\right)v\left(x|dq^{n}\right)}dx$
$\displaystyle=\frac{\left(abcd\right)_{\infty}}{\left(q,ab,ac,ad,bc,bd,cd\right)_{\infty}}.$
(2.13) is nothing else but the celebrated AW integral. Notice also that
recently there appeared at least two papers [15], [14] where (2.13) was
derived from much more advanced theorems.
###### Remark 9.
Notice also that (2.12) allows calculation of all moments of AW density. This
is so since one knows the form of polynomials $h_{n}.$ Moments of AW density
were calculated by Corteel et. al. in 2010 in [5] using combinatorial means.
For complex $a,$ $b,$ $c,$ $d$ but forming conjugate pairs this formula was
also obtained independently about the same time. Namely it was done in [4]
where also an elegant expansion of
$\sigma_{n}^{\left(4\right)}\left(\rho_{1}e^{i\eta},\rho_{1}e^{-i\eta},\rho_{2}e^{i\theta},\rho_{2}e^{-i\theta}|q\right)$
in terms of $h_{n}\left(y|q\right)$ and $h_{n}\left(z|q\right),$ where
$\cos\eta\allowbreak=\allowbreak y$ and $\cos\theta\allowbreak=\allowbreak z\
$was presented.
## 3\. Generalization and open questions
### 3.1. Generalization
The presented above results allow the following generalization. The cases
$\left|q\right|<1$ and $q\allowbreak=\allowbreak 1$ will be treated
separately. First let us consider $\left|q\right|<1$.
Let us denote
$\mathbf{a}^{\left(k\right)}\allowbreak=\allowbreak\left(a_{1},\ldots,a_{k}\right),$
$k\allowbreak=\allowbreak 0,1,\ldots$ . We will assume that
$\left|x\right|\leq 1$ and that all parameters $a_{i}$ have absolute values
less that $1.$ Let us denote
$g_{n}\left(x|\mathbf{a}^{\left(n\right)},q\right)\allowbreak=\allowbreak
f_{h}\left(x|q\right)\prod_{j=1}^{n}\varphi_{h}\left(x|a_{i},q\right),$
where functions $f_{h}$ and $\varphi_{h}$ were defined by (1.16) and (1.14)
respectively.
We have the following general result.
###### Lemma 3.
For every $n\geq 0$ , there exist
$A_{n}\left(\mathbf{a}^{\left(n\right)},q\right)$ a symmetric function of
$\mathbf{a}^{\left(n\right)}$ and a sequence of symmetric in
$\mathbf{a}^{\left(n\right)}$ functions
$\left\\{T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)\right\\}_{j\geq
0}$ such that for $\left|a_{k}\right|<1,$ $k\allowbreak=\allowbreak
1,\ldots,n:$
(3.1)
$g_{n}\left(x|\mathbf{a}^{\left(n\right)},q\right)=A_{n}\left(\mathbf{a}^{\left(n\right)},q\right)f_{h}\left(x|q\right)\sum_{j\geq
0}\frac{T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)}{\left(q\right)_{j}}h_{j}\left(x|q\right).$
Moreover
(3.2) $\sum_{j\geq
0}\left(T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)\right)^{2}<\infty.$
###### Proof.
Let
$\mathcal{G\allowbreak=\allowbreak}L_{2}\left(<-1,1>,\mathcal{F},f_{h}\right)$
be the space of functions $h:<-1,1>\allowbreak\longmapsto\mathbb{R}$ such that
$\int_{-1}^{1}h^{2}\left(x\right)f_{h}\left(x|q\right)dx.$ Notice that this
space is spanned by the polynomials
$\left\\{h_{j}\left(x|q\right)\right\\}_{j\geq 0}.$ Visibly, under our
assumptions and by (1.15),
$\prod_{j=1}^{n}\varphi_{h}\left(x|a_{i},q\right)\allowbreak\in\allowbreak\mathcal{G}.$
Now notice that
$\left\\{T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)\right\\}_{j\geq
0}$ are coefficients of the Fourier expansion of the function
$\prod_{j=1}^{n}\varphi_{h}\left(x|a_{i},q\right)$ in $\mathcal{G}$ with
respect to $\left\\{h_{j}\left(x|q\right)\right\\}_{j\geq 0}.$ Since
$\int_{-1}^{1}f_{h}\left(x|q\right)\sum_{j\geq
0}\frac{T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)}{\left(q\right)_{j}}h_{j}\left(x|q\right)dx\allowbreak=\allowbreak
1,$
$A_{n}\left(\mathbf{a}^{\left(n\right)},q\right)$ is the value of
$\int_{-1}^{1}g_{n}\left(x|\mathbf{a}^{\left(n\right)},q\right)dx.$ (3.2)
follow properties of the Fourier expansion more precise the Perseval’s
identity. The fact that $A_{n}$ and
$\left\\{T_{j}^{\left(n\right)}\right\\}_{j\geq 0}$ are symmetric follows the
observations that $\prod_{j=1}^{n}\varphi_{h}\left(x|a_{i},q\right)$ is
symmetric.
Using formula (1.9) we can write $g_{n}$ in the following way where $h_{j}$
are $q-$Hermite polynomials defined by (1.4). Functions
$A_{n}\left(\mathbf{a}^{\left(n\right)},q\right)$ and
$\left\\{T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)\right\\}_{j\geq
0}$ have the following interpretation:
$\int_{[-1,1]}h_{j}\left(x|q\right)g_{n}\left(x|\mathbf{a}^{\left(n\right)},q\right)dx=A_{n}\left(\mathbf{a}^{\left(n\right)},q\right)T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right),$
for $n,j\geq 0.$
We have the following easy Proposition giving recursions that are satisfied by
functions $A_{n}$ and $T_{j}^{\left(n\right)}.$
###### Proposition 3.
Let us define new sequence of functions
$\left\\{H_{s}\left(\mathbf{a}^{\left(n\right)},q\right)\right\\}_{n,s\geq 0}$
of $n$ variables:
$\sum_{m\geq
0}\frac{a_{n}^{m}}{\left(q\right)_{m}}T_{s+m}^{\left(n-1\right)}\left(\mathbf{a}^{\left(n-1\right)},q\right)=H_{s}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)\sum_{m\geq
0}\frac{a_{n}^{m}}{\left(q\right)_{m}}T_{m}^{\left(n-1\right)}\left(\mathbf{a}^{\left(n-1\right)},q\right).$
Then i)
$A_{n}\left(\mathbf{a}^{\left(n\right)},q\right)=A_{n-1}\left(\mathbf{a}^{\left(n-1\right)},q\right)\sum_{m\geq
0}\frac{a_{n}^{m}}{\left(q\right)_{m}}T_{m}^{\left(n-1\right)}\left(\mathbf{a}^{\left(n-1\right)},q\right),$
ii)
$T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)=\sum_{s=0}^{j}\QATOPD[]{j}{s}_{q}H_{s}^{\left(n-1\right)}\left(\mathbf{a}^{\left(n-1\right)},q\right)\left(a_{n}\right)^{j-s}.$
###### Proof.
Proof is shifted to section 4.
###### Remark 10.
The integral
$\int_{-1}^{1}g_{n}\left(x|\mathbf{a}^{\left(n\right)},q\right)dx$ has been
calculated in [9] (see also theorem 15.3.1 in [8]) by combinatorial methods.
Obtained formula is however very complicated. Besides above mentioned Theorem
15.3.1 of [8] does not provide expansion (3.1) which is automatically obtained
in our approach.
###### Remark 11.
Notice also that following Proposition 3, i) we get for
$\left|a_{j}\right|<1,$ $j\allowbreak=\allowbreak 1,\ldots,5:$
(3.3)
$\int_{-1}^{1}g_{5}\left(x|\mathbf{a}^{\left(5\right)},q\right)\allowbreak=\allowbreak\frac{\left(\prod_{j}^{4}a_{j}\right)_{\infty}}{(q)_{\infty}\prod_{1\leq
k<m\leq 4}\left(a_{k}a_{m}\right)_{\infty}}\sum_{j\geq
0}\frac{a_{5}^{j}}{\left(q\right)_{j}}\sigma_{j}^{\left(4\right)}\left(a_{1},a_{2},a_{3},a_{4}|q\right)$
For $q\allowbreak=\allowbreak 0$ the calculations presented in (3.3) can be
carried out completely and the concise form can be obtained. This is possible
due to the following simplified form of (2.11).
###### Theorem 2.
Let $\mathbf{a}^{\left(5\right)}\allowbreak.$ Under $\left|a_{j}\right|<1,$
$j\allowbreak=\allowbreak 1,\ldots,5$ we have:
i)
$\displaystyle\sigma_{n}^{\left(4\right)}\allowbreak(a_{1},a_{2},a_{3},d|0)=\allowbreak
S_{n}^{\left(2\right)}(a_{2},a_{4}|0)\allowbreak+\allowbreak\frac{(1-a_{2}d)(1-a_{1}a_{4})}{(1-a_{1}a_{2}a_{3}a_{4})}a_{3}S_{n-1}^{(3)}(a_{2},a_{3},a_{4}|0)+$
$\displaystyle\frac{(1-a_{2}a_{4})(1-a_{3}a_{2})}{(1-a_{1}a_{2}a_{3}a_{4})}a_{1}S_{n-1}^{(3)}(a_{1},a_{2},a_{4}|0)+$
$\displaystyle\allowbreak\frac{(1-a_{2}a_{4})(1-a_{2}a_{3})(1-a_{1}a_{4})a_{1}a_{3}}{(1-a_{1}a_{2}a_{3}a_{4})}S_{n-2}^{(4)}(a_{1},a_{2},a_{3},a_{4}|0)\allowbreak,$
ii)
$\int_{-1}^{1}g_{5}\left(x|\mathbf{a}^{\left(5\right)},0\right)\allowbreak
dx=\allowbreak\frac{1-\chi_{4}\left(\mathbf{a}^{\left(5\right)}\right)+\chi_{5}\left(\mathbf{a}^{\left(5\right)}\right)\chi_{1}\left(\mathbf{a}^{\left(5\right)}\right)-\chi_{5}^{2}\left(\mathbf{a}^{\left(5\right)}\right)}{\prod_{1\leq
j<k\leq 5}(1-a_{j}a_{k})},$
where $\chi_{1},\ldots,\chi_{5}$ denote respectively first five elementary
symmetric functions of vector $\mathbf{a}^{\left(5\right)}.$ That is
$\chi_{j}\left(\mathbf{a}^{\left(k\right)}\right)\allowbreak=\allowbreak\sum_{1\leq
n_{1}<n_{2}\ldots<n_{j}\leq k}\prod_{m=1}^{j}a_{n_{m}}.$
###### Proof.
Is shifted to Section 4.
For $q\allowbreak=\allowbreak 1$ the problem of finding sequences
$A_{n}\left(\mathbf{a}^{\left(n\right)}|1\right)$ and
$\left\\{T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},1\right)\right\\}_{j\geq
0}$ can be solved completely and trivially. Namely we have:
###### Proposition 4.
$\displaystyle A_{n}\left(\mathbf{a}^{\left(n\right)}|1\right)$
$\displaystyle=$ $\displaystyle\exp\left(\sum_{1\leq j<k\leq
n}a_{j}a_{k}\right),$ $\displaystyle
T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},1\right)$
$\displaystyle=$ $\displaystyle\left(\sum_{k=1}^{n}a_{k}\right)^{j}.$
###### Proof.
Using Remark 3 we get:
$\displaystyle
g_{n}\left(x|\mathbf{a}^{\left(n\right)},1\right)=\exp\left(-x^{2}/2+x\sum_{j=1}^{n}a_{j}-\frac{1}{2}\sum_{j=1}^{n}a_{j}^{2}\right)/\sqrt{2\pi}\allowbreak$
$\displaystyle\allowbreak=\frac{1}{\sqrt{2\pi}}\exp\left(\frac{1}{2}\left(\left(\sum_{j=1}^{n}a_{j}\right)^{2}-\sum_{j=1}^{n}a_{j}^{2}\right)\right)\exp\left(-x^{2}/2+x\sum_{j=1}^{n}a_{j}-\frac{1}{2}\left(\sum_{j=1}^{n}a_{j}\right)^{2}\right)$
$\displaystyle=\exp\left(\sum_{1\leq j<k\leq
n}a_{j}a_{k}\right)\frac{\exp\left(-x^{2}/2\right)}{\sqrt{2\pi}}\sum_{j\geq
0}\frac{\left(\sum_{k=1}^{n}a_{k}\right)^{j}}{j!}H_{j}\left(x\right).$
### 3.2. Unsolved Problems & Open Questions
#### 3.2.1. Questions
* •
What are the compact forms of functions
$\left\\{T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)\right\\}_{j\geq
0,n\geq 5}$ and
$\left\\{A_{n}\left(\mathbf{a}^{\left(n\right)},q\right)\right\\}_{n\geq 5}$ ?
* •
What is the compact form of these functions for $q\allowbreak=\allowbreak 0$
(free probability case) ?
* •
Following formula for
$\int_{-1}^{1}g_{5}\left(x|\mathbf{a}^{\left(5\right)},0\right)\allowbreak dx$
given in assertion ii) of Theorem 2 is it true that:
$\int_{-1}^{1}g_{5}\left(x|\mathbf{a}^{\left(5\right)},q\right)\allowbreak
dx\allowbreak=\allowbreak\frac{\left(\chi_{4}\left(\mathbf{a}^{\left(5\right)}\right)-\chi_{5}\left(\mathbf{a}^{\left(5\right)}\right)\chi_{1}\left(\mathbf{a}^{\left(5\right)}\right)+\chi_{5}^{2}\left(\mathbf{a}^{\left(5\right)}\right)\right)_{\infty}}{\prod_{1\leq
j<k\leq 5}(a_{j}a_{k})_{\infty}}?$
Notice that for $a_{5}\allowbreak=\allowbreak 0$ it would reduce to AW
integral.
* •
It would be valuable to get values
$\left\\{A_{n}\left(\mathbf{a}^{\left(n\right)},q\right)\right\\}$ for
$n\allowbreak=\allowbreak 8,12$ and so on for complex values of parameters
$\mathbf{a}^{\left(n\right)}$ but forming conjugate pairs. It would be also
fascinating to find polynomials that would be orthogonalized by so obtained
densities.
This problem follows the probabilistic interpretation of Askey–Wilson density
rescaled, with complex parameters. Such interpretation for finite Markov
chains of length at least $3$ was presented in [4], [13]. Let
$\left\\{X_{1},X_{2},X_{3}\right\\}$ denote this finite Markov chain. Then
recall that then AW density can be interpreted as the conditional density of
$X_{2}|X_{1},X_{3}$.
It would be exciting to find out if for say $n\allowbreak=\allowbreak 8$
similar probabilistic interpretation could be established. That is if we could
have defined $5-$dimensional random vector $(X_{1},\ldots,X_{5})$ with
normalized function $g_{8}\left(x|\mathbf{a}^{\left(8\right)},q\right)$ as the
conditional density $X_{3}|X_{1},X_{2},X_{4},X_{5}.$ Note that then the chain
$(X_{1},\ldots,X_{5})$ could not be Markov.
Similar questions apply to the case $n\allowbreak=\allowbreak 12,16,...$ .
#### 3.2.2. Unsolved related problems and direction of further research.
In [10] we find Theorem 10.8.2 which is due Gasper and Rahman (1990) and which
can be stated in our notation. For $\max_{1\,\leq j\leq
5}\left|a_{j}\right|<1,$ $|q|<1$ we have:
$\int_{-1}^{1}\frac{g_{5}\left(x|\mathbf{a}^{\left(5\right)},q\right)}{\varphi_{h}\left(x|\prod_{j=1}^{5}a_{j},q\right)}dx\allowbreak=\allowbreak\frac{\prod_{j=1}^{5}\left(\prod_{k=1,k\neq
j}^{5}a_{k}\right)_{\infty}}{\prod_{1\leq j<k\leq
5}\left(a_{j}a_{k}\right)_{\infty}}.$
This result suggests considering the following functions
$G_{n,m}\left(x|\mathbf{a}^{\left(n\right)},\mathbf{b}^{\left(m\right)},q\right)\allowbreak=\allowbreak
f_{h}\left(x|q\right)\frac{\prod_{j=1}^{n}\varphi_{h}\left(x|a_{i},q\right)}{\prod_{k=1}^{m}\varphi_{h}\left(x|b_{k},q\right)},$
where $\mathbf{a}^{\left(n\right)}$ and $\mathbf{b}^{\left(m\right)}$ are
certain vectors of dimensions respectively $n$ and $m,$ find its integrals
over $[-1,1]$ and expansions similar to (3.1).
## 4\. Proofs
###### Proof of Lemma 2.
We have
$\displaystyle\sum_{k,n,m\geq
0}\frac{a^{n}}{\left(q\right)_{n}}\frac{b^{m}}{\left(q\right)_{m}}\frac{c^{k}}{\left(q\right)_{k}}h_{n}\left(x|q\right)h_{m}\left(x|q\right)h_{k}\left(x|q\right)\allowbreak$
$\displaystyle=\frac{1}{\left(ab\right)_{\infty}}\sum_{j\geq
0}\frac{h_{j}\left(x|q\right)}{\left(q\right)_{j}}\sum_{m=0}^{\infty}\frac{c^{m}}{\left(q\right)_{m}}\sum_{k=0}^{j}\QATOPD[]{j}{k}_{q}c^{k}S_{m+j-k}^{\left(2\right)}\left(a,b|q\right).$
Since obviously
$S_{n}^{\left(2\right)}\left(a,b|q\right)\allowbreak=\allowbreak
a^{n}w_{n}\left(b/a|q\right)\allowbreak$ we get:
$\displaystyle\sum_{k,n,m\geq
0}\frac{a^{n}}{\left(q\right)_{n}}\frac{b^{m}}{\left(q\right)_{m}}\frac{c^{k}}{\left(q\right)_{k}}h_{n}\left(x|q\right)h_{m}\left(x|q\right)h_{k}\left(x|q\right)$
$\displaystyle=$ $\displaystyle\frac{1}{\left(ab\right)_{\infty}}\sum_{j\geq
0}\frac{h_{j}\left(x|q\right)}{\left(q\right)_{j}}\sum_{m=0}^{\infty}\frac{c^{m}}{\left(q\right)_{m}}\sum_{k=0}^{j}\QATOPD[]{j}{k}_{q}c^{k}a^{m+j-k}w_{m+j-k}\left(b/a|q\right)$
$\displaystyle\frac{1}{\left(ab\right)_{\infty}}\sum_{j\geq
0}\frac{h_{j}\left(x|q\right)}{\left(q\right)_{j}}\sum_{k=0}^{j}\QATOPD[]{j}{k}_{q}c^{k}a^{j-k}\sum_{m=0}^{\infty}\frac{(ac)^{m}}{\left(q\right)_{m}}w_{m+j-k}\left(b/a|q\right).$
Now we apply formula (1.7) and get:
$\displaystyle\sum_{k,n,m\geq
0}\frac{a^{n}}{\left(q\right)_{n}}\frac{b^{m}}{\left(q\right)_{m}}\frac{c^{k}}{\left(q\right)_{k}}h_{n}\left(x|q\right)h_{m}\left(x|q\right)h_{k}\left(x|q\right)$
$\displaystyle=$
$\displaystyle\allowbreak\frac{1}{\left(ab,bc,ac\right)_{\infty}}\sum_{j\geq
0}\frac{h_{j}\left(x|q\right)}{\left(q\right)_{j}}\sum_{k=0}^{j}\QATOPD[]{j}{k}_{q}c^{k}a^{j-k}\mu_{j-k}\left(b/a|ac,q\right)\frac{1}{\left(bc\right)_{\infty}\left(bc\right)_{\infty}}$
$\displaystyle=$
$\displaystyle\frac{1}{\left(ab,bc,ac\right)_{\infty}}\sum_{j\geq
0}\frac{h_{j}\left(x|q\right)}{\left(q\right)_{j}}\sum_{l=0}^{j}\QATOPD[]{n}{l}_{q}c^{n-l}a^{l}\left(\frac{b}{a}\right)^{j}\left(\frac{a}{b}\right)^{j}\mu_{j}\left(\left(\frac{a}{b}\right)^{-1}|ac,q\right).$
Now we use (1.12)and Proposition 2, ii) and get:
$\displaystyle\sum_{k,n,m\geq
0}\frac{a^{n}}{\left(q\right)_{n}}\frac{b^{m}}{\left(q\right)_{m}}\frac{c^{k}}{\left(q\right)_{k}}h_{n}\left(x|q\right)h_{m}\left(x|q\right)h_{k}\left(x|q\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\left(ab,bc,ac\right)_{\infty}}\sum_{j\geq
0}\frac{h_{j}\left(x|q\right)}{\left(q\right)_{j}}\sum_{l=0}^{j}\QATOPD[]{j}{l}_{q}c^{j-l}b^{l}\sum_{k=0}^{l}\QATOPD[]{l}{k}_{q}\left(-ac\right)^{k}q^{\binom{k}{2}}w_{l-k}\left(\frac{a}{b}|q\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\left(ab,bc,ac\right)_{\infty}}\sum_{j\geq
0}\frac{h_{j}\left(x|q\right)}{\left(q\right)_{j}}\sum_{k=0}^{j}\QATOPD[]{j}{k}_{q}\left(-ac\right)^{k}q^{\binom{k}{2}}\sum_{l=k}^{j}\QATOPD[]{j-k}{l-k}_{q}c^{j-l}b^{l}w_{l-k}\left(\frac{a}{b}|q\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\left(ab,bc,ac\right)_{\infty}}\sum_{j\geq
0}\frac{h_{j}\left(x|q\right)}{\left(q\right)_{j}}\sum_{k=0}^{j}\QATOPD[]{j}{k}_{q}\left(-ac\right)^{k}q^{\binom{k}{2}}\sum_{m=0}^{j-k}\QATOPD[]{j-k}{m}_{q}c^{j-k-m}b^{k+m}w_{m}\left(a/b|q\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\left(ab,bc,ac\right)_{\infty}}\sum_{j\geq
0}\frac{h_{j}\left(x|q\right)}{\left(q\right)_{j}}\sum_{k=0}^{j}\QATOPD[]{j}{k}_{q}\left(-abc\right)^{k}q^{\binom{k}{2}}\sum_{m=0}^{j-k}\QATOPD[]{j-k}{m}_{q}c^{j-k-m}S_{m}^{\left(2\right)}\left(a,b|q\right).$
###### Proof of Theorem 1.
We have
$\displaystyle\sum_{k,n,m,j\geq
0}\frac{a^{n}}{\left(q\right)_{n}}\frac{b^{m}}{\left(q\right)_{m}}\frac{c^{k}}{\left(q\right)_{k}}\frac{d^{j}}{\left(q\right)_{j}}h_{n}\left(x|q\right)h_{m}\left(x|q\right)h_{k}\left(x|q\right)h_{j}\left(x|q\right)\allowbreak$
$\displaystyle=$
$\displaystyle\frac{1}{\left(ab,cd\right)_{\infty}}\sum_{m,k\geq
0}\frac{s_{m}\left(a,b|q\right)s_{k}\left(c,d|q\right)}{\left(q\right)_{m}\left(q\right)_{k}}h_{m}\left(x|q\right)h_{k}\left(x|q\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\left(ab,cd\right)_{\infty}}\sum_{m,k\geq
0}\frac{s_{m}\left(a,b|q\right)s_{k}\left(c,d|q\right)}{\left(q\right)_{m}\left(q\right)_{k}}\sum_{j=0}^{\min(m,k)}\QATOPD[]{m}{j}_{q}\QATOPD[]{k}{j}_{q}\left(q\right)_{j}h_{m+k-2j}\left(x|q\right)\allowbreak$
$\displaystyle=$
$\displaystyle\allowbreak\frac{1}{\left(ab,cd\right)_{\infty}}\sum_{j\geq
0}\frac{(ac)^{j}}{\left(q\right)_{j}}\sum_{m,k\geq
j}\frac{a^{m-j}c^{k-j}w_{m}\left(b/a|q\right)w_{k}\left(d/c|q\right)}{\left(q\right)_{m-j}\left(q\right)_{k-j}}h_{m-j+k-j}\left(x|q\right)$
and further
$\displaystyle\sum_{k,n,m,j\geq
0}\frac{a^{n}}{\left(q\right)_{n}}\frac{b^{m}}{\left(q\right)_{m}}\frac{c^{k}}{\left(q\right)_{k}}\frac{d^{j}}{\left(q\right)_{j}}h_{n}\left(x|q\right)h_{m}\left(x|q\right)h_{k}\left(x|q\right)h_{j}\left(x|q\right)$
$\displaystyle=$
$\displaystyle\allowbreak\frac{1}{\left(ab,cd\right)_{\infty}}\sum_{j\geq
0}\frac{(ac)^{j}}{\left(q\right)_{j}}\sum_{s,t\geq
0}\frac{a^{s}c^{t}w_{s+j}\left(b/a|q\right)w_{t+j}\left(d/c|q\right)}{\left(q\right)_{s}\left(q\right)_{t}}h_{s+t}\left(x|q\right)\allowbreak$
$\displaystyle=$
$\displaystyle\allowbreak\frac{1}{\left(ab,cd\right)_{\infty}}\sum_{j\geq
0}\frac{(ac)^{j}}{\left(q\right)_{j}}\sum_{n\geq
0}\frac{h_{n}\left(x|q\right)}{\left(q\right)_{n}}\sum_{k=0}^{n}\QATOPD[]{n}{k}_{q}a^{k}c^{n-k}w_{k+j}\left(b/a|q\right)w_{j+n-k}\left(d/c|q\right)$
$\displaystyle=$
$\displaystyle\allowbreak\frac{1}{\left(ab,cd\right)_{\infty}}\sum_{n\geq
0}\frac{h_{n}\left(x|q\right)}{\left(q\right)_{n}}\sum_{k=0}^{n}\QATOPD[]{n}{k}_{q}a^{k}c^{n-k}\sum_{j\geq
0}\frac{(ac)^{j}}{\left(q\right)_{j}}w_{k+j}\left(b/a|q\right)w_{j+n-k}\left(d/c|q\right)$
Now we apply Carlitz formulae (1.10) and (1.11) getting:
$\displaystyle\sum_{k,n,m,j\geq
0}\frac{a^{n}}{\left(q\right)_{n}}\frac{b^{m}}{\left(q\right)_{m}}\frac{c^{k}}{\left(q\right)_{k}}\frac{d^{j}}{\left(q\right)_{j}}h_{n}\left(x|q\right)h_{m}\left(x|q\right)h_{k}\left(x|q\right)h_{j}\left(x|q\right)$
$\displaystyle=$
$\displaystyle\frac{\left(abcd\right)_{\infty}}{\left(ab,cd,ac,bc,ad,bd\right)}\sum_{n\geq
0}\frac{h_{n}\left(x|q\right)}{\left(q\right)_{n}}\sum_{k=0}^{n}\QATOPD[]{n}{k}_{q}a^{k}c^{n-k}\times$
$\displaystyle\sum_{s=0}^{k}\sum_{t=0}^{n-k}\QATOPD[]{k}{s}_{q}\QATOPD[]{n-k}{t}_{q}\frac{\left(cb\right)_{s}\left(ad\right)_{t}\left(bd\right)_{s+t}}{\left(abcd\right)_{s+t}}\left(\frac{b}{a}\right)^{k-s}\left(\frac{d}{c}\right)^{n-k-t}$
$\displaystyle=$
$\displaystyle\frac{\left(abcd\right)_{\infty}}{\left(ab,cd,ac,bc,ad,bd\right)}\sum_{n\geq
0}\frac{h_{n}\left(x|q\right)}{\left(q\right)_{n}}\sum_{k=0}^{n}\QATOPD[]{n}{k}_{q}\times$
$\displaystyle\sum_{s=0}^{k}\sum_{t=0}^{n-k}\QATOPD[]{k}{s}_{q}\QATOPD[]{n-k}{t}_{q}\frac{\left(cb\right)_{s}\left(ad\right)_{t}\left(bd\right)_{s+t}}{\left(abcd\right)_{s+t}}a^{s}b^{k-s}c^{t}d^{n-k-t}.$
Thus it remains to show that for every $n\geq 0.$
$\displaystyle\sum_{k=0}^{n}\QATOPD[]{n}{k}_{q}\sum_{s=0}^{k}\sum_{t=0}^{n-k}\QATOPD[]{k}{s}_{q}\QATOPD[]{n-k}{t}_{q}\frac{\left(cb\right)_{s}\left(ad\right)_{t}\left(bd\right)_{s+t}}{\left(abcd\right)_{s+t}}a^{s}b^{k-s}c^{t}d^{n-k-t}$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{n}\QATOPD[]{n}{j}_{q}\frac{\left(bd\right)_{j}}{\left(abcd\right)_{j}}S_{n-j}^{\left(2\right)}\left(b,d|q\right)\sum_{k=0}^{j}\QATOPD[]{j}{k}_{q}\left(cb\right)_{k}a^{k}\left(ad\right)_{j-k}c^{j-k}.$
This fact follows the following calculations:
$\displaystyle\sum_{k=0}^{n}\QATOPD[]{n}{k}_{q}\sum_{s=0}^{k}\sum_{t=0}^{n-k}\QATOPD[]{k}{s}_{q}\QATOPD[]{n-k}{t}_{q}\frac{\left(cb\right)_{s}\left(ad\right)_{t}\left(bd\right)_{s+t}}{\left(abcd\right)_{s+t}}a^{s}b^{k-s}c^{t}d^{n-k-t}$
$\displaystyle=$ $\displaystyle\sum_{s,t\geq 0,s+t\leq
n}\frac{\left(q\right)_{n}}{\left(q\right)_{s}\left(q\right)_{t}\left(q\right)_{n-s-t}}\frac{\left(cb\right)_{s}\left(ad\right)_{t}\left(bd\right)_{s+t}}{\left(abcd\right)_{s+t}}a^{s}c^{t}\sum_{k=s\vee
n-t}^{n}\QATOPD[]{n-t-s}{k-s}_{q}b^{k-s}d^{n-k-t}\allowbreak$ $\displaystyle=$
$\displaystyle\sum_{s,t\geq 0,s+t\leq
n}\frac{\left(q\right)_{n}}{\left(q\right)_{s}\left(q\right)_{t}\left(q\right)_{n-s-t}}\frac{\left(cb\right)_{s}\left(ad\right)_{t}\left(bd\right)_{s+t}}{\left(abcd\right)_{s+t}}a^{s}c^{t}\sum_{m=0\vee
n-t-s}^{n-s}\QATOPD[]{n-t-s}{m}_{q}b^{m}d^{n-s-m-t}$ $\displaystyle=$
$\displaystyle\sum_{s,t\geq 0,s+t\leq
n}\frac{\left(q\right)_{n}}{\left(q\right)_{s}\left(q\right)_{t}\left(q\right)_{n-s-t}}\frac{\left(cb\right)_{s}\left(ad\right)_{t}\left(bd\right)_{s+t}}{\left(abcd\right)_{s+t}}a^{s}c^{t}S_{n-t-s}^{\left(2\right)}\left(b,d|q\right).$
Now we introduce new indices of summation: $j=t+s,\allowbreak k=s.$ We have
then
$\displaystyle\sum_{s,t\geq 0,s+t\leq
n}\frac{\left(q\right)_{n}}{\left(q\right)_{s}\left(q\right)_{t}\left(q\right)_{n-s-t}}\frac{\left(cb\right)_{s}\left(ad\right)_{t}\left(bd\right)_{s+t}}{\left(abcd\right)_{s+t}}a^{s}c^{t}S_{n-t-s}^{\left(2\right)}\left(b,d|q\right)$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{n}\QATOPD[]{n}{j}_{q}\frac{\left(bd\right)_{j}}{\left(abcd\right)_{j}}S_{n-j}^{\left(2\right)}\left(b,d|q\right)\sum_{k=0}^{j}\QATOPD[]{j}{k}_{q}\left(cb\right)_{k}a^{k}\left(ad\right)_{j-k}c^{j-k}.$
###### Proof of Proposition 3.
Notice that for $n\allowbreak=\allowbreak 0$ our formulae are true since we
have: $g_{1}\left(x|a_{1},q\right)\allowbreak=\allowbreak
f_{h}\left(x|q\right)\varphi_{h}\left(x|a_{1},q\right)\allowbreak=\allowbreak
f_{h}\left(x|q\right)\sum_{m\geq
0}\frac{a_{1}^{m}}{\left(q\right)_{m}}h_{m}\left(x|q\right),$ So
$T_{m}^{\left(1\right)}\left(a_{1},q\right)\allowbreak=\allowbreak a_{1}^{m}$
and $A_{1}\left(a_{1},q\right)\allowbreak=\allowbreak 1.$ Next notice that:
$g_{n+1}\left(x|\mathbf{a}^{\left(n+1\right)},q\right)=g_{n}\left(x|\mathbf{a}^{\left(n\right)},q\right)\varphi_{h}\left(x|a_{n+1},q\right),$
where we understand
$\mathbf{a}^{\left(n+1\right)}\allowbreak=\allowbreak\left(a_{1},\ldots,a_{n},a_{n+1}\right).$
So by induction assumption the left hand side of (3.1) is equal to:
$A_{n+1}\left(\mathbf{a}^{\left(n+1\right)},q\right)f_{h}\left(x|q\right)\sum_{j\geq
0}\frac{T_{j}^{\left(n+1\right)}\left(\mathbf{a}^{\left(n+1\right)},q\right)}{\left(q\right)_{j}}h_{j}\left(x|q\right),$
while the right hand side to
$A_{n}\left(\mathbf{a}^{\left(n\right)},q\right)f_{h}\left(x|q\right)\sum_{j,k\geq
0}\frac{T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)a_{n+1}^{k}}{\left(q\right)_{j}\left(q\right)_{k}}h_{j}\left(x|q\right)h_{k}\left(x|q\right).$
We apply again (1.6) getting:
$\displaystyle\sum_{j,k\geq
0}\frac{T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)a_{n+1}^{k}}{\left(q\right)_{j}\left(q\right)_{k}}h_{j}\left(x|q\right)h_{k}\left(x|q\right)$
$\displaystyle=$ $\displaystyle\sum_{j,k\geq
0}\frac{T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)a_{n+1}^{k}}{\left(q\right)_{j}\left(q\right)_{k}}\sum_{m=0}^{j\wedge
k}\QATOPD[]{k}{m}_{q}\QATOPD[]{j}{m}_{q}\left(q\right)_{m}h_{j+k-2m}\left(x|q\right)$
$\displaystyle=$ $\displaystyle\sum_{m\geq
0}\frac{a_{n+1}^{m}}{\left(q\right)_{m}}\sum_{k,j\geq
m}\frac{a_{n+1}^{k-m}T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)}{\left(q\right)_{k-m}\left(q\right)_{j-m}}h_{j+k-2m}\left(x|q\right)$
$\displaystyle=$ $\displaystyle\sum_{m\geq
0}\frac{a_{n+1}^{m}}{\left(q\right)_{m}}\sum_{s,t\geq
0}\frac{a_{n+1}^{s}T_{t+m}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)}{\left(q\right)_{s}\left(q\right)_{t}}h_{s+t}\left(x|q\right)$
Next we introduce new indices of summation $r\allowbreak=\allowbreak s+t$ and
$j\allowbreak=\allowbreak s$ and get:
$\displaystyle\sum_{j,k\geq
0}\frac{T_{j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)a_{n+1}^{k}}{\left(q\right)_{j}\left(q\right)_{k}}h_{j}\left(x|q\right)h_{k}\left(x|q\right)=\sum_{m\geq
0}\frac{a_{n+1}^{m}}{\left(q\right)_{m}}\sum_{r=0}^{\infty}\frac{h_{r}\left(x|q\right)}{\left(q\right)_{r}}\sum_{j=0}^{r}\QATOPD[]{r}{j}_{q}a_{n+1}^{j}T_{m+r-j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)$
$\displaystyle=\sum_{r=0}^{\infty}\frac{h_{r}\left(x|q\right)}{\left(q\right)_{r}}\sum_{j=0}^{r}\QATOPD[]{r}{j}_{q}a_{n+1}^{j}\sum_{m\geq
0}\frac{a_{n+1}^{m}}{\left(q\right)_{m}}T_{m+r-j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)$
$\displaystyle=\sum_{m\geq
0}\frac{a_{n+1}^{m}}{\left(q\right)_{m}}T_{m}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)\sum_{r=0}^{\infty}\frac{h_{r}\left(x|q\right)}{\left(q\right)_{r}}\sum_{j=0}^{r}\QATOPD[]{r}{j}_{q}a_{n+1}^{j}H_{r-j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right)$
$\displaystyle=\frac{A_{n+1}\left(\mathbf{a}^{\left(n+1\right)},q\right)}{A_{n}\left(\mathbf{a}^{\left(n\right)},q\right)}\sum_{r=0}^{\infty}\frac{h_{r}\left(x|q\right)}{\left(q\right)_{r}}\sum_{j=0}^{r}\QATOPD[]{r}{j}_{q}a_{n+1}^{j}H_{r-j}^{\left(n\right)}\left(\mathbf{a}^{\left(n\right)},q\right).$
###### Proof of Theorem 2.
We use (2.11) and utilizing Remark 1 we get:
$\displaystyle\sigma_{n}^{\left(4\right)}\left(a,b,c,d|0\right)\allowbreak=\allowbreak
S_{n}^{\left(2\right)}(b,d|0)\allowbreak+\allowbreak\frac{(1-bd)}{(1-abcd)}\sum_{j=1}^{n}S_{n-j}^{\left(2\right)}\left(b,d|q\right)((1-ad)c^{j}\allowbreak+$
$\displaystyle\allowbreak(1-cb)a^{j}\allowbreak+\allowbreak(1-cb)(1-ad)ac\sum_{k=1}^{j-1}a^{k-1}c^{j-1-k}).\allowbreak$
And further
$\displaystyle\sigma_{n}^{\left(4\right)}\left(a,b,c,d|0\right)=\allowbreak
S_{n}^{\left(2\right)}(b,d|0)\allowbreak+\allowbreak\frac{(1-bd)}{(1-abcd)}\sum_{j=1}^{n}S_{n-j}^{\left(2\right)}\left(b,d|q\right)((1-ad)c^{j}\allowbreak+\allowbreak(1-cb)a^{j}\allowbreak+\allowbreak$
$\displaystyle(1-cb)(1-ad)acS_{j-2}^{\left(2\right)}(a,c|0)\allowbreak\allowbreak$
$\displaystyle=S_{n}^{\left(2\right)}(b,d|0)\allowbreak+\allowbreak\frac{(1-bd)}{(1-abcd)}\sum_{j=1}^{n}S_{n-j}^{\left(2\right)}\left(b,d|q\right)((1-ad)c^{j}\allowbreak+\allowbreak(1-cb)a^{j})\allowbreak$
$\displaystyle+\allowbreak\frac{(1-bd)(1-cb)(1-ad)ac}{(1-abcd)}\sum_{j=2}^{n}S_{n-j}^{\left(2\right)}\left(b,d|q\right)S_{j-2}^{\left(2\right)}(a,c|0)\allowbreak.$
Now we use formula (2.3). Then we replace $a$ by $a_{1}$ , $b$ by $a_{2}$ and
so on. Finally we use formulae (3.3) and (2.2) which remembering that
$\left(0\right)_{n}\allowbreak=\allowbreak 1$ leads to our integral formula.
## References
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* [2] Bryc, Włodzimierz. Stationary random fields with linear regressions. _Ann. Probab._ 29 (2001), no. 1, 504–519. MR1825162 (2002d:60014)
* [3] Szabłowski, Paweł J. Expansions of one density via polynomials orthogonal with respect to the other. _J. Math. Anal. Appl._ 383 (2011), no. 1, 35–54. MR2812716, http://arxiv.org/abs/1011.1492
* [4] Szabłowski, Paweł J. On the structure and probabilistic interpretation of Askey-Wilson densities and polynomials with complex parameters. _J. Funct. Anal._ 261 (2011), no. 3, 635–659. MR2799574, http://arxiv.org/abs/1011.1541
* [5] Corteel, Sylvie; Williams, Lauren K. Staircase tableaux, the asymmetric exclusion process, and Askey-Wilson polynomials. Proc. Natl. Acad. Sci. USA 107 (2010), no. 15, 6726–6730. MR2630104
* [6] Carlitz, L. Generating functions for certain $Q$-orthogonal polynomials. _Collect. Math._ 23 (1972), 91–104. MR0316773 (47 #5321)
* [7] Koekoek R. , Swarttouw R. F. (1999) The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, ArXiv:math/9602214
* [8] Ismail, Mourad E. H. Classical and quantum orthogonal polynomials in one variable. With two chapters by Walter Van Assche. With a foreword by Richard A. Askey. Encyclopedia of Mathematics and its Applications, 98. _Cambridge University Press,_ Cambridge, 2005. xviii+706 pp. ISBN: 978-0-521-78201-2; 0-521-78201-5 MR2191786 (2007f:33001)
* [9] Ismail, Mourad E. H.; Stanton, Dennis; Viennot, Gérard. The combinatorics of $q$-Hermite polynomials and the Askey-Wilson integral. _European J. Combin._ 8 (1987), no. 4, 379–392. MR0930175 (89h:33015)
* [10] Andrews, George E.; Askey, Richard; Roy, Ranjan. Special functions. Encyclopedia of Mathematics and its Applications, 71. _Cambridge University Press,_ Cambridge, 1999. xvi+664 pp. ISBN: 0-521-62321-9; 0-521-78988-5 MR1688958 (2000g:33001) (2002k:33011)
* [11] Bryc, Włodzimierz; Matysiak, Wojciech; Szabłowski, Paweł J. Probabilistic aspects of Al-Salam-Chihara polynomials. _Proc. Amer. Math. Soc._ 133 (2005), no. 4, 1127–1134 (electronic). MR2117214 (2005m:33033)
* [12] Bryc, Włodzimierz. Stationary random fields with linear regressions. _Ann. Probab._ 29 (2001), no. 1, 504–519. MR1825162 (2002d:60014)
* [13] Paweł J. Szabłowski. Befriending Askey–Wilson polynomials, submitted, http://arxiv.org/abs/1111.0601.
* [14] Ma, X. R. A new proof of the Askey-Wilson integral via a five-variable Ramanujan’s reciprocity theorem. _Ramanujan J._ 24 (2011), no. 1, 61–65. MR2765601
* [15] Liu, Zhi-Guo. An identity of Andrews and the Askey-Wilson integral. _Ramanujan J._ 19 (2009), no. 1, 115–119. MR2501242 (2010g:33007)
* [16] Carlitz, L. Some polynomials related to theta functions. Ann. Mat. Pura Appl. (4) 41 (1956), 359–373. MR0078510 (17,1205e)
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|
arxiv-papers
| 2011-12-20T20:51:37 |
2024-09-04T02:49:25.547064
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pawe{\\l} J. Szab{\\l}owski",
"submitter": "Pawe{\\l} Szab{\\l}owski J.",
"url": "https://arxiv.org/abs/1112.4830"
}
|
1112.4861
|
# Pseudoscalar decay constants, light-quark masses, and $B_{K}$ from mixed-
action lattice QCD
Jack Laiho
Department of Physics and Astronomy, University of Glasgow, Glasgow, Scotland,
UK
E-mail Funded by STFC and the Scottish Universities Physics Alliance.
jlaiho@fnal.gov
Physics Department, Brookhaven National Laboratory, Upton, New York, USA
E-mail This manuscript has been authored by employees of Brookhaven Science
Associates, LLC under Contract No. DE-AC02-98CH10886 with the U.S. Department
of Energy. Computations for this work were carried out with resources provided
by the USQCD Collaboration, the Argonne Leadership Computing Facility, and the
New York Center for Computational Sciences, which are funded by the Office of
Science of the U.S. Department of Energy and by New York State.
###### Abstract:
We present updated results for the leptonic decay constants $f_{\pi}$ and
$f_{K}$, the light $u$, $d$, and $s$-quark masses, and the neutral kaon mixing
parameter $B_{K}$ from mixed-action lattice simulations with staggered sea
quarks and domain-wall valence quarks. We use the publicly-available 2+1
flavor MILC asqtad-improved staggered gauge configurations with multiple light
sea-quark masses and three lattice spacings, and compute the kaon mixing
matrix element with several partially-quenched valence-quark masses. We then
extrapolate to the physical light-quark masses and the continuum using
partially-quenched chiral perturbation theory formulated for mixed-action
lattice simulations. For $B_{K}$ we match the lattice four-fermion operator to
the continuum using the nonperturbative method of Rome-Southampton. Our new
results benefit from two significant improvements over our published work: (1)
we have added a third lattice spacing of a$\approx$0.06 fm to better control
the continuum extrapolation, and (2) we have implemented a new lattice
renormalization scheme (the RI/SMOM${}_{\gamma_{\mu}}$ scheme developed by
Sturm et al.) that suppresses chiral-symmetry breaking and other infrared
effects and, in practice, also shrinks the size of the 1-loop perturbative
coefficient needed to match to the continuum $\overline{\textrm{MS}}$ scheme.
When combined with the use of volume-averaged momentum sources and twisted-
boundary conditions, this significantly reduces the systematic uncertainty in
the renormalization factor $Z_{B_{K}}$.
## 1 Motivation
Lattice-QCD calculations of pseudoscalar decay constants, light-quark masses,
and other kaon weak matrix elements are important ingredients for
understanding the Standard Model and for constraining physics beyond the
Standard Model. For example, the $u$, $d$, and $s$-quark masses are parametric
inputs to calculations of Standard Model and new physics processes. The ratio
of pseudoscalar decay constants $f_{K}/f_{\pi}$, when combined with
experimental measurements of the leptonic decay rates, allows a precise
determination of the ratio of CKM matrix elements $|V_{ud}|/|V_{us}|$ [1]. The
neutral kaon mixing parameter $B_{K}$, when combined with experimental
measurements of indirect $CP$-violation in the kaon sector ($\varepsilon_{K}$)
constrains the apex of the CKM unitarity triangle. Finally, penguin-dominated
$K\to\pi\pi$ and $K\to\pi\nu\overline{\nu}$ decays may be particularly
sensitive probes of new physics once lattice weak matrix element calculations
are sufficiently precise.
Lattice QCD has played a key role in establishing that the CKM paradigm of
$CP$-violation describes experimental observations at the $\sim\\!\\!10\%$
level [2]. Many new-physics scenarios, however, predict new interactions
between quarks and non-standard $CP$-violating phases. Given sufficient
theoretical and experimental precision, these would lead to inconsistencies
between measurements that are expected to be the same within the Standard
Model CKM framework. Recent improvements in lattice weak matrix element
calculations, especially of $B_{K}$ and the $B$-mixing $SU(3)$-breaking ratio
$\xi$, have revealed a $\sim\\!\\!3\sigma$ tension that may indicate the
presence of a non-Standard Model source of $CP$-violation (see Fig. 1) [4].
Given this tension, it is crucial to continue precision studies of kaon
physics using multiple methods including our mixed-action approach.
Figure 1: Global fit of the CKM unitarity triangle [3].
## 2 Overview of mixed-action calculation
Table 1 shows the parameters of our numerical mixed-action lattice
simulations. We analyze the 2+1 flavor asqtad-improved staggered gauge
configurations generated by the MILC Collaboration [5]. Use of this large
suite of ensembles with multiple spatial volumes $\sim$ (2.5 – 4 fm)3,
multiple light sea-quark masses $\sim m_{s}/10$ – $m_{s}$, and three lattice
spacings $\sim$ 0.06 – 0.12 fm gives us good control over the systematic
uncertainties associated with the chiral-continuum extrapolation. We generate
domain-wall valence quarks at several partially-quenched masses $\sim
m_{s}/10$ – $m_{s}$. We apply HYP-smearing [7] to the valence domain-wall
action in order to reduce the size of explicit chiral symmetry breaking [8]
($m_{\rm res}~{}\approx~{}3$ MeV at our coarsest lattice spacing and is $\sim
30$ times smaller at our finest lattice spacing). The use of domain-wall
valence quarks makes the chiral-continuum extrapolation more continuum-like
[9, 10] and simplifies the nonperturbative operator matching via the method of
Rome-Southampton [11]. Hence our mixed-action approach is well-suited for weak
matrix element calculations, as we show empirically in the next section.
$a$(fm) | $L^{3}\times T$ | $m_{l}$ | $m_{h}$ | $m_{\rm val.}^{\rm dwf}$ | # configs.
---|---|---|---|---|---
$\approx$ 0.06 | 64${}^{3}\times$ 144 | 0.0018 | 0.018 | 0.0026, 0.0469, 0.0108, 0.033 | 96
$\approx$ 0.06 | 48${}^{3}\times$ 144 | 0.0036 | 0.018 | 0.0036, 0.0072, 0.0108, 0.033 | 128
$\approx$ 0.09 | 40${}^{3}\times$ 96 | 0.0031 | 0.0031 | 0.004, 0.0124, 0.0186, 0.046 | 102
$\approx$ 0.09 | $40^{3}\times 96$ | 0.0031 | 0.031 | 0.004, 0.0124, 0.0186, 0.046 | 150
$\approx$ 0.09 | $28^{3}\times 96$ | 0.0062 | 0.031 | 0.0062, 0.0124, 0.0186, 0.046 | 374
$\approx$ 0.09 | 28${}^{3}\times$ 96 | 0.0093 | 0.031 | 0.0062, 0.0124, 0.0186, 0.046 | 198
$\approx$ 0.09 | $28^{3}\times 96$ | 0.0124 | 0.031 | 0.0062, 0.0124, 0.0186, 0.046 | 198
$\approx$ 0.09 | $28^{3}\times 96$ | 0.0062 | 0.0186 | 0.0062, 0.0124, 0.0186, 0.046 | 160
$\approx$ 0.125 | 32${}^{3}\times$ 64 | 0.005 | 0.005 | 0.007, 0.02, 0.03, 0.05 | 175
$\approx$ 0.125 | $24^{3}\times 64$ | 0.005 | 0.05 | 0.007, 0.02, 0.03, 0.05, 0.065 | 216
$\approx$ 0.125 | $20^{3}\times 64$ | 0.007 | 0.05 | 0.01, 0.02, 0.03, 0.04, 0.05, 0.065 | 268
$\approx$ 0.125 | $20^{3}\times 64$ | 0.01 | 0.05 | 0.01, 0.02, 0.03, 0.05, 0.065 | 220
$\approx$ 0.125 | $20^{3}\times 64$ | 0.02 | 0.05 | 0.01, 0.03, 0.05, 0.065 | 117
$\approx$ 0.125 | $20^{3}\times 64$ | 0.01 | 0.03 | 0.01, 0.02, 0.03, 0.05, 0.065 | 160
Table 1: Sea-quark ensembles and valence-quark masses used to obtain the
preliminary results presented in this work. Ensembles shown in bold are new
since our 2009 $B_{K}$ publication [6].
## 3 Testing the mixed-action method: decay constants and quark masses
We compute the decay constant using a Ward identity to relate the axial
current to the pseudoscalar density; this method has the advantage that the
renormalization factor is unity up to corrections of ${\cal O}(am_{\rm
res})\sim 10^{-3}$–$10^{-5}$. Once we have the decay constant for all
combinations of valence- and sea-quark masses, we perform a combined chiral-
continuum extrapolation via a simultaneous correlated fit of the full data set
using NLO $SU(3)$ mixed-action $\chi$PT [9] supplemented with higher-order
analytic terms. Because the kaon mass is so heavy and our data is so precise,
these terms are needed to interpolate about the strange-quark mass and obtain
good confidence levels. We correct the numerical data for the known one-loop
finite volume effects, and estimate the systematic uncertainty due to the
chiral-continuum extrapolation by varying the fit function. As shown in Fig.
2, the value of $f_{\pi}$ in the continuum limit at the physical quark masses
agrees with experiment. Our preliminary results for the pseudoscalar decay
constants and their ratio are
$\displaystyle f_{\pi}$ $\displaystyle=$ $\displaystyle 130.53(0.87)_{\rm
stat}(1.68)_{\chi{\rm PT}}(0.80)_{\rm
FV}(0.93)_{r_{1}}(0.25)_{m_{q}}~{}\textrm{MeV}$ (1) $\displaystyle f_{K}$
$\displaystyle=$ $\displaystyle 156.8(1.0)_{\rm stat}(1.1)_{\chi{\rm
PT}}(0.6)_{\rm FV}(0.8)_{r_{1}}(0.8)_{m_{q}}~{}\textrm{MeV}$ (2)
$\displaystyle f_{K}/f_{\pi}$ $\displaystyle=$ $\displaystyle
1.202(0.011)_{\rm stat}(0.009)_{\chi{\rm PT}}(0.008)_{\rm
FV}(0.002)_{r_{1}}(0.005)_{m_{q}}\,,$ (3)
where the errors are from statistics, the chiral-continuum extrapolation,
finite volume effects, the scale uncertainty, and the light-quark mass
uncertainties, respectively. We convert lattice quantities into physical units
with the value of the scale $r_{1}$ obtained by the HPQCD collaboration from
several quantities including pseudoscalar decay constants and $\Upsilon$
splittings [12].
Figure 2: Chiral-continuum extrapolation of the pseudoscalar decay constant.
Only data points with degenerate valence-quark masses are shown, but the fit
includes all nondegenerate points. The circles denote “coarse” ($a\approx
0.12$ fm) data, the squares denote “fine” ($a\approx 0.09$ fm) data, and the
triangles denote “superfine” data. The cyan band shows the continuum full QCD
curve. The black “$\times$” shows $f_{\pi}$ at the physical $ud$ quark mass,
where the inner solid error bar is statistical and the outer dashed error bar
is the total systematic. For comparison, the red star shows the experimental
result for $f_{\pi}$, with which we are in good agreement.
We determine the bare average $ud$ and $s$ domain-wall valence-quark masses by
requiring that the pion and kaon masses in the continuum at these quark masses
take their experimental values [13]. We compute the quark-mass renormalization
factor $Z_{m}=1/Z_{S}$ using a partly-nonperturbative approach in which we
combine a nonperturbative determination of $Z_{A}$ with a one-loop tadpole-
improved lattice perturbation theory [14] calculation of $(1-Z_{A}/Z_{S})$.
Because the ratio $Z_{A}/Z_{S}$ is close to unity, the size of the one-loop
correction is small; nevertheless, the truncation error is still the largest
source of uncertainty in $Z_{m}$. We multiply the bare-quark masses by $Z_{m}$
to obtain the continuum $\overline{\textrm{MS}}$ masses; our preliminary
results are
$\displaystyle m_{s}^{\overline{\textrm{MS}}(\textrm{2 GeV})}$
$\displaystyle=$ $\displaystyle 94.2(1.4)_{\rm stat}(3.2)_{\rm sys}(4.7)_{\rm
match}~{}\textrm{MeV}$ (4) $\displaystyle
m_{ud}^{\overline{\textrm{MS}}(\textrm{2 GeV})}$ $\displaystyle=$
$\displaystyle 3.31(0.07)_{\rm stat}(0.20)_{\rm sys}(0.17)_{\rm
match}~{}\textrm{MeV}$ (5) $\displaystyle
m_{d}^{\overline{\textrm{MS}}(\textrm{2 GeV})}$ $\displaystyle=$
$\displaystyle 4.73(0.09)_{\rm stat}(0.27)_{\rm sys}(0.24)_{\rm
match}~{}\textrm{MeV}$ (6) $\displaystyle
m_{u}^{\overline{\textrm{MS}}(\textrm{2 GeV})}$ $\displaystyle=$
$\displaystyle 1.90(0.08)_{\rm stat}(0.21)_{\rm sys}(0.10)_{\rm
match}~{}\textrm{MeV}$ (7) $\displaystyle\left(m_{s}/m_{ud}\right)$
$\displaystyle=$ $\displaystyle 28.4(0.5)_{\rm stat}(1.3)_{\rm
sys}\qquad\left(m_{u}/m_{d}\right)=0.401(0.013)_{\rm stat}(0.045)_{\rm
sys}\,,$ (8)
where the errors are from statistics, all systematics except for
renormalization, and renormalization, respectively. Although we do not include
isospin-breaking in our lattice simulations, we obtain $m_{u}$ and $m_{d}$
separately following the method adopted by MILC [5], which uses the difference
between the $K^{+}$ and $K^{0}$ meson masses and continuum estimates for the
violation of Dashen’s theorem.
Our preliminary determinations of the decay constants and light-quark masses
agree well with $N_{f}=2+1$ lattice QCD calculations by other collaborations
[15] (see also the recent review by Wittig [16]). This gives us confidence
that the mixed-action method is reliable and can be used to obtain $B_{K}$ and
$K\to\pi\pi$ matrix elements precisely with controlled systematic
uncertainties.
## 4 The neutral kaon mixing parameter: $B_{K}$
In these proceedings we focus on improvements to our published calculation of
$B_{K}$ in Ref. [6]. Since 2009 we have addressed the two largest sources of
uncertainty in our earlier work:
1. 1.
The chiral-continuum extrapolation: We have added ensembles with larger
volumes and lighter pions to shorten the chiral extrapolation. We have also
added data at a third finer lattice spacing to reduce taste-breaking and
shorten the continuum extrapolation.
2. 2.
The $\Delta S=2$ operator renormalization: We now compute the renormalization
factor $Z_{B_{K}}$ using volume-averaged momentum sources to reduce
statistical errors and twisted boundary conditions to eliminate scatter from
$O(4)$ symmetry breaking [17]. We also now compute $Z_{B_{K}}$ with non-
exceptional kinematics to reduce chiral symmetry breaking [18].
We compute $Z_{B_{K}}$ via the nonperturbative method of Rome-Southampton
[11]. We use an intermediate lattice scheme with symmetric momentum
$p_{1}^{2}=p_{2}^{2}=(p_{1}-p_{2})^{2}$ developed by Sturm et al. [18]; this
suppresses chiral symmetry breaking and leads to a more convergent
perturbative series for the conversion factor between the intermediate lattice
RI/(S)MOM scheme and the continuum $\overline{\textrm{MS}}$ scheme.
Specifically, we use the RI/SMOM${}_{\gamma_{\mu}}$ scheme (which has the same
projectors as the RI/MOM scheme), for which the 1-loop correction to the
continuum $\overline{\textrm{MS}}$ scheme at 2 GeV is $\sim$ 4 times smaller
than for the standard RI/MOM scheme [19]. Because the
RI/SMOM${}_{\gamma_{\mu}}$ scheme is defined for massless quarks, we compute
the bilinear and four-fermion vertex functions for several values of the
valence, light, and strange sea-quark masses and extrapolate to the chiral
limit ${m_{\rm val.}^{\rm dwf},m_{l},m_{h}}\to 0$. The largest source of
uncertainty in our calculation of $Z_{B_{K}}$ is the residual perturbative
truncation error from conversion to the continuum $\overline{\textrm{MS}}$
scheme. We estimate this in several ways including the size of 1-loop term
(0.5%), comparison with lattice perturbation theory (2.0%), and the residual
slope in $(r_{1}p)^{2}$ (2.2%), and choose the largest error estimate to be
conservative.
We compute the $B_{K}$ matrix element using symmetric and antisymmetric linear
combinations of periodic and antiperiodic B.C. quark propagators to minimize
finite temporal size effects. We obtain $\sim$0.5-1.5% statistical errors in
the lattice matrix element. We use the same approach for the chiral-continuum
extrapolation as for the pseudoscalar masses and decay constants. We fit the
renormalized $B_{K}$ data to the NLO $SU(3)$ mixed-action $\chi$PT expression
[10] supplemented by higher-order analytic terms (see Fig. 3). We estimate the
systematic uncertainty due to the chiral extrapolation by varying the fit
function in several ways including using an analytic function without chiral
logarithms, adding higher-order analytic and logarithm terms, and changing the
value of $f_{\pi}$ in the coefficient of the 1-loop chiral logarithms. We
conservatively take the largest difference from our preferred fit as the
error.
We obtain the following preliminary result for $B_{K}$ in the
$\overline{\textrm{MS}}$ scheme at 2 GeV:
$B^{\overline{\textrm{MS}}\textrm{(2 GeV)}}_{K}=0.5572(28)_{\rm
stat}(45)_{\chi{\rm PT}}(33)_{\rm FV}(39)_{r_{1},m_{q}}(6)_{\rm EM}(134)_{\rm
match}\,,$ (9)
where the error labels are the same as in previous equations; the total
uncertainty is 2.8%. The largest contribution to the error is from the
renormalization factor; we aim to improve the uncertainty in $Z_{B_{K}}$ due
to the chiral extrapolation with additional data before publication.
Ultimately, the dominant truncation error in $Z_{B_{K}}$ may be reduced with a
2-loop continuum perturbative QCD calculation. Our new result is $\sim
1.1\sigma$ higher than our published value, and agrees well with calculations
from several other lattice methods [20] (see also the recent review by
Mawhinney [21]).
Figure 3: Chiral-continuum extrapolation of $B_{K}$. The fit includes all
available data, but only the nondegenerate data points in which one valence-
quark mass is close to the strange quark and the other valence-quark mass is
the lightest on that ensemble are shown. The black “$\times$” shows $B_{K}$ in
the continuum at the physical $d$ and $s$ quark masses, where the inner solid
error bar is statistical and the outer dashed error bar includes the chiral
extrapolation systematic. The star with dotted error bar, slightly offset,
shows the total error in $B_{K}$.
## 5 Summary and outlook
Mixed-action lattice QCD simulations are well-suited to the calculation of
weak matrix elements, as shown by agreement with experiment and independent
lattice results for pseudoscalar decay constants and light-quark masses, and
also by our precise determination of $B_{K}$. Given this and other recent
improvements in lattice QCD calculations of $B_{K}$, it is no longer the
dominant source of uncertainty in the $\varepsilon_{K}$ band. Hence, although
recent $B_{K}$ results slightly reduce the tension in the global CKM unitarity
triangle fit, the tension remains at the 2-3$\sigma$ level [22]. Given our
success with $B_{K}$, we eventually aim to pursue $K\to\pi\pi$ decays with the
mixed-action approach.
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|
arxiv-papers
| 2011-12-20T21:51:46 |
2024-09-04T02:49:25.557502
|
{
"license": "Public Domain",
"authors": "Jack Laiho and Ruth S. Van de Water",
"submitter": "Ruth Van de Water",
"url": "https://arxiv.org/abs/1112.4861"
}
|
1112.4896
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
LHCb-PAPER-2011-035 CERN-PH-EP-2011-226 December 20, 2011; rev. January
25, 2012
Measurement of $b$-hadron masses
The LHCb Collaboration111Authors are listed on the following pages.
Measurements of $b$-hadron masses are performed with the exclusive decay modes
$B^{+}\rightarrow J/\psi K^{+}$, $B^{0}\\!\rightarrow{J/\psi}{K^{*0}}$ ,
$B^{0}\rightarrow J/\psi K^{0}_{\rm S}$, $B_{s}^{0}\rightarrow J/\psi\phi$ and
$\Lambda^{0}_{b}\rightarrow J/\psi\Lambda$ using an integrated luminosity of
$35~{}\mathrm{pb}^{-1}$ collected in $pp$ collisions at a centre-of-mass
energy of 7$\mathrm{\,Te\kern-1.00006ptV}$ by the LHCb experiment. The
momentum scale is calibrated with $J/\psi\rightarrow\mu^{+}\mu^{-}$ decays and
verified to be known to a relative precision of 2 $\times 10^{-4}$ using other
two-body decays. The results are more precise than previous measurements,
particularly in the case of the $B^{0}_{s}$ and $\Lambda^{0}_{b}$ masses.
Published in Phys. Lett. B 708 (2012) 241–248
The LHCb Collaboration
R. Aaij23, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi42, C. Adrover6, A.
Affolder48, Z. Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J.
Anderson39, R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L.
Arrabito53, A. Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16,
R.J. Barlow50, C. Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10,
Th. Bauer23, A. Bay38, I. Bediaga1, S. Belogurov30, K. Belous34, I.
Belyaev30,37, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson46, J.
Benton42, R. Bernet39, M.-O. Bettler17, M. van Beuzekom23, A. Bien11, S.
Bifani12, T. Bird50, A. Bizzeti17,h, P.M. Bjørnstad50, T. Blake37, F. Blanc38,
C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V. Bocci22, A. Bondar33, N.
Bondar29, W. Bonivento15, S. Borghi47,50, A. Borgia52, T.J.V. Bowcock48, C.
Bozzi16, T. Brambach9, J. van den Brand24, J. Bressieux38, D. Brett50, M.
Britsch10, T. Britton52, N.H. Brook42, A. Büchler-Germann39, I. Burducea28, A.
Bursche39, J. Buytaert37, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez35,n, A. Camboni35, P. Campana18,37, A. Carbone14, G. Carboni21,k, R.
Cardinale19,i,37, A. Cardini15, L. Carson49, K. Carvalho Akiba2, G. Casse48,
M. Cattaneo37, Ch. Cauet9, M. Charles51, Ph. Charpentier37, N. Chiapolini39,
K. Ciba37, X. Cid Vidal36, G. Ciezarek49, P.E.L. Clarke46,37, M. Clemencic37,
H.V. Cliff43, J. Closier37, C. Coca28, V. Coco23, J. Cogan6, P. Collins37, A.
Comerma-Montells35, F. Constantin28, A. Contu51, A. Cook42, M. Coombes42, G.
Corti37, G.A. Cowan38, R. Currie46, C. D’Ambrosio37, P. David8, P.N.Y.
David23, I. De Bonis4, S. De Capua21,k, M. De Cian39, F. De Lorenzi12, J.M. De
Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H.
Degaudenzi38,37, L. Del Buono8, C. Deplano15, D. Derkach14,37, O. Deschamps5,
F. Dettori24, J. Dickens43, H. Dijkstra37, P. Diniz Batista1, F. Domingo
Bonal35,n, S. Donleavy48, F. Dordei11, A. Dosil Suárez36, D. Dossett44, A.
Dovbnya40, F. Dupertuis38, R. Dzhelyadin34, A. Dziurda25, S. Easo45, U.
Egede49, V. Egorychev30, S. Eidelman33, D. van Eijk23, F. Eisele11, S.
Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch. Elsasser39, D. Elsby55, D.
Esperante Pereira36, L. Estève43, A. Falabella16,14,e, E. Fanchini20,j, C.
Färber11, G. Fardell46, C. Farinelli23, S. Farry12, V. Fave38, V. Fernandez
Albor36, M. Ferro-Luzzi37, S. Filippov32, C. Fitzpatrick46, M. Fontana10, F.
Fontanelli19,i, R. Forty37, M. Frank37, C. Frei37, M. Frosini17,f,37, S.
Furcas20, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini51, Y.
Gao3, J-C. Garnier37, J. Garofoli52, J. Garra Tico43, L. Garrido35, D.
Gascon35, C. Gaspar37, N. Gauvin38, M. Gersabeck37, T. Gershon44,37, Ph.
Ghez4, V. Gibson43, V.V. Gligorov37, C. Göbel54, D. Golubkov30, A.
Golutvin49,30,37, A. Gomes2, H. Gordon51, M. Grabalosa Gándara35, R. Graciani
Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E.
Greening51, S. Gregson43, B. Gui52, E. Gushchin32, Yu. Guz34, T. Gys37, G.
Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42, S. Hansmann-Menzemer11, R.
Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44, T. Hartmann56, J. He7,
V. Heijne23, K. Hennessy48, P. Henrard5, J.A. Hernando Morata36, E. van
Herwijnen37, E. Hicks48, K. Holubyev11, P. Hopchev4, W. Hulsbergen23, P.
Hunt51, T. Huse48, R.S. Huston12, D. Hutchcroft48, D. Hynds47, V. Iakovenko41,
P. Ilten12, J. Imong42, R. Jacobsson37, A. Jaeger11, M. Jahjah Hussein5, E.
Jans23, F. Jansen23, P. Jaton38, B. Jean-Marie7, F. Jing3, M. John51, D.
Johnson51, C.R. Jones43, B. Jost37, M. Kaballo9, S. Kandybei40, M. Karacson37,
T.M. Karbach9, J. Keaveney12, I.R. Kenyon55, U. Kerzel37, T. Ketel24, A.
Keune38, B. Khanji6, Y.M. Kim46, M. Knecht38, A. Kozlinskiy23, L. Kravchuk32,
K. Kreplin11, M. Kreps44, G. Krocker11, P. Krokovny11, F. Kruse9, K.
Kruzelecki37, M. Kucharczyk20,25,37,j, T. Kvaratskheliya30,37, V.N. La Thi38,
D. Lacarrere37, G. Lafferty50, A. Lai15, D. Lambert46, R.W. Lambert24, E.
Lanciotti37, G. Lanfranchi18, C. Langenbruch11, T. Latham44, C. Lazzeroni55,
R. Le Gac6, J. van Leerdam23, J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J.
Lefrançois7, O. Leroy6, T. Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M.
Liles48, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, J. von Loeben20, J.H.
Lopes2, E. Lopez Asamar35, N. Lopez-March38, H. Lu38,3, J. Luisier38, A. Mac
Raighne47, F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc10, O. Maev29,37,
J. Magnin1, S. Malde51, R.M.D. Mamunur37, G. Manca15,d, G. Mancinelli6, N.
Mangiafave43, U. Marconi14, R. Märki38, J. Marks11, G. Martellotti22, A.
Martens8, L. Martin51, A. Martín Sánchez7, D. Martinez Santos37, A.
Massafferri1, Z. Mathe12, C. Matteuzzi20, M. Matveev29, E. Maurice6, B.
Maynard52, A. Mazurov16,32,37, G. McGregor50, R. McNulty12, M. Meissner11, M.
Merk23, J. Merkel9, R. Messi21,k, S. Miglioranzi37, D.A. Milanes13,37, M.-N.
Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran12, P. Morawski25, R.
Mountain52, I. Mous23, F. Muheim46, K. Müller39, R. Muresan28,38, B. Muryn26,
B. Muster38, M. Musy35, J. Mylroie-Smith48, P. Naik42, T. Nakada38, R.
Nandakumar45, I. Nasteva1, M. Nedos9, M. Needham46, N. Neufeld37, C. Nguyen-
Mau38,o, M. Nicol7, V. Niess5, N. Nikitin31, A. Nomerotski51, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero23, S. Ogilvy47, O.
Okhrimenko41, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen49, K. Pal52, J. Palacios39, A. Palano13,b, M. Palutan18, J. Panman37, A.
Papanestis45, M. Pappagallo47, C. Parkes50,37, C.J. Parkinson49, G.
Passaleva17, G.D. Patel48, M. Patel49, S.K. Paterson49, G.N. Patrick45, C.
Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino23, G.
Penso22,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez
Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G.
Pessina20, A. Petrella16,37, A. Petrolini19,i, A. Phan52, E. Picatoste
Olloqui35, B. Pie Valls35, B. Pietrzyk4, T. Pilař44, D. Pinci22, R.
Plackett47, S. Playfer46, M. Plo Casasus36, G. Polok25, A. Poluektov44,33, E.
Polycarpo2, D. Popov10, B. Popovici28, C. Potterat35, A. Powell51, J.
Prisciandaro38, V. Pugatch41, A. Puig Navarro35, W. Qian52, J.H. Rademacker42,
B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk40, G. Raven24, S. Redford51,
M.M. Reid44, A.C. dos Reis1, S. Ricciardi45, K. Rinnert48, D.A. Roa Romero5,
P. Robbe7, E. Rodrigues47,50, F. Rodrigues2, P. Rodriguez Perez36, G.J.
Rogers43, S. Roiser37, V. Romanovsky34, M. Rosello35,n, J. Rouvinet38, T.
Ruf37, H. Ruiz35, G. Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P.
Sail47, B. Saitta15,d, C. Salzmann39, M. Sannino19,i, R. Santacesaria22, C.
Santamarina Rios36, R. Santinelli37, E. Santovetti21,k, M. Sapunov6, A.
Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina30, P.
Schaack49, M. Schiller24, S. Schleich9, M. Schlupp9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B.
Sciascia18, A. Sciubba18,l, M. Seco36, A. Semennikov30, K. Senderowska26, I.
Sepp49, N. Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval40,37,
P. Shatalov30, Y. Shcheglov29, T. Shears48, L. Shekhtman33, O. Shevchenko40,
V. Shevchenko30, A. Shires49, R. Silva Coutinho44, T. Skwarnicki52, A.C.
Smith37, N.A. Smith48, E. Smith51,45, K. Sobczak5, F.J.P. Soler47, A.
Solomin42, F. Soomro18, B. Souza De Paula2, B. Spaan9, A. Sparkes46, P.
Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S.
Stone52,37, B. Storaci23, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, S.
Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26, S. T’Jampens4, E.
Teodorescu28, F. Teubert37, C. Thomas51, E. Thomas37, J. van Tilburg11, V.
Tisserand4, M. Tobin39, S. Topp-Joergensen51, N. Torr51, E. Tournefier4,49,
M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23, M. Ubeda Garcia37, A. Ukleja27,
P. Urquijo52, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez35, P.
Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis42, M. Veltri17,g, B. Viaud7, I.
Videau7, X. Vilasis-Cardona35,n, J. Visniakov36, A. Vollhardt39, D.
Volyanskyy10, D. Voong42, A. Vorobyev29, H. Voss10, S. Wandernoth11, J.
Wang52, D.R. Ward43, N.K. Watson55, A.D. Webber50, D. Websdale49, M.
Whitehead44, D. Wiedner11, L. Wiggers23, G. Wilkinson51, M.P. Williams44,45,
M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25, W. Witzeling37, S.A.
Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Xing52, Z. Yang3, R. Young46,
O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3, L. Zhang52, W.C. Zhang12, Y.
Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin37.
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
23Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH University of Science and Technology, Kraków, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55University of Birmingham, Birmingham, United Kingdom
56Physikalisches 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
Within the Standard Model of particle physics, mesons and baryons are
colourless objects composed of quarks and gluons. These systems are bound
through the strong interaction, described by quantum chromodynamics (QCD). A
basic property of hadrons that can be compared to theoretical predictions is
their masses. The most recent theoretical predictions based on lattice QCD
calculations can be found in Refs. [1, 2]. The current experimental knowledge
of the $b$-hadron masses as summarized in Ref. [3] is dominated by results
from the CDF collaboration [4]. In this Letter precision measurements of the
masses of the $B^{+}$, $B^{0}$, $B_{s}^{0}$ and $\Lambda^{0}_{b}$ are
presented as well as the mass splittings with respect to the $B^{+}$. The
results are based on a data sample of proton-proton collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ at the Large Hadron Collider
collected by the LHCb experiment, corresponding to an integrated luminosity of
$\rm 35~{}pb^{-1}$.
The LHCb detector [5] is a forward spectrometer providing charged particle
reconstruction in the pseudorapidity range $2<\eta<5$. The most important
elements for the analysis presented here are precision tracking and excellent
particle identification. The tracking system consists of a silicon strip
vertex detector (VELO) surrounding the $pp$ interaction region, a large area
silicon strip detector located upstream of a dipole magnet with a bending
power of about 4 Tm, and a combination 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${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Pion, kaon and proton separation is
provided by two ring imaging Cherenkov (RICH) detectors whilst muons are
identified by a muon system consisting of alternating layers of iron and
multi-wire proportional chambers.
The data used for this analysis were collected in 2010. The trigger system
consists of two levels. The first stage is implemented in hardware and uses
information from the calorimeters and the muon system. The second stage is
implemented in software and runs on an event filter farm. Dedicated trigger
lines collect events containing $J/\psi$ mesons. For this analysis all events
are used regardless of which trigger line fired.
Simulation samples used are based on the Pythia 6.4 generator [6] configured
with the parameters detailed in Ref. [7]. QED final state radiative
corrections are included using the Photos package [8]. The EvtGen [9] and
Geant4 [10] packages are used to generate hadron decays and simulate
interactions in the detector, respectively.
The alignment of the tracking system, as well as the calibration of the
momentum scale based on the $J/\psi\rightarrow\mu^{+}\mu^{-}$ mass peak, were
carried out in seven time periods corresponding to different running
conditions. The procedure takes into account the effects of QED radiative
corrections which are important in the $J/\psi\rightarrow\mu^{+}\mu^{-}$
decay. Figure 1 shows that the reconstructed $J/\psi$ mass after alignment and
calibration is stable in time to better than $0.02\%$ throughout the data-
taking period. The validity of the momentum calibration has been checked using
samples of $K^{0}_{\rm S}\rightarrow\pi^{+}\pi^{-}$, $D^{0}\rightarrow
K^{-}\pi^{+}$, $\bar{D}^{0}\rightarrow K^{+}\pi^{-}$,
$\psi(2S)\rightarrow\mu^{+}\mu^{-}$, $\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$
and $\Upsilon(2S)\rightarrow\mu^{+}\mu^{-}$ decays. In each case the mass
distribution is modelled taking into account the effect of radiative
corrections, resolution and background, and the mean mass value extracted. To
allow comparison between the decay modes, the deviation of the measured mass
from the expected value [3] is converted into an estimate of the momentum
scale bias, referred to as $\alpha$. This is defined such that the measured
mass is equal to the expected value if all particle momenta are multiplied by
$1-\alpha$. Figure 2 shows the resulting values of $\alpha$. The deviation for
the considered modes is $\pm 0.02\%$, which is taken as the systematic
uncertainty on the momentum scale.
Figure 1: Reconstructed $J/\psi\rightarrow\mu^{+}\mu^{-}$ fitted mass as a
function of run number after the momentum calibration procedure discussed in
the text. The vertical dashed lines indicate the boundaries of the seven
calibration periods. A fit of a constant function (horizontal line) has a
$\chi^{2}$ probability of $6\%$. The shaded area corresponds to the assigned
uncertainty on the momentum scale of $0.02\%$. Figure 2: Momentum scale bias
$\alpha$, extracted from the reconstructed mass of various two-body decays
after the momentum calibration procedure described in the text. By
construction one expects $\alpha=0$ for the $J/\psi\rightarrow\mu^{+}\mu^{-}$
calibration mode. The black error bars represent the statistical uncertainty
whilst the (yellow) shaded areas include contributions to the systematic error
from the fitting procedure, the effect of QED radiative corrections and the
uncertainty quoted by the PDG [3] on the mass of the decaying meson. The (red)
dashed lines correspond to the assigned uncertainty on the momentum scale of
$0.02\%$.
## 2 Event selection
A common strategy, aiming at high signal purity, is adopted for the
reconstruction and selection of $B^{+}\rightarrow J/\psi K^{+}$,
$B^{0}\\!\rightarrow{J/\psi}{K^{*0}}$, $B^{0}\rightarrow J/\psi K^{0}_{\rm
S}$, $B_{s}^{0}\rightarrow J/\psi\phi$ and $\Lambda^{0}_{b}\rightarrow
J/\psi\Lambda$ candidates (the inclusion of charge-conjugated modes is implied
throughout). In general, only tracks traversing the whole spectrometer are
used; however, since $K^{0}_{\rm S}$ and $\Lambda$ particles may decay outside
of the VELO, pairs of tracks without VELO hits are also used to build
$K^{0}_{\rm S}$ and $\Lambda$ candidates. The $\chi^{2}$ per number of degrees
of freedom ($\chi^{2}/{\rm ndf}$) of the track fit is required to be smaller
than four. The Kullback-Leibler (KL) distance [11, *kl2, *kl3] is used to
identify pairs of reconstructed tracks that are very likely to arise from hits
created by the same charged particle: if two reconstructed tracks have a
symmetrized KL divergence less than 5000, only that with the higher fit
quality is considered.
$J/\psi\rightarrow\mu^{+}\mu^{-}$ candidates are formed from pairs of
oppositely-charged muons with a transverse momentum ($p_{\rm T}$) larger than
$0.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, originating from a common vertex
with $\chi^{2}/{\rm ndf}<11$, and satisfying $|M_{\mu\mu}-M_{J/\psi}|<3\sigma$
where $M_{\mu\mu}$ is the reconstructed dimuon mass, $M_{J/\psi}$ is the known
$J/\psi$ mass world average value [3], and $\sigma$ is the estimated event-by-
event uncertainty on $M_{\mu\mu}$. The selected $J/\psi$ candidates are then
combined with one of $K^{+}$, $K^{*0}\rightarrow K^{+}\pi^{-}$,
$\phi\rightarrow K^{+}K^{-}$, $K^{0}_{\rm S}\rightarrow\pi^{+}\pi^{-}$ or
$\Lambda\rightarrow p\pi^{-}$ to create $b$-hadron candidates. Mass windows of
$\pm 70{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, $\pm
12{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, $\pm
12{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ($\pm
21{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) and $\pm
6{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ($\pm
6{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) around the world averages [3] are
used to select the $K^{*0}$, $\phi$, $K_{\rm S}^{0}$ and $\Lambda$ candidates
formed from tracks with (without) VELO hits, respectively. Kaons are selected
by cutting on the difference between the log-likelihoods of the kaon and pion
hypotheses provided by the RICH detectors ($\Delta\ln{\cal L}_{K-\pi}>0$). To
eliminate background from $B_{s}^{0}\rightarrow J/\psi\phi$ in the
$B^{0}\rightarrow J/\psi K^{*0}$ channel, the pion from the $K^{*0}$ candidate
is required to be inconsistent with the kaon hypothesis ($\Delta\ln{\cal
L}_{K-\pi}<0$). To further improve the signal purity, a requirement of $p_{\rm
T}>1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ is applied on the particle
associated with the $J/\psi$ candidate. For final states including a $V^{0}$
($K_{\rm S}^{0}$ or $\Lambda$), an additional requirement of $L/\sigma_{L}>5$
is made, where $L$ is the distance between the $b$-hadron and the $V^{0}$
decay vertex, and $\sigma_{L}$ is the uncertainty on this quantity.
Each $b$-hadron candidate is associated with the reconstructed $pp$ primary
interaction vertex with respect to which it has the smallest impact parameter
significance, and this significance is required to be less than five. As there
is a large combinatorial background due to particles originating directly from
the $pp$ primary vertex, only $b$-hadron candidates with a reconstructed decay
time greater than 0.3 ps are considered for subsequent analysis. A decay chain
fit [14] is performed for each candidate, which constrains the reconstructed
$J/\psi$ mass and, if applicable, the reconstructed $K_{\rm S}^{0}$ or
$\Lambda$ mass to their nominal values [3]. The $\chi^{2}/{\rm ndf}$ of the
fit is required to be smaller than five. The mass of the $b$-hadron candidate
is obtained from this fit and its estimated uncertainty is required to be
smaller than 20${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
## 3 Results
The $b$-hadron masses are determined by performing unbinned maximum likelihood
fits to the invariant mass distributions, in which the signal and background
components are described by a Gaussian and an exponential function,
respectively. Alternative models for both the signal and background components
are considered as part of the systematic studies. Figure 3 shows the invariant
mass distributions and fits for the five modes considered in this study. The
signal yields, mass values and resolutions resulting from the fits are given
in Table 1.
Figure 3: Invariant mass distributions for (a) $B^{+}\rightarrow J/\psi K^{+}$, (b) $B^{0}\rightarrow J/\psi K^{*0}$, (c) $B^{0}\rightarrow J/\psi K^{0}_{\rm S}$, (d) $\Lambda^{0}_{b}\rightarrow J/\psi\Lambda$, and (e) $B_{s}^{0}\rightarrow J/\psi\phi$ candidates. In each case the result of the fit described in the text is superimposed (solid line) together with the background component (dotted line). Table 1: Signal yields, mass values and mass resolutions obtained from the fits shown in Fig. 3 together with the values corrected for the effect of QED radiative corrections as described in the text. The quoted uncertainties are statistical. | | Fitted mass | Corrected mass | Resolution
---|---|---|---|---
Decay mode | Yield | [$\\!{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$] | [$\\!{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$] | [$\\!{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$]
$B^{+}\rightarrow J/\psi K^{+}$ | $11151\pm 115$ | $5279.24\pm 0.11$ | $5279.38\pm 0.11$ | $10.5\pm 0.1$
$B^{0}\rightarrow J/\psi K^{*0}$ | $\,~{}3308\pm\,~{}65$ | $5279.47\pm 0.17$ | $5279.58\pm 0.17$ | $\,~{}7.7\pm 0.2$
$B^{0}\rightarrow J/\psi K^{0}_{\rm S}$ | $\,~{}1184\pm\,~{}38$ | $5279.58\pm 0.29$ | $5279.58\pm 0.29$ | $\,~{}8.6\pm 0.3$
$B_{s}^{0}\rightarrow J/\psi\phi$ | $\,~{}\,~{}816\pm\,~{}30$ | $5366.90\pm 0.28$ | $5366.90\pm 0.28$ | $\,~{}7.0\pm 0.3$
$\Lambda^{0}_{b}\rightarrow J/\psi\Lambda$ | $\,~{}\,~{}279\pm\,~{}19$ | $5619.19\pm 0.70$ | $5619.19\pm 0.70$ | $\,~{}9.0\pm 0.6$
The presence of biases due to neglecting QED radiative corrections in the mass
fits is studied using a simulation based on Photos [8]. The fitted masses
quoted in Table 1 for the $B^{+}\rightarrow J/\psi K^{+}$ and
$B^{0}\rightarrow J/\psi K^{*0}$ are found to be underestimated by $0.14\pm
0.01{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $0.11\pm
0.01{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively, when radiative
corrections are ignored; they are therefore corrected for these biases, and
the uncertainty is propagated as a systematic effect. The bias for the
$B^{0}_{s}\rightarrow J/\psi\phi$ mode is negligible due to the restricted
phase space for the kaons from the $\phi$ decay. There is no bias for the
$B^{0}\rightarrow J/\psi K^{0}_{\rm S}$ and $\Lambda^{0}_{b}\rightarrow
J/\psi\Lambda$ modes since the $J/\psi$, $K^{0}_{\rm S}$ and $\Lambda$ masses
are constrained in the vertex fits.
## 4 Systematic studies and checks
To evaluate the systematic error, the complete analysis is repeated (including
the track fit and the momentum scale calibration when needed), varying within
their uncertainties the parameters to which the mass determination is
sensitive. The observed changes in the central values of the fitted masses
relative to the nominal results are then assigned as systematic uncertainties.
The dominant source of uncertainty is the limited knowledge of the momentum
scale. The mass fits are repeated with the momentum scale varied by $\pm
0.02\%$. After the calibration procedure a $\pm 0.07\%$ variation of the
momentum scale remains as a function of the particle pseudorapidity $\eta$. To
first order the effect of this averages out in the mass determination. The
residual impact of this variation is evaluated by parameterizing the momentum
scale as a function of $\eta$ and repeating the analysis. The amount of
material traversed in the tracking system by a particle is known to 10%
accuracy [15]; the magnitude of the energy loss correction in the
reconstruction is therefore varied by 10%. To ensure the detector alignment is
well understood a further test is carried out: the horizontal and vertical
slopes of the tracks close to the interaction region, which are determined by
measurements in the VELO, are changed by 1$\times 10^{-3}$, corresponding to
the precision with which the length scale along the beam axis is known [16].
Other uncertainties arise from the fit modelling: a double Gaussian function
(with common mean) for the signal resolution and/or a flat background
component are used instead of the nominal Gaussian and exponential functions.
The effect of possible reflections due to particle mis-identification is small
and can be neglected. Finally, a systematic uncertainty related to the
evaluation of the effect of the radiative corrections is assigned. Tables 3
and 3 summarize the systematic uncertainties assigned on the measured masses
and mass differences.
Table 2: Systematic uncertainties (in $\rm\,MeV\\!/\\!{\it c}^{2}$) on the
mass measurements.
Source of uncertainty | $B^{+}\rightarrow$ | $B^{0}\rightarrow$ | $B^{0}\rightarrow$ | $B_{s}^{0}\rightarrow$ | $\Lambda^{0}_{b}\rightarrow$
---|---|---|---|---|---
| $J/\psi K^{+}$ | $J/\psi K^{*0}$ | $J/\psi K^{0}_{\rm S}$ | $J/\psi\phi$ | $J/\psi\Lambda$
Mass fitting: | | | | |
– Background model | 0.04 | 0.03 | $<$0.01 | 0.01 | $<$0.01
– Resolution model | 0.01 | 0.02 | 0.06 | 0.02 | 0.07
– Radiative corrections | 0.01 | 0.01 | – | – | –
Momentum calibration: | | | | |
– Average momentum scale | 0.30 | 0.27 | 0.30 | 0.22 | 0.27
– $\eta$ dependence of momentum scale | 0.04 | $<$0.01 | 0.09 | 0.03 | 0.02
Detector description: | | | | |
– Energy loss correction | 0.10 | $<$0.01 | 0.05 | 0.03 | 0.09
Detector alignment: | | | | |
– Vertex detector (track slopes) | 0.05 | 0.04 | 0.04 | 0.03 | 0.04
Quadratic sum | 0.33 | 0.27 | 0.33 | 0.23 | 0.30
Table 3: Systematic uncertainties (in $\rm\,MeV\\!/\\!{\it c}^{2}$) on the
differences of mass measurements, expressed with respect to the
$B^{+}\rightarrow J/\psi K^{+}$ mass (e.g. the last column gives the
systematic uncertainties on $M(\Lambda_{b}^{0}\rightarrow
J/\psi\Lambda)-M(B^{+}\rightarrow J/\psi K^{+})$).
Source of uncertainty | | $B^{0}\rightarrow$ | $B^{0}\rightarrow$ | $B_{s}^{0}\rightarrow$ | $\Lambda^{0}_{b}\rightarrow$
---|---|---|---|---|---
| | $J/\psi K^{*0}$ | $J/\psi K^{0}_{\rm S}$ | $J/\psi\phi$ | $J/\psi\Lambda$
Mass fitting: | | | | |
– Background model | | 0.05 | 0.04 | 0.04 | 0.04
– Resolution model | | 0.02 | 0.06 | 0.02 | 0.07
– Radiative corrections | | $<$0.01 | 0.01 | 0.01 | 0.01
Momentum calibration: | | | | |
– Average momentum scale | | 0.03 | $<$0.01 | 0.08 | 0.03
– $\eta$ dependence of momentum scale | | 0.04 | 0.05 | 0.01 | 0.02
Detector description: | | | | |
– Energy loss correction | | 0.10 | 0.05 | 0.07 | 0.01
Detector alignment: | | | | |
– Vertex detector (track slopes) | | 0.01 | 0.01 | 0.02 | 0.01
Quadratic sum | | 0.12 | 0.10 | 0.12 | 0.09
The stability of the measured $b$-hadron masses is studied by dividing the
data samples according to the polarity of the spectrometer magnet, final state
flavour (for modes where the final state is flavour specific), as well as
whether the $K^{0}_{\rm S}$ and $\Lambda$ daughter particles have VELO hits.
As a cross-check the analysis is repeated ignoring the hits from the tracking
station before the magnet. This leads to an average shift in measured masses
compatible with statistical fluctuations. In addition, for the $B^{+}$ and
$B^{0}$ modes where the event samples are sizable, the measurements are
repeated in bins of the $b$-hadron kinematic variables. None of these checks
reveals a systematic bias.
## 5 Conclusions
The $b$-hadron masses are measured using data collected in 2010 at a centre-
of-mass energy of $\sqrt{s}=7$$\mathrm{\,Te\kern-1.00006ptV}$. The results are
$M(B^{+}$ $\,\rightarrow\,$ | $J/\psi K^{+})$ $~{}=~{}$ | 5279.38 $\,\,\pm\,$ | 0.11 (stat) $\pm\,$ | 0.33 (syst) ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ | ,
---|---|---|---|---|---
$M(B^{0}$ $\,\rightarrow\,$ | $J/\psi K^{(*)0})$ $~{}=~{}$ | 5279.58 $\,\,\pm\,$ | 0.15 (stat) $\pm\,$ | 0.28 (syst) ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ | ,
$M(B_{s}^{0}$ $\,\rightarrow\,$ | $J/\psi\phi)$ $~{}=~{}$ | 5366.90 $\,\,\pm\,$ | 0.28 (stat) $\pm\,$ | 0.23 (syst) ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ | ,
$M(\Lambda^{0}_{b}$ $\,\rightarrow\,$ | $J/\psi\Lambda)$ $~{}=~{}$ | 5619.19 $\,\,\pm\,$ | 0.70 (stat) $\pm\,$ | 0.30 (syst) ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ | ,
where the $B^{0}$ result is obtained as a weighted average of
$M(B^{0}\rightarrow J/\psi K^{*0})=5279.58\pm 0.17\pm
0.27{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $M(B^{0}\rightarrow J/\psi
K^{0}_{\rm S})=5279.58\pm 0.29\pm
0.33{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ assuming all systematic
uncertainties to be correlated, except those related to the mass model. The
dominant systematic uncertainty is related to the knowledge of the average
momentum scale of the tracking system. It largely cancels in the mass
differences. We obtain
$M(B^{0}\rightarrow J/\psi K^{(*)0})$ $\,-M(B^{+}\rightarrow J/\psi K^{+})=\,\,$ | 0.20$\,\,\pm\,$ | 0.17 (stat) $\pm\,$ | 0.11 (syst) ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ | ,
---|---|---|---|---
$M(B^{0}_{s}\rightarrow J/\psi\phi)$ $\,-M(B^{+}\rightarrow J/\psi K^{+})=\,\,$ | 87.52$\,\,\pm\,$ | 0.30 (stat) $\pm\,$ | 0.12 (syst) ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ | ,
$M(\Lambda^{0}_{b}\rightarrow J/\psi\Lambda)$ $\,-M(B^{+}\rightarrow J/\psi K^{+})=\,\,$ | 339.81$\,\,\pm\,$ | 0.71 (stat) $\pm\,$ | 0.09 (syst) ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ | ,
where the $B^{0}$ result is a combination of $M(B^{0}\rightarrow J/\psi
K^{*0})-M(B^{+}\rightarrow J/\psi K^{+})=0.20\pm 0.20\pm
0.12{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $M(B^{0}\rightarrow J/\psi
K^{0}_{\rm S})-M(B^{+}\rightarrow J/\psi K^{+})=0.20\pm 0.31\pm
0.10{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ under the same hypothesis as
above.
As shown in Table 4, our measurements are in agreement with previous
measurements [3, 4]. Besides the difference between the $B^{+}$ and $B^{0}$
masses they are the most accurate to date, with significantly improved
precision over previous measurements in the case of the $B^{0}_{s}$ and
$\Lambda^{0}_{b}$ masses.
Table 4: LHCb measurements, compared to both the best previous measurements and the results of a global fit to available $b$-hadron mass data [3]. The quoted errors include statistical and systematic uncertainties. All values are in ${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$. | LHCb | Best previous |
---|---|---|---
Quantity | measurement | measurement | PDG fit
$M(B^{+})$ | $5279.38\pm 0.35$ | $5279.10\pm 0.55$ [4] | $5279.17\pm 0.29$
$M(B^{0})$ | $5279.58\pm 0.32$ | $5279.63\pm 0.62$ [4] | $5279.50\pm 0.30$
$M(B_{s}^{0})$ | $5366.90\pm 0.36$ | $5366.01\pm 0.80$ [4] | $5366.3~{}\,\pm 0.6~{}\,$
$M(\Lambda^{0}_{b})$ | $5619.19\pm 0.76$ | $5619.7~{}\,\pm 1.7~{}\,$ [4] | –
$M(B^{0})-M(B^{+})$ | $~{}~{}~{}0.20\pm 0.20$ | $~{}~{}~{}~{}~{}0.33\pm 0.06$ [17] | $~{}~{}~{}~{}0.33\pm 0.06$
$M(B^{0}_{s})-M(B^{+})$ | $87.52\pm 0.32$ | – | –
$M(\Lambda^{0}_{b})-M(B^{+})$ | $339.81\pm 0.72$ | – | –
## Acknowledgements
We would like to thank our colleague Adlène Hicheur who made, as a member of
our collaboration, significant contributions to the tracking alignment
algorithms and provided the first realistic version of the magnetic field map.
He is currently unable to continue his work, and we hope that this situation
will be resolved soon. 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
| 2011-12-21T00:40:06 |
2024-09-04T02:49:25.565181
|
{
"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 G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis, J.\n Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, D.\n S. Bailey, 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, 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, F.\n Constantin, A. Contu, A. Cook, M. Coombes, G. Corti, 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, F. De Lorenzi, J. M. De Miranda, L. De Paula, P. De Simone, D. Decamp,\n M. 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, L. Est\\`eve, A. Falabella, E.\n Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, 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, N. Gauvin, M.\n Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, G. Haefeli, C.\n Haen, S. 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, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, T.\n Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles,\n R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E. Lopez\n Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I.\n V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R. M. D. Mamunur,\n G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, 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, R. Messi, S. Miglioranzi, D. A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Nedos, M.\n Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. 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Szumlak, S.\n T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg,\n V. Tisserand, M. Tobin, S. Topp-Joergensen, N. Torr, E. Tournefier, M. T.\n Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, P. Urquijo,\n U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S.\n Vecchi, J. J. Velthuis, M. Veltri, B. Viaud, I. Videau, X. Vilasis-Cardona,\n J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, H. Voss, 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. Zavertyaev, F.\n Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, E. Zverev, A.\n Zvyagin",
"submitter": "Olivier Schneider",
"url": "https://arxiv.org/abs/1112.4896"
}
|
1112.4907
|
Luisiana X. Cundin is a contractor for Naval Medical Research Unit - San
Antonio (NAMRU-SA), contracted through Conceptual Mindworks, Inc.
# An electromagnetic model for biological tissue
Luisiana X. Cundin
###### Abstract
This essay is a recapitulation of an earlier Kramers-Krönig analysis of
biological tissue, published in 2010 [1]. The intent is to both complement and
bolster the antecedent analysis by furnishing supplemental clarification on
the electromagnetic model employed and provide further technical details.
Biological tissue is modeled a dielectric embedded in a conductor, which
necessitates using both a quotient and product subspace to form a complete
topological cover space. Discerning a suitable quotient enables decisive
separation of tissue’s dielectric behavior from excess conductivity. The
residual dielectric behavior revealed conforms with expectations based upon
electromagnetic theory and those commonly held for dielectric materials. An
appreciation for conflicting experimental absorption measurements spanning the
optical spectrum, reported for biological skin, engenders conceptualizing a
damage dependent attenuation coefficient. Lastly, descriptive codes are
provided for numerical algorithms implemented in the original analysis [1].
## 1 Introduction
Over one hundred years of empirical research has supported high permittivity
measurements for biological tissue, and this would _prima facie_ not be a
source of contention, if it were not for biological tissue being classified a
dielectric. Dielectric materials are polarizable insulators and, by
definition, insulators are poor conductors; yet, experimentation has proven
biological tissue a fairly good conductor, of course, tissue is still a poor
conductor relative to a metal. Pure liquid water is a classic dielectric
material and known for its large dielectric constant; yet, experiment supports
a dielectric constant for biological tissue some five orders of magnitude
greater than that of pure liquid water. Augmented polarizability for tissue is
associated with electric double layers surrounding cellular membranes.
Electrolytic solutions, such as saline solutions, admit a depressed
polarizability relative to pure liquid water and is due to shielding by ions;
albeit, the same solvated ions increase the conductivity of the media,
attributable to electric double layers formed by solvated ionic cages.
What is meant by ’tissue’ is in truth many different types of tissue found
within a biological body. The specific permittivity measured for each
respective tissue type can vary, due to variations in chemical composition,
density, salinity, &c. The intrinsic properties of tissue types can vary
across the taxonomic hierarchy for biology, including lower subclasses. In
addition, the resultant dielectric behavior can be further affected by such
variables as age, disease, environment and so on. With so many dependent
variables defining ’tissue’, its intrinsic properties, its _resultant_
dielectric behavior; it seems any unified treatment of tissue would be
prohibited.
A biological body has a fractal nature, starting with the cellular structure
at the mesoscopic scale, the cellular structure subdivides space into repeated
isolated enclaves translated throughout the entire biological body; further
still, the body is broken into several distinct organs and this lends to many
particular tissue types. There are several different cell types, each based
upon the intended function for that cell; furthermore, this is throughout the
whole tissue field. Organs are a collection of cells with the same or similar
functionalities, e.g. liver. A biological cell differentiates space and is
delineated by the plasma membrane, which separates the entire space into two
disjoint subspaces, namely, the intra and extra-cellular subspaces.
The primary implication, of present interest concerning the plasma membrane,
despite having multiple functionalities, is the creation and maintenance of an
electric potential across the membrane. Biological activity is responsible for
the maintenance of the membrane potential, which is typically maintained
somewhere between -50 to -170 millivolts; specific potentials depend upon the
function of the cell and many other factors. Maintenance requires the
continual ejection of positive ions from the intra-cellular region, for
smaller ions, like sodium ions, may slip through the membrane with relative
ease. The electric potential across the membrane is negative, indicating the
direction of the potential is pointed inward towards the center of the cell.
Positive ions, such as sodium ions, congregate near the outer surface of the
plasma membrane. Negative ions, such as chloride ions, congregate around the
cell, somewhere in the vicinity to balance charge. The separation of charged
ions around the outer portion of the cellular membrane creates a thin
conductive layer referred to as either the ’Stern layer’, ’electric double
layer’ or, as simply, the ’double layer’ [2].
An electric double layer is an interfacial phenomenon occurring in fluids and
consists of like charged particles collecting on the surface of some host
particle; while, opposingly charged particles congregate in the vicinity to
maintain a balance of electric charge. There is a net charge that is null for
the entire solution, but locally, there are sustained electric potentials, as
in the case of cellular membranes. The layer of oppositely charged particles
is referred to as the diffuse layer; this layer is comprised of charged
species freely moving in and around the vicinity of the host particle, yet the
net effect is to maintain a proper charge balance. This phenomenon occurs in
many different fluids, such as air, liquids and even molten metals.
If it were not for the cellular structure, susceptible augmentation of tissue
would not be expected and experiment bears this out. The dielectric constant
for tissue fluid, that is fluid absent any cellular structure, generally shows
a maximum dielectric constant in the neighborhood of two orders of magnitude,
roughly that of pure liquid water; conversely, tissue containing cells, such
as liver or gallbladder, exhibit extremely large dielectric constants,
typically somewhere in the neighborhood of six to eight orders of magnitude.
The increase in polarizability of biological tissue is directly attributable
to the double layer, where charged particles are free to migrate to antipodal
points on the surface of a cell. Given the typical diameter of a cell, _circa_
ten microns, it is not surprising to see such large dielectric constants.
Besides contributing to the creation of an electric double layer surrounding
the cellular membrane, solvated ions also disassociate in solution and thereby
give up a weakly held valence electron, called a solvated electron. There are
many descriptions for the electric properties of electrolytic solutions, but
the electric double layer model can be used with great effect. The ion’s
potential preferentially orients water molecules to surround the ion in a
quasi-lattice structure, referred to as a solvated cage; the solvated cage
structure itself constitutes an electric double layer. The advent of solvated
cages throughout the biological fluid provides a quasi-periodic structure
allowing the propagation of valence electrons throughout the media. As a
consequence, electrolytic solutions admit higher conductivity than pure liquid
water, e.g. saline water and biological tissue. The induced conductive
property is proportional to the ionic concentration and it is known that at
saturation the conductive properties saturate; thus, at some point, there is
no increase in conduction, because there is no longer an increase in solvated
electrons, i.e. there is no further dissolution of salt material.
Electric double layers account for excess conductivity in biological tissue,
but how does excess conductivity relate to embedded dielectric materials?
Given that dielectrics are poor conductors, should bulk conductivity values
measured for tissue be ascribed to all embedded dielectric materials? If
dielectric materials are poor conductors, then shouldn’t excess conductivity
exhibited by tissue be considered a separate phenomenon all together?
Considering all embedded dielectric materials as just that, embedded materials
within a fluid conductor, leads to the postulate that tissue’s dielectric
property is a topological quotient subspace. In this model, dielectric
materials are considered to be a closed set, that is, some particle with
finite dimensions and physical extent; hence, the outer boundary of such
particles come into contact with the surrounding media, which in the case of
tissue is a conductive media. Based upon electromagnetic boundary conditions
along an interface between two separate mediums, an equivalence class is
defined to maintain continuity of all fields across interfaces, known
collectively as _field continuity conditions_. In the case of tissue, the
_field continuity conditions_ impose a quotient subspace topology with regard
to excess conductivity and a product subspace with regard to all embedded
dielectric materials. Once a suitable quotient is realized, the excess
conductivity associated with electric double layers can be parsed out,
rendering the covered dielectric behavior exposed. The residual dielectric
behavior aligns with common expectations for dielectric materials; moreover,
predicted influences from such processes as electrophoresis and ionic
shielding become clearly evident.
The combined or _effective_ intrinsic property of a heterogeneous material is
often modeled using Maxwell’s ’mixing rules’, where such rules are subordinate
to a wider theory called the _effective medium approximation_. The theory
states the combined or overall intrinsic property of a heterogeneous material
is equal to a weighted sum of all the constituents making up that compound.
This is akin to a _resultant_ property; wherefore, the resultant vector in a
vector field is dominated by the largest magnitude and direction of all the
vectors in that field. Considering the chemical makeup of tissue, made
primarily of liquid water, it is certainly reasonable to expect the dielectric
behavior of tissue to mimic that of pure liquid water. Admittedly, there
should be deviations from pure liquid water; but, it should equally be
possible to account for all such deviations through common understanding of
electromagnetic phenomena.
After applying a suitable quotient transforming experimental conductivity
values for biological tissue, expected absorption characteristics for tissue
are clearly visible in the residual absorption curve. Such characteristics as
the expected contraction of the thermal resonance peak, the raised absorption
peak due to electrophoresis and the overall resemblance to pure liquid water
are all displayed nicely enough. The expected absorption characteristics of
tissue has been discussed in the original analysis, but the transformation of
experimental radio frequency measurements requires further clarification.
Expectations are based upon electrolytes in solution, which should increase
the absorption for slowly varying fields, increasing monotonically to peak
around the resonant frequency associated with electrophoresis. Another
consequence of electrolytes in solution is the shielding of water molecules
from external fields; thereby, forming ’bound’ water. Bound water refers to
the phenomenon whereby water molecules are impeded in their free rotation and
leads to contraction of the thermal resonance peak, that is, dilation of the
relaxation time. A reduction in absorption is expected for those frequencies
far removed from the resonant frequency; yet, in the neighborhood of the
resonance frequency, rotational excitement of water molecules is expected to
occur in spite of ionic shielding.
The discussion above covers mostly the lower frequency range, in the case of
the optical frequency range, specifically, the near-infrared, visible and
near-ultraviolet spectrum, an overall increase in absorption of tissue is
justly expected, for common experience proves tissue is not transparent at
these frequencies, unlike pure liquid water. An increase in absorption is not
what is at contention, contention exists over exactly how much of an increase
is expected; worse yet, experimentation has shown two separate magnitudes,
generally, for tissue’s electromagnetic attenuation. Even though there are
many differences in experimental techniques used to measure the optical
property of tissue, the one characteristic common to each separate group of
measurements is that the group admitting a lower absorption is, generally
speaking, derived from _in vivo_ samples, while the other group is derived
from _ex vivo_ sample preparation. As a consequence, it is postulated that the
differences in absorption measurements are attributable to tissue denaturation
and coagulation, that is, the difference is between living and dead tissue.
## 2 Theoretical models
In conjunction with Maxwell’s electromagnetic field equations, appropriate
boundary conditions must be imposed at all interfaces to realize a complete
solution. Each constitutive relation relates a fundamental field to its
corresponding perturbation field, e.g. the displacement field $\vec{\mbox{D}}$
is related to the electric field $\vec{\mbox{E}}$ through the permittivity
$\hat{\epsilon}$ thus $\vec{\mbox{D}}=\hat{\epsilon}\,\vec{\mbox{E}}$. At an
interface possessing charge density, the _field continuity conditions_ impose
certain constraints on the fundamental, as well as, perturbation fields. At
all interfaces, each adjoining region in a domain must be glued along the
boundaries, which results in a quotient mapping for each interface, sending
each boundary point to its equivalence class [3]. An example of the boundary
principle is a finite dielectric body surrounded by a vacuum, to attain the
effective dielectric constant, one must divide the permittivity of the
dielectric material by the vacuum permittivity thus
$\hat{\epsilon}_{\mbox{\scriptsize{\it{eff}}}}=\hat{\epsilon}/\epsilon_{\mbox{\scriptsize{0}}}$
[4, 5].
Biological tissue experiences a conductivity augmented from pure liquid water,
attributable to both electrolytes in solution and the double layer surrounding
each biological cell; as a consequence, it is postulated that all dielectric
materials in tissue are embedded in a conductive media. This means we must
adjoin to the outer boundary of all dielectric regions the inner boundary of a
circumscribing conductor.
###### Postulate (dielectric embedded in a conductor)
Because of excess conductivity exhibited by biological tissue, all embedded
dielectric materials must be surrounded by a conductive media; as a
consequence, each constitutive relation for an embedded dielectric (d) is
divided by that of a conductor (c).
$\sigma^{\prime}_{\mbox{\scriptsize{eff}}}=\sigma^{\prime}_{\mbox{\scriptsize{d}}}/\sigma^{\prime}_{\mbox{\scriptsize{c}}}$
(1)
Where the ratio equals the effective (eff) constitutive relation, in this
case, conductivity.
The process of equating fields along an interface is a quotient mapping that
sends each boundary point to its equivalence class. For example, the quotient
mapping, $\hat{\epsilon}:\vec{\mbox{D}}\mapsto\vec{\mbox{D}}/\sim$, represents
a partition of $\vec{\mbox{D}}$ and together with the quotient topology
determined by the constitutive relation is called the _quotient_ space of the
perturbation field. This process is repeated for all other fields, such as the
induction $\vec{\mbox{B}}$, magnetic $\vec{\mbox{M}}$ and electric
$\vec{\mbox{E}}$ fields; moreover, the field’s orientation, either tangential
or normal to the surface, must also be considered.
In the case of dielectric materials comprising tissue, they do not contain one
another; as a result, the _effective_ constitutive relation would form a
product space. Each material would form a basis $\mathscr{B}$ for the
resulting topology $\mathscr{X}$, where the union of all constitutive
relations form an effective mapping [3]. In keeping with the _effective medium
approximation_ , a weighted sum is applied to each mapping in proportion to
each material’s concentration.
###### Postulate (product space for dielectrics)
A finite set of dielectrics form a basis $\mathscr{B}$ for the topology
$\mathscr{X}$ known as biological tissue. The union of all such mappings
(constitutive relations) form a product topology for tissue, where each
mapping ($\hat{\epsilon}_{i}$) is weighted ($a_{i}$) proportionally to each
constituent’s aliquot part.
$\bigcup_{i}a_{i}\hat{\epsilon}_{i}\subseteq\hat{\epsilon}_{\mbox{\scriptsize{{d}}}}$
(2)
Where each mapping
$\hat{\epsilon}_{i}:\vec{\emph{\mbox{E}}}_{i}\mapsto\vec{\emph{\mbox{D}}}_{i}$
forms a subset basis for the topology $\mathscr{X}$, the union of all weights
($a_{i}$) is unity and the representative constitutive relation chosen is
permittivity.
Modification of the basis set $\mathscr{B}$ amounts to a shift in the partial
molar concentration of participating constituents; the resulting topology
would change proportionately. All tissue types are spanned by suitable
variation of all basis sets comprising biological tissue. Additionally,
variation in the intersection of all possible basis sets, that is, introducing
or removing particular chemical constituents, would equally change the
resultant topology. Postulate 2 is an equivalent statement of the _effective
medium approximation_ theory.
With respect to conductivity, it is a topological property stemming not from a
product space, but a quotient space; thus, resolving the dielectric behavior
from excess conductivity attributable to electrolytes and electric double
layers would require multiplication by a quotient. Once the quotient is
accomplished, the revealed dielectric behavior for biological tissue could
theoretically be deconstructed by aliquot subtraction of each chemical
constituent comprising the mixture. It is at this point of the analysis that
one may apply the concept of an _effective medium approximation_ and predict
the dielectric constant for nearly static fields, adequately explain observed
deviations of the absorptive behavior from pure liquid water and, finally,
provide some insight in to how electromagnetic fields might interact with
biological tissue.
In addition to contraction of the thermal resonance peak, another absorption
peak is expected to be raised due to an increase in absorption from
electrophoresis, which is the transportation of ionic species through a
liquid. Consider a static field, we can imagine the nucleus moving through the
medium under the force of an applied field, similarly, the electron cloud
would move opposingly to the direction of the nucleus. This electronic
displacement sets up a local electric potential, where the attraction between
the electron cloud and nucleus retards the movement of the ion through the
medium. Because of these forces, ion mobility is limited in speed and
therefore represents a loss of energy to the medium. Consider now an
oscillating field, we can imagine now the nucleus and electron cloud shifting
relative positions proportionate to the applied field; furthermore, at some
frequency, both will oscillate so rapidly as to vibrate the nucleus around
some central point in space. Assume the displacement of the nucleus is on the
order of a nucleic width, then the applied field creates a resonant frequency
associated with electrophoresis phenomenon.
Given a set of ionic species, there will be a distribution of corresponding
ionic mobilities, with an average velocity of $10^{-8}$ meters per second [6].
The resonant frequency $f$ can be calculated by assuming the oscillations to
occur in one-dimension, with velocity $v$ and width $d$; then, there will be
$N$ normal modes, where $N$ equal to unity corresponds to the fundamental
frequency, see equation (3). Given that the typical width of an ionic nucleus
is on the order of femtometers, a fundamental frequency can be calculated as
roughly 10 megahertz.
$f=\frac{Nv}{2d},N=\\{1,2,3,\ldots\\}$ (3)
For an externally applied field, as the frequency of oscillation increases,
absorptive loss associated with electrophoresis is expected to monotonically
rise and peak around 10 megahertz. Given that a distribution of ionic species
are present in biological tissue, with corresponding distributions in ionic
mobilities, velocities and nucleic widths, the absorption curve associated
with electrophoresis should be broad and smooth, for there are in addition, a
range of normal modes to be considered as well.
Lastly, damaged tissue is characterized by unfurled proteins and this is
expected to increase the opacity of tissue. Experiment has shown in the near-
infrared, visible and near-ultraviolet spectrum that tissue exhibits either
very low absorption of electromagnetic energy, if the sample tested is _in
vivo_ ; in contrast, if _ex vivo_ sample preparation is used, much higher
absorption is generally measured. It is postulated that the range measured for
tissue’s electromagnetic attenuation is caused by damage related to
temperature, for temperature directly affects the degree of coagulation,
denaturation or unfolding of proteins within tissue, thus increasing the
opacity of biological tissue.
###### Postulate (damage dependent absorption coefficient)
There is a mapping $\mu_{a}$ that maps tissue attenuation and is a function of
temperature ($T$), laser power density ($P_{d}$) and possibly other
parameters, including time ($t$). The mapping sends the absorption ($k_{1}$)
for undamaged tissue continuously to the absorption ($k_{2}$) of damaged
tissue.
$\mu_{a}\left(\mbox{T},\mbox{P}_{\mbox{\scriptsize{d}}},\ldots;\mbox{t}\right):k_{1}\mapsto
k_{2}$ (4)
## 3 Technical notes
The intent of the present essay is not to relive all the details relayed in
the original analysis, but to complement and bolster the antecedent analysis
[1]. Three main topics are to be covered in what follows: the implications of
Postulate 2 are explored for radio frequency conductivity data, then a brief
discussion of optical frequency data shows that the penetration depth is a
function of time, finally, four numerical algorithms are briefly discussed.
For the sake of brevity, formulas already extant in the original paper will be
referenced throughout what follows.
One definitive source suffices for experimental conductivity data covering the
radio frequency range and it is to this set of conductivity data a suitable
quotient must be multiplied in order to parse out the covered dielectric
behavior of tissue [7]. All relevant constitutive relations are linked
together into one succinct formulae, the complex index of refraction
$\hat{\mbox{N}}$, see equation (2) found in the original analysis [1].
Using the definition of $\hat{\mbox{N}}$, equation (5) can be deduced with the
aid of Minkowski’s inequality, Theorem 2.1 in the original paper, and
approximates the absorption behavior in the lower frequency range [1].
Equation (5) states the absorption $k$ of electromagnetic energy is
approximately equal to the square root of the relative conductivity
$\sigma_{r}^{\prime}$, divided by the angular frequency $\omega$, multiplied
by the algebraic term ($4\pi+1$). It is by this formulae absorption is often
related to conductivity in the lower frequency range.
$k\approx\sqrt{\frac{\sigma_{r}^{\prime}}{\omega}(4\pi+1)}$ (5)
Now, the entire biological body is a finite body and is eventually surrounded
by air; therefore, the conductivity for the conductor
$\sigma_{\mbox{\scriptsize{{c}}}}^{\prime}$, found in Postulate 2, is relative
to a vacuum. For the conductor contains all dielectrics in tissue, but air
contains the conductor. Starting with the relative conductivity term in
equation (5), it is possible to trace this term through and finally relate it
directly to the experimental conductivity
$\sigma^{\prime}_{\mbox{\scriptsize{exp}}}$ measured for tissue, _viz_. :
$\frac{\sigma^{\prime}_{r}}{\omega}\equiv\frac{1}{\epsilon_{\mbox{\scriptsize{0}}}\omega}\sigma^{\prime}\equiv\frac{1}{\epsilon_{\mbox{\scriptsize{0}}}\omega}\sigma^{\prime}_{\mbox{\scriptsize{{eff}}}}\equiv\frac{1}{\epsilon_{\mbox{\scriptsize{0}}}\omega}\frac{\sigma_{\mbox{\scriptsize{{d}}}}^{\prime}}{\sigma_{\mbox{\scriptsize{{c}}}}^{\prime}/(\epsilon_{\mbox{\scriptsize{0}}}\omega)}\equiv\frac{\sigma_{\mbox{\scriptsize{{d}}}}^{\prime}}{\sigma_{\mbox{\scriptsize{{c}}}}^{\prime}}\equiv\sigma^{\prime}_{\mbox{\scriptsize{exp}}}$
(6)
Thus, the experimental conductivity
$\sigma^{\prime}_{\mbox{\scriptsize{exp}}}$ values reported for biological
tissue is, by Postulate 2, equated as the ratio of the conductivity for a
dielectric over that of a conductor. Since the dielectric behavior of
biological tissue is what is of interest, a quotient is required to remove the
conductivity ($\sigma_{\mbox{\scriptsize{{c}}}}^{\prime}$) term found in the
denominator; furthermore, this term represents the conductivity of the inner
edge of a conductor that is intended to be glued to the outer edge of a
dielectric material.
The conductivity attributed to any metal is truly confined to a thin region
skirting the outer edge of the metal and is known as the skin depth. As for
inside a metal, the minimum conductivity possible is equal to the _vacuum
displacement current_ ; because, even in a vacuum, a current exists, also, the
inner conductivity of a metal would rise from the floor value proportional to
how poor a conductor the metal should be.
If the minimum conductivity is assumed along the inner edge for all electric
double layers, then the _vacuum displacement current_ becomes the suitable
quotient that must be applied to experimental conductivity measurements. Thus,
multiplying the experimental conductivity
$\sigma^{\prime}_{\mbox{\scriptsize{exp}}}$ values reported for biological
tissue by the _vacuum displacement current_ , then substitution of this
product into equation (5) yields the following:
$k\approx\sqrt{\epsilon_{\mbox{\scriptsize{0}}}\omega\sigma^{\prime}_{\mbox{\scriptsize{exp}}}(4\pi+1)}$
(7)
The approximation for the dielectric absorption for biological tissue,
represented by equation (7), was used to convert experimental conductivity
measurements for several tissue types, so the same in the original analysis.
The result of converting experimental conductivity measurements for several
tissue types are displayed in figure (1). After applying an appropriate
quotient, the dielectric behavior of biological tissue is parsed out. The
residual dielectric behavior aligns with theory and expectation, for the
absorption curves shown in figure (1) clearly show the effect of
electrophoresis, the contraction of the thermal resonance peak, also, all
tissue types conform to common expectations for dielectric materials.
Capitalizing on the transformation of experimental conductivity values for
biological tissue, one may immediately notice that the approximate dielectric
behavior of tissue mimics that of pure liquid water; yet, tissue does diverge
from pure liquid water for very important reasons. The absorption exhibited by
tissue is raised above that of pure liquid water, ranging from static fields
to megahertz frequencies. Absorptive additions from proteins and the like are
responsible for the increase in the extremely low frequency range; conversely,
the increase within the megahertz range is primarily due to electrophoresis.
Another reason absorption is raised and differentiated across tissue types is
the presence of other dielectric materials, such as proteins, amino acids, &c.
The absorption rises monotonically to peak around 10 megahertz, which was
predicted. Despite the range of ionic concentrations for different tissue
types, the absorption behavior for a range of tissue types is seen to converge
onto one another as the resonance frequency for electrophoresis is approached.
Figure 1: Dielectric behavior of several tissue types after removing media
conductivity.
Another important feature clearly visible in figure (1) is contraction of the
absorption peak around the thermal maximum, where the thermal maximum
represents energy loss due to rotational modes for bound water. Bound water is
impeded in its free rotation by ionic electric fields; thus, shielding forces
a reduction in absorption. The relaxation time for pure liquid water has been
calculated to be roughly 5 picoseconds; in the case of tissue, it has been
calculated to be around 17 picoseconds. The relaxation time for tissue has
been dilated, indicating the presence of bound water. The relaxation time is
approximated by measuring the Full Width Half Maximum (FWHM) for the
absorption peak of interest. Because the relaxation time is dilated, a
contraction of the thermal resonance peak is predicted by the Similarity
Theorem 2.3, which is stated in the original analysis.
A variable penetration depth for tissue has been recorded in the optical
frequency range [8, 9, 10]. In the original analysis, an argument for
measuring differing penetration depths was given, where the blame for the
differences was attributed to experimental technique and numerical model
employed to extract relevant dielectric properties. In retrospection, an
obvious difference between the two main experimental techniques is the fact
that one group measures tissue properties _in vivo_ , while the other group
uses _ex vivo_ samples. The denaturation and coagulation of excised tissue is
inevitable and leads immediately to the assertion that differing penetration
depths are dependent upon tissue health, where Postulate 2 embodies this
assertion. As a consequence, the penetration depth is a function of external
source parameters, such as power density; also, it is a function of material
response to rising temperatures, which too is a function of time. Typical
mappings of attenuation are seen on the order of $\mu_{a}:10^{-6}\mapsto
10^{-4}$. One contradiction still stands, Simpson _et. al._ employed _ex vivo_
samples, yet the calibration method he devised appears to correct for this
fact; see original analysis for details.
All three algorithms employed in the original analysis for interpolation,
extrapolation and the combination thereto are given in Appendix 5. The
combination of Neville’s interpolation method with Richardson’s extrapolation
method is used to form a more accurate approximation from a set of discrete
experimental data points. The only reason interpolation is entertained is that
a regular set of data points is required for the entire interval transformed
by the Kramers-Krönig relation. If multiple sets existed spanning all
frequency bands, then arithmetic averaging would avail; but, for the most
part, duplicate data sets do not exist nor is it acceptable to average data
obtained from different experimental techniques, sample preparations and
numerical methods designed to extract optical properties.
The Kramers-Krönig transform is in actuality just Hilbert’s transform and the
most efficient means by which to emulate the Hilbert transform numerically
over discrete sets is through Discrete Fourier transforms. The original
analysis described in great detail the implementation of the Kramers-Krönig
relation transforming a discrete absorption set to yield a theoretical index
of refraction set, but if a more in-depth discussion is desired concerning the
Hilbert transform, see the paper, ”Stieltjes Integral Theorem & The Hilbert
Transform”, published in 2011 [11].
## 4 Discussion
It was never the intent of either this essay nor the antecedent analysis to
undermine reported conductivity for biological tissue, these values were
determined by experiment; rather, the intent is to complement experimental
measurements, by placing proper emphasis upon material behavior as supported
by electromagnetic theory. Proper emphasis enables decoupling excess
conductivity attributable to electrolytes and electric double layers from
embedded dielectric materials present within biological tissue; moreover, the
residual dielectric behavior revealed after decoupling conforms well with
expectations based upon electromagnetic theory.
Biological tissue is made primarily of liquid water, thus it seems natural to
assume there to be a similarity between the two materials with respect to
exhibited absorption characteristics. This assumption is based upon the well
founded _effective medium approximation_ , of which, Maxwell’s ’mixing rules’
is a subordinate sub-theory. Heterogeneous mixtures are not expected to form
properties completely unrelated to the materials that make it up; rather, it
is elementary logic to expect the composite, effective or resultant properties
to be a weighted sum of all the constituents present in the mixture.
Heterogeneous materials pose a difficult problem when attempting to
reconstruct and predict electromagnetic behavior from axiomatic principles.
Maxwell’s ’mixing rules’ can often provide some means of adequate
representation of the dielectric behavior for a mixture of materials;
although, the mixing must be, in some sense, disjoint, as a result, one may
consider the separate materials as being distinct but properly mixed [12]. A
wider theory containing the concept of the ’mixing rules’ is a theory called
’effective medium approximation’, which states the _effective_ property of a
heterogeneous compound is equal to the ratio of each material’s intrinsic
property comprising the compound [13].
This form of analysis has been applied in the optical frequency range, for at
these frequencies, the excess conductivity caused by electric double layers is
vanishingly small. Thus, it is common to see both experimenters and
theoreticians attempt to deconstruct the resultant absorption properties of
tissue in the optical frequency range. Such chemicals as hemoglobin,
deoxyhemoglobin and melanin are all used to attempt a deconstruction of
tissue’s exhibited absorption characteristics. Attempting to apply an
_effective medium approximation_ to the lower frequency range fails dismally,
for in this frequency range, the dielectric properties of tissue are
confounded by excess conductivity associated with electrolytes in solution and
electric double layers surrounding biological cells. As can be seen, in the
case of biological tissue, not only a mixing of materials do occur, but, in
addition, the property of conduction pervades the entire media.
Regardless of experimental results, biological tissue is classified as a
dielectric material; therefore, excessively high conductivity measured for
tissue proved initially nonplussed. Converting reported experimental
conductivity values by the usual relation [_sic_ , equation (5)] produced
results that were confusing, for the calculated absorption was found to
increase towards the origin. It is by theory and classification that the
absorption of a dielectric material should be vanishingly small for static
fields. Equally troubling are the excessively high permittivity values
indicating very large polarizability of tissue, which are some four to five
orders of magnitude higher than pure liquid water. Mind you, water is a
classic dielectric material, exhibiting very large polarization; it is
difficult to reconcile the excessively large increase in polarizability for
tissue simply because of the presence of electrolytes in solution. In fact,
the presence of electrolytes actually depresses the polarizability in the case
of saline solutions. Worse, if reported permittivity values are taken _prima
facie_ , it is very difficult to attribute excessively large polarizability to
the medium as a continuum. In other words, without attributing these large
values to the cellular structure, specifically, the mesoscopic geometry, it
becomes difficult to justify that the ionic media itself is causing such large
polarizations.
The original analysis performed found it necessary to transform radio
frequency conductivity values for tissue in order to form a set of absorption
values that conformed to electromagnetic theory and expectation. The cause for
a transformation was placed squarely upon the shoulders of Maxwell-Wagner
polarization phenomenon; furthermore, it was found necessary to add greater
detail and theoretical support to uphold that original contention. This essay
reaffirms the postulate that excess conductivity causes tissue to exhibit
excessively high polarization and conductivity measurements; moreover, greater
detail is given for why the excess conductivity hides the dielectric behavior
of tissue.
Conductivity in metals has been successfully described by ”electronic band
theory” and predicts a large band gap for insulators, hence, the reason for
poor conduction in dielectrics [14]. Electrolytic solutions do not provide
much geometric structure, hence, the resistivity would be predictably high
from electronic band theory; yet, solvated electrons would lower the band gap,
as evidenced by experiment, i.e. saline solutions. In the case of biological
tissue, there is a regular geometric lattice, comprised of the cellular
structure; therefore, the double layer surrounding each cellular membrane
would lend yet another source decreasing the _resultant_ band gap. Ultimately,
because of both electrolytic and double layer conduction, a conductive band
permeates throughout biological tissue.
As far as the dielectric behavior of tissue is concerned, it would be expected
to resemble pure liquid water based upon an _effective medium approximation_.
Modification of the resultant dielectric behavior should be predictable with
the introduction of each new chemical species, such as ions, proteins, amino
acids, &c. Modification of the electromagnetic absorption behavior of
biological tissue is shown to adhere to spectroscopic principles and
expectations, including contraction of the thermal resonance peak, increased
absorption in the optical spectrum and an additional absorption peak due to
electrophoresis [1]. Many of these features are not visible before a quotient
is applied to reported experimental conductivity values for biological tissue.
A spectroscopist would expect contraction of the absorption curve if told that
the relaxation time for the material had dilated; this expectation is based
upon Fourier theory, specifically, the Similarity Theorem 2.3 found in the
original analysis [1]. In the case of biological tissue, the introduction of
ions in solution would form what is referred to as ’bound water’, that is, the
electric potential associated with ions in solution will force water molecules
to bind or become bound, see Corollary 2.4 found in the original analysis [1].
This means molecules are shielded from external fields by local electric
fields emanating from ions in solution. Because of shielding from electrolytes
in solution, the dielectric relaxation time for tissue is expected to dilate
with respect to pure liquid water; consequently, contraction of the absorption
curve is equally expected, i.e. the Similarity theorem.
_Copyright Statement_
_I am a military service member (or employee of the U.S. Government). This
work was prepared as part of my official duties. Title 17 U.S.C. §105 provides
that ’Copyright protection under this title is not available for any work of
the United States Government.’ Title 17 U.S.C. §101 defines a U.S. Government
work as a work prepared by a military member or employee of the U.S.
Government as part of that person’s official duties._
## References
* [1] L. X. Cundin and W. P. Roach, “Kramers-Krönig analysis of biological skin,” ArXiv e-prints (2010).
* [2] Ángel V Delgado, Ed., Interfacial electrokinetics and electrophoresis, vol. 106 of Surfactant science series, Marcel Dekkar, Inc. (2002).
* [3] J. M. Lee, Introduction to Topological Manifolds, vol. 202 of Graduate Texts in Mathematics, Springer (2000).
* [4] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, vol. 8 of Course of Theoretical Physics, Elsevier Butterworth-Heinemann, 2 ed. (2006).
* [5] J. D. Jackson, Classical Electrodynamics, John Wiley & Sons Inc., 3 ed. (1998).
* [6] P. Atkins, Physical Chemistry, W.H. Freeman and Company, NY, 5 ed. (1994).
* [7] C. Gabriel, “Compilation of the dielectric properties of body tissues at RF and microwave frequencies,” Tech. Rep. Report N.AL/OE-TR-1996-0037, Occupational and Environmental Health Directorate, Brooks Air Force Base, TX (USA) (1996).
* [8] A. J. Welch, Optical-Thermal Response of Laser-Irradiated Tissue, Springer, 2 ed. (1995).
* [9] J.-P. Ritz, A. Roggan, C. Isbert, G. Müller, H. J. Buhr, and C.-T. Germer, “Optical properties of native and coagulated porcine liver tissue between 400 and 2400 nm,” Lasers in Surgery and Medicine 29(3), 205–212 (2001).
* [10] C.-T. Germer, A. Roggan, J. P. Ritz, C. Isbert, D. Albrecht, G. Müller, and H. J. Buhr, “Optical properties of native and coagulated human liver tissue and liver metastases in the near infrared range,” Lasers in Surgery and Medicine 23(4), 194–203 (1998).
* [11] L. X. Cundin and N. Barsalou, “Stieltjes Integral Theorem & The Hilbert Transform,” ArXiv e-prints (2011).
* [12] J. C. M. Garnett, “Colours in metal glasses and in metallic films,” Phil. Trans. R. Soc. Lond. 203, 385–420 (1904).
* [13] D. A. G. Bruggeman, “Berechnung verschiedener physikalischer konstanten von heterogenen substanzen,” Ann. Phys. 24, 636–679 (1935).
* [14] N. W. Ashcroft and N. D. Mermin, Solid State Physics, Harcourt, Inc. (1976).
* [15] Mathematica, Wolfram Research, Inc., Champaign, Illinois, version 5.2 ed. (2005).
## 5 appendix: Mathematica codes
Three separate codes, written in Mathematica 5.2 from Wolfram Research, are
listed below [15].
1\. Combined interpolation/extrapolation algorithm:
InterpolationExtrapolation[xlistoo_, resolution_, rounds_, richardsonorder_]
:=
Module[{newdata, res = IntegerPart[resolution], up = rounds, i, p1, p2, order
= richardsonorder, temp, x, xlisto},
xlisto = Union[N[xlistoo], SameTest $\rightarrow$ (First[#1] == First[#2] &)];
newdata = List[];
p1 = N[First[xlisto[[All, 1]]]];
p2 = N[Last[xlisto[[All, 1]]]];
baselist = Table[x, {x, p1, p2, (p2 - p1)/(res - 1)}];
For[i = 1, i $<$= up, i++,
newdata = Join[newdata, {Table[ NevilleS[xlisto, x, (i + 1)],
{x, p1, p2, (p2 - p1)/(res - 1)}]}]];
temp = First[Richardson[Apply[List, Reverse[newdata]], order]];
2\. Neville’s interpolation algorithm:
NevilleS[xlisto_, xint_, len_] := Catch[Module[{x = N[xint], q, data =
N[xlisto], max, max2, i}, Off[First::”first”]; Off[Last::”normal”];
max=n=2*len + 1; q = Table[0, {i, 1, n}, {j, 1, n + 1}]; {min2, max2}
={Min[Abs[(data[[#1 + 1, 1]] - data[[#1, 1]])]&/@Range[1,Length[data]-1]],
Max[Abs[(data[[#1 + 1, 1]] - data[[#1, 1]])]&/@Range[1,Length[data] - 1]]}; i
= 1; While[x $>$= data[[i, 1]] && i $<$ Length[data], num = i; i = i + 1];
If[IntervalMemberQ[Interval[{Min[data[[All, 1]]], Max[data[[All, 1]]]}], x] &&
NumberQ[num], If[IntervalMemberQ[Interval[{1, len}], num] $||$
IntervalMemberQ[Interval[{Length[data] - len, Length[data]}], num],
If[IntervalMemberQ[Interval[{1, len}], num], q[[All, 1]] = Take[data[[All,
1]], n]; q[[All, 2]] = Take[data[[All, 2]], n], q[[All, 1]] = Take[data[[All,
1]], -n]; q[[All, 2]] = Take[data[[All, 2]], -n]], q[[All, 1]] =
Take[data[[All, 1]], {IntegerPart[num] - len, IntegerPart[num] + len}];
q[[All, 2]] = Take[data[[All, 2]], IntegerPart[num] - len, IntegerPart[num] +
len]], Throw[”Error: out of bounds”]]; For[i = 2, i $<$= n, For[j = 1, j $<$=
i - 1, q[[i, j+2]] = ((x - q[[i - j, 1]])*q[[i, j + 1]] - (x - q[[i, 1]])*q[[i
- 1, j + 1]])/(q[[i, 1]]-q[[i-j,1]]); j++]; i++]; Last[Last[q]]]];
3\. Richardson Extrapolation algorithm:
Richardson[x__, order_] :=
Module[{p, temp = x}, For[i = 1, Length[x] $>$ i, i++,
temp = Map[((order^(i))*temp[[#1 + 1]] - temp[[#1]])/((order^(i)) -
1) &, Range[1, Length[temp] - 1]]]; temp];
|
arxiv-papers
| 2011-12-21T01:58:28 |
2024-09-04T02:49:25.574913
|
{
"license": "Public Domain",
"authors": "Luisiana X. Cundin",
"submitter": "Luisiana Cundin",
"url": "https://arxiv.org/abs/1112.4907"
}
|
1112.5118
|
# Continuous Coupling of Ultracold Atoms to an Ionic Plasma via Rydberg
Excitation
T.M. Weber tweber@physik.uni-kl.de T. Niederprüm T. Manthey P. Langer V.
Guarrera G. Barontini H. Ott Research Center OPTIMAS, Technische
Universität Kaiserslautern, 67663 Kaiserslautern, Germany
###### Abstract
We characterize the two-photon excitation of an ultracold gas of Rubidium
atoms to Rydberg states analysing the induced atomic losses from an optical
dipole trap. Extending the duration of the Rydberg excitation to several ms,
the ground state atoms are continuously coupled to the formed positively
charged plasma. In this regime we measure the $n$-dependence of the blockade
effect and we characterise the interaction of the excited states and the
ground state with the plasma. We also investigate the influence of the quasi-
electrostatic trapping potential on the system, confirming the validity of the
ponderomotive model for states with $20\leq n\leq 120$.
Several proposals have demonstrated that dressing ultracold atoms with highly
excited Rydberg states can be an extremely powerful tool to tune the
interactions among them 1 ; 2 ; 3 . In particular long-range interactions
would lead to novel quantum phases of matter as super-solid phases or dipolar
crystals, once the dressed atoms are loaded into optical lattices 4 ; 5 . With
respect to the regime of frozen Rydberg gases, where all the relevant physics
can take place within a few $\mu$s 6 ; 7 ; 8 , the onset of such exotic phases
would require a much longer timescale, during which the coherent excitation
must be preserved. A limitation in this case comes from the tendency of
Rydberg gases to spontaneously evolve into a plasma 9 ; 10 . Hence a complete
understanding of this process is fundamental in order to find the appropriate
strategies to preserve the dressed states for sufficiently long times. In this
paper we report on the characterisation of the dynamics originating from the
two-photon excitation of an optically trapped ultracold gas of neutral atoms
to Rydberg states. We implement excitation pulses of at least a few tens of
$\mu$s, i.e., sufficiently long to ensure that the Rydberg gas spontaneously
evolves into a plasma. We exploit the excitation itself to continuously couple
the neutral ultracold atoms to the plasma. We measure the strength and the
shape of the resulting resonance lines varying the final state $nl$, with
$20\leq n\leq 120$ and with $l=0,2$. Moreover, we study the dependence of the
atom-plasma dynamics on different excitation times and trapping potentials.
The comparison of our results with those obtained with a rate equation model
allows us to fully characterise the system and to highlight the role of the
plasma-induced blockade and of the trapping potential.
Our experimental sequence starts by cooling room-temperature vapours of 87Rb
atoms in a 2D-MOT. The atoms are subsequently transferred to the main chamber
by a resonant push beam, loading a 3D-MOT with a rate of $2\times
10^{7}\,$atoms/s, thus allowing to collect $10^{8}$ atoms after $5\,$s. A
single 15 W beam at 10.6$\,\mu$m, produced by a commercial cw CO2 laser (model
C55, Coherent Inc.), is focussed at the center of the 3D-MOT to a waist of
30$\,\mu$m, creating an optical dipole trap. After a dark-MOT phase of 70 ms,
$1.2\times 10^{6}$ atoms at 60$\,\mu$K are left in the trap. By ramping down
the power of the trapping beam to 180 mW in about 6 s, we drive forced
evaporation, ending up with $4\times 10^{4}$ atoms in a cigar shaped spinorial
Bose-Einstein condensate in the $\left|5S_{1/2},F=1\right\rangle$ ground
state. The final trapping frequencies are $\nu=(205,205,17)\,$Hz. Unless
explicitly stated we stop the evaporation at a power of 220 mW obtaining a
thermal cloud of $10^{5}$ atoms at 250 nK with a density of $\cong
10^{14}\,$cm-3.
Figure 1: (Color online) Typical resonance lineshape for a thermal cloud at
250 nK. The atoms are excited to the $27S_{1/2}$ state for 1 ms. The solid
line is a Gaussian fit to the data. The inset shows the two photon Rydberg
excitation scheme that we employ.
After the end of the evaporation we drive the transition from the
$\left|5S_{1/2},F=1\right\rangle$ state to the selected $n$ Rydberg state,
using the two-photon scheme depicted in the inset of Fig. 1. The infra-red
(IR) light at 780 nm is locked to the
$\left|5S_{1/2},F=2\right\rangle\rightarrow\left|5P_{3/2},F=3\right\rangle$
transition. In this way it is always red detuned by $\Delta\simeq 2\pi\times
6.8\,$GHz from the $|5S_{1/2},F=1\rangle\rightarrow|5P_{3/2}\rangle$
transition, thus reducing the single photon scattering rate and allowing for a
longer lifetime of the trapped atoms. The IR beam is collimated into the cold
cloud perpendicularly to the CO2 laser beam with a waist of 1 mm and a typical
power of $\simeq 15\,$mW. The blue light for the second excitation step is
generated by frequency doubling the light produced by a diode laser based MOPA
system using a LBO crystal in a bow-tie ring cavity. The master laser can be
locked in a range of about 20 nm exploiting a combined transfer cavity and
offset locking scheme, generating light from 479 nm to 488 nm. Once combined
with the IR light, it provides the possibility to excite Rydberg states from
$n=20$ to $n=150$. The actual frequency of the beam is determined with a
standard EIT spectroscopy technique in a glass cell, with an accuracy of $\pm$
5 MHz. The blue beam is sent on the atoms collinearly with the IR one and
focussed at the center of the atom cloud with a waist of $40\,\mu$m and a
typical power of $\simeq 120\,$mW. The applied intensities correspond to Rabi
frequencies of $\Omega_{1}=2\pi\times 145\,$MHz and $\Omega_{2}=2\pi\times
44\,$MHz for the first and second excitation steps to the $27S_{1/2}$ state,
which result in an effective overall Rabi frequency of
$\Omega=\Omega_{1}\Omega_{2}/2\Delta=2\pi\times 470\,$kHz. The blue and the IR
light are switched on together with variable pulse lengths, typically on the
order of 1 ms. The excitation of Rydberg atoms is revealed by the reduction of
the number of trapped atoms at the end of the two-photon pulse, when scanning
the frequency of the blue beam across the atomic transition. A typical line
shape is shown in Fig. 1.
Figure 2: (Color online) Measured resonance widths for different $ns$ states.
Data (squares) are compared with the solution of the rate equation model
explained in the text (circles). In the inset the same quantities are shown
for $nd$ states. Note, that for $n<27$ the power of the blue excitation laser
significantly drops.
In Fig. 2 we show the measured widths of the resonances as a function of the
principal quantum number $n$. Notably, they are up to six orders of magnitude
larger than the natural linewidths ($\simeq$ 80 kHz for $n=27$ and $\simeq$
300 Hz for $n=120$ 11 ). Moreover they show a non-trivial dependence on $n$.
These features cannot be attributed to Doppler or saturation effects since,
for the given experimental parameters, they are in the order of 100 kHz.
Furthermore it has been repeatedly demonstrated that the bulk excitation of
Rydberg atoms in the ultracold regime rapidly leads to a series of secondary
effects like fast ionization by collisions or radiation 9 ; 14 , production of
plasmas 21 ; 22 ; 23 or blockade effects 24 ; 33 . In our case, on the
timescale of the excitation this secondary effects are certainly present and a
complete understanding of the observed features necessarily requires to take
into account all of them. For these reasons we analyse our data starting from
the following rate equation model:
$\displaystyle\dot{N}_{g}$ $\displaystyle=$ $\displaystyle
W_{gr}\xi(N_{r}-N_{g})+W_{ge}(N_{e}-N_{g})$ $\displaystyle-$ $\displaystyle
K_{HM}N_{g}N_{r}+K_{PI}N_{e}N_{r}+\Gamma_{e}N_{e}-\gamma_{g}N_{g}$
$\displaystyle\dot{N}_{e}$ $\displaystyle=$ $\displaystyle
W_{ge}(N_{g}-N_{e})+W_{er}(N_{r}-N_{e})$ $\displaystyle-$ $\displaystyle
K_{PI}N_{e}N_{r}-\Gamma_{e}N_{e}$ $\displaystyle\dot{N}_{r}$ $\displaystyle=$
$\displaystyle W_{gr}\xi(N_{g}-N_{r})+W_{er}(N_{e}-N_{r})$ $\displaystyle-$
$\displaystyle K_{BB}N_{r}-K_{RR}N_{r}-K_{HM}N_{g}N_{r}$ $\displaystyle-$
$\displaystyle K_{PI}N_{e}N_{r}-\Gamma_{r}N_{r}$ $\displaystyle\dot{N}_{i}$
$\displaystyle=$ $\displaystyle
K_{BB}N_{r}+\frac{1}{2}K_{RR}N_{r}+\frac{1}{2}K_{HM}N_{g}N_{r}$ (1)
$\displaystyle+$ $\displaystyle K_{PI}N_{e}N_{r}-\chi N_{i},$
where $g$ labels the ground state atoms, $e$ the 5p state atoms, $r$ the
Rydberg atoms and $i$ the ions. With $\Gamma_{e}$ and $\Gamma_{r}$ we indicate
the natural linewidths of the corresponding levels. The loss rate
$\gamma_{g}$, which is due to the high scattering rate from the IR laser, is
measured from the decay of the atom number in dependence of the excitation
pulse time when the blue laser is far detuned from the transition
($\delta\simeq 100\,$MHz), as shown in Fig. 3a. The coefficients $W_{ij}$
represent the excitation rates and, in our experimental regime, they are
$\simeq 2\pi\times$300 Hz for the $g\rightarrow e$ transition while they range
between a few Hz to a few mHz for the transitions $e\rightarrow r$, with
$20<n<120$. We calculate the two-photon coupling between the ground state and
the Rydberg states using the standard formula from second-order perturbation
theory 26 :
$\displaystyle W_{gr}$ $\displaystyle=$
$\displaystyle\frac{8I_{1}I_{2}}{\hbar^{4}c^{2}{\varepsilon_{0}}^{2}}{\left|\sum_{k}\frac{\widehat{d_{3k}}\widehat{d_{kg}}}{\omega_{kg}-\omega_{1}}+\frac{\widehat{d_{ek}}\widehat{d_{kg}}}{\omega_{ek}-\omega_{2}}\right|}^{2}$
(2) $\displaystyle\frac{\Gamma_{r}/2}{(\Gamma_{r}/2)^{2}+\Delta\omega^{2}},$
where $k$ labels every intermediate state, $\widehat{d_{ji}}$ are the dipole
matrix elements, $I_{i}$ the laser intensities and $\Delta\omega$ the overall
laser linewidth. We calculate that $W_{gr}$ ranges between a few MHz ($n$=20)
and a few hundred Hz ($n$=120). All the remaining processes are related to the
production of ions or to the interactions with them. When exciting Rydberg
atoms ions can be produced in several ways: in our experimental circumstances
the main channels are blackbody radiation (BB), Hornbeck-Molnar ionisation
(HM), Penning ionisation (PI) and Rydberg-Rydberg collisions (RR). At 300 K
the rate $K_{BB}$ never exceeds 500 s-1 for all the states that we excite 16 .
The HM ionisation is due to collisions between Rydberg atoms and ground state
atoms that produce one Rb${}_{2}^{+}$ ion and one electron. More complicated
is the PI channel, in which one Rydberg atom collides with one atom in the 5p
state producing one ion, one electron and one atom in the ground state. For
atomic densities in the order of $10^{14}-10^{15}\,$cm-3, the corresponding
rates are $K_{HM}\simeq 10^{5}-10^{6}\,$s-1 and $K_{PI}\simeq
10^{6}-10^{7}\,$s-1 12 ; 13 . The last ionisation channel is characterized by
the well-known $n^{4}$ dependence of the cross-section through the rate
$K_{RR}=\rho_{r}v(\pi a_{0}^{2}n^{4})$, where $\rho_{r}$ is the density of the
Rydberg atoms and $v$ their relative velocity 17 . Every time one ion is
produced, the corresponding electron leaves the trapping region extremely fast
leaving an excess positive charge around the atoms. The positively charged
plasma that originates from the continuous production of ions is then
subjected to the so-called Coulomb explosion: the ions repel each other via
the electrostatic force. The complete description of such a complicated
process lies beyond the purposes of this work and we model the expansion of
the plasma with an effective ion loss rate from the trapping volume
$\chi=v/\sigma+\sqrt{2e^{2}\rho_{i}/(4\pi\varepsilon_{0}m)}/\sigma$, where
$v=\sqrt{3k_{B}T/m}$ is the thermal velocity which we suppose to be the atomic
one, $\rho_{i}$ the plasma density and $\sigma$ the radius of the atomic
distribution. Moreover, the plasma that forms and expands produces an electric
field in the region of the trapped atoms that significantly shifts the Rydberg
levels. The most striking consequence is the blockade effect, i.e., an
effective reduction of the coupling between the ground state and the Rydberg
state. We model this process introducing the coefficient
$\xi=\Delta\omega/(\Delta\omega+B)$ in eqs. (1). Indeed, the spacially varying
electric field produced by the space charge of the plasma induces a broadening
of the Rydberg line that is
$B=(2\sigma\rho_{i}/(4\pi\varepsilon_{0}))^{2}\alpha/(2\hbar)$, where $\alpha$
is the $n$-dependent atomic polarisability given by
$\alpha=h\times(2.202\times 10^{-7}n^{6}+5.53\times 10^{-9}n^{7})\,$Cm2V-1 32
. We find the linewidths calculated solving the rate equation model to be in
excellent agreement with the observed ones (Fig. 2). The simulated dynamics of
the number of trapped atoms, Rydberg atoms and ions are shown in Fig. 3b for
the $27s$ state. Through the competition between the Coulomb expansion and the
blockade effect the atom-plasma system rapidly evolves into a self-balanced
situation where the feeding rate is continuously adjusted in order to
compensate the losses of ions. In practice, for a given value of the principal
quantum number $n$, the number of Rydberg atoms that are on average present in
the volume is kept almost constant. Unfortunately this self-balancing effect
is strongly altered by the huge losses induced by the scattering of photons
from the IR laser (Fig. 3a) which can not be balanced. Due to this effect the
self-balanced phase has only a limited lifetime. From the solution of the rate
equation model we can determine the number of Rydberg atoms $N_{r}$ that are
present in the volume during this reduced lifetime for different $n$ states,
as reported in Fig. 3c. For the given trapping volume the maximum number of
Rydberg atoms excited corresponds to an average distance
$r=1/\sqrt[3]{\rho_{r}}$ that ranges from 1$\,\mu$m to 13$\,\mu$m for $27\leq
n\leq 120$. In Fig. 3d the $n$-dependence of the maximum number of ions
present at a time is shown.
Figure 3: (Color online) a) Typical decay measurements in (downpointing
triangles) and out of resonance (uppointing triangles) together with the
curves obtained as solutions of the rate equation system (2) for the $55s$
state. b) Time evolution of total number of trapped atoms (solid line),
Rydberg atoms (dashed) and ions (dashed-dotted) for the $27s$ state. c)
Calculated average number of Rydberg atoms present as a function of $n$. d)
Calculated maximum number of ions present as a function of $n$.
Finally we investigate the influence of the trapping potential on the Rydberg
atoms and on the plasma dynamics. The CO2 laser is expected to create a
repulsive ponderomotive potential for any Rydberg atom 15 while for the atoms
in the $5S_{1/2}$ and $5P_{3/2}$ states it creates the attractive trapping
potential. We first verify the reliability of the ponderomotive assumption for
the dipole potential, recording the atom losses for different trapping powers
and measuring the relative shifts of the center of the resonances, as reported
in Fig.4a. We observe that the AC Stark shift is effectively the same for
every Rydberg state and that it is compatible with the expected theoretical
value of 179 MHz/W, as shown in Fig.4b.
Figure 4: (Color online) a) Typical resonance lineshapes for different powers
of the CO2 laser for the $27S_{1/2}$ state b) Measured AC Stark shifts for
different values of $n$. The solid line is the expected state independent
theoretical value of 179 MHz/W. c) Resonance lineshapes for $n=27$ for a BEC
and a thermal cloud. d) Measurement of the resonance width as a function of
the CO2 power for $n=43$.
It has been demonstrated that the plasma dynamics is not directly affected by
the density of the ground state atoms since the interaction is mediated mainly
by the Rydberg atoms 9 . We have verified this, measuring the linewidths for a
BEC and a thermal cloud when the density differs by one order of magnitude
while the trapping potential remains almost the same. As can be seen in
Fig.4c, there is only a minimal difference between them. However, as reported
in Fig.4d and as shown in Fig.4a we do observe a dependence of the linewidth
on the power of the trapping laser. This effect is due to the fact that an
increase of the power of the CO2 laser compresses the atoms in the ground
state reducing the excitation volume. In a smaller volume the ionic blockade
effects are even stronger producing a more pronounced broadening and a further
detriment of the transition probability.
In summary, we have reported the two-photon excitation and subsequent fast
ionization of Rydberg atoms in a unprecedented wide range of $n$ states.
Elongating the excitation time to a few ms we have systematically investigated
the interplay between the ultracold atoms and the formed plasma. We have
carried out a detailed yet simple analysis of the system, highlighting the
important role of the plasma on hampering the Rydberg excitation. We have
finally characterized the influence of a quasi-electrostatic trap on Rydberg
atoms and plasma dynamics.
Our results have a direct impact on the schemes that aim at the dressing of
ground state atoms with Rydberg states. Indeed the possible evolution of
Rydberg excited samples into plasma must be taken into account, since it leads
to a huge loss in the transition probability. A possible way to avoid this
dynamics is the application of electric fields to remove the produced ions
fastly. However, in the presence of electric fields Rydberg states are
strongly mixed thus leading to population distribution over the neighbouring
states 31 , and hence to loss of coherence. This, together with our findings,
suggests that the best way to obtain long-living plasma-free Rydberg-dressed
samples would require the use of high-intensity lasers in order to reach the
condition where $\delta$ is larger than the plasma-induced broadening with a
decent amount of admixture.
###### Acknowledgements.
We acknowledge financial support by the DFG within the SFB/TRR 49 and GRK 792.
V. G. and G. B. are supported by Marie Curie Intra-European Fellowships. It is
a pleasure to thank P. Pillet. T. Pohl and H. Hotop for enlightening
discussions. We are grateful to A. Widera for technical support.
## References
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* (17) T. F. Gallagher, Rydberg Atoms (Cambridge University Press, 1994).
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|
arxiv-papers
| 2011-12-21T18:14:36 |
2024-09-04T02:49:25.587250
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "T. M. Weber, T. Niederpr\\\"um, T. Manthey, P. Langer, V. Guarrera, G.\n Barontini, and H. Ott",
"submitter": "Giovanni Barontini",
"url": "https://arxiv.org/abs/1112.5118"
}
|
1112.5218
|
# Patterns of neutral diversity under general models of selective sweeps
Graham Coop1 and Peter Ralph1
1 Department of Evolution and Ecology & Center for Population Biology,
University of California, Davis.
To whom correspondence should be addressed: gmcoop@ucdavis.edu
## Abstract
Two major sources of stochasticity in the dynamics of neutral alleles result
from resampling of finite populations (genetic drift) and the random genetic
background of nearby selected alleles on which the neutral alleles are found
(linked selection). There is now good evidence that linked selection plays an
important role in shaping polymorphism levels in a number of species. One of
the best investigated models of linked selection is the recurrent full sweep
model, in which newly arisen selected alleles fix rapidly. However, the bulk
of selected alleles that sweep into the population may not be destined for
rapid fixation. Here we develop a general model of recurrent selective sweeps
in a coalescent framework, one that generalizes the recurrent full sweep model
to the case where selected alleles do not sweep to fixation. We show that in a
large population, only the initial rapid increase of a selected allele affects
the genealogy at partially linked sites, which under fairly general
assumptions are unaffected by the subsequent fate of the selected allele. We
also apply the theory to a simple model to investigate the impact of recurrent
partial sweeps on levels of neutral diversity, and find that for a given
reduction in diversity, the impact of recurrent partial sweeps on the
frequency spectrum at neutral sites is determined primarily by the frequencies
achieved by the selected alleles. Consequently, recurrent sweeps of selected
alleles to low frequencies can have a profound effect on levels of diversity
but can leave the frequency spectrum relatively unperturbed. In fact, the
limiting coalescent model under a high rate of sweeps to low frequency is
identical to the standard neutral model. The general model of selective sweeps
we describe goes some way towards providing a more flexible framework to
describe genomic patterns of diversity than is currently available.
## 1 Introduction
The high levels of genetic variation within natural populations have long
fascinated population geneticists. One school of thought holds that a
substantial proportion of this molecular polymorphism is neutral or very
weakly deleterious (Kimura and Ohta, 1971; Ohta, 1973; Kimura, 1983). For
neutral polymorphism, the level of genetic diversity results from a balance
between the introduction of alleles through mutation and their stochastic loss
(Kimura and Crow, 1964; Kimura, 1969; Ewens, 1972). Under the neutral theory
of molecular evolution this stochasticity is thought to result mostly from
genetic drift (Kimura, 1983), the random resampling that occurs in finite
populations, an effect that is exaggerated by fluctuating population size and
large variation in reproductive success among individuals (see Charlesworth,
2009, for a recent review). However, selection at linked sites may provide a
major source of stochasticity as the dynamics of a neutral allele can be
strongly influenced by the random genetic background on which selected alleles
arise (Maynard Smith and Haigh, 1974; Kaplan et al., 1989; Charlesworth et
al., 1995; Hudson and Kaplan, 1995b).
In many species examined to date, levels of diversity are substantially lower
in regions of low recombination, as found in multiple species of _Drosophila_
(Aguade et al., 1989; Begun and Aquadro, 1992; Berry et al., 1991; Shapiro et
al., 2007; Begun et al., 2007), _Caenorhabditis_ (Cutter and Payseur, 2003;
Cutter and Choi, 2010), humans (Hellmann et al., 2008; Cai et al., 2009) and
_Saccharomyces cerevisiae_ (Cutter and Moses, 2011); but not in all species,
e.g. _Arabidopsis_ (Nordborg et al., 2005; Wright et al., 2006). Moreover,
levels of diversity are also lower in regions that a priori are expected to
have a higher rate of functional mutations, e.g. near genes and conserved
elements (McVicker et al., 2009; Cai et al., 2009; Hernandez et al., 2011).
Since the rate of neutral genetic drift is independent of recombination rate,
this positive correlation between recombination rates and diversity offers
good evidence that linked selection plays a substantial role in the fate of
alleles, especially in low recombination regions. What is still far from clear
is how different forms of linked selection contribute to this reduction, and
whether linked selection can explain the narrow observed range of genetic
diversity across species with vastly different (census) population sizes
(Lewontin, 1974; Maynard Smith and Haigh, 1974).
Models of the effect of linked selection have often been divided between those
that propose the source of this linked selection to be either the purging of
deleterious variation (background selection) or the selective sweep of
beneficial alleles (hitchhiking). In this paper we explore the consequences of
a generalized model of hitchhiking on patterns on neutral diversity. We first
review some of the key results of models of linked selection. Under the
background selection model, genetic diversity is continuously lost from
natural populations due to the removal of haplotypes that carry deleterious
alleles (Charlesworth et al., 1995; Hudson and Kaplan, 1995b). For strongly
deleterious alleles, this continuous loss acts primarily to increase the rate
of genetic drift at markers closely linked to loci with high deleterious
mutation rates (Hudson and Kaplan, 1995a; Nordborg et al., 1996). Therefore,
this background selection model leads to a reduction in genetic diversity but
no skew in the frequency spectrum. However, a skew towards rare neutral
alleles can result if weakly deleterious mutations are incorporated into the
model (Nordborg et al., 1996; Gordo et al., 2002).
On the other end of the spectrum, Maynard Smith and Haigh (1974) proposed that
local levels of genetic diversity could be reduced by the hitchhiking effect.
The hitchhiking effect results from the fact that when an initially rare,
beneficial allele sweeps rapidly to fixation it carries with it a linked
region of the haplotype on which it arose. The size of genomic region affected
by a recent sweep is proportional to the ratio of the strength of selection to
the rate of recombination (Maynard Smith and Haigh, 1974; Kaplan et al., 1989;
Stephan et al., 1992; Barton, 1998), and so the reduction in levels of
diversity is determined by the distribution of selection coefficients and the
rate of sweeps per unit of the genetic map. Neutral alleles further away from
the selected site may not be pulled all of the way to fixation if
recombination occurs during the sweep, which can lead to a transient excess of
high-frequency derived alleles an intermediate distance away from the selected
site after each sweep (Fay and Wu, 2000; Przeworski, 2002; Kim, 2006). As
neutral diversity levels slowly recover through an influx of new mutations
after the sweep there is a strong skew towards low frequency derived alleles,
a pattern that persists for many generations (Braverman et al., 1995;
Przeworski, 2002; Kim, 2006). In a large population, the rate of sweeps could
be high enough that hitchhiking dominates genetic drift as the source of
stochasticity (Maynard Smith and Haigh, 1974; Kaplan et al., 1989; Gillespie,
2000), an idea which has been termed genetic draft (Gillespie, 2000).
Support for a hitchhiking model over the standard model of background
selection is found in Drosophila, where there is a greater skew towards rare
alleles at putatively neutral sites in regions of low recombination
(Andolfatto and Przeworski, 2001; Shapiro et al., 2007) and regions
surrounding amino-acid substitutions have lower levels of diversity
(Andolfatto, 2007; Macpherson et al., 2007; Sattath et al., 2011). However, in
humans (and other species) there is no strong skew towards rare alleles in low
recombination regions (McVicker et al., 2009; Hernandez et al., 2011;
Lohmueller et al., 2011), which combined with other evidence (Coop et al.,
2009; Hernandez et al., 2011) suggests that full sweeps may have been rare,
and that background selection may be the main mode of linked selection, in
humans and a number of other species.
Although the recurrent full sweep model has been the subject of considerable
theoretical investigation, it may actually be relatively rare for advantageous
alleles to sweep rapidly all the way to fixation. Fluctuating environments
(e.g. Gillespie, 1991; Kopp and Hermisson, 2007, 2009a, 2009b) and changing
genetic backgrounds may often act to prevent alleles achieving rapid fixation
within the population (see Pritchard et al. (2010) for a recent discussion).
For example, if multiple mutations affecting the adaptive phenotype segregate
during the sweep then it may be that no one of these alleles sweeps to
fixation (Pennings and Hermisson, 2006a, b; Chevin and Hospital, 2008; Ralph
and Coop, 2010). Multiple alleles spreading rapidly from low frequency can
lead to either a set of partial sweeps within the population, or a soft sweep
if the alleles are tightly linked. Furthermore, a similar effect can occur
when selection acts on an allele present as standing variation, if the allele
is present on multiple haplotypes when it starts to spread (Innan and Kim,
2004; Hermisson and Pennings, 2005; Przeworski et al., 2005). The fact that,
under these models, no single haplotype goes quickly to fixation acts to
reduce the hitchhiking effect, and alters the effect on the frequency
spectrum.
The genome-wide effect of other modes of linked selection on patterns of
diversity is relatively unexplored. One model that has been investigated is an
infinitesimal model of directional selection, where the aggregated effect of
selection over many loci can be a substantial source of stochasticity at
linked and even unlinked sites (Robertson, 1961; Santiago and Caballero, 1995,
1998; Barton, 2000). Fluctuating selection due to varying environments has
also been shown to lead to reduced levels of diversity at linked neutral sites
(Gillespie, 1994, 1997; Barton, 2000) and simulations of specific models of
fluctuating selection have shown that the same reduction in diversity can
result in a much smaller skew in the frequency spectrum than under the
hitchhiking model (Gillespie, 1994, 1997). However, as yet no coalescent model
of the effect of recurrent incomplete sweeps has been developed.
Here is an outline of how we proceed. First, we develop a coalescent-based
model of patterns of diversity surrounding a selected allele that sweeps into
the population but not necessarily to fixation. We concentrate on the case of
a very large population and sites that are partially linked to this selected
locus. We find that if the initial rise of the selected allele is rapid then
the coalescent process is primarily affected by this stage, and relatively
insensitive to the subsequent dynamics of the selected allele. Using this
intuition, we then develop a coalescent model of recurrent sweeps on patterns
of neutral diversity in which selected alleles may only reach intermediate
frequency. To test the approximations involved in the model we compare the
results at several stages to simulations. Some of the implications of these
results for interpretation of genome-wide diversity patterns are presented in
the discussion.
## 2 Results
### 2.1 Coalescent framework and assumptions
As first described by Kaplan et al. (1988) and Hudson and Kaplan (1988),
patterns of neutral diversity at a neutral locus linked to a selected locus
can be modeled by conditioning on the trajectory of the frequency of the
selected allele through time, and treating the two allelic classes as
subpopulations within each of which the dynamics are neutral, with
recombination moving lineages between the two (see also Barton and Etheridge,
2004; Barton et al., 2004). Consider a locus under selection at which a
derived allele $D$ and an ancestral allele $A$ segregate, and let the
frequency of $D$ at time $t$ be denoted $X(t)$. We will study the coalescent
process at a neutral locus partially linked to our selected locus, with
recombination occurring at rate $r$ per generation between the selected and
the neutral locus. Each ancestor on a given lineage in the coalescent process
carried either the $D$ or the $A$ allele at the selected locus, which we refer
to as the “type” of that lineage.
Throughout we assume that the diploid population size $N$ is large and
constant over time. For simplicity, we assume that the effective population
size is $2N$, (i.e. the neutral coalescence rate of a pair of lineages is
$1/(2N)$) and that no more than two lineages coalesce at once in the absence
of a selective sweep.
Suppose at time $t$ that $k_{D}$ and $k_{A}$ of our lineages are of the
derived and ancestral type respectively. There are $NX(t)$ individuals
carrying the derived allele that could be progenitors of the $k_{D}$ lineages,
so the instantaneous rates of coalescence of pairs of lineages within the two
allelic classes at time $t$ are
${k_{D}\choose 2}\frac{1}{2NX(t)}\qquad\mbox{and}\qquad{k_{A}\choose
2}\frac{1}{2N(1-X(t))},\qquad\mbox{respectively.}$ (1)
The total instantaneous rate of recombination is $(k_{D}+k_{A})r$. If a
recombination event occurs on a lineage at time $t$, it chooses to be of type
$D$ with probability $X(t)$, and chooses to be of type $A$ otherwise.
We will leave the dynamics of the selective sweeps that determine $X(t)$
fairly unspecified, and while stochasticity may play an important role in
shaping the trajectories, in examples we usually treat $X(t)$ as nonrandom. As
we want coalescences caused by a single selective sweep to occur at more or
less the same time, we require that once the selected allele is introduced
into the population it increases in frequency rapidly, and that once the
allele frequency leaves the boundary (e.g. moves above 1%), it does not return
(e.g. drops below 1%) unless it does so on the way to loss (e.g. hits 0 before
returning to 1%). This condition implies that our model applies to alleles
that are at least partially codominant, as fully recessive alleles spend
appreciable time, behaving stochastically, at very low frequencies, which can
lead to different coalescent dynamics at linked loci (Teshima and Przeworski,
2006; Ewing et al., 2011).
### 2.2 Relation to previous models
We describe a simple approximation to the coalescent with recurrent sweeps
that is inspired by similar approximations for a model of recurrent full
sweeps. The approximation postulates two types of coalescent events –
“neutral” events occurring at rate $1/2N$ between any pair of lineages, and
additional coalescent events, involving two or more lineages, due to selective
sweeps. The first class of events can occur at any time, due to random
resampling of lineages. The second class of events, the sweep–induced
coalescent events, can involve more than two lineages, as we assume that
lineages forced to coalesce by a sweep do so instantaneously on the relevant
time scale. We assume that all such lineages coalesce into a single lineage,
and that the distribution of the number of such lineages is binomial, with a
success probability that is a function of the trajectory taken by the selected
allele and the recombination distance to that allele. This framework is a
natural extension of similar approximations used for full sweeps (Barton,
1998; Gillespie, 2000; Kim and Stephan, 2002; Nielsen et al., 2005; Durrett
and Schweinsberg, 2005).
Processes with two classes of coalescent events have previously been developed
to approximate a recurrent full-sweep model (Kaplan et al., 1989; Gillespie,
2000; Durrett and Schweinsberg, 2005). When the transition probabilities can
be written in this binomial form, as they also are in the recurrent full sweep
models of Gillespie (2000) and Durrett and Schweinsberg (2005), the model is
called a $\Lambda$-coalescent (Pitman, 1999; Sagitov, 1999). These also arise
in neutral models where individuals have large variance in reproductive
success (e.g. Sargsyan and Wakeley, 2008; Möhle and S. Sagitov, 2001). As in
other work, we present this model as an approximation not in the sense of
asymptotic convergence, but rather as a simplification, which we show later is
close enough to be useful. We make a number of simplifying assumptions, and
often do not make use of the most accurate analytical forms available, in an
effort to maintain an intuitive form and description of the process obtained.
In particular, Durrett and Schweinsberg (2004) showed that a coalescent
process with simultaneous multiple collisions could provide a better
approximation to the coalescent process during a sweep, a direction we do not
pursue (see also Barton, 1998; Etheridge et al., 2006).
### 2.3 An approximation to the coalescent process during the sweep
Figure 1A shows an example of the relationships between different sampled
individuals at a neutral locus in a finite population undergoing recurrent
selective sweeps. At the times indicated by the lightning bolts, selective
alleles sweep into the population at some locus linked to our neutral site.
All lineages descended from the original carrier of the derived allele
coalesce, nearly instantaneously on this time scale.
Figure 1B zooms in on one of these selective sweeps. The derived allele at the
selected locus ($D$) arose $\tau$ generations ago. The five surviving
ancestral lineages recombine on and off the $D$ background, whose frequency
through time is shown by the dark grey shading. Just after time 0 those
lineages on the $D$ background coalesce as $X$ goes to zero (their coalescent
rate, which is proportional to $1/X$, goes to infinity). We will show that the
complexity of the process shown in Figure 1B can be approximated by a much
simpler multiple merger coalescent process suggested by Figure 1A, in which
lineages coalesce “neutrally” at rate $1/(2N)$, and furthermore, each lineage
flips a coin at each selective sweep to decide which type they are, and those
that are of type $D$ merge simultaneously.
Figure 1: (A) An example of a multiple-merger coalescent genealogy. Eight
alleles have been sampled in the present day, and we trace their lineages
backwards through time, up the page. Lightning bolts indicate the times when a
selected allele has swept into the population. At each sweep, each lineage is
either descended from the original carrier of the derived allele at the
selected site (lineages marked with a black dot) or from some other ancestor
(lineages marked with a white dot). (B) Zooming in on one sweep. The frequency
of the derived allele, $D$, through time, $X(t)$, is shown in dark grey. The
four surviving lineages are shown in different colors as in (A). Horizontal
dotted lines depict recombination events in the history of a lineage. A dot
indicates the oldest recombination event experienced by each of our lineages
before the $D$ allele arose, and the color of the dot indicates where the
allele recombined onto the $D$ background (black) or on to the $A$ background
(white). As we approach the time the selected allele arose, the three lineages
found on the $D$ background coalesce into a single lineage.
Suppose that a derived allele at the selected locus ($D$) arose $\tau$
generations ago, at time $0$. The selected mutation may still segregate within
the population in the present day, or may have gone to fixation or loss
sometime before the present (in which case $X(\tau)=1$ or 0 respectively).
First consider coalescences occurring very close to the origin of a selective
mutation. A lineage can be type $D$ at time $0$ for one of two reasons: either
it was of type $D$ in the present day and not yet recombined off the $D$
background, or at the first recombination after the selected allele arose, the
lineage chose to be of type $D$. The lineage of an individual drawn at random
from the present-day population is therefore of type $D$ at time $0$ with
probability
$q=q(r,X):=X(\tau)e^{-r\tau}+r\int_{0}^{\tau}e^{-rt}X(t)dt.$ (2)
Here the integral is over $t$, the number of generations between the origin of
$D$ and the first subsequent recombination on a lineage ($t$ is marked for the
red lineage in Figure 1B). Note that although many recombination events may
have occurred, since at each recombination event the lineage chooses a new
type independently of its previous type, we need only consider the first after
the sweep. If $\tau$ is much larger than $1/r$ the first term can be ignored,
so we commonly assume that
$q(r,X)=r\int_{0}^{\infty}e^{-rt}X(t)dt,$ (3)
as the allelic state of the sample has long been forgotten. Importantly, we
can see that the dependence of $q$ on $X$ decays exponentially through time at
rate $r$. Therefore, the fate of the selected allele more than a few multiples
of $r$ after it arose, including its presence or absence in the present day,
will have little effect on $q$. Concretely, for two trajectories labeled 1 and
2, if $X_{1}(s)=X_{2}(s)$ for all $0\leq s\leq T$, then regardless of
subsequent differences in the trajectories, $|q_{1}-q_{2}|\leq e^{-rT}$.
We can now approximate the rapid coalescence of lineages that are forced by
the sweep by assuming that all lineages descended from the original carrier of
the $D$ allele coalesce simultaneously when the selected allele appears (a
“multiple merger”). The lineages will actually coalesce at slightly different
times, but the assumption the derived allele increases rapidly implies that
this difference is small on the coalescent time scale $(i.e.\ o(2N))$. As each
lineage takes part in this merger independently with probability $q$, the
probability that $i$ out of $k$ surviving lineages coalesce at time $0$ is
${k\choose i}q^{i}(1-q)^{k-i},\quad\mbox{for}\quad 2\leq i\leq k,$ (4)
reducing the number of lineages from $k$ to $k-i+1$.
This approximation assumes that each lineage makes an independent choice of
whether to recombine off the sweep, which is equivalent to assuming that the
coalescences caused by the sweep form a ‘star’-like tree, with no internal
edges of nonzero length. Therefore, the approximation ignores dependencies
between lineages induced by coalescent events earlier in the sweep, and so is
a poorer approximation for large number of lineages. More sophisticated
approximations have been developed to account for this dependency, which
improve the properties for large samples (Barton, 1998; Durrett and
Schweinsberg, 2004; Etheridge et al., 2006; Pfaffelhuber et al., 2006).
However, we believe this approximation captures many of the important
features.
The other component of our approximation is that at all time, all pairs of
lineages coalesce at rate $1/(2N)$ regardless of their allelic background.
This approximation ignores the fact that lineages that are currently on
different backgrounds cannot coalesce and that lineages on the same background
coalesce at a higher rate (see equation (1)).
We should also note that although large changes in the allele frequency over a
small number of generations represent a large number of children descended
from a smaller number of ancestors, this will not cause rapid coalescence in a
large population if the allele remains at intermediate frequencies.
Concretely, consider a short time interval from generation $t_{1}$ to
generation $t_{2}$, over which interval $X(t)\gg(t_{2}-t_{1})/N$. The chance
that any coalescence occurs during this time interval on the derived
background is small ($O((t_{2}-t_{1})/(X(t)N))$), regardless of how the
frequency $X$ changes. Therefore, large, sudden changes in allele frequencies
will only force coalescence on the derived background if $X(t)$ is of order
$1/N$ (and similarly for the ancestral background). For sites that are only
partially linked to the selected locus, if recombination is moving the
lineages across backgrounds at a sufficiently high rate compared to neutral
coalescent rate ($Nr\gg 1$), then two lineages in this subdivided model
coalesce at a rate close to $1/2N$ (see Hudson and Kaplan (1988); Hey (1991);
Nordborg (1997), and Barton and Etheridge (2004) for a detailed discussion).
As such our approximation will therefore be worse close to the selected site,
but is asymptotically correct for large $r$.
#### 2.3.1 A simple trajectory
To build intuition, we first consider a simple trajectory, making further
approximations to keep the results accessible, and compare the results to full
coalescent simulations. Assume that $D$ arises $\tau$ generations ago at a
site at distance $r$ from the neutral site under consideration, rapidly sweeps
to frequency $x$, and remains close to this frequency for a time much greater
than $1/r$. Under many models of directional selection, most of the time spent
in reaching $x$ is spent at low frequency, so that any recombination that
occurs during this time will likely move a lineage to the ancestral type, and
so only lineages that do not recombine during the initial sweep will coalesce.
If we let $t_{x}$ be the time it takes for the selected allele to sweep to $x$
and assume $r\tau\gg 1$, then a simple approximation to $q(r,X)$ is therefore
(with the subscript emphasizing dependence on $x$)
$q(r,X)\approx q_{x}:=xe^{-rt_{x}}.$ (5)
If the initial increase of $D$ is driven by additive selection of strength $s$
with $Ns>1$, then the initial trajectory of $D$ will be logistic, and it is
reasonable to take $t_{x}=\log\big{(}\alpha x/(1-x)\big{)}/s$, where $\alpha$
is $2N$ or $4Ns$ depending on whether $s$ is of order $1$ or $1/N$ the latter
case corresponding to the case where the selected allele has to rapidly
achieve frequency $1/(Ns)$ to escape loss by drift). Using $q_{x}$ to
approximate the probability that a lineage is caught by the sweep, the
expected pairwise coalescent time is smaller by a factor of
$(1-q_{x}^{2}e^{-\tau/(2N)})$ (6)
which can be found by considering whether a pair of lineages coalesce before,
during, or after the sweep.
If rather than remaining near $x$, the selected allele continues to sweep to
fixation – perhaps it is still under selection with strength $s_{2}\gg r$ –
then $q_{x}\approx e^{-rt_{x}}$ because the selected allele has gone quickly
to fixation as in a full sweep, and the only time for recombination is in the
early phase of the trajectory $t_{x}$. On the other hand, if the allele became
strongly deleterious ($-s_{2}\gg r$), then $q\approx 0$, because there is
little chance of it contributing genetic material to the population. However,
if selection subsequently experienced by $D$ is weak ($|s_{2}|\ll r$), so that
subsequent dynamics of the selected allele are sufficiently slow, then $q$ and
therefore the coalescent process are independent of the eventual fate of the
selected allele. In summary, for $q_{x}$ to be a good approximation to
$q(r,X)$ and for the sweep to have an appreciable effect on the coalescent, we
need $|s_{2}|\ll r<s$.
#### Comparison to simulation
To demonstrate this, we will apply the same approximation to situations with
different long-term behaviors. We consider five different possible trajectory
types. In all cases, the initial rise of $D$ was modeled as deterministic
logistic growth begun at frequency $1/2N$ and adjusted to reach frequency $x$
after $t_{x}$ units of time. In the first case (“balanced”), the allele
remains thereafter at frequency $x$. In the next two cases (Figures 2A–C),
after time $t_{x}$, allele $D$ approaches either frequency 1 (“fixed”) or
frequency 0 (“lost”) logistically, reaching frequency $1-1/2N$ (or $1/2N$
respectively) after the next $\tau$ time units. In the last two cases (Figures
2D–F), the allele $D$ remains at $x$ for $T$ generations, and then proceeds
logistically, in time $t_{x}$, either to frequency $1-1/2N$ (“step”) or
frequency $1/2N$ (“top-hat”).
In each case, we used mssel (a modified version of ms (Hudson, 2002) that
allows an arbitrary trajectory, kindly supplied by Richard Hudson) to simulate
genealogies for a recombining sequence surrounding a selected locus at which a
selected allele performs one of the trajectories shown in Figure 2 . The
average pairwise coalescence time from these simulations was calculated by
dividing the pairwise genetic diversity by the mutation rate, and is shown in
Figure 2 at different distances from the selected locus, compared to the
quantity predicted by equation (6). Close to the selected site (e.g. for
$r<1/T$ in Figure 2E and F) the curves diverge, since the sites represented by
the blue curves see a full sweep, reducing diversity close to the selected
site, while those in the orange curves see a short-term balanced polymorphism,
and hence show a peak in polymorphism near the selected site). As we increase
recombination distance away from the selected site, the three curves are in
good agreement with the black line (equation (6)), indicating that our partial
sweep model captures the main effect on diversity.
Figure 2: The effect of a single partial sweep. (A) Three possible
trajectories followed by the D allele after it arises $\tau$ generations ago,
described in the text: blue is “fixed”, green is “lost”, and orange is
“balanced”. (B) and (C) Mean pairwise coalescent time against recombination
distance away from a selected site that has experienced one of the three types
of sweeps shown in (A), with $x=0.4$ and $0.8$ respectively. The other
parameters were $t_{x}/2N=6.6\times 10^{-3}$ and $\tau/2N=0.05$. (D) Another 3
possible trajectories: green is “top–hat” and blue is “step”. (E) and (F)
Pairwise coalescent time as in (B) and (C), but using the trajectories shown
in (D). The other parameters were $t_{x}/2N=6.1\times 10^{-4}$, $\tau/2N=0.1$
and $T/2N=0.02$. The black line shows the approximation to the pairwise
coalescent time of equation (6). In E and F, the vertical line grey line marks
$r=1/T$.
Our simple approximation describes diversity levels well at partially linked
sites over a range of different scenarios, and works well for a wider range of
parameters (results not shown). We furthermore used equation (4) to predict
the effect of this simple partial sweep on the coalescent process of more than
two lineages, and found close agreement with further mssel simulations for
various summaries of diversity such as the expected number of segregating
sites (results not shown). Overall, these results confirm that for partially
linked sites, the coalescent process is mostly determined by the initial rapid
behavior of the selected allele.
### 2.4 A recurrent sweep coalescent model
We now consider patterns of diversity at a neutral locus affected by many
different selected alleles that sweep into the population at the times of a
homogeneous Poisson process with rate $\nu$. We assume that the sweep rate is
low enough that sweeps do not interfere with each other, and return to discuss
this assumption later. Each sweep occurs at some distance $r$ from the neutral
locus, and as it sweeps its frequency follows some particular trajectory
$X(t)$, which together in equation (3) determine $q$, the probability that a
lineage at the neutral site is caught by the sweep. Rather than try to
explicitly model randomness in these two components, at first we will assume
that each sweep independently chooses its value of $q$ from a probability
distribution with density $f(q)$. This model is exactly a Lambda coalescent,
with $\Lambda(dq)=q^{2}\nu f(q)dq+\delta_{0}(dq)/2N$ (see Berestycki, 2009,
for a recent review), but we leave our discussion in terms of $f$ to make the
results more intuitive.
Following from our assumption that each lineage is affected by a given sweep
independently with probability $q$, when there are $k$ surviving lineages, the
rate at which they coalesce to $k-i+1$ lineages due to sweeps is
$\nu{k\choose i}\int_{0}^{1}q^{i}(1-q)^{k-i}f(q)dq.$ (7)
This follows from our assumption that sweeps occur homogeneously through time
and do not interfere with each other, and properties of marked Poisson
processes. For ease of presentation we denote
$I_{k,i}={k\choose i}\int_{0}^{1}q^{i}(1-q)^{k-i}f(q)dq.$ (8)
Recall that under our model, the rate of coalescence of pairs of lineages due
to genetic drift is $1/(2N)$, so that the rate at which the coalescent process
with $k$ lineages coalesces to $k-i+1$ lineages is
$\lambda_{k,i}={k\choose 2}\frac{1}{2N}\delta_{i,2}+\nu
I_{k,i}\quad\mbox{for}\;2\leq i\leq k,$ (9)
where $\delta_{i,2}=1$ if $i=2$ and $0$ otherwise. The total rate of
coalescent events when there are $k$ lineages is therefore
$\lambda_{k}=\frac{1}{2N}{k\choose
2}+\nu\sum_{i=2}^{k}I_{k,i}\quad\mbox{for}\;k\geq 2,$ (10)
and conditional on a coalescent event the probability that $i$ lineages out of
$k$ coalesce, reducing from $k$ to $k-i+1$ lineages, is
$p_{k,k-i+1}=\frac{\lambda_{k,i}}{\lambda_{k}}=\frac{\frac{1}{2N}{k\choose
2}\delta_{i,2}+\nu I_{k,i}}{\frac{1}{2N}{k\choose
2}+\nu\sum_{i=2}^{k}I_{k,i}},\quad\mbox{for}\;2\leq i\leq k.$ (11)
To simulate from this coalescent process we can simulate an exponential
waiting time with rate $\lambda_{k}$, pick a number of lineages to coalesce
using probabilities $p_{k,k-i+1}$, and run this process until we have a single
lineage remaining.
Note that in deriving this process we have assumed that at all times, lineages
also coalesce at a neutral rate $1/2N$. This can be justified by assuming that
recombination moves lineages between backgrounds at a high enough rate to
allow the effects of the partitioning of the population by segregating alleles
to be ignored. Therefore, the approximation will break down if a typical
neutral site, at any given time, is close enough (e.g. within an $r$ of order
$1/N$) to an allele maintained at intermediate frequency by long-term
balancing selection (e.g. alleles maintained for time scales of order $N$).
Further work is needed to refine the coalescent under those conditions, but
our approximations should be suitable for a broad range of scenarios and
genomic regions.
### 2.5 The coalescent process with homogeneous sweeps
It is natural to examine the case in which selective sweeps occur at uniform
rate along a sequence of total length $L$. We assume that this sequence
recombines at rate $r_{BP}$ per base each generation, and that sweeps enter
the population at a rate $\nu_{BP}$ per base each generation, so that the
total rate of sweeps is $\nu=\nu_{BP}L$. We also assume that the sweeps are
homogeneous, i.e. the trajectory followed by the frequency of the derived
allele, $X$, is independent of the distance between our neutral site and the
site at which a sweep occurs.
We will consider sweeps occurring along a very long chromosome and so will
take $L\to\infty$, but then the total rate of sweeps, $\nu=\nu_{BP}L$, also
goes to infinity. To obtain a meaningful limit, we need that as $L\to\infty$
the rate of sweeps corresponding to each nonzero value of $q$ converges to a
finite value. Recall from (3) that the probability a lineage is caught up in a
given sweep depends on the distance to the sweep (which is $r_{BP}\ell$ for a
site $\ell$ bases away) and the trajectory $X$ taken by the sweep, and is
given by $q(r_{BP}\ell,X)=r_{BP}\ell\int_{0}^{\tau}\exp(-r_{BP}\ell t)X(t)dt$.
In a finite genome of length $L$, the probability distribution on values of
$q$ has density $f(q)=h_{L}(q)/L$, where
$h_{L}(q)=\int_{0}^{L}\mathbb{P}_{X}\\{q(r_{BP}\ell,X)\in dq\\}d\ell$. Here
$h_{L}(q)$ is the rate at which selective sweeps appear at location
$r_{BP}\ell$ and whose trajectory $X$ gives $q(r_{BP}\ell,X)=q$, integrated
across the genome; and $f(q)$ is $h_{L}(q)$ normalized to integrate to 1,
since $\int_{0}^{1}h_{L}(q)dq=L$. The functions $h_{L}$ converge for $q>0$ as
$L$ becomes large as long as the probability that distant sweeps affect the
focal site decays quickly enough. We therefore assume that $h_{L}(q)$
converges to a finite limit $h(q)$, i.e. that the following exists:
$h(q)=\lim_{L\to\infty}Lf(q)~{}~{}\quad\mbox{for}\;0<q\leq 1.$ (12)
This means that although the total rate of sweeps per generation is infinite,
only a finite number happen close enough to our neutral site to potentially
affect our coalescent process. Therefore, the rate at which $k$ lineages
coalesce down to $k-i+1$ due to sweeps converges:
$\nu_{BP}\,L\,I_{k,i}\rightarrow\nu_{BP}{k\choose
i}\int_{0}^{1}q^{i}(1-q)^{k-i}h(q)\,dq\quad~{}~{}\mbox{as}\;~{}~{}L\to\infty.$
(13)
If we take the trajectory $X$ to be fixed, we can rewrite equation (13) as
$\displaystyle\nu_{BP}{k\choose i}\int_{0}^{1}q^{i}(1-q)^{k-i}h(q)dq$
$\displaystyle=\nu_{BP}{k\choose
i}\int_{0}^{\infty}q(r_{BP}\ell,X)^{i}(1-q(r_{BP}\ell,X))^{k-i}d\ell$
$\displaystyle=\frac{\nu_{BP}}{r_{BP}}{k\choose
i}\int_{0}^{\infty}q(r,X)^{i}(1-q(r,X))^{k-i}dr,$ (14)
which decouples the dependency of the rate of sweeps on the recombination rate
$r_{BP}$ from the trajectory $X$. If $X$ is random, then we need to average
over possible trajectories, and so we define
$J_{k,i}={k\choose
i}\mathbb{E}_{X}\left[\int_{0}^{\infty}q(r,X)^{i}(1-q(r,X))^{k-i}dr\right],$
(15)
where $\mathbb{E}_{X}[\cdot]$ denotes the average over possible trajectories.
We will assume that this integral is finite for $2\leq i\leq k$; for further
discussion of these points see Appendix A.1. Importantly, under our assumption
that sweeps do not interfere with each other, $J_{k,i}$ does not depend on the
recombination rate $r_{BP}$ or the rate of sweeps $\nu_{BP}$, but only on the
dynamics of the selective sweeps $X$.
Allowing coalescent events due to drift, $k$ lineages coalesce down to $k-i+1$
at rate
$\lambda_{k,i}=\frac{1}{2N}{k\choose
2}\delta_{i,2}+\frac{\nu_{BP}}{r_{BP}}J_{k,i}\quad\mbox{for}\;2\leq i\leq k,$
(16)
where $\delta_{i,2}=1$ if $i=2$ and is 0 otherwise. As equation (16) follows
from the simple change of variable in equation (14) it will hold under any
homogeneous sweep model where sweeps instantaneously (relative to a time scale
of $2N$) force lineages to coalescence, with $J_{k,i}$ replaced by some
constant that does not depend on $r_{BP}$ or $\nu_{BP}$. This result greatly
generalizes that of Kaplan et al. (1989) who described a similar coalescent
process for a full sweep model.
We can see from equation (16) that $2N\nu_{BP}/r_{BP}$ is the relevant
compound parameter that in a general sweep model determines the rate of sweeps
relative to neutral coalescent events. In small samples, sweep-induced
coalescent events will dominate those due to drift if the population-scaled
rate of sweeps per unit of the genetic map is much greater than one, provided
that not all the $J_{k,i}$ are too small. We revisit this strong sweep limit
in Section 2.7.
#### The coalescent process with homogeneous partial sweeps.
We now return to the setting of section 2.3.1, in which a simple trajectory
rises quickly to frequency $x$, under which assumptions $q(r,X)\approx q_{x}$
(equation (5)). We suppose that the frequency $x$ at which each sweep slows is
chosen independently with probability density $g(x)$. It also seems reasonable
to assume furthermore that $t_{x}$, the time it takes to reach frequency $x$,
does not depend on $x$; we will denote this time by $t$. This is approximately
true for many models of directional selection, since selected alleles move
quickly through intermediate frequencies. In this case, the rate at which $k$
lineages coalesce to $k-i+1$ is
$\lambda_{k,i}\frac{1}{2N}{k\choose 2}\delta_{i,2}+{k\choose
i}\frac{\nu_{BP}}{t\,r_{BP}}\int_{0}^{\infty}\left(\int_{0}^{1}\left(xe^{-r}\right)^{i}\left(1-xe^{-r}\right)^{k-i}g(x)dx\right)dr,$
(17)
suggesting that the important quantity, which acts as a coalescent time
scaling, is $2N\nu_{BP}/(t\,r_{BP})$, with the distribution on $x$ acting to
control how many lineages are forced to coalesce with each sweep. If we
determine $t$ by a simple model of additive selection with selection
coefficient $s$, the key parameter becomes $2N\nu_{BP}s/(\log(Ns)\,r_{BP})$.
This compound parameter, $2N\nu_{BP}s/(\log(Ns)\,r_{BP})$, is also the key
parameter in full sweep models (Kaplan et al., 1989; Stephan et al., 1992).
However, since full sweeps require $x=1$, if diversity is strongly reduced
then numerous lineages must merge at each sweep, which in turn leads to a
strong skew towards rare alleles in the frequency spectrum. We will see that
this relationship between the reduction in diversity and the skew in the
frequency spectrum is substantially weakened under a partial sweep model when
we allow $x\ll 1$.
### 2.6 Summaries of neutral genetic diversity.
#### 2.6.1 Level of neutral diversity.
A key quantity of interest is the level of neutral nucleotide diversity,
$\pi$, the number of differences between randomly sampled alleles at a neutral
locus. Under an infinite sites model of mutation, which we will use here, the
expectation of $\pi$, averaging across sites, is equal to the expected
coalescent time of a pair of lineages multiplied by twice the mutation rate.
If the mutation rate per generation at our neutral locus is $\mu$, in the
absence of sweeps, the level of diversity is $\mathbb{E}[\pi]=\theta$, where
$\theta=4N\mu$ is the population-size scaled mutation rate, and the
expectation is the average across sites. Note that $\theta$ is the level of
diversity under the usual neutral model.
Under our model featuring both sweeps and drift,
$\mathbb{E}[\pi]=\frac{\theta}{1+2NI_{2,2}\nu}.$ (18)
so a key parameter is the population–scaled rate of sweeps $2N\nu$.
To examine the applicability of our approximations we again performed
coalescent simulations with mssel for a selected locus at a fixed location
experiencing recurrent sweeps. In this case, where selected alleles
recurrently sweep into the population at a _fixed_ genetic distance $r$,
following our simple partial sweep trajectory again as characterized by
$q_{x}$ and $2N$, the nucleotide diversity is given by
$\mathbb{E}[\pi]=\frac{\theta}{1+2N\nu x^{2}\exp\left(-2rt_{x}\right)}.$ (19)
We used two types of recurrent trajectory – the recurrent ‘step’ and the
recurrent ‘top-hat’, as described earlier. For the recurrent top-hat
trajectory, we simulated an exponential waiting time with mean $\nu$ between
the end of one ‘top-hat’ and the start of the next (and similarly for the
‘step’ case). In Figure 3 we show diversity levels moving away from the locus
undergoing these two types of recurrent sweeps, as well as the analytical
approximation given by equation (19). Recall that in both types of
trajectories the derived allele pauses at frequency $x$ for time $T$, and
therefore we expect that the fate of the allele will affect diversity at
recombination distances smaller than $1/T$. For distances larger than $1/T$,
equation (19) shows good agreement with our simulations, regardless of whether
the recurrent sweeps go to loss or fixation. The approximation does not
perfectly match our simulations, presumably because $e^{-r2t_{x}}$ is an
imperfect approximation to the probability of recombination during the sweep.
Nevertheless, diversity levels generated by the two types of recurrent
trajectory agree away from the selected site, which importantly confirms that
only the initial rapid stage of the trajectory affects the coalescent process
at partially linked sites.
Figure 3: Reduction in diversity ($\pi/\theta$) as a function of
recombination distance from a site experiencing recurrent sweeps. The three
panels are for different values of the frequency $x$ that each sweep reached
rapidly. The solid line is for recurrent top-hat trajectories and the broken
line for recurrent step trajectories The time that the trajectory pauses is
$T/2N=0.01$ and $t_{x}/2N=0.003$ in both cases. The three colors correspond to
three different population-scaled rates of sweeps: $2N\nu=$ $2$, $4$ and $8$.
The vertical grey line marks recombination distance $r>1/T$ from the selected
locus, above which the dynamics subsequent to reach $x$ should make little
difference. The solid black lines give the prediction of (19).
##### The level of diversity under homogeneous sweeps.
Under the model in which sweeps occur homogeneously along an infinite
sequence, with coalescent rates given by equation (16), the level of
nucleotide diversity is given by
$\mathbb{E}[\pi]=\frac{\theta}{2N\nu_{BP}J_{2,2}/r_{BP}+1}.$ (20)
These results generalize previous results by Kaplan et al. (1989) and Stephan
et al. (1992), who found a relationship of the form (20) for a model of
homogeneous recurrent full sweeps. In fact, since equation (20) follows only
from the assumption that the rate and characteristics of sweeps are
independent of their location along the genome (see equation (14)), this
relationship between diversity, the density of selective targets, and
recombination rate will hold for a wide variety of homogeneous recurrent sweep
models including the homogeneous full sweep model.
#### 2.6.2 Frequency Spectrum.
We now study the effects of recurrent partial sweeps on other properties of
neutral diversity at a locus besides pairwise nucleotide diversity, and
compare our calculations to simulation.
Two commonly studied properties of a sample of neutral diversity at a locus
are the expected number of segregating sites in a sample of size $n$, and the
expected number of singletons in a sample of size $n$. Under the infinite-
sites assumption, these are respectively equal to the mutation rate multiplied
by the expected total length of the genealogical tree of the sample (which we
denote $T_{tot}$) and by the mutation rate multiplied by the expected total
length of the terminal branches ($T_{1}$). We provide recursions that allow
easy calculation of both $\mathbb{E}[T_{tot}]$ and $\mathbb{E}[T_{1}]$ in
Appendix A.2.
We also look more generally at the frequency spectrum of segregating alleles,
which is, in a sample of $n$ individuals, the proportion of segregating sites
at which $k$ derived alleles are found, for each $1\leq k\leq n$. Let
$F_{n,k}$ denote the expected proportion of segregating sites in a sample of
size $n$ at which exactly $k$ samples carry the derived allele under an
infinite sites model of mutation. $F_{n,k}$ is equal to the expected time in
the coalescent tree spent on branches that subtend exactly $k$ tips (those on
which mutation would lead to a site segregating at $k$ out of the $n$
samples), divided by $\mathbb{E}[T_{tot}]$. Under neutrality (Kingman’s
coalescent), this quantity is $F^{N}_{n,k}=(1/k)/\sum_{j=1}^{n-1}(1/j)$. It is
not so easy to find an explicit general expression under the coalescent model
with sweeps that we study, but for the case $k=1$ we have described in
Appendix A.2 how to compute $\mathbb{E}[T_{1}]/\mathbb{E}[T_{tot}]$, and the
general case can be found from simulation of the coalescent process.
Figure 4A shows the ratio of $F_{n,k}/F^{N}_{n,k}$, estimated by direct
simulation of our coalescent process. The rates are given by equation (9),
with $q$ fixed to $q_{x}=xe^{-t_{x}r}$, and $t_{x}r=0.6$ (and various $x$). To
make the simulations comparable, the population scaled rate of sweeps $2N\nu$
was adjusted such that $\pi/\theta=1/2$ in each, i.e. to obtain a $50\%$
reduction in diversity due to sweeps. We see that for partial sweeps at a
fixed site, across a range of $x$, the frequency spectrum is skewed towards
rare alleles and away from intermediate frequency alleles.
To test the degree to which our coalescent matches the full model, in Figure
4B we compare the mean proportion of singleton sites under our coalescent
model to that found from simulation with mssel. We simulated a recurrent top-
hat trajectory of the frequency at a selected locus as before, and used this
trajectory with mssel to simulate the neutral coalescent at a non-recombining
locus a distance $r$ away from this selected locus. We used the three values
$x=0.9$, $0.5$, and $0.2$ for the intermediate frequency the allele reached,
and in each case varied the rate of sweeps, $\nu$ Each combination of $\nu$
and $x$ gives a point in Figure 4B, plotted at its mean reduction in diversity
($\pi/\theta$) and the mean number of singletons divided by the mean number of
segregating sites. These are compared to the analytical values of
$\mathbb{E}[T_{1}]/\mathbb{E}[T_{tot}]$ computed using equations (29) and
(31), with coalescent rates given by equation (9), using a constant
$q=xe^{-rt_{x}}$ and (20) to find the reduction $\pi/\theta$. There is good
agreement between the simulations and the analytical results, showing that our
simplified process approximates the properties of the full coalescent process
at a single site reasonably well.
Figure 4: Properties of the frequency spectrum with sweeps occurring at a
fixed genetic distance Coalescent rates are given by equation (9), with $q$
fixed to $q_{x}=xe^{-t_{x}r}$ and $t_{x}r=0.6$, across a range of $x$. (A) The
percentage of segregating sites found at frequency $1\leq k\leq 20$, relative
to the neutral expectation (i.e. $F_{20,k}/F^{N}_{20,k}$). In these
simulations the rate of sweeps $N\nu$ has been fixed to result in a $50\%$
reduction in diversity. The dotted grey line gives the neutral expectation.
(B) The mean number of singletons divided by mean number of segregating sites,
from mssel simulations with a sample size of $10$ at a neutral site a distance
$2Nr=200$ from a selected site. The selected allele performs a recurrent top-
hat trajectory (with $N=10,000$ and $t_{x}/2N=.003$, giving $rt_{x}=0.6$, and
pausing $T/2N=0.01$) to frequency $x=0.2$, $x=0.5$, or $x=0.9$ across a range
of $2N\nu$. Note the span of $\pi/\theta$ is smaller in the low $x$
simulations as the effect on diversity of a given $2N\nu$ is smaller. Solid
lines show the analytical approximation for
$\mathbb{E}[T_{1}]/\mathbb{E}[T_{tot}]$ of Appendix A.2. The dotted grey line
gives the neutral value of the expected proportion of singletons
$1/\sum_{j=1}^{n-1}1/j$.
Figure 4 studied the effect on the frequency spectrum of recurrent sweeps at a
fixed distance from a neutral site; in Figure 5 we study the frequency
spectrum under the coalescent process with sweeps occurring homogeneously
along the genome. Figures 5A and B show the same quantities as Figure 4A, for
simulations of the homogeneous partial sweep coalescent process with a fixed
value of $x$, using rates given by equation (17), and $2N\nu_{BP}/(tr_{BP})$
chosen so that $\pi$ is 50% and 10% of its value under neutrality
respectively. In Figure 5C, there is no genetic drift and only sweeps force
coalescence, i.e. $N=\infty$ and so we do not need to specify
$2N\nu_{BP}/(tr_{BP})$ as it acts only as a time scaling. In 5D we show our
analytic calculation of $\mathbb{E}[T_{1}]/\mathbb{E}[T_{tot}]$ as a function
of the reduction in $\pi$ caused by selective sweeps.
Figure 5: Properties of the frequency spectrum under a spatially homogeneous
model of sweeps using the coalescent process with rates given by equation
(17). Simulations were performed for a sample size of $20$. For a particular
$x$ we adjusted the value of $N\nu_{BP}/(tr_{BP})$ to achieve the specified
reduction in $\pi$. (A) and (B) The percentage of segregating sites found at
frequency $1\leq k\leq 20$, relative to the neutral expectation for sweeps. In
each panel the reduction in diversity, $\pi/\theta$ is fixed. (C) The same
quantities as in A and B, but for the case where there is no genetic drift,
and sweeps are the only stochastic force affecting allele frequencies. (D) The
fraction of segregating sites that are singletons, for different $x$, as a
function of $\pi/\theta$, calculated using recursions for
$\mathbb{E}[T_{1}]/\mathbb{E}[T_{tot}]$ (Appendix A.2).
The skew in the frequency spectrum depends strongly on the frequency $x$
reached by the selected allele. Sweeps to low frequencies lead to a much
smaller distortion for the same reduction in $\pi$. Therefore, the strong
relationship between the reduction in $\pi$ and the skew in the frequency
spectrum under a model of full sweeps is much weaker if the sweeps do not go
to fixation.
Intriguingly, sweeps that go to intermediate frequency can lead to a greater
proportion of high frequency derived alleles than under a full sweep model.
While a single, recent full sweep leads to high frequency derived alleles
through hitchhiking (Fay and Wu, 2000), under a recurrent full sweep model
these alleles are then fixed in the population by subsequent sweeps and drift
(Kim, 2006), and therefore removed from the frequency spectrum. Further work
would be needed to understand the intuition for the excess of high frequency
derived alleles under a recurrent partial sweep model.
##### Summaries of the frequency spectrum
In Figures 4 and 5, we saw that regardless of whether sweeps occur at a fixed
distance from our neutral site or homogeneously along the sequence, as we
increase the rate of sweeps the frequency spectrum becomes further skewed
towards rare derived alleles at the expense of intermediate frequency alleles.
Here we provide evidence that this will hold for any set of parameter values.
Tajima’s $D$ and Fu and Li’s $D$ (Tajima, 1989; Fu and Li, 1993) are two
common ways of detecting deviations away from the frequency spectrum expected
under a neutral model with a constant population size. Negative values of
Tajima’s $D$ can be thought of as indicating a deficit of intermediate
frequency alleles, and Fu and Li’s $D$ indicates an excess of singleton
alleles. Durrett and Schweinsberg (2005) proved that in large samples, both of
these summary statistics are negative under a multiple mergers coalescent
model of full sweeps, as long as $\lambda_{k}$, the total coalescent rate when
there are $k$ lineages, satisfies
$\sum_{k=2}^{\infty}\left(\lambda_{k}-{k\choose
2}\right)\frac{\log(k)}{k^{2}}<\infty.$ (21)
See equation (4.5) in Durrett and Schweinsberg (2005). Informally, this
condition requires that the total coalescent rate is not too much higher than
the neutral coalescent rate when there are a large number of lineages. Their
methods were not specific to their situation but hold for all multiple merger
coalescent models satisfying equation (21). As above, we argued that a
generalized sweep model can be approximated by a multiple merger coalescent,
and therefore, it seems that reasonable generalized sweep models will, at
least for large samples, have a frequency spectrum that is skewed towards
singletons at the expense of intermediate frequency alleles (a notable
exception is the ‘low frequency’ limit we discuss below).
### 2.7 Limiting processes
Before we move to discuss the implications of these results for data analysis
there are two limiting processes that merit our attention. The first is when
the rate of sweeps is sufficiently high to dominate genetic drift as a source
of stochasticity. The second limit results when sweeps very rarely achieve
high frequency in the population, in which case the resulting coalescent model
is identical to the standard “neutral” coalescent, despite that fact that much
of the stochasticity may be driven by sweeps.
#### The rapid sweep limit
A surprising conclusion from the homogeneous model and equation (16) is that
if all coalescences come from “selective” events, then the frequency spectrum
does not depend on the density of selective targets or on the recombination
rate (although the number of segregating sites certainly does). This effect
can be seen in Figure 5D as the fraction of singleton sites plateaus when the
reduction in $\pi$ is large, i.e. when the population scaled rate of sweeps
per unit of recombination is high, $\nu_{BP}/r_{BP}\gg 1/2N$. The easiest way
to see this is to take $N\to\infty$ while keeping the rate of sweeps and their
trajectory dynamics fixed, so that in a sample of fixed size the coalescence
rate from equation (16) converges to $\lambda_{k,i}\to\nu_{BP}/r_{BP}J_{k,i}$,
where $J_{k,i}$ does not depend on $\nu_{BP}$, $r_{BP}$, or $N$. In this
limit, $\nu_{BP}$ and $r_{BP}$ only affect the process by a time scaling, do
not affect the transition probabilities of equation (11), and so do not affect
the frequency spectrum. Diversity in this limit behaves as
$\mathbb{E}[\pi]=\frac{2\mu r_{BP}}{\nu_{BP}J_{2,2}}.$ (22)
(assuming, as usual, that $\mu$ is sufficiently small) i.e. nucleotide
diversity increases linearly with the recombination rate, if neither
$\nu_{BP}$ or $J_{2,2}$ varies across recombination environments. Similar
limits can also be derived by letting $N\to\infty$ under the more general
coalescent process with rates given by equation (7).
For this limit to be a reasonable approximation for a sample of size $k$ in a
population of size $N$, we need the rate of neutral coalescences to be much
smaller than the rate of selective coalescences, i.e. ${k\choose 2}\ll
N\nu_{BP}/r_{BP}\sum_{i=2}^{k}J_{k,i}$. In sufficiently large samples,
${k\choose 2}$ will be large enough that the coalescence rate due to genetic
drift will be appreciable, at least until the number of lineages surviving
back in time declines. From a technical standpoint, this is related to the
question of whether the coalescent process “comes down from infinity” (for a
review see Berestycki, 2009).
#### The low frequency limit
As noted in our discussion of Figure 5, the frequency spectrum may be close to
neutral in appearance even with large reductions in $\pi$ if selected alleles
sweep only to low frequency. In fact, by taking a limit (satisfying certain
conditions) in which sweeps occur frequently, but each sweep has a small
probability of causing coalescence, we can recover Kingman’s coalescent.
We illustrate this limit by taking $\nu\to\infty$ and allowing $f(q)$ to
depend on $\nu$ in such a way that as $\nu\to\infty$, $I_{k,\ell}/I_{k,2}\to
0$ for all $3\leq\ell\leq k$, and that $\nu\;I_{k,2}\to{k\choose 2}\gamma$,
for some $0<\gamma<\infty$. As shown in Appendix A.3, a sufficient condition
for this is that $\lim_{\nu\to\infty}\nu\,\int_{0}^{1}q^{2}f(q)dq$ is finite.
In this limiting case, the rate of coalescence is
$\lambda_{k}={k\choose 2}\left(\gamma+\frac{1}{2N}\right),$ (23)
so the limiting model behaves exactly as the standard neutral coalescent but
with an effective population size of
$N_{e}=\frac{2N}{2N\gamma+1}.$ (24)
Note that the limiting coalescent process does not satisfy condition (21) of
Durrett and Schweinsberg (2005), and that Tajima’s $D$ and Fu and Li’s $H$
will have mean equal to zero at all sample sizes, as is natural since the
limiting process is just the neutral (Kingman’s) coalescent.
In the case of our simple partial sweep coalescent this limit would occur if
the frequency $x$ reached by sweeps is taken to zero as the rate of sweeps
grows at least as $1/x^{2}$. The simple homogeneous full sweep coalescent
process obviously can not be taken to this limit as there is a proscribed set
of $J_{k,\bullet}$, which feature non-trivial amount of coalescence involving
more than pairs of lineages.
##### Interference
In both limits discussed above the population-scaled rate of sweeps has to be
very high. In the first limit the rate of sweeps has to be high enough to
dominate the rate of neutral coalescence, in the second limit the rate of
sweeps has to be high enough to compensate for the fact that any one sweep is
very unlikely to cause coalescence. The requirement of a high rate of sweeps
implies that interference between the sweeps may occur, thus violating our
assumption that the sweeps are independent. Investigations of the effect of
such interference on the signal of hitchhiking have shown that interference
reduces the impact of any one sweep on patterns of polymorphism (Kim and
Stephan, 2003; Chevin et al., 2008), although to interfere, the sweeps must
begin at very similar times at loci separated by a low recombination rate.
This suggests that a very high rate of sweeps is needed indeed before
interference will have an appreciable impact on the hitchhiking effect, as
would occur in the homogeneous sweep model if $\nu_{BP}/r_{BP}$ is very large.
The limits we describe above only require that the population size-scaled rate
of sweeps ($N\nu$ or $N\nu_{BP}$) be high, and therefore it is possible to
keep the per generation rate of sweeps sufficiently low as to avoid the effect
of interference. Further work is needed to investigate coalescent models under
such high rates of sweeps, and could be useful in understanding genealogical
processes in organisms with low or no recombination that also experience
strong selection pressures.
## 3 Discussion
The prevailing view of adaptation in a population genetics setting is based on
a lone selected allele racing from its introduction into the population to
fixation, carrying with it a chunk of the chromosome on which it arose. This
cartoon has been a very useful prop for developing tests to identify genes
underlying recent adaptations, and for interpreting genome-wide patterns of
polymorphism. However, it seems likely that such full sweeps constitute only a
small proportion of the selected loci whose frequency changes in response to
adaptation (see Pritchard et al., 2010, for a recent discussion). If we are to
develop a better understanding of the full impact of linked selection on
patterns of diversity we need to develop a richer and more flexible set of
models.
The work in this paper was motivated by models in which the external
environment or the genetic background vary on a fast enough time scale that
new alleles rarely reach fixation before selective pressures change, either
slowing their advance or reversing their trajectory. We laid out an
approximation to the coalescent process under such a model, and showed that,
while the initial rapid stage of the trajectory will strongly impact the
coalescent process, subsequent slower dynamics of the selected alleles have a
much smaller effect. We then extended this idea to a recurrent sweep model,
approximating the dynamics by a multiple-merger coalescent. While some of our
results are fairly general, to provide a more intuitive sense we have often
employed simple allele frequency trajectories and made other approximations.
Nonetheless, we expect more realistic models to give rise to qualitatively
equivalent results.
Each sweep we consider consists of a single allele at a locus rising on a
single haplotype from very low frequency in to the population. This contrasts
with many other soft sweep models, under which a sweep starts on multiple
haplotypes, either because multiple different alleles initially segregated at
the locus (Hermisson and Pennings, 2005); or as a result of multiple mutations
occurring after selection pressures switched (Pennings and Hermisson, 2006a,
b; Ralph and Coop, 2010); or because the adaptive allele was previously
neutral and present on multiple haplotypes (Innan and Kim, 2004; Przeworski et
al., 2005). It is likely that recurrent models of such soft sweeps could be
approximated through coalescent models with simultaneous multiple collisions
(Schweinsberg, 2000), to model the simultaneous rise of multiple haplotypes.
This seems like a fruitful area of future work as it would substantially
extend our understanding of the effects of a broad family of recurrent sweep
models on genomic patterns of diversity.
We have also ignored the effect of background selection. To a first
approximation, the effect of background selection can be modeled as an
increase in coalescence rate, which would be a minor modification to equations
(9) and (16). This would alter the predicted relationship between diversity
and recombination (Innan and Stephan, 2003) given by equation (20), and would
offer a simple way to model the genealogical effects of both general models of
hitchhiking and background selection.
#### The interpretation of population genomic patterns
Models in which selective sweeps do not always sweep to fixation have a much
wider spectrum of predictions than the recurrent full sweep model. Three broad
correlations that have been used to argue for the prevalence of linked
selection, and used to potentially discriminate between models invoking
background selection or full sweeps are: 1) correlations between neutral
diversity and the recombination rate; 2) correlations between the frequency
spectrum and the rate of recombination; and 3) correlations between putatively
adaptive divergence and neutral diversity. We now describe some of the
implications of our results for understanding these patterns in population
genomic data.
##### Correlation between recombination and diversity
One of the earliest and most compelling pieces of evidence for the role of
linked selection in the fate of neutral alleles is a positive correlation
between recombination and levels of diversity at putatively neutral sites
(factoring out substitution rates as a proxy for differences in mutation
rate). This pattern is consistent with both full sweeps and background
selection, as both predict positive, albeit differently shaped, relationships
(Innan and Stephan, 2003). The shape of the diversity-recombination curve
under a homogeneous rate of partial sweeps is identical to the full sweep
model, and more generally for a broad class of homogeneous sweep models. In
fact, the relationship under a homogeneous model only depends on
$2N\nu_{BP}J_{2,2}$, as seen in equation (20).
To illustrate this point, in Table 1 we present estimates of
$2N\nu_{BP}J_{2,2}$ for humans and Drosophila melanogaster, assuming a model
with drift and a homogeneous rate of selective sweeps across the genome, and
from equation (20) and data from Hellmann et al. (2008); Shapiro et al.
(2007). Along with these estimates, Table 1 also shows the implied rate of
sweeps per generation per base pair, $\nu_{BP}$, under the simple partial
sweep model, for a variety of values of $x$. These rates are surely
overestimates, are intended for illustrative purposes only, as they ignore the
effect of other forms of linked selection, e.g. background selection.
The strength of the relationship between diversity levels and recombination
varies dramatically between the two species, as indicated by the very
different estimates of $2N\nu_{BP}J_{2,2}$ (note that the estimates of
$\nu_{BP}$ are similar due to the thousand fold difference in $N$). In
Drosophila the positive relationship between recombination and diversity is
strong (e.g. Aguade et al., 1989; Begun and Aquadro, 1992; Berry et al., 1991;
Shapiro et al., 2007; Begun et al., 2007), but in humans the relationship
seems to be weaker and is and complicated by other confounding factors
(Payseur and Nachman, 2002; Hellmann et al., 2003, 2005, 2008; Cai et al.,
2009). However, we should be cautious in the biological interpretation of this
difference, as in humans diversity is usually estimated in large windows (much
of which will be noncoding and far from genes), while in Drosophila neutral
diversity levels are usually estimated from synonymous sites in individual
genes. What is needed is a comparative analysis that studies these patterns at
the same genomic scale and accounts for the profound differences in the
density of functional targets among species.
The fact that the diversity–recombination curve plateaus rapidly in humans is
strong evidence that linked selection does not affect the average neutral site
in regions of high recombination. Technically, this could also occur if the
density of selective targets $\nu_{BP}$ decreases approximately linearly with
recombination rate; however, this option does not seem likely a priori.
Although in Drosophila melanogaster this curve is still concave, it does not
appear to flatten completely in high recombination regions (e.g. Sella et al.,
2009), suggesting that linked selection is an important source of
stochasticity even in these regions. At face value the concave nature of the
curve suggests that both genetic drift and linked selection contribute to
stochasticity, as $N\nu_{BP}\gg r_{BP}$ would lead to an almost linear
relationship across the observed range of recombination rates (see equation
(22)). However, a model with effectively no genetic drift can produce a
concave curve and fit the observed data if $\nu_{BP}J_{2,2}$ is not constant
across recombination environments, e.g. if sweeps occur at a moderately higher
rate or achieve higher frequency in high recombination regions. Neither of
these two options seem particularly unlikely, suggesting that we still have
little unambiguous evidence favoring genetic drift as an important source of
stochasticity in Drosophila.
| | | $\nu_{BP}$ across a range of $x$
---|---|---|---
| $\theta$ | $2N\nu_{BP}J_{2,2}$ | $x=1.0$ | $x=0.5$ | $x=0.2$ | $x=0.05$
Human | $0.0017$ | $6\times 10^{-11}$ | $3.0\times 10^{-12}$ | $1.2\times 10^{-11}$ | $7.5\times 10^{-11}$ | $1.2\times 10^{-9}$
D. mel | $0.025$ | $7.3\times 10^{-9}$ | $3.6\times 10^{-12}$ | $1.5\times 10^{-11}$ | $9.1\times 10^{-11}$ | $1.5\times 10^{-9}$
Table 1: Estimates of sweep parameters from the relationship between
diversity and recombination. The estimate for humans was taken from Hellmann
et al. (2008) who fitted a curve of the form of equation (20). The estimate
from Drosophila melanogaster (D. mel) was obtained from fitting equation (20)
to the synonymous polymorphism and sex-averaged recombination rates of Shapiro
et al. (2007) (kindly provided by Peter Andolfatto, see Sella et al. (2009)
for details) using non-linear least squares via the nls() function in R. These
estimates were converted into estimates of the rate of sweeps per generation
per base pair ($\nu_{BP}$, last four columns) under the simple partial sweep
trajectory model where $J_{2,2}=x^{2}/t_{x}$, assuming $t_{x}=1,000$
generations (equivalent to a selection coefficient of $\sim 0.01$) and that
$N=10^{6}$ in D. mel and $N=10^{4}$ in humans.
##### The frequency spectrum
The recurrent full sweep model predicts a strong positive relationship between
the reduction in neutral diversity and the skew towards rare alleles
(Braverman et al., 1995; Kim, 2006), a pattern not predicted under models of
strong background selection. This relationship has been used to test between
full sweeps and background selection models, although note that as we
discussed in Section 2.7, this relationship is not expected if all coalescence
comes from selective sweeps. Under our simple trajectory model, the distortion
of the frequency spectrum is primarily determined by the frequencies that
sweeps achieve. Therefore, although a lack of a strong skew in the frequency
spectrum is consistent with a low rate of full sweeps, it cannot be used to
rule out a high rate of partial sweeps. A lack of a genomic relationship
between the frequency spectrum and recombination rate is therefore not grounds
for rejecting sweeps as a force in shaping genetic diversity in favor of a
model of background selection. Our results suggest that recurrent partial
sweeps to low frequency in regions of high recombination in D. melanogaster
and in the low recombination regions in humans may be a major source of
stochasticity in allele frequencies.
##### Correlation between divergence and polymorphism.
Attention has recently focused on examining the correlation between neutral
diversity and amino acid substitutions (or other putatively functional
changes) between recently separated species. If a reasonable fraction of amino
acid substitutions are driven by new mutations sweeping to fixation, then
levels of diversity should dip on average around amino-acid substitutions.
This relationship has been tested for by looking for a positive correlation
between diversity levels and amino-acid substitution rates (Macpherson et al.,
2007; Andolfatto, 2007; Cai et al., 2009; Haddrill et al., 2011) or for a dip
in diversity levels around a large set of aggregated amino acid substitutions
(Hernandez et al., 2011; Sattath et al., 2011). If the density of functional
sites is properly controlled for, these types of correlations between amino-
acid substitutions and neutral diversity are not expected under a (simple)
model of background selection. Such correlations have been detected in
_Drosophila_ (Macpherson et al., 2007; Sattath et al., 2011) but in humans the
dip in diversity around non-synonymous substitutions seems to result from the
dip in diversity levels around genes, an observation that seems inconsistent
with a high rate of strong full sweeps (Hernandez et al., 2011). Similarly, it
has been observed that the highest $F_{ST}$ signals between human populations
are not associated with strongly reduced haplotypic diversity (Coop et al.,
2009).
The fact that selected alleles in the partial sweep coalescent model do not
have to sweep all the way to fixation partially decouples the rate of fixation
of adaptive alleles from their effects on patterns of diversity within
populations. Therefore, the strength of the positive relationship between
substitution rates and diversity depends on the fate of alleles that sweep
into the population. For example, this positive relationship may be weak, and
a poor predictor of the total reduction in diversity, if the majority of
adaptive alleles that initially sweep into the population are eventually lost
(e.g. as can be the case for major effect alleles in polygenic models of
adaptation, see Lande, 1983; Chevin and Hospital, 2008).
##### Concluding thoughts
In this article, we have concerned ourselves with patterns of diversity at a
single neutral site. However, partial sweeps also have a strong effect on
linkage disequilibrium and haplotype diversity, a signature that has been
exploited in scans for selection (e.g. Hudson et al., 1994; Sabeti et al.,
2002; Voight et al., 2006). One simple case that we can immediately describe
is the low $q$ limit (section 2.7). In that limit, the coalescent is
equivalent to the standard neutral model and as such the decay of LD will be
the same as the standard neutral model with an $N_{e}$ given by equation (24).
A natural way to extend this exploration would be the genealogical framework
developed by McVean (2007) that has recently been extended to a multiple
mergers coalescent by Eldon and Wakeley (2008).
We will soon have polymorphism data across a broad range of taxa that will
differ dramatically in selection regimes, recombination rates, genome size,
and population size allowing a much fuller picture of how these various
factors interplay to shape genome-wide levels of polymorphism. The results
presented here, however, suggest that we will continue to struggle to
distinguish between modes of selection, as relaxing the assumptions of various
models can generate a broad range of overlapping predictions.
Despite that, our results suggest a promising way forward, since a broad range
of sweep models can be captured by a simple parameterizations of multiple
merger coalescence processes. Importantly, this would allow parameter
inference under a general model of linked selection, rather than focusing on a
limited number of specific models. For example, we could estimate the rate
that selection forces different numbers of lineages to coalesce (parameterized
by $\nu f(q)$) as function of recombination rates and the density of selective
targets. As the multiple–mergers coalescent model is easily simulated under,
it may be readily incorporated into many of our existing genealogical
inference frameworks. It is likely that parameters of such models could be
estimated very precisely from genome–wide data, allowing us to concentrate on
what these high level summaries of polymorphism tell us about linked selection
across genomic environments and species. Such inferences may be important if
we wish to move beyond documenting the presence of linked selection towards
describing the genealogical process in species where selection is a major
source of stochasticity.
#### 3.0.1 Acknowledgements
Thanks to Yaniv Brandvain, Chuck Langley, Molly Przeworski, Josh Schraiber,
Alisa Sedghifar, and Guy Sella for helpful conversations and comments on
previous drafts. We thank the two anonymous reviewers and the Editor for
helpful feedback. This work is supported by a Sloan Fellowship and funds from
UC Davis to GC and a NIH NRSA postdoctoral fellowship to PR.
## Appendix A Appendices
### A.1 $J_{k,i}$ for a generalized trajectory
Recall that we defined in equation (13)
$J_{k,i}={k\choose
i}\mathbb{E}_{X}\left[\int_{0}^{\infty}q(X,r)^{i}(1-q(X,r))^{k-i}dr\right],~{}~{}2\leq
i\leq k,$ (25)
so that the rate at which the coalescent process having $k$ lineages coalesces
down to $i$ lineages from “selective” events is $\nu_{BP}/r_{BP}J_{k,i}$. The
quantity $q(X,r)$ is the pathwise Laplace transform of the process $X$,
defined in equation (3), and consequently
$1-q(X,r)=\int_{0}^{\infty}re^{-rt}(1-X(t))dt.$ (26)
It is useful to note that by changing the order of integration,
$\displaystyle J_{k,i}$ $\displaystyle={k\choose
i}\mathbb{E}_{X}\left[\int_{0}^{\infty}\left(\int_{0}^{\infty}\cdots\int_{0}^{\infty}\prod_{j=1}^{i}X(t_{j})\prod_{\ell=i+1}^{k}(1-X(t_{\ell}))r^{k}\exp\left(-r\sum_{j=1}^{k}t_{j}\right)dt_{1}\cdots
dt_{k}\right)dr\right]$ $\displaystyle=k!{k\choose
i}\mathbb{E}_{X}\left[\int_{0}^{\infty}\cdots\int_{0}^{\infty}\frac{\prod_{j=1}^{i}X(t_{j})\prod_{j=i+1}^{k}(1-X(t_{j}))}{\left(\sum_{j=1}^{n}t_{i}\right)^{k+1}}dt_{1}\cdots
dt_{k}\right]$ (27)
for $2\leq i\leq k$, as long as the integral is finite. In the case of a pair
of lineages $i=2$ and this simplifies to
$J_{2,2}=2\mathbb{E}_{X}\left[\int_{0}^{\infty}\int_{0}^{\infty}\frac{X(\tau-
t_{1})X(\tau-t_{2})}{(t_{1}+t_{2})^{3}}dt_{1}dt_{2}.\right]$ (28)
To briefly explore the conditions for $J$ to be finite, we will suppose that
$X$ leaves zero as a power of $t$, i.e. $X(t)\sim t^{\alpha}$ for some
$\alpha>0$, for small $t$. We see that $J_{k,2}$ increases as $\alpha$
increases, i.e. the rate of sweeps is larger the more rapidly $X$ leaves zero.
In this case, $q(r)\sim C\,r^{-\alpha}$ for large $r$, where $C$ is a
constant. Then since
$\displaystyle J_{k,2}$ $\displaystyle=\lim_{L\to\infty}{k\choose
2}\int_{0}^{L}q(r)^{2}(1-q(r))^{k-2}dr$
$\displaystyle\leq\lim_{L\to\infty}{k\choose 2}\int_{0}^{L}q(r)^{2}dr,$
it can be seen that $J_{k,2}$ is infinite if $\alpha\leq 1/2$, in the limit of
an infinite genome. More generally, if $X$ leaves zero more quickly than
$\sqrt{t}$ (which may be biologically unrealistic), then sweeps occurring
arbitrarily far away along the genome will cause coalescences.
### A.2 Recursions to find $\mathbb{E}[T_{tot}]$ and $\mathbb{E}[T_{1}]$
Two properties of interest are the expected total amount of time in the
genealogy at a neutral locus ($\mathbb{E}[T_{tot}]$) and the expected total
amount of time in terminal branches ($\mathbb{E}[T_{1}]$).
We first derive the expected total time in the genealogy. Recall that if the
coalescent process has $k$ lineages, then it waits an exponentially
distributed amount of time with mean $1/\lambda_{k}$, and then jumps to a
smaller number of lineages chosen with probabilities according to
$p_{k,\ell}$, with $\lambda_{k}$ and $p_{k,\ell}$ given in equations (10) and
(11). Therefore, if we let $G_{n,k}$ be the probability that the coalescent
process that starts from $n$ lineages ever visits the state with $k$ lineages,
then
$\mathbb{E}[T_{tot}]=\sum_{k=2}^{n}\frac{k}{\lambda_{k}}G_{n,k}.$ (29)
By conditioning on the last state visited before dropping to $k$ lineages, we
can see that $G_{n,k}$ satisfies the recursion
$G_{n,k}=\sum_{i=k+1}^{n}G_{n,i}\;p_{i,k},\quad\mbox{for}\;k<n,$ (30)
with $G_{n,n}=1$. This recursion is of upper triangular form, so is easily
solvable, which together with (29) allows us to compute $\mathbb{E}[T_{tot}]$.
We now turn to the expected total time in terminal branches, i.e. those
branches on which mutations will lead to singletons. Note that, since all
lineages are exchangeable, $\mathbb{E}[T_{1}]$ is equal to $n$ times the mean
time until a particular lineage – say, the first one – coalesces with any
other. To find this, let $S_{n,k}$ be the probability that at some point there
are $k$ lineages, and that one of those $k$ lineages is the original first
lineage, still not coalesced with any others. Then the mean time until the
first lineage coalesces is $\sum_{k=2}^{n}\frac{1}{\lambda_{k}}S_{n,k}$, and
hence
$\mathbb{E}[T_{1}]=n\;\sum_{k=2}^{n}\frac{1}{\lambda_{k}}S_{n,k}.$ (31)
As above, we can get a solvable recursion for $S_{n,k}$ by conditioning on the
last coalescent event before reaching $k$ lineages. If the coalescent process
jumps from $\ell$ to $k$ lineages, then the probability that a given lineage
is not part of this coalescent event is $(k-1)/\ell$, and hence
$S_{n,k}=\sum_{\ell=k+1}^{n}S_{n,\ell}\,p_{\ell,k}\frac{k-1}{\ell}\quad\mbox{for}\;k<n,$
(32)
and $S_{n,n}=1$. The recursion is also easily solvable, which lets us obtain
$\mathbb{E}[T_{1}]$.
### A.3 More on the low $q$ limit
We would like to arrange things so that asymptotically, all coalescent events
affect only two lineages. We illustrate this limit by taking $\nu\to\infty$
and allowing $f(q)$ to depend on $\nu$ in such a way that as $\nu\to\infty$,
$I_{k,\ell}/I_{k,2}\to 0$ for all $3\leq\ell\leq k$, and that
$\nu\;I_{k,2}\to{k\choose 2}\gamma$, for some $0<\gamma<\infty$. Since this
model is a Lambda coalescent with $\Lambda(dq)=q^{2}\nu
f(q)dq+\delta_{0}(dq)/2N$, if we rescale time by a factor of $C$, a necessary
and sufficient condition is that $C\Lambda$ converges weakly to a point mass
at 0.
To emphasize the dependence of $f$ on $\nu$ we write $f(q)=f_{\nu}(q)$ and
$I_{k,\ell}=I_{k,\ell}(\nu)$. We would like to find a simple condition under
which the proportion of coalescences involving more than two lineages goes to
zero, i.e. that $I_{k,\ell}(\nu)/I_{k,2}(\nu)\to 0$ as $\nu\to\infty$ if
$\ell>2$. Fix $k$, and suppose for convenience that $f(q)=0$ for all
$q>1-\epsilon$, for some $\epsilon>0$. Then
$\epsilon^{k}\int_{0}^{1}q^{\ell}f_{\nu}(q)dq<\int_{0}^{1}q^{\ell}(1-q)^{k-\ell}f_{\nu}(q)dq<\int_{0}^{1}q^{\ell}f_{\nu}(q)dq,$
so that $I_{k,\ell}(\nu)/I_{k,2}(\nu)\to 0$ if and only if
$\frac{\int_{0}^{1}q^{\ell}f_{\nu}(q)dq}{\int_{0}^{1}q^{2}f_{\nu}(q)dq}\to 0.$
Using Jensen’s inequality,
$\displaystyle\frac{\int_{0}^{1}q^{\ell}f_{\nu}(q)dq}{\int_{0}^{1}q^{2}f_{\nu}(q)dq}$
$\displaystyle\leq\frac{\left(\int_{0}^{1}q^{2}f_{\nu}(q)dq\right)^{\ell/2}}{\int_{0}^{1}q^{2}f_{\nu}(q)dq}$
$\displaystyle=\left(\int_{0}^{1}q^{2}f_{\nu}(q)dq\right)^{(\ell-2)/2},$
so if $\int_{0}^{1}q^{2}f_{\nu}(q)dq\to 0$, this will be achieved. By the same
result,
$\displaystyle\frac{I_{k,2}(\nu)}{\nu{k\choose
2}\int_{0}^{1}q^{2}f_{\nu}(q)dq}\to 1,$
so that, rescaling time by a factor $C_{\nu}$, if
$\nu C_{\nu}\int_{0}^{1}q^{2}f_{\nu}(q)dq\to\gamma\quad\mbox{as}\;L\to\infty,$
then $\nu C_{\nu}I_{k,2}\to{k\choose 2}\gamma$ for all $k$. In this limit, the
rate at which a pair of lineages coalesces converges, and does not depend on
the number of lineages present.
Ideally, we would illustrate this with an stochastic model for $X$. However,
the formula requires the model to be analytically tractable to a degree
satisfied by no population genetics models that we could think of, and it is
much easier to make a concrete choice of $f(q)$. Consider the case where
$f(q)$ is the density of a Beta($1,M$) distribution. The mean of this
distribution is $1/(1+M)$. In that case
$I_{k,\ell}={k\choose\ell}\int_{0}^{1}q^{\ell}(1-q)^{k-\ell+M-1}Mdq=M{k\choose\ell}\bigg{/}{k+M-1\choose\ell},$
(33)
so that as $M\to\infty$,
$MI_{k,2}={k\choose
2}\frac{2M^{2}}{(M+k-1)(M+k-2)}\xrightarrow{L\to\infty}2{k\choose 2},$
so if $\nu=M$, then $\gamma=2$. We can furthermore check that
$\frac{I_{k,\ell}}{I_{k,2}}=\frac{{k\choose\ell}}{{k\choose
2}}\frac{\ell!(k+M-\ell-1)!}{2!(k+M-3)!}\sim\frac{1}{M^{\ell-2}}\xrightarrow{M\rightarrow\infty}0.$
(34)
so that this simple case satisfies our limit.
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|
arxiv-papers
| 2011-12-22T01:35:33 |
2024-09-04T02:49:25.597198
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Graham Coop, Peter Ralph",
"submitter": "Graham Coop",
"url": "https://arxiv.org/abs/1112.5218"
}
|
1112.5310
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
LHCb-PAPER-2011-034 CERN-PH-EP-2011-216 December 20, 2011; rev. March 22,
2012
Observation of $X(3872)$ production in $pp$ collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-2.07413ptV}$
The LHCb collaboration †††Authors are listed on the following pages.
Using 34.7$\mbox{\,pb}^{-1}$ of data collected with the LHCb detector, the
inclusive production of the $X(3872)$ meson in $pp$ collisions at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ is observed for the first time.
Candidates are selected in the $X(3872)\rightarrow J/\psi\pi^{+}\pi^{-}$ decay
mode, and used to measure
$\displaystyle\sigma(pp\rightarrow X(3872)+{\rm
anything})\,\mathcal{B}(X(3872)\rightarrow J/\psi$
$\displaystyle\pi^{+}\pi^{-})=$ $\displaystyle 5.4\pm 1.3\,{\rm(stat)}\pm
0.8\,{\rm(syst)}\rm\,nb\,,$
where $\sigma(pp\rightarrow X(3872)+{\rm anything})$ is the inclusive
production cross-section of $X(3872)$ mesons with rapidity in the range
$2.5-4.5$ and transverse momentum in the range
$5-20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In addition the masses of both the
$X(3872)$ and $\psi(2S)$ mesons, reconstructed in the $J/\psi\pi^{+}\pi^{-}$
final state, are measured to be
$\displaystyle m_{X(3872)}$ $\displaystyle=$ $\displaystyle 3871.95\pm
0.48\,({\rm stat})\pm 0.12\,({\rm
syst}){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}~{}\mbox{and}$ $\displaystyle
m_{\psi(2S)}$ $\displaystyle=$ $\displaystyle 3686.12\pm 0.06\,({\rm stat})\pm
0.10\,({\rm syst}){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}\,.$
(Published in Eur. Phys. J. C 72 (2012) 1972)
The LHCb Collaboration
R. Aaij23, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi42, C. Adrover6, A.
Affolder48, Z. Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J.
Anderson39, R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L.
Arrabito53, A. Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16,
R.J. Barlow50, C. Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10,
Th. Bauer23, A. Bay38, I. Bediaga1, S. Belogurov30, K. Belous34, I.
Belyaev30,37, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson46, J.
Benton42, R. Bernet39, M.-O. Bettler17, M. van Beuzekom23, A. Bien11, S.
Bifani12, T. Bird50, A. Bizzeti17,h, P.M. Bjørnstad50, T. Blake37, F. Blanc38,
C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V. Bocci22, A. Bondar33, N.
Bondar29, W. Bonivento15, S. Borghi47,50, A. Borgia52, T.J.V. Bowcock48, C.
Bozzi16, T. Brambach9, J. van den Brand24, J. Bressieux38, D. Brett50, M.
Britsch10, T. Britton52, N.H. Brook42, H. Brown48, A. Büchler-Germann39, I.
Burducea28, A. Bursche39, J. Buytaert37, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,37, A. Cardini15, L. Carson49, K. Carvalho
Akiba2, G. Casse48, M. Cattaneo37, Ch. Cauet9, M. Charles51, Ph.
Charpentier37, N. Chiapolini39, K. Ciba37, X. Cid Vidal36, G. Ciezarek49,
P.E.L. Clarke46,37, M. Clemencic37, H.V. Cliff43, J. Closier37, C. Coca28, V.
Coco23, J. Cogan6, P. Collins37, A. Comerma-Montells35, F. Constantin28, A.
Contu51, A. Cook42, M. Coombes42, G. Corti37, G.A. Cowan38, R. Currie46, C.
D’Ambrosio37, P. David8, P.N.Y. David23, I. De Bonis4, S. De Capua21,k, M. De
Cian39, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi38,37, L. Del Buono8, C. Deplano15, D.
Derkach14,37, O. Deschamps5, F. Dettori24, J. Dickens43, H. Dijkstra37, P.
Diniz Batista1, F. Domingo Bonal35,n, S. Donleavy48, F. Dordei11, A. Dosil
Suárez36, D. Dossett44, A. Dovbnya40, F. Dupertuis38, R. Dzhelyadin34, A.
Dziurda25, S. Easo45, U. Egede49, V. Egorychev30, S. Eidelman33, D. van
Eijk23, F. Eisele11, S. Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch.
Elsasser39, D. Elsby55, D. Esperante Pereira36, L. Estève43, A.
Falabella16,14,e, E. Fanchini20,j, C. Färber11, G. Fardell46, C. Farinelli23,
S. Farry12, V. Fave38, V. Fernandez Albor36, M. Ferro-Luzzi37, S. Filippov32,
C. Fitzpatrick46, M. Fontana10, F. Fontanelli19,i, R. Forty37, M. Frank37, C.
Frei37, M. Frosini17,f,37, S. Furcas20, A. Gallas Torreira36, D. Galli14,c, M.
Gandelman2, P. Gandini51, Y. Gao3, J-C. Garnier37, J. Garofoli52, J. Garra
Tico43, L. Garrido35, D. Gascon35, C. Gaspar37, N. Gauvin38, M. Gersabeck37,
T. Gershon44,37, Ph. Ghez4, V. Gibson43, V.V. Gligorov37, C. Göbel54, D.
Golubkov30, A. Golutvin49,30,37, A. Gomes2, H. Gordon51, M. Grabalosa
Gándara35, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35, G.
Graziani17, A. Grecu28, E. Greening51, S. Gregson43, B. Gui52, E. Gushchin32,
Yu. Guz34, T. Gys37, G. Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42, S.
Hansmann-Menzemer11, R. Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44,
T. Hartmann56, J. He7, V. Heijne23, K. Hennessy48, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks48, K. Holubyev11, P. Hopchev4, W.
Hulsbergen23, P. Hunt51, T. Huse48, R.S. Huston12, D. Hutchcroft48, D.
Hynds47, V. Iakovenko41, P. Ilten12, J. Imong42, R. Jacobsson37, A. Jaeger11,
M. Jahjah Hussein5, E. Jans23, F. Jansen23, P. Jaton38, B. Jean-Marie7, F.
Jing3, M. John51, D. Johnson51, C.R. Jones43, B. Jost37, M. Kaballo9, S.
Kandybei40, M. Karacson37, T.M. Karbach9, J. Keaveney12, I.R. Kenyon55, U.
Kerzel37, T. Ketel24, A. Keune38, B. Khanji6, Y.M. Kim46, M. Knecht38, P.
Koppenburg23, A. Kozlinskiy23, L. Kravchuk32, K. Kreplin11, M. Kreps44, G.
Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki37, M. Kucharczyk20,25,37,j,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty50, A.
Lai15, D. Lambert46, R.W. Lambert24, E. Lanciotti37, G. Lanfranchi18, C.
Langenbruch11, T. Latham44, C. Lazzeroni55, R. Le Gac6, J. van Leerdam23,
J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J. Lefrançois7, O. Leroy6, T.
Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles48, R. Lindner37, C. Linn11,
B. Liu3, G. Liu37, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-
March38, H. Lu38,3, J. Luisier38, A. Mac Raighne47, F. Machefert7, I.V.
Machikhiliyan4,30, F. Maciuc10, O. Maev29,37, J. Magnin1, S. Malde51, R.M.D.
Mamunur37, G. Manca15,d, G. Mancinelli6, N. Mangiafave43, U. Marconi14, R.
Märki38, J. Marks11, G. Martellotti22, A. Martens8, L. Martin51, A. Martín
Sánchez7, D. Martinez Santos37, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev29, E. Maurice6, B. Maynard52, A. Mazurov16,32,37, G. McGregor50, R.
McNulty12, M. Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S.
Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski25, R. Mountain52, I. Mous23, F. Muheim46, K.
Müller39, R. Muresan28,38, B. Muryn26, B. Muster38, M. Musy35, J. Mylroie-
Smith48, P. Naik42, T. Nakada38, R. Nandakumar45, I. Nasteva1, M. Nedos9, M.
Needham46, N. Neufeld37, C. Nguyen-Mau38,o, M. Nicol7, V. Niess5, N.
Nikitin31, A. Nomerotski51, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen49, K. Pal52, J. Palacios39, A.
Palano13,b, M. Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo47, C.
Parkes50,37, C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49, S.K.
Paterson49, G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos
Alvarez36, A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A.
Petrolini19,i, A. Phan52, E. Picatoste Olloqui35, B. Pie Valls35, B.
Pietrzyk4, T. Pilař44, D. Pinci22, R. Plackett47, S. Playfer46, M. Plo
Casasus36, G. Polok25, A. Poluektov44,33, E. Polycarpo2, D. Popov10, B.
Popovici28, C. Potterat35, A. Powell51, J. Prisciandaro38, V. Pugatch41, A.
Puig Navarro35, W. Qian52, J.H. Rademacker42, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk40, G. Raven24, S. Redford51, M.M. Reid44, A.C. dos Reis1,
S. Ricciardi45, K. Rinnert48, D.A. Roa Romero5, P. Robbe7, E. Rodrigues47,50,
F. Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43, S. Roiser37, V.
Romanovsky34, M. Rosello35,n, J. Rouvinet38, T. Ruf37, H. Ruiz35, G.
Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail47, B. Saitta15,d,
C. Salzmann39, M. Sannino19,i, R. Santacesaria22, C. Santamarina Rios36, R.
Santinelli37, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m,
A. Satta21, M. Savrie16,e, D. Savrina30, P. Schaack49, M. Schiller24, S.
Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt37, O. Schneider38, A.
Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba18,l, M.
Seco36, A. Semennikov30, K. Senderowska26, I. Sepp49, N. Serra39, J. Serrano6,
P. Seyfert11, M. Shapkin34, I. Shapoval40,37, P. Shatalov30, Y. Shcheglov29,
T. Shears48, L. Shekhtman33, O. Shevchenko40, V. Shevchenko30, A. Shires49, R.
Silva Coutinho44, T. Skwarnicki52, A.C. Smith37, N.A. Smith48, E. Smith51,45,
K. Sobczak5, F.J.P. Soler47, A. Solomin42, F. Soomro18, B. Souza De Paula2, B.
Spaan9, A. Sparkes46, P. Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39,
S. Stoica28, S. Stone52,37, B. Storaci23, M. Straticiuc28, U. Straumann39,
V.K. Subbiah37, S. Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26,
S. T’Jampens4, E. Teodorescu28, F. Teubert37, C. Thomas51, E. Thomas37, J. van
Tilburg11, V. Tisserand4, M. Tobin39, S. Topp-Joergensen51, N. Torr51, E.
Tournefier4,49, M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23, M. Ubeda
Garcia37, A. Ukleja27, P. Urquijo52, U. Uwer11, V. Vagnoni14, G. Valenti14, R.
Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis42, M.
Veltri17,g, B. Viaud7, I. Videau7, X. Vilasis-Cardona35,n, J. Visniakov36, A.
Vollhardt39, D. Volyanskyy10, D. Voong42, A. Vorobyev29, H. Voss10, S.
Wandernoth11, J. Wang52, D.R. Ward43, N.K. Watson55, A.D. Webber50, D.
Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G. Wilkinson51, M.P.
Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25, W.
Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Xing52, Z.
Yang3, R. Young46, O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3, L. Zhang52,
W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin37.
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
23Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraców, Poland
26AGH University of Science and Technology, Kraców, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55University of Birmingham, Birmingham, United Kingdom
56Physikalisches 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
The $X(3872)$ particle was discovered in 2003 by the Belle collaboration in
the $B^{\pm}\rightarrow X(3872)K^{\pm}$, $X(3872)\rightarrow
J/\psi\pi^{+}\pi^{-}$ decay chain [1]. Its existence was confirmed by the CDF
[2], DØ [3] and BaBar [4] collaborations. The discovery of the $X(3872)$
particle and the subsequent observation of several other new states in the
mass range $3.9-4.7~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ have led to a
resurgence of interest in exotic meson spectroscopy [5].
Several properties of the $X(3872)$ have been determined, in particular its
mass [6, 7, 8] and the dipion mass spectrum in the decay $X(3872)\rightarrow
J/\psi\pi^{+}\pi^{-}$ [9, 7], but its quantum numbers, which have been
constrained to be either $J^{PC}=2^{-+}$ or $1^{++}$ [10], are still not
established. Despite a large experimental effort, the nature of this new state
is still uncertain and several models have been proposed to describe it. The
$X(3872)$ could be a conventional charmonium state, with one candidate being
the $\eta_{c2}(1D)$ meson [5]. However, the mass of this state is predicted to
be far below the observed $X(3872)$ mass. Given the proximity of the $X(3872)$
mass to the $D^{*0}\bar{D}^{0}$ threshold, another possibility is that the
$X(3872)$ is a loosely bound $D^{*0}\bar{D}^{0}$ ‘molecule’, i.e. a
$((u\overline{c})(c\overline{u}))$ system [5]. For this interpretation to be
valid the mass of the $X(3872)$ should be less than the sum of $D^{*0}$ and
${D}^{0}$ masses. A further, more exotic, possibility is that the $X(3872)$ is
a tetraquark state [11].
Measurements of $X(3872)$ production at hadron colliders, where most of the
production is prompt rather than from $b$-hadron decays, may shed light on the
nature of this particle. In particular, it has been discussed whether or not
the possible molecular nature of the $X(3872)$ is compatible with the
production rate observed at the Tevatron [12, 13]. Predictions for $X(3872)$
production at the LHC have also been published [13].
This paper reports an observation of $X(3872)$ production in $pp$ collisions
at $\sqrt{s}~{}=7~{}\mathrm{\,Te\kern-1.00006ptV}$ using an integrated
luminosity of 34.7 $\mbox{\,pb}^{-1}$ collected by the LHCb experiment. The
$X(3872)\rightarrow J/\psi\pi^{+}\pi^{-}$ selection is optimized on the
similar but more abundant $\psi(2S)\rightarrow J/\psi\pi^{+}\pi^{-}$ decay.
The observed $X(3872)$ signal is used to measure both the $X(3872)$ mass and
the production rate from all sources including $b$-hadron decays, i.e. the
absolute inclusive $X(3872)$ production cross-section in the detector
acceptance multiplied by the $X(3872)\rightarrow J/\psi\pi^{+}\pi^{-}$
branching fraction.
## 2 The LHCb spectrometer and data sample
The LHCb detector is a forward spectrometer [14] at the Large Hadron Collider
(LHC). It provides reconstruction of charged particles in the pseudorapidity
range $2<\eta<5$. The detector elements are placed along the LHC beam line
starting with the vertex detector (VELO), a silicon strip device that
surrounds the proton-proton interaction region. It is used to reconstruct both
the interaction vertices and the decay vertices of long-lived hadrons. It also
contributes to the measurement of track momenta, along with a large area
silicon strip detector located upstream of a dipole magnet and a combination
of silicon strip detectors and straw drift-tubes placed downstream. The magnet
has a bending power of about 4 Tm. 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}$. Two ring imaging Cherenkov (RICH)
detectors are used to identify charged hadrons. The detector is completed by
electromagnetic calorimeters for photon and electron identification, a hadron
calorimeter, and a muon system consisting of alternating layers of iron and
multi-wire 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.
The cross-section analysis described in this paper is based on a data sample
collected in 2010, exclusively using events that passed dedicated $J/\psi$
trigger algorithms. These algorithms selected a pair of oppositely-charged
muon candidates, where either one of the muons had a transverse momentum
$p_{\rm T}$ larger than 1.8 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ or one of
the two muons had $p_{\rm T}>0.56$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
the other $p_{\rm T}>0.48$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The pair of
muons was required to originate from a common vertex and have an invariant
mass in a wide window around the $J/\psi$ mass. The $X(3872)$ mass measurement
also uses events triggered with other algorithms, such as single-muon
triggers. To avoid domination of the trigger CPU time by a few events with
high occupancy, a set of cuts was applied on the hit multiplicity of each sub-
detector used by the pattern recognition algorithms. These cuts reject high-
multiplicity events with a large number of $pp$ interactions.
The accuracy of the $X(3872)$ mass measurement relies on the calibration of
the tracking system [15]. The spatial alignment of the tracking detectors, as
well as the calibration of the momentum scale, are based on the
$J/\psi\rightarrow\mu^{+}\mu^{-}$ mass peak. This was carried out in seven
time periods corresponding to known changes in the detector running
conditions. The procedure takes into account the effects of QED radiative
corrections which are important in this decay.
The analysis uses fully simulated samples based on the Pythia 6.4 generator
[16] configured with the parameters detailed in Ref. [17]. The EvtGen [18],
Photos [19] and Geant4 [20] packages are used to describe the decays of
unstable particles, model QED radiative corrections and simulate interactions
in the detector, respectively. The $X(3872)\rightarrow J/\psi\pi^{+}\pi^{-}$
Monte Carlo events are generated assuming that the $\rho$ resonance dominates
the dipion mass spectrum, as established by the CDF [9] and Belle [7] data.
## 3 Event selection
To isolate the $X(3872)$ signal, tight cuts are needed to reduce combinatorial
background where a correctly reconstructed $J/\psi$ meson is combined with a
random $\pi^{+}\pi^{-}$ pair from the primary $pp$ interaction. The cuts are
defined using reconstructed $\psi(2S)\rightarrow J/\psi\pi^{+}\pi^{-}$ decays,
as well as ‘same-sign pion’ candidates satisfying the same criteria as used
for the $X(3872)$ and $\psi(2S)$ selection but where the two pions have the
same electric charge. The Kullback-Leibler (KL) distance [21, *kl2, *kl3] is
used to suppress duplicated particles created by the reconstruction: if two
particles have a symmetrized KL divergence less than 5000, only that with the
higher track fit quality is considered.
$J/\psi\rightarrow\mu^{+}\mu^{-}$ candidates are formed from pairs of
oppositely-charged particles identified as muons, originating from a common
vertex with a $\chi^{2}$ per degree of freedom ($\chi^{2}/{\rm ndf}$) smaller
than $20$, and with an invariant mass in the range
$3.04-3.14~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The two muons are each
required to have a momentum above $10~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
and a transverse momentum above $1~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. To
reduce background from the decay in flight of pions and kaons, each muon
candidate is required to have a track fit $\chi^{2}/{\rm ndf}$ less than 4.
Finally $J/\psi$ candidates are required to have a transverse momentum larger
than $3.5~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
Pairs of oppositely-charged pions are combined with $J/\psi$ candidates to
build $\psi(2S)$ and $X(3872)$ candidates. To reduce the combinatorial
background, each pion candidate is required to have a transverse momentum
above $0.5~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and a track fit
$\chi^{2}/{\rm ndf}$ less than 4. In addition, kaons are removed using the
RICH information by requiring the likelihood for the kaon hypothesis to be
smaller than that for the pion hypothesis. A vertex fit is performed [24] that
constrains the four daughter particles to originate from a common point and
the mass of the muon pair to the nominal $J/\psi$ mass [25]. This fit both
improves the mass resolution and reduces the sensitivity of the result to the
momentum scale calibration. To further reduce the combinatorial background the
$\chi^{2}/{\rm ndf}$ of this fit is required to be less than 5. Finally, the
requirement $Q<300~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is applied
where $Q=M_{\mu\mu\pi\pi}-M_{\mu\mu}-M_{\pi\pi}$, and $M_{\mu\mu\pi\pi}$,
$M_{\mu\mu}$ and $M_{\pi\pi}$ are the reconstructed masses before any mass
constraint; this requirement removes $35\%$ of the background whilst retaining
$97\%$ of the $X(3872)$ signal.
Figure 1 shows the $J/\psi\pi^{+}\pi^{-}$ mass distribution for the selected
candidates, with clear signals for both the $\psi(2S)$ and the $X(3872)$
mesons, as well as the $J/\psi\pi^{\pm}\pi^{\pm}$ mass distribution of the
same-sign pion candidates.
Figure 1: Invariant mass distribution of $J/\psi\pi^{+}\pi^{-}$ (points with
statistical error bars) and same-sign $J/\psi\pi^{\pm}\pi^{\pm}$ (filled
histogram) candidates. The curves are the result of the fit described in the
text. The inset shows a zoom of the $X(3872)$ region.
## 4 Mass measurements
The masses of the $\psi(2S)$ and $X(3872)$ mesons are determined from an
extended unbinned maximum likelihood fit of the reconstructed
$J/\psi\pi^{+}\pi^{-}$ mass in the interval
$3.60<M_{J/\psi\pi\pi}<3.95~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The
$\psi(2S)$ and $X(3872)$ signals are each described with a non-relativistic
Breit-Wigner function convolved with a Gaussian resolution function. The
intrinsic width of the $\psi(2S)$ is fixed to the PDG value,
$\Gamma_{\psi(2S)}=0.304{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [25]. The
Belle collaboration recently reported [7] that the $X(3872)$ width is less
than $1.2$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ at $90\%$ confidence
level; we fix the $X(3872)$ width to zero in the nominal fit. The ratio of the
mass resolutions for the $X(3872)$ and the $\psi(2S)$ is fixed to the value
estimated from the simulation, $\sigma^{\rm MC}_{X(3872)}/\sigma^{\rm
MC}_{\psi(2S)}=1.31$.
Studies using the same-sign pion candidates show that the background shape can
be described by the functional form $f(M)\propto(M-m_{\rm
th})^{c_{0}}\exp(-c_{1}M-c_{2}M^{2})$, where $m_{\rm
th}=m_{J/\psi}+2m_{\pi}=3376.05~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
[25] is the mass threshold and $c_{0}$, $c_{1}$ and $c_{2}$ are shape
parameters. To improve the stability of the fit, the parameter $c_{2}$ is
fixed to the value obtained from the same-sign pion sample.
In total, the fit has eight free parameters: three yields ($\psi(2S)$,
$X(3872)$ and background), two masses ($\psi(2S)$ and $X(3872)$), one
resolution parameter, and two background shape parameters. The correctness of
the fitting procedure has been checked with simplified Monte Carlo samples,
fully simulated Monte Carlo samples, and samples containing a mixture of fully
simulated Monte Carlo signal events and same-sign background events taken from
the data. The fit results are shown in Fig. 1 and Table 1. The fit does not
account for QED radiative corrections and hence underestimates the masses.
Using a simulation based on Photos [19] the biases on the $X(3872)$ and
$\psi(2S)$ masses are found to be $-0.07\pm
0.02{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $-0.02\pm
0.02{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively. The fitted mass
values are corrected for these biases and the uncertainties propagated in the
estimate of the systematic error.
Table 1: Results of the fit to the $J/\psi\pi^{+}\pi^{-}$ invariant mass distribution of Fig. 1. Fit parameter or derived quantity | $\psi(2S)$ | $X(3872)$
---|---|---
Number of signal events | $3998$ | $\pm\,83$ | $565$ | $\pm\,62$
Mass $m~{}[{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}]$ | $3686.10$ | $\pm\,0.06$ | $3871.88$ | $\pm\,0.48$
Resolution $\sigma~{}[{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}]$ | $2.54$ | $\pm\,0.06$ | $3.33$ | $\pm\,0.08$
Signal-to-noise ratio in $\pm 3\sigma$ window | $1.5$ | | $0.15$ |
Number of background events | $73094\pm 282$
Several other sources of systematic effects on the mass measurements are
considered. For each source, the complete analysis is repeated (including the
track fit and the momentum scale calibration when needed) under an alternative
assumption, and the observed change in the central value of the fitted masses
relative to the nominal results assigned as a systematic uncertainty. The
dominant source of uncertainty is due to the calibration of the momentum
scale. Based on checks performed with reconstructed signals of various mesons
decaying into two-body final states (such as $\pi^{+}\pi^{-}$,
$K^{\mp}\pi^{\pm}$ and $\mu^{+}\mu^{-}$) a relative systematic uncertainty of
0.02% is assigned to the momentum scale [15], which translates into a 0.10
(0.08) ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ uncertainty on the $X(3872)$
($\psi(2S)$) mass. After the calibration procedure with the
$J/\psi\rightarrow\mu^{+}\mu^{-}$ decay, a $\pm 0.07\%$ variation of the
momentum scale remains as a function of the particle pseudorapidity $\eta$. To
first order this effect averages out in the mass determination. The residual
impact of this variation is evaluated by parameterizing the momentum scale as
function of $\eta$ and repeating the analysis. The systematic uncertainty
associated with the momentum calibration indirectly takes into account any
effect related to the imperfect alignment of the tracking stations. However,
the alignment of the VELO may affect the mass measurements through the
determination of the horizontal and vertical slopes of the tracks. This is
investigated by changing the track slopes by amounts corresponding to the 0.1%
relative precision with which the length scale along the beam axis is known
[26]. Other small uncertainties arise due to the limited knowledge of the
$X(3872)$ width and the modelling of the resolution. The former is estimated
by fixing the $X(3872)$ width to 0.7
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ instead of zero, as suggested by
the likelihood published by Belle [7]. The latter is estimated by fixing the
ratio $\sigma_{X(3872)}/\sigma_{\psi(2S)}$ using the covariance estimates
returned by the track fit algorithm on signal events in the data sample,
rather than using the mass resolutions from the simulation. The effect of
background modelling is estimated by performing the fit on two large samples,
one with only Monte Carlo signal events, and one containing a mixture of Monte
Carlo signal events and background candidates obtained by combining a $J/\psi$
candidate and a same-sign pion pair from different data events: the difference
in the fitted mass values is taken as a systematic uncertainty. The amount of
material traversed in the tracking system by a particle is estimated to be
known to a 10% accuracy [27]; the magnitude of the energy loss correction in
the reconstruction is therefore varied by 10%. The assigned systematic
uncertainties are summarized in Table 2 and combined in quadrature.
Table 2: Systematic uncertainties on the $\psi(2S)$ and $X(3872)$ mass measurements. Category | Source of uncertainty | $\Delta m$ [${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$]
---|---|---
$\psi(2S)$ | $X(3872)$
Mass fitting | Natural width | – | 0.01
Radiative tail | 0.02 | 0.02
Resolution | – | 0.01
Background model | 0.02 | 0.02
Momentum calibration | Average momentum scale | 0.08 | 0.10
$\eta$ dependence of momentum scale | 0.02 | 0.03
Detector description | Energy loss correction | 0.05 | 0.05
Detector alignment | Track slopes | 0.01 | 0.01
Total | | 0.10 | 0.12
Systematic checks of the stability of the measured $\psi(2S)$ mass are
performed, splitting the data sample according to different run periods or to
the dipole magnet polarity, or ignoring the hits from the tracking station
before the magnet. In addition, the measurement is repeated in bins of the
$p$, $p_{\rm T}$ and $Q$ values of the $\psi(2S)$ signal. No evidence for a
systematic bias is found.
## 5 Determination of the production cross-section
The observed $X(3872)$ signal is used to measure the product of the inclusive
production cross-section $\sigma(pp\rightarrow X(3872)+{\rm anything})$ and
the branching fraction $\mathcal{B}(X(3872)\rightarrow J/\psi\pi^{+}\pi^{-})$,
according to
$\sigma(pp\rightarrow X(3872)+{\rm anything})\,\mathcal{B}(X(3872)\rightarrow
J/\psi\pi^{+}\pi^{-})=\frac{N^{\rm
corr}_{X(3872)}}{\xi\,\mathcal{B}(J/\psi\rightarrow\mu^{+}\mu^{-})\,\mathcal{L}_{\rm
int}}\,,$ (1)
where $N^{\rm corr}_{X(3872)}$ is the efficiency-corrected signal yield, $\xi$
is a correction factor to the simulation-derived efficiency that accounts for
known differences between data and simulation,
$\mathcal{B}(J/\psi\rightarrow\mu^{+}\mu^{-})=(5.93\pm 0.06)\%$ [25] is the
$J/\psi\rightarrow\mu^{+}\mu^{-}$ branching fraction, and $\mathcal{L}_{\rm
int}$ is the integrated luminosity.
The absolute luminosity scale was measured at specific periods during the 2010
data taking [28] using both Van der Meer scans [29] and a beam-gas imaging
method [30]. The instantaneous luminosity determination is then based on a
continuous recording of the multiplicity of tracks in the VELO, which has been
normalized to the absolute luminosity scale [28]. The integrated luminosity of
the sample used in this analysis is determined to be $\mathcal{L}_{\rm
int}=34.7\pm 1.2\mbox{\,pb}^{-1}$, with an uncertainty dominated by the
knowledge of the beam currents.
Only $X(3872)$ candidates for which the $J/\psi$ triggered the event are
considered, keeping 70% of the raw signal yield used for the mass measurement.
In addition, the candidates are required to lie inside the fiducial region for
the measurement,
$2.5<y<4.5~{}~{}~{}\mbox{and}~{}~{}~{}5<p_{\rm
T}<20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}\,,$ (2)
where $y$ and $p_{\rm T}$ are the rapidity and transverse momentum of the
$X(3872)$. This region provides a good balance between a high efficiency (92%
of the triggered events) and a low systematic uncertainty on the acceptance
correction.
The corrected yield $N^{\rm corr}_{X(3872)}=9140\pm 2224$ is obtained from a
mass fit in the narrow region
$3820-3950{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, with a linear background
model and the same $X(3872)$ signal model as used previously but with the mass
and resolution fixed to the central values presented in Sect. 4. In this fit,
each candidate is given a weight equal to the reciprocal of the total signal
efficiency estimated from simulation for the $y$ and $p_{\rm T}$ of that
candidate. A second method based on the sWeight [31] technique was found to
give consistent results. The average total signal efficiency in the fiducial
region of Eq. 2 is estimated to be $N_{X(3872)}/N^{\rm corr}_{X(3872)}=4.2\%$,
where $N_{X(3872)}$ is the observed signal yield obtained from a mass fit
without weighting the events. This low value of the efficiency is driven by
the geometrical acceptance and the requirement on the $p_{\rm T}$ of the
$J/\psi$ meson.
Table 3: Relative systematic uncertainties on the $X(3872)$ production cross-section measurement. The total uncertainty is the quadratic sum of the individual contributions. Source of uncertainty | $\Delta\sigma/\sigma$ [%]
---|---
$X(3872)$ polarization | $2.1$
$X(3872)$ decay model | $1.0$
$X(3872)$ decay width | $5.0$
Mass resolution | $2.5$
Background model | $6.4$
Tracking efficiency | 7.4
Track $\chi^{2}$ cut | 2.0
Vertex $\chi^{2}$ cut | 3.0
Muon trigger efficiency | 2.9
Hit-multiplicity cuts | 3.0
Muon identification | 1.1
Pion identification | 4.9
Integrated luminosity | 3.5
$J/\psi\rightarrow\mu^{+}\mu^{-}$ branching fraction | 1.0
Total | $14.2$
The quantity $\xi$ of Eq. 1 is the product of three factors. The first two,
$1.024\pm 0.011$ [32] and $0.869\pm 0.043$, account for differences between
the data and simulation for the efficiency of the muon and pion
identifications, respectively. The third factor, $0.92\pm 0.03$, corresponds
to the efficiency of the hit-multiplicity cuts applied in the trigger, which
is not accounted for in the simulation. It is obtained from a fit of the
distribution of the number of hits in the VELO.
The relative systematic uncertainties assigned to the cross-section
measurement are listed in Table 3, and quadratically add up to 14.2%. The
cross-section measurement is performed under the most favoured assumption for
the quantum numbers of the $X(3872)$ particle, $J^{PC}=1^{++}$ [33], which is
used for the generation of Monte Carlo events. No systematic uncertainty is
assigned to cover other cases. Besides the uncertainties already mentioned on
$\mathcal{B}(J/\psi\rightarrow\mu^{+}\mu^{-})$, $\mathcal{L}_{\rm int}$ and
$\xi$, the following sources of systematics on $N^{\rm corr}_{X(3872)}$ are
considered. The dominant uncertainty is due to differences in the efficiency
of track reconstruction between the data and simulation. This is estimated to
be $7.4\%$ using a data driven tag and probe approach based on
$J/\psi\rightarrow\mu^{+}\mu^{-}$ candidates. An additional uncertainty of
0.5% per track is assigned to cover differences in the efficiency of the track
$\chi^{2}/{\rm ndf}$ cut between data and simulation. Similarly, a 3%
uncertainty is assigned due to the effect of the vertex $\chi^{2}$ cuts.
Other important sources of uncertainty are due to the modelling of the signal
and background mass distributions. Repeating the mass fit with the $X(3872)$
decay width fixed to $0.7~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ instead
of zero results in a 5% change of the signal yield. Similarly, the
uncertainties due to the $X(3872)$ mass resolution are estimated by repeating
the mass fit with different fixed mass resolutions: first changing it by the
statistical uncertainty reported in Table 1, and then changing it by the
systematic uncertainty resulting from the knowledge of the resolution ratio
$\sigma_{X(3872)}/\sigma_{\psi(2S)}$, as described in Sect. 4. The combined
effect on the $X(3872)$ signal yield corresponds to a 2.5% systematic
uncertainty.
Using an exponential rather than linear function to describe the background
leads to a change of 6.4% in signal yield, which is taken as an additional
systematic uncertainty.
The unknown $X(3872)$ polarization affects the total efficiency, mainly
through the $J/\psi$ reconstruction efficiency. The dipion system is less
affected, in particular the efficiency is found to be constant as a function
of the dipion mass. The simulation efficiency, determined assuming no $J/\psi$
polarization, is recomputed in two extreme schemes for the $J/\psi$
polarization (fully transverse and fully longitudinal) [32] and the maximum
change of 2.1% is taken as systematic uncertainty. The efficiency of the $Q$
cut depends on the $X(3872)$ decay model. The dipion mass spectrum obtained in
this analysis does not have enough accuracy to discriminate between reasonable
models. Comparing the results obtained with the $X(3872)\rightarrow
J/\psi\rho$ decay models used by CDF [9] and by Belle [7], we evaluated a 1%
systematic uncertainty on the $Q$-cut efficiency.
Finally, differences in the trigger efficiency between data and simulation are
studied using events triggered independently of the $J/\psi$ candidate. Based
on these studies an uncertainty of 2.9% is assigned.
## 6 Results and conclusion
With an integrated luminosity of 34.7$\mbox{\,pb}^{-1}$ collected by the LHCb
experiment, the production of the $X(3872)$ particle is observed in $pp$
collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. The product of the
production cross-section and the branching ratio into $J/\psi\pi^{+}\pi^{-}$
is
$\sigma(pp\rightarrow X(3872)+{\rm anything})\,\mathcal{B}(X(3872)\rightarrow
J/\psi\pi^{+}\pi^{-})=5.4\pm 1.3\,{\rm(stat)}\pm 0.8\,{\rm(syst)}\rm\,nb\,,$
for $X(3872)$ mesons produced (either promptly or from the decay of other
particles) with a rapidity between 2.5 and 4.5 and a transverse momentum
between 5 and 20${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
Predictions for the $X(3872)\rightarrow J/\psi\pi^{+}\pi^{-}$ production at
the LHC are available from a non-relativistic QCD model which assumes that the
cross-section is dominated by the production of charm quark pairs with
negligible relative momentum [13]. The calculations are normalized using
extrapolations from measurements performed at the Tevatron. When restricted to
the kinematic range of our measurement and summed over prompt production and
production from $b$-hadron decays, the results of Ref. [13] yield $13.0\pm
2.7\rm\,nb$, where the quoted uncertainty originates from the experimental
inputs used in the calculation. This prediction exceeds our measurement by
$2.4\sigma$.
After calibration using $J/\psi\rightarrow\mu^{+}\mu^{-}$ decays, the masses
of both the $X(3872)$ and $\psi(2S)$ mesons, reconstructed in the same
$J/\psi\pi^{+}\pi^{-}$ final state, are measured to be
$\displaystyle m_{X(3872)}$ $\displaystyle=$ $\displaystyle 3871.95\pm
0.48\,({\rm stat})\pm 0.12\,({\rm
syst}){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}\,,$ $\displaystyle
m_{\psi(2S)}$ $\displaystyle=$ $\displaystyle 3686.12\pm 0.06\,({\rm stat})\pm
0.10\,({\rm syst}){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}\,,$
in agreement with the current world averages [25], and with the recent
$X(3872)$ mass measurement from Belle [7]. The measurements of the $X(3872)$
mass are consistent, within uncertainties, with the sum of the $D^{0}$ and
$D^{*0}$ masses, $3871.79\pm 0.29~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$,
computed from the results of the global PDG fit of the charm meson masses
[25].
## Acknowledgements
We thank P. Artoisenet and E. Braaten for useful discussions and for
recomputing the numerical prediction of Ref. [13] in the fiducial region of
our measurement. 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
| 2011-12-22T13:51:14 |
2024-09-04T02:49:25.614032
|
{
"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 G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis, J.\n Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, D.\n S. Bailey, 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, F.\n Constantin, A. Contu, A. Cook, M. Coombes, G. Corti, 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, F. De Lorenzi, J. M. De Miranda, L. De Paula, P. De Simone, D. Decamp,\n M. 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, L. Est\\`eve, A. Falabella, E.\n Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, 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, N. Gauvin, M.\n Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, G. Haefeli, C.\n Haen, S. 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, P. Koppenburg, A. Kozlinskiy, L. Kravchuk, K. Kreplin,\n M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, T.\n Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M. Lieng, M. Liles,\n R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E. Lopez\n Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F. Machefert, I.\n V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R. M. D. Mamunur,\n G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, L. Martin, A. Mart\\'in S\\'anchez, 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, R. Messi, S. Miglioranzi, D. A. Milanes, M.-N. Minard, J. Molina\n Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Nedos, M.\n Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora Goicochea, P.\n Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis,\n M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D. Patel, M.\n Patel, S. K. Paterson, G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A.\n Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.\n L. Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrella, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R. Plackett, S.\n Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B.\n Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig\n Navarro, W. Qian, J. H. Rademacker, B. Rakotomiaramanana, M. S. Rangel, I.\n Raniuk, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, K.\n Rinnert, D. A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez\n Perez, G. J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T.\n Ruf, H. Ruiz, G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, C. Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R.\n Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M.\n Savrie, D. Savrina, P. Schaack, M. Schiller, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, A. C. Smith, N. A.\n Smith, E. Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F. Soomro, B. Souza\n 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. Topp-Joergensen, N. Torr, E. Tournefier, M. T.\n Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, P. Urquijo,\n U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S.\n Vecchi, J. J. Velthuis, M. Veltri, B. Viaud, I. Videau, X. Vilasis-Cardona,\n J. Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, H. Voss, 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. Zavertyaev, F.\n Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, E. Zverev, A.\n Zvyagin",
"submitter": "Joel Bressieux Mr.",
"url": "https://arxiv.org/abs/1112.5310"
}
|
1112.5461
|
# A-geometrical approach to Topological Insulators with defects
D. Schmeltzer Physics Department, City College of the City University of New
York
New York, New York 10031
###### Abstract
The study of the propagation of electrons with a varying spinor orientability
is performed using the coordinate transformation method.
Topological Insulators are characterized by an odd number of changes of the
orientability in the Brillouin zone. For defects the change in orientability
takes place for closed orbits in real space. Both cases are characterized by
nontrivial spin connections.
Using this method , we derive the form of the spin connections for topological
defects in three dimensional Topological Insulators.
On the surface of a Topological Insulator, the presence an edge dislocation
gives rise to a spin connection controlled by torsion. We find that electrons
propagate along two dimensional regions and confined circular contours. We
compute for the edge dislocations the tunneling density of states. The edge
dislocations violates parity symmetry resulting in a current measured by the
in-plane component of the spin on the surface.
I Introduction
The propagation of electrons in solids is characterized by the topological
properties of the the electronic band spinors. Topological Insulators Konig ;
Volkov ; Gotelman ; Kreutz ; Mele ; Kane ; More ; Essin ; BernewigZhang ;
Ludwig ; davidtop ; ZhangField ; Zhangnew can be identified by an odd number
of changes of the $orientability$ davidtop of the spinors in the Brillouin
zone. As a results non trivial spin connections with a non- zero curvature
characterized by the Chern numbers can be identified. In time reversal
invariant systems one finds that for Kramer’s states the time reversal
operator $T$ obeys $T^{2}=-1$ and one thus the second Chern number for four
dimensional space is given by $(-1)^{\nu}=-1$, where $\nu$ is an odd number of
orientability changes Nakahara .
Real materials are imperfect and contain topological defects such as
dislocations Ran ; Sinova ,disclinations alberto ; Vozmediano and gauge
fields induced by strain in graphene Baruch ; Guinea ;therefore, a natural
question is to formulate the physics of Topological Insulators in the presence
of such defects davidtop . These topological defects can be analyzed using the
coordinate transformation method given in ref.kleinert which modifies the
Hamiltonian for a Topological Insulator with a defect by the metric tensor and
the spin connection Pnueli ; Green ; Birrell ; Randono ; Ryu .
The effect of strain fields dislocations and disclinations plays an important
role in material science and can be study using Scanning Tunneling Microscopy
($STM$) and Transmission Electron Spectroscopy ($TEM$ ). Therefore we expect
that the chiral metallic boundary Wu will be sensitive to such defects.
In this paper we will introduce the tangent space approach used in
differential geometry Nakahara ; Randono ; Ryu to study propagation of
electrons for a space dependent coordinate kleinert . We find that the
continuum representation of the edge dislocation kleinert generates a spin
connection Pnueli ; Green ; Birrell which is controlled by the $Burger$
vector.
Using this formulation we obtain the form of the topological insulator in
three dimensions which simplifies for the surface Hamiltonian (on the
boundary). For the surface Hamiltonian we find that the electronic excitations
are confined to a two-dimensional region and to a set of circular contours of
radius $R_{g}(n)$.
The contents of this paper is as follows: In chapter $II$ we introduce the
gemetrical method. In section $IIA$ we present the geometrical method for the
edge dislocations and strain fields. In section $IIB$ we consider the effects
of the strain fields on the three- dimensional Topological Insulator ($TI$).
The Chiral model for the boundary surface is presented in section $IIIA$.
Section $IIIB$ is devoted to the derivation of the metric tensor and spin
connection for an edge dislocation kleinert . In section $IIIC$ we identify
the stable solutions. Section $IIID$ is devoted to the stable two dimensional
solutions $n=0$ and section $IIIE$ is devoted to the stable solution for
circular contours $n=\pm 1$. Chapter $IVA$ is devoted to the computation of
the tunneling density of states. In section $IVB$ we present results for the
two dimensional region $n=0$. Section $IVC$ is devoted to a large number of
dislocations. In section $IVD$ we compute the tunneling density of states for
the circular contours $n=\pm 1$. In chapter $V$ we consider the current which
is given by the in-plane spin component. In section $VA$ we show that this
current is zero for a $TI$. In section $VB$ we show that in the presence of an
edge dislocation the parity symmetry is violated, and current, representing
the in-plane spin component, is generated. Chapter $VI$ is devoted to
conclusions.
II-The Geometrical method for dislocations and strain fields
A-General Considerations
A perfect crystal is described by the lattice coordinates $\vec{r}=[x,y,z]$.
For a crystal with a deformation , the coordinates $\vec{r}$ are replaced by
$\vec{r}\rightarrow\vec{R}=\vec{r}+\vec{u}\equiv[X^{1}(\vec{r}),X^{2}(\vec{r}),X^{3}(\vec{r})]$
where $\vec{u}(\vec{r})$ is the local lattice deformation and $X^{a}$,
$a=1,2,3$ is the local coordinate which changes when we move from one point to
another.
In a deformed crystal we introduced a set of local vectors $e_{a}$ which are
orthogonal to each other $(e_{b},e_{a})\equiv<e^{b}|e_{a}>=\delta^{b}_{a}$ and
local coordinates $X^{a}$, $a=1,2,3$. The unit vector $e_{a}$ can be
represented in terms of a Cartesian fixed frame space with the coordinate
basis $\partial_{\mu}$ ,$\mu=x,y,z$. In the fixed Cartesian frame the
coordinates are given by $x^{\mu}$. Using the Cartesian basis $\partial_{\mu}$
we expand the deformed medium in terms of the local tangent vector $e_{a}$ :
$e_{a}=e^{\mu}_{a}\partial_{\mu}$ (for the particular case where vectors
$e_{a}$ are given by $e_{a}=\partial_{a}$, the transformation between the two
basis is $e^{\mu}_{a}=\delta^{\mu}_{a}$). Any vector $\vec{X}$ (in the
deformed space) can be represented in terms of the unit vectors $e_{a}$ or the
$\partial_{\mu}$ (the tangent vectors in the Cartesian fixed coordinates
space). The vector $\vec{X}$ can be represented in two different ways,
$\vec{X}=X^{a}e_{a}=X^{\mu}\partial_{\mu}$ (when an index appear twice is
understood as a summation, $X^{a}e_{a}\equiv\sum_{a=1,2,3}X^{a}e_{a}$). The
dual vector $e^{a}$ is a $one$ $form$ and can be expanded in terms of the one
forms $dx^{\mu}$. We have: $e^{a}=e^{a}_{\mu}dx$, where $e^{a}_{\mu}$
represents the matrix transformation
$e^{a}\equiv(\partial_{\mu}X^{a})dx^{\mu}$. The scalar product of the
components $e^{a}_{\mu}e^{a}_{\nu}=g_{\mu,\nu}$,
$e^{\nu}_{a}e^{\nu}_{b}=\delta_{a,b}$ defines the metric tensors,
$g_{\mu,\nu}$ (in the Cartesian frame ) and $\delta_{a,b}$ in the local medium
frame.
B-Application to the Topological insulators in three dimensions
The three dimensional electronic $TI$ bands for $Bi_{2}Se_{3}$ and
$Bi_{2}Te_{3}$ can be represented using four projected states Chao ,
$|orbital=1,2>\otimes|spin=\uparrow,\downarrow>$ (the Pauli matrix $\tau$
describes the orbital states and the Pauli matrix $\sigma$ describes the
spin). The effective $h^{3D}$ Hamiltonian in the first quantized form is given
by:
$h^{3D}=\hbar
v_{0}[k_{y}(\sigma_{1}\otimes\tau_{1})-k_{x}(\sigma_{2}\otimes\tau_{1})+\epsilon
k_{z}(\sigma_{3}\otimes\tau_{1})+M(\vec{k})(I\otimes\tau_{3})]$ (1)
The parameter $M(\vec{k})$ determines if the insulator is trivial or
topological. For $Bi_{2}Se_{3}$ and $Bi_{2}Te_{3}$ the gap is inverted, namely
$M(\vec{k})=-M_{0}+B_{1}k_{z}^{2}+B_{2}k_{\bot}^{2}$ with
$M_{0}>0,B_{1}>0,B_{2}>0$ and therefore topological ZhangField ; Chao ;
Zhangnew .
Using the metric tensor $g_{\mu,\nu}$ given by the coordinate transformation (
the transformation between the two sets of coordinates - the one without the
dislocation and the second with the dislocation )
$e^{a}_{\mu}e^{a}_{\nu}=g_{\mu,\nu}$, defines the Jacobian $\sqrt{G}$ where
$G=det[g_{\mu,\nu}]$. We find that the derivative for a spinor component
$\Psi^{(\alpha)}(\vec{r})$,
$\alpha=[1=1\uparrow;2=1\downarrow;3=2\uparrow;4=2\downarrow]$ is replaced by
the $covariant$ derivative Green :
$\nabla_{\mu}\Psi^{(\alpha)}(\vec{r})=\partial_{\mu}\Psi^{(\alpha)}(\vec{r})+\frac{1}{8}\omega^{(a,b)}_{\mu}[\hat{\Gamma}^{a},\hat{\Gamma}^{b}]^{\alpha}_{\beta}\Psi^{(\beta)}(\vec{r})$
(2)
where $\hat{\Gamma}^{a}$ ,$a=1,2,3,4,5$ are the matrixes:
$\hat{\Gamma}^{1}=-\Gamma^{2}\equiv-(\sigma_{2}\otimes\tau_{1})$;
$\hat{\Gamma}^{2}=\Gamma^{1}\equiv(\sigma_{1}\otimes\tau_{1})$;
$\hat{\Gamma}^{3}=\Gamma^{3}\equiv(\sigma_{3}\otimes\tau_{1})$;
$\hat{\Gamma}^{4}=\Gamma^{4}\equiv(I\otimes\tau_{2})$;$\hat{\Gamma}^{5}=\Gamma^{5}\equiv(I\otimes\tau_{3})$.
The $spin$ $connection$ $\omega^{a,b}_{\mu}$ determines the covariant
derivative Green is given in terms of the tangent vectors $e^{a}_{\mu}$:
$e^{a}_{\mu}=\partial_{\mu}X^{a}(\vec{r})$; $a=1,2,3$ ; $\mu=x,y,z$.
$\displaystyle\omega^{a,b}_{\mu}=\frac{1}{2}e^{\nu,a}(\partial_{\mu}e^{b}_{\nu}-\partial_{\nu}e^{b}_{\mu})-\frac{1}{2}e^{\nu,b}(\partial_{\mu}e^{a}_{\nu}-\partial_{\nu}e^{a}_{\mu})$
$\displaystyle-\frac{1}{2}e^{\rho,a}e^{\sigma,b}(\partial_{\rho}e_{\sigma,c}-\partial_{\sigma}e_{\rho,c})e^{c}_{\mu}$
(3)
We notice the asymmetry between $e^{\nu,a}$ and $e_{a,\nu}$: $e^{\nu,a}\equiv
g^{\nu,\lambda}e^{a}_{\lambda}$ and $e_{a,\nu}\equiv\delta_{a,b}e^{b}_{\nu}$.
As a result the Hamiltonian in eq.$(1)$ in the second quantized form is
replaced by:
$\displaystyle H^{(3D)}=\hbar
v_{0}\int\,d^{3}r\sqrt{G}[\Psi^{\dagger}(\vec{r})[e^{\mu}_{a}\hat{\Gamma}^{a}(-i\nabla_{\mu})-E_{F}(I\otimes
I)+\hat{\Gamma}^{5}(-M_{0})]\Psi(\vec{r})$
$\displaystyle+B_{1}g^{\mu,\nu}(\nabla_{\mu}\Psi^{\dagger}_{1}(\vec{r}))(\nabla_{\nu}\Psi_{1}(\vec{r}))-B_{1}g^{\mu,\nu}(\nabla_{\mu}\Psi^{\dagger}_{2}(\vec{r})\nabla_{\nu}\Psi_{2})]$
where
$e^{\mu}_{a}\hat{\Gamma}^{a}=\sum_{a}e^{\mu}_{a}\hat{\Gamma}^{a}\equiv\hat{\Gamma}^{\mu}(\vec{r})$,
$[\hat{\Gamma}^{\mu}(\vec{r})\hat{\Gamma}^{\nu}(\vec{r})+\hat{\Gamma}^{\mu}(\vec{r})\hat{\Gamma}^{\nu}(\vec{r})]=2g^{\mu,\nu}(\vec{r})$
, $det[g^{\mu,\nu}(\vec{r})]\equiv G$ and $\nabla_{\mu}$ is the covariant
derivative given in terms of the spin connection given in equation $(2)$:
$\nabla_{\mu}\Psi^{(\alpha)}(\vec{r})=\partial_{\mu}\Psi^{(\alpha)}(\vec{r})+\frac{1}{8}\omega^{(a,b)}_{\mu}[\hat{\Gamma}^{a},\hat{\Gamma}^{b}]^{\alpha}_{\beta}\Psi^{(\beta)}(\vec{r})$
C-The Mechanical strain effect on $H^{(3D)}$
From the work of young we learn that the effect of the strained field is
different on $Bi_{2}Se_{3}$ than on $Bi_{2}Te_{3}$. In $Bi_{2}Se_{3}$ the
$compressive$ strain decreases the Coulombic gap while increasing the inverted
gap strength induced by the spin-orbit interaction. We will use the result in
equation $(4)$ to analyze the effect of strain. The strain field
$\epsilon_{i,j}$ (symmetric in $i,j$) is related to the stress field
$\sigma_{i,j}$ and elastic stiffness $Lame$ constant $\lambda$ and $\mu$:
$\sigma_{i,j}=\lambda\delta_{i,j}\epsilon_{k,k}+2\mu\epsilon_{i,j}$. Applying
a constant stress $\sigma_{i,j}$ one can determine the value of the constant
strain field $\epsilon_{i,j}$ which is related to the tangent vectors
$e^{i}_{j}\equiv\delta_{i,j}+\epsilon_{i,j}$. In the present case the spin
connection and the Christofel tensor vanish. The metric tensor $g_{i,j}$ is
given by :$g_{i,j}=\delta_{i,j}+2\epsilon_{i,j}$. Using this formulation we
can investigate the effect of the stress on the $Bi_{2}Se_{3}$ at the $\Gamma$
point $\vec{k}=0$. The TI Hamiltonian given in eq.$(4)$
,$M(\vec{k})=-M_{0}+B_{1}k^{2}_{z}+B_{2}(k^{2}_{x}+k^{2}_{y})$ with the
inverted case $M_{0}>0$ Zhangnew . The Hamiltonian in eq. $(4)$ is replaced
by:
$\displaystyle H^{(3D-strain)}=\hbar
v_{0}\int\,d^{3}r\sqrt{G}[\Psi^{\dagger}(\vec{r})[\hat{\Gamma}^{a}(\delta_{\mu,a}+\epsilon_{\mu,a})(-i\partial_{\mu}))+\hat{\Gamma}^{4}(-M_{0})+B(1-2\epsilon_{\mu,\nu})\partial_{\mu}\hat{\Gamma}^{4}\partial_{\nu}]\Psi(\vec{r})$
$\displaystyle\approx\hbar
v_{0}\int\,d^{3}r\sqrt{G}[\Psi^{\dagger}(\vec{r})[\hat{\Gamma}^{a}[(\delta_{\mu,a}(1+<\epsilon>)(-i\partial_{\mu}))]+\hat{\Gamma}^{4}(-M_{0})+B(1-2<\epsilon>\delta_{\mu,\nu})\partial_{\mu}\hat{\Gamma}^{4}\partial_{\nu}]\Psi(\vec{r})$
In equation $(5)$ we have used the average strain field $<\epsilon>$,
$<\epsilon>\equiv\frac{\epsilon_{1,1}+\epsilon_{2,2}+\epsilon_{3,3}}{3}$. We
replace the spinor field $\Psi(\vec{r})$ by
$\Psi(\vec{r})\sqrt{(1+<\epsilon>)}\equiv\hat{\Psi}(\vec{r})$. As a result we
obtain:
$\displaystyle H^{(3D-strain)}\approx\hbar
v_{0}\int\,d^{3}r\sqrt{G}[\hat{\Psi}^{\dagger}(\vec{r})\hat{\Gamma}^{\mu}(-i\partial_{\mu})+\hat{\Gamma}^{4}\frac{(-M_{0})}{(1+<\epsilon>)}+B\frac{(1-2<\epsilon>)}{1+<\epsilon>}\partial_{\mu}\hat{\Gamma}^{4}\partial_{\mu}]\hat{\Psi}(\vec{r})$
For the compressive case $<\epsilon>$ is negative,
$<\epsilon>\equiv-|<\epsilon>|$ . As a result we observe that the inverted gap
is enhanced
$\frac{|M_{0}|}{(1+<\epsilon>)}=\frac{|M_{0}|}{(1-|<\epsilon>|)}>|M_{0}|$.
In the same way we can show that the Coulomb interaction is reduced: We
introduce the Hubbard Stratonovici field $a_{0}$ to describe the Coulomb
interactions.
$\displaystyle H^{e-e}=\int\,d^{3}r\sqrt{G}[I(-e)\cdot
a_{0}\Psi^{\dagger}(\vec{r})\Psi(\vec{r})+\frac{(1-2\epsilon_{\mu,\nu})}{2}a_{0}\partial_{\mu}\partial_{\nu}a_{0}]$
$\displaystyle\approx\int\,d^{3}r\sqrt{G}[I(-e)\cdot
a_{0}\Psi^{\dagger}(\vec{r})\Psi(\vec{r})+\frac{(1-2<\epsilon>)}{2}a_{0}\partial_{\mu}\partial_{\nu}a_{0}]$
Next we rescale $a_{0}=\frac{A_{0}}{\sqrt{(1-2<\epsilon>)}}$ and obtain:
$H^{e-e}\approx\int\,d^{3}r\sqrt{G}[I\frac{(-e)}{\sqrt{1-2<\epsilon>}}A_{0}\Psi^{\dagger}(\vec{r})\Psi(\vec{r})+A_{0}\partial_{\mu}\partial_{\mu}A_{0}]$
(8)
We observe that for the compressive case the effective charge
$e_{eff.}\equiv\frac{(-e)}{\sqrt{1-2<\epsilon>}}=\frac{(-e)}{\sqrt{1+2|<\epsilon>|}}$
is reduced and therefore the Coulomb gap decreases, while at the same time the
inverted gap increases,
$\frac{|M_{0}|}{(1+<\epsilon>)}=\frac{|M_{0}|}{(1-|<\epsilon>|)}>|M_{0}|$ in
qualitative agreement with young .
III-The chiral metal with an edge dislocation
A-Description of the Chiral model
The low energy Hamiltonian for the bulk $3D$ $TI$ in the $Bi_{2}Se_{3}$ family
was shown to behave on the boundary surface (the $x,y$\- plane) as a two
dimensional chiral metal nature .
$\displaystyle
H=\int\,d^{2}r\Psi^{\dagger}(\vec{r})[h^{T.I}-\mu]\Psi(\vec{r})]\equiv\hbar
v_{F}\int\,d^{2}r\Psi^{\dagger}(\vec{r})[i\sigma^{1}\partial_{y}-i\sigma^{2}\partial_{x}-\mu]\Psi(\vec{r})$
$h^{T.I}=\hbar v_{F}[i\sigma^{1}\partial_{y}-i\sigma^{2}\partial_{x}]$ is the
chiral Dirac Hamiltonian in the first quantized language. $v_{F}\approx 5\cdot
10^{5}\frac{m}{sec}$ is the Fermi velocity, $\sigma$ is the Pauli matrix
describing the electron spin and $\mu$ is the chemical potential measured
relative to the Dirac $\Gamma$ point. The Hamiltonian for the two dimensional
surface $L\times L$ describes well the excitations smaller than the bulk gap
of the $3D$ $TI$ at $0.3$ $eV$. Moving away from the $\Gamma$ point, the Fermi
velocity becomes momentum dependent; therefore, we will introduce a momentum
cut off $\Lambda$ to restrict the validity of the Dirac model. The chiral
Dirac model in the Bloch representation takes the form: $h=\hbar
v_{F}(\vec{K}\times\vec{\sigma})\cdot\hat{z}\equiv\hbar
v_{F}(-\sigma^{1}k_{y}+\sigma^{2}k_{x})$ The eigen-spinors for this
Hamiltonian are :
$|u(\vec{K})>=[|u_{\uparrow}(\vec{K})>,|u_{\downarrow}(\vec{K})>]^{T}=|\vec{K}>\otimes[1,ie^{i\chi(k_{x},k_{y})}]^{T}$
where $\chi(k_{x},k_{y})=tan^{-1}(\frac{k_{y}}{k_{x}})$ is the spinor phase
and $\epsilon=\hbar v_{F}\sqrt{k^{2}_{x}+k^{2}_{y}}$ is the eigenvalue for
particles . For holes we have the eigenvalue $\epsilon=-\hbar
v_{F}\sqrt{k^{2}_{x}+k^{2}_{y}}$ and eigenvectors
$|v(\vec{K})>=[|v_{\uparrow}(\vec{K})>,|v_{\downarrow}(\vec{K})>]^{T}=|\vec{K}>\otimes[-1,ie^{i\chi(k_{x},k_{y})}]^{T}$.
The chirality operator is defined in terms of the chiral phase
$\chi(k_{x},k_{y})$:
$(\vec{\sigma}\times\frac{\vec{K}}{|\vec{K}|})\cdot\hat{z}\equiv\sin[\chi(k_{x},k_{y})]\sigma^{1}-\cos[\chi(k_{x},k_{y})]\sigma^{2}$
(10)
The chirality operator takes the eigenvalue $-$ (counter-clockwise) for
particles
$[\sin(\chi(k_{x},k_{y}))\sigma^{1}-\cos(\chi(k_{x},k_{y}))\sigma^{2}]|\vec{K}>\otimes[1,ie^{i\chi(k_{x},k_{y})}]^{T}=-|\vec{K}>\otimes[1,ie^{i\chi(k_{x},k_{y})}]^{T}$
and $+$ (clockwise) for holes
$[\sin(\chi(k_{x},k_{y}))\sigma^{1}-\cos(\chi(k_{x},k_{y}))\sigma^{2}]|\vec{K}>\otimes[-1,ie^{i\chi(k_{x},k_{y})}]^{T}=|\vec{K}>\otimes[-1,ie^{i\chi(k_{x},k_{y})}]^{T}$.
B-The effect of edge dislocation on a two dimensional chiral surface
Hamiltonian
We use the notation $x^{\mu}$ ,$\mu=x,y$ and $X^{a}$ ,$a=1,2$ to describe the
media with dislocations. For an edge dislocation in the $x$ direction the
$Burger$ vector $B^{(2)}$ is in the $y$ direction . The value of the burger
vector $B^{(2)}$ is given by the shortest translation lattice vector in the
$y$ direction. (For the $TI$ $Bi_{2}Se_{3}$ the length of the vector $B^{(2)}$
is $5$ times the inter atomic distance ). Following kleinert we introduce the
coordinate transformation for an edge dislocation:
$\vec{r}=(x,y)\rightarrow[X(\vec{r})=x,Y(\vec{r})=y+\frac{B^{(2)}}{2\pi}\tan^{-1}(\frac{y}{x})]$
with the core of the dislocation centered at $\vec{r}=(0,0)$. The matrix
elements fields $e^{a}_{\mu}$ for the edge dislocation is given by :
$e^{a}_{\mu}=\partial_{\mu}X^{a}(\vec{r});\hskip 7.22743pta=1,2;\hskip
7.22743pt\mu=x,y$ (11)
We express the Burger vector in terms of the the partial derivatives with
respect the coordinates $a=1,2$ in the dislocation frame and $\mu=x,y$ for the
fixed Cartesian frame kleinert :
$\partial_{x}e^{2}_{y}-\partial_{y}e^{2}_{x}=B^{(2)}\delta^{2}(\vec{r})$ (12)
Using Stokes theorem, we replace the line integral $\displaystyle\oint
dx^{\mu}e^{2}_{\mu}(\vec{r})$ by the surface integral $\int\int
dx^{\mu}dx^{\nu}[\partial_{x}e^{2}_{y}-\partial_{y}e^{2}_{x}]$. For a system
with zero $curvature$ and non zero $torsion$ $T^{(2)}_{\mu,\nu}$ we find that
the surface torsion tensor integral $\int\int
dx^{\mu}dx^{\nu}T^{(2)}_{\mu,\nu}$ is equal to $\int\int
dx^{\mu}dx^{\nu}[\partial_{x}e^{2}_{y}-\partial_{y}e^{2}_{x}]$, and therefore
both integrals are equal to the Burger vector.
$\displaystyle\displaystyle\oint dx^{\mu}e^{2}_{\mu}(\vec{r})=\int\int
dx^{\mu}dx^{\nu}[\partial_{\mu}e^{2}_{\nu}-\partial_{\nu}e^{2}_{\mu}]=B^{(2)};$
$\displaystyle\int\int dx^{\mu}dx^{\nu}T^{(2)}_{\mu,\nu}=\int\int
dx^{\mu}dx^{\nu}[\partial_{\mu}e^{2}_{\nu}-\partial_{\nu}e^{2}_{\mu}]=B^{(2)};$
where $dx^{\mu}dx^{\nu}$ represents the surface element. The tangent
components $e^{a}_{\mu}$ can be expressed in terms of the Burger vector
density $B^{(2)}\delta^{2}(\vec{r})$ kleinert :
$\displaystyle e^{2}_{x}=(\frac{B^{(2)}}{2\pi})\frac{y}{(x^{2}+y^{2})};\hskip
21.68121pte^{2}_{y}=1-(\frac{B^{(2)}}{2\pi})\frac{x}{(x^{2}+y^{2})}$
$\displaystyle e^{1}_{x}=1;\hskip 28.90755pte^{1}_{y}=0$ (14)
Using the tangent components, we obtain the metric tensor $g_{\mu,\nu}$.
$e^{a}_{\mu}e^{a}_{\nu}\equiv
e^{1}_{\mu}e^{1}_{\nu}+e^{2}_{\mu}e^{2}_{\nu}=g_{\mu,\nu}(\vec{r});\hskip
7.22743pte^{a}_{\mu}e^{b}_{\mu}\equiv
e^{a}_{x}e^{b}_{x}+e^{a}_{y}e^{b}_{y}=\delta_{a,b}$ (15)
The inverse of the metric tensor $g_{\mu,\nu}(\vec{r})$ is the tensor
$g^{\nu,\mu}(\vec{r})$ defined trough the equation
$g_{\mu,\tau}(\vec{r})g^{\tau,\nu}(\vec{r})=\delta_{\mu}^{\nu}$. Using the
tangent vectors, we find $to$ $first$ $order$ in the Burger vector the metric
tensor $g_{\mu,\nu}$ and the Jacobian transformation $\sqrt{G}$:
$g_{x,x}=1;\hskip
7.22743ptg_{x,y}=\frac{B^{(2)}}{2\pi}\frac{y}{x^{2}+y^{2}};\hskip
7.22743ptg_{y,y}=1-\frac{B^{(2)}}{2\pi}\frac{y}{x^{2}+y^{2}};\hskip
7.22743ptg_{y,x}=0;\hskip
7.22743ptG=det[g_{\mu,\nu}]=1-\frac{B^{(2)}}{2\pi}\frac{y}{x^{2}+y^{2}}$ (16)
The inverse tensor is given by:$g^{x,x}\approx 1$,
$g^{x,y}=g^{y,x}=-\frac{B^{(2)}}{2\pi}\frac{y}{x^{2}+y^{2}}$,
$g^{y,y}=1+\frac{B^{(2)}}{\pi}\frac{x}{x^{2}+y^{2}}$. Using the inverse tensor
$g^{\mu,\nu}$ we obtain the inverse matrix $e^{\mu}_{a}$ which is given by:
$e^{\mu}_{a}=e_{a,\nu}g^{\nu,\mu}=(\delta_{a,b}e^{b}_{\nu})g^{\nu,\mu}=e^{a}_{\nu}g^{\nu,\mu}$
(17)
Using the components $e^{\mu}_{a}$ we compute the the transformed Pauli
matrices. The Hamiltonian in the absence of the edge dislocation is given by
$h^{T.I.}=i\gamma^{a}\partial_{a}\equiv\sum_{a=1,2}i\gamma^{a}\partial_{a}$
where the Pauli matrices are given by $\gamma^{1}=-\sigma^{2}$ ,
$\gamma_{2}=\sigma_{1}$ and $\gamma^{3}=\sigma^{3}$. (We will use the
convention that when an index appears twice we perform a summation over this
index.) In the presence of the edge dislocation, the term
$\gamma^{a}\partial_{a}$ must be expressed in terms of the Cartesian fixed
coordinates $\mu=x,y$. As a result, the spinor $\Psi(\vec{r})$ transforms
accordingly to the $SU(2)$ transformation . If $\widetilde{\Psi}(\vec{R})$ is
the spinor for the deformed lattice, it can be related with the help of an
$SU(2)$ transformation to the spinor $\Psi(\vec{r})$ in the undeformed
lattice:
$\widetilde{\Psi}(X,Y)=e^{-i\frac{\delta\varphi(x,y)}{2}\sigma^{3}}\Psi(x,y)$
. Where $\delta\varphi(x,y)$ is the rotation angle between the two set of
coordinates: $\delta\varphi(x,y)=tan^{-1}(\frac{Y}{X})-tan^{-1}(\frac{y}{x})$.
Using the relation between the coordinates $X=x$, and
$Y=y+\frac{B^{(2)}}{2\pi}tan^{-1}(\frac{y}{x})$ with the singularity at
$x=y=0$ gives us that the derivative of the phase which is a delta function,
$\partial_{x}\delta\varphi(x,y)=-\partial_{y}\delta\varphi(x,y)\propto\delta^{2}(x,y)$.
Combining the transformation of the derivative with the $SO(2)$ rotation in
the plane, we obtain the form of the chiral Dirac equation in the Cartesian
space (see Appendix A) given in terms of the $spin$ $connection$
$\omega_{\mu}^{1,2}$ Nakahara :
$i\gamma_{a}\partial_{a}\widetilde{\Psi}(\vec{R})=i\delta_{a,b}\gamma^{b}\partial_{a}\widetilde{\Psi}(\vec{R})=i\gamma^{a}e^{\mu}_{a}[\partial_{\mu}+\frac{1}{4}[\gamma^{b},\gamma^{c}]\omega_{\mu}^{bc}]\Psi(\vec{r})$
(18)
The Hamiltonian $h^{T.I.}\rightarrow h^{edge}$ is transformed to the
dislocation edge Hamiltonian with the explicit form given by:
$\displaystyle
h^{edge}=i\sigma^{1}\partial_{2}-i\sigma^{2}\partial_{1}=i\sigma^{1}e^{\mu}_{2}[\partial_{\mu}+\frac{1}{8}[\sigma^{1},\sigma^{2}]\omega^{1,2}_{\mu}]-i\sigma^{2}e^{\mu}_{1}[\partial_{\mu}+\frac{1}{8}[\sigma^{1},\sigma^{2}]\omega^{1,2}_{\mu}]$
$\displaystyle=i(\sigma^{1}e^{\mu}_{2}-\sigma^{2}e^{\mu}_{1})(\partial_{\mu}+\frac{1}{8}[\sigma^{1},\sigma^{2}]\omega^{1,2}_{\mu})$
To first order in the Burger vector we find :
$\omega_{x}^{12}=-\omega_{x}^{21}=0$ and
$-\omega_{y}^{21}=\omega_{y}^{1,2}=-\frac{B^{(2)}}{2}\delta^{2}(\vec{r})$, see
eqs. $(72-74)$ in Appendix A.
$h^{edge}\approx
i\sigma^{1}(\partial_{y}-\frac{i}{2}\sigma^{3}B^{(2)}\delta^{2}(\vec{r}))-i\sigma^{2}\partial_{x}$
(20)
In the second quantized form the chiral Dirac Hamiltonian in the presence of
an edge dislocations is given by :
$\displaystyle
H^{edge}\approx\int\,d^{2}r\sqrt{G}\Psi^{\dagger}(\vec{r})[h^{edge}-\mu]\Psi(\vec{r})$
$\displaystyle\equiv\hbar
v_{F}\int\,d^{2}r\sqrt{G}\Psi^{\dagger}(\vec{r})[i\sigma^{1}(\partial_{y}-\frac{i}{2}\sigma^{3}B^{(2)}\delta^{2}(\vec{r}))-i\sigma^{2}\partial_{x}-\mu]\Psi(\vec{r})$
$h^{edge}$ is the Hamiltonian in the first quantized language, $\mu$ is the
chemical potential and
$\Psi(\vec{r})=[\Psi_{\uparrow}(\vec{r}),\Psi_{\downarrow}(\vec{r})]^{T}$ is
the two component spinor field.
C- The Identification of the physical contours for the edge Hamiltonian
$h^{edge}$
In order to identify the solutions, we will use the complex representation.
The coordinates in the complex representation are given by,
$z=\frac{1}{2}(x+iy)$, $\overline{z}=\frac{1}{2}(x-iy)$,
$\partial_{z}=\partial_{x}-i\partial_{y}$,
$\partial_{\overline{z}}=\partial_{x}+i\partial_{y}$. In this representation
the two dimensional delta function $\delta^{2}(\vec{r})$ is given by
$\delta^{2}(\vec{r})\equiv\frac{1}{\pi}\partial_{z}(\frac{1}{\overline{z}})=\frac{1}{\pi}\partial_{\overline{z}}(\frac{1}{z})$
Conformal ; Nair . We will use the edge Hamiltonian $h^{edge}$ and will
compute the eigenfunctions
$u_{\epsilon}(z,\overline{z})=[U_{\epsilon\uparrow}(z,\overline{z}),U_{\epsilon\downarrow}(z,\overline{z})]^{T}$
and
$v_{\epsilon}(z,\overline{z})=V_{\epsilon\uparrow}(z,\overline{z}),V_{\epsilon\downarrow}(z,\overline{z})]^{T}$.
The eigenvalue equation is given by:
$\displaystyle\epsilon
U_{\epsilon\uparrow}(z,\overline{z})=-[\partial_{z}+(\frac{B^{(2)}}{\sqrt{2}\pi})\partial_{z}(\frac{1}{\overline{z}})]U_{\epsilon\downarrow}(z,\overline{z})$
$\displaystyle\epsilon
U_{\epsilon\downarrow}(z,\overline{z})=[\partial_{\overline{z}}+(\frac{B^{(2)}}{\sqrt{2}\pi})\partial_{\overline{z}}(\frac{1}{z})]U_{\epsilon\uparrow}(z,\overline{z})$
The eigenfunctions $u_{\epsilon}(z,\overline{z})$ and
$v_{\epsilon}(z,\overline{z})$ can be written with the help of a singular
matrix $M(z,\overline{z})$ Ezawa :
$u_{\epsilon}(z,\overline{z})=M(z,\overline{z})\hat{F}_{\epsilon}(z,\overline{z})\equiv\left[\begin{array}[]{rrr}e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{z})}&0\\\
0&e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{\overline{z}})}\\\
\end{array}\right]\left(\begin{array}[]{cc}F_{\epsilon\uparrow}(z,\overline{z})\\\
F_{\epsilon\downarrow}(z,\overline{z})\end{array}\right)$
($F_{\epsilon}(z,\overline{z})$ and $F_{-\epsilon}(z,\overline{z})$ are the
transformed eigenfunctions for $\epsilon>0$ and $\epsilon<0$ respectively .)
In terms of the transformed spinors the eigenvalue equation
$h^{edge}(z,\overline{z})u_{\epsilon}(z,\overline{z})=\epsilon
u_{\epsilon}(z,\overline{z})$ and $F_{\epsilon\downarrow}(z,\overline{z})$
becomes:
$\epsilon\left(\begin{array}[]{cc}F_{\epsilon\uparrow}(z,\overline{z})\\\
F_{\epsilon\downarrow}(z,\overline{z})\end{array}\right)=\left[\begin{array}[]{rrr}I(z,\overline{z})&0\\\
0&(I(z,\overline{z})^{*}\\\
\end{array}\right]\left[\begin{array}[]{rrr}-\partial_{z}&0\\\
0&\partial_{\overline{z}}\\\
\end{array}\right]\left(\begin{array}[]{cc}F_{\epsilon\uparrow}(z\overline{z})\\\
F_{\epsilon\downarrow},(z,\overline{z})\end{array}\right)$
where
$I(z,\overline{z})=e^{-\frac{B^{(2)}}{2\pi}(\frac{\overline{z}-z}{z\overline{z}})}\equiv
e^{2\frac{B^{(2)}}{\pi}(\frac{iy}{x^{2}+y^{2}})}$ ,
$(I(z,\overline{z}))^{*}=e^{2\frac{B^{(2)}}{\pi}(\frac{-iy}{x^{2}+y^{2}})}$,
$|I(z,\overline{z})|=1$. We search for zero modes $\epsilon=0$ and find :
$\partial_{z}F_{\epsilon\downarrow}(z,\overline{z})=0\hskip
36.135pt\partial_{\overline{z}}F_{\epsilon\uparrow}(z,\overline{z})=0$ (23)
The solutions are given by the holomorphic representation
$F_{\epsilon=0\uparrow}(z,\overline{z})=f_{\uparrow}(z)$ and the anti-
holomorphic function
$F_{\epsilon=0\downarrow}(z,\overline{z})=f_{\downarrow}(\overline{z})$. The
zero mode eigenfunctions are given by :
$u_{\epsilon=0,\uparrow}(z)=e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{z})}f_{\uparrow}(z),\hskip
36.135ptu_{\epsilon=0,\downarrow}(\overline{z})=e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{\overline{z}})}f_{\downarrow}(\overline{z})$
(24)
Due to the presence of the essential singularity at $z=0$ it is not possible
to find analytic functions $f_{\uparrow}(z)$ and
$f_{\downarrow}(\overline{z})$ which vanish fast enough around $z=0$ such that
$\int d^{2}z(u_{\epsilon=0,\lambda}(z))^{*}u_{\epsilon=0,\lambda}(z)<\infty$.
Therefore, we conclude that zero mode solution does not exists. The only way
to remedy the problem is to allow for states with finite energy.
In the next step we look for finite energy states. We perform a coordinate
transformation :
$z\rightarrow W[z,\overline{z}];\hskip
36.135pt\overline{z}\rightarrow\overline{W}[z,\overline{z}]$ (25)
We demand that the transformation is conformal and preserve the orientation.
This restricts the transformations to holomorphic and anti-holomorphic
functions Conformal . This means that we have the conditions
$\partial_{\overline{z}}W[z,\overline{z}]=0$ and
$\partial_{z}\overline{W}[z,\overline{z}]=0$. As a result we obtain
$W[z,\overline{z}]=W[z]$ and
$\overline{W}[z,\overline{z}]=\overline{W}[\overline{z}]$, which obey the
eigenvalue equations:
$\displaystyle\epsilon
F_{\epsilon\uparrow}(W,\overline{W})=-\partial_{W}F_{\epsilon\downarrow}(W,\overline{W})$
$\displaystyle\epsilon
F_{\epsilon\downarrow}(W,\overline{W})=\partial_{\overline{W}}F_{\epsilon\uparrow}(W,\overline{W})$
This implies the conditions $\frac{dW[z]}{dz}=(I(z,\overline{z}))^{*}$ and
$\frac{d\overline{W}[\overline{z}]}{dz}=I(z,\overline{z})$. Since
$I(z,\overline{z})$ is neither holomorphic or anti-holomorphic and satisfy
$|I(z,\overline{z})|=1$, the only solutions for $W[z]$ and
$\overline{W}[\overline{z}]$ must obey $I(z,\overline{z})=1$:
$I(z,\overline{z})\equiv
e^{2\frac{B^{(2)}}{\pi}(\frac{iy}{x^{2}+y^{2}})}=e^{i2\pi n};\hskip
14.45377ptn=0,\pm 1,\pm 2....$ (27)
For $I(z,\overline{z})\neq 1$ one obtains solutions which are unstable . The
stable solutions will be given by a one parameter $s$ curve ($s$ is the length
of the curve) $\vec{r}(s)\equiv[x(s),y(s)]$ which obey the equation
$I(z,\overline{z})=1$. The curve $\vec{r}(s)$ allows us to define the
$tangent$ $\vec{t}(s)$ and the $normal$ vectors $\vec{N}(s)$. This allows us
to introduce a two- dimensional region in the vicinity of the contour of
$\vec{r}(s)\rightarrow\vec{R}(s,u)=\vec{r}(s)+u\vec{N}(s)$.
IID- The wave function for the edge dislocation-the $n=0$ contour
The condition
$I(z,\overline{z})=e^{2\frac{B^{(2)}}{\pi}(\frac{iy}{x^{2}+y^{2}})}=1$ for
$n=0$ is satisfied for $y=0$ and large value of $y$ which obey
$2\frac{B^{(2)}}{\pi}(\frac{y}{x^{2}+y^{2}})<<1$ . The values of $y$ which
satisfy this conditions are restricted to
$I(z,\overline{z})=e^{2\frac{B^{(2)}}{\pi}(\frac{iy}{x^{2}+y^{2}})}\approx 1$.
This condition is satisfied for values of $y$ in the range:
$2\frac{B^{(2)}}{\pi}(\frac{y}{x^{2}+y^{2}})\leq\eta<\frac{\pi}{4}<1$ (28)
We introduce the radius $R_{g}=\frac{B^{2}}{2\pi^{2}}$ and find that the
condition $I(z,\overline{z})\approx 1$ gives rise to the equation for $y$. The
solution is given by
$x^{2}+(y\pm\frac{2\pi}{\eta}R_{g})^{2}=(\frac{2\pi}{\eta}R_{g})^{2}$.
Therefore, for $|y|>|d|\geq(\frac{2\pi}{\eta})2R_{g}>2R_{g}$ we have $I\approx
1$ which corresponds to a free particle eigenvalue equations.
$\displaystyle\epsilon
F_{\epsilon\uparrow}(x,y)=e^{\frac{B^{(2)}}{\pi}\frac{i2y}{(x^{2}+y^{2})}}[-\partial_{x}+i\partial_{y}]F_{\epsilon\downarrow}(x,y)$
$\displaystyle\approx[-\partial_{x}+i\partial_{y}]F_{\epsilon\downarrow}(x,y);$
$\displaystyle\epsilon
F_{\epsilon\downarrow}(x,y)=e^{\frac{B^{(2)}}{\pi}\frac{-i2y}{(x^{2}+y^{2})}}[\partial_{x}+i\partial_{y}]F_{\epsilon\uparrow}(x,y)$
$\displaystyle\approx[\partial_{x}+i\partial_{y}]F_{\epsilon\uparrow}(x,y)$
For $|y|>d$ the eigenfunctions are given by:
$U_{\epsilon,\uparrow}(x,y)=e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+iy})}F_{\epsilon,\uparrow}(x,y)$,
$U_{\epsilon,\downarrow}(x,y)=e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-iy})}F_{\epsilon,\downarrow}(x,y)$
where $F_{\epsilon\uparrow}(x,y)$ and $F_{\epsilon\downarrow}(x,y)$ are the
eigenfunctions of equation (21). The envelope functions
$e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+iy})}$,
$e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-iy})}$ which multiply the wave functions
$F_{\epsilon\uparrow}(x,y)$ , $F_{\epsilon\downarrow}(x,y)$ impose vanishing
boundary conditions for the eigenfunctions $U_{\epsilon,\downarrow}(x,y)$ and
$U_{\epsilon,\uparrow}(x,y)$ at $y\rightarrow\pm\infty$ . therefore, we demand
that the eigenfunctions $U_{\epsilon,\uparrow}(x,y)$,
$U_{\epsilon,\downarrow}(x,y)$ should vanish at the boundaries
$y=\pm\frac{L}{2}$. Since the multiplicative envelope functions for opposite
spins is complex conjugate to each other we have to make the choice that one
of the spin components vanishes at one side and the other component at the
opposite side. Two possible choices can be made:
$U_{\epsilon,\uparrow}(x,y=\frac{L}{2})\equiv
e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+i\frac{L}{2}})}F_{\epsilon\uparrow}(x,\frac{L}{2})=U_{\epsilon,\downarrow}(x,y=-\frac{L}{2})\equiv
e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-i(-\frac{L}{2})})}F_{\downarrow}(x,-\frac{L}{2})=0$
or
$U_{\epsilon,\uparrow}(x,y=-\frac{L}{2})\equiv
e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+i(-\frac{L}{2})})}F_{\epsilon\uparrow}(x,-\frac{L}{2})=U_{\epsilon,\downarrow}(x,y=\frac{L}{2})\equiv
e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-i\frac{L}{2}})}F_{\downarrow}(x,\frac{L}{2})=0$
Making the first choice, (both choices give the same eigenvalues and
eigenfunction) we compute the eigenfunctions $F_{\epsilon\uparrow}(x,y)$ and
$F_{\epsilon\downarrow}(x,y)$ for $|y|>d$ using the boundary conditions :
$F_{\epsilon\uparrow}(x,y=\frac{L}{2})=0;\hskip
28.90755ptF_{\epsilon\downarrow}(x,y=-\frac{L}{2})=0$ (30)
Due to the fact that the solutions are restricted to $|y|>d$ no conditions
need to be imposed at $x=y=0$. In the present case we consider a situation
with a single dislocation. This is justified for a dilute concentration of
dislocations typically separated by a distance $l\approx 10^{-6}m$. ( In
principle we need at least two dislocations in order to satisfy the condition
that the sum of the Burger vectors is zero.) The eigenvalues are given by
$\epsilon=\pm\hbar v_{F}\sqrt{p^{2}+q^{2}}$. The value of $p$ is determined by
the periodic boundary condition in the $x$ direction
$p(m)=\frac{2\pi}{L}m\equiv\frac{2\pi}{Na}m$, $m=0,1,...,(N-2),(N-1)$ and $a$
is the lattice constant $a\approx\frac{2\pi}{\Lambda}$. The value of $q$ will
be obtained from the vanishing boundary conditions at $y=\pm\frac{L}{2}$. The
eigenfunctions $F_{\epsilon,\sigma}(x,y)$ will be obtained using the linear
combination of the spinors introduced in chapter $III$. In the Cartesian
representation we can build four spinors $\Gamma_{p,q}(x,y)$,
$\Gamma_{p,-q}(x,y)$,$\Gamma_{-p,q}(x,y)$,$\Gamma_{-p,-q}(x,y)$ which are
eigenstates of the chirality operator and are given by:
$\Gamma_{p,q}(x,y)=e^{ipx}e^{iqy}\left(\begin{array}[]{cc}1\\\
ie^{i\chi(p,q)}\end{array}\right)$
$\Gamma_{p,-q}(x,y)=e^{ipx}e^{-iqy}\left(\begin{array}[]{cc}1\\\
ie^{-i\chi(p,q)}\end{array}\right)$
$\Gamma_{-p,q}(x,y)=e^{ipx}e^{iqy}\left(\begin{array}[]{cc}1\\\
-ie^{-i\chi(p,q)}\end{array}\right)$
$\Gamma_{-p,-q}(x,y)=e^{-ipx}e^{-iqy}\left(\begin{array}[]{cc}1\\\
-ie^{i\chi(p,q)}\end{array}\right)$ (31)
where $tan[\chi(p,q)]=\frac{q}{p}$.
The $TI$ Hamiltonian $h^{T.I}(x,y)=\hbar
v_{F}[i\sigma^{1}\partial_{y}-i\sigma^{2}\partial_{x}]$ is invariant under the
symmetry the operation $x\rightarrow-x$ which is described by the
transformation $P_{x}$ : $P_{x}xP^{-1}_{x}=-x$;
$P_{x}\sigma^{2}P^{-1}_{x}=-\sigma^{2}$; $P_{x}yP^{-1}_{x}=y$;
$P_{x}\sigma^{1}P^{-1}_{x}=\sigma^{1}$.
The edge Hamiltonian $h^{edge}$ contains in addition the term
$\sigma^{2}\delta(\vec{r})$ which changes sign under the symmetry operation
$P_{x}$ . As a result the symmetry operation does not commute with the edge
Hamiltonian $[h^{edge},P_{x}]\neq 0$. This result demands that we construct
two independent eigenfunctions $F^{(n=0,R)}_{p>0,q}(x,y)$ ($right-mover$) for
$p>0$ and $F^{(n=0,L)}_{-p>0,q}(x,y)$ ($left-mover$) $p<0$.
$\displaystyle
F^{(n=0,R)}_{p>0,q}(x,y)=A(q)\Gamma_{p,q}(x,y)+B(q)\Gamma_{p,-q}(x,y)$
$\displaystyle
F^{(n=0,L)}_{-p>0,q}(x,y)=C(q)\Gamma_{-p,q}(x,y)+D(q)\Gamma_{-p,-q}(x,y)$
Employing the boundary conditions given in equation $(29)$ we obtain the
amplitudes $\frac{D(q)}{C(q)}$ , $\frac{B(q)}{A(q)}$ and the discrete momenta
$q_{+}$. Using the pair $\Gamma_{p,q}(x,y)$ , $\Gamma_{p,-q}(x,y)$ $p>0$ we
obtain :
$\displaystyle
F^{(n=0,R)}_{\epsilon(p>0,q_{+}),\uparrow}(x,y)=e^{ipx}e^{\frac{i}{2}\chi(p,q_{+})}[e^{i(q_{+}y-\frac{1}{2}\chi(p,q_{+}))}+(-1)^{k+1}e^{-i(q_{+}y-\frac{1}{2}\chi(p,q_{+}))}];|y|>d$
$\displaystyle
F^{(n=0,R)}_{\epsilon(p>0,q_{+}),\downarrow}(x,y)=ie^{ipx}e^{\frac{i}{2}\chi(p,q_{+})}[e^{i(q_{+}y+\frac{1}{2}\chi(p,q_{+}))}+(-1)^{k+1}e^{-i(q_{+}y+\frac{1}{2}\chi(p,q_{+}))}];|y|>d$
$\displaystyle q\equiv
q_{+}=\frac{\pi}{L}k+\frac{1}{L}\tan^{-1}(\frac{q_{+}}{p});k=1,2,3...;\tan[\chi(p,q_{+})]=(\frac{q_{+}}{p})$
$\displaystyle\epsilon(p,q_{+})=\pm\hbar
v_{F}\sqrt{(\frac{2\pi}{L}m)^{2}+q_{+}^{2}}$
Similarly, for the second pair $\Gamma_{-p,q}(x,y)$,$\Gamma_{-p,-q}(x,y)$,
$p>0$ we obtain:
$\displaystyle
F^{(n=0,L)}_{\epsilon(-p>0,q_{-}),\uparrow}(x,y)=e^{-ipx}e^{-\frac{i}{2}\chi(p,q_{-})}[e^{i(q_{-}y+\frac{1}{2}\chi(p,q_{-}))}+(-1)^{k+1}e^{-i(q_{-}y+\frac{1}{2}\chi(p,q_{-}))}];|y|>d$
$\displaystyle
F^{(n=0,L)}_{\epsilon(-p>0,q_{-}),\downarrow}(x,y)=-ie^{-ipx}e^{-\frac{i}{2}\chi(p,q_{-})}[e^{i(q_{-}y-\frac{1}{2}\chi(p,q_{-}))}+(-1)^{k+1}e^{-i(q_{-}y-\frac{1}{2}\chi(p,q_{-}))}];|y|>d$
$\displaystyle q\equiv
q_{-}=\frac{\pi}{L}k-\frac{1}{L}\tan^{-1}(\frac{q_{-}}{p});k=1,2,3...;\tan[\chi(p,q_{-})]=(\frac{q_{-}}{p})$
$\displaystyle\epsilon(-p,q_{-})=\pm\hbar
v_{F}\sqrt{(\frac{2\pi}{L}m)^{2}+q_{-}^{2}}$
For the state with zero momentum $p=0$ we find:
$\displaystyle
F^{(n=0,0)}_{\epsilon(p=0,q),\uparrow}(x,y)=2e^{\frac{-i\pi}{4}}\cos[qy+\frac{\pi}{4}];|y|>d$
$\displaystyle
F^{(n=0,0)}_{\epsilon(p=0,q),\downarrow}(x,y)=i2e^{\frac{-i\pi}{4}}\cos[qy-\frac{\pi}{4}];|y|>d$
$\displaystyle q=\frac{\pi}{2L}+\frac{\pi}{L}k;k=0,1,2,3...$
$\displaystyle\epsilon(p=0,q)=\pm\hbar v_{F}|q|$
The eigenfunctions for the dislocation problem for $|y|>d$ will be given in
terms of the envelope functions $e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+iy})}$ ,
$e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-iy})}$
($U_{\epsilon,\uparrow}(x,y)=e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+iy})}F_{\epsilon,\uparrow}(x,y)$,
$U_{\epsilon,\downarrow}(x,y)=e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-iy})}F_{\epsilon,\downarrow}(x,y)$).
The explicit solutions are given by :
$u^{(n=0,R)}_{\epsilon}(x,y)\equiv[U^{(n=0,R}_{\epsilon\uparrow}(x,y),U^{(n=0,R)}_{\epsilon\downarrow}(x,y)]^{T}$;
$u^{(n=0,L)}_{\epsilon}(x,y)\equiv[U^{(n=0,L}_{\epsilon\uparrow}(x,y),U^{(n=0,L)}_{\epsilon\downarrow}(x,y)]^{T}$.
The components of the spinor are given by:
$\displaystyle
U^{(n=0,R)}_{\uparrow}(x,y)\approx\frac{2const.(B^{(2)})}{G^{\frac{1}{4}}(x,y)L}e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+iy})}F^{(n=0,R)}_{\epsilon(p>0,q_{+}),\uparrow}(x,y)$
$\displaystyle
U^{(n=0,R)}_{\downarrow}(x,y)\approx\frac{2const.(B^{(2)})}{G^{\frac{1}{4}}(x,y)L}e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-iy})}F^{(n=0,R)}_{\epsilon(p>0,q_{+}),\downarrow}(x,y)$
$\displaystyle
U^{(n=0,L)}_{\uparrow}(x,y)\approx\frac{2const.(B^{(2)})}{G^{\frac{1}{4}}(x,y)L}e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+iy})}F^{(n=0,L)}_{\epsilon(-p>0,q_{-}),\uparrow}(x,y)$
$\displaystyle
U^{(n=0,L)}_{\downarrow}(x,y)\approx\frac{2const.(B^{(2)})}{G^{\frac{1}{4}}(x,y)L}e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-iy})}F^{(n=0,L)}_{\epsilon(-p>0,q_{-}),\downarrow}(x,y)$
$\displaystyle U^{(n=0,0)}_{\uparrow}(x,y)\approx
e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+iy})}F^{(n=0,0)}_{\epsilon(p=0,q),\uparrow}(x,y)$
$\displaystyle U^{(n=0,0)}_{\downarrow}(x,y)\approx
e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-iy})}F^{(n=0,0)}_{\epsilon(p=0,q),\downarrow}(x,y)$
where $G(x,y)=1-\frac{B^{(}2)}{2\pi}\frac{y}{\sqrt{2}(x^{2}+y^{2})}$ is the
Jacobian introduced by the edge dislocation. The eigenstates are normalized
and
obey:$\int\,dx\int\,dy\sqrt{G(x,y)}(U^{(n=0,R)}_{\sigma}(x,y))^{*}U^{(n=0,R)}_{\sigma^{\prime}}(x,y)\approx\delta_{\sigma,\sigma^{\prime}}$,
$\int\,dx\int\,dy\sqrt{G(x,y)}(U^{(n=0,L)}_{\sigma}(x,y))^{*}U^{(n=0,L)}_{\sigma^{\prime}}(x,y)\approx\delta_{\sigma,\sigma^{\prime}}$.
The normalization factor $\frac{2const.(B^{(2)})}{L}\approx\frac{2}{L}$, has a
weak dependence on the Burger vector $B^{(2)}$ . This dependence is a
consequence of the Jacobian $\sqrt{G}$ which affects the normalization
constant.
(The multiplicative factor $e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x\pm iy})}$
gives rise to a weak non-orthogonality between the states. This non-
orthogonality of the linear independent eigenfunctions can be corrected with
the help of the Grahm-Shmidt method.)
For the present case, backscattering is allowed but it is much weaker in
comparison to regular metals. This is seen as follows: Time reversal is not
violated; due to the parity violation, the eigenstates
$u^{(n=0,R)}_{\epsilon}(x,y)$ ,$u^{(n=0,L)}_{\epsilon}(x,y)$ are not related
by a time reversal symmetry ($Tu^{(n=0,R)}_{\epsilon}(x,y)\neq
u^{(n=0,L)}_{\epsilon}(x,y)$) . As a result, the backscattering potential
$V_{p,-p}$ is controlled by a finite matrix element between states with
different eigenvalues $\epsilon(-p,q_{-})\neq\epsilon(p,q_{+})$ (contrary to
regular metals where the impurity potential $V_{p,-p}$ connects states with
the same energy). In the present case
$|\epsilon(-p,q_{-})-\epsilon(p,q_{+})|=\hbar
v_{F}|[\sqrt{(\frac{2\pi}{L}m)^{2}+q_{-}^{2}}-\sqrt{(\frac{2\pi}{L}m)^{2}+q_{+}^{2}}]|\neq
0$ the eigenvalues are not equal, therefore the finite matrix element
controlled by the backscattering potential $V_{p,-p}$ gives rise only to a
second order backscattering effect!
IIIE- The circular contours-the wave function for $n\neq 0$
The equation
$I(z,\overline{z})=e^{2\frac{B^{(2)}}{\pi}(\frac{iy(s)}{x^{2}(s)+y^{2}(s)})}=e^{i2\pi
n}$ gives the set of ring contours for $n=\pm 1,\pm 2,\pm 3,...$ shown in
figure 1. The radius $R_{g}$ for the fundamental contour($n=1$) is represented
in terms of the Burger vector $B^{(2)}$, $R_{g}=\frac{B^{(2)}}{2\pi^{2}}$ and
$R_{g}(n)=\frac{R_{g}}{|n|}$.
$(x(s))^{2}+(y(s)\pm R_{g}(n))^{2}=(R_{g}(n))^{2}$ (37)
The centers of the contours are given by :$[\bar{x},\bar{y}]=[0,R_{g}(n)]$ for
$n\neq 0$. When $n>0$ the center of the contours has positive coordinates
(upper contour) and for $n<0$ the center has negative coordinates (lower
contour). Each contour is characterized by a circle with a radius
$R_{g}(n)\equiv\frac{R_{g}}{|n|}$ centered at $[\bar{x}=0,\bar{y}=R_{g}(n)]$.
The contour is parametrized in terms of the arc length $0\leq
s<2\pi\frac{R_{g}}{|n|}$ which is equivalent to $0\leq\varphi<2\pi$ . Each
contour is parametrized by $\vec{r}(s)\equiv[x(s),R_{g}(n)+y(s)]$ where
$x(s)=R_{g}(n)\cos[\frac{s}{R_{g}(n)}]\equiv\ R_{g}(n)\cos[\varphi]$ and
$y(s)=R_{g}(n)\sin[\frac{s}{R_{g}(n)}]\equiv\ R_{g}(n)\sin[\varphi]$. We will
extend this curve to a two dimensional strip with the coordinate $u$ in the
normal direction: For the curve curve $\vec{r}(s)=[x(s),y(s)]$ we will use the
tangent $\vec{t}(s)$ and the normal vector $\vec{N}(s)$ Therefore, the two
dimensional region in the vicinity of the one parameter curve $\vec{r}(s)$ is
replaced by $\vec{r}(s)\rightarrow\vec{R}(s,u)=\vec{r}(s)+u\vec{N}(s)$.
$\displaystyle
x(s,u)=R_{g}(n)\cos[\frac{s}{R_{g}(n)}]+u\cos[\frac{s}{R_{g}(n)}]$
$\displaystyle
y(s,u)=R_{g}(n)\sin[\frac{s}{R_{g}(n)}]+u\sin[\frac{s}{R_{g}(n)}]$
We will restrict the width $|u|$ such that $e^{i2\pi n}e^{\pm i\eta}\approx 1$
where $\eta$ obeys $\eta<\frac{\pi}{4}<1$ ,
$|u|\leq\frac{R_{g}(n)}{1-\frac{\eta}{2\pi n}}-R_{g}(n)\approx
R_{g}(n)(\frac{\eta}{2\pi n})<\frac{R_{g}(n)}{8n}$. In these new coordinates,
the Dirac equation is approximated for $|u|\leq R_{g}(n)(\frac{\eta}{2\pi
n})=\frac{D(n)}{2}$ by :
$\displaystyle\epsilon
F_{\epsilon\uparrow}(s,u)=-I(s,u)e^{-i\frac{s}{R_{g}(n)}}[\partial_{u}-\frac{i}{1+\frac{u}{R_{g}(n)}}\partial_{s}]F_{\epsilon\downarrow}(s,u)\approx-e^{-i\frac{s}{R_{g}(n)}}[\partial_{u}-i\partial_{s}]F_{\epsilon\downarrow}(s,u)$
$\displaystyle\epsilon
F_{\epsilon\downarrow}(s,u)=(I(s,u))^{*}e^{i\frac{s}{R_{g}(n)}}[\partial_{u}+\frac{i}{1+\frac{u}{R_{g}(n)}}\partial_{s}]F_{\epsilon\uparrow}(s,u)\approx
e^{i\frac{s}{R_{g}(n)}}[\partial_{u}+i\partial_{s}]F_{\epsilon\uparrow}(s,u)$
The solution for the contour $n\neq 0$, $0\leq s<2\pi R_{g}(n)$;
$|u|\leq\frac{D(n)}{2}$
The periodicity in $s$ allows us to represent the eigenfunctions in the form:
$F_{\epsilon\uparrow}(s,u)=\sum_{j=-\infty}^{\infty}\sum_{q}e^{ij(\frac{s}{R_{g}(n)})}e^{iqu}F_{\epsilon\uparrow}(j,q)$
and
$F_{\epsilon\downarrow}(s,u)=\sum_{j=-\infty}^{\infty}\sum_{q}e^{i(j+1)(\frac{s}{R_{g}(n)})}e^{iqu}F_{\epsilon\downarrow}(j,q)$.
We find:
$\displaystyle\epsilon
F_{\uparrow}(\epsilon;j,q)=(iq+\frac{j}{R_{g}(n)})F_{\downarrow}(\epsilon;j,q)$
$\displaystyle\epsilon
F_{\downarrow}(\epsilon;j,q)=(iq+\frac{j+1}{R_{g}(n)})F_{\uparrow}(\epsilon;j,q)$
The determinant of the two equations determines the relation between the
eigenvalue $\epsilon$, the transverse momentum $Q(\epsilon)$ and the
eigenfunctions $F_{\epsilon\downarrow}(j,q)$,$F_{\epsilon\uparrow}(j,q)$. The
eigenvalues are degenerate and obey : $\epsilon(j=l;k)=\epsilon(j=-(l+1);k)$
,where $l\geq 0$.
$\displaystyle q\equiv\frac{-i}{2R_{g}(n)}\pm Q(\epsilon);\hskip
14.45377ptQ(\epsilon)=\sqrt{\epsilon^{2}-(\frac{l+\frac{1}{2}}{R_{g}(n)})^{2}}$
$\displaystyle
F_{\epsilon}(l,q)\equiv[F_{\epsilon\uparrow}(l,q),F_{\epsilon\downarrow}(l,q)]^{T}\propto[1,e^{-i\kappa(Q,l)}]^{T};\hskip
7.22743pt\kappa(Q,l)=tan^{-1}(\frac{QR_{g}(n)}{l+\frac{1}{2}})$
The value of the transversal momentum $Q(\epsilon)$ will be determined from
the boundary conditions at $\pm\frac{D(n)}{2}$. We will introduce a polar
angle $\theta$ measured with respect the Cartesian axes: The angle
$0<\varphi(n=1)\leq 2\pi$ for the upper contour $n=1$ centered at
$[\overline{x}=0,\overline{y}=R_{g}]$ is described by the polar coordinate
$0<\theta\leq\pi$ measured from the center of the Cartesian coordinate
$[0,0]$. The lower contour centered at $[\overline{x}=0,\overline{y}=-R_{g}]$
characterized by the angle $0<\varphi(n=-1)\leq 2\pi$ is described by the
polar angle $\theta$ restricted to $\pi<\theta\leq 2\pi$. We establish the
correspondence between $\varphi(n=\pm 1)$ and $\theta$:
$\displaystyle\varphi(n=1)=2\theta+\frac{3\pi}{2}\hskip 7.22743ptfor\hskip
7.22743ptthe\hskip 7.22743ptupper\hskip 7.22743ptcontour\hskip
7.22743ptn=1,\hskip 7.22743pt0<\theta\leq\pi$
$\displaystyle\varphi(n=-1)=2\theta+\frac{3\pi}{2}+\pi\hskip
7.22743ptfor\hskip 7.22743ptthe\hskip 7.22743ptlower\hskip
7.22743ptcontour\hskip 7.22743ptn=-1,\hskip 7.22743pt0<\theta\leq\pi$
Following the discussion from the previous chapter we will introduce the
following boundary conditions:
$\displaystyle F^{(n=1)}_{\epsilon\uparrow}(s,u=\frac{D}{2})=0;\hskip
28.90755ptF^{(n=1)}_{\epsilon\downarrow}(s,y=-\frac{D}{2})=0$ $\displaystyle
F^{(n=-1)}_{\epsilon\uparrow}(s,u=-\frac{D}{2})=0;\hskip
28.90755ptF^{(n=-1)}_{\epsilon\downarrow}(s,y=\frac{D}{2})=0$ $\displaystyle
D(n=\pm 1)\equiv D$ (43)
For the two contours $n=\pm 1$ we introduce eight spinors $\Gamma^{(n=\pm
1)}_{l,Q}(\varphi(n=\pm 1),u)$,$\Gamma^{(n=\pm 1)}_{l,-Q}(\varphi(n=\pm
1),u)$, $\Gamma^{(n=\pm 1)}_{-l,Q}(\varphi(n=\pm 1),u)$, $\Gamma^{(n=\pm
1)}_{-l,-Q}(\varphi(n=\pm 1),u)$. Using this spinor we will compute the
eigenfunctions. For the case $n=0$ we had only four spinors given in equation
$(30)$. The four spinors have been used to construct the eigenfunctions
$F^{(n=0,R)}_{p>0,q}(x,y)$ for $p>0$ and $F^{(n=0,L)}_{-p>0,q}(x,y)$ . Due to
the fact that for each $n\neq 0$ we have two contours $n=\pm$ we have eight
spinors which will be used to construct the eigenfunctions.
$\Gamma^{(n=\pm 1)}_{l,Q}(\varphi(n=\pm 1),u)=e^{il(\varphi(n=\pm
1))}e^{iQu}\left(\begin{array}[]{cc}1\\\ e^{i(\varphi(n=\pm
1))}e^{-i\kappa(l,Q)}\end{array}\right)$ $\Gamma^{(n=\pm
1)}_{l,-Q}(\varphi(n=\pm 1),u)=e^{il(\varphi(n=\pm
1))}e^{-iQu}\left(\begin{array}[]{cc}1\\\ e^{i(\varphi(n=\pm
1))}e^{i\kappa(l,Q)}\end{array}\right)$ $\Gamma^{(n=\pm
1)}_{-l,Q}(\varphi(n=\pm 1),u)=e^{-il(\varphi(n=\pm
1))}e^{iQu}\left(\begin{array}[]{cc}1\\\ -e^{-i(\varphi(n=\pm
1))}e^{i\kappa(l,Q)}\end{array}\right)$ $\Gamma^{(n=\pm
1)}_{-l,-Q}(\varphi(n=\pm 1),u)=e^{-il(\varphi(n=\pm
1))}e^{-iQu}\left(\begin{array}[]{cc}1\\\ -e^{-i(\varphi(n=\pm
1))}e^{-i\kappa(l,Q)}\end{array}\right)$ (44)
Using the vanishing boundary condition given in equation $(42)$ we construct
for this case similar spinors as the one given in equation $(31)$. In the
present case we have for each $n\neq 0$ two contours, therefore the number of
spinors will be doubled. We find instead of the eigenfunction given in
equation $(33)$ two sets of eigenfunctions with momentum $Q_{-}$ (which
replaces $q_{-}$ , see $(33)$) and $Q_{+}$ (which replaces $q_{+}$ , see
$(32)$) .
Using the boundary conditions given in eq.$(35)$ we determine the quantization
conditions $Q_{-}$,$Q_{+}$ and the eigenfunctions for the $n=1$ and $n=-1$
contours.
$\displaystyle
Q_{-}=\frac{\pi}{D}k-\frac{1}{D}\tan^{-1}(\frac{Q_{-}R_{g}(1)}{l+\frac{1}{2}}),k=1,2,3...;\tan[\kappa(l,Q_{-})]=(\frac{Q_{-}R_{g}(1)}{l+\frac{1}{2}})$
$\displaystyle\epsilon(l,Q_{-})=\pm\hbar
v_{F}\sqrt{(\frac{l+\frac{1}{2}}{R_{g}(1)})^{2}+Q_{-}^{2}}$ $\displaystyle
Q_{+}=\frac{\pi}{D}k+\frac{1}{D}\tan^{-1}(\frac{Q_{+}R_{g}(1)}{l+\frac{1}{2}}),k=1,2,3...$
$\displaystyle\tan[\kappa(l,Q_{+})]=(\frac{Q_{+}R_{g}(1)}{l+\frac{1}{2}})$
$\displaystyle\epsilon(l,Q_{+})=\pm\hbar
v_{F}\sqrt{(\frac{l+\frac{1}{2}}{R_{g}(n)})^{2}+Q_{+}^{2}}$ (45)
Using the fact that the combined wave function on the contours $n=1$ and
$n=\pm 1$ must be finite we obtain two sets of wave functions. We include the
envelope function and obtain the wave function for $Q_{-}$ and $Q_{+}$: The
envelope functions $e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x+iy})}$ ,
$e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{x-iy})}$ when projected to the contours
take a complicated form. The envelope functions can be expressed in terms of
the functions $\eta(u)$ and $\zeta(\theta,u)$:
$\displaystyle\eta(u)=\frac{R_{g}(1)}{R_{g}(1)+u},\frac{|u|}{R_{g}(1)}<1$
$\displaystyle\zeta(\theta,u)=\frac{-B^{(2)}}{2\pi(R_{g}(1)+u)((\sin[2\theta])^{2}+(\eta(u)-\cos[2\theta])^{2})}$
We find for $Q_{-}$:
$\displaystyle
U_{\epsilon(l,Q_{-})\uparrow}(\theta,u)=G^{\frac{-1}{4}}(\theta,u)\cdot[U^{(even,k)}_{\epsilon(l,Q_{-})\uparrow}(\theta,u)+U^{(odd,k)}_{\epsilon(l,Q_{-})\uparrow}(\theta,u)];$
$\displaystyle
U^{(even,k)}_{\epsilon(l,Q_{-})\uparrow}(\theta,u)=2ie^{\frac{-i}{2}\kappa(l,Q_{-})}[e^{\zeta(\theta,u)\sin[2\theta]}e^{-i\zeta(\theta,u)(\eta(u)-\cos[2\theta])}e^{il(2\theta+\frac{3\pi}{2})}\sin[Q_{-}u+\frac{1}{2}\kappa(l,Q_{-})]$
$\displaystyle+(-1)^{l}e^{-\zeta(\theta,u)\sin[2\theta]}e^{-i\zeta(\theta,u)(-\eta(u)+\cos[2\theta])}e^{-il(2\theta+\frac{3\pi}{2})}\sin[Q_{-}u-\frac{1}{2}\kappa(l,Q_{-})]];$
$\displaystyle
U^{(odd,k)}_{\epsilon(l,Q_{-})\uparrow}(\theta,u)=2e^{\frac{-i}{2}\kappa(l,Q_{-})}[e^{\zeta(\theta,u)\sin[2\theta]}e^{-i\zeta(\theta,u)(\eta(u)-\cos[2\theta])}e^{il(2\theta+\frac{3\pi}{2})}\cos[Q_{-}u+\frac{1}{2}\kappa(l,Q_{-})]$
$\displaystyle+(-1)^{l}e^{-\zeta(\theta,u)\sin[2\theta]}e^{-i\zeta(\theta,u)(-\eta(u)+\cos[2\theta])}e^{-il(2\theta+\frac{3\pi}{2})}\cos[Q_{-}u-\frac{1}{2}\kappa(l,Q_{-})]];$
$\displaystyle
U_{\epsilon(l,Q_{-})\downarrow}(\theta,u)=G^{\frac{-1}{4}}(\theta,u)\cdot[U^{(even,k)}_{\epsilon(l,Q_{-})\downarrow}(\theta,u)+U^{(odd,k)}_{\epsilon(l,Q_{-})\uparrow}(\theta,u)];$
$\displaystyle
U^{(even,k)})_{\epsilon(l,Q_{-})\downarrow}(\theta,u)=2ie^{\frac{-i}{2}\kappa(l,Q_{-})}[e^{\zeta(\theta,u)\sin[2\theta]}e^{i\zeta(\theta,u)(\eta(u)-\cos[2\theta])}e^{i((l+1)(2\theta+\frac{3\pi}{2}))}\sin[Q_{-}u-\frac{1}{2}\kappa(l,Q_{-})]$
$\displaystyle-(-1)^{l}e^{-\zeta(\theta,u)\sin[2\theta]}e^{i\zeta(\theta,u)(-\eta(u)+\cos[2\theta])}e^{-i((l+1)(2\theta+\frac{3\pi}{2}))}\sin[Q_{-}u+\frac{1}{2}\kappa(l,Q_{-})]];$
$\displaystyle
U^{(odd,k)}_{\epsilon(l,Q_{-})\downarrow}(\theta,u)=2e^{\frac{-i}{2}\kappa(l,Q_{-})}[e^{\zeta(\theta,u)\sin[2\theta]}e^{i\zeta(\theta,u)(\eta(u)-\cos[2\theta])}e^{i((l+1)(2\theta+\frac{3\pi}{2}))}\cos[Q_{-}u-\frac{1}{2}\kappa(l,Q_{-})]$
$\displaystyle-(-1)^{l}e^{-\zeta(\theta,u)\sin[2\theta]}e^{i\zeta(\theta,u)(-\eta(u)+\cos[2\theta])}e^{-i((l+1)(2\theta+\frac{3\pi}{2}))}\cos[Q_{-}u+\frac{1}{2}\kappa(l,Q_{-})]];$
Similarly for $Q_{+}$we obtain the wave function:
$\displaystyle
U_{\epsilon(l,Q_{+})\uparrow}(\theta,u)=G^{\frac{-1}{4}}(\theta,u)\cdot[U^{(even,k)}_{\epsilon(l,Q_{+})\uparrow}(\theta,u)+U^{(odd,k)}_{\epsilon(l,Q_{+})\uparrow}(\theta,u)];$
$\displaystyle
U^{(even,k)}_{\epsilon(l,Q_{+})\uparrow}(\theta,u)=2ie^{\frac{-i}{2}\kappa(l,Q_{+})}[(-1)^{l}e^{-\zeta(\theta,u)\sin[2\theta]}e^{-i\zeta(\theta,u)(-\eta(u)+\cos[2\theta])}e^{il(2\theta+\frac{3\pi}{2})}\sin[Q_{+}u+\frac{1}{2}\kappa(l,Q_{+})]$
$\displaystyle+e^{\zeta(\theta,u)\sin[2\theta]}e^{-i\zeta(\theta,u)(\eta(u)-\cos[2\theta])}e^{-il(2\theta+\frac{3\pi}{2})}\sin[Q_{+}u-\frac{1}{2}\kappa(l,Q_{+})]];$
$\displaystyle
U^{(odd,k)}_{\epsilon(l,Q_{+})\uparrow}(\theta,u)=2e^{\frac{-i}{2}\kappa(l,Q_{+})}[(-1)^{l}e^{-\zeta(\theta,u)\sin[2\theta]}e^{-i\zeta(\theta,u)(-\eta(u)+\cos[2\theta])}e^{il(2\theta+\frac{3\pi}{2})}\cos[Q_{+}u+\frac{1}{2}\kappa(l,Q_{+})]$
$\displaystyle+e^{\zeta(\theta,u)\sin[2\theta]}e^{-i\zeta(\theta,u)(\eta(u)-\cos[2\theta])}e^{-il(2\theta+\frac{3\pi}{2})}\cos[Q_{+}u-\frac{1}{2}\kappa(l,Q_{+})]];$
$\displaystyle
U_{\epsilon(l,Q_{+})\downarrow}(\theta,u)=G^{\frac{-1}{4}}(\theta,u)\cdot[U^{(even,k)}_{\epsilon(l,Q_{+})\downarrow}(\theta,u)+U^{(odd,k)}_{\epsilon(l,Q_{+})\downarrow}(\theta,u)];$
$\displaystyle
U^{(even,k)}_{\epsilon(l,Q_{+})\downarrow}(\theta,u)=2ie^{\frac{-i}{2}\kappa(l,Q_{+})}[-(-1)^{l}e^{-\zeta(\theta,u)\sin[2\theta]}e^{i\zeta(\theta,u)(-\eta(u)+\cos[2\theta])}e^{i(l+1)(2\theta+\frac{3\pi}{2})}\sin[Q_{+}u-\frac{1}{2}\kappa(l,Q_{+})]$
$\displaystyle+e^{\zeta(\theta,u)\sin[2\theta]}e^{i\zeta(\theta,u)(\eta(u)-\cos[2\theta])}e^{-i(l+1)(2\theta+\frac{3\pi}{2})}\sin[Q_{+}u+\frac{1}{2}\kappa(l,Q_{+})]];$
$\displaystyle
U^{(odd,k)}_{\epsilon(l,Q_{+})\downarrow}(\theta,u)=2e^{\frac{-i}{2}\kappa(l,Q_{+})}[-(-1)^{l}e^{-\zeta(\theta,u)\sin[2\theta]}e^{i\zeta(\theta,u)(-\eta(u)+\cos[2\theta])}e^{i(l+1)(2\theta+\frac{3\pi}{2})}\cos[Q_{+}u-\frac{1}{2}\kappa(l,Q_{+})]$
$\displaystyle+e^{\zeta(\theta,u)\sin[2\theta]}e^{i\zeta(\theta,u)(\eta(u)-\cos[2\theta])}e^{-i(l+1)(2\theta+\frac{3\pi}{2})}\cos[Q_{+}u+\frac{1}{2}\kappa(l,Q_{+})]];$
where $G^{\frac{-1}{4}}(\theta,u)$ is the Jacobian transformation induced by
the metric tensor.
IV -Computation of the STM density of states
A-Description of the STM method
The STM tunneling current $I$ is a function of the bias voltage $V$ which
gives spatial and spectroscopic information about the electronic surface
states. At zero temperature, the derivative of the current with respect the
bias voltage $V$ is given in term of the single particles eigenvalues:
$\epsilon(m,q_{-})=\pm\hbar v_{F}\sqrt{(\frac{2\pi}{L}m)^{2}+q_{-}^{2}}$,
$\epsilon(m,q_{+})=\pm\hbar v_{F}\sqrt{(\frac{2\pi}{L}m)^{2}+q_{-}^{2}}$
,$m=0,1,2,3...$ for contour $n=0$. For the upper and lower circular contours
$n=\pm 1$, we have :$\epsilon(l,Q_{-})=\pm\hbar
v_{F}\sqrt{(\frac{l+\frac{1}{2}}{R_{g}(1)})^{2}+Q_{-}^{2}}$
,$\epsilon(l,Q_{+})=\pm\hbar
v_{F}\sqrt{(\frac{l+\frac{1}{2}}{R_{g}(1)})^{2}+Q_{+}^{2}}$ ,$l=0,1,2,3..$.
The $STM$ density of states is computed for a voltage $V$ between the $STM$
tip and the sample. The tunneling current is a function of the bias voltage
$V$ and the chemical potential $\mu>0$ kittel :
$\displaystyle\frac{dI}{dV}\propto
D(E=eV;s,u)\equiv\sum_{n}D^{(n)}(E=eV;s,u)=$
$\displaystyle=\sum_{\eta=\pm}[\sum_{m}\sum_{q_{r}=q_{+},q_{-}}\sum_{\sigma}|U^{(n=0;m,q_{r})}_{\sigma}(x,y)|^{2}\delta[eV+\mu-\eta\hbar
v_{F}\sqrt{(\frac{2\pi}{L}m)^{2}+q_{r}^{2}}]$ $\displaystyle+\sum_{n=\pm
1}\sum_{l}\sum_{Q_{r}=Q_{+},Q_{-}}\sum_{\sigma}|U^{(n=\pm
1;l,Q_{r})}_{\sigma}(\theta,u)|^{2}\delta[eV+\mu-\eta\hbar
v_{F}\sqrt{(\frac{l+\frac{1}{2}}{R_{g}(1)})^{2}+Q_{r}^{2}}]]$
($\eta=+$ corresponds to electrons with energy $0<\epsilon\leq\mu$ and
$\eta=-$ corresponds to electrons below the Dirac point $\epsilon<0$. For the
rest of this paper we will take the chemical potentials to be $\mu=120mV$
(this is typical value for the $TI$ ). We will neglect the states with
$\eta=-$ which correspond to particles below the Dirac cone. The density of
states at the tunneling energy $eV$ is weighted by the probability density of
the $STM$ tip at position $[x,y]$ for n=0. The contours for $n=\pm 1$ will be
parametrized in terms of the polar angle $\theta$ and transverse coordinate
$u$. The proportionality factor $J$ for the tunneling probability (not shown
in the equation ) $\frac{dI}{dV}=JD(V;x,y)$ is a function of the distance
between the tip and the sample. The notation $D^{(n)}(V;x,y)$ represents the
tunneling density for the different contours.
IVB-The tunneling density of states $D^{(n=0)}(V;x,y)$ for $n=0$
Summing up the single particle states weighted with occupation probability
$|U^{(n=0;m,q_{r})}_{\sigma}(x,y)|^{2}$, we obtain a space dependent density
of states for the two dimensional boundary surface ,$\frac{-L}{2}\leq
x\leq\frac{L}{2}$ and the coordinate $y$ is restricted to the regions
$\frac{d}{2}<y\leq\frac{L}{2}$ and $\frac{-L}{2}<y\leq\frac{d}{2}$. We will
perform the computation at the thermodynamic limit, namely we replace the
discrete momentum $\frac{\pi}{L}k$ by $Y=\frac{k}{N}$ and $\frac{2\pi}{L}m$ by
$X=\frac{m}{N}$ where $N=\frac{L}{a}$. We find for the dimensionless momentum
$\hat{q}\equiv qa$ the equations : $\hat{q}_{\pm}(Y)=\pi
Y\pm\frac{1}{N}\tan^{-1}[\frac{\hat{q}_{\pm}(Y)}{2\pi X}]$ where $2\pi
X=pa=\hat{p}$. As a result we obtain the following density of states
$\frac{\partial\hat{q}_{\pm}}{\partial Y}$
$\displaystyle[\frac{\partial\hat{q}_{+}}{\partial
Y}]^{-1}=\frac{1}{\pi}\frac{\hat{q}_{+}^{2}+\hat{p}^{2}-\frac{1}{N}\hat{p}}{\hat{q}_{+}^{2}+\hat{p}^{2}}$
$\displaystyle[\frac{\partial\hat{q}_{-}}{\partial
Y}]^{-1}=\frac{1}{\pi}\frac{\hat{q}_{-}^{2}+\hat{p}^{2}+\frac{1}{N}\hat{p}}{\hat{q}_{-}^{2}+\hat{p}^{2}}$
Using this results, we compute the tunneling density of states in terms of the
energy $\mu+eV$ measured with respect the chemical potential $\mu$ and the
transverse energy $\epsilon_{\bot}\equiv\hbar v_{F}q_{\pm}$.
$\displaystyle
D^{(n=0)}(V;x,y)=(\frac{L}{hv_{F}})^{2}(\frac{B^{(2)}}{L})^{2}\frac{1}{4\sqrt{G(x,y)}}e^{\frac{-B^{(2)}}{\pi}(\frac{x}{x^{2}+y^{2}+a^{2}})}[\int_{0}^{E_{max.}}\,d\epsilon_{\bot}\frac{(\mu+eV)}{\sqrt{(\mu+eV)^{2}-\epsilon_{\bot}^{2}}}\dot{}$
$\displaystyle[\frac{1}{2}(1+\frac{1}{\pi}\frac{hv_{F}}{L(\mu+V)}\sqrt{1-(\frac{\epsilon_{\bot}}{\mu+V})^{2}}\hskip
7.22743pt)+\frac{1}{2}(1-\frac{1}{\pi}\frac{hv_{F}}{L(\mu+V)}\sqrt{1-(\frac{\epsilon_{\bot}}{\mu+V})^{2}}\hskip
7.22743pt)]$
$\displaystyle+\frac{hv_{F}}{L}(H[\mu+V-\frac{hv_{F}}{2L}]-H[\mu+eV-
E_{max}])\cdot((\cos[\frac{(\mu+eV)}{\hbar
v_{F}}y-\frac{\pi}{4}])^{2}+(\cos[\frac{(\mu+eV)}{\hbar
v_{F}}y-\frac{\pi}{4}])^{2})]$
$\displaystyle=(\frac{L}{hv_{F}})^{2}(\frac{B^{(2)}}{L})^{2}\frac{1}{4\sqrt{G(x,y)}}e^{\frac{-B^{(2)}}{\pi}(\frac{x}{x^{2}+y^{2}+a^{2}})}[\int_{0}^{E_{max.}}\,d\epsilon_{\bot}\frac{(\mu+eV)}{\sqrt{(\mu+eV)^{2}-\epsilon_{\bot}^{2}}}$
$\displaystyle+\frac{hv_{F}}{L}(H[\mu+V-\frac{hv_{F}}{2L}]-H[\mu+eV-
E_{max}])\cdot((\cos[\frac{(\mu+eV)}{\hbar
v_{F}}y-\frac{\pi}{4}])^{2}+(\cos[\frac{(\mu+eV)}{\hbar
v_{F}}y-\frac{\pi}{4}])^{2})]=$
$\displaystyle(\frac{L}{hv_{F}})^{2}(\frac{B^{(2)}}{L})^{2}\frac{1}{4\sqrt{G(x,y)}}e^{\frac{-B^{(2)}}{\pi}(\frac{x}{x^{2}+y^{2}+a^{2}})}[\frac{\pi}{2}(\mu+eV)+\frac{hv_{F}}{L}(H[\mu+V-\frac{hv_{F}}{2L}]-H[\mu+eV-
E_{max}])]$ $\displaystyle for\hskip 14.45377pt|y|>d$
$H[\mu+eV-\frac{hv_{F}}{2L}]$ is the step function which is one for
$\mu+eV-\frac{hv_{F}}{2L}\geq 0$ and zero otherwise. $a=\frac{2\pi}{\Lambda}$
is the short distance cut-off and $E_{max}=\hbar v_{F}\Lambda<0.3eV$ is the
maximal energy which restricts the validity of the Dirac model. We observe in
the second line that the asymmetry in the density of states
$1\pm\frac{1}{\pi}\frac{hv_{F}}{L(\mu+V)}\sqrt{1-(\frac{\epsilon_{\bot}}{\mu+V})^{2}}\hskip
7.22743pt)$ cancels.
Equation $(51)$ shows that the tunneling density of states is linear in the
energy $\mu+eV$ (in the present case we have looked only for energies above
the Dirac cone ). For the chemical potential $\mu=120mV$, the zero energy
corresponds to the Voltage $V=-120mV$. The tunneling density of states has a
constant part at energies $\frac{hv_{F}}{2L}\approx 0.2mV$ for
$-120mV<V<-119.8mV$. For $V>-119.8mV$ the density of states is proportional to
$\mu+eV$.
In figure $2$ we have plotted the tunneling density of states as a function of
the coordinates $x$ and $y$. The shape of the plot is governed by the the
multiplicative factor $e^{-\frac{B^{(2)}}{\pi}(\frac{x}{x\pm iy})}$ which
governs the solutions in eq.$(35)$. We observe that the density of state is
maximal in the region $|y|<10B^{(2)}$.
Figure $3$ shows the dependence on the voltage $V$ and coordinate $y$. We
observe the linear increase in the tunneling density of states which is
maximal in the region $|y|<10B^{(2)}$.
IVC-The tunneling density of states
$D^{(n=0)}(V,x,y;\vec{r}_{1},..\vec{r}_{2M})$ for $2M$ dislocations.
For many dislocations which satisfy $\sum_{w=1}^{2M}B^{(2,w)}=0$ ( sum of the
Burger vectors is zero ) with the core centered at $[x_{w},y_{w}]$
,$w=1,2..2M$ the coordinate $\vec{r}=(x,y)\rightarrow[X(\vec{r}),Y(\vec{r})]$
is replaced by
$[X(\vec{r})=x,Y(\vec{r})=y+\sum_{w}\frac{B^{(2,w)}}{2\pi}\tan^{-1}(\frac{y-y_{w}}{x-x_{w}})]$.
Following the method used previously, we find the edge Hamiltonian with many
dislocations takes the form:
$h^{edge}(w=1,2...2M)\approx
i\sigma^{1}[\partial_{y}-\frac{i}{2}\sum_{w=1}^{2M}\sigma^{3}B^{(2,w)}\delta^{2}(\vec{r}-\vec{r}_{w})]-i\sigma^{2}\partial_{x}$
(52)
As a result, the wave functions are given by:
$\displaystyle
U^{(n=0,w=1,2...2M)}_{\uparrow}(x,y)\propto\prod_{w=1,2...2M}e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{(x-x_{w})+i(y-y_{w})})}F^{(n=0)}_{\uparrow}(x,y)$
$\displaystyle
U^{(n=0,w=1,2...2M)}_{\downarrow}(x,y)\propto\prod_{w=1,2...2M}e^{-\frac{B^{(2)}}{2\pi}(\frac{1}{(x-x_{w})-i(y-y_{w})})}F^{(n=0)}_{\downarrow}(x,y)$
(53)
Using these wave functions, we find that the tunneling density of states is
given by:
$D^{(n=0)}(V,x,y;\vec{r}_{1},..\vec{r}_{2M})\propto\prod_{w=1,2...2M}e^{-\frac{B^{(2)}}{\pi}(\frac{(x-x_{w})}{(x-x_{w})^{2}+(y-y_{w})^{2}+a^{2}})}$
(54)
In figure $4$ we show the tunneling density of states for an even number of
dislocations in the $y$ directions which have the core on the $y=0$ axes
($\vec{r}_{w}=[x_{w},y_{w}=0]$, $w=1,2,3,...2M$). When the coordinate of the
$w=1,2,3,...2M$ dislocations is replaced by a continuum variable $w$ which can
be described by a domain a wall model: $h^{domain-wall}(x,y)=\hbar
v_{F}[-i\sigma^{1}\partial_{y}+i\sigma^{2}\partial_{x}-\sigma^{3}\kappa M(y)]$
where $M(y)=sgn[y]|M(y)|$ Jackiw .
Using this model find that the tunneling density of states density $D^{domain-
wall}(V;x,y)$ confined to $|y|<W$ (the width $W$ depends on the explicit form
of the domain wall function $M(y)$ and strength $\kappa$) is given by:
$D^{domain-
wall}(V;x,y)\propto(\frac{L}{hv_{F}})^{2}e^{-2\kappa\int_{0}^{|y|}\,dy^{\prime}M(y^{\prime})}$.
This show the similarity between the result obtain from the $domain-wall$
model and the large numbers of of dislocations given in equation $(54)$.
IVD-The tunneling density of states $D^{(n=\pm 1)}(V;\theta,u)$ for the $n=\pm
1$ contours.
Following the same procedure as used for the $n=0$ and using the
eigenfunctions for $n=\pm 1$ we find :
$D^{(n=\pm 1)}(V;\theta,u)\equiv D^{(n=\pm
1)}(\mu,V;\theta,u)_{even}+D^{(n=\pm 1)}(\mu,V;\theta,u,\mu)_{odd}$ (55)
For the even $k$’s, we solve for the momentum $Q_{+}$ and $Q_{-}$ and find:
$\displaystyle D^{(n=\pm 1)}(\mu,V;\theta,u)_{even}=\frac{(B^{(2)})^{2}}{2\pi
R_{g}(1)D(1)\sqrt{G(\theta,u)}}\sum_{Q_{r}=Q_{+},Q_{-}}\sum_{l=0}^{\infty}\delta[eV+\mu-\hbar
v_{F}\sqrt{(\frac{l+\frac{1}{2}}{R_{g}(1)})^{2}+Q_{r}^{2}}]$
$\displaystyle[(e^{-2\zeta(\theta,u)\sin[2\theta]}+e^{2\zeta(\theta,u)\sin[2\theta]})((\sin[Q_{r}u-\frac{1}{2}\kappa(l,Q_{r})])^{2}+(\sin[Q_{r}u+\frac{1}{2}\kappa(l,Q_{r})])^{2})+$
$\displaystyle
2(-1)^{l}\sin[Q_{r}u+\frac{1}{2}\kappa(l,Q_{r})]\sin[Q_{r}u-\frac{1}{2}\kappa(l,Q_{r})]\cdot$
$\displaystyle(\cos[l(\theta+\frac{3\pi}{2})-\zeta(\theta,u)(-\eta(u)+\cos[2\theta])]-\cos[(l+1)(\theta+\frac{3\pi}{2})+\zeta(\theta,u)(-\eta(u)+\cos[2\theta])]];$
Similarly for the odd $k$’s we find:
$\displaystyle D^{(n=\pm 1)}(\mu,V;\theta,u)_{odd}=\frac{(B^{(2)})^{2}}{2\pi
R_{g}(1)D(1)\sqrt{G(\theta,u)}}\sum_{Q_{r}=Q_{+},Q_{-}}\sum_{l=0}^{\infty}\delta[eV+\mu-\hbar
v_{F}\sqrt{(\frac{l+\frac{1}{2}}{R_{g}(1)})^{2}+Q_{r}^{2}}]$
$\displaystyle[(e^{-2\zeta(\theta,u)\sin[2\theta]}+e^{2\zeta(\theta,u)\sin[2\theta]})((\cos[Q_{r}u-\frac{1}{2}\kappa(l,Q_{r})])^{2}+(\cos[Q_{r}u+\frac{1}{2}\kappa(l,Q_{r})])^{2})+$
$\displaystyle
2(-1)^{l}\cos[Q_{r}u+\frac{1}{2}\kappa(l,Q_{r})]\cos[Q_{r}u-\frac{1}{2}\kappa(l,Q_{r})]\cdot$
$\displaystyle(\cos[l(\theta+\frac{3\pi}{2})-\zeta(\theta,u)(-\eta(u)+\cos[2\theta])]-\cos[(l+1)(\theta+\frac{3\pi}{2})+\zeta(\theta,u)(-\eta(u)+\cos[2\theta])]]$
For the present case the energy scale of the excitations is governed by the
radius $R_{g}(1)$ and width $D$. The spectrum is discrete and we can’t replace
it by a continuum density of states as we did for the case $n=0$.
In figure $5$ we show the tunneling density of states at a fixed polar angle
$\theta=\frac{\pi}{2}$ as a function of the voltage $V$. We observe that the
density of states is dominated by high energy eigenvalues. This solutions are
localized in energy. The range of the spectrum is above $\mu+eV>200mV$ which
is well separated from the low energy spectrum controlled by the $n=0$ contour
(which ranges from $-120mV$ to $70mV$).
Figure $6$ shows the tunneling density of states as a function of the polar
angle $\theta$ for a fixed energy . The periodicity in $\theta$ is controlled
by the discrete energy eigenvalues.
In figure $7$ we show the tunneling density of states at a fixed voltage $V$
as a function of the polar angle $0<\theta<\pi$ and width $|u|<0.1$.
V-The charge current-the in plane spin on the surface of the $h^{T.I.}$
Hamiltonian
A-The current in the absence of the edge dislocation for the $h^{T.I.}$
From the Hamiltonian given in equation $1$ we compute the equation of motion
for the velocity operator:
$\frac{dx}{dt}=\frac{1}{i\hbar}[x,h]=v_{F}\sigma^{y}$ ,
$\frac{dy}{dt}=\frac{1}{i\hbar}[y,h]=-v_{F}\sigma^{x}$. We multiply the
velocity operator by the charge $(-e)$ and identify the charge current
operators : $\hat{J}_{x}=(-e)v_{F}\sigma^{2}$,
$\hat{J}_{y}=(-e)(-v_{F})\sigma^{1}$. This also represent the ”‘real”’ spin on
the surface. Therefore, the charge current is a measure of the in-plane spin
on the surface.
Integrating over the $y$ coordinate we obtain the current $I_{x}^{T.I.}$ in
the $x$ direction. Using the eigenstates $\Gamma_{p,q}(x,y)$ and
$\Gamma_{-p,q}(x,y)$ of the $h^{T.I.}$ Hamiltonian
$\Gamma_{p,q}(x,y)=e^{ipx}e^{iqy}\left(\begin{array}[]{cc}1\\\
ie^{i\chi(p,q)}\end{array}\right)$
$\Gamma_{-p,q}(x,y)=e^{-ipx}e^{iqy}\left(\begin{array}[]{cc}1\\\
-ie^{-i\chi(p,q)}\end{array}\right)$
we find
$(\Gamma_{p,q}(x,y))(\sigma^{2})(\Gamma_{p,q}(x,y))=-(\Gamma_{-p,q}(x,y))(\sigma^{2})(\Gamma_{-p,q}(x,y))$
therefore, we conclude that the current $I_{x}^{T.I.}=0$ is zero.
VB-The current in the presence of the edge dislocation
We will compute the current in the presence of the edge dislocation. The
current operator $\hat{J}^{edge}_{x}(x,y)$ will be given in terms of the
transformed currents. We find that the current density operator
$J^{edge}_{x}(x,y)$ is given by:
$\hat{J}^{edge}_{x}(x,y)=(-e)v_{F}[\sigma^{2}e^{x}_{1}-\sigma^{1}e^{x}_{2}]=(-e)v_{F}\sigma^{2}-(-e)v_{F}\frac{B^{(2)}}{2\pi}(\frac{y\sigma^{1}+x\sigma^{2}}{x^{2}+y^{2}})\approx(-e)v_{F}\sigma^{2}$
(58)
We use the zero order current operator
$\hat{J}^{edge}_{x}(x,y)\approx(-e)v_{F}\sigma^{2}$ to construct the second
quantization form for the current density. The operator is defined with
respect the to shifted ground state $|\mu>\equiv|\tilde{0}>$ with the energy
$E=\epsilon-\mu$ measured with respect the chemical potential and spinor field
$\Psi_{n=0}(x,y)$.
$J^{edge}_{x}(x,y)=<\mu|\Psi_{n=0}^{\dagger}(x,y)\hat{J}^{edge}_{x}(x,y)\Psi_{n=0}(x,y)|\mu>$
(59)
Using the spinor eigenfunction given in equation $(35)$ and the second
quantized form with the electron like operators $\alpha_{E,R}$,$\alpha_{E,L}$
and hole like $\beta_{E,R}$,$\beta_{E,L}$ we find :
$\Psi_{n=0}(x,y;t)\approx\sum_{E>0}[\alpha_{E,R}\left(\begin{array}[]{cc}U^{(n=0,R)}_{\uparrow}(x,y)\\\
U^{(n=0,R)}_{\downarrow}(x,y)\end{array}\right)_{E+\mu}{e^{-i\frac{E}{\hbar}t}}+\beta^{\dagger}_{E,R}\left(\begin{array}[]{cc}U^{(n=0,R)}_{\uparrow}(x,y)\\\
U^{(n=0,R)}_{\downarrow}(x,y)\end{array}\right)_{-E+\mu}e^{i\frac{E}{\hbar}t}$
$+\alpha_{E,L}\left(\begin{array}[]{cc}U^{(n=0,L)}_{\uparrow}(x,y)\\\
U^{(n=0,L)}_{\downarrow}(x,y)\end{array}\right)_{E+\mu}e^{-i\frac{E}{\hbar}t}+\beta^{\dagger}_{E,L}\left(\begin{array}[]{cc}U^{(n=0,L)}_{\uparrow}(x,y)\\\
U^{(n=0,L)}_{\downarrow}(x,y)\end{array}\right)_{-E+\mu}e^{i\frac{E}{\hbar}t}]$
(60)
The current is a sum of two terms computed with the eigen spinor obtained in
equation $(35)$:
$[U^{(n=0,R)}_{\uparrow}(x,y),U^{(n=0,R)}_{\downarrow}(x,y)]^{T}\sigma^{2}[U^{(n=0,R)}_{\uparrow}(x,y),U^{(n=0,R)}_{\downarrow}(x,y)]$
and
$[U^{(n=0,L)}_{\uparrow}(x,y),U^{(n=0,L)}_{\downarrow}(x,y)]^{T}\sigma^{2}[U^{(n=0,L)}_{\uparrow}(x,y),U^{(n=0,L)}_{\downarrow}(x,y)]$
which have opposite signs. Due to the parity violation caused by the
dislocation, the density of states is asymmetric
$1\pm\frac{1}{\pi}\frac{hv_{F}}{L(\mu+V)}\sqrt{1-(\frac{\epsilon_{\bot}}{\mu+V})^{2}}\hskip
7.22743pt)$ resulting in a finite current. We integrate over the transversal
direction $y$ and obtain the edge current $I_{x}^{n=0,edge}$.
$\displaystyle
I_{x}^{n=0,edge}=(-e)v_{F}\int_{-\frac{L}{2}}^{\frac{L}{2}}\frac{dx}{L}\int_{-\frac{L}{2}}^{\frac{L}{2}}\,dy<\mu|J^{edge}_{x}(x,y)|\mu>=$
$\displaystyle\frac{(-e)v_{F}}{4\pi}(\frac{L}{hv_{F}})^{2}(\frac{1}{L})\int_{-\frac{L}{2}}^{\frac{L}{2}}\frac{dx}{L}\int_{-\frac{L}{2}}^{\frac{L}{2}}\frac{dy}{L}\frac{e^{\frac{-B^{(2)}}{\pi}(\frac{x}{x^{2}+y^{2}+a^{2}})}}{\sqrt{G(x,y)}}\int\,d\epsilon_{||}\int
d\epsilon_{\bot}H[\mu-\sqrt{(\epsilon_{||})^{2}+(\epsilon_{\bot})^{2}}\hskip
7.22743pt]\frac{(hv_{F}/L)\cdot\epsilon_{||}}{(\epsilon_{||})^{2}+(\epsilon_{\bot})^{2}}$
$\displaystyle=\frac{1}{4\pi}(\frac{-ev_{F}}{L})(\frac{\mu}{hv_{F}/L})f[\frac{B^{(2)}}{L}]\cdot(H[\mu+eV-\frac{hv_{F}}{L}]-H[\mu+eV-
E_{max.}]);\hskip 14.45377ptf[\frac{B^{(2)}}{L}]\approx 6.22$
$H[\mu-\sqrt{(\epsilon_{||})^{2}+(\epsilon_{\bot})^{2}}\hskip 7.22743pt]$ is
the step function which is one for
$\sqrt{(\epsilon_{||})^{2}+(\epsilon_{\bot})^{2}}\leq\mu$. The single particle
energies are $\epsilon_{\bot}=\hbar v_{F}q_{\pm}$ and $\epsilon_{||}=\hbar
v_{F}p$. For $L\approx 10^{-6}m$, chemical potential $\mu=120mV$ and
$\frac{L}{B^{(2)}}\approx 100$ we find that the current $I_{x}^{n=0,edge}$ is
in the range of $mA$.
To conclude, we have shown that the presence of an edge dislocation gives rise
to a non-zero current which is a manifestation of the in-plane component of
the spin on the two dimensional surface . Therefore a nonzero value
$I_{x}^{n=0,edge}\neq 0$ will be an indication of the presence of the edge
dislocation. This effect might be measured using a coated tip with magnetic
material used by the technique of Magnetic Force Microscopy.
VI-Conclusions
We have used the coordinate transformation method to investigate $TI$ in the
presence of deformations. We have computed the spin connection and the metric
tensor for the three dimensional $TI$. This theory has been applied to the
surface of a $TI$ with an edge dislocation. We have shown that the tunneling
density of states is confined to two dimensional region $n=0$ and to high
energy circular contours with $n=\pm 1$. The edge dislocations violate the
parity symmetry. As a result a current which is a manifestation of in plane
spin orientation is generated. The in plane spin orientation is a
manifestation of the parity violation induced by the edge dislocation. We
propose that scanning tunneling methods might be able to verify our
prediction.
Appendix -A
We consider that a two dimensional manifold with a mapping from the curved
space $X^{a}$, $a=1,2$, to the $local$ $flat$ space $x^{\mu}$, $\mu=x,y$
exists. We introduce the tangent vector Green
$e^{a}_{\mu}(\vec{x})=\frac{\partial X^{a}(\vec{x})}{\partial x^{\mu}}$,
$\mu=x,y$ which satisfies the orthonormality relation
$e^{a}_{\mu}(\vec{x})e^{b}_{\mu}(\vec{x})=\delta_{a,b}$ (here we use the
convention that we sum over indices which appear twice). The metric tensor for
the curved space is given in terms of the flat metric $\delta_{a,b}$ and the
scalar product of the tangent vectors:
$e^{a}_{\mu}(\vec{x})e^{a}_{\nu}(\vec{x})=g_{\mu,\nu}(\vec{x})$. The linear
connection is determined by the Christoffel tensor
$\Gamma^{\lambda}_{\mu,\nu}$ :
$\nabla_{\partial_{\mu}}\partial_{\nu}=-\Gamma^{\lambda}_{\mu,\nu}\partial_{\lambda}$
(62)
The Christoffel tensor is constructed from the metric tensor
$g_{\mu,\nu}(\vec{x})$.
$\Gamma^{\lambda}_{\mu,\nu}=-\frac{1}{2}\sum_{\tau=x,y}g^{\lambda,\tau}(\vec{x})[\partial_{\nu}g_{\nu,\tau}(\vec{x})+\partial_{\mu}g_{\nu,\tau}(\vec{x})-\partial_{\tau}g_{\mu,\nu}(\vec{x})]$
(63)
Next, we introduce the vector field
$\vec{V}=V^{a}\partial_{a}=V^{\mu}\partial_{\mu}$ where $a=1,2$ are the
components in the curved space and $\mu=x,y$ represents the coordinate in the
fixed cartesian frame. The covariant derivative of the vector field $V^{a}$ is
determined by the spin connection $\omega_{q,b}^{\mu}$ which needs to be
computed:
$D_{\mu}V^{a}(\vec{x})=\partial_{\mu}V^{a}(\vec{x})+\omega_{a,b}^{\mu}V^{b}$
(64)
For a two component spinor, we can identify the spin connection in the
following way: The spinor in the the curved space (generated by the
dislocation) is represented by $\widetilde{\Psi}(\vec{X})$ and in the
Cartesian space it is given by is given by $\Psi(\vec{x})$ Maggiore . The two
component spinor represents a chiral fermion which transform under spatial
rotation as spin half fermion:
$\displaystyle\widetilde{\Psi}(\vec{X})=e^{\frac{-i}{2}\omega_{1,2}\sigma_{3}}\Psi(\vec{x})$
$\displaystyle e^{\frac{-i}{2}\omega_{1,2}\sigma_{3}}\equiv
e^{\frac{1}{2}\omega_{a,b}\Sigma^{a,b}}\equiv
e^{\sum_{a=1,2}\sum_{b=1,2}\frac{1}{2}\omega_{a,b}\Sigma^{a,b}}$
$\displaystyle\omega_{a,b}\equiv-\omega_{b,a}$
$\displaystyle\Sigma^{a,b}\equiv\frac{1}{4}[\sigma^{a},\sigma^{b}]$
We have used the anti symmetric property of the rotation matrix
$\omega_{a,b}\equiv-\omega_{b,a}$, and the representation of the generator
$\Sigma^{a,b}$ in terms of the Pauli matrices.
Therefore for a two component spinor we obtain the connection:
$D_{\mu}\Psi(\vec{x})=(\partial_{\mu}+\frac{1}{2}\omega^{a,b}_{\mu}\Sigma_{a,b})\Psi(\vec{x})\equiv(\partial_{\mu}+\frac{1}{8}\omega^{a,b}_{\mu}[\sigma_{a},\sigma_{b}])\Psi(\vec{x})$
(66)
Next we will compute the spin connection $\omega^{a,b}_{\mu}$ using the
Christoffel tensor. In the physical coordinate basis $x^{\mu}$ the covariant
derivative $D_{\mu}V^{\nu}(\vec{x})$ is determined by the Christoffel tensor:
$D_{\mu}V^{\nu}(\vec{x})=\partial_{\mu}V^{\nu}(\vec{x})+\Gamma^{\lambda}_{\mu,\nu}V^{\lambda}$
(67)
The relation between the spin connection and the linear connection can be
obtained from the fact that the two covariant derivative of the vector
$\vec{V}$ are equivalent.
$D_{\mu}V^{a}=e^{a}_{\nu}D_{\mu}V^{\nu}$ (68)
Since we have the relation $V^{a}=e^{a}_{\nu}V^{\nu}$ it follows from the last
equation
$D_{\mu}[e^{a}_{\nu}]=D_{\mu}\partial_{\nu}e^{a}=(D_{\mu}\partial_{\nu})e^{a}+\partial_{\nu}(D_{\mu}e^{a})=0$
(69)
Using the definition of the Christoffel index and the differential geometry
relation
$\nabla_{\partial_{\mu}}\partial_{\nu}=-\Gamma^{\lambda}_{\mu,\nu}\partial_{\lambda}$
Green , we obtain the relation between the spin connection and the linear
connection:
$D_{\mu}[e^{a}_{\nu}]=\partial_{\mu}e^{a}_{\nu}(\vec{x})-\Gamma^{\lambda}_{\mu,\nu}e^{a}_{\lambda}(\vec{x})+\omega^{a}_{\mu,b}e^{b}_{\nu}(\vec{x})\equiv
0$ (70)
Solving this equation, we obtain the spin connection given in terms of the
Burger vector. We multiply from left equation $(70)$ by the tangent vector
$e^{a}_{\nu}$ and replace $\Gamma^{\lambda}_{\mu,\nu}$ with the representation
given in equation $(63)$. We use the metric tensor relations
$e^{a}_{\mu}(\vec{x})e^{b}_{\mu}(\vec{x})=\delta_{a,b}$,
$e^{a}_{\mu}(\vec{x})e^{a}_{\nu}(\vec{x})=g_{\mu,\nu}(\vec{x})$. and find
Green :
$\displaystyle\omega^{a,b}_{\mu}=\frac{1}{2}e^{\nu,a}(\partial_{\mu}e^{b}_{\nu}-\partial_{\nu}e^{b}_{\mu})-\frac{1}{2}e^{\nu,b}(\partial_{\mu}e^{a}_{\nu}-\partial_{\nu}e^{a}_{\mu})$
$\displaystyle-\frac{1}{2}e^{\rho,a}e^{\sigma,b}(\partial_{\rho}e_{\sigma,c}-\partial_{\sigma}e_{\rho,c})e^{c}_{\mu}$
(71)
We notice the asymmetry between $e^{\nu,a}$ and $e_{a,\nu}$:
$e^{\nu,a}\equiv g^{\nu,\lambda}e^{a}_{\lambda}$ and
$e_{a,\nu}\equiv\delta_{a,b}e^{b}_{\nu}$
For our case we have a two component the spin connection $\omega_{x}^{12}$ and
$\omega_{y}^{12}$
$\displaystyle\omega_{x}^{12}=\frac{1}{2}e^{\nu,1}(\partial_{x}e^{2}_{\nu}-\partial_{\nu}e^{2}_{x})-\frac{1}{2}e^{\nu,2}(\partial_{x}e^{1}_{\nu}-\partial_{\nu}e^{1}_{x})-\frac{1}{2}e^{\rho,a}e^{\sigma,b}(\partial_{\rho}e_{\sigma,c}-\partial_{\sigma}e_{\rho,c})e^{c}_{x};$
$\displaystyle\omega_{y}^{12}=\frac{1}{2}e^{\nu,1}(\partial_{y}e^{2}_{\nu}-\partial_{\nu}e^{2}_{y})-\frac{1}{2}e^{\nu,2}(\partial_{y}e^{1}_{\nu}-\partial_{\nu}e^{1}_{y})-\frac{1}{2}e^{\rho,a}e^{\sigma,b}(\partial_{\rho}e_{\sigma,c}-\partial_{\sigma}e_{\rho,c})e^{c}_{y}$
These equations are further simplified with the help of equations $(11-17)$
with $e^{1}_{y}=0$ , $e^{1}_{x}=1$ and the Burger tensor
$\partial_{x}e^{2}_{y}-\partial_{y}e^{2}_{x}=B^{(2)}\delta^{2}(\vec{r})$ .
$\displaystyle\omega_{x}^{12}=\frac{1}{2}g^{\nu,\lambda}e^{1}_{\lambda}(\partial_{x}e^{2}_{\nu}-\partial_{\nu}e^{2}_{x})-\frac{1}{2}g^{\rho,r}e^{1}_{r}g^{\rho,s}e^{2}_{s}[\partial_{\rho}(\delta_{c,b}e^{b}_{\sigma})-\partial_{\sigma}(\delta_{c,d}e^{d}_{\rho})]e^{c}_{x}=$
$\displaystyle\frac{1}{2}B^{(2)}\delta^{(2)}(\vec{r})[g^{y,x}e^{1}_{x}+g^{y,y}e^{1}_{y}-(g^{x,r}g^{y,s}-g^{y,r}g^{x,s})(e^{1}_{r}e^{2}_{s}e^{2}_{x})]=$
$\displaystyle\frac{1}{2}B^{(2)}\delta^{(2)}(\vec{r})[g^{y,x}e^{1}_{x}-(g^{x,x}g^{y,y}-g^{y,x}g^{x,y})e^{1}_{x}e^{2}_{y}e^{2}_{x}]\approx$
$\displaystyle\frac{1}{2}B^{(2)}\delta^{(2)}(\vec{r})[-\frac{B^{(2)}}{2\pi}\frac{y}{y^{2}+x^{2}}-(1-(\frac{B^{(2)}}{2\pi}\frac{y}{y^{2}+x^{2}})^{2})(\frac{B^{(2)}}{2\pi}\frac{y}{x^{2}+y^{2}})(1-\frac{B^{(2)}}{2\pi}\frac{x}{x^{2}+y^{2}})]\approx$
$\displaystyle\frac{1}{2}B^{(2)}\delta^{(2)}(\vec{r})[-\frac{B^{(2)}}{2\pi}\frac{2y-x}{y^{2}+x^{2}}]$
and
$\displaystyle\omega_{y}^{12}=\frac{1}{2}e^{\nu,1}(\partial_{y}e^{2}_{\nu}-\partial_{\nu}e^{2}_{y})-\frac{1}{2}e^{\nu,2}(\partial_{y}e^{1}_{\nu}-\partial_{\nu}e^{1}_{y})-\frac{1}{2}e^{\rho,1}e^{\sigma,2}[\partial_{\rho}(\delta_{c,b}e_{\sigma}^{b})-\partial_{\sigma}(\delta_{c,d}e_{\rho}^{d})]e^{c}_{y}=$
$\displaystyle\frac{1}{2}g^{\nu,\lambda}e^{1}_{\lambda}[\partial_{y}e^{2}_{\nu}-\partial_{\nu}e^{2}_{y}]-\frac{1}{2}g^{\nu,r}e^{1}_{r}[\partial_{y}e^{1}_{\nu}-\partial_{\nu}e^{1}_{y}]-\frac{1}{2}g^{\rho,r}e^{1}_{r}g^{\sigma,s}e^{2}_{s}[\partial_{\rho}e^{c}_{\sigma}-\partial_{\sigma}e^{c}_{\rho}]e^{c}_{y}=$
$\displaystyle-\frac{B^{(2)}}{2}\delta^{(2)}(\vec{r})g^{x,\lambda}e^{1}_{\lambda}-\frac{B^{(2)}}{2}\delta^{(2)}(\vec{r})[g^{x,r}g^{y,s}-g^{y,r}g^{x,s}]e^{1}_{r}e^{2}_{s}e^{2}_{y}\approx-\frac{B^{(2)}}{2}\delta^{(2)}(\vec{r})$
To first order first the Burger vector $B^{(2)}$ the spin connections are
given by : $\omega_{x}^{12}=-\omega_{x}^{21}\approx 0$ and
$\omega_{y}^{12}=-\omega_{y}^{21}\approx-\frac{1}{2}B^{(2)}\delta^{2}(\vec{r})$.
Figure 1: The contours
$(x(s))^{2}+(y(s)-\frac{R_{g}}{n})^{2}=(\frac{R_{g}}{n})^{2}$ for $n=\pm 1,\pm
2,\pm 3$(in decreasing size ),$R_{g}(n)=\frac{R_{g}}{n}$. $n=0$ corresponds to
the equation $y(s)=0$ and $|y|>d$ (see the text). The the distance is measured
in units of the Burger vector $B^{(2)}$.
Figure 2: The tunneling density of states for $n=0$ , $\frac{dI}{dV}\propto
D^{(n=0)}(\frac{x}{B^{(2)}},\frac{y}{B^{(2)}};\mu=120mV)$. The right corner
represents the intersection of the $x$ coordinate which runs from $30$ (right
corner) to $-30$ and the $y$ coordinate which runs from $-30$ (right corner)
to $30$ in units of the Burger vector.
Figure 3: The tunneling density of states for $n=0$ as a function of $y$ and
$V$ $\frac{dI}{dV}\propto
D^{(n=0)}(\frac{x}{B^{(2)}}=-2,\frac{y}{B^{(2)}};\mu=120mV)$. The voltage
range is $-120\leq V\leq 50$ and the $y$ coordinate is in the range
$-30\leq\frac{y}{B^{(2)}}\leq 30$.
Figure 4: Many Dislocations - with the core of the dislocations at
$[x_{w},y=0]$ , $w=1,2...2M$; The maximum of the tunneling density of states
is confined along $y=0$. The coordinates of the tunneling density of states
are restricted to : $-40\leq\frac{x}{B^{(2)}}\leq 40$ and
$-20\leq\frac{y}{B^{(2)}}\leq 20$.
Figure 5: The discrete tunneling density of states for $n=1$, as a function of
the voltage $V$
$D^{(n=1)}(V;\theta=\frac{\pi}{2},\frac{u}{B^{(2)}},\mu=120mV)$
Figure 6: The tunneling density of states as a function of $\theta$
$D^{(n=1)}(\theta;\frac{u}{B^{(2)}}=0.01,V=280mV,\mu=120mV)$
Figure 7: The tunneling density of states as a function of $\theta$ and $u$ at
a fixed voltage $V=280mV$
$D^{(n=1)}(\theta,\frac{u}{B^{(2)}};V=280mV,\mu=120mV)$
## References
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* (2) B.A.Volkov and O.A. Pankratov, JETP LETT. vol.42,179(1985)
* (3) M.F.L.Gotelman, K.Jansen and D.B. Kaplan ”‘Chern-Simons Currents and Chiral Fermions on the Lattice ”‘ Phys.Lett.B301,219(1993).
* (4) Michael Creutz and Ivan Horwath ”‘Surface States and Chiral Symmetry On The Lattice”’ Phys.Rev.50,2297(1994)
* (5) C.L. Kane and E.J. Mele Phys.Rev. Lett. 95 226801 (2005)
* (6) C.L. Kane and E.J. Mele ,‘Phys.Rev.Lett. 95,146802(2005).
* (7) H.Zhang et al. nature physics 5,438,(2009).
* (8) D. Schmeltzer, “Topological Insulators-transport in curved space” , arXiv:1012.5871 and Advances in Condensed Matter and Materials Research ,volume 10 Editors:Hans Geelvinck and Sjaak Reyst ,chapter 9, pages 379-403(2011).
* (9) J.E.Moore and L.Balents, cond-mat/0607314
* (10) Andrew M.Essin and J.E.More, cond-mat/0705.0172.
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* (12) Xiao-Liang Qi and Shou-Cheng Zhang cond-mat/1008.2026
* (13) Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki, Andreas W.W. Ludwig cond-mat/0803.2786.
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* (24) M.Nakahara ”‘Geometry,Topology And Physics ”‘ Taylor Francis Press 2003.
* (25) R.Jackiw and J.R. Schrieffer,Nuclear Physics B190,253-265,(1981)
* (26) H.Kleinert ”‘Multivalued Fields in Condensed Mattewr ,Electromagnetism, and Gravitation”’ World Scientific (2008) pages 348-350.
* (27) Z.F.Ezawa,”’Quantum Hall effects”’ World Scientific (2008).
* (28) B.Andrea Bernevig Taylor L.Hughes, and Shou-Cheng Zhang cond-mat/061139.
* (29) C. Wu,B. A. Bernevig and Shou-Cheng Zhang Phys.Rev.Lett. 96 106401 (2006).
* (30) Ayelet Pnueli ‘J.Phys. A: Math.Gen.27 1345-1352 (1994)
* (31) M.B.Green,J.H. Schwartz and E.Witten, ”‘Some Differential Geometry”’Superstrings Theory vol.2 Cambridge Monographs on Mathematical Physics pages $271-277$ (1987).
* (32) N.D.Birrell and P.C. Davies ”‘Quantum Fields in Curved Space”’ pages $81-85$ Cambridge University Press (1982).
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|
arxiv-papers
| 2011-12-22T21:11:24 |
2024-09-04T02:49:25.627442
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. Schmeltzer",
"submitter": "David Schmeltzer",
"url": "https://arxiv.org/abs/1112.5461"
}
|
1112.5465
|
# The $p_{x}+ip_{y}$ Chiral Superconductor wire weakly coupled to two metallic
rings pierced by an external flux
D. Schmeltzer Physics Department, City College of the City University of New
York
New York, New York 10031
###### Abstract
We consider a p-wave superconductor wire coupled to two metallic rings.
confined to a one-dimensional wire. At the two interface between the the wire
and the metallic rings the pairing order parameter vanishes, as result two
zero modes Majorana fermion appear. The two metallic rings are pierced by
external magnetic fluxes. The special features of the Majorana Fermions can be
deduced from the correlation between the currents in the two rings.
I Introduction
Topological Superconductors are characterized by the invariance under charge
conjugation symmetry. As a result of this invariance zero modes Majorana
fermion appear at the interfaces between the superconductor a metal.
$Sr_{2}Ru$ is the material where Majorana fermion might be observed since the
pairing order parameter is characterized by $p_{x}+ip_{y}$ symmetry. The
electronic excitations for the ground state pairing $p_{x}+ip_{y}$ are given
by half vortices which are zero mode Majorana fermions Kitaev ; DunghaiLee ;
Oreg1 ; Oreg2 .
The realization of the p-wave superconductors physics and the formation of the
zero mode Majorana Fermions at the edges of the wire can be achieved using a
p-wave wire of length $L$ coupled to two metallic rings which are pierced by
external magnetic fields.The current in the rings is coupled to the p-wave
wire trough the Majorana modes which are bound at the interface. For different
fluxes in the rings we find that the excitations in the wires become imaginary
resulting in an unstable vanishing current. The only stable current are
obtained for the case that the imaginary part vanishes. The stable current are
obtained for special relations between the magnetic fluxes and the wire
excitations energy. This feature is attributed to the existence of the
Majorana fermions. We find that the correlation between the currents in the
two rings is affected by the presence of the Majorana Fermions. Therefore the
model introduced here can be used for the identification of the Majorana
fermions.
The content of this paper is as follows: In chapter $II$ we present the model
of the the $p-wave$ wire coupled to two metallic rings. Using the left and
right mover we obtain the continuum representation of the superconductor wire.
At the two edges of the wire we obtain the two zero modes of the wire. The
effect of the coupling between the wire and the metallic rings is dominated by
the zero modes of the wire. As a result we derive the effective Hamiltonian
between the zero modes and the metallic rings. This effective Hamiltonian is
controlled by the low energy excitation in the wire (the coupling energy
between the two edges) $\epsilon\approx|\Delta_{0}|e^{-L|\Delta_{0}|}$
,$L|\Delta_{0}|>>1$ where $\Delta_{0}$ is the pairing field and $L$ is the
length of the wire. In section $III$ we consider the case that the wire energy
$\epsilon\rightarrow 0$. In section $IV$ we consider the case where the wire
energy is finite. Section $V$ is devoted to conclusions.
II- The model for the p-wave wire weakly coupled to two rings pierced by
external fluxes
The p-wave wire of length $L$ is given by the Hamiltonian $H_{P-W}$ :
$H_{P-W}=-t\sum_{x,x^{\prime}}(C^{+}(x)C(x^{\prime})+h.c.)-\mu_{F}\sum_{x}C^{+}(x)C(x)-\hat{\Delta}\sum_{x,x^{\prime}}[\gamma_{x,x^{\prime}}C^{+}(x)C^{+}(x^{\prime})+\gamma_{x,x^{\prime}}C(x^{\prime})C(x)]$
(1)
The pairing gap is given by $\hat{\Delta}$ and the polarized fermion operator
is given by $C(x)\equiv C_{\sigma=\uparrow}(x)$. The matrix elements
$\gamma_{x,x^{\prime}}$ obey obey the p-wave symmetry:
$\gamma_{x,x^{\prime}=x-a}=-\gamma_{x,x^{\prime}=x+a}$,
$|\gamma_{x,x^{\prime}}|=1$, therefore the time reversal and parity symmetry
are both broken and obeys the pairing boundary conditions
$\hat{\Delta}(x=0)=\hat{\Delta}(x=L)=0$ and $\hat{\Delta}(x)=\hat{\Delta}_{0}$
for $0<x<L$.
We introduce the right and left fermions in the continuum representation for
the fermions in the wire : $C(x=na)\sqrt{a}\rightarrow
C(x)=e^{ik_{F}x}\hat{C}_{R}(x)+e^{-ik_{F}x}\hat{C}_{L}(x)$ and find that
equation $1$ is replaced by the Hamiltonian :
$H_{P-W}=\int
dx[v_{F}\Psi^{\dagger}(x)\sigma^{z}(-i\partial_{x})\Psi(x)+\Delta(x)\Psi^{\dagger}(x)\sigma^{x}\Psi(x)]$
(2)
This Hamiltonian is invariant under the charge conjugation symmetry. The
spinor
$\Psi^{\dagger}(x)=[C^{\dagger}_{R}(x),C_{L}(x)]\equiv[e^{\frac{i\pi}{4}}\hat{C}^{\dagger}_{R}(x),e^{\frac{-i\pi}{4}}\hat{C}_{L}(x)]$
satisfies the reality constraint condition $K\Psi^{\dagger}(x)=\Psi(x)$, where
$K$ is the charge conjugation operator.
The pairing field $\Delta(x)\equiv 4\hat{\Delta}(x)\sin(k_{F}a)$ can be
written as $\Delta(x)=M_{L}(x)+M_{R}(x-L)$ where
$M_{L}(x)=\frac{\Delta(0)}{2}sgn(x)$ and
$M_{R}(x-L)=\frac{\Delta(0)}{2}sgn(x-L)$ obeys the domain wall property:
$M_{L}(-x)=-M_{L}(x)$ (at x=0) , $M_{R}(-(x-L))=-M_{R}(x-L)$ (at x=L) . The
zero modes eigenfunctions are given by
$\eta_{\lambda}(x)=[\eta_{1}(x),\eta_{2}(x)]^{T}$ and eigenstates of the
operator $\sigma^{y}\eta_{\lambda}(x)=\lambda\eta_{\lambda}(x)$ with
$\lambda=\pm 1$. The zero mode spinor which is localized around $x=0$ is
identified with $\lambda=-1$ and the second one which localized around $x=L$
is identified with $\lambda=1$
$\displaystyle\eta_{Left}(x)\equiv\eta_{\lambda=-1}(x)=e^{\frac{-1}{v_{F}}\int_{0}^{x}\Delta(x^{\prime})\,dx^{\prime}}\frac{e^{i\frac{\pi}{4}}}{\sqrt{2}}[1,-i]^{T}$
$\displaystyle\eta_{Right}(x)\equiv\eta_{\lambda=1}(x)=e^{\frac{1}{v_{F}}\int_{L}^{x}\Delta(x^{\prime})\,dx^{\prime}}\frac{e^{-i\frac{\pi}{4}}}{\sqrt{2}}[1,i]^{T}$
The spinor operator $\Psi(x)$ with the two zero mode Majorana operators
$\alpha_{l}$ (at the left edge) and $\alpha_{r}$ (at the right edge),
$(\alpha_{r})^{2}=(\alpha_{l})^{2}=\frac{1}{2}$ takes the form :
$\Psi(x)\rightarrow\Psi(x)+\alpha_{r}\eta_{\lambda=1}(x)+\alpha_{l}\eta_{\lambda=-1}(x)$
(4)
As a result the low energy of the p-wave wire is given by:
$H_{P-W}=\int
dx[v_{F}\Psi^{\dagger}(x)\sigma^{z}(-i\partial_{x})\Psi(x)+\Delta(x)\Psi^{\dagger}(x)\sigma^{x}\Psi(x)]\approx\frac{i}{2}\epsilon\alpha_{l}\alpha_{r}$
(5)
where $\epsilon\approx|\Delta_{0}|e^{-L|\Delta_{0}|}$ ,$L|\Delta_{0}|>>1$.
At this stage we include the two rings Hamiltonian pierced by the fluxes
$\hat{\varphi}_{i}$ , $i=1,2$ and length $l_{ring}<<L$. The left ring is
restricted to the region $-l_{ring}\leq x\leq 0$ and the right ring is
restricted to $L\leq x\leq L+l_{ring}$. Since only the wire fields at $x=0$
and $x=L$ are involved we fold the space of the right ring $i=2$ such that
both rings are restricted to the region $-l_{ring}\leq x\leq 0$. As a result
the external fluxes obey: $\hat{\varphi}_{1}\rightarrow\hat{\varphi}_{1}$ and
$\hat{\varphi}_{2}\rightarrow-\hat{\varphi}_{2}$. In addition we replace for
each ring the fermion operator $\psi_{i}(x)$,i=1,2 by the $right$ $R_{i}(x)$
and $left$ fermions $L_{i}(x)$:
$\psi_{i}(x)=R_{i}(x)e^{ik_{F}x}+L_{i}(x)e^{-ik_{F}x}$ (6)
We replace for each ring the $right$ and $left$ movers by four Mayorana
operators:
$\displaystyle r_{i}\equiv(R_{i}(0)-R^{\dagger}_{i}(0))(-i)$ $\displaystyle
l_{i}\equiv L_{i}(0)+L^{\dagger}_{i}(0)$
The the matrix element between the wire and rings is given by $-g$. As a
result the low energy hopping Hamiltonian is given by:
$H_{T}=\frac{-ig}{\sqrt{2}}[\alpha_{l}(r_{1}+l_{1})+\alpha_{r}(r_{2}-l_{2})]$
(8)
We replace the Majorana zero modes $\alpha_{l}$ and $\alpha_{r}$ by the
fermion pair, $q=\alpha_{l}+i\alpha_{r}$, $q^{\dagger}=\alpha_{l}-i\alpha_{r}$
which obey: $[q,q^{\dagger}]_{+}=1$, $q^{\dagger}|0>=|1>$ and $q|1>=|0>$ where
$|0>$ is ground state of wire and rings:
$R_{i,p}(x)|0>=L_{i,p}(x)|0>=R^{\dagger}_{i,h}(x)|0>=L^{\dagger}_{i,h}(x)|0>=q|0>=0$.
Where $R_{i,p}(x)$, $L_{i,p}(x)$ are the particle operators and
$R^{\dagger}_{i,h}(x)$, $L^{\dagger}_{i,h}(x)$ are the holes operators
$R^{\dagger}_{i,h}(x)$, $L^{\dagger}_{i,h}(x)$. The right and left mover are
given as a linear combination of particles and holes operators. $R_{i,p}(x)$
$L_{i,p}(x)$ represent the annihilation of particles and
$R^{\dagger}_{i,h}(x)$, $L^{\dagger}_{i,h}(x)$ are the creation operators for
holes.
$R_{i}(x)=R_{i,p}(x)+R^{\dagger}_{i,h}(x);\hskip
7.22743ptL_{i}(x)=L_{i,p}(x)+L^{\dagger}_{i,h}(x)$ (9)
Using the Fermionic representation we replace $H_{P-W}$ given in equation $5$
and $H_{T}$ given in equation $(8)$ by :
$H_{P-W}+H_{T}\equiv\epsilon
q^{\dagger}q-\frac{ig}{2\sqrt{2}}[(q+q^{\dagger})(r_{1}+l_{1})-i(q-q^{\dagger})(r_{2}-l_{2})]$
(10)
The value of the wire energy $\epsilon$ in equations $5,10$ is based on the
projection of the spinor (in equation $4$) on the zero modes
$\eta_{\lambda=1}(x)$ and $\eta_{\lambda=-1}(x)$. Since the leads couple to
the modes in the wire, we expect that the non-zero modes will give rise to a
finite width of the energy $\epsilon$. Therefore for finite energies we will
replace $\epsilon$ by $\hat{\epsilon}\equiv\epsilon-i\Gamma$, where the width
$\Gamma\propto g^{4}$.
We perform an exact integration over the fermion operators $q$,$q^{\dagger}$
and find the time dependent effective interaction $H_{eff}(t)$:
$\displaystyle
H_{eff}(t)=\frac{-ig^{2}}{2}\int\,dt^{\prime}\mu[t,t^{\prime}]e^{-i\frac{\hat{\epsilon}}{\hbar}(t-t^{\prime})}[r_{2}(t)-l_{2}(t)+i(r_{1}(t)+l_{1}(t))][r_{2}(t^{\prime})-l_{2}(t^{\prime})-i(r_{1}(t^{\prime})+l_{1}(t^{\prime}))]$
where $\mu[t,t^{\prime}]$ is the step function which is one for
$t>t^{\prime}$.
III-The effective interaction $H_{eff}(t)$ in the limit $\epsilon\rightarrow
0$
When $\epsilon\rightarrow 0$ equation $11$ is replaced by :
$\displaystyle
H_{eff}(t)=\frac{-ig^{2}}{2}\int\,dt^{\prime}\mu[t,t^{\prime}][r_{2}(t)-l_{2}(t)+i(r_{1}(t))+l_{1}(t)][r_{2}(t^{\prime})-l_{2}(t^{\prime})-i(r_{1}(t^{\prime})+l_{1}(t^{\prime})]$
Using the scaling analysis given in davidimpurity we observe that the
effective interaction flows to the strong coupling limit and find
$g^{2}(b)=g^{2}b^{2-\alpha}$, $\alpha\approx 1$ where $b>1$. Since the
coupling constant $g(b)$ flows to infinity, the only way a solution will
exists if the effective interaction annihilates the ground state:
$H_{eff}(t)|0>=0$. Therefore the physical solution is given by the
$constraint$ rings equation:
$[r_{2}(t)-l_{2}(t)-i(r_{1}(t)+l_{1}(t)]|0>=0$ (13)
Since
$R_{i,p}(x)|0>=L_{i,p}(x)|0>=R^{\dagger}_{i,h}(x)|0>=L^{\dagger}_{i,h}(x)|0>=0$
the constraint condition implies for particles ($p$ stands for particles) the
equation :
$[i(R^{\dagger}_{2,p}-L^{\dagger}_{1,p})-(L^{\dagger}_{2,p}-R^{\dagger}_{1,p})]|0>=0$;
and for holes ($h$ stands for holes)
$[i(R_{2,h}-L_{1,h})-(L_{2,h}-R_{1,h})]|0>=0$. We find the constraint
equation:
$\psi_{1}(x=0)\equiv
R_{1}(x=0)+L_{1}(x=0)=e^{-i\frac{\pi}{2}}[R_{2}(x=0)+L_{2}(x=0)]\equiv
e^{-i\frac{\pi}{2}}\psi_{2}(x=0)=\widetilde{\psi}_{2}(x=0)$ (14)
The explicit identity contains the phase factor $e^{-i\frac{\pi}{2}}$ which is
obtained from Bosonization (see below). Following equation $14$ we find the
constraint condition for the ground state $|0>$:
$\kappa\equiv[\psi_{1}(x)-\psi_{2}(x)]|_{x=0},\hskip 14.45377pt\kappa|0>=0$
(15)
Following rings we find two additional constraints equation :
$\mathcal{E}\equiv[(-i\partial_{x}-\frac{2\pi}{l_{ring}}\hat{\varphi}_{1})^{2}\psi_{1}(x)-(-i\partial_{x}+\frac{2\pi}{l_{ring}}\hat{\varphi}_{2})^{2}\psi_{2}(x)]|_{x=0},\hskip
14.45377pt\mathcal{E}|0>=0$ (16)
$\mathcal{J}\equiv[(-i\partial_{x}-\frac{2\pi}{l_{ring}}\hat{\varphi}_{1})\psi_{1}(x)+(-i\partial_{x}+\frac{2\pi}{l_{ring}}\hat{\varphi}_{2})\psi_{2}(x)]|_{x=0},\hskip
14.45377pt\mathcal{J}|0>=0$ (17)
Any eigenstate of N particles must satisfy the set of equations $15-17$ with
the periodic boundary condition $\psi_{i}(x)=\psi_{i}(x+l_{ring})$. For the
case $N=1$ (one particle) we have :
$|N=1>=\int_{-l_{ring}}^{0}\,dx[f_{1}(x)\psi^{\dagger}_{i}(x)+f_{2}(x)\psi^{\dagger}_{2}(x)]$
(18)
where $f_{i}(x)$ are the amplitudes . Using equations $15-17$ we find finite
solutions for the amplitudes $f_{i}(x)$ only when the fluxes are equal.
IV-The finite limit $\epsilon\neq 0$
In order to study the finite limit $\epsilon\neq 0$ we will use the zero mode
Bosonization method davidimpurity ; Berkovits . The right $R_{i}(x)$ and left
$L_{i}(x)$ fermions for each ring $i=1,2$ is given by:
$\displaystyle
R_{i}(x)=\sqrt{\frac{\Lambda}{2\pi}}Z_{i}e^{i\alpha_{R,i}}e^{\frac{2\pi}{l_{ring}}(N_{R,i}-\frac{1}{2})x}e^{i\sqrt{4\pi}\vartheta_{R,i}(x)}$
$\displaystyle
L_{i}(x)=\sqrt{\frac{\Lambda}{2\pi}}Z_{i}e^{i\alpha_{L,i}}e^{\frac{2\pi}{l_{ring}}(N_{L,i}-\frac{1}{2})x}e^{i\sqrt{4\pi}\vartheta_{L,i}(x)}$
Where $Z_{1}Z_{2}=-Z_{2}Z_{1}$ are Majorana variable which ensure the anti-
commutation between the two rings in the bosonic representation. In the
bosonic representation we have the zero modes $\alpha_{R,i}$ $\alpha_{L,i}$
bosons and their conjugates ,$N_{R,j}$,$N_{L,j}$. The zero modes obey the
commutation rules :$[-\alpha_{L,i},N_{L,j}]=i\delta_{i,j}$ and
$[\alpha_{R,i},N_{R,j}]=i\delta_{i,j}$.
As a result we obtain the zero mode representation in terms of the fermion
numbers $N_{L,i}$,$N_{R,i}$ and fluxes $\hat{\varphi}_{i}$ in each ring:
$\displaystyle H_{0}=\frac{\pi
v_{F}\hbar}{2l_{ring}}[(N_{L,1}-N_{R,1}+2\hat{\varphi}_{1})^{2}+(N_{L,1}+N_{R,1})^{2}]+\frac{\pi
v_{F}\hbar}{2l_{ring}}[(N_{L,2}-N_{R,2}+2\hat{\varphi}_{2})^{2}+(N_{L,1}+N_{R,1})^{2}]$
Using the equations of motion
$i\hbar\frac{d\alpha_{R,i}}{dt}=[\alpha_{R,i},H_{0}]$ and
$i\hbar\frac{d\alpha_{L,i}}{dt}=[\alpha_{L,i},H_{0}]$ ,$i=1,2$ we obtain the
zero mode representation in the interaction picture. We will use the zero mode
representation $\alpha^{I}_{R,i}(t)$, $\alpha^{I}_{L,i}(t)$ in the interaction
picture in order to evaluate equation $15$. We find that $H_{eff}(t)$ is given
in terms of the zero mode functions $F(t+\frac{\tau}{2})$ and
$G(t-\frac{\tau}{2})$ :
$H_{eff}(t)=-2i\hat{g}^{2}\int_{0}^{\infty}\,d\tau
F(t+\frac{\tau}{2})e^{-i\frac{\hat{\epsilon}}{\hbar}\tau}G(t-\frac{\tau}{2})$
(21)
where $\hat{g}^{2}=g^{2}\frac{\Lambda}{2\pi}$ is the coupling constant and
$\nu_{0}\equiv\frac{\epsilon}{\hbar}$, $\nu_{i}\equiv\frac{2\pi
v_{F}}{l_{ring}}\hat{\varphi}_{i}$ are the equivalent wire and rings
frequencies and $\hat{\Gamma}=\frac{\Gamma}{\hbar}$ is the width. The
functions $F(t+\frac{\tau}{2})$ and $G(t-\frac{\tau}{2})$ are given by:
$\displaystyle
F(t+\frac{\tau}{2})=Z_{2}((\sin\alpha^{I}_{R,2}(t+\frac{\tau}{2})-\cos\alpha^{I}_{L,2}(t+\frac{\tau}{2}))+iZ_{1}((\sin\alpha^{I}_{R,1}(t+\frac{\tau}{2})-\cos\alpha^{I}_{L,1}(t+\frac{\tau}{2}))=$
$\displaystyle
Z_{2}[\sin\alpha^{I}_{R,2}(t)\cos(\nu_{2}\tau)+\cos\alpha^{I}_{R,2}(t)\sin(\nu_{2}\tau)-\cos\alpha^{I}_{L,2}(t)\cos(\nu_{2}\tau)+\sin\alpha^{I}_{L,2}(t)\sin(\nu_{2}\tau)]$
$\displaystyle+iZ_{1}[\sin\alpha^{I}_{R,1}(t)\cos(\nu_{1}\tau)-\cos\alpha^{I}_{R,1}(t)\sin(\nu_{1}\tau)+\cos\alpha^{I}_{L,1}(t)\cos(\nu_{1}\tau)-\sin\alpha^{I}_{L,1}(t)\sin(\nu_{1}\tau)]$
$\displaystyle
G(t-\frac{\tau}{2})=Z_{2}((\sin\alpha^{I}_{R,2}(t-\frac{\tau}{2})-\cos\alpha^{I}_{L,2}(t-\frac{\tau}{2}))-iZ_{1}((\sin\alpha^{I}_{R,1}(t-\frac{\tau}{2})+\cos\alpha^{I}_{L,1}(t-\frac{\tau}{2}))=$
$\displaystyle
Z_{2}[\sin\alpha^{I}_{R,2}(t)\cos(\nu_{2}\tau)-\cos\alpha^{I}_{R,2}(t)\sin(\nu_{2}\tau)-\cos\alpha^{I}_{L,2}(t)\cos(\nu_{2}\tau)-\sin\alpha^{I}_{L,2}(t)\sin(\nu_{2}\tau)]$
$\displaystyle-
iZ_{1}[\sin\alpha^{I}_{R,1}(t)\cos(\nu_{1}\tau)+\cos\alpha^{I}_{R,1}(t)\sin(\nu_{1}\tau)+\cos\alpha^{I}_{L,1}(t)\cos(\nu_{1}\tau)-\sin\alpha^{I}_{L,1}(t)\sin(\nu_{1}\tau)]$
We perform the integration with respect $\tau$ and find:
$H_{eff}\approx H^{real}_{eff}+iH^{Im.}_{eff}$ (24)
where $H^{real}_{eff}$ is the real part of the effective action:
$\displaystyle H^{real}_{eff}=$
$\displaystyle\frac{\hbar\hat{g}^{2}\nu_{0}}{\nu_{0}^{2}+\hat{\Gamma}^{2}}[\cos(2\alpha_{R,2})-\cos(2\alpha_{L,2})-\cos(2\alpha_{L,1})+\cos(2\alpha_{R,1})+2(\sin(\alpha_{R,2}+\alpha_{L,2})-\sin(\alpha_{R,1}+\alpha_{L,1}))]$
$\displaystyle-2\hbar\hat{g}^{2}(\frac{\nu_{0}-\nu_{1}}{(\nu_{0}-\nu_{1})^{2}+\hat{\Gamma}^{2}}+\frac{\nu_{0}+\nu_{1}}{(\nu_{0}+\nu_{1})^{2}+\hat{\Gamma}^{2}})[1+\sin(\alpha_{R,1}-\alpha_{L,1})]$
$\displaystyle-2\hbar\hat{g}^{2}(\frac{\nu_{0}-\nu_{2}}{(\nu_{0}-\nu_{2})^{2}+\hat{\Gamma}^{2}}+\frac{\nu_{0}+\nu_{2}}{(\nu_{0}+\nu_{2})^{2}+\hat{\Gamma}^{2}})[1+\sin(\alpha_{R,2}-\alpha_{L,2})]$
The imaginary part $H^{Im.}_{eff}$ of the action causes the current to vanish.
Finite solutions will be obtained for the ground states $|0>$ which obey
$H^{Im.}_{eff}|0>=0$. Therefore the solutions for finite currents are
equivalent to a constraint condition for the ground state $|0>$.
$\displaystyle
H^{Im.}_{eff}=Z_{2}Z_{1}\hat{g}^{2}[\frac{\hat{\Gamma}}{(\nu_{0}-\frac{\nu_{2}-\nu_{1}}{2})^{2}+\hat{\Gamma}^{2}}+\frac{\hat{\Gamma}}{(\nu_{0}+\frac{\nu_{2}-\nu_{1}}{2})^{2}+\hat{\Gamma}^{2}}]$
$\displaystyle[(\sin(\alpha_{R,2})-\cos(\alpha_{L,2}))(\cos(\alpha_{L,1})+\sin(\alpha_{R,1}))+(\sin(\alpha_{R,2})+\sin(\alpha_{L,2}))$
$\displaystyle(\cos(\alpha_{R,1})+\sin(\alpha_{L,1}))]$
$\displaystyle+Z_{2}Z_{1}\hat{g}^{2}[\frac{\hat{\Gamma}}{(\nu_{0}-\frac{\nu_{2}+\nu_{1}}{2})^{2}+\hat{\Gamma}^{2}}+\frac{\hat{\Gamma}}{(\nu_{0}+\frac{\nu_{2}+\nu_{1}}{2})^{2}+\hat{\Gamma}^{2}}]$
$\displaystyle[(\sin(\alpha_{R,2})-\cos(\alpha_{L,2}))(\cos(\alpha_{L,1})+\sin(\alpha_{R,1}))-(\cos(\alpha_{R,2})+\sin(\alpha_{L,2}))$
$\displaystyle(\cos(\alpha_{R,1})-\sin(\alpha_{L,1}))]$
We find two cases for which a finite solution exist.
The first case corresponds to $\nu_{0}\approx(|\frac{\nu_{2}-\nu_{1}}{2}|)$
with the solution:
$\displaystyle H^{Im.}_{eff}|0>=0$ $\displaystyle\alpha_{R,2}=\alpha_{R,1}+u$
$\displaystyle\alpha_{L,2}=\alpha_{L,1}-u$
The second case corresponds to $\nu_{0}\approx(\frac{\nu_{2}+\nu_{1}}{2})$
with the solution:
$\displaystyle H^{Im.}_{eff}|0>=0$ $\displaystyle\alpha_{R,2}=-\alpha_{R,1}+u$
$\displaystyle\alpha_{L,2}=-\alpha_{L,1}-u$
We introduce the definitions for the zero mode fields :
$\alpha_{1}=\alpha_{2}\equiv\alpha;\hskip
14.45377pt\beta_{2}=\beta_{1}-2u,\beta_{1}\equiv\beta$ (29)
The solutions are independent on the arbitrary field $u$ which plays the role
of a gauge condition and has to be integrated out. We integrated over the
field $u$ we find the effective Hamiltonian for the conditions
$\nu_{0}\approx(\frac{\nu_{2}-\nu_{1}}{2})$ and
$\nu_{0}\approx(\frac{\nu_{2}+\nu_{1}}{2})$. We introduce the magnetic flux
the variables $\bar{\varphi}$ and $\Delta$:
$\bar{\varphi}=\frac{\hat{\varphi}_{1}+\hat{\varphi}_{2}}{2};\Delta=\frac{\hat{\varphi}_{1}-\hat{\varphi}_{2}}{2}$
(30)
We can write both cases in a closed form:
$\displaystyle\frac{H}{\hbar}\approx\frac{\pi
v_{F}}{2l_{ring}}[(-i\frac{d}{d\alpha}+2\hat{\varphi}_{1})^{2}+(-i\frac{d}{d\beta})^{2}]+\frac{\pi
v_{F}}{2l_{ring}}[(-i\frac{d}{d\alpha}+2\hat{\varphi}_{2})^{2}+(-i\frac{d}{d\beta})^{2}]+$
$\displaystyle[\frac{2\hat{g}^{2}\nu_{0}}{\nu_{0}^{2}+\hat{\Gamma}^{2}}\sin(\alpha)\sin(\beta)-2\hat{g}^{2}[\frac{\nu_{0}-(\bar{\varphi}+\Delta)}{(\nu_{0}-(\bar{\varphi}+\Delta)^{2}+\hat{\Gamma}^{2}}+\frac{\nu_{0}+(\bar{\varphi}+\Delta)}{(\nu_{0}+(\bar{\varphi}+\Delta)^{2}+\hat{\Gamma}^{2}}]\sin(\beta)$
$\displaystyle-2\hat{g}^{2}[\frac{\nu_{0}-(\bar{\varphi}+\Delta)}{(\nu_{0}-(\bar{\varphi}+\Delta)^{2}+\hat{\Gamma}^{2}}+\frac{\nu_{0}+(\bar{\varphi}+\Delta)}{(\nu_{0}+(\bar{\varphi}+\Delta)^{2}+\hat{\Gamma}^{2}}+\frac{\nu_{0}-(\bar{\varphi}-\Delta)}{(\nu_{0}-(\bar{\varphi}-\Delta)^{2}+\hat{\Gamma}^{2}}+\frac{\nu_{0}+(\bar{\varphi}-\Delta)}{(\nu_{0}+(\bar{\varphi}-\Delta)^{2}+\hat{\Gamma}^{2}}]]$
$\displaystyle(\delta_{\frac{\hat{\varphi}_{1}-\hat{\varphi}_{2}}{2},\nu_{0}}+\delta_{\frac{\hat{\varphi}_{1}+\hat{\varphi}_{2}}{2},\nu_{0}})$
The first line of equation $31$ represents the the Hamiltonian for the two
metallic rings pierced by the external fluxes expressed in therms of the zero
mode of the metallic rings. The second part of equation $31$ represents the
coupling between the wire and the two rings. We observe that this part is
restricted by the constraint condition
$\frac{\hat{\varphi}_{1}-\hat{\varphi}_{2}}{2}=\nu_{0}$ or
$\frac{\hat{\varphi}_{1}+\hat{\varphi}_{2}}{2}=\nu_{0}$. This constraints
represents the effect of the Majorana fermions on the p-wave wire.
In order two investigate the Hamiltonian in equation $31$ we will use the
algebra of the zero modes david ; davidDirac ; Berkovits :
$\hat{J}=N_{R}-N_{L}\equiv-2i\frac{d}{d\alpha};\hskip
14.45377pt\hat{Q}=N_{R}+N_{L}\equiv-2i\frac{d}{d\beta}$ (32)
with the eigenvalues and commutation rules:
$\displaystyle\hat{J}|J,Q>=J|J,Q>;J=0,\pm 1,\pm 2,..$
$\displaystyle\hat{Q}|J,Q>=Q|J,Q>;Q=0,\pm 1,\pm 2,..$
From the commutation relations $[\alpha,\hat{J}]=2i$, $[\beta,\hat{Q}]=2i$ we
establish the relations :
$\displaystyle e^{i\alpha}|J,Q>=|J+1,Q>;e^{-i\alpha}|J,Q>=|J-1,Q>$
$\displaystyle e^{i\beta}|J,Q>=|J,Q+1>;e^{-i\beta}|J,Q>=|J,Q-1>$
The eigenfunctions are given by:
$\displaystyle<\alpha|J,Q=0>=\frac{1}{\sqrt{4\pi}}e^{i\alpha
J};<\beta|J=0,Q>=\frac{1}{\sqrt{4\pi}}e^{i\alpha J}$
Using the algebra of the zero modes we compute to lowest order (in
perturbation theory) the energy for the ground state of the two rings coupled
to the wire. As a function of the coupling constant
$\lambda\equiv\frac{2\hat{g}^{2}}{\nu_{max}}<1$ and maximum frequency
$\nu_{max}$ which is given by the electronic bandwidth frequency. We find for
the ground state energy $E(\hat{\varphi}_{1},\hat{\varphi}_{2})$:
$\displaystyle E(\hat{\varphi}_{1},\hat{\varphi}_{2})=[\frac{2\hbar\pi
v_{F}}{l_{ring}}((\bar{\varphi}+\Delta)^{2}+(\bar{\varphi}-\Delta)^{2})$
$\displaystyle-\lambda(\frac{\nu_{max}}{\nu_{0}})(\frac{\nu_{0}-(\bar{\varphi}+\Delta)}{(\nu_{0}-(\bar{\varphi}+\Delta)^{2}+\hat{\Gamma}^{2}}+\frac{\nu_{0}+(\bar{\varphi}+\Delta)}{(\nu_{0}+(\bar{\varphi}+\Delta)^{2}+\hat{\Gamma}^{2}}+$
$\displaystyle\frac{\nu_{0}-(\bar{\varphi}-\Delta)}{(\nu_{0}-(\bar{\varphi}-\Delta)^{2}+\hat{\Gamma}^{2}}+\frac{\nu_{0}+(\bar{\varphi}-\Delta)}{(\nu_{0}+(\bar{\varphi}-\Delta)^{2}+\hat{\Gamma}^{2}})](\delta_{\nu_{0},\bar{\varphi}}+\delta_{\nu_{0},\Delta})$
Using eq.36 we compute the currents $I_{i}=\frac{\partial
E(\hat{\varphi}_{1},\hat{\varphi}_{2})}{\partial\hat{\varphi}_{i}}$ for the
two rings $i=1,2$ using the conditions:
$\nu_{0}\approx\frac{\hat{\varphi}_{1}-\hat{\varphi}_{2}}{2}\equiv\Delta$.
The current in ring one $I_{1}$ and ring two $I_{2}$ are represented in terms
of $\bar{\varphi}=\frac{\hat{\varphi}_{1}+\hat{\varphi}_{2}}{2}$,
$\nu_{0}=\Delta$ and the current amplitude $I_{0}=\frac{2\pi v_{F}}{l_{ring}}$
$\displaystyle\frac{I_{1}}{I_{0}}=\bar{\varphi}+\nu_{0}-\lambda[\frac{\bar{\varphi}^{2}-\hat{\Gamma}^{2}}{(\bar{\varphi}^{2}+\hat{\Gamma}^{2})^{2}}+\frac{-(2\nu_{0}+\bar{\varphi})^{2}+\hat{\Gamma}^{2}}{(2\nu_{0}+\bar{\varphi})^{2}+\hat{\Gamma}^{2})^{2}}]$
$\displaystyle\frac{I_{2}}{I_{0}}=\bar{\varphi}-\nu_{0}-\lambda[\frac{\bar{\varphi}^{2}-\hat{\Gamma}^{2}}{(\bar{\varphi}^{2}+\hat{\Gamma}^{2})^{2}}+\frac{-(2\nu_{0}+\bar{\varphi})^{2}+\hat{\Gamma}^{2}}{(2\nu_{0}+\bar{\varphi})^{2}+\hat{\Gamma}^{2})^{2}}]$
$\displaystyle\frac{I_{1}+I_{2}}{2I_{0}}=[\bar{\varphi}-\lambda(\frac{\bar{\varphi}^{2}-\hat{\Gamma}^{2}}{(\bar{\varphi}^{2}+\hat{\Gamma}^{2})^{2}}+\frac{-(2\nu_{0}+\bar{\varphi})^{2}+\hat{\Gamma}^{2}}{(2\nu_{0}+\bar{\varphi})^{2}+\hat{\Gamma}^{2})^{2}})]$
In figure $1$ we have plotted the current $\frac{I_{1}+I_{2}}{2I_{0}}$ as a
function of the flux $\bar{\varphi}$ for the case
$\nu_{0}\approx\frac{\hat{\varphi}_{1}-\hat{\varphi}_{2}}{2}\equiv\Delta=0.01$.
We observe that for $\bar{\varphi}>0.1$ the current is proportional to
$\bar{\varphi}$. From other-hand when the $\bar{\varphi}<0.1$ the current in
each ring is affected by the flux in the other ring. This is seen from the
negative contribution of the current shown in figure $1$.The negative current
contribution might be related to the Andreev reflection which occurs at the
interfaces between the superconductor and the metal.
In figure $2$ we have plotted the current $\frac{I_{1}+I_{2}}{2I_{0}}$ as a
function of the flux $\bar{\varphi}$ for the case $\epsilon=0$ considered in
chapter $III$. Due to the constraint condition
$\nu_{0}=\frac{\hat{\varphi}_{1}-\hat{\varphi}_{2}}{2}$ we have the relation
$\nu_{0}=\frac{\hat{\varphi}_{1}-\hat{\varphi}_{2}}{2}=\Delta=0.$. We find a
stable current in agreement with chapter $III$ where the current scales
linearly with the flux $\hat{\varphi}_{1}=\hat{\varphi}_{2}=\bar{\varphi}$.
V-Conclusion
We have investigate the dependence of the current on the fluxes for the entire
regime of parameters davidimpurity ; berkovits In the limit of large $L$ ,
$\epsilon\rightarrow 0$ the current vanishes in both rings when the two fluxes
are different. We observe that for a finite energy $\epsilon$ and different
fluxes the current dependence is more complex. When the two fluxes are almost
equal the current is a function of the averaged flux. For the case that the
flux difference is comparable to the flux average, the current changes sign.
We can interpret this effect as an Andreev reflection and represents a finger
print of the Majorana fermions.
Figure 1: The average current as a function of $\bar{\varphi}$ for the
condition
$\nu_{0}\approx\frac{\hat{\varphi}_{1}-\hat{\varphi}_{2}}{2}\equiv\Delta=0.01$
Figure 2: The average current as a function of $\bar{\varphi}$ for the case
$\epsilon=0$,
$\nu_{0}=\frac{\hat{\varphi}_{1}-\hat{\varphi}_{2}}{2}\equiv\Delta=0.$
## References
* (1) Alexei Yu Kitaev ”‘Unpaired Majorana Fermions In Quantum Wires”’ cond-mat/ 0010440 and Alexei Yu Kitaev ,Ann.Phys.303,2 (2003).
* (2) D.Schmeltzer ”‘Topological Insulators”’-Transport In Curved Space”’ Advances in Condensed Matter and Material Research Volume 10 pp. 379-402 Editors: Hans Geelvinck and Sjaak Reynst and cond-mat/1012.5876.
* (3) D.Schmeltzer , J. Phys:Condens Matter 20 335205(2008).
* (4) D.Schmeltzer and A.Saxena ,Phys.Rev.B 81 ,195310 (2010).
* (5) S.Tewari,S.Das Sarma and Dung-Hai Lee”’ An index Theorem For The Majorana Zero Modes, In Chiral P-Wave Superconductors”’ cond-ma/0609556
* (6) Y.Oreg ,Gil Refaeli and Felix von Oppen ”‘Helical Liquids and Majorana Bound States In Quantum Wires”’ cond-mat/1003.1145
* (7) L.Jiang, D.Peccker,J.Alice,Gil Refaeli, Y.Oreg and Felix von Oppen cond-mat/1107.4102
* (8) D.Schmeltzer and R.Berkovits Physics Letters A 253 341-344(1999).
* (9) D.Schmeltzer ”‘Dirac’s Method For Constraints Quantum Wires”’ J.Phys:Condens.Matter 23 155601 (2011).
* (10) Gordon W.Semenoff and Pasquale Sodano ”‘ Teleportation By A Majorana Medium”’ cond-mat/0601261.
* (11) Daniel Boyanovsky Phys.Rev.B.39, 6744(1989).
* (12) D.Schmeltzer et al.,Phys.Rev.Lett90,116802(2003)
* (13) D.Schmeltzer et. al J.Phys.Condens. Matter 22, 095301 (2010)
* (14) D.Schmeltzer J.Phys.Condens. Matter 23,155601 (2011)
* (15) D.Schmeltzer and R.Berkovits Phys.Lett.A243341,(1999)
|
arxiv-papers
| 2011-12-22T21:23:43 |
2024-09-04T02:49:25.638988
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. Schmeltzer",
"submitter": "David Schmeltzer",
"url": "https://arxiv.org/abs/1112.5465"
}
|
1112.5472
|
[by] Gerth Stølting Brodal and Casper Kejlberg-Rasmussen STACS 2012
# Cache-Oblivious Implicit Predecessor Dictionaries with the Working-Set
Property
Gerth Stølting Brodal MADALGO111Center for Massive Data Algorithmics, a
Center of the Danish National Research Foundation, Department of Computer
Science, Aarhus University, Denmark Casper Kejlberg-Rasmussen
MADALGO111Center for Massive Data Algorithmics, a Center of the Danish
National Research Foundation, Department of Computer Science, Aarhus
University, Denmark
###### Abstract.
In this paper we present an implicit dynamic dictionary with the working-set
property, supporting insert($e$) and delete($e$) in $\mathcal{O}(\log n)$
time, predecessor($e$) in $\mathcal{O}(\log\ell_{\textsf{p}(e)})$ time,
successor($e$) in $\mathcal{O}(\log\ell_{\textsf{s}(e)})$ time and search($e$)
in $\mathcal{O}(\log\min(\ell_{\textsf{p}(e)},\ell_{e},\ell_{\textsf{s}(e)}))$
time, where $n$ is the number of elements stored in the dictionary, $\ell_{e}$
is the number of distinct elements searched for since element $e$ was last
searched for and $\textsf{p}(e)$ and $\textsf{s}(e)$ are the predecessor and
successor of $e$, respectively. The time-bounds are all worst-case. The
dictionary stores the elements in an array of size $n$ using _no_ additional
space. In the cache-oblivious model the $\log$ is base $B$ and the cache-
obliviousness is due to our black box use of an existing cache-oblivious
implicit dictionary. This is the first implicit dictionary supporting
predecessor and successor searches in the working-set bound. Previous implicit
structures required $\mathcal{O}(\log n)$ time.
###### Key words and phrases:
working-set property, dictionary, implicit, cache-oblivious, worst-case,
external memory, I/O efficient
###### 1991 Mathematics Subject Classification:
Algorithms and data structures, E.1 Data Structures
## 1\. Introduction
In this paper we consider the problem of maintaining a cache-oblivious
implicit dictionary [13] with the working-set property over a dynamically
changing set $P$ of $|P|=n$ distinct and totally ordered elements. We define
the _working-set number_ of an element $e\in P$ to be
$\ell_{e}=|\\{e^{\prime}\in P\mid$ we have searched for $e^{\prime}$ after we
last searched for $e\\}|$. An implicit dictionary maintains $n$ distinct keys
without using any other space than that of the $n$ keys, i.e. the data
structure is encoded by permuting the $n$ elements. The fundamental trick in
the implicit model, [12], is to encode a bit using two distinct elements $x$
and $y$: if $\min(x,y)$ is before $\max(x,y)$ then $x$ and $y$ encode a 0 bit,
else they encode a 1 bit. This can then be used to encode $l$ bits using $2l$
elements. The implicit model is a restricted version of the unit cost RAM
model with a word size of $\mathcal{O}(\log n)$. The restrictions are that
between operations we are only allowed to use an array of the $n$ input
elements to store our data structures by permuting the input elements, i.e.,
there can be used _no_ additional space between operations. In operations we
are allowed to use $\mathcal{O}(1)$ extra words. Furthermore we assume that
the number of elements $n$ in the dictionary is externally maintained. Our
structure will support the following operations:
* •
Search$(e)$ determines if $e$ is in the dictionary, if so its working-set
number is set to $0$.
* •
Predecessor$(e)$ will find $\max\\{e^{\prime}\in P\cup\\{-\infty\\}\mid
e^{\prime}<e\\}$, without changing any working-set numbers.
* •
Successor$(e)$ will find $\min\\{e^{\prime}\in P\cup\\{\infty\\}\mid
e<e^{\prime}\\}$, without changing any working-set numbers.
* •
Insert$(e)$ inserts $e$ into the dictionary with at working-set number of $0$,
all other working-set numbers are increased by one.
* •
Delete$(e)$ deletes $e$ from the dictionary, and does not change the working-
set number of any element.
There are numerous data structures and algorithms in the implicit model which
range from binary heaps [16] to in-place 3-D convex hull algorithms [6]. There
has been a continuous development of implicit dictionaries, the first
milestone was the implicit AVL-tree [12] having bounds of
$\mathcal{O}(\log^{2}n)$. The second milestone was the implicit B-tree [7]
having bounds of $\mathcal{O}(\log^{2}n/\log\log n)$ the third was the flat
implicit tree [9] obtaining $\mathcal{O}(\log n)$ worst-case time for
searching and amortized bounds for updates. The fourth milestone is the
optimal implicit dictionary [8] obtaining worst-case $\mathcal{O}(\log n)$ for
search, update, predecessor and successor.
Numerous non-implicit dictionaries attain the working-set property; splay
trees [15], skip list variants [2], the working-set structure in [11], and two
structures presented in [3]. All achieve the property in the amortized,
expected or worst-case sense. The unified access bound, which is achieved in
[1], even combines the working-set property with finger search. In finger
search we have a finger located on an element $f$ and the search cost of
finding say element $e$ is a function of $d(f,e)$ which is the rank distance
between elements $f$ and $e$. The unified bound combines these two to obtain a
bound of $\mathcal{O}(\min_{e\in P}\\{\log(\ell_{e}+d(e,f)+2)\\})$. Table 1
gives an overview of previous results, and our contribution.
Ref. | WS prop. | Insert/ Delete$(e)$ | Search$(e)$ | Pred$(e)$/ Succ$(e)$ | Additional words
---|---|---|---|---|---
[12] | – | $\mathcal{O}(\log^{2}n)$ | $\mathcal{O}(\log^{2}n)$ | – | None
[7] | – | $\mathcal{O}\left(\frac{\log^{2}n}{\log\log n}\right)$ | $\mathcal{O}\left(\frac{\log^{2}n}{\log\log n}\right)$ | – | None
[9] | – | $\mathcal{O}(\log n)$ amor. | $\mathcal{O}(\log n)$ | $\mathcal{O}(\log n)$ | None
[8] | – | $\mathcal{O}(\log n)$ | $\mathcal{O}(\log n)$ | $\mathcal{O}(\log n)$ | None
[11] | + | $\mathcal{O}(\log n)$ | $\mathcal{O}(\log\ell_{e})$ | $\mathcal{O}(\log\ell_{e^{*}})$ | $\mathcal{O}(n)$
[3, Sec. 2] | + | $\mathcal{O}(\log n)$ | $\mathcal{O}(\log\ell_{e})$ exp. | $\mathcal{O}(\log n)$ | $\mathcal{O}(\log\log n)$
[3, Sec. 3] | + | $\mathcal{O}(\log n)$ | $\mathcal{O}(\log\ell_{e})$ exp. | $\mathcal{O}(\log\ell_{e^{*}})$ exp. | $\mathcal{O}(\sqrt{n})$
[4] | + | $\mathcal{O}(\log n)$ | $\mathcal{O}(\log\ell_{e})$ | $\mathcal{O}(\log n)$ | None
This paper | + | $\mathcal{O}(\log n)$ | $\mathcal{O}(\log\min(\ell_{\textsf{p}(e)},\ell_{\textsf{s}(e)},\ell_{e}))$ | $\mathcal{O}(\log\ell_{e^{*}})$ | None
Table 1. The operation time and space overhead of important structures for the
dictionary problem. Here $e^{*}$ is the predecessor or successor in the given
context. In a search for an element $e$ that is not present in the dictionary
$\ell_{e}$ is $n$.
The dictionary in [8] is, in addition to being implicit, also designed for the
cache-oblivious model [10], where all the operations imply
$\mathcal{O}(\log_{B}n)$ cache-misses. Here $B$ is the cache-line length which
is unknown to the algorithm. The cache-oblivious property also carries over
into our dictionary. Our structure combines the two worlds of implicit
dictionaries and dictionaries with the working-set property to obtain the
first implicit dictionary with the working-set property supporting search,
predecessor and successor queries in the working-set bound. The result of this
paper is summarized in Theorem 1.
###### Theorem 1.
There exists a cache-oblivious implicit dynamic dictionary with the working-
set property that supports the operations insert and delete in time
$\mathcal{O}(\log n)$ and $\mathcal{O}(\log_{B}n)$ cache-misses, search,
predecessor and successor in time
$\mathcal{O}(\log\min(\ell_{\textsf{p}(e)},\ell_{e},\ell_{\textsf{s}(e)}))$,
$\mathcal{O}(\log\ell_{\textsf{p}(e)})$ and
$\mathcal{O}(\log\ell_{\textsf{s}(e)})$, and cache-misses
$\mathcal{O}(\log_{B}\min(\ell_{\textsf{p}(e)},\ell_{e},\ell_{\textsf{s}(e)}))$,
$\mathcal{O}(\log_{B}\ell_{\textsf{p}(e)})$ and
$\mathcal{O}(\log_{B}\ell_{\textsf{s}(e)})$, respectively, where
$\textsf{p}(e)$ and $\textsf{s}(e)$ are the predecessor and successor of $e$,
respectively.
Similarly to previous work [1, 4] we partition the dictionary elements into
$\mathcal{O}(\log\log n)$ blocks $B_{0},\ldots,B_{m}$, of double exponential
increasing sizes, where $B_{0}$ stores the most recently accessed elements.
The structure in [4] supports predecessors and successors queries, but there
is no way of knowing if an element is actually the predecessor or successor,
without querying all blocks, which results in $\mathcal{O}(\log n)$ time
bounds. We solve this problem by introducing the notion of _intervals_ and
particularly a dynamic implicit representation of these. We represent the
whole interval $[\min(P);\max(P)]$ by a set of disjoint intervals spread
across the different blocks. Any point that intersects an interval in block
$B_{i}$ will lie in block $B_{i}$ and have a working-set number of at least
$2^{2^{i}}$. This way when we search for the predecessor or successor of an
element and hit an interval, then no more points can be contained in the
interval in higher blocks, and we can avoid looking at these, which give
working-set bounds for the search, predecessor and successor queries.
## 2\. Data structure
We now describe our data structure and its invariants. We will use the
moveable dictionary from [4] as a black box. The dictionary over a point set
$S$ is laid out in the memory addresses $[i;j]$. It supports the following
operations in $\mathcal{O}(\log n^{\prime})$ time and
$\mathcal{O}(\log_{B}n^{\prime})$ cache-misses, where $n^{\prime}=j-i+1$:
* •
Insert-left$(e)$ inserts $e$ into $S$ which is now laid out in the addresses
$[i-1;j]$.
* •
Insert-right$(e)$ inserts $e$ into $S$ which is now laid out in the addresses
$[i;j+1]$.
* •
Delete-left$(e)$ deletes $e$ from $S$ which is now laid out in the addresses
$[i+1;j]$.
* •
Delete-right$(e)$ deletes $e$ from $S$ which is now laid out in the addresses
$[i;j-1]$.
* •
Search$(e)$ determines if $e\in S$, if so the address of element $e$ is
returned.
* •
Predecessor$(e)$ returns the address of the element $\max\\{e^{\prime}\in
S\mid e^{\prime}<e\\}$ or that no such element exists.
* •
Successor$(e)$ returns the address of the element $\min\\{e^{\prime}\in S\mid
e<e^{\prime}\\}$ or that no such element exists.
From these operations we notice that we can move the moveable dictionary, say
left, by performing a delete-right operation for an arbitrary element and re-
inserting the element again by an insert-left operation. Similarly we can also
move the dictionary one position to the right.
Our structure consists of $m=\Theta(\log\log n)$ blocks $B_{0},\ldots,B_{m}$,
each block $B_{i}$ is of size $\mathcal{O}(2^{2^{i+k}})$, where $k$ is a
constant. Elements in $B_{i}$ have a working-set number of at least
$2^{2^{i+k-1}}$. The block $B_{i}$ consists of an array $D_{i}$ of
$w_{i}=d\cdot 2^{i+k}$ elements, where $d$ is a constant, and moveable
dictionaries $A_{i},R_{i},W_{i},H_{i},C_{i}$ and $G_{i}$, for
$i=0,\ldots,m-1$, see Figure 1. For block $B_{m}$ we only have $D_{m}$ if
$|B_{m}\backslash\\{\min(P),\max(P)\\}|\leq w_{m}$, otherwise we have the same
structures as for the other blocks. We use the block $D_{i}$ to encode the
sizes of the movable dictionaries $A_{i},R_{i},W_{i},H_{i},C_{i}$ and $G_{i}$
so that we can locate them. Discussion of further details of the memory layout
is postponed to Section 3.
Figure 1. Overview of how the working set dictionary is laid out in memory.
The dictionary grows and shrinks to the right when elements are inserted and
deleted.
We call elements in the structures $D_{i}$ and $A_{i}$ for _arriving_ points,
and when making a non-arriving point arriving, we will put it into $A_{i}$
unless specified otherwise. We call elements in $R_{i}$ for _resting_ points,
elements in $W_{i}$ for _waiting_ points, elements in $H_{i}$ for _helping_
points, elements in $C_{i}$ for _climbing_ points and elements in $G_{i}$ for
_guarding_ points.
Crucial to our data structure is the partitioning of $[\min(P);\max(P)]$ into
_intervals_. Each interval is assigned to a _level_ and level $i$ corresponds
to block $B_{i}$. Consider an interval lying at level $i$. The endpoints
$e_{1}$ and $e_{2}$ will be guarding points stored at level $0,\ldots,i$. All
points inside of this interval will lie in level $i$ and cannot be guarding
points, i.e. $]e_{1};e_{2}[\cap(\bigcup_{j\neq i}{B_{j}\cup
G_{i}})=\emptyset$. We do not allow intervals defined by two consecutive
guarding points to be empty, they must contain at least one non-guarding
point. We also require $\min(P)$ and $\max(P)$ to be guarding points in
$G_{0}$ at level $0$, but they are special as they do not define intervals to
their left and right, respectively. A query considers $B_{0},B_{1},\ldots$
until $B_{i}$ where the query is found to be in a level $i$ interval where the
answer is guaranteed to have been found in blocks $B_{0},\ldots,B_{i}$.
The basic idea of our construction is the following. When searching for an
element it is moved to level $0$. This can cause block overflows (see
invariants I.5–I.9 in Section 2.2), which are handled as follows. The arriving
points in level $i$ have just entered from level $i-1$, and when there are
$2^{2^{i+k}}$ of them in $A_{i}$ they become resting. The resting points need
to charge up their working-set number before they can begin their journey to
level $i+1$. They are charged up when there have come $2^{2^{i+k}}$ further
arriving points to level $i$, then the resting points become waiting points.
Waiting points have a high enough working-set number to begin the journey to
level $i+1$, but they need to wait for enough points to group up so that they
can start the journey. When a waiting point is picked to start its journey to
level $i+1$ it becomes a helping or climbing point, and every time enough
helping points have grouped up, i.e. there is at least $c=5$ consecutive of
them, then they become climbing points and are ready to go to level $i+1$. The
climbing points will then incrementally be going to level $i+1$. See Figure 2
for an example of the structure of the intervals.
Figure 2. The structure of the levels for a dictionary. The levels are
indicated to the left.
### 2.1. Notation
Before we introduce the invariants we need to define some notation. For a
subset $S\subseteq P$, we define $\textsf{p}_{S}(e)=\max\\{s\in
S\cup\\{-\infty\\}\mid s<e\\}$ and $\textsf{s}_{S}(e)=\min\\{s\in
S\cup\\{\infty\\}\mid e<s\\}$. When we write $S_{\leq i}$ we mean
$\bigcup_{j=0}^{i}{S_{j}}$ where $S_{j}\subseteq P$ for $j=0,\ldots,i$.
For $S\subseteq P$, we define
$\textsf{GIL}_{S}(e)=S\cap]\textsf{p}_{P\backslash S}(e);e[$ to be the Group
of Immediate Left points of $e$ in $S$ which does not have any other point of
$P\backslash S$ in between them, see Figure 3. Similarly we define
$\textsf{GIR}_{S}(e)=S\cap]e;\textsf{s}_{P\backslash S}(e)[$ to the right of
$e$. We will notice that we will never find all points of
$\textsf{GIL}_{S}(e)$ unless $|\textsf{GIL}_{S}(e)|<c$, the same applies for
$\textsf{GIR}_{S}(e)$. For $S\subseteq P$, we define
$\textsf{FGL}_{S}(e)=S\cap]\textsf{p}_{P\backslash
S}(\textsf{p}_{S}(e));\textsf{p}_{S}(e)]$ to be the First Group of points from
$S$ Left of $e$, i.e. the group does not have any points of $P\backslash S$ in
between its points, see Figure 3. Similarly we define
$\textsf{FGR}_{S}(e)=S\cap[\textsf{s}_{S}(e);\textsf{s}_{P\backslash
S}(\textsf{s}_{S}(e))[$. We will notice that we will never find all points of
$\textsf{FGL}_{S}(e)$ unless $|\textsf{FGL}_{S}(e)|<c$, the same applies for
$\textsf{FGR}_{S}(e)$.
Figure 3. Here is a illustration of FGL and GIL. Notice that
$\textsf{GIL}_{S}(e_{1})=\emptyset$ whereas
$\textsf{FGL}_{S}(e_{1})\neq\emptyset$.
We will sometimes use the phrasings _a group of points_ or _$e$ ’s group of
points_. This refers to a group of points of the same type, i.e. arriving,
resting, etc., and with no other types of points in between them. Later we
will need to move elements around between the structures $D_{i}$, $A_{i}$,
$R_{i}$, $W_{i}$, $H_{i}$, $C_{i}$ and $G_{i}$. For this we have the notation
$X\stackrel{{\scriptstyle h}}{{\rightarrow}}Y$, meaning that we move $h$
arbitrary points from $X$ into $Y$, where $X$ and $Y$ can be one of $D_{i}$,
$A_{i}$, $R_{i}$, $W_{i}$, $H_{i}$, $C_{i}$ and $G_{i}$ for any $i$.
When we describe the intervals we let $]a;b]$ be an interval from $a$ to $b$
that is open at $a$ and closed at $b$. We let $(a;b)$ be an interval from $a$
to $b$ that can be open or closed at $a$ and $b$. We use this notation when we
do not care if $a$ and $b$ are open or closed. In the methods updating the
intervals we will sometimes branch depending on which type an interval is. For
clarity we will explain how to determine this given the level $i$ of the
interval and its two endpoints $e_{1}$ and $e_{2}$. The interval
$(e_{1};e_{2})$ is of type $[e_{1};e_{2})$ if $e_{1}\in G_{i}$, else $e_{1}\in
G_{\leq i-1}$ and the interval is of type $]e_{1};e_{2})$. This is symmetric
for the other endpoint $e_{2}$.
### 2.2. Invariants
We will now define the invariants which will help us define and prove
correctness of our interface operations: insert$(e)$, delete$(e)$,
search$(e)$, predecessor$(e)$ and successor$(e)$. We maintain the following
invariants which uniquely determine the intervals222We assume that $|P|=n\geq
2$ at all times if this is not the case we only store $G_{0}$ which contains a
single element and we ignore all invariants.:
1. I.1
A guarding point is part of the definition of at most two intervals333Only the
smallest and largest guarding points will not participate in the definition of
two intervals, all other guarding points will., one to the left at level $i$
and/or one to the right at level $j$, where $i\neq j$. The guarding point $e$
lies at level $\min(i,j)$. The interval at level $\min(i,j)$ is closed at $e$,
and the interval at level $\max(i,j)$ is open at $e$. We also require that
$\min(P)$ and $\max(P)$ are guarding points stored in $G_{0}$, but they do not
define an interval to their left and right, respectively, and the intervals
they help define are open in the end they define. A non-guarding point
intersecting an interval at level $i$, lies in level $i$. Each interval
contains at least one non-guarding point. The union of all intervals give
$]\min(P);\max(P)[$.
2. I.2
Any climbing point, which lies in an interval with other non-climbing points,
is part of a group of at least $c$ points. In intervals of type
$[e_{1};e_{2}]$ which only contain climbing points, we allow there to be less
than $c$ of them.
3. I.3
Any helping point is part of a group of size at most $c-1$. A helping point
cannot have a climbing point as a predecessor or successor. An interval of
type $[e_{1};e_{2}]$ cannot contain only helping points.
We maintain the following invariants for the working-set numbers:
1. I.4
Each arriving point in $D_{i}$ and $A_{i}$ has a working set value of at least
$2^{2^{i-1+k}}$, arriving points in $D_{0}$ and $A_{0}$ have a working-set
value of at least $0$. Each resting point in $R_{i}$ will have a working-set
value of at least $2^{2^{i-1+k}}+|A_{i}|$, resting points in $R_{0}$ have a
working-set value of at least $|A_{0}|$. Each waiting, helping or climbing
point in $W_{i},H_{i}$ and $C_{i}$, respectively, will have a working-set
value of at least $2^{2^{i+k}}$. Each guarding point in $G_{i}$, who’s left
interval lies at level $i$ and right interval lies at level $j$, has a working
set value of at least $2^{2^{\max(i,j)-1+k}}$.
We maintain the following invariants for the size of each block and their
components:
1. I.5
$|D_{0}|=\min(|B_{0}|-2,w_{0})$ and $|D_{i}|=\min(|B_{i}|,w_{i})$ for
$i=1,\ldots,m$.
2. I.6
$|R_{i}|\leq 2^{2^{i+k}}$ and $|W_{i}|+|H_{i}|+|C_{i}|\neq
0\Rightarrow|R_{i}|=2^{2^{i+k}}$ for $i=0,\ldots,m$.
3. I.7
$|A_{i}|+|W_{i}|=2^{2^{i+k}}$ for $i=0,\ldots,m-1$, and $|A_{m}|+|W_{m}|\leq
2^{2^{m+k}}$.
4. I.8
$|A_{i}|<2^{2^{i+k}}$ for $i=0,\ldots,m$.
5. I.9
$|H_{i}|+|C_{i}|=4c2^{2^{i+k}}+c_{i}$, where $c_{i}\in[-c;c]$, for
$i=0,\ldots,m-1$.
From the above invariants we have the following observation:
1. O.1
From I.1 all points in $G_{i}$ are endpoints of intervals in level $i$, and
each interval has at most two endpoints. Hence for $i=0,\ldots,m$ we have that
$|G_{i}|\leq
2(|D_{i}|+|A_{i}|+|R_{i}|+|W_{i}|+|H_{i}|+|C_{i}|)\stackrel{{\scriptstyle(*)}}{{\leq}}(4+2d+8c)\cdot
2^{2^{i+k}}+2c\;,$
where we in $(*)$ we have used I.5, I.6, I.7 and I.9.
From I.1 we have the following lemma.
###### Lemma 1.
Let $e$ be an element, $e_{1}=\textsf{p}_{G_{\leq i}}(e)$,
$e_{2}=\textsf{s}_{G_{\leq i}}(e)$ and $i$ be the smallest integer for which
$I(e_{1},e_{2},i)=]e_{1};e_{2}[\cap\bigcup_{j=0}^{i}{B_{j}}\neq\emptyset$.
Then 1) $(e_{1};e_{2})$ is an interval at level $i$ if $e$ is non-guarding and
2) $(e_{1};e)$ or $(e;e_{2})$ is an interval at level $i$ if $e$ is guarding.
###### Proof.
Assume that $i$ is the minimal $i$ that fulfills
$I(e_{1},e_{2},i)\neq\emptyset$, where $e_{1}=\textsf{p}_{G_{\leq i}}(e)$ and
$e_{2}=\textsf{s}_{G_{\leq i}}(e)$. We will have two cases depending on if $e$
is guarding or not.
Lets first handle case 2) where $e$ is guarding and hence in the dictionary:
Since $e$ is in the dictionary and $e_{1}<e<e_{2}$ we have from the minimality
of $i$ that $e$ lies in level $i$, and from I.1 $e$ is then part of an
interval lying in level $i$ either to the left or to the right. Say $e$ is
part of an interval to the left i.e. the interval $(e^{\prime}_{1};e)$. If
$e_{1}<e^{\prime}_{1}$ then this would contradict that
$e_{1}=\textsf{p}_{G_{\leq i}}(e)$ hence $e^{\prime}_{1}\leq e_{1}$, but since
$e^{\prime}_{1}$ is the predecessor of $e$ we have that
$e^{\prime}_{1}=e_{1}$. So we know that $(e_{1};e)$ defines an interval at
level $i$. The argument for $(e;e_{2})$ is symmetric.
In the case 1) $e$ is non-guarding and $e$ may lie in the dictionary or not:
Since $e_{1}<e<e_{2}$ we have from the minimality of $i$ that $e$ lies in
level $i$, hence from I.1 we have that the interval $(e_{1};e_{2})$ lies at
level $i$. ∎
### 2.3. Operations
We will briefly give an overview of the helper operations and state their
requirements (R) and guarantees (G), then we will describe the helper and
interface operations in details. Search$(e)$ uses the helper operations as
follows: when a search for element $e$ is performed then the level $i$ where
$e$ lies is found using find, then $e$ and $\mathcal{O}(1)$ of its surrounding
elements are moved into level $0$ by use of move-down while maintaining
I.1–I.4. Calls to fix for the levels we have altered will ensure that I.5–I.8
will be maintained, finally a call to rebalance-below$(i-1)$ will ensure that
I.9 is maintained by use of shift-up$(j)$ which will take climbing points from
level $j$ and make them arriving in level $j+1$ for $j=0,\ldots,i-1$.
Insert$(e)$ uses find to find the level where $e$ intersects, then it uses fix
to ensure the size constraints and finally $e$ is moved to level $0$ by use of
search.
* •
Find$(e)$ \- returns the level $i$ of the interval that $e$ intersects along
with $e$’s type and whatever $e$ is in the dictionary or not. [R&G: I.1–I.9]
* •
Fix$(i)$ \- moves points around inside of $B_{i}$ to ensure the size
invariants for each type of point. Fix$(i)$ might violate I.9 for level $i$.
[R: I.1–I.4 and that there exist $\tilde{c}_{1},\ldots,\tilde{c}_{6}$ such
that
$|D_{i}|+\tilde{c}_{1},|A_{i}|+\tilde{c}_{2},|R_{i}|+\tilde{c}_{3},|W_{i}|+\tilde{c}_{4},|H_{i}|+\tilde{c}_{5},|C_{i}|+\tilde{c}_{6}$
fulfill I.5–I.8, where $|\tilde{c}_{i}|=\mathcal{O}(1)$ for $i=1,\ldots,6$. G:
I.1–I.8].
* •
Shift-down$(i)$ \- will move at least $1$ and at most $c$ points from level
$i$ into level $i-1$. [R: I.1–I.8 and
$|H_{i}|+|C_{i}|=4c2^{2^{i+k}}+c^{\prime}_{i}$, where $0\leq
c^{\prime}_{i}=\mathcal{O}(1)$. G: I.1–I.8].
* •
Shift-up$(i)$ \- will move at least $1$ and at most $c$ points from level $i$
into level $i+1$. [R: I.1–I.8 and
$|H_{i}|+|C_{i}|=4c2^{2^{i+k}}+c^{\prime}_{i}$, where $c\leq
c^{\prime}_{i}=\mathcal{O}(1)$. G: I.1–I.8].
* •
Move-down$(e,i,j,t_{\text{before}},t_{\text{after}})$ \- If $e$ is in the
dictionary at level $i$ it is moved from level $i$ to level $j$, where $i\geq
j$. The type $t_{\text{before}}$ is the type of $e$ before the move and
$t_{\text{after}}$ is the type that $e$ should have after the move, unless
$i=j$ in which case $e$ will be made arriving in level $j$. [R&G: I.1–I.8].
* •
Rebalance-below$(i)$ \- If any $c<c_{l}$ for $l=0,\ldots,i$ rebalance-
below$(i)$ will correct it so I.9 will be fulfilled again for $l=0,\ldots,i$.
[R: I.1–I.8 and $\sum_{l=0}^{i}{\textsf{slack}(c_{l})}=\mathcal{O}(1)$, where
$\textsf{slack}(c_{l})=\left\\{\begin{array}[]{cl}0&\textit{if}\quad
c_{l}\in[-c;c]\;,\\\ |c_{l}|-c&\textit{otherwise}\;.\end{array}\right.$
G: I.1–I.9].
* •
Rebalance-above$(i)$ \- If any $c_{l}<-c$ for $l=i,\ldots,m-1$ rebalance-
above$(i)$ will correct it so I.9 will be fulfilled again for
$l=i,\ldots,m-1$. [R: I.1–I.8 and
$\sum_{l=i}^{m-1}{\textsf{slack}(c_{l})}=\mathcal{O}(1)$. G: I.1–I.9].
#### Find$(e)$
We start at level $i=0$. If $e<\min(P)$ or $\max(P)<e$ we return false and
$0$. For each level we let $e_{1}=\textsf{p}_{G_{\leq i}}(e)$,
$e_{2}=\textsf{s}_{G_{\leq i}}(e)$, $p=\textsf{p}_{B_{i}\backslash G_{i}}(e)$
and $s=\textsf{s}_{B_{i}\backslash G_{i}}(e)$. We find $p$ and $s$ by querying
each of the structures $D_{i},A_{i},R_{i},W_{i},H_{i}$ and $C_{i}$, we find
$e_{1}$ and $e_{2}$ by querying $G_{i}$ and comparing with the values of
$e_{1}$ and $e_{2}$ from level $i-1$. While $p<e_{1}$ and $e_{2}<s$ we
continue to the next level, that is we increment $i$. Now outside the loop, if
$e\in B_{i}$ we return $i$, the type of $e$ and the boolean true as we found
$e$, else we return $i$ and false as we did not find $e$. See Figure 4 for an
example of the execution.
#### Predecessor$(e)$ (successor$(e)$)
We start at level $i=0$. If $e<\min(P)$ then return $-\infty$ ($\min(P)$). If
$\max(P)<e$ then return $\max(P)$ ($\infty$). For each level we let
$e_{1}=\textsf{p}_{G_{\leq i}}(e)$, $p=\textsf{p}_{B_{i}}(e)$,
$e_{2}=\textsf{s}_{G_{\leq i}}(e)$ and $s=\textsf{s}_{B_{i}}(e)$. While
$p<e_{1}$ and $e_{2}<s$ we continue to the next level, that is we increment
$i$. When the loop breaks we return $\max(e_{1},p)$ ($\min(s,e_{2})$). See
Figure 4 for an example of the execution.
Figure 4. The last three iterations of the while-loop of find$(e)$,
predecessor$(e)$ and successor$(e)$.
#### Insert$(e)$
If $e<\min(P)$ we swap $e$ and $\min(P)$, call fix$(0)$, rebalance-below$(m)$
and return. If $\max(P)<e$ we swap $e$ and $\max(P)$, call fix$(0)$,
rebalance-below$(m)$ and return.
Let $c_{l}=\textsf{GIL}_{C_{i}}(e)$, $c_{r}=\textsf{GIR}_{C_{i}}(e)$,
$h_{l}=\textsf{GIL}_{H_{i}}(e)$ and $h_{r}=\textsf{GIR}_{H_{i}}(e)$. We find
the level $i$ of the interval $(e_{1};e_{2})$ which $e$ intersects using
find$(e)$.
If $e$ is already in the dictionary we give an error. If $|c_{l}|>0$ or
$|c_{r}|>0$ or $(e_{1};e_{2})$ is of type $[e_{1};e_{2}]$ and does not contain
non-climbing points then insert $e$ as climbing at level $i$. Else if
$|h_{l}|+1+|h_{r}|\geq c$ then insert $e$ as climbing at level $i$ and make
the points in $h_{l}$ and $h_{r}$ climbing at level $i$. Else insert $e$ as
helping at level $i$. Finally we call rebalance-below$(m)$ and then
search$(e)$ to move $e$ from the current level $i$ down to level $0$.
#### Search$(e)$
We first find $e$’s current level $i$ and its type $t$, by a call to
find$(e)$. If $e$ is in the dictionary then we call move-
down$(e,i,0,t,\text{arriving})$ which will move $e$ from level $i$ down to
level $0$ and make it arriving, while maintaining I.1–I.8, but I.9 might be
broken so we finally call rebalance-below$(i-1)$ to fix this.
#### Fix$(i)$
In the following we will be moving elements around between $D_{i}$, $A_{i}$,
$R_{i}$, $W_{i}$, $H_{i}$ and $C_{i}$. The moves $A_{i}\rightarrow R_{i}$ and
$R_{i}\rightarrow W_{i}$, i.e. between structures which are next to each other
in the memory layout, are simply performed by deleting an element from the
left structure and inserting it into the right structure. The moves
$W_{i}\rightarrow H_{i}\cup C_{i}$ and the other way around $H_{i}\cup
C_{i}\rightarrow W_{i}$ will be explained below.
If $|D_{i}|>w_{i}$ then perform $D_{i}\stackrel{{\scriptstyle
h}}{{\rightarrow}}A_{i}$ where $h=|D_{i}|-w_{i}$. If $|D_{i}|<w_{i}$ and
$|B_{i}\backslash\\{\min(P),$ $\max(P)\\}|>|D_{i}|$ then perform $H_{i}\cup
C_{i}\stackrel{{\scriptstyle h_{1}}}{{\rightarrow}}W_{i}$,
$W_{i}\stackrel{{\scriptstyle h_{2}}}{{\rightarrow}}R_{i}$,
$R_{i}\stackrel{{\scriptstyle h_{3}}}{{\rightarrow}}A_{i}$ and
$A_{i}\stackrel{{\scriptstyle h_{4}}}{{\rightarrow}}D_{i}$ where
$h_{1}=\min(w_{i}-|D_{i}|,|H_{i}|+|C_{i}|)$,
$h_{2}=\min(w_{i}-|D_{i}|,|W_{i}|+h_{1})$,
$h_{3}=\min(w_{i}-|D_{i}|,|R_{i}|+h_{2})$ and
$h_{4}=\min(w_{i}-|D_{i}|,|A_{i}|+h_{3})$.
If $|W_{i}|+|H_{i}|+|C_{i}|\neq 0$ and $|R_{i}|<2^{2^{i+k}}$ then perform
$H_{i}\cup C_{i}\stackrel{{\scriptstyle h_{1}}}{{\rightarrow}}W_{i}$ and
$W_{i}\stackrel{{\scriptstyle h_{2}}}{{\rightarrow}}R_{i}$ where
$h_{1}=\min(2^{2^{i+k}}-|R_{i}|,|H_{i}|+|C_{i}|)$ and
$h_{2}=\min(2^{2^{i+k}}-|R_{i}|,|W_{i}|+h_{1})$. If $|R_{i}|>2^{2^{i+k}}$ then
perform $R_{i}\stackrel{{\scriptstyle h_{1}}}{{\rightarrow}}A_{i}$ where
$h_{1}=|R_{i}|-2^{2^{i+k}}$.
If $i<m$ and $|A_{i}|+|W_{i}|<2^{2^{i+k}}$ then perform $H_{i}\cup
C_{i}\stackrel{{\scriptstyle h_{1}}}{{\rightarrow}}W_{i}$, where
$h_{1}=\min(2^{2^{i+k}}-(|A_{i}|+|W_{i}|),|H_{i}|+|C_{i}|)$. If
$|A_{i}|+|W_{i}|>2^{2^{i+k}}$ then perform $W_{i}\stackrel{{\scriptstyle
h_{1}}}{{\rightarrow}}H_{i}\cup C_{i}$ where
$h_{1}=\min(|A_{i}|+|W_{i}|-2^{2^{i+k}},|W_{i}|)$.
If $|A_{i}|\geq 2^{2^{i+k}}$ then let $h_{1}=|A_{i}|-2^{2^{i+k}}$, delete
$W_{i}$ as it is empty and rename $R_{i}$ to $W_{i}$. Now move $h_{1}$
elements from $A_{i}$ into a new moveable dictionary $X$, rename $A_{i}$ to
$R_{i}$, rename $X$ to $A_{i}$ and perform $W_{i}\stackrel{{\scriptstyle
h_{1}}}{{\rightarrow}}H_{i}\cup C_{i}$.
Performing $W_{i}\rightarrow H_{i}\cup C_{i}$: Let
$w=\textsf{s}_{W_{i}}(-\infty)$, $c_{l}=\textsf{GIL}_{C_{i}}(w)$,
$c_{r}=\textsf{GIR}_{C_{i}}(w)$, $h_{l}=\textsf{GIL}_{H_{i}}(w)$ and
$h_{r}=\textsf{GIR}_{H_{i}}(w)$. If $|c_{l}|>0$ or $|c_{r}|>0$ or
$(e_{1};e_{2})$ is of type $[e_{1};e_{2}]$ and only contains climbing points
then make $w$ climbing at level $i$. Else if $|h_{l}|+1+|h_{r}|\geq c$ then
make $h_{l}$, $w$ and $h_{r}$ climbing at level $i$. Else make $w$ helping at
level $i$.
Performing $H_{i}\cup C_{i}\rightarrow W_{i}$: Let $w$ be the minimum element
of $\textsf{s}_{H_{i}}(-\infty)$ and $\textsf{s}_{C_{i}}(-\infty)$, and let
$c_{r}=\textsf{GIR}_{C_{i}}(w)$. Make $w$ waiting at level $i$. If $w$ was
climbing and $|c_{r}|<c$ then make $c_{r}$ helping at level $i$.
#### Shift-down$(i)$
We move at least one element from level $i$ into level $i-1$, see Figure 4. If
$|D_{i}|<w_{i}$ then we let $a$ be some element in $D_{i}$. If
$|D_{i}|<|B_{i}|$ then: if $|A_{i}|=0$ we perform444The move $H_{i}\cup
C_{i}\stackrel{{\scriptstyle l}}{{\rightarrow}}W_{i}$ will be performed the
same way as we did it in fix. $H_{i}\cup C_{i}\stackrel{{\scriptstyle
h_{1}}}{{\rightarrow}}W_{i}$, $W_{i}\stackrel{{\scriptstyle
h_{2}}}{{\rightarrow}}R_{i}$ and $R_{i}\rightarrow A_{i}$, where
$h_{1}=\min(1,|H_{i}|+|C_{i}|)$ and $h_{2}=\min(1,|W_{i}|+h_{1})$, now we know
that $|A_{i}|>0$ so let $a=\textsf{s}_{A_{i}}(-\infty)$, i.e., $a$ is the
leftmost arriving point in $A_{i}$ at level $i$. We call move-
down$(a,i,i-1,\text{arriving},\text{climbing})$.
#### Shift-up$(i)$
Assume we are at level $i$, we want to move at least one and at most $c$
arbitrary points from $B_{i}$ into $B_{i+1}$. Let555See the analysis in
Section 4 for a proof that $|C_{i}|>0$. $s_{1}=\textsf{s}_{C_{i}}(-\infty)$,
$e_{1}=\textsf{p}_{G_{\leq i}}(s_{1})$ and $e_{2}=\textsf{s}_{G_{\leq
i}}(s_{1})$, and let $s_{2}=\textsf{s}_{C_{i}\cap[e_{1};e_{2}]}(s_{1})$,
$s_{3}=\textsf{s}_{C_{i}\cap[e_{1};e_{2}]}(s_{2})$,
$s_{4}=\textsf{s}_{C_{i}\cap[e_{1};e_{2}]}(s_{3})$ and
$s_{5}=\textsf{s}_{C_{i}\cap[e_{1};e_{2}]}(s_{4})$, if they exist, also let
$c_{r}=\textsf{GIR}_{C_{i}}(s_{4})$ be the group of climbing elements to the
immediate right of $s_{4}$, if they exist, see Figure 5. We will now move one
or more climbing points from $B_{i}$ into $B_{i+1}$ where they become arriving
points. If $i=m-1$ or $i=m$ then we put arriving points into $D_{i+1}$, which
we might have to create, instead of $A_{i+1}$.
We now deal with the case where $(e_{1};e_{2})$ is of type $[e_{1};e_{2}]$ and
only contains climbing points. Let $l$ be the level of $e_{1}$’s left
interval, and $r$ the level of $e_{2}$’s right interval, also let $c_{I}$ be
the number of climbing points in the interval. If $l=i+1$ we make $e_{1}$
arriving, else we make it guarding, at level $i+1$. Make the points of
$s_{1},s_{2},s_{3}$ and $s_{4}$ that exist arriving at level $i+1$. If
$c_{I}\leq c$ then make $s_{5}$ arriving at level $i+1$ if it exists, also if
$r=i+1$ we make $e_{2}$ arriving, else we make it guarding, at level $i+1$.
Else make $s_{5}$ guarding at level $i$.
We now deal with the cases where $(e_{1};e_{2})$ might contain non-climbing
points. If $\textsf{p}(s_{1})=e_{1}$ we make $s_{1}$ and $s_{2}$ waiting and
guarding at level $i$, respectively, else we make $s_{1}$ guarding at level
$i$ and $s_{2}$ arriving at level $i+1$. Now in both cases we make $s_{3}$
arriving at level $i+1$ and $s_{4}$ guarding at level $i$. If
$\langle(s_{4};e_{2})$ is not of type $[s_{4};e_{2}]$ or contains non-climbing
points$\rangle$ and $|c_{r}|<c$, i.e. there are less than $c$ consecutive
climbing points to the right of $s_{4}$, then we make the points $c_{r}$
helping at level $i$.
We have moved climbing points from $B_{i}$ into $B_{i+1}$, and made them
arriving. Finally we call fix$(i+1)$.
Figure 5. Here we see illustrations of how we maintain the intervals when
updating the intervals. These only show single cases of each of the update
methods many cases.
#### Move-down$(e,i,j,t_{\text{before}},t_{\text{after}})$
Depending on the type $t_{\text{before}}$ of point $e$ we have different
cases, see Figure 5.
Non-guarding Let $e_{1}=\textsf{p}_{G_{\leq i}}(e)$,
$e_{2}=\textsf{s}_{G_{\leq i}}(e)$ and let $l$ be the level of the left
interval of $e_{1}$ and $r$ the level of the right interval of $e_{2}$. Also
let $p_{2}=\textsf{p}_{B_{i}\backslash G_{i}\cap[e_{1};e_{2}]}(p_{1})$,
$p_{1}=\textsf{p}_{B_{i}\backslash G_{i}\cap[e_{1};e_{2}]}(e)$,
$s_{1}=\textsf{s}_{B_{i}\backslash G_{i}\cap[e_{1};e_{2}]}(e)$ and
$s_{2}=\textsf{s}_{B_{i}\backslash G_{i}\cap[e_{1};e_{2}]}(s_{1})$, also let
$c_{l}=\textsf{FGL}_{C_{i}\cap[e_{1};e_{2}]}(e)$ be the elements in the first
climbing group left of $e$, likewise let
$c_{r}=\textsf{FGR}_{C_{i}\cap[e_{1};e_{2}]}(e)$ be the elements in the first
climbing group right of $e$.
Case $i=j$: make $e$ arriving in level $j$, if $|c_{l}|<c$ then make the
points in $c_{l}$ helping at level $j$, if $|c_{r}|<c$ then make the points in
$c_{r}$ helping at level $j$. Finally call fix$(j)$.
Case $i>j$: If both $p_{2}$ and $p_{1}$ exists we make $p_{1}$ guarding in
level $j$ and let $e_{1}^{\prime}$ denote $p_{1}$, else if only $p_{1}$ exists
we make $e_{1}$ guarding at level $\min(l,j)$ and $p_{1}$ of type
$t_{\text{after}}$ at level $j$ and let $e_{1}^{\prime}$ denote $e_{1}$, else
we make $e_{1}$ guarding in level $\min(l,j)$, and let $e_{1}^{\prime}$ denote
$e_{1}$. If both $s_{1}$ and $s_{2}$ exists we make $s_{1}$ guarding at level
$j$, and let $e_{2}^{\prime}$ denote $s_{1}$, else if only $s_{1}$ exists we
make $s_{1}$ of type $t_{\text{after}}$ at level $j$ and make $e_{2}$ guarding
at level $\min(j,r)$ and let $e_{2}^{\prime}$ denote $e_{2}$, else we make
$e_{2}$ guarding at level $\min(j,r)$ and let $e_{2}^{\prime}$ denote $e_{2}$.
Lastly we make $e$ of type $t_{\text{after}}$ in level $j$. Now let
$c_{l}^{\prime}$ denote the elements of $c_{l}$ which we have not moved in the
previous steps, likewise let $c_{r}^{\prime}$ denote the elements of $c_{r}$
which we have not moved. If $\langle(e_{1};e_{1}^{\prime}]$ is not of type
$[e_{1};e_{1}^{\prime}]$ or contains non-climbing points$\rangle$ and
$|c_{l}^{\prime}|<c$ then make $c_{l}^{\prime}$ helping at level $i$. If
$\langle[e_{2}^{\prime};e_{2})$ is not of type $[e_{2}^{\prime};e_{2}]$ or
contains non-climbing points$\rangle$ and $|c_{r}^{\prime}|<c$ then make
$c_{r}^{\prime}$ helping at level $i$. Call fix$(i)$, fix$(j)$,
fix$(\min(l,i))$ and fix$(\min(i,r))$.
Guarding If $e=\min(P)$ or $e=\max(P)$ we simply do nothing and return. Let
$e_{1}=\textsf{p}_{G_{\leq h}}(e)$ be the left endpoint of the left interval
$(e_{1};e[$ lying at level $h$ and $e_{2}=\textsf{s}_{G_{\leq h}}(e)$ be the
right endpoint of the right interval $[e;e_{2})$ lying at level $i$, we assume
w.l.o.g. that $h>i$, the case $h<i$ is symmetric. Also let $l$ be the level of
the left interval of $e_{1}$ and $r$ the level of the right interval of
$e_{2}$. Let $p_{2}=\textsf{p}_{B_{h}\backslash G_{h}\cap[e_{1};e]}(p_{1})$
and $p_{1}=\textsf{p}_{B_{h}\backslash G_{h}\cap[e_{1};e]}(e)$ be the two left
points of $e$, if they exists, $s_{1}=\textsf{s}_{B_{i}\backslash
G_{i}\cap[e;e_{2}]}(e)$ and $s_{2}=\textsf{s}_{B_{i}\backslash
G_{i}\cap[e;e_{2}]}(s_{1})$ the two right points of $e$, if they exits. Also
let $c_{l}=\textsf{FGL}_{C_{i}\cap[e_{1};e]}(e)$ and
$c_{r}=\textsf{FGR}_{C_{i}\cap[e;e_{2}]}(e)$.
If $p_{2}$ does not exist we make $e_{1}$ guarding at level $\min(l,j)$, we
make $p_{1}$ of type $t_{\text{after}}$ at level $j$ and let $e_{1}^{\prime}$
denote $e_{1}$, else we make $p_{1}$ guarding at level $j$ and let
$e_{1}^{\prime}$ denote $p_{1}$. If it is the case that $i>j$ then we check:
if $s_{2}$ does not exist then we make $s_{1}$ of type $t_{\text{after}}$ at
level $j$, $e_{2}$ guarding at level $\min(j,r)$ and let $e_{2}^{\prime}$
denote $e_{2}$, else we make $s_{1}$ guarding at level $j$ and let
$e_{2}^{\prime}$ denote $s_{1}$. We make $e$ of type $t_{\text{after}}$ at
level $j$.
Now let $c_{l}^{\prime}$ be the points of $c_{l}$ which was not moved and
$c_{r}^{\prime}$ the points of $c_{r}$ which was not moved. If
$|c_{l}^{\prime}|<c$ then make $c_{l}^{\prime}$ helping at level $h$. We now
have two cases if $e_{2}^{\prime}$ exists: then if $|c_{r}^{\prime}|<c$ then
make $c_{r}^{\prime}$ helping at level $i$. The other case is if
$e_{2}^{\prime}$ does not exist: then if $\langle(e_{1}^{\prime};e_{2})$ is
not of type $[e_{1}^{\prime};e_{2}]$ or contains non-climbing points$\rangle$
and $|c_{r}^{\prime}|<c$ then make $c_{r}^{\prime}$ helping at level $i$. In
all cases call fix$(\min(l,h))$, fix$(h)$ and fix$(i)$. If $i>j$ then call
fix$(j)$ and fix$(\min(j,r))$.
#### Delete$(e)$
We first call find$(e)$ to get the type of $e$ and its level $i$, if $e$ is
not in the dictionary we just return. If $e$ is in the dictionary we have two
cases, depending on if $e$ is guarding or not.
Non-guarding Let $c_{l}=\textsf{GIL}_{C_{i}}(e)$ be the elements in the
climbing group immediately left of $e$, let $c_{r}=\textsf{GIR}_{C_{i}}(e)$ be
the elements in the climbing group immediately right of $e$, let
$h_{l}=\textsf{GIL}_{H_{i}}(e)$ be the elements in the helping group
immediately left of $e$, and let $h_{r}=\textsf{GIR}_{H_{i}}(e)$ be the
elements in the helping group immediately right of $e$. Let
$e_{1}=\textsf{p}_{G_{\leq i}}(e)$ and let $e_{2}=\textsf{s}_{G_{\leq i}}(e)$.
Let $l$ be the level of the interval left of $e_{1}$ and $r$ the level of the
interval right of $e_{2}$.
We have two cases, the first is $|]e_{1};e_{2}[\cap B_{i}|=1$: if $l>r$ make
$e_{1}$ guarding and $e_{2}$ arriving at level $r$, if $l<r$ then make $e_{1}$
arriving and $e_{2}$ guarding at level $l$. If $l=r$ and $|P|=n\geq 4$ then
make $e_{1}$ and $e_{2}$ arriving at level $l=r$. Delete $e$, call fix$(r)$,
fix$(l)$, fix$(i)$ and rebalance-above$(1)$.
The other case is $|]e_{1};e_{2}[\cap B_{i}|>1$: If $\langle(e_{1};e_{2})$ is
not of type $[e_{1};e_{2}]$ or contains non-climbing points$\rangle$ and
$|c_{l}|+|c_{r}|<c$ then make $c_{l}$ and $c_{r}$ helping at level $i$. If
$|h_{l}|+|h_{r}|\geq c$ then make $h_{l}$ and $h_{r}$ climbing at level $i$.
Delete $e$, call fix$(i)$ and rebalance-above$(1)$.
Min-guarding If $e=\min(P)$ then let $e^{\prime}=\textsf{s}_{G_{\leq m}}(e)$
and $e^{\prime\prime}=\textsf{s}_{G_{\leq m}}(e^{\prime})$ where $0$ is the
level of $(e;e^{\prime})$ and $i$ is the level of
$(e^{\prime};e^{\prime\prime})$. The case of $e=\max(P)$ is symmetric. Also
let $s_{1}=\textsf{s}_{B_{0}\backslash G_{0}\cap[e;e^{\prime}]}(e)$,
$s_{2}=\textsf{s}_{B_{0}\backslash G_{0}\cap[e;e^{\prime}]}(s_{1})$,
$t_{1}=\textsf{s}_{B_{i}\backslash
G_{i}\cap[e^{\prime};e^{\prime\prime}]}(e^{\prime})$ and
$t_{2}=\textsf{s}_{B_{i}\backslash
G_{i}\cap[e^{\prime};e^{\prime\prime}]}(t_{1})$.
If $s_{2}$ exists then delete $e$ make $s_{1}$ guarding at level $0$ and call
fix$(0)$. If $s_{2}$ does not exist and $t_{2}$ exists then delete $e$ make
$s_{1}$ and $t_{1}$ guarding and $e^{\prime}$ arriving at level $0$ and
finally call fix$(0)$ and fix$(i)$. If $s_{2}$ does not exist and $t_{2}$ does
not exist then delete $e$, make $s_{1}$ and $e^{\prime\prime}$ guarding and
$e^{\prime}$ and $t_{1}$ arriving at level $0$ and finally call fix$(0)$ and
fix$(i)$. In all the previous cases return.
Guarding Let $h$ be the level of the left interval $(e_{1}:e[$, let $i$ the
level of the right interval $[e:e_{2})$ that $e$ participates in. We assume
w.l.o.g. that $h>i$, the case $h<i$ is symmetric. Let $l$ the level of the
left interval that $e_{1}$ participates in, where $e_{1}=\textsf{p}_{G_{\leq
h}}(e)$ and $e_{2}=\textsf{s}_{G_{\leq h}}(e)$. Let
$p_{2}=\textsf{p}_{B_{h}\backslash G_{h}\cap[e_{1};e]}(p_{1})$ and
$p_{1}=\textsf{p}_{B_{h}\backslash G_{h}\cap[e_{1};e]}(e)$. Let
$c_{l}=\textsf{FGL}_{C_{i}}(e)$ be the points in the first group of climbing
points left of $e$.
If $p_{2}$ exist we make $p_{1}$ guarding at level $i$, and let $e^{\prime}$
denote $p_{1}$, else we make $e_{1}$ guarding at level $\min(l,i)$, let
$e^{\prime}$ denote $e_{1}$ and if $[e^{\prime};e_{2})$ is of type
$[e^{\prime};e_{2}]$ and contains only climbing points then we make $p_{1}$
climbing at level $i$ else we make $p_{1}$ waiting at level $i$. Let
$c_{l}^{\prime}$ be the points in $c_{l}$ which was not moved in the previous
movement of points. If $|c_{l}^{\prime}|<c$ make $c_{l}^{\prime}$ helping at
level $h$. If $e^{\prime}$ is $e_{1}$ then call fix$(l)$. Delete $e$, call
fix$(h)$, fix$(i)$ and rebalance-above$(1)$.
#### Rebalance-below$(i)$
For each level $l=0,\ldots,i$ we perform a shift-up$(l)$ while $c<c_{l}$.
#### Rebalance-above$(i)$
For each level $l=i,\ldots,m-1$ we perform shift-down$(l+1)$ while $c_{l}<-c$.
## 3\. Memory management
We will now deal with the memory layout of the data structure. We will put the
blocks in the order $B_{0},\ldots,B_{m}$, where block $B_{i}$ further has its
dictionaries in the order $D_{i},A_{i},R_{i},W_{i},H_{i},C_{i}$ and $G_{i}$,
see Figure 1. Block $B_{m}$ grows and shrinks to the right when elements are
inserted and deleted from the working set dictionary.
The $D_{i}$ structure is not a moveable dictionary as the other structures in
a block are, it is simply an array of $w_{i}=d2^{i+k}$ elements which we use
to encode the size of each of the structures $A_{i},R_{i},W_{i},H_{i},C_{i}$
and $G_{i}$ along with their own auxiliary data, as they are not implicit and
need to remember $\mathcal{O}(2^{i+k})$ bits which we store here. As each of
the moveable dictionaries in $B_{i}$ have size $\mathcal{O}(2^{2^{i+k}})$ we
need to encode numbers of $\mathcal{O}(2^{i+k})$ bits in $D_{i}$.
We now describe the memory management concerning the movement, insertion and
deletion of elements from the working-set dictionary. First notice that the
methods find, predecessor and successor do not change the working-set
dictionary, and layout in memory. Also the methods shift-down, search,
rebalance-below and rebalance-above only calls other methods, hence their
memory management is handled by the methods they call. The only methods where
actual memory management comes into play are in insert, shift-up, fix, move-
down and delete. We will now describe two methods internal-movement – which
handles movement inside a single block/level – and external-movement – which
handles movement across different blocks/levels. Together these two methods
handle all memory management.
#### Internal-movement$(m_{1},\ldots,m_{l})$
Internal-movement in level $i$ takes a list of _internal moves_
$m_{1},\ldots,m_{l}$ to be performed on block $B_{i}$, where
$l=\mathcal{O}(1)$ and move $m_{j}$ consists of:
* •
the index $\gamma=D_{i},A_{i},R_{i},W_{i},H_{i},C_{i},G_{i}$ of the dictionary
to change, where we assume666We will misuse notation and let $\gamma+1$ denote
the next in the total order $D,A,R,W,H,C,G$. We will also compare
$m_{j}.\gamma$ and $m_{h}.\gamma$ with $\leq$ in this order. that
$m_{j}.\gamma\leq m_{h}.\gamma$, for $j\leq h$,
* •
the set of elements $S_{\text{in}}$ to put into $\gamma$, where
$|S_{\text{in}}|=\mathcal{O}(1)$,
* •
the set of elements $S_{\text{out}}$ to take out of $\gamma$, where
$|S_{\text{out}}|=\mathcal{O}(1)$ and
* •
the total size difference $\delta=|S_{\text{in}}|-|S_{\text{out}}|$ of
$\gamma$ after the move.
For $j=1,\ldots,l$ do: if $m_{j}.\delta<0$ then remove $S_{\text{out}}$ from
$\gamma$, insert $S_{\text{in}}$ into $\gamma$ and move $\gamma+1,\ldots,G$
left $|m_{j}.\delta|$ positions, where we move them in the order
$\gamma+1,\ldots,G$. If $m_{j}.\delta>0$ then move $\gamma+1,\ldots,G$ right
$m_{j}.\delta$ positions, where we move them in the order $G,\ldots,\gamma+1$,
remove $S_{\text{out}}$ from $\gamma$ and insert $S_{\text{in}}$ into
$\gamma$. See Figure 6.
It takes $\mathcal{O}(\log(2^{2^{i+k}}))=\mathcal{O}(2^{i+k})$ time and
$\smash{\mathcal{O}(\log_{B}(2^{2^{i+k}}))=\mathcal{O}(\frac{2^{i+k}}{\log
B})}$ cache-misses to perform move $j$. In total all the moves
$m_{1},\ldots,m_{l}$ use $\mathcal{O}(2^{i+k})$ time and
$\mathcal{O}(\frac{2^{i+k}}{\log B})$ cache-misses, as $l=\mathcal{O}(1)$.
Figure 6. (Left) Memory movement of internal-movement inside of a block
$B_{i}$. (Right) Memory movement of external-movement across multiple blocks
$B_{M_{1}.\gamma},\ldots,B_{M_{l}.\gamma}$.
#### External-movement$(M_{1},\ldots,M_{l})$
External-movement takes a list of _external moves_ $M_{1},\ldots,$ $M_{l}$,
where $l=\mathcal{O}(1)$. Move $M_{j}$ consists of:
* •
the index $0\leq\gamma\leq m$ of the block/level to perform the internal moves
$m_{1},\ldots,m_{q}$ on, where $M_{j}.\gamma<M_{h}.\gamma$ for $j<h$,
* •
the list of internal moves $m_{1},\ldots,m_{q}$ to perform on block $\gamma$,
where $q=\mathcal{O}(1)$, and
* •
the total size difference $\Delta=\sum_{h=1}^{q}{m_{h}.\delta}$ of block
$\gamma$ after all the internal moves $m_{1},\ldots,m_{q}$ have been
performed.
Let $\overline{\Delta}=\sum_{i=1}^{l}{M_{i}.\Delta}$ be the total size change
of the dictionary after the external-moves have been performed. If
$\overline{\Delta}=0$ then we let $\gamma_{\text{end}}=M_{l}.\gamma$ else we
let $\gamma_{\text{end}}=m$. Let
$p_{\text{end}}=\sum_{j=0}^{\gamma_{\text{end}}}{|B_{j}|}+\overline{\Delta}$
be the last address of the right most block that we need to alter. Let
$s_{1},\ldots,s_{k}$ be the sublist of the indexes $\\{1,\ldots,l\\}$ where
$M_{s_{i}}.\Delta\leq 0$ for $i=1,\ldots,k$. Let $a_{1},\ldots,a_{h}$ be the
sublist of the indexes $\\{1,\ldots,l\\}$ where $M_{a_{i}}.\Delta>0$ for
$i=1,\ldots,h$.
We first perform all the internal moves of each of the external moves
$M_{s_{1}},\ldots,M_{s_{k}}$. Then we compact all the blocks with index $i$
where $M_{1}.\gamma\leq i\leq\gamma_{\text{end}}$ so the rightmost block ends
at position $p_{\text{end}}$. Finally for each external move $M_{a_{i}}$ for
$i=1,\ldots,h$: move $B_{M_{a_{i}}.\gamma}$ left so it aligns with
$B_{M_{a_{i}}.\gamma-1}$ and perform all the internal moves of $M_{a_{i}}$,
then compact the blocks
$B_{M_{a_{i}}.\gamma+1},\ldots,B_{M_{a_{i+1}}.\gamma-1}$ at the left end so
they align with block $B_{M_{a_{i}}.\gamma}$.
It takes
$\mathcal{O}\left(l\log\left(2^{2^{i+k}}\right)\right)=\mathcal{O}\left(l2^{i+k}\right)$
time and
$\mathcal{O}\left(l\log_{B}\left(2^{2^{i+k}}\right)\right)=\mathcal{O}\left(l\frac{2^{i+k}}{\log
B}\right)$ cache-misses to perform the internal moves on level $i$. In total
all the external moves $M_{1},\ldots,M_{l}$ use
$\mathcal{O}(2^{\gamma_{\text{end}}+k})$ time and
$\mathcal{O}\left(\frac{2^{\gamma_{\text{end}}+k}}{\log B}\right)$ cache-
misses, as the external move at level $\gamma_{\text{end}}$ dominates the rest
and $l=\mathcal{O}(1)$.
### 3.1. Memory management in updates of intervals
With the above two methods we can perform the memory management when updating
the intervals in Section 2.3: Whenever an element moves around, is deleted or
inserted, it is simply put in one or two internal moves. All internal moves in
a single block/level are grouped into one external move. Since all updates of
intervals only move around a constant number of elements, the requirements for
internal/external-movement that $l=\mathcal{O}(1)$ and $q=\mathcal{O}(1)$ are
fulfilled. From the above time and cache bounds for the memory management the
bounds in Theorem 1 follows.
## 4\. Analysis
We will leave it for the reader to check that the pre-conditions for each
methods in Section 2.3 are fulfilled and that the methods maintains all
invariants. We will instead concentrate on using the invariants to prove
correctness of the find, predecessor, successor and shift-up operations along
with proving time and cache-miss bounds for these. We will leave the time and
cache-miss bounds of search, rebalance-above, rebalance-below, shift-down,
insert, delete and fix for the reader as they are all similarly in nature.
#### Find$(e)$
We only consider the cases where $\min(P)<e<\max(P)$, the other cases
trivially gives the correct answer in $\mathcal{O}(1)$ time and cache-misses
as $\min(P),\max(P)\in G_{0}$. Assume that find$(e)$ stops at level $i$, then
we have that $e_{1}\leq p$ or $s\leq e_{2}$ so $I(e_{1},e_{2},i)\neq\emptyset$
and $i$ is the minimal $i$ where this happens, see lemma 1. Notice that
$e_{1}=\textsf{p}_{G_{\leq i}}(e)$ and $e_{2}=\textsf{s}_{G_{\leq i}}(e)$, so
$e_{1}$ and $e_{2}$ are the same as in lemma 1. When the while loop breaks we
have all the preconditions for lemma 1. Now $e$ is either in the dictionary,
or not, and if $e$ is in the dictionary it is either guarding or not, so we
have three cases.
Case 1) $e$ is in the dictionary and is non-guarding: then we have from lemme
1 that $(e_{1};e_{2})$ is a interval at level $i$ and $e\in B_{i}$. From this
we also have that $\log(\ell_{e})\geq\log(2^{2^{i+k-1}})$.
Case 2) $e$ is not in the dictionary: from lemma 1 $(e_{1};e_{2})$ lie at
level $i$ and we know that $e$ intersects it. Since $e$ is not in the
dictionary $\ell_{e}=n$ and then $\log(\ell_{e})\geq\log(2^{2^{i+k-1}})$.
Case 3) $e$ is in the dictionary and is guarding: from lemma 1 we have that
either $(e_{1};e)$ or $(e;e_{2})$ lie in level $i$, hence $e\in G_{i}\subseteq
B_{i}$. From this we also have that
$\log(\ell_{e})\geq\log(2^{2^{\max(i,j)+k-1}})\geq\log(2^{2^{i+k-1}})$.
From the above we see that find$(e)$ runs in
$\mathcal{O}(\log(2^{2^{i+k-1}}))=\mathcal{O}(\log\min(\ell_{\textsf{p}(e)},\ell_{e},\ell_{\textsf{s}(e)}))$
time. When we look at the cache-misses we will first notice that the first
$\left\lfloor\log\log B\right\rfloor$ levels will fit in a single cache-line
because all levels are next to each other in the memory layout, so the total
cache-misses will be
$\mathcal{O}\left(1+\sum_{j=\left\lfloor\log\log
B\right\rfloor+1}^{i}{\left(1+\log_{B}\left(2^{2^{j+k}}\right)\right)}\right)=\mathcal{O}\left(\frac{2^{i+k}}{\log
B}\right)=\mathcal{O}(\log_{B}\min(\ell_{\textsf{p}(e)},\ell_{e},\ell_{\textsf{s}(e)})).$
#### Predecessor$(e)$ (and successor$(e)$)
We will only handle the predecessor operation, the case for the successor is
symmetric. Since we have the same condition in the while loop as for find, we
know that when it breaks it implies that $I(e_{1},e_{2},i)\neq\emptyset$. So
from lemma 1, $e$ intersects a interval at level $i$ and the predecessor of
$e$ is now $\max(e_{1},p)$.
From I.4 we know that $\log(\ell_{p})\geq\log(2^{2^{i+k-1}})$ and the total
time usage is $\sum_{j=0}^{i}{\mathcal{O}(\log(2^{2^{i+k}}))}$
$=\mathcal{O}(2^{i+k})=\mathcal{O}(\log(\ell_{p}))$. Like in find, the first
$\left\lfloor\log\log B\right\rfloor$ levels fit into one block/cache-line
hence the total cache-misses will be $\mathcal{O}(\log_{B}(\ell_{p}))$.
#### Shift-up$(i)$
For shift-up to work for level $i$ it is mandatory that $|C_{i}|>0$ so that
$\textsf{s}_{C_{i}}(-\infty)$ will return a element which can be moved to
level $i+1$. From the precondition that
$|H_{i}|+|C_{i}|=4c2^{2^{i+k}}+c^{\prime}_{i}$, where $c\leq
c^{\prime}_{i}=\mathcal{O}(1)$, we have that
$|C_{i}|=4c2^{2^{i+k}}+c^{\prime}_{i}-|H_{i}|\geq 4c2^{2^{i+k}}-c-|H_{i}|$
so proving that $|H_{i}|<4c2^{2^{i+k}}-c$ is enough. From I.3 we can at most
have $c-1$ helping points in a helping group, so for every $c-1$ helping
points we need a separating point, the role of the separating point can be
played by a point from $D_{i},A_{i},R_{i},W_{i}$ or $G_{\leq i-1}$. These are
the only ways to contribute points to $H_{i}$ hence for $i\geq 1$ we have this
bound
$\displaystyle|H_{i}|$ $\displaystyle\leq$
$\displaystyle(c-1)(|D_{i}|+|A_{i}|+|R_{i}|+|W_{i}|+|G_{\leq i-1}|)$
$\displaystyle\stackrel{{\scriptstyle(*)}}{{\leq}}$
$\displaystyle(c-1)\left(w_{i}+2\cdot
2^{2^{i+k}}+\sum_{j=0}^{i-1}{\left((4+2d+8c)2^{2^{j+k}}+2c\right)}\right)$
$\displaystyle\stackrel{{\scriptstyle(**)}}{{\leq}}$
$\displaystyle(c-1)\left(d\cdot 2^{i+k}+2\cdot 2^{2^{i+k}}+(4+2d+8c)\cdot
2\cdot 2^{2^{i+k-1}}+2ci\right)$
Where we in $(*)$ have used I.5, I.6 I.7 and O.1, and in $(**)$ have used that
$2^{2^{l}}=2^{2^{l-1}}\cdot 2^{2^{l-1}}$ and $2^{2^{l-1}}\geq l$ for $l\geq
1$. If we use that $c=5$ then for $k>\log\log(380+20d)+1$ we have that
$|C_{i}|\geq 4c2^{2^{i+k}}-c-|H_{i}|>0$ for $i=1,\ldots,m-1$.
For $i=0$ we have a different bound as $G_{\leq i-1}$ is empty, we get the
bound
$\displaystyle|H_{0}|$ $\displaystyle\leq$
$\displaystyle(c-1)(|D_{i}|+|A_{i}|+|R_{i}|+|W_{i}|)$ $\displaystyle\leq$
$\displaystyle(c-1)\left(d\cdot 2^{i+k}+2\cdot 2^{2^{i+k}}\right)$
but for $k>\log\log(380+20d)+1$ this is of course still sufficient as
$|H_{0}|$ only got smaller. So we have proved that $|C_{i}|>0$ for level
$i=0,\ldots,m-1$.
#### Move-down$(e,i,j,t_{\text{before}},t_{\text{after}})$
Move-down moves a constant number of points around and into level $j$ from
$i$. If $e$ is non-guarding we call fix$(i)$, fix$(j)$, fix$(\min(l,i))$ and
fix$(\min(i,r))$. If $e$ is guarding we call fix$(\min(l,h))$, fix$(h)$ and
fix$(i)$, and if $i>j$ we also call fix$(j)$ and fix$(\min(j,r))$. In the non-
guarding case the time is bounded by $\mathcal{O}(\log
2^{2^{i+k}})=\mathcal{O}(\log\ell_{e})$ and the cache-miss bounds are
dominated by $\mathcal{O}(\log_{B}2^{2^{i+k}})=\mathcal{O}(\log_{B}\ell_{e})$.
In the guarding case the time is bounded by $\mathcal{O}(\log
2^{2^{h+k}})=\mathcal{O}(\log\ell_{e})$ and the cache-miss bounds are
dominated by $\mathcal{O}(\log_{B}2^{2^{h+k}})=\mathcal{O}(\log_{B}\ell_{e})$.
## 5\. Further work
We still have some open problems. Is it possible to change the insert
operation such that when we insert a new point it will get a working-set value
of $n+1$ instead of $0$? We can actually achieve this in our structure by
loosening the invariant on the working-set number of guarding points to only
require that they have a working-set number of at least
$2^{2^{\min(i,j)+k-1}}$, but then for search the time will increase to
$\mathcal{O}(\log\min(\ell_{e},\max(\ell_{\textsf{p}(e)},\ell_{\textsf{s}(e)})))$
and the cache-misses to
$\mathcal{O}(\log_{B}\min(\ell_{e},\max(\ell_{\textsf{p}(e)},\ell_{\textsf{s}(e)})))$
and the bounds for predecessor and successor queries would increase to
$\mathcal{O}(\log\max(\ell_{\textsf{p}(e)},\ell_{\textsf{s}(e)}))$ time and
$\mathcal{O}(\log_{B}\max(\ell_{\textsf{p}(e)},\ell_{\textsf{s}(e)}))$ cache-
misses.
Another interesting question is if we can have a dynamic dictionary supporting
efficient finger searches [5] in the implicit model, i.e., we have a finger
$f$ located at a element and then we want to find an element $e$ in time
$\mathcal{O}(\log d(f,e))$, where $d(f,e)$ is the rank distance between $f$
and $e$. But very recently [14] have shown that finger search in
$\mathcal{O}(\log d(e,f))$ time is not possible in the implicit model. They
give a lower bound of $\Omega(\log n)$. Now we could instead separate the
finger search and the update of the finger, say we allow the finger search to
use $\mathcal{O}(q(d(e,f)))$ time for some function $q$. In this setting they
also prove a lower of $\Omega(q^{-1}(\log n))$ for the update finger
operation, where $q^{-1}$ is the inverse function of $q$. They also give
almost tight upper bounds for this setting, in the form of a trade-off bound
between the finger search and the update finger operations. The finger search
operation uses $\mathcal{O}(\log d(e,f))+q(d(e,f))$ time, and the update
finger operation uses $\mathcal{O}(q^{-1}(\log n)\log n)$ time. But even given
their result it still remains an open problem whatever dynamic finger search
with an externally maintained finger is possible in $\mathcal{O}(\log d(e,f))$
time. So in other words is it possible to do finger search in
$\mathcal{O}(\log d(e,f))$ time if we allow the data structure to store
$\mathcal{O}(\log n)$ bits of data that can store the finger?
## References
* [1] Mihai Bǎdoiu, Richard Cole, Erik D. Demaine, and John Iacono. A unified access bound on comparison-based dynamic dictionaries. Theoretical Computer Science, 382(2):86–96, 2007.
* [2] Prosenjit Bose, Karim Douïeb, and Stefan Langerman. Dynamic optimality for skip lists and B-trees. In Proc. 19th Annual ACM-SIAM Symposium on Discrete algorithms, pages 1106–1114. SIAM, 2008.
* [3] Prosenjit Bose, John Howat, and Pat Morin. A distribution-sensitive dictionary with low space overhead. In Proc. 11th International Symposium on Algorithms and Data Structures, volume 5664 of LNCS, pages 110–118. Springer-Verlag, 2009\.
* [4] Gerth Brodal, Casper Kejlberg-Rasmussen, and Jakob Truelsen. A cache-oblivious implicit dictionary with the working set property. In Proc. 12th International Symposium on Algorithms and Data Structures, volume 6507 of LNCS, pages 37–48. Springer-Verlag, 2010.
* [5] Gerth Stølting Brodal. Finger search trees. In Dinesh Mehta and Sartaj Sahni, editors, Handbook of Data Structures and Applications, chapter 11. CRC Press, 2005.
* [6] Timothy Moon-Yew Chan and Eric Y. Chen. Optimal in-place algorithms for 3-D convex hulls and 2-D segment intersection. In Proc. 25th Annual Symposium on Computational Geometry, pages 80–87. ACM, 2009.
* [7] G. Franceschini, R. Grossi, J.I. Munro, and L. Pagli. Implicit $B$-trees: New results for the dictionary problem. In Proc. 43rd Annual IEEE Symposium on Foundations of Computer Science, pages 145–154, 2002.
* [8] Gianni Franceschini and Roberto Grossi. Optimal worst-case operations for implicit cache-oblivious search trees. In Proc. 8th International Workshop on Algorithms and Data Structures, volume 2748 of LNCS, pages 114–126. Springer-Verlag, 2003\.
* [9] Gianni Franceschini and Roberto Grossi. Optimal implicit dictionaries over unbounded universes. Theory of Computing Systems, 39:321–345, 2006.
* [10] Matteo Frigo, Charles Eric Leiserson, Harald Prokop, and Sridhar Ramachandran. Cache-oblivious algorithms. In Proc. 40th Annual IEEE Symposium on Foundations of Computer Science, pages 285–297. IEEE, 1999.
* [11] John Iacono. Alternatives to splay trees with $\mathcal{O}(\log n)$ worst-case access times. In Proc. 12th Annual ACM-SIAM Symposium on Discrete algorithms, pages 516–522. SIAM, 2001.
* [12] James Ian Munro. An implicit data structure supporting insertion, deletion, and search in $\mathcal{O}(\log^{2}n)$ time. Journal of Computer and System Sciences, 33(1):66–74, 1986.
* [13] James Ian Munro and Hendra Suwanda. Implicit data structures for fast search and update. Journal of Computer and System Sciences, 21(2):236–250, 1980.
* [14] Jesper Sindahl Nielsen and Jakob Truelsen. Finger search in the implicit model. Work in progress, 2011.
* [15] Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. J. ACM, 32(3):652–686, 1985.
* [16] John William Joseph Williams. Algorithm 232: Heapsort. Communications of the ACM, 7(6):347–348, 1964.
|
arxiv-papers
| 2011-12-22T21:45:16 |
2024-09-04T02:49:25.646817
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gerth St{\\o}lting Brodal and Casper Kejlberg-Rasmussen",
"submitter": "Casper Kejlberg-Rasmussen",
"url": "https://arxiv.org/abs/1112.5472"
}
|
1112.5480
|
# A Posteriori Error Estimates for Energy-Based Quasicontinuum Approximations
of a Periodic Chain
Hao Wang Hao Wang, Oxford University Mathematical Institute, 24-29 St Giles’,
Oxford, OX1 3LB, UK wangh@maths.ox.ac.uk
###### Abstract.
We present a posteriori error estimates for a recently developed
atomistic/continuum coupling method, the Consistent Energy-Based QC Coupling
method. The error estimate of the deformation gradient combines a residual
estimate and an a posteriori stability analysis. The residual is decomposed
into the residual due to the approximation of the stored energy and that due
to the approximation of the external force, and are bounded in negative
Sobolev norms. In addition, the error estimate of the total energy using the
error estimate of the deformation gradient is also presented. Finally,
numerical experiments are provided to illustrate our analysis.
## 1\. Introduction
Quasicontinuum (QC) methods, or in general atomistic/continuum coupling
methods, are a class of multiscale methods for coupling an atomistic model of
a solid with a continuum model. These methods have been widely employed in
computational nano-technology, where a fully atomistic model will result in a
prohibitive computational cost but an exact configuration is required in a
certain region of the material. In this situation, atomistic model is applied
in the region which contains the defect core to retain certain accuracy, while
continuum model is applied in the far field to reduce the computational cost.
A number of QC methods have been developed in the past decades and are
classified in two groups: energy-based coupling methods and force-based
coupling methods. Despite the fact that the force-based methods are easy to
implement and extend to higher dimensional cases, energy-based methods have
certain advantages. For example, the forces derived from an energy potential
are conservative which could leads to a faster convergence rate in
computation, and the energy of an atomistic system can also be a quantity of
interest in real application. However, consistent energy-based coupling
methods can be tedious and restrictive on the shape of the coupling interface
in more than one dimension (see [11, 4]) and it was not until recent that a
practical consistent energy-based coupling method was created by Shapeev [10],
which is the Consistent Energy-Based QC Coupling method that we analyze in the
present paper.
A number of literature on the rigorous analysis of different QC methods have
been proposed since the first one by Lin [5]. However, most of the analysis
are on the a priori error analysis, and only a few are on the a posteriori
error analysis. Arndt and Luskin give a posteriori error estimates for the QC
approximation of a Frenkel-Kontorova model [2, 3, 1]. A goal-oriented approach
is used and error estimates on different quantity of interests, each of which
is essentially the difference between the values of a linear functional at the
atomistic solution and the QC solution, are proposed. The estimates are
decomposed into two parts, one is used to correctly chose the atomistic region
and another is used to optimally choose the mesh in the continuum region.
Serge et al. [9] give error estimates, also through a goal-oriented approach,
of the original energy-based QC approximation, whose consistency is not
guaranteed. Both of the above works employ the technique of deriving and
solving dual problems as a result of the goal-oriented approach. Ortner and
Süli [7] derive an a posteriori error indicator for a global norm through a
similar approach as ours. However, the QC method analyzed there does not
contain an approximation of the stored energy which is essentially different
from the QC method we are interested.
The present paper provides the a posteriori error analysis for the Consistent
Energy-Based QC Coupling method [10] for a one dimensional periodic chain with
nearest and next nearest neighbour interactions. The formulation of the QC
approximation has the feature that the finite element nodes in the continuum
region are not restricted to reside at the atomistic positions, which creates
the situation that interaction bonds often cross the element boundaries, which
is common in two dimensional formulation. We then derive the residual in
negative Sobolev norms and then the a posteriori stability constant as a
function of the QC solution. The error estimator of the deformation gradient
in $L_{2}$-norm is then obtained by combining these two analysis. In addition,
we derive an error estimator for the total energy difference by using that of
the deformation gradient. It should be remarked that though both of the error
estimators are global quantities, they consist of contributions from element.
As a result, an adaptive mesh refinement algorithm is developed and applied to
a problem that mimics the vacancy in the two dimensional case, and the
numerical results are presented.
### 1.1. Outline
In Section 2, we first formulate the atomistic model through both a continuous
approach, i.e., the deformation and the displacement are considered as
continuous functions on the reference lattice, and a discrete approach, which
is always taken in previous literature. We then formulate the Consistent
Energy-Based QC Coupling method in one dimensional setting.
In Section 3, we derive the residual estimates for the Consistent Energy-Based
QC Coupling method in a negative Sobolev norm. The residual is split into two
part, one is due to the approximation of the stored energy and the other is
due to the approximation of the external force.
In Section 4, we give the a posteriori stability analysis.
In Section 5, we combine the residual estimate and the stability analysis to
give the a posteriori error estimate of the deformation gradient in
$L_{2}$-norm and that of the total energy.
In Section 6, we present a numerical example to complement our analysis.
## 2\. Model Problem and QC Approximation
### 2.1. Atomistic Model
As opposed to taking only a discrete point of view in many QC researches, we
use both continuous functions and discretized vectors to denote the
displacement and the deformation. The reason for doing this is that the
Consistent Energy-Based QC coupling method, which we analyze in this paper, is
easily formulated through the continuous approach, while discrete formulations
could make the residual analysis of the external forces much easier.
For an infinite reference lattice with atomistic spacing $\varepsilon$, we
make the partition
$\mathcal{T}^{\varepsilon}=\\{T^{\varepsilon}_{\ell}\\}_{\ell=-\infty}^{\infty}$
of the domain $\mathbb{R}$ such that
$\mathbb{R}=\cup_{\ell=-\infty}^{\infty}T^{\varepsilon}_{\ell}$ and
$T^{\varepsilon}_{\ell}=[(\ell-1)\varepsilon,\ell\varepsilon]$. We then define
the displacement and deformation of this infinite lattice to be continuous
piecewise linear functions $u$,
$y\in\mathcal{P}_{1}(\mathcal{T}^{\varepsilon})\cap C^{0}(\mathbb{R})$. We use
$\boldsymbol{u}$ and $\boldsymbol{y}$ to denote the vectorizations of $u$ and
$y$ such that $u_{\ell}=u(\ell\varepsilon)$ and $y_{\ell}=y(\ell\varepsilon)$.
We know that $u_{\ell}$ and $y_{\ell}$ are the physical displacement and
deformation of atom $\ell$ respectively.
To avoid technical difficulties with boundaries, we apply periodic boundary
conditions. We rescale the problem so that there are $N\in\mathbb{N}$ atoms in
each period and $\varepsilon=1/N$, which implies that $u$ and $y$ are
1-periodic functions and $\boldsymbol{u}$ and $\boldsymbol{y}$ are
$N$-periodic vectors. We also impose a zero-mean condition to the admissible
space of displacements, which is defined to be
$\mathcal{U}=\big{\\{}u\in\mathcal{P}_{1}(\mathcal{T}^{\varepsilon})\cap
C^{0}(\mathbb{R}):u(x+1)=u(x)\text{ and }\int_{0}^{1}u(x)\,{\rm
d}x=0\big{\\}}.$ (2.1)
The set of admisible deformations is given by
$\mathcal{Y}=\big{\\{}y\in\mathcal{P}_{1}(\mathcal{T}^{\varepsilon})\cap
C^{0}(\mathbb{R}):y(x)=Fx+u(x),u\in\mathcal{U}\big{\\}},$ (2.2)
where $F>0$ is a given macroscopic deformation gradient.
As we mentioned above, it is necessary in the analysis of the external forces
to employ the discretization of the displacement and the deformation.
Therefore, by the relationship between $u,y$ and their vectorizations
$\boldsymbol{u},\boldsymbol{y}$, the discrete space of displacement and the
admissible set of deformation are defined by
$\mathcal{U}^{\varepsilon}=\\{\boldsymbol{u}\in\mathbb{R}^{\mathbb{Z}}:u_{\ell+N}=u_{\ell},\varepsilon\sum_{\ell=1}^{N}u_{\ell}=0\\},$
(2.3)
and
$\mathcal{Y}^{\varepsilon}=\\{\boldsymbol{y}\in\mathbb{R}^{\mathbb{Z}}:y_{\ell+N}=F\ell\varepsilon+u_{\ell},\boldsymbol{u}\in\mathcal{U}^{\varepsilon}\\},$
(2.4)
where the zero-mean condition on the displacements, i.e.,
$\varepsilon\sum_{\ell=1}^{N}u_{\ell}=0$ is obatined by applying the
trapezoidal rule to evaluate the integration $\int_{0}^{1}u(x)\,{\rm d}x$ with
respect to the partition $\mathcal{T}^{\varepsilon}$ and using the periodicity
of $u$.
For simplicity of analysis, we adopt a pair interaction model and assume that
only nearest neighbours and the next-nearest neighbours interact. With a
slight abuse of notation, the stored atomistic energy (per period) of an
admissible deformation is then given by
$\displaystyle\mathcal{E}_{\rm a}(y)$
$\displaystyle:=\varepsilon\sum_{\ell=1}^{N}\phi\Big{(}\frac{y(\ell\varepsilon)-y((\ell-1)\varepsilon)}{\varepsilon}\Big{)}+\varepsilon\sum_{\ell=1}^{N}\phi\Big{(}\frac{y(\ell\varepsilon)-y((\ell-2)\varepsilon)}{\varepsilon}\Big{)}$
$\displaystyle=\varepsilon\sum_{\ell=1}^{N}\phi\Big{(}\frac{y_{\ell}-y_{\ell-1}}{\varepsilon}\Big{)}+\varepsilon\sum_{\ell=1}^{N}\phi\Big{(}\frac{y_{\ell}-y_{\ell-2}}{\varepsilon}\Big{)}=:\mathcal{E}_{\rm
a}(\boldsymbol{y}),$ (2.5)
where $\phi\in C^{3}((0,+\infty))$ is a Lennard-Jones type interaction
potential. We assume that there exists $r_{\ast}>0$ such that $\phi$ is convex
in $(0,r_{\ast})$ and concave in $(r_{\ast},+\infty)$.
For the formulation of the external energy, we first define the linear nodal
interpolation operator
$I_{\varepsilon}:C^{0}(\mathbb{R})\rightarrow\mathcal{P}_{1}(\mathcal{T}^{\varepsilon})\cap
C^{0}(\mathbb{R})$ such that
$I_{\varepsilon}g(\ell\varepsilon)=g(\ell\varepsilon)\quad\forall g\in
C^{0}(\mathbb{R}).$ (2.6)
Then given a dead load $f\in\mathcal{U}$, we define the external energy (per
period) caused by $f$ to be
$\langle
f,u\rangle_{\varepsilon}:=\int_{0}^{1}I_{\varepsilon}(fu)dx=\sum_{\ell=1}^{N}\varepsilon
f_{\ell}u_{\ell}=:\langle\boldsymbol{f},\boldsymbol{u}\rangle_{\varepsilon},$
(2.7)
where $\boldsymbol{f}$ and $\boldsymbol{u}$ are the vectorizations of the
external force $f$ and the displacement $u$ according to
$\mathcal{T}^{\varepsilon}$.
Thus, the total energy (per period) under a deformation $y\in\mathcal{Y}$ is
given by
$E_{\rm a}(y;F)=\mathcal{E}_{\rm a}(y)-\langle f,u\rangle_{\varepsilon},$
as $u$ is determined by $y$ and $F$. However, in our analysis, we always
assume that $F$ is given and as a result, we simply write $E_{\rm a}(y;F)$ as
$E_{\rm a}(y)$.
The problem we wish to solve is to find
$y_{\rm a}\in{\rm argmin}E_{\rm a}(\mathcal{Y}),$ (2.8)
where ${\rm argmin}$ denotes the set of local minimizers.
### 2.2. Notation of Partitions, Norms and Discrete Derivatives
Though it is natural to introduce the QC approximation after the atomistic
model, we decide to pause here and introduce some important notation that are
used throughout the paper in order to make the flow of the paper more smooth
and save some space.
In Section 2.1, we have introduced the partition $\mathcal{T}^{\varepsilon}$
of the domain $\mathbb{R}$. We now fix the notation for a generalized
partition.
Let $\mathcal{T}^{m}=\\{T_{k}^{m}\\}_{k=-\infty}^{\infty}$ be a given
partition such that $T_{k}^{m}=[x^{m}_{k-1},x^{m}_{k}]$, where
$x^{m}_{k}>x^{m}_{k-1}$ are the nodes of the partition. We denote the size (or
the length) of the $k$’th element by
$\varepsilon^{m}_{k}:=|T_{k}^{m}|=x^{m}_{k}-x^{m}_{k-1}$. We also define the
mesh size vector $\boldsymbol{\varepsilon}^{m}$ such that
$\boldsymbol{\varepsilon}^{m}:=(\varepsilon^{m}_{k})_{k=-\infty}^{\infty}\in(\mathbb{R}^{+})^{\mathbb{Z}}$.
Given a partition $\mathcal{T}^{m}$ and a function $g\in C^{0}(R)$, we define
the $\mathcal{P}_{1}$ direct interpolation
$I_{m}:C^{0}(\mathbb{R})\rightarrow\mathcal{P}_{1}(\mathcal{T}^{m})\cap
C^{0}(\mathbb{R})$ by
$(I_{m}g)(x^{m}_{i})=g(x^{m}_{i})\quad\forall g\in C^{0}(\mathbb{R}),$ (2.9)
and $I_{m}g$ is often denoted by $g_{m}$. We also denote the vectorization of
$g\in C^{0}(\mathbb{R})$ with respect to $\mathcal{T}^{m}$ by
$\boldsymbol{g}^{m}$ such that
$g^{m}_{j}=g(x^{m}_{j}).$ (2.10)
Let $\mathcal{D}$ be a subset of $\mathbb{Z}$. For a vector
$\boldsymbol{v}\in\mathbb{R}^{\mathbb{Z}}$ and a partition $\mathcal{T}^{m}$,
we define the (semi-)norms
$\displaystyle\|\boldsymbol{v}\|_{\ell^{p}_{\boldsymbol{\varepsilon}^{m}}(\mathcal{D})}=\left\\{\begin{array}[]{l
l}\Big{(}\sum_{\ell\in\mathcal{D}}\varepsilon^{m}_{\ell}|v_{\ell}|^{p}\Big{)}^{1/p},&1\leq
p<\infty,\\\ \max_{\ell\in\mathcal{D}}|v_{\ell}|,&p=\infty.\end{array}\right.$
In particular, if $n_{m}$ is the number of the nodes of $\mathcal{T}^{m}$ that
are in $[0,1]$, we simply define
$\displaystyle\|\boldsymbol{v}\|_{\ell^{p}_{\boldsymbol{\varepsilon}^{m}}}=\left\\{\begin{array}[]{l
l}\Big{(}\sum_{\ell=1}^{n_{m}}\varepsilon^{m}_{\ell}|v_{\ell}|^{p}\Big{)}^{1/p},&1\leq
p<\infty,\\\
\max_{\ell=1,\ldots,n_{m}}|v_{\ell}|,&p=\infty.\end{array}\right.$
We now define discrete derivatives. Suppose $v\in C^{0}(\mathbb{R})$ and
$\boldsymbol{v}^{m}$ is its vectorization according to $\mathcal{T}^{m}$. We
define the first and second order discrete derivative
${\boldsymbol{v}^{m}}^{\prime}$ by
${v^{m}}^{\prime}_{k}=\frac{v^{m}_{j}-v^{m}_{j-1}}{\varepsilon^{m}_{j}},\text{
and
},{v^{m}}^{\prime\prime}_{k}=\frac{{v^{m}}^{\prime}_{j+1}-{v^{m}}^{\prime}_{j}}{\bar{\varepsilon}^{m}_{j}},$
(2.11)
where
$\bar{\varepsilon}^{m}_{j}:={\textstyle\frac{1}{2}}(\varepsilon^{h}_{j}+\varepsilon^{h}_{j+1})$.
It can be proved that for $v^{m}\in
C^{0}(\mathbb{R})\cap\mathcal{P}_{1}(\mathcal{T}^{m})$ and
$\boldsymbol{v}^{m}$ being its vectorization, we have the identity
$\|{v^{m}}^{\prime}\|_{L^{p}[0,1]}=\|{\boldsymbol{v}^{m}}^{\prime}\|_{\ell^{p}_{\boldsymbol{\varepsilon}^{m}}}.$
(2.12)
Since $\mathcal{T}^{\varepsilon}$ is special and uniform, we simply use
$\boldsymbol{\varepsilon}$ and $\varepsilon$ to denote its mesh size vector
and mesh size without superscripts and subscripts.
In addition to these, we denote the left and the right limit of an open
interval $\omega$ by $L_{\omega}$ and $R_{\omega}$, which are also used later
in our analysis.
### 2.3. QC Approximation
The QC approximation we analyze in this paper is essentially the Consistent
Atomistic/Continuum Coupling method developed in [10]. We briefly redevelop
this method in 1D so that it is easily understood and enough for us to carry
out the analysis.
We first decompose the reference lattice, which occupies $\mathbb{R}$, into an
atomistic region $\Omega_{\rm a}$, which should contain any ’defects’, and a
continuum region $\Omega_{c}$, where the solution is expected to be smooth.
Moreover, we assume $\Omega_{\rm a}$ to be a union of open intervals and
$\Omega_{c}$ to be a union of closed intervals, and $\Omega_{\rm
a}\cup\Omega_{c}=\mathbb{R}$. Since we impose periodic boundary conditions on
the displacement and notationally it is easier to assume the atomistic region
is away from the boundary of the period we analyze, we make the following
assumptions on $\Omega_{\rm a}$ and $\Omega_{c}$:
* •
$\Omega_{\rm a}$ and $\Omega_{c}$ appear periodically with exactly period of
$1$, i.e, if $x\in\Omega_{\rm a}$ then $x+1\in\Omega_{\rm a}$ and the same for
$\Omega_{c}$.
* •
$\exists\delta>2\varepsilon$ such that
$\big{(}(0,\delta)\cup(1-\delta,1)\big{)}\subset\Omega_{c}$, i.e., the
atomistic region is contained in the ’middle’ of the chain.
Note that, there is no such a restriction, that the interfaces where different
regions meet should lie on the positions of the atoms, i.e., it is not
necessary that $\Omega_{c}\cap\Omega_{\rm a}\in\varepsilon\mathbb{Z}$, which
was always assumed in previous modeling and analysis of 1D QC method.
Then, in order to reduce the number of degrees of freedom, we make the
partition $\mathcal{T}^{h}=\\{T^{h}_{k}\\}_{k=-\infty}^{\infty}$ of the domain
$\mathbb{R}$ according to the above region decomposition of $\mathbb{R}$ as
follows:
* •
$T_{k}^{h}=[x^{h}_{k},x^{h}_{k-1}]$ and
$T_{k+K}=[x^{h}_{k}+1,x^{h}_{k-1}+1]=[x^{h}_{k+K},x^{h}_{k-1+K}]$, which
implies that the partition is $K$-periodic with $|\cup_{k=1}^{K}T_{k}|=1$, and
there are $K$ elements in each $[0,1]$. We also assume that $x^{h}_{1}$ is the
left most node and $x^{h}_{K}$ is the right most node in $[0,1]$.
* •
If $\ell\varepsilon\in\Omega_{\rm a}$, then $\exists i\in\mathbb{Z}$ such that
$x_{i}=\ell\varepsilon$, i.e., every position of an atom in the atomistic
region is a node of this partition.
* •
$\partial\Omega_{c}$ is a node in this partition which means that each element
is contained in only one of the two regions.
* •
$|T_{k}^{h}|=\varepsilon^{h}_{k}\geq 2\varepsilon$ if
$T^{h}_{k}\subset\Omega_{c}$, i.e., the size of each element in the continnum
region is larger than or equal to $2\varepsilon$.
We emphasize two definitions
$\ell_{k}:=\max_{\ell}\\{\ell:\ell\varepsilon\leq x^{h}_{k}\\}\text{ and
}\theta_{k}:=\frac{x^{h}_{k}-\ell_{k}\varepsilon}{\varepsilon},$ (2.13)
which are extensively used in the analysis and significantly simply the
notation. Note that $0\leq\theta_{k}\leq 1$.
Based on this partition of the domain, the QC space of displacement and the QC
set of admissible deformation are defined by
$\mathcal{U}_{{\rm qc}}=\big{\\{}u\in\mathcal{P}_{1}(\mathcal{T}^{h})\cap
C^{0}(\mathbb{R}):u(x+1)=u(x)\text{ and }\int_{0}^{1}u(x)\,{\rm
d}x=0\big{\\}},$ (2.14)
and
$\mathcal{Y}_{{\rm qc}}=\big{\\{}y\in\mathcal{P}_{1}(\mathcal{T}^{h})\cap
C^{0}(\mathbb{R}):y(x)=Fx+u(x),u\in\mathcal{U}_{{\rm qc}}\big{\\}}.$ (2.15)
The discrete QC space of displacement and the QC set of admissible deformation
are defined by
$\mathcal{U}^{h}_{{\rm
qc}}=\big{\\{}\boldsymbol{u}^{h}\in\mathbb{R}^{\mathbb{Z}}:u^{h}_{k}=u^{h}_{k+K},\forall
k\in\mathbb{Z},\text{ and
}\sum_{k=1}^{K}\frac{1}{2}(x^{h}_{k+1}-x^{h}_{k-1})u^{h}_{k}=0\big{\\}},$
(2.16)
and
$\mathcal{Y}^{h}_{{\rm
qc}}=\big{\\{}\boldsymbol{y}^{h}\in\mathbb{R}^{\mathbb{Z}}:y^{h}_{k}=Fx_{k}+u^{h}_{k},\boldsymbol{u}\in\mathcal{U}^{h}_{{\rm
qc}}\big{\\}}.$ (2.17)
Note that unlike $\mathcal{U}^{\varepsilon}$ and $\mathcal{Y}^{\varepsilon}$,
in which every vector has the physical displacements and deformations of the
atoms as its components, $\mathcal{U}^{h}$ and $\mathcal{Y}^{h}$ only contain
vectors whose components are the values of displacements and deformations at
the nodes of $\mathcal{T}^{h}$.
The approach to couple the atomistic and continuum energy is to associate the
energy with interaction bonds. The term bond between atoms $i\in\mathbb{Z}$
and $i+r\in\mathbb{Z}$ refer to the open interval
$b=(i\varepsilon,(i+r)\varepsilon)$. In our case, since only nearest neighbour
and next nearest neighbour bonds are taken into account, $r=1,2$ only.
To develop the coupling method, we define the operator $D_{\omega}y$ for an
open interval $\omega=(L_{\omega},R_{\omega})\subset\mathbb{R}$ and $y\in
C^{0}(\mathbb{R})$ such that
$D_{\omega}y:=\frac{1}{|\omega|}\big{(}y(R_{\omega})-y(L_{\omega})\big{)}.$
(2.18)
If we take any $y\in C^{0}(\mathbb{R})\cap W^{1,\infty}(\mathbb{R})$ as a
deformation (note that this ’deformation’ might be non-physical) and a bond
$b=(i\varepsilon,(i+r_{b})\varepsilon)$, we can define the atomistic energy
contribution of bond $b$ to the stored energy to be
$a_{b}(y)=\frac{|b\cap\Omega_{\rm
a}|}{r_{b}}\phi\big{(}r_{b}D_{b\cap\Omega_{\rm a}}y\big{)},$ (2.19)
and its continuum energy contribution to the stored energy to be
$c_{b}(y)=\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x,$ (2.20)
where $\nabla_{r_{b}}y=r_{b}y^{\prime}(x)$.
Since we are only interested in the situation in $[0,1]$, which is extended
periodically to the whole domain, the set of bonds that we will consider is
$\mathcal{B}=\big{\\{}(i\varepsilon,(i+r)\varepsilon):r=1,2,i=0,1,\ldots,N-1\big{\\}}.$
(2.21)
Therefore, coupling the two energy contributions together, the stored QC
energy (per period) of a deformation $y\in C^{0}(\mathbb{R})\cap
W^{1,\infty}(\mathbb{R})$ is then given by
$\mathcal{E}_{{\rm qc}}(y)=\sum_{b\in B}\big{[}a_{b}(y)+c_{b}(y)\big{]},$
(2.22)
which was shown in [10] to be a consistent coupling method, where the
definition of consistency is as follows:
$\mathcal{E}^{\prime}_{\rm a}(Fx)[v]=\mathcal{E}^{\prime}_{{\rm
qc}}(Fx)[v]=0\quad\forall v\in C^{0}(\mathbb{R})\cap
W^{1,\infty}(\mathbb{R}).$ (2.23)
Given a dead load $f\in\mathcal{U}$ the QC approximation of the external
energy (per period) caused by $f$ is given by
$\langle f,u_{h}\rangle_{h}:=\int_{0}^{1}I_{h}(fu_{h})\,{\rm
d}x=\sum_{k=1}^{K}\frac{1}{2}(x^{h}_{k+1}-x^{h}_{k-1})f^{h}_{k}u^{h}_{k}=:\langle\boldsymbol{f}^{h},\boldsymbol{u}^{h}\rangle_{h},$
(2.24)
where $I_{h}$ is the linear nodal interpolation with respect to
$\mathcal{T}^{h}$ , and $\boldsymbol{f}^{h}$ and $\boldsymbol{u}^{h}$ are the
vectorizations of $f$ and $u_{h}$.
Thus, the total energy (per period) of a deformation
$y_{h}\in\mathcal{Y}_{{\rm qc}}$ is given by
$E_{{\rm qc}}(y_{h};F)=\mathcal{E}_{{\rm qc}}(y_{h})-\langle
f,u_{h}\rangle_{h}.$
For the same reason that $F$ is given, we write $E_{{\rm qc}}(y_{h};F)$ as
$E_{{\rm qc}}(y_{h})$. The problem we wish to solve is to find
$y_{{\rm qc}}\in{\rm argmin}E_{{\rm qc}}(\mathcal{Y}_{{\rm qc}}).$ (2.25)
## 3\. Residual Analysis
In this section, we bound the residual in a negative Sobolev norms. We equip
the space $\mathcal{U}$ with the Sobolev norm
$\|v\|_{\mathcal{U}^{1,2}}=\|v^{\prime}\|_{L^{2}[0,1]},\quad\text{ for
}v\in\mathcal{U},$
and denote it by $\mathcal{U}^{1,2}$. The norm on the dual
$\mathcal{U}^{-1,2}:=(\mathcal{U}^{1,2})^{\ast}$ is defined by
$\|T\|_{\mathcal{U}^{-1,2}}:=\sup_{\begin{subarray}{c}v\in\mathcal{U}\\\
\|v\|_{\mathcal{U}^{1,2}}=1\end{subarray}}T[v],\quad\text{ for
}T\in\mathcal{U}^{-1,2}.$
In the following sections, we formulate the problems in variational forms and
then analyze the residual.
### 3.1. Variational Formulation and Residual
Let $y_{\rm a}$ be a solution of the atomistic problem (2.8). If ${y_{\rm
a}}^{\prime}(x)>0$ on $[0,1]$, $\mathcal{E}_{\rm a}(y)$ has the variational
derivative at $y_{\rm a}$ and therefore, the first order optimality condition
for (2.8) in variational form is
$\mathcal{E}^{\prime}_{\rm a}(y_{\rm a})[v]=\langle
f,v\rangle_{\varepsilon}\qquad\forall v\in\mathcal{U},$ (3.1)
where
$\mathcal{E}^{\prime}_{\rm a}(y_{\rm
a})[v]=\varepsilon\sum_{b\in\mathcal{B}}\phi^{\prime}(r_{b}D_{b}y_{\rm
a})r_{b}D_{b}v.$ (3.2)
Let $y_{{\rm qc}}$ be a solution of the QC problem (2.25). If ${y_{{\rm
qc}}}^{\prime}(x)>0$ on $[0,1]$, then $\mathcal{E}_{{\rm qc}}(y)$ has the
variational derivative at $y_{{\rm qc}}$ and the first order optimality
condition for (2.25) in variational form is
$\mathcal{E}^{\prime}_{{\rm qc}}(y_{{\rm qc}})[v_{h}]=\langle
f,v_{h}\rangle_{h}\quad\forall v_{h}\in\mathcal{U}_{{\rm qc}},$ (3.3)
where
$\displaystyle\mathcal{E}^{\prime}_{{\rm qc}}(y_{h})[v_{h}]$
$\displaystyle=\sum_{b\in
B}\big{[}a^{\prime}_{b}(y_{h})[v_{h}]+c_{b}(y_{h})[v_{h}]\big{]}$
$\displaystyle=|b\cap\Omega_{\rm a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm
a}}y_{h})D_{b\cap\Omega_{\rm
a}}v_{h}+\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v_{h}\,{\rm
d}x\quad\forall v_{h}\in\mathcal{U}_{{\rm qc}}.$ (3.4)
In conforming finite element analysis, where the finite element solution space
is a subspace of the original solution space, the residual is defined as the
quantity we obtain by inserting the computed solution to the equation which
the real solution satisfies. However, in our case, $\mathcal{Y}_{{\rm qc}}$ is
in general not a subspace of $\mathcal{Y}_{\rm a}$, and hence the functional
$\mathcal{E}_{\rm a}(\cdot)$ is not defined on $\mathcal{Y}_{{\rm qc}}$ in
general and $\mathcal{E}_{{\rm qc}}(\cdot)$ is not defined on
$\mathcal{Y}_{\rm a}$ either. The way through which we circumvent this
difficulty is to define mappings between the solution spaces so that the
residual could be well defined. In concrete, we define
$J_{\mathcal{U}}:\mathcal{U}\rightarrow\mathcal{U}_{{\rm qc}}$ and
$J_{\mathcal{U}_{{\rm qc}}}:\mathcal{U}_{{\rm qc}}\rightarrow\mathcal{U}$ such
that
$J_{\mathcal{U}}u=I_{h}u-\frac{1}{2}\sum_{\ell=1}^{K}(x^{h}_{k+1}-x^{h}_{k-1})u(x^{h}_{k})\quad\forall
u\in\mathcal{U},$ (3.5)
and
$J_{\mathcal{U}_{{\rm
qc}}}u_{h}=I_{\varepsilon}u_{h}-\varepsilon\sum_{\ell=1}^{N}u_{h}(\ell\varepsilon)\quad\forall
u_{h}\in\mathcal{U}_{{\rm qc}}.$ (3.6)
It is easy to check that $J_{\mathcal{U}}u$ and $J_{\mathcal{U}_{{\rm
qc}}}u_{h}$ satisfy the corresponding mean zero condition of $\mathcal{U}$ and
$\mathcal{U}_{{\rm qc}}$, which implies that
$J_{\mathcal{U}}u\in\mathcal{U}_{{\rm qc}}$ and $J_{\mathcal{U}_{{\rm
qc}}}u_{h}\in\mathcal{U}$. With a slight abuse of notation, we define
$J_{\mathcal{U}}y=Fx+J_{\mathcal{U}}u=Fx+I_{h}u-\frac{1}{2}\sum_{\ell=1}^{K}(x^{h}_{k+1}-x^{h}_{k-1})u(x^{h}_{k})=I_{h}y-\frac{1}{2}\sum_{\ell=1}^{K}(x^{h}_{k+1}-x^{h}_{k-1})u(x^{h}_{k})\quad\forall
y\in\mathcal{U},$ (3.7)
and
$J_{\mathcal{U}_{{\rm qc}}}y_{h}=Fx+J_{\mathcal{U}_{{\rm
qc}}}u_{h}=Fx+I_{\varepsilon}u_{h}-\frac{1}{2}\varepsilon\sum_{\ell=1}^{N}u_{h}(\ell\varepsilon)=I_{\varepsilon}y-\frac{1}{2}\varepsilon\sum_{\ell=1}^{N}u_{h}(\ell\varepsilon)\quad\forall
y_{h}\in\mathcal{U}_{{\rm qc}}.$ (3.8)
We then define the residual (at the solution $y_{{\rm qc}}$) to be
$\displaystyle R[v]$ $\displaystyle=E^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{{\rm qc}})[v]$ $\displaystyle=E^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{{\rm qc}})[v]-E^{\prime}_{{\rm qc}}(y_{{\rm qc}})[v]$
$\displaystyle=\big{[}\mathcal{E}^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{{\rm qc}})[v]-\langle
f,v\rangle_{\varepsilon}\big{]}-\big{[}\mathcal{E}^{\prime}_{{\rm qc}}(y_{{\rm
qc}})[J_{\mathcal{U}}v]-\langle f,J_{\mathcal{U}}v\rangle_{h}\big{]}$
$\displaystyle=\big{[}\mathcal{E}^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{{\rm qc}})[v]-\mathcal{E}^{\prime}_{{\rm qc}}(y_{{\rm
qc}})[J_{\mathcal{U}}v]\big{]}+\big{[}\langle
f,J_{\mathcal{U}}v\rangle_{h}-\langle f,v\rangle_{\varepsilon}\big{]}.$ (3.9)
and understand $R$ as a functional in $\mathcal{U}^{-1,2}$. By this
formulation, we essentially split the residual into two parts: the first part
is the residual of the stored energy and the second part is the residual of
the external force. We will bound these two parts in the following sections.
### 3.2. Estimate of the Residual of the Stored Energy
In this section, we analyze the first part of (3.1), which is the residual of
the stored energy.
Before we give the theorem, we make several definitions that simplify our
notation.
First, we define the set $\mathcal{K}_{c}$ to be
$\mathcal{K}_{c}:=\big{\\{}k:k\in\\{1,\ldots,K\\}\text{ such that
}T_{k}\cap[0,1]\neq\emptyset\text{ but }T_{k}\cap(1,+\infty)=\emptyset\text{
and }T_{k}\subset\Omega_{c}\big{\\}},$ (3.10)
which is essentailly the set of indices of the elements in the continuum
region in $[0,1]$.
Second, suppose the atomistic region consists of $M$ disjoint subregions in
$[0,1]$, i.e., $\Omega_{\rm a}\cap[0,1]=\cup_{i=1}^{M}\Omega^{i}_{\rm a}$
among which $\Omega^{i}_{\rm a}\cap\Omega^{j}_{\rm a}=\emptyset$ if $i\neq j$,
we define the nodes lie on the atomistic-continuum interface of the atomistic
regions be $x^{h}_{La_{i}}$, $i=1,\ldots,M$ and those lie on the right
interface be $x^{h}_{Rc_{i}}$, $i=1,\ldots,M$.
Third, we define $\mathcal{K}^{\prime}_{c}\subset\mathcal{K}_{c}$ to be the
set of indices of the elements in the continuum region but not adjacent to an
atomistic region, i.e., $\forall k\in\mathcal{K}^{\prime}_{c}$, $k\neq
L_{a_{i}}$ and $k-1\neq R_{a_{i}}$, $\forall i\in\\{1,2,\ldots,M\\}$.
Using these definitions, we have the following theorem.
Theorem 1. For $y_{h}\in\mathcal{Y}$ with $y_{h}^{\prime}(x)>0$, we have
$\big{\|}\mathcal{E}^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})[\cdot]-\mathcal{E}^{\prime}_{{\rm
qc}}(y_{h})[J_{\mathcal{U}}\cdot]\big{\|}_{\mathcal{U}^{-1,2}}\leq\big{\\{}\sum_{k\in\mathcal{K}_{c}}{\eta_{k}^{e}}^{2}\big{\\}}^{\frac{1}{2}}=:\mathscr{E}_{\rm
store}(y_{h}),$ (3.11)
where
$\eta_{k}^{e}=\big{(}\frac{1}{2}\sum_{j=0}^{2}[[\phi^{\prime}]]_{\ell_{k-1}+j}^{2}+\frac{1}{2}\sum_{j=0}^{2}[[\phi^{\prime}]]_{\ell_{k}+j}^{2}\big{)}^{\frac{1}{2}},$
(3.12)
if $k\in\mathcal{K}^{\prime}_{c}$,
$\eta_{k}^{e}=\big{(}\varepsilon\sum_{j=0}^{2}[[\phi^{\prime}]]_{\ell_{La_{i}}+j}^{2}+\frac{1}{2}\varepsilon\sum_{j=0}^{2}[[\phi^{\prime}]]_{\ell_{k-1}+j}^{2}\big{)}^{\frac{1}{2}},$
(3.13)
if $k=L_{a_{i}}$ for some $i\in\\{1,2,\ldots,M\\}$, i.e., $T_{k}$ is adjacent
to and to the left of an atomistic region, and
$\eta_{k}^{e}=\big{(}\varepsilon\sum_{j=0}^{2}[[\phi^{\prime}]]_{\ell_{Ra_{i}}+j}^{2}+\frac{1}{2}\varepsilon\sum_{j=0}^{2}[[\phi^{\prime}]]_{\ell_{k}+j}^{2}\big{)}^{\frac{1}{2}},$
(3.14)
if $k-1=R_{a_{i}}$ for some $i\in\\{1,2,\ldots,M\\}$, i.e., $T_{k}$ is
adjacent to and to the right of an atomistic region.
$[[\phi^{\prime}]]_{\ell}$’s will be defined in the proof.
###### Proof.
By (3.2) and (3.1), we have
$\displaystyle\mathcal{E}^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})[v]-\mathcal{E}^{\prime}_{{\rm qc}}(y_{h})[J_{\mathcal{U}}v]=$
$\displaystyle\varepsilon\sum_{b\in\mathcal{B}}\phi^{\prime}(r_{b}D_{b}J_{\mathcal{U}_{{\rm
qc}}}y_{h})r_{b}D_{b}v-\sum_{b\in\mathcal{B}}|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}J_{\mathcal{U}}v$
$\displaystyle-\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}J_{\mathcal{U}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\sum_{b\in\mathcal{B}}\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-\sum_{b\in\mathcal{B}}|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}I_{h}v$
$\displaystyle-\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}I_{h}v\,{\rm
d}x,$ (3.15)
since $D_{b}J_{\mathcal{U}_{{\rm
qc}}}y_{h}=D_{b}I_{\varepsilon}y_{h}=D_{b}y_{h}$,
$D_{\omega}J_{\mathcal{U}}v=D_{\omega}I_{h}v$ for any $\omega$ being an open
interval, and $(J_{\mathcal{U}}v)^{\prime}=(I_{h}v)^{\prime}$, which can be
easily verified by noting that $J_{\mathcal{U}_{{\rm qc}}}y_{h}$ and
$J_{\mathcal{U}}v$ are $I_{\varepsilon}y_{h}$ and $I_{h}v$ shifted by some
constants.
To make further analysis of (3.2), we subtract and add the same terms
$\sum_{b\in\mathcal{B}}|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v\text{ and
}\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$
to get
$\displaystyle\mathcal{E}^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})[v]-\mathcal{E}^{\prime}_{{\rm qc}}(y_{h})[J_{\mathcal{U}}v]=$
$\displaystyle\sum_{b\in\mathcal{B}}\bigg{\\{}\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm a}}v$
$\displaystyle-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x\bigg{\\}}$
$\displaystyle-\bigg{\\{}\sum_{b\in\mathcal{B}}|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})\big{[}D_{b\cap\Omega_{\rm
a}}I_{h}v\,{\rm d}x-D_{b\cap\Omega_{\rm a}}v\big{]}\bigg{\\}}$
$\displaystyle-\bigg{\\{}\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\big{[}\nabla_{r_{b}}I_{h}v-\nabla_{r_{b}}v\big{]}\,{\rm
d}x\bigg{\\}}.$ (3.16)
We first analyze the second and third groups, which turn out to be $0$ as we
will see immediately.
For the second group, we have,
$\displaystyle\quad\sum_{b\in\mathcal{B}}|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})\big{[}D_{b\cap\Omega_{\rm
a}}I_{h}v\,{\rm d}x-D_{b\cap\Omega_{\rm a}}v\big{]}$
$\displaystyle=\sum_{b\in\mathcal{B}}|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm
a}}y_{h})\bigg{[}\frac{I_{h}v(R_{b\cap\Omega_{\rm
a}})-I_{h}v(L_{b\cap\Omega_{\rm a}})}{R_{b\cap\Omega_{\rm
a}}-L_{b\cap\Omega_{\rm a}}}-\frac{v(R_{b\cap\Omega_{\rm
a}})-v(L_{b\cap\Omega_{\rm a}})}{R_{b\cap\Omega_{\rm a}}-L_{b\cap\Omega_{\rm
a}}}\bigg{]}.$ (3.17)
We define the above to be $0$ if $b\cap\Omega_{\rm a}=\emptyset$. If
$b\cap\Omega_{\rm a}\neq\emptyset$, since both $R_{b\cap\Omega_{\rm a}}$ and
$L_{b\cap\Omega_{c}}$ are either at atomistic postions in $\Omega_{\rm a}$ or
on $\partial\Omega_{c}$, they must be nodes in $\mathcal{T}^{h}$. Therefore,
by the definition of $I_{h}v$, the following holds
$I_{h}v(L_{b\cap\Omega_{\rm a}})=v(L_{b\cap\Omega_{\rm a}})\text{ and
}I_{h}v(R_{b\cap\Omega_{\rm a}})=v(R_{b\cap\Omega_{\rm a}}),$
which implies that (3.2) is $0$.
For the third group, upon defining $\chi_{\mathcal{S}}$ to be the
characteristic function of a set $\mathcal{S}$, we can rewrite it as
$\displaystyle\quad\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\big{[}\nabla_{r_{b}}I_{h}v-\nabla_{r_{b}}v\big{]}\,{\rm
d}x$
$\displaystyle=\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\int_{\Omega_{c}}\chi_{b}\phi^{\prime}(\nabla_{r_{b}}y_{h})\big{[}\nabla_{r_{b}}I_{h}v-\nabla_{r_{b}}v\big{]}\,{\rm
d}x$
$\displaystyle=\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\sum_{k\in\mathcal{K}_{c}}\int_{T_{k}}\chi_{b}\phi^{\prime}(\nabla_{r_{b}}y_{h})\big{[}\nabla_{r_{b}}I_{h}v-\nabla_{r_{b}}v\big{]}\,{\rm
d}x$
$\displaystyle=\sum_{r=1}^{2}\sum_{b\in\mathcal{B},r_{b}=r}\sum_{k\in\mathcal{K}_{c}}\frac{1}{r_{b}}\int_{T_{k}}\chi_{b}\phi^{\prime}(\nabla_{r_{b}}y_{h})\big{[}\nabla_{r_{b}}I_{h}v-\nabla_{r_{b}}v\big{]}\,{\rm
d}x$
$\displaystyle=\sum_{r=1}^{2}\sum_{k\in\mathcal{K}_{c}}\sum_{b\in\mathcal{B},r_{b}=r}\frac{1}{r_{b}}\int_{T_{k}}\chi_{b}\phi^{\prime}(\nabla_{r}y_{h})\big{[}\nabla_{r}I_{h}v-\nabla_{r}v\big{]}\,{\rm
d}x$
$\displaystyle=\sum_{r=1}^{2}\sum_{k\in\mathcal{K}_{c}}\phi^{\prime}(\nabla_{r}y_{h}|_{T_{k}})\int_{T_{k}}\big{[}\sum_{b\in\mathcal{B},r_{b}=r}\frac{1}{r_{b}}\chi_{b}\big{]}\big{[}\nabla_{r}I_{h}v-\nabla_{r}v\big{]},$
(3.18)
since $\nabla_{r}y_{h}|_{T_{k}}$ is a constant on each element. By the 1D bond
density lemma[10, Lemma 3.4],
$\sum_{b\in\mathcal{B},r_{b}=r}\frac{1}{r}\chi_{b}(x)=_{a.e.}1,$
we have
$\displaystyle\sum_{r=1}^{2}\sum_{k\in\mathcal{K}_{c}}\phi^{\prime}(\nabla_{r}y_{h}|_{T_{k}})\int_{T_{k}}\big{[}\sum_{b\in\mathcal{B},r_{b}=r}\frac{1}{r_{b}}\chi_{b}\big{]}\big{[}\nabla_{r}I_{h}v-\nabla_{r}v\big{]}$
$\displaystyle=$
$\displaystyle\sum_{r=1}^{2}\sum_{k\in\mathcal{K}_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h}|_{T_{k}})\bigg{[}r\big{(}I_{h}v(x_{k})-I_{h}v(x_{k-1})\big{)}-r\big{(}v(x_{k})-v(x_{k-1})\big{)}\bigg{]}.$
(3.19)
Again by the definition of $I_{h}v$,
$I_{h}v(x^{h}_{k})=v(x^{h}_{k})\text{ and
}I_{h}v(x^{h}_{k-1})=v(x^{h}_{k-1}),$
and thus (3.19) is $0$.
Now we turn to the analysis of the first group and analyze
$\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ (3.20)
for each interaction bond $b$.
If $b\subset\Omega_{\rm a}$, we have $|b\cap\Omega_{c}|=r_{b}\varepsilon$,
$|b\cap\Omega_{\rm a}|=0$ and the equivalence of the operators
$D_{b}=D_{b\cap\Omega_{\rm a}}$. We know that (3.20) is $0$ by substituting
these equivalences.
If $b\subset\Omega_{c}\cap T_{k}$ for some $k\in\mathcal{K}_{c}$, then
$|b\cap\Omega_{c}|=r_{b}\varepsilon$ and $|b\cap\Omega_{\rm a}|=0$. We also
note that $\nabla_{r_{b}}y_{h}(x)=r_{b}D_{b}y_{h}$, as $y_{h}$ is affine on
$T_{k}$, and $\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\nabla_{r_{b}}v=\varepsilon
r_{b}D_{b}v$. Using these equivalences, we know that (3.20) is again $0$.
Therefore, we only need to analyze the bonds crossing the atomistic-continuum
interface or the boundaries of two adjacent elements in $\Omega_{c}$. Because
of its tediousness, we leave the detailed analysis to the Appendix but just
present the result here. Employing the notation often adopted by a posteriori
error analysis for elliptic equations, we have the following result
$\displaystyle\quad\varepsilon\sum_{b\in\mathcal{B}}\phi^{\prime}(r_{b}D_{b}I_{\varepsilon}y_{h})r_{b}D_{b}v-\sum_{b\in\mathcal{B}}|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v+\sum_{b\in\mathcal{B}}\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$
$\displaystyle=\sum_{i=1}^{M}\varepsilon\bigg{\\{}[[\phi^{\prime}]]_{\ell_{La_{i}}}v^{\prime}_{\ell_{La_{i}}}+[[\phi^{\prime}]]_{\ell_{La_{i}}+1}v^{\prime}_{\ell_{La_{i}}+1}+[[\phi^{\prime}]]_{\ell_{La_{i}}+2}v^{\prime}_{\ell_{La_{i}}+2}\bigg{\\}}$
$\displaystyle\quad+\sum_{i=1}^{M}\varepsilon\bigg{\\{}[[\phi^{\prime}]]_{\ell_{Ra_{i}}}v^{\prime}_{\ell_{Ra_{i}}}+[[\phi^{\prime}]]_{\ell_{Ra_{i}}+1}v^{\prime}_{\ell_{Ra_{i}}+1}+[[\phi^{\prime}]]_{\ell_{Ra_{i}}+2}v^{\prime}_{\ell_{Ra_{i}}+2}\bigg{\\}}$
$\displaystyle\quad+\sum_{k\in\mathcal{K}^{\prime}_{c}}\varepsilon\bigg{\\{}[[\phi^{\prime}]]_{\ell_{k}}v^{\prime}_{\ell_{k}}+[[\phi^{\prime}]]_{\ell_{k}+1}v^{\prime}_{\ell_{k}+1}+[[\phi^{\prime}]]_{\ell_{k}+2}v^{\prime}_{\ell_{k}+2}\bigg{\\}},$
(3.21)
where for $k=La_{i}$,
$\displaystyle[[\phi^{\prime}]]_{\ell}$
$\displaystyle=\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}}\big{)}-\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k}}\big{)},$
(3.22) $\displaystyle[[\phi^{\prime}]]_{\ell+1}$
$\displaystyle=\bigg{[}\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}-(1-\theta_{k})\phi^{\prime}\big{(}y_{h}^{\prime}|_{T_{k+1}}\big{)}-\theta_{k}\phi^{\prime}\big{(}y_{h}^{\prime}|_{T_{k}}\big{)}\bigg{]}$
$\displaystyle\quad+\bigg{[}(1-\theta_{k})\phi^{\prime}\big{(}\frac{2}{2-\theta_{k}}y_{h}^{\prime}|_{T_{k+2}}+\frac{2(1-\theta_{k})}{2-\theta_{k}}y_{h}^{\prime}|_{T_{k}+1}\big{)}$
$\displaystyle\quad\quad+\theta_{k}\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k}}\big{)}-\phi^{\prime}\big{(}y_{h}^{\prime}|_{T_{k+2}}+(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}\bigg{]}$
$\displaystyle\quad+\bigg{[}(1-\theta_{k})\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k+1}}\big{)}+\theta_{k}\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k}}\big{)}-\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}}\big{)}\bigg{]}$
(3.23) $\displaystyle[[\phi^{\prime}]]_{\ell+2}$
$\displaystyle=\phi^{\prime}(\frac{2(1-\theta_{k})}{2-\theta_{k}}y_{h}^{\prime}|_{T_{k+1}}+\frac{2}{2-\theta_{k}}y_{h}^{\prime}|_{T_{k+2}}\big{)}-\phi^{\prime}\big{(}y_{h}^{\prime}|_{T_{k+2}}+(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)},$
(3.24)
for $k=Ra_{i}$
$\displaystyle[[\phi^{\prime}]]_{\ell}$
$\displaystyle=\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}+y_{h}^{\prime}|_{T_{k-1}}\big{)}-\phi^{\prime}\bigg{(}\frac{2\theta_{k}}{(1+\theta_{k})}y_{h}^{\prime}|_{T_{k}}+\frac{2}{1+\theta_{k}}y_{h}^{\prime}|_{T_{k-1}}\bigg{)},$
(3.25) $\displaystyle[[\phi^{\prime}]]_{\ell+1}$
$\displaystyle=\bigg{[}\theta_{k}\phi^{\prime}\big{(}y_{h}^{\prime}|_{T_{k}}\big{)}+(1-\theta_{k})\phi^{\prime}\big{(}y_{h}^{\prime}|_{T_{k+1}}\big{)}-\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}\bigg{]}$
$\displaystyle\quad+\bigg{[}\theta_{k}\phi^{\prime}\big{(}\frac{2\theta_{k}}{1+\theta_{k}}y_{h}^{\prime}|_{T_{k}}+\frac{2}{1+\theta_{k}}y_{h}^{\prime}|_{T_{k-1}}\big{)}+(1-\theta_{k})\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k+1}}\big{)}$
$\displaystyle\quad-\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}+y_{h}^{\prime}|_{T_{k-1}}\big{)}\bigg{]}$
$\displaystyle\quad+\bigg{[}(1-\theta_{k})\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k+1}}\big{)}+\theta_{k}\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k}}\big{)}-\phi^{\prime}\big{(}(2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}\bigg{]}$
(3.26) $\displaystyle[[\phi^{\prime}]]_{\ell+2}$
$\displaystyle=\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k+1}}\big{)}-\phi^{\prime}\big{(}(2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)},$
(3.27)
and for $k\in\mathcal{K}^{\prime}_{c}$
$\displaystyle[[\phi^{\prime}]]_{\ell}$
$\displaystyle=\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k}}\big{)}-\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y^{\prime}_{h}|_{T_{k}}\big{)},$
(3.28) $\displaystyle[[\phi^{\prime}]]_{\ell+1}$
$\displaystyle=\bigg{[}(1-\theta_{k})\phi^{\prime}\big{(}y^{\prime}_{h}|_{T_{k+1}}\big{)}+\theta_{k}\phi^{\prime}\big{(}y^{\prime}_{h}|_{T_{k}}\big{)}-\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}\bigg{]}$
$\displaystyle+\bigg{[}2(1-\theta_{k})\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k+1}}\big{)}+2\theta_{k}\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k}}\big{)}$
$\displaystyle-\phi^{\prime}\big{(}(2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}-\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}}\big{)}\bigg{]},$
(3.29) $\displaystyle[[\phi^{\prime}]]_{\ell+2}$
$\displaystyle=\phi^{\prime}\big{(}2y_{h}^{\prime}|_{T_{k+1}}\big{)}-\phi^{\prime}\big{(}(2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}.$
(3.30)
Distributing the contribution of (3.2) to each element and applying Cauchy-
Schwarz inequality, we obtain the estimate stated in the theorem. ∎
### 3.3. Estimate of the Residual of the External Force
We now turn to the estimate of the residual of the external energy. Upon
defining $J_{\mathcal{U}}v:=v_{h}$, the residual of the external force is
given by
$\langle f,v_{h}\rangle_{h}-\langle f,v\rangle_{\varepsilon},$ (3.31)
where $f,v\in\mathcal{U}$.
To further analyze (3.31), we introduce a new partition
$\mathcal{T}^{r}=\\{T^{r}_{j}\\}_{j=-\infty}^{+\infty}$ of the domain
$\mathbb{R}$, such that all the nodes in partition $\mathcal{T}^{\varepsilon}$
and partition $\mathcal{T}^{h}$ are included in this partition. The indexing
of the nodes in $\mathcal{T}^{r}$ follow the rule that the node $x^{h}_{k}$ in
$\mathcal{T}^{h}$ is labeled as $x_{j_{k}}^{r}$ in $\mathcal{T}^{r}$. We also
assume there are $n$ nodes in $\mathcal{T}^{r}$ in $[0,1]$, i.e.,
$n=\big{|}\\{\varepsilon,2\varepsilon,\ldots,N\varepsilon\\}\cup\\{x_{1},x_{2},\ldots,x_{K}\\}\big{|},$
where $|\mathcal{A}|$ denote the cardinality of a finite set $\mathcal{A}$.
The inner product associated with $\mathcal{T}^{r}$ partition is then defined
by
$\langle f,g\rangle_{r}:=\int_{0}^{1}I_{r}(fg)\,{\rm
d}x=\sum_{j=1}^{n}\frac{1}{2}(x^{r}_{j+1}-x^{r}_{j-1})f^{r}_{j}g^{r}_{j}=:\langle\boldsymbol{f}^{r},\boldsymbol{g}^{r}\rangle_{r}\quad\forall
f,g\in C^{0}(\mathbb{R}),$ (3.32)
where $I_{r}$ is the linear nodal interpolation operator with respect to
$\mathcal{T}^{r}$, and $\boldsymbol{f}^{r}$ and $\boldsymbol{g}^{r}$ are the
vectorizations of $f$ and $g$ with respect to $\mathcal{T}^{r}$.
Now we decompose the residual of the external force into three parts by adding
and subtracting the same terms,
$\displaystyle\langle f,v_{h}\rangle_{h}-\langle
f,v\rangle_{\varepsilon}=\big{[}\langle f,v\rangle_{r}-\langle
f,v\rangle_{\varepsilon}\big{]}+\big{[}\langle f,v_{h}\rangle_{r}-\langle
f,v\rangle_{r}\big{]}+\big{[}\langle f,v_{h}\rangle_{h}-\langle
f,v_{h}\rangle_{r}\big{]}.$ (3.33)
The following three lemma are derived to give the estimates of the three
parts.
Lemma 2. Let
$\boldsymbol{f},\boldsymbol{v},\boldsymbol{f}^{r},\boldsymbol{v}^{r}$ be the
vectorizations of $f,v\in C^{0}(\mathbb{R})$ according to
$\mathcal{T}^{\varepsilon}$ and $\mathcal{T}^{r}$. Then the following
inequality holds
$\big{|}\langle f,v\rangle_{r}-\langle
f,v\rangle_{\varepsilon}\big{|}=\big{|}\langle\boldsymbol{f}^{r},\boldsymbol{v}^{r}\rangle_{r}-\langle\boldsymbol{f},\boldsymbol{v}\rangle_{\varepsilon}\big{|}\leq\frac{1}{8}\varepsilon^{2}\|\boldsymbol{f}^{\prime}\|_{\ell_{\varepsilon}^{2}(\mathcal{K}_{U})}\|\boldsymbol{v}^{\prime}\|_{\ell_{\varepsilon}^{2}},$
(3.34)
where
$\mathcal{K}_{U}=\big{\\{}k\in\\{1,\ldots,K\\}:x_{k}\neq\ell_{k}\varepsilon\big{\\}}$,
in other words, $\mathcal{K}_{U}$ is the set of indices of the nodes
$x^{h}_{k}$ in $\mathcal{T}^{h}$ such that $x^{h}_{k}$ does not coincide with
any of the nodes in $\mathcal{T}^{\varepsilon}$.
###### Proof.
We first write out the two inner products and eliminate the terms that are the
same
$\displaystyle\langle\boldsymbol{f}^{r},\boldsymbol{v}^{r}\rangle_{r}-\langle\boldsymbol{f},\boldsymbol{v}\rangle_{\varepsilon}$
$\displaystyle=\sum_{\ell=1}^{n}\varepsilon^{r}_{\ell}\frac{1}{2}(f^{r}_{\ell}v^{r}_{\ell}+f^{r}_{\ell+1}v^{r}_{\ell+1})-\sum_{\ell=1}^{N}\varepsilon\frac{1}{2}(f_{\ell}v_{\ell}+f_{\ell+1}v_{\ell+1})$
$\displaystyle=\sum_{k\in\mathcal{K}_{U}}\big{(}\varepsilon_{j_{k}}\frac{1}{2}(f^{r}_{j_{k}-1}v^{r}_{j_{k}-1}+f^{r}_{j_{k}}v^{r}_{j_{k}})+\varepsilon_{j_{k}+1}\frac{1}{2}(f^{r}_{j_{k}}v^{r}_{j_{k}}+f^{r}_{j_{k}+1}v^{r}_{j_{k}+1})\big{)}$
$\displaystyle\quad-\sum_{k\in\mathcal{K}_{U}}\varepsilon\big{(}f_{\ell_{k}}v_{\ell_{k}}+f_{\ell_{k}+1}v_{\ell_{k}+1}\big{)},$
(3.35)
as
$\varepsilon_{j_{k}+2}=\varepsilon_{j_{k}+3}=\ldots=\varepsilon_{j_{k+1}-1}=\varepsilon$
and $f_{\ell_{k}+i}v_{\ell_{k}+i}=f^{r}_{j_{k}+i}v^{r}_{j_{k}+i},\quad
i=1,2,\ldots,\ell_{k+1}-\ell_{k}$, if $\ell_{k}\varepsilon\neq x_{k}$ and
$\ell_{k+1}\varepsilon\neq x_{k+1}$.
For $k$ such that $\ell_{k}\varepsilon\neq x_{k}$, by the definition of
$\boldsymbol{f}$, $\boldsymbol{v}$, $\boldsymbol{f}^{r}$ and
$\boldsymbol{v}^{r}$, we have $f_{\ell_{k}}=f^{r}_{j_{k}-1}$,
$v_{\ell_{k}}=v^{r}_{j_{k}-1}$, $f_{\ell_{k}+1}=f^{r}_{j_{k}+1}$ and
$v_{\ell_{k}+1}=v^{r}_{j_{k}+1}$. We also have
$f^{r}_{j_{k}}=(1-\theta_{k})f_{\ell_{k}}+\theta_{k}f_{\ell_{k}+1}$ and
$v^{r}_{j_{k}}=(1-\theta_{k})v_{\ell_{k}}+\theta_{k}v_{\ell_{k}+1}$. Inserting
these equalities, (3.35) can be estimated as
$\displaystyle\big{|}\langle\boldsymbol{f}^{r},\boldsymbol{v}^{r}\rangle_{r}-\langle\boldsymbol{f},\boldsymbol{v}\rangle_{\varepsilon}\big{|}$
$\displaystyle=$
$\displaystyle\bigg{|}\sum_{k\in\mathcal{K}_{U}}\bigg{\\{}\frac{1}{2}\theta_{k}\varepsilon
f_{\ell_{k}}v_{\ell_{k}}+\frac{1}{2}\theta_{k}\varepsilon\big{[}(1-\theta_{k})f_{\ell_{k}}+\theta_{k}f_{\ell_{k+1}}\big{]}\big{[}(1-\theta_{k})v_{\ell_{k}}+\theta_{k}v_{\ell_{k}+1}\big{]}$
$\displaystyle+\frac{1}{2}(1-\theta_{k})\varepsilon
f_{\ell_{k}+1}v_{\ell_{k}+1}+\frac{1}{2}(1-\theta_{k})\varepsilon\big{[}(1-\theta_{k})f_{\ell_{k}}+\theta_{k}f_{\ell_{k}+1}\big{]}\big{[}(1-\theta_{k})v_{\ell_{k}}+\theta_{k}v_{\ell_{k}+1}\big{]}$
$\displaystyle-\frac{1}{2}\varepsilon
f_{\ell_{k}}v_{\ell_{k}}-\frac{1}{2}\varepsilon
f_{\ell_{k}+1}v_{\ell_{k}+1}\bigg{\\}}\bigg{|}$ $\displaystyle=$
$\displaystyle\big{|}\sum_{k\in\mathcal{K}_{U}}\frac{1}{2}\varepsilon\bigg{\\{}\big{[}\theta_{k}(\theta_{k}-1)(f_{\ell_{k}+1}-f_{\ell_{k}})v_{\ell_{k}+1}\big{]}-\big{[}\theta_{k}(\theta_{k}-1)(f_{\ell_{k}+1}-f_{\ell_{k}})v_{\ell_{k}}\big{]}\bigg{\\}}\big{|}$
$\displaystyle=$
$\displaystyle\sum_{k\in\mathcal{K}_{U}}\frac{1}{2}\varepsilon^{3}\big{|}\theta_{k}(1-\theta_{k})f^{\prime}_{\ell_{k}+1}v^{\prime}_{\ell_{k}+1}\big{|}$
$\displaystyle\leq\frac{1}{8}\varepsilon^{2}\bigg{(}\sum_{k\in\mathcal{K}_{U}}\varepsilon|f^{\prime}_{\ell_{k}+1}|^{2}\bigg{)}^{\frac{1}{2}}\bigg{(}\sum_{k\in\mathcal{K}_{U}}\varepsilon|v^{\prime}_{\ell_{k}+1}|^{2}\bigg{)}^{\frac{1}{2}}\leq\frac{1}{8}\varepsilon^{2}\|\boldsymbol{f}^{\prime}\|_{\ell_{\varepsilon}^{2}(\mathcal{K}_{U})}\|\boldsymbol{v}^{\prime}\|_{\ell_{\varepsilon}^{2}},$
(3.36)
which concludes the proof. ∎
Remark 1. If $\mathcal{K}=\emptyset$, i.e., every node in $T^{h}$ is also in
$T^{\varepsilon}$, then this part of the residual is $0$. ∎
Lemma 3. Let $f,v\in
C^{0}(\mathbb{R})\cap\mathcal{P}_{1}(\mathcal{T}^{\varepsilon})$ and
$v_{h}=I_{h}v\in C^{0}(\mathbb{R})\cap\mathcal{P}_{1}(\mathcal{T}^{h})$ be the
$\mathcal{P}_{1}$ interpolation of $v$ according to $\mathcal{T}^{h}$
partition. Let $\boldsymbol{f}^{r},\boldsymbol{v}^{r}$ and
$\boldsymbol{v}_{h}^{r}$ be the vectorizations of $f,v,v_{h}$ respectively
according to $\mathcal{T}^{r}$, and $\mathcal{K}_{c}$ is defined in (3.10).
Then we have the following estimate
$\langle f,v_{h}\rangle_{r}-\langle
f,v\rangle_{r}=\langle\boldsymbol{f}^{r},\boldsymbol{v}_{h}^{r}\rangle_{r}-\langle\boldsymbol{f}^{r},\boldsymbol{v}^{r}\rangle_{r}\leq\bigg{[}\sum_{k=\mathcal{K}_{c}}\tilde{h}_{k}^{2}\|\boldsymbol{f}^{r}\|^{2}_{\ell^{2}_{\bar{\varepsilon}^{r}}(\mathcal{D}^{2}_{k})}\bigg{]}^{\frac{1}{2}}\|\boldsymbol{v}^{\prime}\|_{\ell^{2}_{\varepsilon}},$
(3.37)
$\bar{\varepsilon}^{r}_{j}=\frac{1}{2}(\varepsilon^{r}_{j}+\varepsilon^{r}_{j+1})$,
$\tilde{h}_{k}=\frac{1}{2}(j_{k+1}-j_{k})\varepsilon$ and
$\mathcal{D}^{2}_{k}=\\{j_{k}+1,\ldots,j_{k+1}-1\\}$.
###### Proof.
Using the fact that $(v_{h}^{r})_{j_{k}}=v^{r}_{j_{k}}$ and by Cauchy-Schwarz
inequality, we have
$\displaystyle\langle\boldsymbol{f}^{r},\boldsymbol{v}_{h}^{r}\rangle_{r}-\langle\boldsymbol{f}^{r},\boldsymbol{v}^{r}\rangle_{r}=$
$\displaystyle\sum_{j=1}^{n}\frac{1}{2}(\varepsilon^{r}_{j}+\varepsilon^{r}_{j+1})(f^{r}_{j}(v_{h}^{r})_{j}-f^{r}_{j}v^{r}_{j})$
$\displaystyle=$
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\sum_{j=j_{k-1}}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}f^{r}_{j}\big{[}(v_{h}^{r})_{j}-v^{r}_{j}\big{]}$
$\displaystyle\leq$
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\bigg{[}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}(f^{r}_{j})^{2}\bigg{]}^{\frac{1}{2}}\bigg{\\{}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}\big{[}(v_{h}^{r})_{j}-v^{r}_{j}\big{]}^{2}\bigg{\\}}^{\frac{1}{2}},$
(3.38)
where
$\bar{\varepsilon}^{r}_{j}=\frac{1}{2}(\varepsilon^{r}_{j}+\varepsilon^{r}_{j+1})$.
Upon defining $\boldsymbol{g}$ such that $g_{j}=(v_{h}^{r})_{j}-v^{r}_{j}$
(note $g_{j_{k}}=g_{j_{k+1}}=0$) and by Lemma C in Appendix C (Discrete
Friedrich’s Inequality) and Rieze-Thorin Theorem,
$\displaystyle\bigg{\\{}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}\big{[}(v_{h}^{r})_{j}-v^{r}_{j}\big{]}^{2}\bigg{\\}}^{\frac{1}{2}}=$
$\displaystyle\bigg{\\{}\sum_{j=j_{k-1}}^{j_{k}}\bar{\varepsilon}^{r}_{j}g_{j}^{2}\bigg{\\}}^{\frac{1}{2}}\leq\frac{1}{2}(j_{k}-j_{k-1})\varepsilon\bigg{\\{}\sum_{j=j_{k-1}+1}^{j_{k}}\varepsilon^{r}_{j}{g^{\prime}_{j}}^{2}\bigg{\\}}^{\frac{1}{2}},$
(3.39)
where
$g_{j}^{\prime}={\textstyle\frac{g_{j}-g_{j-1}}{\varepsilon^{r}_{j}}}=(v^{r})^{\prime}_{j}-(v_{h}^{r})^{\prime}_{j}$.
$\varepsilon$ appears in the last inequality since
$\max_{j}\bar{\varepsilon}^{r}_{j}\leq\varepsilon$. Since $v_{h}^{r}$ and
$v^{r}$ are both piecewise linear on $\mathcal{T}^{r}$, we have
$(v_{h}^{r})^{\prime}_{j}-(v^{r})^{\prime}_{j}=(v^{\prime}-v_{h}^{\prime})(x)\
\forall x\in(x^{r}_{j-1},x^{r}_{j})$, and as a result,
$\bigg{\\{}\sum_{j=j_{k-1}+1}^{j_{k}}\varepsilon^{r}_{j}\big{[}(v_{h}^{r})^{\prime}_{j}-(v^{r})^{\prime}_{j}\big{]}^{2}\bigg{\\}}^{\frac{1}{2}}=\int_{x^{r}_{j_{k-1}}}^{x^{r}_{j_{k}}}|(v^{\prime}-v_{h}^{\prime})(x)|^{2}\,{\rm
d}x=\|v^{\prime}-v_{h}^{\prime}\|_{L^{2}[x^{r}_{j_{k-1}},x^{r}_{j_{k}}]}.$
By Lemma B in Appendix B,
$\|v^{\prime}-v_{h}^{\prime}\|_{L^{2}[x^{r}_{j_{k-1}},x^{r}_{j_{k}}]}\leq\|v^{\prime}\|_{L^{2}[x^{r}_{j_{k-1}},x^{r}_{j_{k}}]}.$
Put all the results above together and apply Cauchy-Schwarz inequality, we
obtain
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\bigg{[}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}(f^{r}_{j})^{2}\bigg{]}^{\frac{1}{2}}\bigg{\\{}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}\big{[}(v_{h}^{r})_{j}-v^{r}_{j}\big{]}^{2}\bigg{\\}}^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\Bigg{\\{}\tilde{h}_{k}\Big{(}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}{f^{r}_{j}}^{2}\Big{)}^{\frac{1}{2}}\|v^{\prime}\|_{L^{2}(x^{r}_{j_{k-1}},x^{r}_{j_{k}})}\Bigg{\\}}\leq\bigg{[}\sum_{k=\mathcal{K}_{c}}\tilde{h}_{k}^{2}\Big{(}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}{f^{r}_{j}}^{2}\Big{)}\bigg{]}^{\frac{1}{2}}\|v^{\prime}\|_{L^{2}{[0,1]}}.$
(3.40)
The eatimate in the theorem holds as
$\|v^{\prime}\|_{L^{2}{[0,1]}}=\|\boldsymbol{v}^{\prime}\|_{\ell^{2}_{\varepsilon}}$
for $v\in C^{0}(\mathbb{R})\cap\mathcal{P}_{1}(\mathcal{T}^{\varepsilon})$. ∎
Lemma 4. Let $f,v\in
C^{0}(\mathbb{R})\cap\mathcal{P}_{1}(\mathcal{T}^{\varepsilon})$ and
$v_{h}=I_{h}v\in C^{0}(\mathbb{R})\cap\mathcal{P}_{1}(\mathcal{T}^{h})$ be the
$\mathcal{P}_{1}$ interpolation of $v$ according to the $\mathcal{T}^{h}$. Let
$\boldsymbol{f}^{r},\boldsymbol{v}^{r}$ and $\boldsymbol{v}_{h}^{r}$ be the
vectorizations $f,v$ and $v_{h}$ according to $\mathcal{T}^{r}$. If
$\int_{0}^{1}v_{h}=0$, then we have the following estimate
$\langle f,v_{h}\rangle_{h}-\langle
f,v_{h}\rangle_{r}\leq\Bigg{\\{}\frac{1}{8}\bigg{[}(n\varepsilon)^{4}\sum_{k=\mathcal{K}_{c}}\hat{h}_{k+1}^{4}\|{\boldsymbol{f}^{r}}^{\prime\prime}\|^{2}_{\ell^{2}_{\bar{\varepsilon}^{r}}(\mathcal{D}^{2}_{k})}\bigg{]}^{\frac{1}{2}}+\bigg{[}\sum_{k=\mathcal{K}_{c}}\hat{h}_{k+1}^{4}\|{\boldsymbol{f}^{r}}^{\prime}\|^{2}_{\ell^{2}_{\varepsilon^{r}(\mathcal{D}^{1}_{k})}}\bigg{]}^{\frac{1}{2}}\Bigg{\\}}\|\boldsymbol{v}^{\prime}\|_{\ell^{2}_{\varepsilon}}.$
(3.41)
where $\mathcal{K}_{c}$ is defined in (3.10),
$\mathcal{D}_{k}^{1}=\\{j_{k}+1,\ldots,j_{k+1}\\}$ and $\hat{h}_{k}$ will be
defined in the proof.
###### Proof.
Since $I_{h}(fv_{h})$ is also piecewise linear with respect to the
$\mathcal{T}^{r}$ partition, we apply the trapezoidal rule here to evaluate
$\langle f,v_{h}\rangle_{h}=\int_{0}^{1}I_{h}(fv_{h})\,{\rm d}x$ to obtain
$\displaystyle\langle f,v_{h}\rangle_{h}-\langle f,v_{h}\rangle_{r}$
$\displaystyle=$
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\Bigg{\\{}\bigg{[}\frac{1}{2}\varepsilon^{r}_{j_{k-1}+1}I_{h}(fv_{h})(x^{r}_{k})+\sum_{j=j_{k-1}+1}^{j_{k}-1}\frac{1}{2}(\varepsilon^{r}_{j}+\varepsilon^{r}_{j+1})I_{h}(fv_{h})(x^{r}_{j})+\frac{1}{2}\varepsilon^{r}_{j_{k}}I_{h}(fv_{h})(x^{r}_{j_{k}})\bigg{]}$
$\displaystyle-\bigg{[}\frac{1}{2}\varepsilon^{r}_{j_{k-1}+1}(fv_{h})(x^{r}_{j_{k-1}+1})+\sum_{j=j_{k-1}+1}^{j_{k}-1}\frac{1}{2}(\varepsilon^{r}_{j}+\varepsilon^{r}_{j+1})(fv_{h})(x^{r}_{j})+\frac{1}{2}\varepsilon^{r}_{j_{k}}(fv_{h})(x^{r}_{j_{k}})\bigg{]}\Bigg{\\}}.$
(3.42)
We define $\boldsymbol{g}$ and $\boldsymbol{G}$ such that
$g_{j}=(fv_{h})(x^{r}_{j})$ and $G_{j}=(I_{h}(fv_{h}))(x^{r}_{j})$. It is easy
to check that $g_{j_{k}}=G_{j_{k}}$ and
$G_{j_{k-1}+i}=g_{j_{k-1}}+\frac{\sum_{\ell=1}^{i}\varepsilon^{r}_{j_{k-1}+\ell}}{\varepsilon^{h}_{k}}(g_{j_{k}}-g_{j_{k-1}})\quad\forall
k\in\mathcal{K}_{c}\quad and\quad i=1,\ldots,j_{k}-j_{k-1},$ (3.43)
where
$\varepsilon^{h}_{k}=\sum_{j=j_{k-1}+1}^{j_{k}}\varepsilon^{r}_{j}=x_{k}-x_{k-1}$.
Therefore, by Theorem C, we obatin the following estimate
$\displaystyle|\langle
f,v_{h}\rangle_{h}-\langle\boldsymbol{f}^{r},I_{r}\boldsymbol{v}^{h}\rangle_{r}|$
$\displaystyle=$
$\displaystyle\Bigg{|}\sum_{k\in\mathcal{K}_{c}}\bigg{[}\frac{1}{2}\varepsilon^{r}_{j_{k-1}}(g_{j_{k-1}}-G_{j_{k-1}})+\sum_{j=j_{k-1}+1}^{j_{k}-1}\frac{1}{2}(\varepsilon^{r}_{j}+\varepsilon^{r}_{j+1})(g_{j}-G_{j})+\frac{1}{2}\varepsilon^{r}_{j_{k}}(g_{j_{k}}-G_{j_{k}})\bigg{]}\Bigg{|}$
$\displaystyle\leq$
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\frac{1}{4}\frac{\big{(}(j_{k}-j_{k-1})\varepsilon\big{)}\big{(}(j_{k}-j_{k-1}+1)\varepsilon\big{)}^{2}}{\varepsilon^{h}_{k}}\|\boldsymbol{g}^{\prime\prime}\|_{\ell_{\bar{\boldsymbol{\varepsilon}}^{r}}^{1}(\mathcal{D}^{2}_{k})},$
(3.44)
where $\boldsymbol{g}^{\prime\prime}$ is the second finite difference
derivative with respect to the $\mathcal{T}^{r}$.
By the definition of $\boldsymbol{f}^{r}$ and $\boldsymbol{v}_{h}^{r}$,
$g_{j}=(fv_{h})(x^{r}_{j})=f^{r}_{j}(v_{h}^{r})_{j}$. Using
$(v_{h}^{r})^{\prime\prime}_{j}=0\ \forall j\in\mathcal{D}^{2}_{k}$,
$g_{j}^{\prime\prime}$ can be written as
$g_{j}^{\prime\prime}=(fv_{h}^{r})^{\prime\prime}_{j}=(f^{r})^{\prime\prime}_{j}(v_{h}^{r})_{j}+\frac{\varepsilon^{r}_{j}}{\bar{\varepsilon}^{r}_{j}}(f^{r})^{\prime}_{j}(v_{h}^{r})^{\prime}_{j}+\frac{\varepsilon^{r}_{j+1}}{\bar{\varepsilon}^{r}_{j}}(f^{r})^{\prime}_{j+1}(v_{h}^{r})^{\prime}_{j+1}.$
(3.45)
Noting that $\frac{\varepsilon^{r}}{\bar{\varepsilon}_{j}}\leq 2$ and
$\frac{\varepsilon^{r}_{j+1}}{\bar{\varepsilon}_{j}}\leq 2$ and defining
$\hat{h}_{k}:=\bigg{[}\frac{((j_{k}-j_{k-1})\varepsilon)(j_{k}-j_{k-1}+1)\varepsilon\big{)}^{2}}{\varepsilon^{h}_{k}}\bigg{]}^{\frac{1}{2}}$,
we have the following estimate
$\displaystyle\langle f,v_{h}\rangle_{h}-\langle f,v_{h}\rangle_{r}\leq$
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\frac{1}{4}\hat{h}_{k}^{2}\bigg{[}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}\big{|}(fv_{h}^{r})^{\prime\prime}_{j}\big{|}\bigg{]}$
$\displaystyle\leq$
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\frac{1}{4}\hat{h}_{k}^{2}\bigg{[}\sum_{j=j_{k-1}+1}^{j_{k}-1}\bar{\varepsilon}^{r}_{j}\big{|}(f^{r})^{\prime\prime}_{j}\big{|}\big{|}(v_{h}^{r})_{j}\big{|}+4\sum_{j=j_{k-1}+1}^{j_{k}}\varepsilon^{r}_{j}\big{|}(f^{r})^{\prime}_{j}\big{|}\big{|}(v_{h}^{r})^{\prime}_{j}\big{|}\bigg{]}$
$\displaystyle\leq$
$\displaystyle\sum_{k\in\mathcal{K}_{c}}\frac{1}{4}\hat{h}_{k}^{2}\bigg{[}\|{\boldsymbol{f}^{r}}^{\prime\prime}\|_{\ell^{2}_{\bar{\varepsilon}^{r}}(\mathcal{D}_{k}^{2})}\|\boldsymbol{v}_{h}^{r}\|_{\ell^{2}_{\bar{\varepsilon}^{r}}(\mathcal{D}_{k}^{2})}+4\|{\boldsymbol{f}^{r}}^{\prime}\|_{\ell^{2}_{\boldsymbol{\varepsilon}^{r}(\mathcal{D}_{k}^{1})}}\|(\boldsymbol{v}_{h}^{r})^{\prime}\|_{\ell^{2}_{\boldsymbol{\varepsilon}^{r}(\mathcal{D}_{k}^{1})}}\bigg{]}$
$\displaystyle\leq$
$\displaystyle\frac{1}{4}\bigg{[}\sum_{k\in\mathcal{K}_{c}}\hat{h}_{k}^{4}\|{\boldsymbol{f}^{r}}^{\prime\prime}\|^{2}_{\ell^{2}_{\bar{\boldsymbol{\varepsilon}}^{r}}(\mathcal{D}_{k}^{2})}\bigg{]}^{\frac{1}{2}}\|\boldsymbol{v}_{h}^{r}\|_{\ell^{2}_{\bar{\boldsymbol{\varepsilon}}^{r}}}+\bigg{[}\sum_{k\in\mathcal{K}_{c}}\hat{h}_{k}^{4}\|{\boldsymbol{f}^{r}}^{\prime}\|^{2}_{\ell^{2}_{\boldsymbol{\varepsilon}^{r}(\mathcal{D}_{k}^{1})}}\bigg{]}^{\frac{1}{2}}\|(\boldsymbol{v}_{h}^{r})^{\prime}\|_{\ell^{2}_{\boldsymbol{\varepsilon}^{r}}}.$
(3.46)
For further estimate, we first bound
$\|\boldsymbol{v}_{h}^{r}\|_{\ell^{2}_{\bar{\boldsymbol{\varepsilon}}^{r}}}$
by
$\|(\boldsymbol{v}_{h}^{r})^{\prime}\|_{\ell^{2}_{\boldsymbol{\varepsilon}^{r}}}$.
Since $v_{h}(x)$ is piecewise linear with respect to $\mathcal{T}^{r}$
partition, we can apply the trapezoidal rule to the integration on each
element to get
$\sum_{j=1}^{n}\bar{\varepsilon}^{r}_{j}(v_{h}^{r})_{j}=\sum_{j=1}^{n}\frac{1}{2}(\varepsilon^{r}_{j}+\varepsilon^{r}_{j-1})(v^{r}_{h})_{j}=\sum_{j=1}^{n}\varepsilon^{r}_{j}\frac{1}{2}\big{[}(v^{r}_{h})_{j}+(v^{r}_{h})_{j+1}\big{]}=\int_{0}^{1}v_{h}(x)\,{\rm
d}x=0.$ (3.47)
The last equality holds by the periodic condition on $v_{h}$. Thus, we can
apply Lemma C in Appendix C and Riez-Thorin Theorem to obtain
$\|\boldsymbol{v}_{h}^{r}\|_{\ell^{2}_{\bar{\varepsilon}^{r}}}\leq\frac{1}{2}n\varepsilon\bigg{(}\sum_{j=1}^{n}\varepsilon^{r}_{j}{(v_{h}^{r})^{\prime}_{j}}^{2}\bigg{)}^{\frac{1}{2}}.$
(3.48)
Since $v_{h}^{\prime}(x)=(v_{h}^{r})^{\prime}_{j}$ on
$(x^{r}_{j-1},x^{r}_{j})$,
$\sum_{j=1}^{n}\varepsilon^{r}_{j}{(v_{h}^{r})^{\prime}_{j}}^{2}=\int_{0}^{1}(v_{h}^{\prime})^{2}\,{\rm
d}x=\|v_{h}^{\prime}\|_{L^{2}[0,1]}.$ (3.49)
By Lemma B in Appendix B,
$\|v_{h}^{\prime}\|_{L^{2}[0,1]}\leq\|v^{\prime}\|_{L^{2}[0,1]}=\|\boldsymbol{v}^{\prime}\|_{\ell^{2}_{\varepsilon}}.$
(3.50)
Combine these results, the estimate stated in the theorem is easy to
establish. ∎
Having the three lemma and distribute the contribution to each element, we now
give the theorem which essentially gives the estimate of the residual due to
the external force.
Theorem 5. For $f,v\in\mathcal{U}$ and $J_{\mathcal{U}}$ defined in (3.5), we
have
$\|\langle f,J_{\mathcal{U}}\cdot\rangle_{h}-\langle
f,\cdot\rangle_{\varepsilon}\|_{\mathcal{U}^{-1,2}}\leq\big{\\{}\sum_{k\in\mathcal{K}_{c}}{\eta_{k}^{f}}^{2}\big{\\}}^{\frac{1}{2}}=:\mathscr{E}_{\rm
ext}(f),$ (3.51)
where
$\displaystyle\eta_{k}^{f}=$
$\displaystyle\bigg{\\{}\frac{1}{128}\big{[}\varepsilon^{3}(f^{\prime}_{\ell_{k-1}+1})^{2}+\varepsilon^{3}(f^{\prime}_{\ell_{k}+1})^{2}\big{]}^{2}+\tilde{h}_{k}^{2}\|\boldsymbol{f}^{r}\|^{2}_{\ell^{2}_{\bar{\boldsymbol{\varepsilon}}^{r}}(\mathcal{D}^{2}_{k})}$
$\displaystyle+\frac{1}{64}(n\varepsilon)^{4}\hat{h}_{k+1}^{4}\|{\boldsymbol{f}^{r}}^{\prime\prime}\|^{2}_{\ell^{2}_{\bar{\boldsymbol{\varepsilon}}^{r}}(\mathcal{D}^{2}_{k})}+\hat{h}_{k+1}^{4}\|{\boldsymbol{f}^{r}}^{\prime}\|^{2}_{\ell^{2}_{\boldsymbol{\varepsilon}^{r}(\mathcal{D}^{1}_{k})}}\bigg{\\}}^{\frac{1}{2}},$
(3.52)
and $\mathcal{K}_{c}$ is defined in (3.10), $\mathcal{D}_{k}^{2}$ is defined
in Lemma 3.3, $\tilde{h}_{k}$, $\mathcal{D}_{k}^{1}$ is defined in Lemma 3.3,
and $\hat{h}_{k+1}$ is defined in Lemma 3.3.
###### Proof.
We can not directly apply the three lemma to estimate the three parts in
(3.33). The reason is that $J_{\mathcal{U}}v\neq v_{h}$, which is the direct
interpolation of $v$ according to $\mathcal{T}^{h}$. The way to circumvent
this difficulty is by defining
$w:=v-\sum_{k=1}^{K}\frac{1}{2}(x_{k+1}-x_{k-1})v(x_{k})\text{ and
}w_{h}:=I_{h}w=J_{\mathcal{U}}v,$ (3.53)
and noting that
$\langle f,J_{\mathcal{U}}v\rangle_{h}-\langle
f,v\rangle_{\varepsilon}=\langle f,w_{h}\rangle_{h}-\langle
f,w\rangle_{\varepsilon}-\langle f,C\rangle_{\varepsilon}=\langle
f,w_{h}\rangle_{h}-\langle f,w\rangle_{\varepsilon},$ (3.54)
and $w^{\prime}(x)=v^{\prime}(x)\ \forall x\in\mathbb{R}$. Then by the three
lemma, we have
$\big{|}\langle f,J_{\mathcal{U}}v\rangle_{h}-\langle
f,v\rangle_{\varepsilon}\big{|}=\big{|}\langle f,w_{h}\rangle_{h}-\langle
f,w\rangle_{\varepsilon}\big{|}\leq\big{\\{}\sum_{k\in\mathcal{K}_{c}}{\eta_{k}^{f}}^{2}\big{\\}}^{\frac{1}{2}}\|w^{\prime}\|_{L^{2}[0,1]}=\big{\\{}\sum_{k\in\mathcal{K}_{c}}{\eta_{k}^{f}}^{2}\big{\\}}^{\frac{1}{2}}\|v^{\prime}\|_{L^{2}[0,1]},$
(3.55)
which establishes the estimate in the theorem. ∎
## 4\. Stability
Stability of the QC approximation is the second key ingredient for deriving an
a posteriori error bounds. Since we would like to bound the error of the
deformation gradient in $L^{2}$-norm, we derive the $L^{2}$ stability estimate
in this section. The procedure of deriving the a posteriori stability
condition largely follows that of the a priori stability condition in [8].
For an a posteriori error analysis, the natural notion of stability for energy
minimization problem is the coercivity(or, positivity) of the atomistic
Hessian at the projected QC solution $J_{\mathcal{U}_{{\rm qc}}}y_{{\rm qc}}$:
$E^{\prime\prime}_{\rm a}(J_{\mathcal{U}_{{\rm qc}}}y_{{\rm qc}})[v,v]\geq
c_{\rm a}(y_{{\rm qc}})\|v\|_{L^{2}[0,1]}^{2}\quad\forall v\in\mathcal{U},$
(4.1)
for some constant $c_{\rm a}(y_{{\rm qc}})>0$. To avoid notational difficulty,
we vectorize the above inequality and work on $\mathcal{U}^{\varepsilon}$
instead. Let $J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm qc}}$ be the
vectorization of $J_{\mathcal{U}_{{\rm qc}}}y_{{\rm qc}}$, then (4.1) is
equivalent to
$E^{\prime\prime}_{\rm a}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})[\boldsymbol{v},\boldsymbol{v}]\geq c_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm
qc}})\|\boldsymbol{v}\|_{\ell^{2}_{\boldsymbol{\varepsilon}}}^{2}\quad\forall\boldsymbol{v}\in\mathcal{U}.$
(4.2)
In the remainder of this section, we derive the explicit condition on
$\boldsymbol{y}^{{\rm qc}}$ such that (4.2) holds.
The Hessian operator of the atomistic model is given by
$E^{\prime\prime}_{\rm
a}(\boldsymbol{y})[\boldsymbol{v},\boldsymbol{v}]=\varepsilon\sum_{\ell=1}^{N}\phi^{\prime\prime}(y_{\ell}^{\prime})|v_{\ell}^{\prime}|^{2}+\varepsilon\sum_{\ell=1}^{N}\phi^{\prime\prime}(y_{\ell}^{\prime}+y_{\ell+1}^{\prime})|v_{\ell}^{\prime}+v^{\prime}_{\ell+1}|^{2}\quad\forall
y\in\mathcal{Y}.$
We note that the ’non-local’ Hessian terms
$|v^{\prime}_{\ell}+v^{\prime}_{\ell+1}|^{2}$ can be rewritten in terms of the
’local’ terms $|v^{\prime}_{\ell}|^{2}$ and $|v^{\prime}_{\ell+1}|^{2}$ and a
strain-gradient correction,
$|v^{\prime}_{\ell}+v^{\prime}_{\ell+1}|^{2}=2|v^{\prime}_{\ell}|^{2}+2|v^{\prime}_{\ell+1}|^{2}-\varepsilon^{2}|v^{\prime\prime}_{\ell}|^{2}.$
Using this formula, we can rewrite the Hessian in the form
$E_{\rm
a}^{\prime\prime}(\boldsymbol{y})[\boldsymbol{v},\boldsymbol{v}]=\varepsilon\sum_{\ell=1}^{N}A_{\ell}|v_{\ell}^{\prime}|^{2}+\varepsilon\sum_{\ell=1}^{N}B_{\ell}|v_{\ell}^{\prime\prime}|^{2},$
where
$\displaystyle
A_{\ell}(\boldsymbol{y})=\phi^{\prime\prime}(y_{\ell}^{\prime})+2\phi^{\prime\prime}(y_{\ell-1}^{\prime}+y_{\ell}^{\prime})+2\phi^{\prime\prime}(y_{\ell}^{\prime}+y_{\ell+1}^{\prime})$
(4.3) $\displaystyle
B_{\ell}(\boldsymbol{y})=-\phi^{\prime\prime}(y^{\prime}_{\ell}+y^{\prime}_{\ell+1}).$
Recall our assumption that $\phi$ is convex in $(0,{r_{\ast}})$ and concave in
$({r_{\ast}},+\infty)$. For typical pair interaction potentials,
$y^{\prime}_{\ell}<{r_{\ast}}/2$ can only be achieved under extreme
compressive forces. Since, under such extreme conditions a pair potential may
be an inappropriate model to employ anyhow, it is not too restrictive to
assume that the the projected QC solution $J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm qc}}$ satisfies
$(J_{\mathcal{U}_{{\rm qc}}}y^{{\rm
qc}})^{\prime}_{\ell}\geq{r_{\ast}}/2\qquad\forall\ell\in\mathbb{Z}.$
As a result of this assumption, and the properties of $\phi$, we have
$-\phi^{\prime\prime}(y^{\prime}_{\ell}+y^{\prime}_{\ell+1})\geq 0\
\forall\ell\in\mathbb{Z}$ and thus $B_{\ell}\geq 0\ \forall\ell\in\mathbb{Z}$.
As an immediate consequence we obtain the following lemma, which gives
sufficient conditions under which the a posteriori stability of QC
approximation can be guaranteed.
Lemma 6. Let $J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}}\in\mathcal{Y}^{\varepsilon}$ satisfies $\min_{\ell}(J_{\mathcal{U}_{{\rm
qc}}})^{\prime}_{\ell}\geq{r_{\ast}}/2$; then,
$E^{\prime\prime}_{\rm a}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})[\boldsymbol{v},\boldsymbol{v}]\geq A_{\ast}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm
qc}})\|\boldsymbol{v}^{\prime}\|_{\ell_{\varepsilon}^{2}}^{2}\qquad\forall\boldsymbol{v}\in\mathcal{U},\quad\text{where}\quad
A_{\ast}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})=\min_{\ell=1,\dots,N}A_{\ell}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm qc}}).$
The coefficients $A_{\ell}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})$ are defined in (4.3).
###### Proof.
If $\min_{\ell}(J_{\mathcal{U}_{{\rm qc}}}y^{{\rm
qc}})^{\prime}_{\ell}\geq{r_{\ast}}/2$, then
$\displaystyle\qquad E^{\prime\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm
qc}})[\boldsymbol{v},\boldsymbol{v}]\geq\varepsilon\sum_{\ell=1}^{N}A_{\ell}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm qc}})|v^{\prime}_{\ell}|^{2}\geq
A_{\ast}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})\varepsilon\sum_{\ell=1}^{N}|v^{\prime}_{\ell}|^{2}=A_{\ast}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm
qc}})\|\boldsymbol{v}^{\prime}\|_{\ell^{2}_{\varepsilon}}^{2}.\qquad\qquad\qquad\qed$
Before we present the main theorem and its proof in the next section, we state
a useful auxiliary result: a local Lipschitz bound on $E^{\prime\prime}_{\rm
a}$. The proof of this Lipschitz bound is straightforward and is therefore
omitted.
Lemma 7. Let $\boldsymbol{y},\boldsymbol{z}\in\mathcal{Y}^{\varepsilon}$ such
that $\min_{\ell}y_{\ell}^{\prime}\geq\mu$ and
$\min_{\ell}z_{\ell}^{\prime}\geq\mu$ for some constant $\mu>0$, then
$\big{|}\\{\mathcal{E}_{\rm a}^{\prime\prime}(\boldsymbol{y})-\mathcal{E}_{\rm
a}^{\prime\prime}(\boldsymbol{z})\\}[\boldsymbol{v},\boldsymbol{w}]\big{|}\leq
C_{\rm
Lip}\|\boldsymbol{y}^{\prime}-\boldsymbol{z}^{\prime}\|_{\ell_{\varepsilon}^{\infty}}\|\boldsymbol{v}^{\prime}\|_{\ell^{2}_{\varepsilon}}\|\boldsymbol{w}^{\prime}\|_{\ell^{2}_{\varepsilon}}\qquad\forall\boldsymbol{v},\boldsymbol{w}\in\mathcal{U},$
where $C_{\rm Lip}=M_{3}([\mu,+\infty))+{8M_{3}([2\mu,+\infty))}$ and
$M_{i}(S)=\max_{\xi\in\mathcal{S}}|\phi^{i}(\mathcal{S})|$.
## 5\. A Posteriori Error Estimates
### 5.1. The a posterior error estimates for the deformation gradient
The error we estimate is $e:=y_{\rm a}-J_{\mathcal{U}_{{\rm qc}}}y_{{\rm qc}}$
in the $\mathcal{U}^{1,2}$-norm, as for $y_{1},y_{2}\in\mathcal{Y}$,
$y_{1}-y_{2}\in\mathcal{U}$. To avoid technicalities associated with the
nonlinearity of our models, we make an a priori assumption: we assume the
existence of the atomistic and QC solutions and make a mild requirement on
their smoothness and closeness (cf. (5.1)).
Theorem 8. Let $y_{{\rm qc}}$ be a solution of the QC problem (2.25) whose
gradients are such that $\min_{\ell}\big{(}J_{\mathcal{U}_{{\rm qc}}}y^{{\rm
qc}}\big{)}^{\prime}_{\ell}\geq r_{*}/2\ \forall\ell\in\mathbb{Z}$ and
$A_{*}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm qc}})>0$, where $A_{*}$
is defined in the statement of Lemma 4. Suppose, further, that $y^{{\rm a}}$
is a solution of the atomistic model (2.8) such that, for some $\tau>0$,
$\|(y^{{\rm a}}-J_{\mathcal{U}_{{\rm qc}}}y_{{\rm
qc}})^{\prime}\|_{L^{\infty}[0,1]}=\|(\boldsymbol{y}^{{\rm
a}}-J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})^{\prime}\|_{\ell^{\infty}_{\varepsilon}}\leq\tau.$ (5.1)
Then, if $\tau$ is sufficiently small, we have the error estimate
$\|y^{\rm a}-J_{\mathcal{U}_{{\rm qc}}}y_{{\rm
qc}}\|_{L^{2}[0,1]}=\|(\boldsymbol{y}^{{\rm a}}-J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm
qc}})^{\prime}\|_{\ell^{2}_{\varepsilon}}=\leq{{\textstyle\frac{2}{A_{*}(\boldsymbol{y}^{{\rm
a}})}}}\big{(}\mathscr{E}_{\rm store}(y_{{\rm qc}})+\mathscr{E}_{\rm
ext}(f)\big{)},$ (5.2)
where the functional of the residual of the stored energy $\mathscr{E}_{\rm
store}(\cdot)$ is defined in (3.11) and the functional of the approximation
error for the external forces $\mathscr{E}_{\rm ext}(\cdot)$ is defined in
(3.51).
###### Proof.
From the mean value theorem we deduce that there exists
$\boldsymbol{\theta}\in{\rm conv}\\{\boldsymbol{y}^{{\rm
a}},J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm qc}}\\}$ such that
$\displaystyle E^{\prime\prime}_{\rm
a}(\boldsymbol{\theta})[\boldsymbol{e},\boldsymbol{e}]=$ $\displaystyle
E^{\prime}_{\rm a}(\boldsymbol{y}^{{\rm a}})[\boldsymbol{v}]-E^{\prime}_{{\rm
qc}}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})[J_{\mathcal{U}}\boldsymbol{v}]$ $\displaystyle=$
$\displaystyle\big{(}\mathcal{E}^{\prime}_{\rm a}(\boldsymbol{y}^{{\rm
a}})[\boldsymbol{v}]-\mathcal{E}^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm qc}})[J_{\mathcal{U}}\boldsymbol{v}]\big{)}$
$\displaystyle-$
$\displaystyle\big{(}\langle\boldsymbol{f},\boldsymbol{v}\rangle_{\varepsilon}-\langle\boldsymbol{f},J_{\mathcal{U}}\boldsymbol{v}\rangle_{h}\big{)}.$
The first group was analyzed in section 3.2 Theorem 3.2 and the second group
was analyzed in section 3.3 Theorem 3.3. Inserting these estimates we arrive
at
$E^{\prime\prime}_{\rm
a}(\boldsymbol{\theta})[\boldsymbol{e},\boldsymbol{e}]\leq\big{(}\mathscr{E}_{\rm
store}(y^{{\rm qc}})+\mathscr{E}_{\rm
ext}(f)\big{)}\|\boldsymbol{e}^{\prime}\|_{\ell^{2}_{\varepsilon}}.$ (5.3)
It remains to prove a lower bound on $\mathcal{E}^{\prime\prime}_{\rm
a}(\boldsymbol{\theta})[\boldsymbol{e},\boldsymbol{e}]$. From our assumption
that $\min(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})^{\prime}_{\ell}\geq r_{*}/2$, and from (5.1) it follows that
$\min_{\ell}\theta^{\prime}_{\ell}\geq r_{*}/2-\tau.$
Assuming that $\tau$ is sufficiently small, e.g.,
$\tau\leq\tau_{1}:=\frac{1}{4}\min_{\ell}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm qc}})^{\prime}_{\ell}$, we can apply Lemma 4 to
deduce that
$\displaystyle\mathcal{E}^{\prime\prime}_{\rm
a}(\boldsymbol{\theta})[\boldsymbol{e},\boldsymbol{e}]\geq$
$\displaystyle\mathcal{E}^{\prime\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm qc}})[\boldsymbol{e},\boldsymbol{e}]-C_{\rm
Lip}\|(\boldsymbol{\theta}-\boldsymbol{y}^{{\rm
a}})\|_{\ell^{\infty}_{\varepsilon}}\|\boldsymbol{e}^{\prime}\|^{2}_{\ell^{2}_{\varepsilon}}$
$\displaystyle\geq$ $\displaystyle\mathcal{E}^{\prime\prime}_{\rm
a}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})[\boldsymbol{e},\boldsymbol{e}]-C_{\rm
Lip}\tau\|\boldsymbol{e}^{\prime}\|^{2}_{\ell^{2}_{\varepsilon}},$ (5.4)
where $C_{\rm Lip}$ may depend on $\tau_{1}$.
We can now apply our stability analysis in Section 4. Since
$(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm qc}})_{\ell}^{\prime}\geq
r_{*}/2$ for all $\ell$, Lemma 4 implies that
$\mathcal{E}^{\prime\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm qc}})[\boldsymbol{e},\boldsymbol{e}]\geq
A_{*}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})\|\boldsymbol{e}^{\prime}\|_{\ell^{2}_{\varepsilon}}^{2},$
which, combined with (5.3) and (5.1) , yields
$\big{(}A_{*}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm qc}})-C_{\rm
Lip}\tau\big{)}\|\boldsymbol{e}^{\prime}\|_{\ell^{2}_{\varepsilon}}^{2}\leq\mathcal{E}^{\prime\prime}_{\rm
a}(\theta)[\boldsymbol{e},\boldsymbol{e}]\leq\big{(}\mathscr{E}_{\rm
store}(y_{{\rm qc}})+\mathscr{E}_{\rm
ext}(f)\big{)}\|\boldsymbol{e}^{\prime}\|_{\ell^{2}_{\varepsilon}}.$
Dividing through by $\|\boldsymbol{e}^{\prime}\|_{\ell^{2}_{\varepsilon}}$,
and assuming that $\tau\leq\min(\tau_{1},\tau_{2})$ where
$\tau_{2}=A_{*}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm qc}})/(2C_{\rm
Lip})$, we deduce that
${\textstyle\frac{A_{*}(J_{\mathcal{U}_{{\rm qc}}}\boldsymbol{y}^{{\rm
qc}})}{2}}\|\boldsymbol{e}^{\prime}\|_{\ell^{2}_{\varepsilon}}\leq\big{(}\mathscr{E}_{\rm
store}(y_{{\rm qc}})+\mathscr{E}_{\rm ext}(f)\big{)},$
which concludes the proof of the a posteriori error estimate for the
deformation gradient. ∎
### 5.2. The a posterior error estimate for the energy
Besides the deformation gradient, the energy of the system is another quantity
of interest. In this section, we derive an a posteriori error estimator for
the energy difference between the atomistic model and the QC approximation,
namely,
$E_{\rm a}(y^{{\rm a}})-E_{{\rm qc}}(y_{{\rm qc}}).$ (5.5)
To analyze this difference, we decompose (5.5) as
$\big{|}E_{\rm a}(y^{{\rm a}})-E_{{\rm qc}}(y_{{\rm qc}})\big{|}=\big{|}E_{\rm
a}(y^{{\rm a}})-E_{\rm a}(J_{\mathcal{U}_{{\rm qc}}}y_{{\rm
qc}})\big{|}+\big{|}E_{\rm a}(J_{\mathcal{U}_{{\rm qc}}}y_{{\rm qc}})-E_{{\rm
qc}}(y_{{\rm qc}})\big{|}.$ (5.6)
We then analyze the two groups separately.
To analyze the first group, we have the following Lemma.
Lemma 9. Let $y,z\in\mathcal{Y}$ and
$\boldsymbol{y},\boldsymbol{z}\in\mathcal{Y}^{\varepsilon}$ be their
vectorizations, such that $\min_{\ell}y_{\ell}^{\prime}\geq\mu$ and
$\min_{\ell}z_{\ell}^{\prime}\geq\mu$ for some constant $\mu>0$, and $y\in{\rm
argmin}E_{\rm a}(\mathcal{Y})$. Let
$\boldsymbol{e}=\boldsymbol{y}-\boldsymbol{z}$, then
$\big{|}E_{\rm a}(\boldsymbol{y})-E_{\rm a}(\boldsymbol{z})\big{|}\leq
C^{E}_{\rm Lip}\|\boldsymbol{e}^{\prime}\|^{2}_{\ell^{2}_{\varepsilon}},$
(5.7)
where $C^{E}_{\rm
Lip}=\frac{1}{2}M_{2}([\mu,+\infty))+{2M_{3}([2\mu,+\infty))}$, where
$M_{i}(S)=\max_{\xi\in\mathcal{S}}|\phi^{i}(\mathcal{S})|$.
###### Proof.
We first rewrite the difference of the total energy as the summation of the
differences of the stored energy and that of the external energy:
$E_{\rm a}(\boldsymbol{y})-E_{\rm a}(\boldsymbol{z})=\big{(}\mathcal{E}_{\rm
a}(\boldsymbol{y})-\mathcal{E}_{\rm
a}(\boldsymbol{z})\big{)}-\big{(}\langle\boldsymbol{f},\boldsymbol{z}\rangle_{\varepsilon}-\langle\boldsymbol{f},\boldsymbol{y}\rangle_{\varepsilon}\big{)}$
For the difference of the stored energy, we have
$\displaystyle\mathcal{E}_{\rm a}(\boldsymbol{y})-\mathcal{E}_{\rm
a}(\boldsymbol{z})=$
$\displaystyle\varepsilon\sum_{\ell=1}^{N}\big{[}\phi(y^{\prime}_{\ell})-\phi(z^{\prime}_{\ell})\big{]}+\varepsilon\sum_{\ell=1}^{N}\big{[}\phi(y^{\prime}_{\ell}+y^{\prime}_{\ell+1})-\phi(z^{\prime}_{\ell}+z^{\prime}_{\ell+1})\big{]}$
$\displaystyle=$
$\displaystyle\varepsilon\sum_{\ell=1}^{N}\big{[}\phi^{\prime}(y^{\prime}_{\ell})e^{\prime}_{\ell}-\frac{1}{2}\phi^{\prime\prime}(\xi^{1}_{\ell}){e^{\prime}_{\ell}}^{2}\big{]}$
$\displaystyle+$
$\displaystyle\varepsilon\sum_{\ell=1}^{N}\big{[}\phi^{\prime}(y^{\prime}_{\ell}+y^{\prime}_{\ell+1})(e^{\prime}_{\ell}+e^{\prime}_{\ell+1})-\frac{1}{2}\phi^{\prime\prime}(\xi^{2}_{\ell})(e^{\prime}_{\ell}+e^{\prime}_{\ell+1})^{2}\big{]}$
$\displaystyle=$ $\displaystyle\mathcal{E}_{\rm
a}^{\prime}(\boldsymbol{y})[\boldsymbol{e}]-\frac{1}{2}\varepsilon\sum_{\ell=1}^{N}\phi^{\prime\prime}(\xi^{1}_{\ell}){e^{\prime}_{\ell}}^{2}-\frac{1}{2}\varepsilon\sum_{\ell=1}^{N}\phi^{\prime\prime}(\xi^{2}_{\ell})(e^{\prime}_{\ell}+e^{\prime}_{\ell+1})^{2},$
where $\xi^{1}_{\ell}\in{\rm conv}\\{y^{\prime}_{\ell},z^{\prime}_{\ell}\\}$
and $\xi^{2}_{\ell}\in{\rm
conv}\\{y^{\prime}_{\ell}+y^{\prime}_{\ell+1},z^{\prime}_{\ell}+z^{\prime}_{\ell+1}\\}$.
For the difference of the energy caused by the external forces, we have
$\langle\boldsymbol{f},\boldsymbol{z}\rangle_{\varepsilon}-\langle\boldsymbol{f},\boldsymbol{y}\rangle_{\varepsilon}=-\langle\boldsymbol{f},\boldsymbol{e}\rangle_{\varepsilon}=-\mathcal{E}_{\rm
a}^{\prime}(\boldsymbol{y})[\boldsymbol{e}],$
by the first optimality condition of $y\in{\rm argmin}E_{\rm a}(\mathcal{Y})$.
It is then easy to obtain the estimate stated in the Lemma by using Cauchy-
Schwaz inequality to the non-local term. ∎
Lemma 10. For $y_{h}\in\mathcal{Y}_{{\rm qc}}$ and $y^{\prime}_{h}(x)>0$, we
have
$\big{|}E_{\rm a}(J_{\mathcal{U}_{{\rm qc}}}y_{h})-E_{{\rm
qc}}(y_{h})\big{|}\leq\sum_{k\in\mathcal{K}_{c}}{\eta_{E}^{e}}_{k}+{\eta_{E}^{f}}_{k},$
(5.8)
where
$\eta_{k}^{e}=\frac{1}{2}\sum_{j=-1}^{1}[[\phi]]_{\ell_{k-1}+j}+\frac{1}{2}\sum_{j=-1}^{1}[[\phi]]_{\ell_{k}+j},$
(5.9)
if $k\in\mathcal{K}^{\prime}_{c}$,
$\eta_{k}^{e}=\sum_{j=-1}^{1}[[\phi]]_{\ell_{La_{i}}+j}+\frac{1}{2}\sum_{j=-1}^{1}[[\phi]]_{\ell_{k-1}+j},$
(5.10)
if $k=L_{a_{i}}$ for some $i\in\\{1,2,\ldots,M\\}$, and
$\eta_{k}^{e}=\sum_{j=-1}^{1}[[\phi]]_{\ell_{Ra_{i}}+j}+\frac{1}{2}\sum_{j=-1}^{1}[[\phi]]_{\ell_{k}+j},$
(5.11)
if $k=R_{a_{i}}$ for some $i\in\\{1,2,\ldots,M\\}$. $[[\phi]]_{\ell_{k}}$’s
and ${\eta_{E}^{f}}_{k}$’s will be defined in the proof.
###### Proof.
We first decompose the energy difference to two parts:
$E_{\rm a}(J_{\mathcal{U}_{{\rm qc}}}y_{h})-E_{{\rm
qc}}(y_{h})=\big{(}\mathcal{E}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})-\mathcal{E}_{{\rm qc}}(y_{h})\big{)}-\big{(}\langle
f,J_{\mathcal{U}_{{\rm qc}}}y_{h}\rangle_{\varepsilon}-\langle
f_{h},y_{h}\rangle_{h}\big{)}.$
We first analyze the energy difference of the stored energy. Since
$r_{b}D_{b}J_{\mathcal{U}_{{\rm qc}}}y_{h}=r_{b}D_{b}y_{h}$, we have
$\mathcal{E}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})=\sum_{b\in\mathcal{B}}a_{b}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})=\sum_{b\in\mathcal{B}}\varepsilon\phi(r_{b}D_{b}J_{\mathcal{U}_{{\rm
qc}}}y_{h})=\sum_{b\in\mathcal{B}}\varepsilon\phi(r_{b}D_{b}y_{h}),$ (5.12)
and
$\mathcal{E}_{{\rm
qc}}(y_{h})=\sum_{b\in\mathcal{B}}\big{[}a_{b}(y_{h})+c_{b}(y_{h})\big{]}=\sum_{b\in\mathcal{B}}\bigg{[}\frac{|b\cap\Omega_{\rm
a}|}{r_{b}}\phi\big{(}r_{b}D_{b\cap\Omega_{\rm
a}}y\big{)}+\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x\bigg{]}.$ (5.13)
We analyze the energy difference bond by bond,
If $b\subset\Omega_{\rm a}$, then
$a_{b}(y_{h})+c_{b}(y_{h})=a_{b}(y_{h})=\varepsilon\phi(r_{b}D_{b}y_{h})=a_{b}(J_{\mathcal{U}_{{\rm
qc}}}y_{h}),$
and the energy difference in this bond is thus $0$.
If $b\subset\Omega_{c}\cap T_{k}$ for some $k\in\mathcal{K}_{c}$, then
$|b\cap\Omega_{\rm a}|=0$ and
$\displaystyle a_{b}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})-\big{[}a_{b}(y_{h})+c_{b}(y_{h})\big{]}=$ $\displaystyle
a_{b}(J_{\mathcal{U}_{{\rm qc}}}y_{h})-c_{b}(y_{h})$ $\displaystyle=$
$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y_{h}(x))\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\frac{1}{r_{b}}\int_{b}\big{[}\phi(r_{b}D_{b}y_{h})-\phi(\nabla_{r_{b}}y_{h}(x))\big{]}\,{\rm
d}x.$ (5.14)
Since $y_{h}$ is affine on $T_{k}$, $\nabla_{r_{b}}y(x))=r_{b}D_{b}y_{h}$ and
subsequently, (5.2) is $0$.
We are left with the interaction bonds crossing the atomistic-continuum
interface and the boundaries of the elements in the continuum region. Again
because of its tediousness, we leave the detail of this analysis to the
Appendix and only give the results here:
$\displaystyle\mathcal{E}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})-\mathcal{E}_{{\rm
qc}}(y_{h})=\sum_{i=1}^{M}\sum_{j=-1}^{1}[[\phi]]_{\ell_{La_{i}}+j}+\sum_{i=1}^{M}\sum_{j=-1}^{1}[[\phi]]_{\ell_{Ra_{i}}+j}+\sum_{k\in\mathcal{K}^{\prime}_{c}}\sum_{j=-1}^{1}[[\phi]]_{\ell_{k}+j}.$
(5.15)
For $k=L_{a_{i}}$ where $i\in\\{1,2,\ldots,M\\}$, we have
$\displaystyle[[\phi]]_{\ell_{k}}=$
$\displaystyle\varepsilon\big{\\{}\phi((1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}})-(1-\theta_{k})\phi(y_{h}^{\prime}|_{T_{k+1}})-\theta_{k}\phi(y_{h}^{\prime}|_{T_{k}})\big{\\}},$
(5.16)
$\displaystyle[[\phi]]_{\ell_{k-1}}=2\varepsilon\big{\\{}\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})\theta_{k}y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k+1}})-(1+\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k}})\big{\\}}.$
(5.17)
and
$\displaystyle[[\phi]]_{\ell_{k+1}}=$ $\displaystyle
2\varepsilon\big{\\{}\phi(y_{h}^{\prime}|_{T_{k+2}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}+(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}})$
$\displaystyle-(2-\theta_{k})\phi\big{(}\frac{2}{2-\theta_{k}}y_{h}^{\prime}|_{T_{k+2}}+\frac{2(1-\theta_{k})}{2-\theta_{k}}y_{h}^{\prime}|_{T_{k+1}}\big{)}-\theta_{k}\phi(2y_{h}^{\prime}|_{T_{k}})\big{\\}}.$
(5.18)
For $k=R_{a_{i}}$ where $i\in\\{1,2,\ldots,M\\}$, we have
$\displaystyle[[\phi]]_{\ell_{k}}=\varepsilon\big{\\{}\phi((1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}})-\theta_{k}\phi(y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(y_{h}^{\prime}|_{T_{k+1}})\big{\\}},$
(5.19) $\displaystyle[[\phi]]_{\ell_{k-1}}=$ $\displaystyle
2\varepsilon\big{\\{}\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}+y_{h}^{\prime}|_{T_{k-1}})$
$\displaystyle-(1-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k+1}})-(1+\theta_{k})\phi\big{(}\frac{2\theta_{k}}{1+\theta_{k}}y_{h}^{\prime}|_{T_{k}}+\frac{2}{1+\theta_{k}}y_{h}^{\prime}|_{T_{k-1}}\big{)}\big{\\}},$
(5.20)
and
$\displaystyle[[\phi]]_{\ell_{k+1}}=2\varepsilon\big{\\{}\phi((2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}})-(2-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k+1}})-\theta_{k}\phi\big{(}2y_{h}^{\prime}|_{T_{k}}\big{)}\big{\\}}.$
(5.21)
For $k\in\mathcal{K}^{\prime}_{c}$, we have
$\displaystyle[[\phi]]_{\ell_{k}}=\varepsilon\big{\\{}\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(y_{h}^{\prime}|_{T_{k+1}})-\theta_{k}\phi(y_{h}^{\prime}|_{T_{k}})\big{\\}},$
(5.22)
$\displaystyle[[\phi]]_{\ell_{k}-1}=\frac{1}{2}\varepsilon\big{\\{}2\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k+1}})-(1+\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k}})\big{\\}},$
(5.23)
and
$\displaystyle[[\phi]]_{\ell_{k}+1}=\frac{1}{2}\varepsilon\big{\\{}2\phi((2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}})-\theta_{k}\phi(2y_{h}^{\prime}|_{T_{k+1}})-(2-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k}})\big{\\}}.$
(5.24)
We then analyze the energy difference caused by the external forces. The
energy difference is given by
$\langle f,J_{\mathcal{U}_{{\rm qc}}}u_{h}\rangle_{\varepsilon}-\langle
f,u_{h}\rangle_{h}=\langle f,u_{h}\rangle_{\varepsilon}-\langle
f,u_{h}\rangle_{h},$
since $J_{\mathcal{U}_{{\rm qc}}}u_{h}=u_{h}+C$ for some contant $C$ and
$\langle f,C\rangle_{\varepsilon}=0\ \forall C$. We decompose this energy
difference to each element and write it as
$\big{|}\langle f,u_{h}\rangle_{\varepsilon}-\langle
f,u_{h}\rangle_{h}\big{|}\leq\sum_{k=1}^{K}{\eta_{E}^{f}}_{k},$ (5.25)
where
$\displaystyle{\eta_{E}^{f}}_{k}=$
$\displaystyle\bigg{|}(1-\theta_{k})\frac{1}{2}\varepsilon(f_{\ell_{k-1}}u_{\ell_{k-1}}+f_{\ell_{k-1}+1}u_{\ell_{k-1}+1})+\frac{1}{2}\sum_{\ell=\ell_{k-1}+2}^{\ell_{k}-1}\varepsilon(f_{\ell}u_{\ell}+f_{\ell+1}u_{\ell+1})$
$\displaystyle+\theta\frac{1}{2}\varepsilon(f_{\ell_{k}}u_{\ell_{k}}+f_{\ell_{k}+1}u_{\ell_{k}+1})\big{\\}}-\sum_{k=1}^{K}\frac{1}{2}(x_{k}-x_{k-1})\big{[}f(x_{k-1})y(x_{k-1})+f(x_{k})y(x_{k})\big{]}\bigg{|},$
(5.26)
where $f_{\ell}=f(\ell\varepsilon)$ and $u_{\ell}=u(\ell\varepsilon)$. ∎
## 6\. Numerical Experiments
In this section, we present numerical experiments to illustrate our analysis.
Throughout this section we fix $F=1$, $N=8193$, and let $\phi$ be the Morse
potential
$\phi(r)=\exp(-2\alpha(r-1))-2\exp(-\alpha(r-1)),$
with the parameter $\alpha=5$.
For our benchmark problem, we defined the external force $\boldsymbol{f}$ to
be
$\displaystyle
f_{\ell}=\left\\{\begin{array}[]{rl}-0.1\big{\\{}1-\frac{|\ell-\frac{N-1}{2}|}{\frac{N-1}{2}}\big{\\}}\frac{N}{|\ell-\frac{N-1}{2}-0.5|},&\text{for
$\ell\leq\frac{N-1}{2}$},\\\
0.1\big{\\{}1-\frac{\ell-\frac{N-1}{2}-1}{\frac{N-1}{2}}\big{\\}}\frac{N}{|\ell-\frac{N-1}{2}-0.5|},&\text{for
$\ell\geq\frac{N-1}{2}+1$.}\end{array}\right.$
We briefly explain the meaning of the external force. On each atom, the
external force is a product of three components. The third component, namely
$\frac{N}{|\ell-\frac{N-1}{2}-0.5|}$ is essentially $\frac{1}{r_{\ell}}$ where
$r_{\ell}$ is the distance between an atom and the center of this atomistic
chain located at $\frac{N-1}{2}+0.5$. This non-linear force will create a
defect in the middle of the chain but affect little in the far field. The
second component, namely $1-\frac{|\ell-\frac{N-1}{2}|}{\frac{N-1}{2}}$, adds
a decay of the first component and in particular, it is $0$ when $\ell=N$,
which prevents the ’kink’ of the force on the boundary due to a rapid change
of the sign of the force that will leads to non-smooth deformation gradient
that should be contained in the atomistic region. The first component, which
is the constant $0.1$, is to rescale the force so that the solution of this
problem is stable.
We solve for the atomistic problem and consider the solution to be the
accurate solution. We then solve for the QC problem on different meshes
generated by the mesh refinement schemes.
We show two relative errors against the number of degrees of freedom. The
first one is the error of the deformation gradient in $L_{2}$-norm over the
$L_{2}$-norm of the difference between the deformation gradient of the
atomistic solution and the homogeneous state, which is defined by
$e_{deformation}:=\frac{\|y_{{\rm qc}}^{\prime}-y_{\rm
a}^{\prime}\|_{L^{2}[0,1]}}{\|y_{\rm a}^{\prime}-Fx\|_{L^{2}[0,1]}}.$ (6.1)
The second relative error is the absolute value of the energy difference of
the atomistic solution and the QC solution over the absolute value of the
energy change of the atomistic solution from the homogeneous state, which is
defined by
$e_{energy}:=\frac{|E_{\rm a}(y_{\rm a})-E_{{\rm qc}}(y_{{\rm qc}})|}{|E_{\rm
a}(y_{\rm a})-E_{\rm a}(Fx)|}.$ (6.2)
Before we present the plots of the errors, we first introduce the mesh
generating schemes.
### 6.1. Mesh Construction
To avoid unnecessary technical difficulty in the mesh refinement algorithm, we
assume that the defect core is already captured in the middle of the chain.
There are three mesh generating schemes we use.
The first mesh generating scheme is derived in Section 7.1 of [6] using
calculus of variations. From this analysis, we get that the (quasi-)optimal
mesh size in the continuum region, with the restriction that the atomistic
region is symmetric and has $K$ atoms on each side, is given by
$h(r)=\big{(}\frac{f(K\varepsilon)}{f(r)}\frac{r}{K\varepsilon}\big{)}^{\frac{2}{3}}.$
(6.3)
Since the mesh size can not change continuously and we restrict the smallest
mesh size in the continuum region to be $2\varepsilon$, we use the following
algorithm to generate this mesh (we only list the case on the right hand side
of the atomistic region):
Algorithm 1.
1. (1)
Set atom $\frac{N-1}{2}+1$ to be the middle of the atomistic region.
2. (2)
Choose $K$ so that there are $K$ atoms on each side of the atomistic region.
3. (3)
Choose $h$ to be $2\varepsilon$ for every element on the right hand side of
the atomistic region until $h(r)>2\varepsilon$, where $r$ is the distance
between the right boundary of the previous element and the middle of the
atomistic region.
4. (4)
Choose $h$ according to (6.3) until the right boundary of the newly created
element is out of the right limit of the chain.
∎
The second mesh generating scheme is essentially a mesh refinement process
according to the error estimator with respect to the deformation gradient
according to Lemma 3.3, Lemma 3.3 and Lemma 3.3. The mesh refinement algorithm
is stated as follows:
Algorithm 2.
1. (1)
Set atom $\frac{N-1}{2}+1$ to be the middle of the atomistic region.
2. (2)
Choose $5$ atoms on each side of the atomistic region.
3. (3)
Divide the left and the right part of the continuum region into two equally
large element
4. (4)
Compute the QC solution on this mesh and then compute the squared error
indicator of each element $\eta_{i}$ and sort these indicators according to
its value.
5. (5)
Bisect the first $M$ sorted elements such that
$\sum_{i=1}^{M-1}\eta^{2}_{i}\leq 0.5\eta^{2}\text{ and
}\sum_{i=1}^{M}\eta^{2}_{i}\geq 0.5\eta^{2},$ (6.4)
where $\eta_{i}$ is the error estimator of each element defined by
$\eta^{deformation}_{i}=\big{[}(\eta^{e}_{i})^{2}+(\eta^{f}_{i})^{2}\big{]})^{\frac{1}{2}}\big{/}{\textstyle\frac{A_{*}(J_{\mathcal{U}_{{\rm
qc}}}\boldsymbol{y}^{{\rm qc}})}{2}}.$ (6.5)
If the element is near the atomistic region, merge the element into the
atomistic region.
6. (6)
If the resulting mesh reaches the maximal number of degrees of freedom, stop
the process, else, go to Step 4.
∎
The third mesh generating scheme is the mesh refinement process according to
the error estimator with respect to the energy which is defined by
$\eta^{energy}_{i}=C^{E}_{Lip}\big{(}\eta^{deformation}_{i}\big{)}^{2}+{\eta^{e}_{E}}_{k}+{\eta^{f}_{E}}_{k},$
(6.6)
for each element and the refinement algorithm is exactly the same.
In short, the first and second mesh generating schemes tend to minimize the
error in the deformation gradient and the third one tends to minimize the
error in the total energy.
### 6.2. Numerical Results
We compare the relative errors defined in 6.1 and 6.2. We plot the relative
errors against the number of degrees of freedom with respect to the meshes
generated.
Figure 1. Relative Error of the Gradient Figure 2. Efficiency Factor of the
Gradient
Figure 1 shows that the pre-defined optimal mesh performs better than the two
mesh refinement strategies for a fix number of degrees of freedom. The
possible reason for this is that, due to some technical difficulty in coding,
both of the mesh refinement algorithms tend to produce larger atomistic region
by merging the elements in the continuum region to the atomistic region and
create some unnecessary degrees of freedom. For the two mesh refinement
strategies, the one according to the gradient error indicator perform better
asymptotically.
Figure 2 shows the efficiency factor of the error estimator of the deformation
gradient. It shows that the efficiency factor is comparatively large but
decreases as the number of degrees of freedom increases and finally become
stable. The reason for this phenomenon lies in the form of the external force.
One can show that if the external force takes the form of $f(r)=\frac{1}{r}$,
where $r$ is the distance to the centre of the defect, then the residual due
to the external force is of order $h^{2}$ as opposed to order $h$ in general
which is achieved by our analysis. As a result, our estimate exaggerate the
real error by $\frac{1}{h}$ for this particular external force. This
phenomenon gradually disappear as the continuum region moves apart from the
centre of the defect since the influence of this exaggeration is eliminated as
the external force tends to $0$ when it is away from the centre of the defect,
which makes the residual of the sotred energy become the leading error term.
It can also well explain the fact that the efficiency of the estimate is
better for the mesh refinement strategies than the pre-defined mesh for a
certain number of degrees of freedom, as the two mesh refinement algorithms
tend to put more atoms in the atomistic region, i.e., the continuum region is
further away from the centre of defect than that of the pre-defined mesh.
Figure 3. Relative Error of the Total Energy Figure 4. Efficiency Factor of
the Energy
Figure 3 shows that the refinement based on the energy error performs the best
among all the three mesh generating schemes.
Figure 4 shows the efficiency factor of the error estimator of the energy. For
the same reason, this factor decreases as the number of degrees of freedom
increases and finally becomes stable.
## 7\. Conclusion
We have presented the a posteriori error estimates for the Consistent Energy-
Based QC method in one dimension. The procedure of the estimate is the same as
that in [8]. However, since the formulation of the QC problem is newly
developed and is totally different from previous ones, new techniques have
been developed and applied to deal with the difficulty in the analysis.
Several results derived may be of independent interest and usefulness. In
addition, the error estimate of the total energy is also derived. Numerical
experiments are also implemented to illustrate our analysis.
Particular interesting future work are the extension and the implementation of
the a posteriori error estimate in higher dimensional problems. The difficulty
lies in the complication of the formulation and the varied location of the
interaction bonds. However, since a priori analysis for the two dimensional
problem has been proposed [6], ways of circumventing these difficulties could
be a source of reference.
## Appendix A Detailed Analysis for the Residuals of the Stored Energy
In this section, we provide the omitted detailed analysis for the residuals of
the stored energy, namely
$\mathcal{E}^{\prime}_{\rm a}(J_{\mathcal{U}_{{\rm
qc}}}y_{h})[v]-\mathcal{E}^{\prime}_{{\rm qc}}(y_{h})[J_{\mathcal{U}}[v]\text{
and }\mathcal{E}_{\rm a}(J_{\mathcal{U}_{{\rm qc}}}y_{h})-\mathcal{E}_{{\rm
qc}}(y_{h}),$
where $y_{h}\in\mathcal{Y}_{{\rm qc}}$, $y^{\prime}_{h}(x)>0\ \forall
x\in\mathbb{R}$ and $v\in\mathcal{U}$.
The idea is to find the differences defined by
$\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x,$ (A.1)
and
$\varepsilon\phi(r_{b}D_{b}y_{h})-|b\cap\Omega_{\rm
a}|{r_{b}}\phi\big{(}r_{b}D_{b\cap\Omega_{\rm
a}}y\big{)}-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x,$ (A.2)
for each interaction bond $b$.
We have analyzed the cases that $b\in\Omega_{\rm a}$ and $b\in
T_{k}\cap\Omega_{c}$ and are left with the analysis for the cases that $b$ is
across the atomistic-continuum interface and the boundaries of the elements in
the continuum region. There are three cases and in each case there are three
subcases to be considered.
Case 1: $b$ is across two adjacent elements $T_{k},T_{k+1}\in\Omega_{c}$. In
this case $|b\cap\Omega_{\rm a}|=0$ and the atomistic contribution of the
interaction bond in the QC energy is $0$.
Subcase 1: If $b=\big{(}\ell_{k}\varepsilon,(\ell_{k}+1)\varepsilon\big{)}$,
then $r_{b}=1$, $b\cap T_{k}=[\ell_{k}\varepsilon,x^{h}_{k}]$, $b\cap
T_{k+1}=[x^{h}_{k},(\ell_{k}+1)\varepsilon]$,
$r_{b}D_{b}v=v^{\prime}_{\ell_{k}+1}$ and
$r_{b}D_{b}y_{h}=\frac{y_{h}((\ell_{k}+1)\varepsilon)-y_{h}(\ell_{k}\varepsilon)}{\varepsilon}=(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}.$
We have
$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}v^{\prime}_{\ell_{k}+1}-\frac{1}{r_{b}}\phi^{\prime}(r_{b}y_{h}|_{T_{k+1}})\int_{b\cap
T_{k}}r_{b}v^{\prime}\,{\rm
d}x-\frac{1}{r_{b}}\phi^{\prime}(r_{b}y_{h}|_{T_{k}})\int_{b\cap
T_{k+1}}r_{b}v^{\prime}\,{\rm d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{\\{}\phi^{\prime}(r_{b}D_{b}y_{h})-(1-\theta_{k})\phi^{\prime}(y_{h}^{\prime}|_{T_{k+1}})-\theta_{k}\phi^{\prime}(\theta_{k}y_{h}^{\prime}|_{T_{k}})\bigg{\\}}v^{\prime}_{\ell_{k}+1},$
and
$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}})-\int_{b\cap
T_{k}}\phi(y_{h}^{\prime}|_{T_{k}})\,{\rm d}x-\int_{b\cap
T_{k+1}}\phi(y_{h}^{\prime}|_{T_{k+1}})\,{\rm d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{(}\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}})-\theta_{k}\phi(y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(y_{h}^{\prime}|_{T_{k+1}})\bigg{)}.$
(A.3)
Subcase 2: If
$b=\big{(}(\ell-1)_{k}\varepsilon,(\ell_{k}+1)\varepsilon\big{)}$, then
$r_{b}=2$, $b\cap T_{k}=\big{[}(\ell_{k}-1)\varepsilon,x^{h}_{k}\big{]}$,
$b\cap T_{k+1}=\big{[}x^{h}_{k},(\ell_{k}+1)\varepsilon\big{]}$,
$r_{b}D_{b}v=v^{\prime}_{\ell_{k}+1}+v^{\prime}_{\ell_{k}}$ and
$r_{b}D_{b}y_{h}=\frac{y_{h}(\ell_{k}\varepsilon)-y_{h}((\ell_{k}-1)\varepsilon)}{\varepsilon}=(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}}.$
We have
$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}}\big{)}(v^{\prime}_{\ell_{k}+1}+v^{\prime}_{\ell_{k}})$
$\displaystyle-\frac{1}{r_{b}}\phi^{\prime}(r_{b}y_{h}|_{T_{k+1}})\int_{b\cap
T_{k}}r_{b}v^{\prime}\,{\rm
d}x-\frac{1}{r_{b}}\phi^{\prime}(r_{b}y_{h}|_{T_{k}})\int_{b\cap
T_{k+1}}r_{b}v^{\prime}\,{\rm d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{\\{}\big{[}\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}}\big{)}-\phi^{\prime}(2y^{\prime}_{h}|_{T_{k}})\big{]}v^{\prime}_{\ell_{k}}$
$\displaystyle+\big{[}\phi^{\prime}\big{(}(1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}}\big{)}-(1-\theta_{k})\phi^{\prime}(2y^{\prime}_{h}|_{T_{k+1}})-\theta_{k}\phi^{\prime}(2y^{\prime}_{h}|_{T_{k}})\big{]}v^{\prime}_{\ell_{k}+1}\bigg{\\}},$
and
$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varepsilon\bigg{(}2\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})y_{h}^{\prime}|_{T_{k}})-(1+\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k+1}})\bigg{)}.$
(A.4)
Subcase 3: If $b=\big{(}\ell_{k}\varepsilon,(\ell_{k}+2)\varepsilon\big{)}$,
then $r_{b}=2$, $b\cap T_{k}=\big{[}\ell_{k}\varepsilon,x^{h}_{k}\big{]}$,
$b\cap T_{k+1}=\big{[}x^{h}_{k},(\ell_{k}+2)\varepsilon\big{]}$,
$r_{b}D_{b}v=v^{\prime}_{\ell_{k}+2}+v^{\prime}_{\ell_{k}+1}$ and
$r_{b}D_{b}y_{h}=\frac{y_{h}((\ell_{k}+2)\varepsilon)-y_{h}(\ell_{k}\varepsilon)}{\varepsilon}=(2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}.$
We have
$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\phi^{\prime}\big{(}(2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}})(v^{\prime}_{\ell_{k}+2}+v^{\prime}_{\ell_{k}+1})$
$\displaystyle-\frac{1}{r_{b}}\phi^{\prime}(r_{b}y_{h}|_{T_{k+1}})\int_{b\cap
T_{k}}r_{b}v^{\prime}\,{\rm
d}x-\frac{1}{r_{b}}\phi^{\prime}(r_{b}y_{h}|_{T_{k}})\int_{b\cap
T_{k+1}}r_{b}v^{\prime}\,{\rm d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{\\{}\big{[}\phi^{\prime}\big{(}(2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}-(1-\theta_{k})\phi^{\prime}(2y^{\prime}_{h}|_{T_{k+1}})-\theta_{k}\phi^{\prime}(2y^{\prime}_{h}|_{T_{k}})\big{]}v^{\prime}_{\ell_{k}+1}$
$\displaystyle+\big{[}\phi^{\prime}\big{(}(2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}\big{)}-\phi^{\prime}(2y^{\prime}_{h}|_{T_{k+1}})\big{]}v^{\prime}_{\ell_{k}+2}\bigg{\\}},$
and
$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varepsilon\bigg{(}2\phi((2-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}})-\theta_{k}\phi(2y_{h}^{\prime}|_{T_{k+1}})-(2-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k}})\bigg{)}.$
(A.5)
Case 2: $b$ is across the left atomistic-continuum interface of an atomistic
region.
Subcase 1: If $b=(\ell_{k}\varepsilon,\ell_{k+1}\varepsilon)$, then $r_{b}=1$,
$b\cap\Omega_{c}=(\ell_{k}\varepsilon,x^{h}_{k})$, $b\cap\Omega_{\rm
a}=\big{(}x^{h}_{k},(\ell_{k}+1)\varepsilon\big{)}$,
$r_{b}D_{b}v=v_{\ell_{k}+1}^{\prime}$ and
$r_{b}D_{b}y_{h}=(1-\theta_{k})y_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}.$
We have
$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{[}\phi^{\prime}\big{(}(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}\big{)}-(1-\theta_{k})\phi^{\prime}(y^{\prime}_{h}|_{T_{k+1}})-\theta_{k}\phi^{\prime}(y^{\prime}_{h}|_{T_{k}})\bigg{]}v^{\prime}_{\ell_{k}+1},$
(A.6)
and
$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-|b\cap\Omega_{\rm
a}|{r_{b}}\phi\big{(}r_{b}D_{b\cap\Omega_{\rm
a}}y\big{)}-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\big{(}\phi((1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}})-\theta_{k}\phi(y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(y_{h}^{\prime}|_{T_{k+1}})\big{)}$
(A.7)
Subcase 2: If
$b=\big{(}(\ell_{k}-1)\varepsilon,(\ell_{k}+1)\varepsilon\big{)}$, then
$r_{b}=2$, $b\cap\Omega_{c}=\big{(}(\ell_{k}-1)\varepsilon,x^{h}_{k})$,
$b\cap\Omega_{\rm a}=\big{(}x^{h}_{k},(\ell_{k}+1)\varepsilon\big{)}$,
$r_{b}D_{b}v=v_{\ell_{k}+1}^{\prime}+v_{\ell_{k}}^{\prime}$ and
$r_{b}D_{b}y_{h}=(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+(1+\theta_{k})y^{\prime}_{h}|_{T_{k}},\quad
r_{b}D_{b\cap\Omega_{\rm a}}y_{h}=2y^{\prime}_{h}|_{T_{k+1}}$
We have
$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{\\{}\big{[}\phi^{\prime}\big{(}(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+(1+\theta_{k})y^{\prime}_{h}|_{T_{k}}\big{)}-(1-\theta_{k})\phi^{\prime}(2y^{\prime}_{h}|_{T_{k+1}})-\theta_{k}\phi^{\prime}(2y^{\prime}_{h}|_{T_{k}})\big{]}v^{\prime}_{\ell_{k}}$
$\displaystyle+\big{[}\phi^{\prime}\big{(}(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+(1+\theta_{k})y^{\prime}_{h}|_{T_{k}}\big{)}-\phi^{\prime}(2y^{\prime}_{h}|_{T_{k}})\big{]}v^{\prime}_{\ell_{k}}-\theta_{k}\phi^{\prime}(y^{\prime}_{h}|_{T_{k}})v^{\prime}_{\ell_{k}+1},$
(A.8)
and
$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-|b\cap\Omega_{\rm
a}|{r_{b}}\phi\big{(}r_{b}D_{b\cap\Omega_{\rm
a}}y\big{)}-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x$ $\displaystyle=$ $\displaystyle
2\varepsilon\big{(}\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+(1+\theta_{k})\theta_{k}y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k+1}})-(1+\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k}})\big{)}.$
(A.9)
Subcase 3: If $b=(\ell_{k}\varepsilon,(\ell_{k}+2)\varepsilon)$, then
$r_{b}=2$, $b\cap\Omega_{c}=\big{(}\ell_{k}\varepsilon,x^{h}_{k})$,
$b\cap\Omega_{\rm a}=\big{(}x^{h}_{k},(\ell_{k}+2)\varepsilon\big{)}$,
$r_{b}D_{b}v=v_{\ell_{k}+2}^{\prime}+v_{\ell_{k}+1}^{\prime}$,
$D_{b\cap\Omega_{\rm
a}}v=\frac{1}{2-\theta_{k}}v^{\prime}_{\ell_{k}+2}+\frac{1-\theta_{k}}{2-\theta_{k}}v^{\prime}_{\ell_{k}+1}$,
and
$r_{b}D_{b}y_{h}=y^{\prime}_{h}|_{T_{k+2}}+(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}\quad
r_{b}D_{b\cap\Omega_{\rm
a}}y_{h}=\frac{2}{2-\theta_{k}}y^{\prime}_{h}|_{T_{k+2}}+\frac{2(1-\theta_{k})}{2-\theta_{k}}y^{\prime}_{h}|_{T_{k+1}}.$
We have
$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{\\{}\big{[}\phi^{\prime}\big{(}y^{\prime}_{h}|_{T_{k+2}}+(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}\big{)}-\phi^{\prime}(\frac{2}{2-\theta_{k}}y^{\prime}_{h}|_{T_{k+2}}+\frac{2(1-\theta_{k})}{2-\theta_{k}}y^{\prime}_{h}|_{T_{k+1}})\big{]}v^{\prime}_{\ell_{k}+2}$
$\displaystyle+\big{[}\phi^{\prime}\big{(}y^{\prime}_{h}|_{T_{k+2}}+(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}\big{)}-(1-\theta_{k})\phi^{\prime}(\frac{2}{2-\theta_{k}}y^{\prime}_{h}|_{T_{k+2}}+\frac{2(1-\theta_{k})}{2-\theta_{k}}y^{\prime}_{h}|_{T_{k+1}})$
$\displaystyle\quad\
-\theta_{k}\phi^{\prime}(2y^{\prime}_{h}|_{T_{k}})\big{]}v^{\prime}_{\ell_{k}+1},$
(A.10)
Case 2: $b$ is across the right atomistic-continuum interface of an atomistic
region.
Subcase 1: If $b=(\ell_{k}\varepsilon,\ell_{k+1}\varepsilon)$, then $r_{b}=1$,
$b\cap\Omega_{c}=\big{(}x^{h}_{k},(\ell_{k}+1)\varepsilon\big{)}$,
$b\cap\Omega_{\rm a}=(\ell_{k}\varepsilon,x^{h}_{k})$,
$r_{b}D_{b}v=v_{\ell_{k}+1}^{\prime}$ and
$r_{b}D_{b}y_{h}=(1-\theta_{k})y_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}.$
We have
$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{[}\phi^{\prime}\big{(}(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}\big{)}-(1-\theta_{k})\phi^{\prime}(y^{\prime}_{h}|_{T_{k+1}})-\theta_{k}\phi^{\prime}(y^{\prime}_{h}|_{T_{k}})\bigg{]}v^{\prime}_{\ell_{k}+1},$
(A.11)
and
$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}})-\int_{b\cap
T_{k}}\phi(y_{h}^{\prime}|_{T_{k}})\,{\rm d}x-\int_{b\cap
T_{k+1}}\phi(y_{h}^{\prime}|_{T_{k+1}})\,{\rm d}x$ $\displaystyle=$
$\displaystyle\varepsilon\big{(}\phi((1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}})-\theta_{k}\phi(y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(y_{h}^{\prime}|_{T_{k+1}})\big{)}.$
(A.12)
Subcase 2: If $b=\big{(}\ell_{k}\varepsilon,(\ell_{k}+2)\varepsilon\big{)}$,
then $r_{b}=2$,
$b\cap\Omega_{c}=\big{(}x^{h}_{k},(\ell_{k}+2)\varepsilon\big{)}$,
$b\cap\Omega_{\rm a}=(\ell_{k}\varepsilon,x^{h}_{k})$,
$r_{b}D_{b}v=v^{\prime}_{\ell_{k}+2}+v^{\prime}_{\ell_{k}+1}$,
$r_{b}D_{b\cap\Omega_{\rm a}}v=v^{\prime}_{\ell_{k}+1}$ and
$\displaystyle
r_{b}D_{b}y_{h}=(2-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}\text{
and }r_{b}D_{b\cap\Omega_{\rm a}}y_{h}=y^{\prime}_{h}|_{T_{k}}.$ (A.13)
We have
$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{\\{}\big{[}\phi^{\prime}\big{(}(2-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}\big{)}-\theta_{k}\phi^{\prime}\big{(}2y^{\prime}_{h}|_{T_{k}})-(1-\theta_{k})\phi^{\prime}\big{(}2y^{\prime}_{h}|_{T_{k+1}}\big{]}v^{\prime}_{\ell_{k}+1}$
$\displaystyle+\big{[}\phi^{\prime}\big{(}(2-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}\big{)}-\phi^{\prime}(2y^{\prime}_{h}|_{T_{k+1}})\big{]}v^{\prime}_{\ell_{k}+2},$
(A.14)
and
$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\big{(}\phi((1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}})-\theta_{k}\phi(y_{h}^{\prime}|_{T_{k}})-(1-\theta_{k})\phi(y_{h}^{\prime}|_{T_{k+1}})\big{)}.$
(A.15)
Subcase 3: If
$b=\big{(}(\ell_{k}-1)\varepsilon,(\ell_{k}+1)\varepsilon\big{)}$, then
$r_{b}=2$, $b\cap\Omega_{c}=\big{(}x^{h}_{k},(\ell_{k}+1)\varepsilon\big{)}$,
$b\cap\Omega_{\rm a}=\big{(}(\ell_{k}-1)\varepsilon,x^{h}_{k}\big{)}$,
$r_{b}D_{b}v=v^{\prime}_{\ell_{k}+1}+v^{\prime}_{\ell_{k}}$,
$r_{b}D_{b\cap\Omega_{\rm
a}}v=\frac{\theta_{k}}{1+\theta_{k}}v^{\prime}_{\ell_{k}+1}+\frac{1}{1+\theta_{k}}v^{\prime}_{\ell_{k}}$
and
$\displaystyle
r_{b}D_{b}y_{h}=(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}+y^{\prime}_{h}|_{T_{k-1}}\text{
and }r_{b}D_{b\cap\Omega_{\rm
a}}y_{h}=\frac{2\theta_{k}}{1+\theta_{k}}y^{\prime}_{h}|_{T_{k}}+\frac{2}{1+\theta_{k}}y^{\prime}_{h}|_{T_{k-1}}.$
(A.16)
We have
$\displaystyle\varepsilon\phi^{\prime}(r_{b}D_{b}y_{h})r_{b}D_{b}v-|b\cap\Omega_{\rm
a}|\phi^{\prime}(r_{b}D_{b\cap\Omega_{\rm a}}y_{h})D_{b\cap\Omega_{\rm
a}}v-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi^{\prime}(\nabla_{r_{b}}y_{h})\nabla_{r_{b}}v\,{\rm
d}x$ $\displaystyle=$
$\displaystyle\varepsilon\bigg{\\{}\big{[}\phi^{\prime}\big{(}(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}+y^{\prime}_{h}|_{T_{k-1}}\big{)}-\phi^{\prime}(\frac{2\theta_{k}}{1+\theta_{k}}y^{\prime}_{h}|_{T_{k}}+\frac{2}{1+\theta_{k}}y^{\prime}_{h}|_{T_{k-1}})\big{]}v^{\prime}_{\ell_{k}}$
$\displaystyle+\big{[}\phi^{\prime}\big{(}(1-\theta_{k})y^{\prime}_{h}|_{T_{k+1}}+\theta_{k}y^{\prime}_{h}|_{T_{k}}+y^{\prime}_{h}|_{T_{k-1}}\big{)}-\theta_{k}\phi^{\prime}(\frac{2\theta_{k}}{1+\theta_{k}}y^{\prime}_{h}|_{T_{k}}+\frac{2}{1+\theta_{k}}y^{\prime}_{h}|_{T_{k-1}})$
$\displaystyle-(1-\theta_{k})\phi^{\prime}(y^{\prime}_{h}|_{T_{k+1}})\big{]}v^{\prime}_{\ell_{k}+1},$
(A.17)
and
$\displaystyle\varepsilon\phi(r_{b}D_{b}y_{h})-\frac{1}{r_{b}}\int_{b\cap\Omega_{c}}\phi(\nabla_{r_{b}}y(x))\,{\rm
d}x$ $\displaystyle=$ $\displaystyle
2\varepsilon\big{(}\phi((1-\theta_{k})y_{h}^{\prime}|_{T_{k+1}}+\theta_{k}y_{h}^{\prime}|_{T_{k}}+y_{h}^{\prime}|_{T_{k-1}})$
$\displaystyle-(1+\theta_{k})\phi\big{(}\frac{2\theta_{k}}{1+\theta_{k}}y_{h}^{\prime}|_{T_{k}}+\frac{2}{1+\theta_{k}}y_{h}^{\prime}|_{T_{k-1}}\big{)}-(1-\theta_{k})\phi(2y_{h}^{\prime}|_{T_{k+1}})\big{)}.$
(A.18)
## Appendix B Approximation Properties
In this section, we prove some approximation properties which we have used but
are hardly found in standard text books.
Lemma 11. Let $v\in C^{0}(\mathbb{R})\cap W^{1,2}(\mathbb{R})$ be a periodic
function with $[a,b]$ being one of its period. Let $v_{h}$ be a
$\mathcal{P}_{1}$ interpolation of $v$ with respect to the nodes $a\leq
x_{0}<x_{1}<\cdots<x_{n}\leq b\leq x_{n+1}=x_{0}+(b-a)$ in $[x_{0},x_{n+1}]$,
subject to a constant, i.e., $v_{h}(x_{k})=v(x_{k})+C$, for
$k=\\{1,2,\ldots,n+1\\}$, and is extended periodically with period $b-a$. Then
the following estimate holds:
$\|v_{h}^{\prime}\|_{L^{2}_{[a,b]}}\leq\|v^{\prime}\|_{L^{2}_{[a,b]}},$ (B.1)
where $v^{\prime}$ and $v_{h}^{\prime}$ denote the weak derivatives of $v$ and
$v_{h}$ respectively.
###### Proof.
First we note that, since $v\in C^{0}(\mathbb{R})$ and $v_{h}$ is a
$\mathcal{P}_{1}$ interpolation of $v$, the weak derivative of $v_{h}$ on
$(x_{k},x_{k+1})$ is defined by
$v_{h}^{\prime}(x)=\frac{v(x_{k+1})-v(x_{k})}{x_{k+1}-x_{k}}.$
Since $v\in C^{0}(\mathbb{R})$ is piecewise differentiable, we have
$v(x_{k+1})-v(x_{k})=\int_{x_{k}}^{x_{k+1}}v^{\prime}(t)\,{\rm d}t,$
where $v^{\prime}$ is the weak derivative of $v$. By the periodicity of
$v_{h}^{\prime}$ and $v^{\prime}$, and Cauchy-Schwarz Inequality, we have
$\displaystyle\|v_{h}^{\prime}\|^{2}_{L^{2}_{[a,b]}}$
$\displaystyle=\int_{a}^{b}\big{[}v_{h}^{\prime}(x)\big{]}^{2}\,{\rm d}x$
$\displaystyle=\int_{x_{0}}^{x_{n+1}}\big{[}v_{h}^{\prime}(x)\big{]}^{2}\,{\rm
d}x$
$\displaystyle=\sum_{k=0}^{n}\int_{x_{k}}^{x_{k+1}}(\frac{v(x_{k+1})-v(x_{k})}{x_{k+1}-x_{k}})^{2}\,{\rm
d}x$
$\displaystyle=\sum_{k=0}^{n}\int_{x_{k}}^{x_{k+1}}\frac{(\int_{x_{k}}^{x_{k+1}}v^{\prime}(t)\,{\rm
d}t)^{2}}{(x_{k+1}-x_{k})^{2}}\,{\rm d}x$
$\displaystyle\leq\sum_{k=0}^{n}\frac{1}{x_{k+1}-x_{k}}(\int_{x_{k}}^{x_{k+1}}\,{\rm
d}t)(\int_{x_{k}}^{x_{k+1}}|v^{\prime}(t)|^{2}\,{\rm d}t)$
$\displaystyle=\sum_{k=0}^{n}\int_{x_{k}}^{x_{k+1}}|v^{\prime}(t)|^{2}\,{\rm
d}t$ $\displaystyle=\int_{a}^{b}|v^{\prime}(t)|^{2}\,{\rm d}t$
$\displaystyle=\|v^{\prime}\|^{2}_{L^{2}_{[a,b]}}.$
Taking the square root on both sides gives the stated result. ∎
Lemma 12. Let $v\in C^{0}([a,b])\cap W^{1,2}([a,b])$ and $I_{h}v$ is the
$\mathcal{P}_{1}$ function that interpolates $v$ at the points $a$ and $b$. We
have the following inequality:
$\|v^{\prime}-(I_{h}v)^{\prime}\|^{2}_{L^{2}_{(a,b)}}\leq\|v^{\prime}\|^{2}_{L^{2}_{(a,b)}}.$
(B.2)
###### Proof.
Since $v(a)=I_{h}v(a)$ and $v(b)=I_{h}v(b)$, by the definition of $I_{h}v$, we
have
$\int_{a}^{b}v^{\prime}\,{\rm d}x=\int_{a}^{b}(I_{h}v)^{\prime}dx,$
and equivalently,
$\int_{a}^{b}\big{(}v^{\prime}-(I_{h}v)^{\prime}\big{)}\cdot 1\,{\rm d}x=0,$
where $v^{\prime}$ denotes the weak derivative of $v$ on $[a,b]$. This shows
that $(I_{h}v)^{\prime}$ is the best $L^{2}$ approximation of $v^{\prime}$ in
the space of $\mathcal{P}_{0}$ functions as $(I_{h}v)^{\prime}$ is a constant.
Therefore, by the property of best approximation,
$\|v^{\prime}-(I_{h}v)^{\prime}\|^{2}_{L^{2}_{(a,b)}}\leq\|v^{\prime}-C\|^{2}_{L^{2}_{(a,b)}},$
(B.3)
for any constant $C$. In particular, if we choose $C$ to be $0$, the stated
result holds. ∎
## Appendix C Discrete Sobolev Inequalities on Non-uniform mesh
In this section, we prove some discrete Soblev inequalities on non-uniform
mesh that are used in the residual analysis for the external force. These
results are extensions to the inequalities proved in [7, Lemma A.1, Lemma A.2,
Theorem A.4] on non uniform mesh.
Lemma 13. Let $\boldsymbol{g}\in\mathbb{R}^{L}$,
$\boldsymbol{\varepsilon}^{0},\boldsymbol{\varepsilon}^{1}\in\mathbb{R}^{L}$
and $\varepsilon^{0}_{i},\varepsilon^{1}_{i}>0\ \forall i=1,\ldots,L$,
$\boldsymbol{g}^{\prime}=(g^{\prime}_{i})_{i=2}^{L}\in\mathbb{R}^{L-1}$,
$g_{i}^{\prime}:=\frac{g_{i}-g_{i-1}}{\varepsilon^{1}}\ i=2,\ldots,L$. If
$\sum_{i=1}^{L}\varepsilon^{0}_{i}g_{i}=0$, then
$|g_{i}|\leq\frac{1}{h}\sum_{i=2}^{L}\bar{\varepsilon}^{1}_{k}|g_{k}^{\prime}|\phi_{i,k},$
(C.1)
where, $h=\sum_{i=1}^{L}\varepsilon^{0}_{i}$,
$\phi_{i,k}=\sum_{\ell=1}^{k-1}\varepsilon^{0}_{\ell}$ for $k=2,\ldots,i$ and
$\phi_{i,k}=\sum_{\ell=k}^{L}\varepsilon^{0}_{\ell}$ for $k=i+1,\ldots,L$.
###### Proof.
Let $i\in\\{1,\ldots,L\\}$, then
$\displaystyle h|g_{i}|$
$\displaystyle=|hg_{i}-\sum_{j=1}^{L}\varepsilon^{0}_{j}g_{j}|$
$\displaystyle=|\sum_{j=1}^{L}\varepsilon^{0}_{j}g_{i}-\sum_{j=1}^{L}\varepsilon^{0}_{j}g_{j}|$
$\displaystyle\leq\sum_{j=1}^{i-1}\varepsilon^{0}_{j}|g_{i}-g_{j}|+\sum_{j=i+1}^{L}\varepsilon^{0}_{j}|g_{i}-g_{j}|.$
Since
$|g_{i}-g_{j}|=|\sum_{k=j+1}^{i}\varepsilon^{1}_{k}g^{\prime}_{k}|,$
we have
$\displaystyle h|g_{i}|$
$\displaystyle\leq\sum_{j=1}^{i-1}\varepsilon^{0}_{j}\sum_{k=j+1}^{i}\varepsilon^{1}_{k}|g^{\prime}_{k}|+\sum_{j=i+1}^{L}\varepsilon^{0}_{j}\sum_{k=i+1}^{j}\varepsilon^{1}_{k}|g^{\prime}_{k}|$
$\displaystyle=\sum_{k=2}^{i}\varepsilon^{1}_{k}|g^{\prime}_{k}|\big{(}\sum_{j=1}^{k-1}\varepsilon^{0}_{j}\big{)}+\sum_{k=i+1}^{L}\varepsilon^{1}_{k}|g^{\prime}_{k}|\big{(}\sum_{j=k}^{L}\varepsilon^{0}_{j}\big{)}$
$\displaystyle=\sum_{k=2}^{L}\varepsilon^{1}_{k}|g^{\prime}_{k}|\phi_{i,k}.$
Divide both sides by $h$, we obtain the stated result. ∎
Lemma 14. (Discrete Poincare’s Inequality) Suppose that $L\geq 1$,
$\boldsymbol{\varepsilon}^{0},\boldsymbol{\varepsilon}^{1}\in\mathbb{R}^{L}$
with $\varepsilon^{0}_{i},\varepsilon^{1}_{i}>0$, $\forall i=1,\ldots,L$. Let
$\boldsymbol{g}\in\mathbb{R}^{L}$ such that
$\sum_{i=1}^{L}\varepsilon^{0}_{i}g_{i}=0$ and
$\boldsymbol{g}^{\prime}=(g^{\prime}_{i})_{i=2}^{L}\in\mathbb{R}^{L-1}$ such
that $g_{i}^{\prime}=\frac{g_{i}-g_{i-1}}{\varepsilon^{1}_{i}}$. Define
$\mathcal{D}_{0}$ to be the set $\\{1,\ldots,L\\}$ and $\mathcal{D}_{1}$ to be
the set $\\{2,\ldots,L\\}$, then
$\|\boldsymbol{g}\|_{\ell^{p}_{\boldsymbol{\varepsilon}^{0}(\mathcal{D}_{0})}}\leq\frac{1}{2}\frac{L^{2}\max\\{\max_{1\leq
i\leq L}\varepsilon^{0}_{i},\max_{2\leq k\leq
L}\varepsilon^{1}_{k}\\}^{2}}{h}\|\boldsymbol{g}^{\prime}\|_{\ell^{p}_{\boldsymbol{\varepsilon}^{1}(\mathcal{D}_{1})}},$
(C.2)
for $p\in\\{1,\infty\\}$, where $\ h=\sum_{i=1}^{L}\varepsilon^{0}_{i}$.
###### Proof.
Using the result of Lemma C, we have
$\displaystyle\sum_{i=1}^{L}\varepsilon^{0}_{i}|g_{i}|$
$\displaystyle\leq\sum_{i=1}^{L}\frac{\varepsilon^{0}_{i}}{h}\sum_{k=2}^{i}\varepsilon^{1}_{k}|g_{k}^{\prime}|\phi_{i,k}+\sum_{i=1}^{L}\frac{\varepsilon^{0}_{i}}{h}\sum_{k=i+1}^{L}\varepsilon^{1}_{k}|g_{k}^{\prime}|\phi_{i,k}$
$\displaystyle=\frac{1}{h}\bigg{[}\sum_{k=2}^{L}\big{(}\sum_{i=1}^{L}\varepsilon^{0}_{i}\phi_{i,k}\big{)}\varepsilon^{1}_{k}|g_{k}^{\prime}|\bigg{]}.$
Since
$\sum_{i=1}^{L}\varepsilon^{0}_{i}\phi_{i,k}\leq\max_{1\leq i\leq
L}\varepsilon^{0}_{i}\sum_{i=1}^{L}\phi_{i,k}=\max_{1\leq i\leq
L}\varepsilon^{0}_{i}\bigg{[}\sum_{i=1}^{k-1}\phi_{i,k}+\sum_{i=k}^{L}\phi_{i,k}\bigg{]},$
and
$\displaystyle\sum_{i=1}^{k-1}\phi_{i,k}+\sum_{i=k}^{L}\phi_{i,k}$
$\displaystyle\leq(k-1)\sum_{\ell=k}^{L}\varepsilon^{0}_{\ell}+\big{(}L-(k-1)\big{)}\sum_{\ell=1}^{k-1}\varepsilon^{0}_{\ell}$
$\displaystyle\leq\big{[}(k-1)\big{(}L-(k-1)\big{)}+\big{(}L-(k-1)\big{)}(k-1)\big{]}\max_{1\leq
i\leq L}\varepsilon^{0}_{i}$ $\displaystyle\leq\frac{1}{2}\max_{1\leq i\leq
L}\varepsilon^{0}_{i}L^{2}.$
Put these results together, we obtain the stated result for $p=1$. For
$p=\infty$,
$\displaystyle|g_{i}|$
$\displaystyle\leq\frac{1}{h}\sum_{k=2}^{L}\varepsilon^{1}_{k}|g_{k}^{\prime}|\phi_{i,k}$
$\displaystyle\leq\frac{1}{h}\bigg{[}\sum_{k=2}^{i}\varepsilon^{1}_{k}|g_{k}^{\prime}|\phi_{i,k}+\sum_{k=i+1}^{L}\varepsilon^{1}_{k}|g_{k}^{\prime}|\phi_{i,k}\bigg{]}$
$\displaystyle\leq\frac{1}{h}\sum_{k=2}^{L}\phi_{i,k}\max_{2\leq k\leq
L}\varepsilon^{1}_{k}|g_{k}^{\prime}|$
$\displaystyle\leq\frac{1}{2}\frac{L^{2}\max_{1\leq i\leq
L}\varepsilon^{0}_{i}}{h}\max_{2\leq k\leq
L}\varepsilon^{1}_{k}|g_{k}^{\prime}|.$
The stated result is obtained by taking the maximum of $\varepsilon^{0}_{i}$
and $\varepsilon^{1}_{k}$ over $\mathcal{D}_{0}$ and $\mathcal{D}_{1}$. ∎
Lemma 15. (Discrete Friedrichs’ Inequality) Suppose that $L\geq 1$,
$\boldsymbol{\varepsilon}^{0}$, $\boldsymbol{\varepsilon}^{1}$,
$\mathcal{D}_{0}$, $\mathcal{D}_{2}$ are the same as in Lemma C. Let
$\boldsymbol{f}\in\mathbb{R}^{L}$ such that $f_{1}=f_{L}=0$, and
$\boldsymbol{f}^{\prime}=(f_{i}^{\prime})_{i=2}^{L}\in\mathbb{R}^{L-1}$ such
that $f_{i}^{\prime}=\frac{f_{i}-f_{i-1}}{\varepsilon^{1}_{i}}$, then
$\|\boldsymbol{f}\|_{\ell^{p}_{\boldsymbol{\varepsilon}^{0}(\mathcal{D}_{0})}}\leq\frac{1}{2}(L-1)\max_{2\leq
i\leq
L-1}\max\\{\varepsilon^{0}_{i},\varepsilon^{1}_{i}\\}\|\boldsymbol{f}^{\prime}\|_{\ell^{p}_{\boldsymbol{\varepsilon}^{1}(\mathcal{D}_{1})}},$
(C.3)
for $p\in\\{1,\infty\\}$.
###### Proof.
For $p=1$,
$\displaystyle\sum_{i=1}^{L}\varepsilon^{0}_{i}|f_{i}|$
$\displaystyle=\sum_{i=2}^{L-1}\varepsilon^{0}_{i}|f_{i}|$
$\displaystyle=\frac{1}{2}\sum_{i=2}^{L-1}\varepsilon^{0}_{i}\big{[}|\sum_{j=2}^{i}(f_{j}-f_{j-1})|+|\sum_{j=i+1}^{L}(f_{j}-f_{j-1})|\big{]}$
$\displaystyle\leq\frac{1}{2}\sum_{i=2}^{L-1}\varepsilon^{0}_{i}\big{[}\sum_{j=2}^{i}\varepsilon^{1}_{j}|f_{j}^{\prime}|+\sum_{j=i+1}^{L}\varepsilon^{1}_{j}|f_{j}^{\prime}|\big{]}$
$\displaystyle=\frac{1}{2}\sum_{i=2}^{L-1}\varepsilon^{0}_{i}\sum_{j=1}^{L}\varepsilon^{1}_{j}|f_{j}^{\prime}|$
$\displaystyle\leq\frac{1}{2}(L-1)\max_{2\leq i\leq
L-1}\varepsilon^{0}_{i}\sum_{j=1}^{L}\varepsilon^{1}_{j}|f_{j}^{\prime}|.$
For $p=\infty$,
$|f_{i}|\leq\sum_{j=2}^{i}\varepsilon^{1}_{j}|f_{j}^{\prime}|=(i-1)\max_{2\leq
j\leq L}\varepsilon^{1}_{j}\max_{2\leq j\leq L}|f_{j}^{\prime}|,$
and
$|f_{i}|\leq\sum_{j=i+1}^{L}\varepsilon^{1}_{j}|f_{j}^{\prime}|=(L-i)\max_{2\leq
j\leq L}\varepsilon^{1}_{j}\max_{2\leq j\leq L}|f_{j}^{\prime}|.$
Thus we have
$\displaystyle\max_{i\in\mathcal{D}_{0}}|f_{i}|$
$\displaystyle\leq\min(i-1,L-i)\max_{2\leq j\leq
L}\varepsilon^{1}_{j}\max_{2\leq j\leq L}|f_{j}^{\prime}|$
$\displaystyle\leq\frac{1}{2}(L-1)\max_{2\leq j\leq
L}\varepsilon^{1}_{j}\max_{2\leq j\leq L}|f_{j}^{\prime}|.$
∎
Remark 2. The bounds we have got here are not optimal as if
$\varepsilon_{i}$’s and $\bar{\varepsilon}_{j}$’s vary too much, taking the
maximum of them in the inequalities could significantly reduce the sharpness
of the estimate. However, for the analysis of this paper, such a bound is
optimal enough to produce efficient error estimators and we leave the work of
looking for optimal bounds to future work. ∎
Theorem 16. (bounds on the interpolation error) Let $L\geq 1$,
$\boldsymbol{\varepsilon}^{0},\boldsymbol{\varepsilon}^{1},\boldsymbol{\varepsilon}^{2}\in\mathbb{R}^{L}$,
with $\varepsilon^{0}_{i},\varepsilon^{1}_{i},\varepsilon^{2}_{i}>0\ \forall
i=1,\ldots,L$. Let $\boldsymbol{f}\in\mathbb{R}^{L}$ and
$\boldsymbol{F}=\in\mathbb{R}^{L}$ such that $F_{1}=f_{1}$ and
$F_{i}=f_{1}+\frac{\sum_{j=2}^{i}\varepsilon^{0}_{i}}{h}(f_{L}-f_{1})\quad
i=2,\ldots,L,$ (C.4)
where $h=\sum_{i=2}^{L}\varepsilon^{0}_{i}$. Define
$\boldsymbol{f}^{\prime}=(f_{i}^{\prime})_{i=2}^{L}\in\mathbb{R}^{L-1}$ such
that $f_{i}^{\prime}=\frac{f_{i}-f_{i-1}}{\varepsilon^{1}_{i}}$ and
$\boldsymbol{f}^{\prime\prime}=(f_{i}^{\prime\prime})_{i=2}^{L-1}\in\mathbb{R}^{L-2}$
such that
$f_{i}^{\prime\prime}=\frac{f^{\prime}_{i+1}-f^{\prime}_{i}}{\varepsilon^{2}_{i}}$,
and $\boldsymbol{F}^{\prime}$ and $\boldsymbol{F}^{\prime\prime}$ are defined
in the same way. Let $\mathcal{D}_{0}$, $\mathcal{D}_{1}$ be the same sets
defined in Lellmma C and $\mathcal{D}_{2}$ be the set $\\{2,\ldots,L-1\\}$.
Then, for $p\in\\{1,\infty\\}$,
$\|\boldsymbol{f}-\boldsymbol{F}\|_{\ell_{\boldsymbol{\varepsilon}^{0}}^{p}(\mathcal{D}_{0})}\leq\frac{1}{4}\frac{L^{3}\max_{2\leq
i\leq L-1}\varepsilon_{i}^{0}\max_{2\leq j\leq
L-1}\varepsilon^{1}_{j}\max_{2\leq k\leq
L-1}\varepsilon^{2}_{j}}{h}\|\boldsymbol{f}^{\prime\prime}\|_{\ell_{\boldsymbol{\varepsilon}^{2}}^{p}(\mathcal{D}_{2})}.$
(C.5)
###### Proof.
Let $\boldsymbol{g}=\boldsymbol{f}-\boldsymbol{F}$, by the definition of
$\boldsymbol{F}$, we have $g_{1}=g_{L}=0$ and
$\sum_{i=2}^{L}\varepsilon_{i}g_{i}^{\prime}=\sum_{i=2}^{L}(f_{i}-f_{i-1})-\sum_{i=2}^{L}(F_{i}-F_{i-1})=0.$
By Lemma C,
$\|\boldsymbol{g}\|_{\ell_{\boldsymbol{\varepsilon}^{1}}^{p}(\mathcal{D}_{0})}\leq\frac{1}{2}(L-1)\max_{2\leq
i\leq
L-1}\max\\{\varepsilon^{0}_{i},\varepsilon^{1}_{i}\\}\|\boldsymbol{g}^{\prime}\|_{\ell_{\boldsymbol{\varepsilon}}^{p}(\mathcal{D}_{1})},$
as $g_{1}=g_{L}=0$, and by Lemma C,
$\|\boldsymbol{g}^{\prime}\|_{\ell_{\boldsymbol{\varepsilon}}^{p}(\mathcal{D}_{1})}\leq\frac{1}{2}\frac{L^{2}\max\\{\max_{1\leq
i\leq L}\varepsilon^{1}_{i},\max_{2\leq k\leq
L-1}\varepsilon^{2}_{k}\\}^{2}}{h}\|\boldsymbol{g}^{\prime\prime}\|_{\ell_{\bar{\boldsymbol{\varepsilon}}}^{p}(\mathcal{D}_{2})},$
as $\sum_{i=2}^{L}\varepsilon_{i}g_{i}^{\prime}=0$. Since
$\boldsymbol{F}^{\prime\prime}=0$, from which we know
$\boldsymbol{g}^{\prime\prime}=\boldsymbol{f}^{\prime\prime}$, the stated
estimate holds. ∎
## References
* [1] Marcel Arndt and Mitchell Luskin. Goal-oriented atomistic-continuum adaptivity for the quasicontinuum approximation. Int. J. Multiscale Comput. Engrg., 5(49-50):407–415, 2007.
* [2] Marcel Arndt and Mitchell Luskin. Error estimation and atomistic-continuum adaptivity for the quasicontinuum approximation of a Frenkel-Kontorova model. Multiscale Model. Simul., 7(1):147–170, 2008.
* [3] Marcel Arndt and Mitchell Luskin. Goal-oriented adaptive mesh refinement for the quasicontinuum approximation of a Frenkel-Kontorova model. Comput. Methods Appl. Mech. Engrg., 197(49-50):4298–4306, 2008\.
* [4] W. E and P. Ming. Analysis of the local quasicontinuum method. In Frontiers and prospects of contemporary applied mathematics, volume 6 of Ser. Contemp. Appl. Math. CAM, pages 18–32. Higher Ed. Press, Beijing, 2005.
* [5] P. Lin. Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp., 72(242):657–675, 2003.
* [6] C. Ortner and A.V. Shapeev. A priori error analysis of an energy-based atomistic/continuum coupling method for pair interactions in two dimensions. arXiv:1104.0311v1.
* [7] C. Ortner and E. Süli. Analysis of a quasicontinuum method in one dimension. M2AN Math. Model. Numer. Anal., 42(1):57–91, 2008.
* [8] C. Ortner and H. WANG. A priori error estimates for energy-based quasicontinuum approximations of a periodic chain. 2010\. to appear in M3AS Math. Model. Meth. Appl. Sci.
* [9] Serge. Prudhomme, Paul. Bauman, and Tinsley Oden. Error control for molecular statics problems. Int. J. Multiscale Comput. Engrg., 4:647–662, 2007.
* [10] A.V. Shapeev. Consistent energy-based coupling of atomistic and continuum static models for two-body potential. arXiv:1010.0512.
* [11] T. Shimokawa, J.J. Mortensen, J. Schiotz, and K.W. Jacobsen. Matching conditions in the quasicontinuum method: Removal of the error introduced at the interface between the coarse-grained and fully atomistic region. Phys. Rev. B, 69(21):214104, 2004.
|
arxiv-papers
| 2011-12-22T22:28:04 |
2024-09-04T02:49:25.658936
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hao Wang",
"submitter": "Hao Wang",
"url": "https://arxiv.org/abs/1112.5480"
}
|
1112.5775
|
# A single trapped ion in a finite range trap
M. Bagheri Harouni m-bagheri@phys.ui.ac.ir M. Davoudi Darareh
m.davoudi@sci.ui.ac.ir Department of Physics, Faculty of Science, University
of Isfahan, Hezar Jerib, Isfahan, 81746-73441, Iran
###### Abstract
This paper presents a method to describe dynamics of an ion confined in a
realistic finite range trap. We model this realistic potential with a solvable
one and we obtain dynamical variables (raising and lowering operators) of this
potential. We consider coherent interaction of this confined ion in a finite
range trap and we show that its center-of-mass motion steady state is a
special kind of nonlinear coherent states. Physical properties of this state
and their dependence on the finite range of potential are studied.
###### keywords:
Nonclassical property , Finite range trap , Trapped ion , Nonlinear coherent
state
###### PACS:
42.50.Dv, 42.50.Gy
## 1 Introduction
Single trapped ions represent elementary quantum systems that are
approximately isolated from the environment [1]. In these systems both
internal electronic states and external center-of-mass motional states
(external states) can be coupled to and manipulated by light fields. This
makes the trapped ion systems suited for quantum optical and quantum dynamical
studies under well-controlled conditions. Motivated by the strong analogy
between cavity quantum electrodynamics and the trapped ion system, various
theoretical and experimental proposals have been made on how to create
nonclassical and arbitrary states of motion of trapped ions. Preparation of
the number states [2], coherent, quadrature squeezed number states and
superposition of the number states were considered in this system
experimentally [3] and theoretically [4]. Experimental preparation of the
Schrödinger-cat state was considered [5]. Theoretical schemes for generation
of arbitrary center-of-mass motional states of a trapped ion is described in
[6]. Moreover, the possibility of generation of even and odd coherent states
of the center-of-mass motion of a trapped ion is considered [7]. A new scheme
for preparation of nonclassical motional states of trapped ions is
investigated in [8]. Recently, Preparation of Dicke states in an ion chain is
considered theoretically and experimentally [9]. In addition to the above-
mentioned attempts, preparation of different family of nonlinear coherent
states [10], is also studied theoretically [11, 12].
On the other hand, the trapped ion system has found some applications in
quantum information and quantum computation [13]. For quantum information
processing by trapped ion, preparation of some special states is important.
Among these states, entangled states have found crucial importance.
Preparation of the entangled states of trapped ion is considered recently
[14]. For quantum computation applications, preparation of the two-dimensional
cluster-state is considered [15]. Because of the some similarities between the
trapped ion system and the Jaynes-Cummings model [16], the trapped ion system
is used to realize different generalizations of the Jaynes-Cummings model
which have found some applications in quantum information [17].
In all of the above-mentioned efforts on the trapped ion system, it is assumed
that the ion is confined in a harmonic oscillator-shaped potential while the
dimension of this potential extended to infinity. Hence, the range of the
confining trap is infinity. However, in the realistic experimental setup, the
dimension of trap is finite and the realistic trapping potential is not a
harmonic oscillator potential but the truncated and modified one within the
extension of the trap. In this paper, we assume that the confining potential
for ion has finite range. We will model this confining potential with a
solvable one. By using the concept of the $f$-deformed oscillator [10], we try
to consider the trapped ion in confining potential with finite range as an
$f$-deformed oscillator and in this context we obtain raising and lowering
operators (dynamical variables) of this potential. The finite range effects of
this model can be used in traps of the order of nano-scale, called nano Paul
traps, that are attracted a great deal of attention recently [18]. It is worth
to note that the confining model potential which is considered here is used
for other confined physical systems, such as the Bose-Einstein condensate
[19], and carriers in a quantum well [20]. The $f$-deformed oscillator
approach where we have considered here, has been used before for some other
confined systems [21].
This paper presents a method to describe dynamics of an ion confined in a
finite range trap. We will show that stationary state of the center-of-mass
motion of the trapped ion is a special kind of nonlinear coherent states where
its properties depend on the range of the confining potential. The outline of
the paper is as follows. Section 2 deals with scheme for model potential and
in this section we will obtain dynamical variables of this potential in the
context of the $f$-deformed oscillator. In Sec. 3 we propose coherent
interaction of an ion confined in a finite range potential and we consider its
dynamics in the steady state. In this section we will obtain an eigenvalue
equation for the state of the center-of-mass motion of the ion. In Sec. 4 we
summarize definition of the nonlinear coherent states and we will show that
the steady state of the ion motion can be considered as a nonlinear coherent
state. Physical properties of this system are investigated in this section.
Section 5 is devoted to the conclusion.
## 2 Algebraic approach for a particle in a finite range potential
To consider an ion in a finite range trap, we try to model the potential
energy function of the realistic trap by an analytically solvable potential.
For comparing new results with previous ones we are looking for a potential
which reduces to the harmonic oscillator potential in a specific limit of its
parameters. A potential which has this property is the modified Pöschl-Teller
(MPT) potential [22]. The MPT potential has the following form
$V(x)=D\,tanh^{2}(\frac{x}{\delta}),$ (1)
where $D$ is the depth of the well, $\delta$ determines the range of the
potential and $x$ gives the relative distance from the equilibrium position.
The well depth, D, can be defined as $D=\frac{1}{2}m\omega^{2}\delta^{2}$,
with mass of the particle $m$ and angular frequency $\omega$ of the harmonic
oscillator, so that, in the limiting case $D\rightarrow\infty$(or
$\delta\rightarrow\infty)$, but keeping the product $m\omega^{2}$ finite, the
MPT potential energy reduces to the harmonic potential energy,
$\lim_{D\rightarrow\infty}V(x)=\frac{1}{2}m\omega^{2}x^{2}$. Solving the
Schrödinger equation, the energy eigenvalues for the MPT potential are
obtained as [23]
$E_{n}=D-\frac{\hbar^{2}\omega^{2}}{4D}(s-n)^{2},\hskip
28.45274ptn=0,1,2,\cdots,[s]$ (2)
in which $s=(\sqrt{1+(\frac{4D}{\hbar\omega})^{2}}-1)/2$, and $[s]$ stands for
the closest integer to $s$ that is smaller than $s$. The MPT oscillator
quantum number $n$ can not be larger than the maximum number of bound states
$[s]$, because of the dissociation condition $s-n\geq 0$. Detailed description
about this energy spectrum can be found in [24]. By introducing a
dimensionless parameter
$N=\frac{4D}{\hbar\omega}=\frac{2m\omega\delta^{2}}{\hbar}$, the bound energy
spectrum in equation (2) can be rewritten as
$E_{n}=\hbar\omega[-\frac{n^{2}}{N}+(\sqrt{1+\frac{1}{N^{2}}}-\frac{1}{N})n+\frac{1}{2}(\sqrt{1+\frac{1}{N^{2}}}-\frac{1}{N})].$
(3)
The relation (3) shows a nonlinear dependence on the quantum number $n$, so
that, different energy levels are not equally spaced. As is evident, $N$ is a
dimensionless parameter and from now we refer this parameter as the depth of
the trap. It is clear that, in the limiting case $D\rightarrow\infty$ (or
$N\rightarrow\infty$), the energy spectrum for the quantum harmonic oscillator
will be obtained, i.e., $E_{n}=\hbar\omega(n+\frac{1}{2})$. This means that
for finite values of $D$ (or finite values of $\delta$), we have a deformed
quantum oscillator, which its natural deformation from the quantum harmonic
oscillator can be amplified by decreasing $D$ or $N$. Thus, the well depth of
this potential that identifies its range, is used to approximate the harmonic
oscillator potential and it can also be considered as a controllable physical
deformation parameter. It is interesting to note that the dimensionless
parameter $N$ can also be written as $N=\frac{\delta}{\Delta x}$. Here $\Delta
x=\sqrt{\frac{\hbar}{2m\omega}}$, is the ground state wave function spread
which for typical traps is of the order of nanometer $(nm)$ [1]. $\delta$,
that determines the range of the potential would be of the same order of
magnitude as the ion-electrode distance in a Paul trap system. It results that
if trap size be of the order of $nm$, the finite range effects of the trap
would be important. Such kind of the Paul traps have considered recently [18].
It is shown that [21], each quantum system which has an unequal spaced energy
spectrum can be considered as an $f$-deformed oscillator. Therefore, according
to the energy spectrum of the MPT potential, this system can be considered as
an $f$-deformed oscillator [24]. On the other hand, the $f$-deformed quantum
oscillator [10], as a nonlinear oscillator with a specific kind of
nonlinearity, is characterized by the following deformed dynamical variables
$\hat{A}$ and $\hat{A}^{\dagger}$
$\displaystyle\hat{A}$ $\displaystyle=$
$\displaystyle\hat{a}f(\hat{n})=f(\hat{n}+1)\hat{a},$
$\displaystyle\hat{A}^{\dagger}$ $\displaystyle=$ $\displaystyle
f(\hat{n})\hat{a}^{\dagger}=\hat{a}^{\dagger}f(\hat{n}+1),\hskip
42.67912pt\hat{n}=\hat{a}^{\dagger}\hat{a},$ (4)
where $\hat{a}$ and $\hat{a}^{\dagger}$ are usual boson annihilation and
creation operators $([\hat{a},\hat{a}^{\dagger}]=1)$, respectively. The real
deformation function $f(\hat{n})$ is a nonlinear operator-valued function of
the harmonic number operator $\hat{n}$, which introduces some nonlinearities
to the system. From equation (2), it follows that the $f$-deformed operators
$\hat{A}$, $\hat{A}^{\dagger}$ and $\hat{n}$ satisfy the following closed
algebra
$\displaystyle[\hat{A},\hat{A}^{\dagger}]=$
$\displaystyle(\hat{n}+1)f^{2}(\hat{n}+1)-\hat{n}f^{2}(\hat{n}),$ (5)
$\displaystyle[\hat{n},\hat{A}]=$ $\displaystyle-\hat{A},\hskip
42.67912pt[\hat{n},\hat{A}^{{\dagger}}]=\hat{A}^{{\dagger}}.$
The above-mentioned algebra, represents a deformed Heisenberg-Weyl algebra
whose nature depends on the nonlinear deformation function $f(\hat{n})$. An
$f$-deformed oscillator is a nonlinear system characterized by a Hamiltonian
of the harmonic oscillator form
$\hat{H}=\frac{\hbar\omega}{2}(\hat{A}^{\dagger}\hat{A}+\hat{A}\hat{A}^{\dagger}).$
(6)
Using equation (2) and the number state representation
$\hat{n}|n\rangle=n|n\rangle$, the eigenvalues of the Hamiltonian (6) can be
written as
$E_{n}=\frac{\hbar\omega}{2}[(n+1)f^{2}(n+1)+nf^{2}(n)].$ (7)
It is worth noting that in the limiting case $f(n)\rightarrow 1$, the deformed
algebra (5) and the deformed energy eigenvalues (7) will reduce to the
conventional Heisenberg-Weyl algebra and the harmonic oscillator spectrum,
respectively.
Comparing the bound energy spectrum of the MPT oscillator, equation (3), and
the energy spectrum of an $f$-deformed oscillator, equation (7), we obtain the
corresponding deformation function for the MPT oscillator as
$f^{2}(\hat{n})=\sqrt{1+\frac{1}{N^{2}}}-\frac{\hat{n}}{N}.$ (8)
Furthermore, the ladder operators of the bound eigenstates of the MPT
Hamiltonian can be written in terms of the conventional operators $\hat{a}$
and $\hat{a}^{\dagger}$ as follows
$\hat{A}=\hat{a}\sqrt{\sqrt{1+\frac{1}{N^{2}}}-\frac{\hat{n}}{N}},\hskip
14.22636pt\hat{A}^{\dagger}=\sqrt{\sqrt{1+\frac{1}{N^{2}}}-\frac{\hat{n}}{N}}\hat{a}^{\dagger}.$
(9)
These two operators satisfy the deformed Heisenberg-Weyl commutation relation
$[\hat{A},\hat{A}^{\dagger}]=\sqrt{1+\frac{1}{N^{2}}}-\frac{2\hat{n}+1}{N},$
(10)
As is clear, in the limiting case $f(n)\rightarrow 1\;(N\rightarrow\infty)$
this deformed commutation relation will reduce to the conventional commutation
relation, $[\hat{a},\hat{a}^{{\dagger}}]=1$.
As a result, in this section we conclude that the trapped ion in MPT potential
can be considered as an $f$-deformed oscillator with specific kind of the
$f$-deformed Heisenberg-Weyl algebra.
In the following, we will consider coherent interaction of a single trapped
ion in a finite range trap with light fields. Then, we will generate the
nonlinear coherent states of ionic vibrational motion in a finite range trap
and finally we will investigate some physical properties of these states such
as, their number distribution, quadrature squeezing and their phase-space
distribution.
## 3 Ion dynamics in a finite range trap
As is usual in theoretical consideration of trapped ion systems, the confining
potential is assumed to be a spatial varying high-frequency time-dependent
field, the so-called Paul trap, $V(\vec{r},t)$. It is shown that, motion of a
particle inside a such high-frequency trap can be treated by averaging over
the fast motion (part of the particle displacement that its frequency is the
same as frequency of trap fields). In this approach a confined particle in
such a trap experiences a spatial static effective potential [25]. Usually
this static potential is assumed to be a three dimensional harmonic
oscillator-like potential so that in one direction ($x$-direction) can be
written as $V(x)=\frac{1}{2}m\omega^{2}x^{2}$ [1]. As is conventional, ion is
cooled to the ground-state of the trap and in this situation due to smallness
of the ratio of trap height to other energy scales, such as energy distance
between two adjacent energy levels of the trap, the trap is assumed extend to
infinity. However, in the realistic experimental setup, the dimension of the
trap is finite and the realistic trapping potential is not a harmonic
oscillator potential extending to infinity but the truncated and modified one
within the extension of the trap. Thus, the realistic confining potential
becomes flat near the edge of the trap and can be simulated by the tanh-shaped
potential, so that in one dimension ($x$-direction) can be written as
$V(x)=D\tanh^{2}(\frac{x}{\delta})$. In this paper, we try to investigate some
effects which originate from finite range property of the trap.
According to the previous section, we model this trapped ion as an
$f$-deformed quantum oscillator. Therefore, the oscillator-like Hamiltonian of
this system can be written as
$\hat{H}_{t}=\frac{\hbar\omega}{2}(\hat{A}\hat{A}^{\dagger}+\hat{A}^{\dagger}\hat{A}),$
(11)
where we interpret the operator $\hat{A}\;(\hat{A}^{\dagger})$ as the operator
whose action causes the transition of the ion center-of-mass motion to the
lower (upper) energy state of the trap. These operators are given in Eq. (9).
In fact, the Hamiltonian (11) is related to the external degrees of freedom of
the ion. According to the resonant condition, the ion is assumed as a two-
level system with the ground state $|g\rangle$ and the excited state
$|e\rangle$. Then, internal degrees of freedom of the ion can be expressed
with electronic flip operators $\hat{S}_{z}=|e\rangle\langle
e|-|g\rangle\langle g|$, $\hat{S}^{+}=|e\rangle\langle g|$ and
$\hat{S}^{-}=|g\rangle\langle e|$ which satisfy the usual $su(2)$ algebra. On
the other hand, with the help of the suitable laser fields, the internal
levels of the trapped ion can be coherently coupled to each other and to the
external motional degrees of freedom of the ion. Therefore, the total
Hamiltonian of the system may be given as
$\hat{H}=\hat{H}_{0}+\hat{H}_{int}(t),$ (12)
where $\hat{H}_{0}=\hat{H}_{t}+\hbar\omega_{i}\hat{S}_{z}$, with $H_{t}$ given
in Eq. (11), describes the free motion of the internal and external degrees of
freedom of the ion. Here, $\hbar\omega_{i}$ refers to the energy difference of
internal states of the ion, $\hbar\omega_{i}=E_{e}-E_{g}$. The interaction of
the ion with the laser fields is described by $\hat{H}_{int}(t)$ and is
written as
$\hat{H}_{int}(t)=g\left[E_{0}e^{-i(k_{0}\hat{x}-\omega_{i}t)}+E_{1}e^{-i[k_{1}\hat{x}-(\omega_{i}-\omega_{n})t]}\right]\hat{S}^{+}+\;H.c.\;,$
(13)
in which $g$ is coupling constant, $k_{0}$ and $k_{1}$ are the wave numbers of
the driving laser fields and $\omega_{n}$ refers to the energy of the lower
vibrational side-band with respect to the electronic transition of the ion.
$\omega_{n}$ is the frequency of the ion transition between energy levels of
the finite range trap. Because energy spectrum of the trap depends on the
energy level numbers and we consider a transition between specific side-band
levels, hence, we show the transition frequency with definite dependence to
$n$. In the above Hamiltonian, $\hat{H}_{int}(t)$, $\hat{x}$ is the operator
of the center-of-mass position and may be defined as [21]
$\hat{x}=\frac{\eta}{k_{l}}(\hat{A}+\hat{A}^{\dagger}),$ (14)
where $\eta$ being the Lamb-Dicke parameter and $k_{l}$ is associated wave
number to the characteristic length of the trap and assume to be $k_{l}\simeq
k_{0}\simeq k_{1}$. The interaction Hamiltonian (13) can be written as
$\hat{H}_{int}(t)=\hbar
e^{i\omega_{i}t}\left[\Omega_{0}+\Omega_{1}e^{-i\omega_{n}t}\right]e^{i\eta(\hat{A}+\hat{A}^{\dagger})}\hat{S}^{+}+H.c.\;,$
(15)
$\Omega_{0}=\frac{gE_{0}}{\hbar}$ and $\Omega_{1}=\frac{gE_{1}}{\hbar}$ are
the Rabi frequencies of the laser fields tuned to the electronic transition
and the lower sideband, respectively. The interaction Hamiltonian in the
interaction picture with respect to the $\hat{H}_{0}$ can be written as
$\hat{H}_{I}=\hbar\Omega_{1}\hat{S}^{+}\left[\frac{\Omega_{0}}{\Omega_{1}}+e^{-i\omega_{n}t}\right]\exp\left[i\eta\left(e^{-i\hat{\nu}_{n}t}\hat{A}+\hat{A}^{\dagger}e^{i\hat{\nu}_{n}t}\right)\right]+H.c.\;,$
(16)
where
$\hat{\nu}_{n}=\frac{\omega}{2}[(\hat{n}+2)f^{2}(\hat{n}+2)-\hat{n}f^{2}(\hat{n})]$.
In this relation the function $f(\hat{n})$ is given by Eq. (8).
By using the vibrational rotating-wave approximation [11] and applying the
disentangling approach in [26] for the exponential term which appeared in
equation (16), the interaction Hamiltonian (16) may be written as
$\hat{H}_{I}=\hbar\Omega_{1}\hat{S}^{+}\left[\frac{\Omega_{0}}{\Omega_{1}}F_{0}(\hat{n},\eta)+g(\eta)F_{1}(\hat{n},\eta)\hat{a}\right]+H.c.\;,$
(17)
where the function $F_{j}(\hat{n},\eta)\;(j=0,1)$ is defined by
$F_{j}(\hat{n},\eta)=\sum_{l=0}^{n}\frac{[g(\eta)]^{2l}}{l!(l+j)!}\frac{f(\hat{n})!f(\hat{n}+j)!}{[f(\hat{n}-l)!]^{2}}\frac{\hat{n}!}{(\hat{n}-l)!}M(\hat{n}-l).$
(18)
In this equation different functions are appeared which are defined as follows
$\displaystyle g(\eta)$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{\gamma}}\tan(\sqrt{\gamma}\eta),\hskip
28.45274ptX_{n}=\beta-\gamma(2n+1),$ $\displaystyle M(n)$ $\displaystyle=$
$\displaystyle e^{-\frac{X_{n}}{\gamma}\ln(\cos(\sqrt{\gamma}\eta))},$ (19)
where $\gamma=\frac{1}{N}$, $\beta=\sqrt{1+\frac{1}{N^{2}}}$ and $\hat{n}$ is
an operator whose eigenvalues, $n$, refer to the excitation energy level
number inside the trap. It is worth to note that in the limiting case
$N\rightarrow\infty$ which is equivalent to $f(n)\rightarrow 1$, the system
will reduce to the confined ion in the harmonic oscillator-shaped trap, which
has been considered in [11]. The function $F_{j}(\hat{n},\eta)$, given in Eq.
(18), will reduce to its counterpart in the harmonic oscillator-shaped trap
[11].
The time evolution of the system is characterized by the master equation
$\frac{d\hat{\rho}}{dt}=-\frac{i}{\hbar}[\hat{H}_{I},\hat{\rho}]+\frac{\Gamma}{2}(2\hat{S}^{-}\hat{\rho}^{{}^{\prime}}\hat{S}^{+}-\hat{S}^{+}\hat{S}^{-}\hat{\rho}-\hat{\rho}\hat{S}^{+}\hat{S}^{-}),$
(20)
where $\Gamma$ is the spontaneous emission rate. To account for the recoil of
spontaneously emitted photons the first term of the damping part of the master
equation contains
$\hat{\rho}^{{}^{\prime}}=\frac{1}{2}\int_{-1}^{1}dzY(z)e^{ik_{l}\hat{x}z}\hat{\rho}e^{-ik_{l}\hat{x}z},$
(21)
$Y(z)$ is the angular distribution of the spontaneous emission and
$\hat{\rho}$ is the vibronic density operator.
In the long-time limit, the ion will be populated in the ground state
$|g\rangle$ as a consequence of atomic spontaneous emission. In this case, the
steady-state solution of the master equation (20) can be assumed to be
$\hat{\rho}_{ss}=|g\rangle|\psi\rangle\langle\psi|\langle g|$, where
$|\psi\rangle$ stands for the vibronic motion of the ion. The stationary
solution of Eq. (20) can be found by setting $\frac{d\hat{\rho}}{dt}=0$ and
since
$\hat{S}^{-}|g\rangle\langle g|=\hat{S}^{+}\hat{S}^{-}|g\rangle\langle
g|=|g\rangle\langle g|\hat{S}^{+}\hat{S}^{-}=0,$ (22)
we obtain
$[\hat{H}_{I},\hat{\rho}_{ss}]=0.$ (23)
From this equation, we find that the vibronic state $|\psi\rangle$ satisfies
the following equation
$\hat{a}h(\hat{n})|\psi\rangle=\chi|\psi\rangle,\hskip
42.67912pt\chi=-\frac{\Omega_{0}}{g(\eta)\Omega_{1}}.$ (24)
In this equation $h(\hat{n})=F_{1}(\hat{n}-1,\eta)/F_{0}(\hat{n}-1,\eta)$.
## 4 Nonlinear coherent states of ionic vibrational motion and their physical
properties
Similar to the definition of the canonical coherent states [27], the coherent
state of a generalized $f$-deformed oscillator is defined as a right-hand
eigenstate of the generalized annihilation operator
$(\hat{A}=\hat{a}f(\hat{n}))$ as follows
$\hat{A}|\alpha,f\rangle=\alpha|\alpha,f\rangle.$ (25)
Due to the appearance of nonlinear deformation function, $f(\hat{n})$, in
definition of these states, they are called nonlinear coherent states.
According to this definition, vibronic state of the ion in the steady state,
Eq. (24), is a nonlinear coherent state with
$\displaystyle f(\hat{n})$ $\displaystyle=$ $\displaystyle
h(\hat{n})=\frac{F_{1}(\hat{n}-1,\eta)}{F_{0}(\hat{n}-1,\eta)},$
$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\chi=-\frac{\Omega_{0}}{g(\eta)\Omega_{1}}.$ (26)
Nonlinear coherent states can be expanded in terms of the usual Fock states
$(\hat{n}|n\rangle=n|n\rangle)$ as follows
$|\alpha,f\rangle=N_{f}\sum_{n}\frac{\alpha^{n}}{\sqrt{n!}f(n)!}|n\rangle,\hskip
28.45274ptN_{f}=\left[\sum_{n}\frac{|\alpha|^{2n}}{n![f(n)!]^{2}}\right]^{-\frac{1}{2}},$
(27)
where $f(n)!=f(n)f(n-1)\cdots f(0)$. Thus, the steady state of the ion in Eq.
(24) is a special kind of the nonlinear coherent state where its properties
are defined by the function $h(\hat{n})$. This function is characterized by
the Lamb-Dicke parameter $\eta$ and quantum number $n$ which refers to the
level of vibronic excitation. Moreover, according to the Eq. (24), nonlinear
coherent state of the ion depends on the complex parameter $\chi$, which is
controlled by the Rabi frequencies of the lasers, the Lamb-Dicke parameter and
$\gamma$ parameter that governed by the range of the trap. In order to get
some insight about physical properties of this family of nonlinear coherent
state, we consider some statistical properties of this state. In Fig.
(LABEL:f1) we show the vibrational number distribution of this state,
$p(n)=|\langle n|\psi\rangle|^{2}$. In all of the plots in this figure, Lamb-
Dicke parameter and the ratio $\frac{\Omega_{0}}{\Omega_{1}}$ are chosen as
$\eta=0.22$ and $\frac{\Omega_{0}}{\Omega_{1}}=0.85$, respectively. It can be
seen that the vibrational number distribution depends sensitively on the depth
(or range) of the trap. In some cases it is possible to prepare a
superposition of several Fock states. Another feature of this figure is that
by choosing the proper values of the depth of the trap, such as $(N=30)$, it
is possible to prepare a superposition of two or three Fock states. An
interesting property of this vibrational number distribution is that we can
prepare a highly excited Fock state for external motion of the ion $(N=45)$
[28]. In this case with most probability we can claim that one Fock state is
prepared. By increasing the depth of the trap $(N=75)$, the vibrational number
distribution will reduce to a superposition of Fock states again. In this case
the distribution of the Fock states is approximately symmetric about the most
probable number state. Thus, it is shown that for definite values of physical
parameters, $\eta$ and $\frac{\Omega_{0}}{\Omega_{1}}$ and for different
values of the trap depth, we can prepare different states even a highly
excited Fock state.
In Fig. (LABEL:f2) we have plotted quadrature squeezing of the state
$|\psi\rangle$, Eq. (24). Physical parameters for this plot are chosen as
$\eta=0.25,\;\frac{\Omega_{0}}{\Omega_{1}}=0.31$ and the phase of the
quadrature operator is chosen as $\frac{\pi}{4}$. This figure depicts
squeezing behavior versus the depth of the trap. It is evident that for some
values of the depth, the state (24) exhibits quadrature squeezing. Hence, in
addition to the remarkable properties of the vibrational number distribution,
this state has other nonclassical property. The non-classical properties of
nonlinear coherent states is one of their most important properties [29].
In Fig. (LABEL:f3), we have shown the contour plots of the $Q$ function of the
state (24). In this figure, different plots belong to different depths of the
trap with $\eta=0.75$ and $\frac{\Omega_{0}}{\Omega_{1}}=0.9$. In the case of
$N=7$ (plot (a)) the plot contains contribution at several amplitudes. This
feature implies occurrence of quantum interference effects inherent in this
state. It displays several localized regions where it becomes extremely small.
This phenomena is related to the separate peaks of the number distribution of
state (24) which are rather close together. By increasing the depth of the
trap, in plot (b), $N=26$, and plot (c), $N=45$, this strong structure of the
$Q$ function is disappeared. In these cases the $Q$ function has one peak and
this shows that the peaks of the number distribution are decreased. On the
other hand, the cross section of the $Q$ function is not symmetric and this
shows that for selected values of the parameters, the associated quadrature
operator exhibits quadrature squeezing. With more increasing the depth of the
trap, in plot (d), $N=75$, structure of the $Q$ function becomes stronger than
plots (b) and (c). In this case, the state exhibits quadrature squeezing and
we expect that quantum interference occurs again.
To obtain more information about the nature of the state (24), we have
considered its associated Wigner function, $W(\alpha)$. The Wigner function
for different values of the Lamb-Dicke parameter and the depth of the trap is
shown in Fig. (LABEL:f4). In this figure the ratio
$\frac{\Omega_{0}}{\Omega_{1}}$ is chosen equal to $0.9$. The negative values
of the Wigner function are a signature of the nonclassical nature of the
associated state. As is seen, in all cases the Wigner function has negative
values. To consider the Lamb-Dicke parameter effects, in plots
LABEL:f4(a)-LABEL:f4(c), we have decreased the Lamb-Dicke parameter while the
depth of the trap is chosen constant. The Wigner function in plot LABEL:f4(a)
shows occurrence of the quantum interference. Decreasing of the Lamb-Dicke
parameter splits peaks of the Wigner function in two groups. This yields a
coherent superposition of two quantum states. It is evident that decreasing of
the Lamb-Dicke parameter will decrease the amplitude of the Wigner function.
In addition to the Lamb-Dicke parameter effects, dependence of the Wigner
function to the depth of the trap is considered in plots
LABEL:f4(d)-LABEL:f4(f). It is seen that in plot LABEL:f4(d), for selected
parameters, the Wigner function is split into two parts which is signature of
superposition of two coherent states, because each part consists of several
peaks. By increasing the depth of the trap, these two parts are going to be
mixed and the quantum interference will be occurred.
## 5 Conclusion
We have studied dynamics of a single trapped ion in a finite range trap. In
the context of the $f$-deformed oscillators, we have shown that the confined
ion in a finite range trap can be assumed as an $f$-deformed oscillator. By
modelling the realistic potential with the modified Pöschl-Teller potential,
we have obtained dynamical variables (raising and lowering operators) of this
system. Moreover, we have proposed a scheme for preparation of a special
family of nonlinear coherent states. Such states could be generated as
stationary states of the center-of-mass motion of a laser-driven trapped ion
in a finite range trap while interacts with a bichromatic laser field. When
the motional state is nonlinear coherent state, the ion is decoupled from the
driving laser field. Then, any perturbation of this motional state leads to
the switching of the interaction and this leads to a self-stabilization of the
state. We have shown that the prepared motional state of the ion has some
nonclassical features which strongly depend on the depth of the trap. These
states show some coherence effects such as localization of their phase-space
distribution and splitting to two or more sub-states which the latter leads to
quantum interference. According to the profile of the $Q$ function of these
states, they exhibit quadrature squeezing and for specific values of the
physical parameters we have calculated their quadrature squeezing. It is shown
that the nonclassical nature of the prepared states depends on the depth of
the trap so that for specific values of the depth, both quantum interference
and quadrature squeezing will occur but for some other values, this state
exhibits quadrature squeezing only.
In view of interesting properties of generated states in this paper, states of
this type and physical system under consideration might to be of more general
interest. First of all, the single trapped ion in finite range trap has a
finite dimensional Hilbert space. As mentioned before, the number of energy
levels in this system is controlled by the depth of the trap. As we know, size
of the Hilbert space (dimension of the Hilbert space) has a crucial importance
in some quantum phenomena, such as decoherence. Due to the development in
experimental set ups of trapped ion, it seems possible to organize an
experiment to consider Hilbert space size effects for this system. Then, our
system can be considered as an experimental set up to investigate Hilbert
space size effects. Second, this system turn out to be of interest for
realization of the quantum groups. If we take a look at Hamiltonian (13), it
seems that in the Lamb-Dicke regime $(\eta\ll 1)$, this system can be
considered as a realization of a deformed Jaynes-Cummings model. By
considering the Lamb-Dicke limit, the exponential in Eq. (13) can be expanded
to lowest order, resulting in the operator
$g^{\prime}(\hat{A}\hat{S}^{+}+\hat{A}^{{\dagger}}\hat{S}^{-})$, which
corresponds to the deformed Jaynes-Cummings model (in this relation
$g^{\prime}=\eta g$). In addition, it is shown that there is a relation
between the operators $\hat{A}$ and $\hat{A}^{\dagger}$ in Eq. (9) and the
$q$-deformed algebra [24]. Therefore, our model can be considered as a
realization of $q$-deformed and general deformed Jaynes-Cummings model where
Lamb-Dicke parameter plays an important role on this issue. Third, in recent
types of the Paul traps, the so-called nano Paul traps [18], the finite range
effects of trapping potential are more important. It seems that our model
which tries to consider finite range effects can provide a theoretical
description for investigating the nano Paul traps. To put every things in a
nut shell, our model in this paper provides an experimental set up to consider
Hilbert space size effects and realization of $q$-deformed and general
$f$-deformed algebras.
Acknowledgments
The authors wish to thank The Office of Graduate Studies and Research Vice
President of The University of Isfahan for their support.
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Figure captions
Fig. 1. The vibrational number distribution is shown for four values of the
depth of the trap. The values of the depth, $N$, are written on each plot and
$\eta=0.22$ and $\frac{\Omega_{0}}{\Omega_{1}}=0.85$.
Fig. 2. Plot of quadrature squeezing versus depth of the trap. In this plot
$\eta=0.25$, $\frac{\Omega_{0}}{\Omega_{1}}=0.31$ and quadrature operator
phase is selected as $\frac{\pi}{4}$.
Fig. 3. Contour plots of the $Q$ function for $\eta=0.75$ and
$\frac{\Omega_{0}}{\Omega_{1}}=0.9$. In this figure light region indicates
large values of the function. Each plot belongs to specific values of the
depth of the trap. In plot (a) $N=7$, plot (b) $N=26$, plot (c) $N=45$ and in
plot (d) the depth of the trap is selected as $N=75$.
Fig. 4. Plots of the Wigner function for different values of the Lamb-Dicke
parameter and the depth of the trap which are shown on each plot. In all plots
the ratio $\frac{\Omega_{0}}{\Omega_{1}}$ is selected equal to $0.9$.
|
arxiv-papers
| 2011-12-25T11:09:55 |
2024-09-04T02:49:25.685718
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Bagheri Harouni, M. Davoudi Darareh",
"submitter": "Malek Bagheri",
"url": "https://arxiv.org/abs/1112.5775"
}
|
1112.5798
|
# Validity of nonequilibrium work relations for the rapidly expanding quantum
piston
H. T. Quan Department of Chemistry and Biochemistry,
University of Maryland, College Park, MD 20742 USA Christopher Jarzynski
Department of Chemistry and Biochemistry, and Institute for Physical Science
and Technology,
University of Maryland, College Park, MD 20742 USA
###### Abstract
Recent work by Teifel and Mahler [Eur. Phys. J. B 75, 275 (2010)] raises
legitimate concerns regarding the validity of quantum nonequilibrium work
relations in processes involving moving hard walls. We study this issue in the
context of the rapidly expanding one-dimensional quantum piston. Utilizing
exact solutions of the time-dependent Schrödinger equation, we find that the
evolution of the wave function can be decomposed into static and dynamic
components, which have simple semiclassical interpretations in terms of
particle-piston collisions. We show that nonequilibrium work relations remains
valid at any finite piston speed, provided both components are included, and
we study explicitly the work distribution for this model system.
###### pacs:
05.70.Ln, 05.30.-d, 05.40.-a. 05.90.+m.
In the past two decades, much attention has been devoted to theoretical
predictions and experimental investigations regarding the fluctuations of
small systems away from thermal equilibrium. These predictions include the
nonequilbrium work relation Jarzynski (1997a, b)
$\langle e^{-\beta W}\rangle=e^{-\beta\Delta F},$ (1)
and the corresponding fluctuation theorem derived by Crooks Crooks (1998,
1999, 2000)
$\frac{\rho_{F}(+W)}{\rho_{R}(-W)}=e^{\beta(W-\Delta F)},$ (2)
which pertain to the work ($W$) performed on a system driven out of
equilibrium. (See Ref. Jarzynski (2011) for details and a recent review of
these and related results.) Most of the research in this area has concerned
systems evolving under classical deterministic or stochastic dynamics.
However, the past few years have seen increased focus on the fluctuations of
quantum systems driven away from equilibrium Campisi et al. (2011).
While the derivation of Eq. 1 for an isolated quantum system is
straightforward and rests on familiar properties of unitary evolution Kurchan
(2000); Tasaki (2000); Mukamel (2003); Talkner et al. (2007), Teifel and
Mahler (TM) Teifel and Mahler (2010) have recently presented a calculation
suggesting that Eq. 1 (and by extension, Eq. 2) might be violated for the one-
dimensional quantum piston. In this familiar model system, the wavefunction
describing a particle inside a box evolves in time as the length of the box is
increased (Fig. 1) or decreased. Although TM focus specifically on this simple
model, their analysis has broader implications, raising the possibility that
Eqs. 1 and 2 might generically be violated for processes involving the motion
of hard walls. In such situations the system’s Hilbert space changes with
time, and questions of unitarity must be handled with care. This feature has a
classical counterpart, emphasized by Sung Sung (2008): the phase space
accessible to a classical particle confined by hard walls changes with time as
those walls move.
In the classical setting, processes involving moving boundaries have proven to
be instructive Lua and Grosberg (2005); Presse and Silbey (2006); Jarzynski
(2006); Crooks and Jarzynski (2007); Sung (2008), deepening our understanding
of nonequilibrium work relations by highlighting apparent paradoxes and
counterintuitive features. In this paper we use exact solutions of the time-
dependent Schrödinger equation Doescher and Rice (1969) to investigate the
validity of Eq. 1 for an expanding quantum piston.
In what follows, we first sketch the usual derivation of Eq. 1 for an isolated
quantum system (Eqs. 3 \- 8), as well as an apparent counter-argument which
suggests that Eq. 1 is violated for the quantum piston (Eqs. 9 \- 11). We then
apply the exact results of Ref. Doescher and Rice (1969) to the case in which
the piston moves outward at speed $v$. We find that Eq. 1 is valid for any
finite pulling speed, which seems to contradict the analysis in Eqs. 9 \- 11.
We then consider the limit $v\rightarrow\infty$, and we find that the apparent
discrepancy has an appealing semiclassical interpretation that parallels the
purely classical analyses of Refs. Lua and Grosberg (2005); Presse and Silbey
(2006); Jarzynski (2006); Sung (2008).
Figure 1: Schematic depiction of a quantum piston. A quantum particle is
confined by hard walls, one of which acts as an externally controlled piston.
We focus on the case in which the piston is pulled outward at a speed that is
much greater than the initial thermal speed of the particle.
Consider a quantum system whose parameter-dependent Hamiltonian
$\hat{H}^{\lambda}$ has a discrete energy spectrum:
$\hat{H}^{\lambda}\,|m^{\lambda}\rangle=E_{m}^{\lambda}\,|m^{\lambda}\rangle,$
(3)
with $m=0,1,2\cdots$. We use superscripts to indicate the value of the
externally controlled parameter, $\lambda$, which for the case of the quantum
piston is the position of the piston itself, equivalently the length of the
box. Now imagine that this system is subjected to the following process. (1)
With the parameter fixed at $\lambda=A$, the system is equilibrated with a
reservoir at temperature $\beta^{-1}$; the reservoir is then disconnected and
the energy of the system is measured. At this point the system is in a pure
state $|m^{A}\rangle$, set by the outcome of the energy measurement. (2) The
system now evolves under Schrödinger’s equation, from time $t=0$ to $t=\tau$,
as the parameter is varied from $A$ to $B$ according to a schedule, or
protocol, $\lambda_{t}$. The energy is then measured once more, resulting in
“collapse” into an eigenstate $|n^{B}\rangle$ of $\hat{H}^{B}$. Following
Refs. Kurchan (2000); Tasaki (2000); Mukamel (2003); Talkner et al. (2007) we
identify the work performed on the system with the net change in its energy:
$W\equiv E_{n}^{B}-E_{m}^{A}.$ (4)
By repeating this process, we generate an ensemble of realizations, each
defined by an initial state $|m^{A}\rangle$ and a final state $|n^{B}\rangle$.
The initial states are distributed according to
$P_{m}^{{\rm eq},A}=\frac{1}{Z_{A}}e^{-\beta E_{m}^{A}},$ (5)
where $Z_{A}=\sum_{m}e^{-\beta E_{m}^{A}}=e^{-\beta F_{A}}$ is the partition
function, and the final states according to the conditional distribution
$P(n^{B}|m^{A})=\Bigl{|}\langle n^{B}|\hat{U}|m^{A}\rangle\Bigr{|}^{2},$ (6)
where $\hat{U}$ is the time-evolution operator that describes evolution under
Schrödinger’s equation from $t=0$ to $\tau$. Combining Eqs. 4 \- 6, the left
side of Eq. 1 can now be evaluated:
$\langle e^{-\beta W}\rangle=\sum_{m}P_{m}^{{\rm
eq},A}\sum_{n}P(n^{B}|m^{A})\,e^{-\beta W}=\frac{1}{Z_{A}}\sum_{n}e^{-\beta
E_{n}^{B}}\,s_{n},$ (7)
where
$s_{n}\equiv\sum_{m}P(n^{B}|m^{A})=\sum_{m}\langle
n^{B}|\hat{U}|m^{A}\rangle\,\langle m^{A}|\hat{U}^{\dagger}|n^{B}\rangle.$ (8)
At this point, one normally argues that the sum
$\sum_{m}|m^{A}\rangle\,\langle m^{A}|$ is the identity operator, hence
$s_{n}=1$ and the right side of Eq. 7 becomes $Z_{B}/Z_{A}=e^{-\beta\Delta
F}$, completing the proof.
Teifel and Mahler Teifel and Mahler (2010) correctly point out that this
argument requires care if the eigenstates of $\hat{H}^{A}$ do not span the
Hilbert space of $\hat{H}^{B}$. For a quantum piston whose length is increased
from $\lambda_{0}=A$ to $\lambda_{\tau}=B$ at speed $v$, the states
$|m^{A}\rangle$ are restricted to the interval $0<x<A$, whereas the final
Hilbert space supports states extending over the wider interval $0<x<B$. If
$\psi(x)=\langle x|\psi\rangle$ is a wave function belonging to the Hilbert
space of $\hat{H}^{B}$, then the operator $\sum_{m}|m^{A}\rangle\,\langle
m^{A}|$ effectively “chops off” a portion of this wavefunction:
$\sum_{m}\langle x|m^{A}\rangle\langle
m^{A}|\psi\rangle=\theta(A-x)\,\psi(x)\,,$ (9)
where $\theta(\cdot)$ is the unit step function. We conclude that
$\sum_{m}|m^{A}\rangle\,\langle m^{A}|$ is not the identity operator when it
acts in the Hilbert space spanned by eigenstates of $\hat{H}^{B}$. Hence the
derivation described in the previous paragraph does not automatically apply to
the quantum piston, and this raises concerns regarding the validity of Eq. 1
in that context.
As a limiting case, let us analyze the infinitely fast expansion of the
piston, $v\rightarrow\infty$. The sudden approximation Messiah (1966) suggests
that the wave function then remains in its initial state,
$\lim_{v\rightarrow\infty}\hat{U}|m^{A}\rangle=|m^{A}\rangle.$ (10)
Combining Eqs. 8-10 leads to
$\lim_{v\rightarrow\infty}s_{n}\overset{?}{=}\sum_{m=1}^{\infty}\langle
n^{B}|m^{A}\rangle\,\langle m^{A}|n^{B}\rangle=\int_{0}^{A}{\rm
d}x\,\left|\phi_{n}(x;B)\right|^{2}=\frac{1}{r}-\frac{\sin(2\pi n/r)}{2\pi
n}<1,$ (11)
where $r\equiv B/A$ and the wavefunction
$\phi_{n}(x;\lambda)=\sqrt{\frac{2}{\lambda}}\sin\left(\frac{n\pi
x}{\lambda}\right)$ (12)
describes the $n$’th eigenstate of $\hat{H}^{\lambda}$. (The notation
$\overset{?}{=}$ indicates that we question the validity of the first step in
Eq. 11.) Substitution of Eq. 11 ($s_{n}<1$) into Eq. 7 implies a violation of
Eq. 1. In the opposite limit, namely adiabatic expansion, $v\rightarrow 0$, TM
find that Eq. 1 is satisfied. These considerations suggest that for the
expansion of a quantum piston at finite speed $v$, Eq. 1 is only approximately
valid, but the approximation becomes exact in the adiabatic limit,
$v\rightarrow 0$.
In what follows we will argue that in fact $s_{n}=1$ for all finite values of
$n$ and $v$, and therefore
$\lim_{v\rightarrow\infty}s_{n}=1,$ (13)
in contradiction with Eq. 11. By Eq. 7, our conclusion implies that Eq. 1 is
valid for any finite piston speed.
For a quantum piston expanding at speed $v$ from an initial length
$\lambda_{0}=A$, a set of independent solutions to the time-dependent
Schrödinger equation can be written as Doescher and Rice (1969)
$\Phi_{l}(x,t)=\exp\left[\frac{i}{\hbar\lambda_{t}}\left(\frac{1}{2}Mvx^{2}-E_{l}^{A}At\right)\right]\,\phi_{l}(x;\lambda_{t}),\quad\quad
l=1,2,\cdots,$ (14)
where $M$ denotes the mass of the particle, and
$E_{l}^{A}=l^{2}\pi^{2}\hbar^{2}/2MA^{2}$ is the $l$’th eigenenergy of the
system at $t=0$. The wavefunctions $\Phi_{l}(x,t)$ form a complete orthonormal
set, $\langle\Phi_{k}|\Phi_{l}\rangle=\delta_{kl}$, but are not eigenstates of
$\hat{H}^{\lambda_{t}}$. (The $\phi_{l}$’s defined in Eq. 12 are the
eigenstates.) A general solution of the time-dependent Schrödinger equation
takes the form
$\Psi(x,t)=\sum_{l=1}^{\infty}c_{l}\,\Phi_{l}(x,t),$ (15)
where the time-independent coefficients $c_{l}$ are set by the initial wave
function:
$c_{l}=\int_{0}^{A}\Phi_{l}^{*}(x,0)\Psi(x,0)\,{\rm d}x.$ (16)
For initial conditions $|\Psi(0)\rangle=|m^{A}\rangle$ these coefficients are
(setting $\hbar=M=1$)
$c_{l}(m)=\frac{2}{A}\int_{0}^{A}e^{-ivx^{2}/2A}\,\sin\left({\frac{l\pi
x}{A}}\right)\sin\left({\frac{m\pi x}{A}}\right)\,{\rm d}x,$ (17a) and the
transition matrix element to the state $|n^{B}\rangle$ at the final time
$\tau$ is $\langle n^{B}|\hat{U}|m^{A}\rangle=\left\langle
n^{B}|\Psi(\tau)\right\rangle=\sum_{l=1}^{\infty}c_{l}(m)\int_{0}^{B}\phi^{*}_{n}(x;B)\,\Phi_{l}(x,\tau)\,{\rm
d}x.$ (17b)
Eqs. 6 and 17 give the transition probability $P(n^{B}|m^{A})$, in terms of
one-dimensional integrals that are easily computed numerically. This
transition probability satisfies normalization:
$\sum_{n}P(n^{B}|m^{A})=\int_{0}^{B}{\rm d}x\,\left|\langle
x|\hat{U}|m^{A}\rangle\right|^{2}=1.$ (18)
Although we have considered the expansion of a quantum piston, Eq. 14 is
equally valid for compression Doescher and Rice (1969). By reversing the roles
of $A$ and $B$ and the roles of $m$ and $n$ and by replacing $v$ with $-v$ (in
Eq. 17) we obtain the transition probability $\bar{P}(m^{A}|n^{B})$ from the
$n$’th eigenstate of $\hat{H}^{B}$ to the $m$’th eigenstate of $\hat{H}^{A}$,
where the notation $\bar{P}$ indicates the compression process. This
transition probability also satisfies normalization:
$\sum_{m}\bar{P}(m^{A}|n^{B})=\int_{0}^{A}{\rm d}x\,\left|\langle
x|\hat{U}^{\prime}|n^{B}\rangle\right|^{2}=1,$ (19)
where $\hat{U}^{\prime}$ is the time-evolution operator for the compression
process.
In the Appendix, we provide explicit expressions for $P(n^{B}|m^{A})$ and
$\bar{P}(m^{A}|n^{B})$, and using these expressions we directly verify the
relation
$P(n^{B}|m^{A})=\bar{P}(m^{A}|n^{B}).$ (20)
It should be clear that this relation is precisely what we expect from time-
reversal invariance ($\hat{U}^{\prime}=\hat{U}^{\dagger}$), see e.g. Eq. 56 of
Ref. Campisi et al. (2011). Using Eq. 20 we can now transform the sum over
initial states in Eq. 8 into a sum over final states:
$s_{n}\equiv\sum_{m}P(n^{B}|m^{A})=\sum_{m}\bar{P}(m^{A}|n^{B})=1$ (21)
using Eq. 19 in the last step. Since this result is independent of $v$, we
conclude that Eq. 1 is valid at any finite speed of expansion.
To obtain Eq. 2 by similar means, we follow Tasaki Tasaki (2000) and write
explicit expressions for the forward and reverse work distributions
(corresponding to piston expansion and compression, respectively):
$\begin{split}\rho_{F}(W)&=Z_{A}^{-1}\sum_{m}e^{-\beta
E_{m}^{A}}\sum_{n}P(n^{B}|m^{A})\,\delta\left(W-E_{n}^{B}+E_{m}^{A}\right),\\\
\rho_{R}(W)&=Z_{B}^{-1}\sum_{n}e^{-\beta
E_{n}^{B}}\sum_{m}\bar{P}(m^{A}|n^{B})\,\delta\left(W-E_{m}^{A}+E_{n}^{B}\right).\end{split}$
(22)
For every realization $m^{A}\rightarrow n^{B}$ that gives a particular work
value during the forward process, there is a corresponding realization
$n^{B}\rightarrow m^{A}$ that gives the opposite work value during the reverse
process. Combining this observation with Eqs. 20 and 22 we obtain Eq. 2 Tasaki
(2000).
Up to this point we have used the symmetry relation, Eq. 20, to show that
$s_{n}=1$ for any finite speed $v$, and therefore that Eqs. 1 and 2 remain
valid for the quantum piston. However, this analysis does not yet explain why
Eq. 11 gives a contradictory result in the limit $v\rightarrow\infty$. To
address this issue, in the following paragraphs we present numerical evidence
that the value of $s_{n}$ is naturally expressed as the sum of a static and a
dynamic contribution, reflected in the two-peak structure seen in Figs. 2(a)
\- 2(c). The sum of these contributions is unity for any finite $v$ (as per
Eq. 21), but Eq. 11 accounts only for the static contribution, thus giving
$s_{n}^{\rm Eq.(\ref{eq:questionableLimit})}<1$. Here and in the following
discussion, we use the notation $s_{n}^{\rm Eq.(\ref{eq:questionableLimit})}$
to denote the value for $s_{n}$ predicted (incorrectly!) by Eq. 11, in the
limit $v\rightarrow\infty$. After presenting the numerical results, we suggest
a semiclassical interpretation in terms of piston-particle collisions.
(a) $\,\,v=10$
(b) $\,\,v=100$
(c) $\,\,v=500$
Figure 2: $P(n^{B}|m^{A})$ is plotted as a function of $m$ at fixed $n=3$
(open circles), revealing a two-peak structure, with the left peak around
$m=2$ and the right peak near $m=2vA/\pi$. We refer to these peaks as the
static and dynamic components, respectively. Also plotted is the quantity
$O(n^{B}|m^{A})$ (red points), which displays only a single peak around $m=2$.
Note that for $v=100$ and $v=500$ the single peak of $O(n^{B}|m^{A})$ is
virtually identical to the static component of $P(n^{B}|m^{A})$.
We have used Eq. 17 to evaluate $P(n^{B}|m^{A})$ numerically. In Fig. 2, this
quantity is plotted for fixed final state $n=3$, as a function of initial
state $m=1,2,\cdots$, for piston expansion from $A=1.0$ to $B=2.0$ at various
speeds: $v=10$, 100 and 500. The plot reveals a two-peak structure. The left
peak, near $m=2$, remains approximately independent of $v$, whereas the right
peak is located near $m=2vA/\pi$; thus with increasing $v$ the right peak
shifts further rightward. (Note the change of scale in the plots.) We will
refer to the left and right peaks as the static and dynamic components,
respectively. We can decompose the value of $s_{n}$, with $n=3$ in our case,
into contributions from these components:
$s_{n}^{L}=\sum_{m\leq m^{*}}P(n^{B}|m^{A}),\quad\quad
s_{n}^{R}=\sum_{m>m^{*}}P(n^{B}|m^{A}).$ (23)
Here $m^{*}$ is the value of $m$ at which $P(n^{B}|m^{A})$ is minimized in the
region between the two peaks. Table 1 lists the values of these contributions,
obtained by numerical evaluation of the integrals in Eq. 17, as well as their
sum, $s_{n}$. Note that $s_{n}=1.000$ at all three speeds, in agreement with
Eq. 21.
| $v=10$ | $v=100$ | $v=500$ | $v\rightarrow\infty$
---|---|---|---|---
$s_{n}^{\rm Eq.(\ref{eq:questionableLimit})}$ | | | | 0.500
$s_{n}^{L}$ | 0.644 | 0.499 | 0.500 |
$s_{n}^{R}$ | 0.356 | 0.501 | 0.500 |
$s_{n}=s_{n}^{L}+s_{n}^{R}$ | 1.000 | 1.000 | 1.000 |
Table 1: Static ($L$) and dynamic ($R$) contributions to $s_{n=3}$, as well as
the asymptotic value of $s_{n}$ predicted by Eq. 11, for piston expansion from
$A=1.0$ to $B=2.0$.
Let us now rewrite Eq. 11 as
$\lim_{v\rightarrow\infty}s_{n}\overset{?}{=}\sum_{m}\langle
n^{B}|m^{A}\rangle\,\langle m^{A}|n^{B}\rangle=\sum_{m}\left|\int_{0}^{A}{\rm
d}x\,\phi_{n}^{*}(x;B)\phi_{m}(x;A)\right|^{2}\equiv\sum_{m}O(n^{B}|m^{A}).$
(24)
We can interpret the overlap $O(n^{B}|m^{A})=\left|\langle
n^{B}|m^{A}\rangle\right|^{2}$ as the probability to end in state
$|n^{B}\rangle$ after the measurement of the final energy, when starting from
state $|m^{A}\rangle$, under the assumption that the wave function remains
unchanged during the sudden expansion. This assumption amounts to a literal
intepretation of the sudden approximation, Eq. 10. Using Eq. 12 to evaluate
the integral, in Fig. 2 we have also plotted $O(n^{B}|m^{A})$, which exhibits
a single peak around $m=2$. We observe that the larger the value of $v$, the
more closely $O(n^{B}|m^{A})$ resembles the left peak of $P(n^{B}|m^{A})$;
indeed at $v=100$ and 500 they are virtually identical. These empirical
observations suggest that Eq. 11 captures only the contribution to $s_{n}$
from the static component $s_{n}^{L}$, while missing the contribution from the
dynamic component $s_{n}^{R}$.
Quantitatively, $s_{n}^{\rm Eq.(\ref{eq:questionableLimit})}=0.5$ for $A=1.0$,
$B=2.0$, and $n=3$, whereas the data in Table 1 suggest that the static
contribution $s_{n}^{L}$ approaches $0.5$ as $v\rightarrow\infty$. Moreover,
Table 2 lists these quantities for the case $A=1.0$, $B=1.485$, and $n=3$,
with $s_{n}^{L}$ and $s_{n}^{R}$ again calculated using Eq. 17. Once again we
find that $s_{n}^{L}+s_{n}^{R}=1.000$ at all speeds, and $s_{n}^{L}\rightarrow
s_{n}^{\rm Eq.(\ref{eq:questionableLimit})}\approx 0.667$ as
$v\rightarrow\infty$. These findings are consistent with our hypothesis that
Eq. 11 reflects only the static and not the dynamic contribution to $s_{n}$.
| $v=10$ | $v=100$ | $v=500$ | $v\rightarrow\infty$
---|---|---|---|---
$s_{n}^{\rm Eq.(\ref{eq:questionableLimit})}$ | | | | 0.667
$s_{n}^{L}$ | 0.638 | 0.667 | 0.667 |
$s_{n}^{R}$ | 0.362 | 0.333 | 0.333 |
$s_{n}=s_{n}^{L}+s_{n}^{R}$ | 1.000 | 1.000 | 1.000 |
Table 2: Same as Table 1, but for expansion from $A=1.0$ to $B=1.485$.
We now build a semiclassical interpretation to reinforce these conclusions.
For $A=1.0$, $B=2.0$, and piston speed $v=100$, consider the value
$P(3^{B}|64^{A})$, corresponding to the right peak in Fig. 2(b). This gives
the probability to end in state $|3^{B}\rangle$, starting from state
$|64^{A}\rangle$, during the expansion process. Semiclassically, the initial
state $|\Phi(0)\rangle=|64^{A}\rangle$ can be imagined as a particle moving
with speed
$|u|=\sqrt{2E_{m=64}^{A}}=\frac{m\pi}{A}\approx 200$ (25)
between two hard walls. At $t=0$, when the piston begins to move rightward
with speed $v=100$, the particle is moving either leftward ($u\approx-200$) or
rightward ($u\approx+200$), with equal likelihood. In the latter case, the
particle will collide once with the receding piston, losing approximately all
of its kinetic energy. The final state $|\Phi(\tau)\rangle$ will then contain
a substantial component of low-energy states (including $|3^{B}\rangle$)
reflecting this one-collision scenario. In other words, $P(3^{B}|64^{A})$ is
non-negligible because a single collision with the piston scatters the
particle from the high-energy state $|64^{A}\rangle$ to the low-energy state
$|3^{B}\rangle$. The same argument explains, quantitatively, why the right
peak occurs at $m^{A}\approx 320$ in Fig. 2(c).
Alternatively, we can use Eq. 20 to rewrite $P(3^{B}|64^{A})$ as
$\bar{P}(64^{A}|3^{B})$, which is the probability to end in state
$|64^{A}\rangle$, starting from state $|3^{B}\rangle$ when compressing at
piston speed $v=100$. Here we imagine a particle initially moving with speed
$|u|=\sqrt{2E_{n=3}^{B}}\approx 5.$ (26)
As the piston moves from $B=2.0$ to $A=1.0$, the particle might suffer a
single collision with the piston, imparting a leftward velocity $\Delta
u\approx-2v=-200$ to the particle. Thus for the initial state
$|\Phi(0)\rangle=|3^{B}\rangle$ we expect the final state $|\Phi(\tau)\rangle$
to be a superposition of low-energy states (corresponding to no collisions)
and high-energy states near $|64^{A}\rangle$ (one collision). This is indeed
the spectrum seen in Fig. 2(b). This interpretation suggests that $s_{n}^{R}$
is equal to the probability that the particle suffers a collision with the
piston during the compression process, and $s_{n}^{L}$ is the probability it
avoids a collision, when starting from state $|n^{B}\rangle$. Semiclassically
and in the limit $v\rightarrow\infty$, the probability to avoid a collision
during compression is just the probability to find the particle in the region
$0<x<A$ at time $t=0$ (when the piston is at location $B$), which leads to
$\lim_{v\rightarrow\infty}s_{n}^{L,{\rm sc}}=\frac{A}{B}=\frac{1}{r}.$ (27)
The superscript “sc” emphasizes that this is a semiclassical approximation.
Eq. 27 agrees with the term $1/r$ in the expression appearing in Eq. 11 (just
before the inequality); the oscillatory term there, $\sin(2\pi n/r)/2\pi n$,
is quantum-mechanical in origin.
In either case – expansion or compression – the dynamic component is
associated semiclassically with a collision between the particle and the
piston. We conclude that Eq. 11 underestimates $s_{n}$ because it neglects the
contribution due to a particle-piston collision.
These considerations relate to the ordering of limits. Fig. 2 suggests that
$\lim_{v\rightarrow\infty}P(n^{B}|m^{A})=O(n^{B}|m^{A}),$ (28)
for any fixed initial state $|m^{A}\rangle$. Now, Eq. 11 implicitly contains a
double limit, namely,
$\lim_{v\rightarrow\infty}s_{n}=\lim_{v\rightarrow\infty}\lim_{K\rightarrow\infty}\sum_{m=1}^{K}P(n^{B}|m^{A}).$
(29)
If we take the limit $K\rightarrow\infty$ first (with $v$ fixed), then both
the static and dynamic components $s_{n}^{L}$ and $s_{n}^{R}$ are included in
the sum, and the right side of Eq. 29 sums to unity (Eq. 21):
$\lim_{v\rightarrow\infty}\lim_{K\rightarrow\infty}\sum_{m=1}^{K}P(n^{B}|m^{A})=1.$
(30)
However, if we reverse the ordering of limits and first take
$v\rightarrow\infty$ (with $K$ fixed), then the dynamic component gets pushed
beyond the value of $K$, and only the static component contributes:
$\lim_{K\rightarrow\infty}\lim_{v\rightarrow\infty}\sum_{m=1}^{K}P(n^{B}|m^{A})=\lim_{K\rightarrow\infty}\sum_{m=1}^{K}O(n^{B}|m^{A})=\frac{1}{r}-\frac{\sin(2\pi
n/r)}{2\pi n}.$ (31)
The physical interpretation should be clear. For any fixed piston speed $v$,
the sudden approximation breaks down if $m^{A}\pi/A\gtrsim v$; for such
initial states the evolving wavefunction catches up with the moving piston.
Therefore if we sum over all initial states at fixed $v$, then this sum
necessarily includes states that violate the sudden approximation. Conversely,
the use of the sudden approximation in Eq. 11 is equivalent to imposing a
cutoff $K$ on the sum over initial states: the effect of this cutoff is to
exclude those states that give rise to the dynamic component, $s_{n}^{R}$.
The result appearing in Eq. 31 is the same as that obtained for the process of
sudden expansion into a vacuum, in which the length of the box increases
instantaneously from $A$ to $B$. This case, considered explicitly by TM (see
Eq. 23 of Ref. Teifel and Mahler (2010)) and for the classical piston by Sung
Sung (2008), highlights the importance of the ordering of limits for the
validity of Eq. 1. This issue is discussed in detail by Pressé and Silbey
Presse and Silbey (2006). See also Kurchan’s lectures Kurchan (2009) for an
alternative analysis of the sudden expansion process.
(a) $\,\,v=10$
(b) $\,\,v=100$
(c) $\,\,v=500$
Figure 3: The probability to generate a realization from $|m^{A}\rangle$ to
$|3^{B}\rangle$ is plotted, for the same parameters as in Fig. 2, and taking
$\beta=0.01$.
While our arguments establish that Eq. 1 is valid for any piston expansion
speed $v$, they also imply that for large $v$, transitions $\langle
n^{B}|\hat{U}|m^{A}\rangle$ from high-lying initial energy eigenstates make a
large contribution to $s_{n}$ and ultimately to $\langle e^{-\beta W}\rangle$
(Eq. 7). When the energies of such high-lying states are much greater than
$\beta^{-1}$, then the probability to sample these states from the initial
canonical distribution, $P_{m}^{{\rm eq},A}\propto e^{-\beta E_{m}^{A}}$,
becomes exceedingly small. In this case, even though Eq. 1 is valid, the
number of realizations required to confirm its validity is prohibitively
large. Fig. 3 illustrates this point by displaying the product $P_{m}^{{\rm
eq},A}\,P(n^{B}|m^{A})$, that is the net probability to generate a realization
with initial and final states $|m^{A}\rangle$ and $|n^{B}\rangle$,
respectively, setting $\beta=0.01$ and $n=3$. Comparing Figs. 2 and 3, we see
that although realizations that correspond to the dynamic component represent
an important contribution to $s_{n}$, the probability to observe these
realizations is vanishingly small. This conclusion is mirrored in the
classical version of this expanding piston Lua and Grosberg (2005); Jarzynski
(2006), where a substantial contribution to $\langle e^{-\beta W}\rangle$
arises from single-collision events, in which the particle loses energy as it
strikes the rapidly receding piston. If $Mv^{2}\gg\beta^{-1}$, then many
realizations of the process are needed in order to stand a decent chance of
sampling initial conditions in which the particle is moving sufficiently fast
to collide with the piston. By analogy with the classical calculations of Ref.
Lua and Grosberg (2005); Jarzynski (2006), we expect that the number of
realizations needed for the convergence of the exponential average in Eq. 1
scales like $\exp(\beta Mv^{2})$, for large $v$.
We note in passing that in Figs. 2(b) and 2(c) the right peak itself exhibits
a double-peak structure. This too has a semiclassical interpretation, which is
easiest to explain in terms of the compression process. At $t=0$ in the state
$|3^{B}\rangle$, the particle is moving with speed $|u|\approx 5$ (Eq. 26).
Its speed after a collision with the leftward-moving piston is greater if the
particle was moving toward the piston just before the collision ($u\approx+5$)
than if it was moving away from the piston ($u\approx-5$). A back-of-the
envelope calculation suggests that this difference splits the right peak into
two sub-components separated by $\Delta m=2A|u|/\pi\approx 3$, in agreement
with what we see in Figs. 2(b) and 2(c).
(a) $\,\,v=0.1$
(b) $\,\,v=1$
(c) $\,\,v=2$
(d) $\,\,v=4$
(e) $\,\,v=8$
(f) $\,\,v=1000$
Figure 4: Work distribution for the expanding quantum piston. Here $A=1$,
$B=2$, $\beta=0.01$, and the piston speed ranges from $v=0.1$ to $v=1000$. The
free energy difference is $\Delta F=-\beta^{-1}\ln(B/A)\approx-30.10$. The
left tail of the distributions in the region $W<-100$ is not shown.
Finally, since this model provides a useful pedagogical illustration of a
quantum nonequilibrium process (see also Refs. Quan et al. (2008); Deffner and
Lutz (2008); Deffner et al. (2010); Talkner et al. (2008)), we briefly discuss
the work distribution $\rho_{F}(W)$ for the expanding quantum piston (see Eq.
22), plotted in Fig. 4 for various piston speeds. In the limit $v\rightarrow
0$, the quantum adiabatic theorem gives us
$P(n^{B}|m^{A})\rightarrow\delta_{mn}$. Thus the work distribution in Fig.
4(a) reflects the initial thermal energy distribution: the largest peak
corresponds to the situation in which the system begins and ends in the ground
state, the next largest corresponds to the first excited state, and so on. In
the opposite limit of large $v$, $\rho_{F}(W)$ approaches an asymptotic
distribution, obtained by replacing $P(n^{B}|m^{A})$ with its static component
$O(n^{B}|m^{A})$ in Eq. 22. (However, the dynamic component, which gets pushed
off to infinity as discussed earlier, remains essential for the the validity
of Eq. 1.) There are two uniquely quantal features of the distributions shown
in Fig. 4. First, for $v\geq 2$ we can clearly see a nonzero probability to
obtain a positive value of work. This is forbidden in the classical case, as
the particle loses energy each time it collides with the piston. Second, for
the classical expanding piston the probability to obtain $W=0$ approaches
unity as $v\rightarrow\infty$, whereas for the quantum piston with $A=1.0$ and
$B=2.0$ this probability approaches 1/2, as illustrated by the peak at $W=0$
in Fig. 4(f). Finally, although it might not be obvious from Fig. 4, the
average work performed in the limit $v\rightarrow\infty$ is zero for the
quantum piston Schlitt and Stutz (1970), just as it is for the classical
piston.
To conclude, we have used exact solutions of the time-dependent Schrödinger
equation to study the validity of nonequilibrium work relations (Eqs. 1, 2)
for the quantum piston, focusing on the limit of a rapidly expanding piston,
$v\rightarrow\infty$. Our investigation was motivated by Teifel and Mahler’s
study Teifel and Mahler (2010), which highlighted the subtleties that arise
when the system’s Hilbert space changes due to the motion of hard boundaries.
As in the classical case, we found that both Eqs. 1 and 2 remain valid for any
finite piston speed, but the convergence of $\left\langle e^{-\beta
W}\right\rangle$ to $e^{-\beta\Delta F}$ requires a sum over all possible
realizations. In particular, when $v\gg\beta^{-1/2}$ important contributions
arise from those rare realizations in which the particle begins with a
sufficiently high energy to collide with the piston. These realizations show
up as the dynamic component (the right peak) in Fig. 2. Although we have
considered only the one-dimensional quantum piston, we speculate that similar
conclusions will apply to more complicated quantum systems involving moving
hard boundaries, for which exact solutions of the Schrödinger equation are
unavailable.
###### Acknowledgements.
We gratefully acknowledge support from the National Science Foundation (USA)
under grant DMR-0906601. HTQ thanks Prof. Jaeyoung Sung for stimulating
discussions and Andy Ballard for help with computational matters.
## Appendix A
Eq. 17 gives the following expression for the transition probability from
$|m^{A}\rangle$ to $|n^{B}\rangle$ during the expansion process:
$\begin{split}&P(n^{B}|m^{A})=\left|\sum_{l=1}^{\infty}\frac{2}{A}\int_{0}^{A}e^{-ivx^{2}/2A}\sin\left(\frac{l\pi
x}{A}\right)\sin\left(\frac{m\pi x}{A}\right){\rm d}x\right.\\\
&\times\left.\exp\left[-i\frac{\pi^{2}l^{2}(B-A)}{2ABv}\right]\,\frac{2}{B}\int_{0}^{B}e^{ivx^{2}/2B}\sin\left(\frac{n\pi
x}{B}\right)\sin\left(\frac{l\pi x}{B}\right){\rm d}x\right|^{2}.\end{split}$
(32)
For the contraction process, the transition probability from $|n^{B}\rangle$
to $|m^{A}\rangle$ is obtained from this result by making the replacements
$m\leftrightarrow n$, $A\leftrightarrow B$, and $v\rightarrow-v$:
$\begin{split}&\bar{P}(m^{A}|n^{B})=\left|\sum_{l=1}^{\infty}\frac{2}{B}\int_{0}^{B}e^{ivx^{2}/2B}\sin\left(\frac{l\pi
x}{B}\right)\sin\left(\frac{n\pi x}{B}\right){\rm d}x\right.\\\
&\times\left.\exp\left[i\frac{\pi^{2}l^{2}(A-B)}{2BAv}\right]\,\frac{2}{A}\int_{0}^{A}e^{-ivx^{2}/2A}\sin\left(\frac{m\pi
x}{A}\right)\sin\left(\frac{l\pi x}{A}\right){\rm d}x\right|^{2}.\end{split}$
(33)
Comparing these expressions, it is straightforward to verify that they are
equal:
$P(n^{B}|m^{A})=\bar{P}(m^{A}|n^{B})$ (34)
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|
arxiv-papers
| 2011-12-25T22:03:39 |
2024-09-04T02:49:25.693949
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. T. Quan and Christopher Jarzynski",
"submitter": "Haitao Quan",
"url": "https://arxiv.org/abs/1112.5798"
}
|
1112.5845
|
# Influence of phonons on exciton-photon interaction and photon statistics of
a quantum dot
M. Bagheri Harouni, R. Roknizadeh and M. H. Naderi 1Quantum Optics Group,
Department of physics, University of Isfahan, Isfahan, Iran
###### Abstract
In this paper, we investigate, phonon effects on the optical properties of a
spherical quantum dot. For this purpose, we consider the interaction of a
spherical quantum dot with classical and quantum fields while the exciton of
quantum dot interacts with a solid state reservoir. We show that phonons
strongly affect the Rabi oscillations and optical coherence on first
picoseconds of dynamics. We consider the quantum statistics of emitted photons
by quantum dot and we show that these photons are anti-bunched and obey the
sub-Poissonian statistics. In addition, we examine the effects of detuning and
interaction of quantum dot with the cavity mode on optical coherence of energy
levels. The effects of detuning and interaction of quantum dot with cavity
mode on optical coherence of energy levels are compared to the effects of its
interaction with classical pulse.
###### pacs:
42.50.Ct, 42.50.Ar, 73.21.La, 63.20.kd
††preprint: APS/123-QED
## I Introduction
The fundamental system in cavity quantum electrodynamics (cavity-QED) is a two
level atom interacting with a single-cavity mode qo1 -qo2 . Recent
developments in semiconductor nano-technology have shown that excitons in
quantum dots (QDs) constitute an alternative two-level system for cavity-QED
application gerard . There are many similarities between the excitons in QDs
and atomic systems, such as the discrete level structures which is subsequent
of three-dimensional confinement of electrons. On the other hand, there are
also important differences, for example coupling to phonons, carrier-carrier
interaction and surface fluctuation. Coupling of electrons to phonons plays a
major role in QDs. The coupling of phonons to the QD provides a basic
dephasing mechanism and thus marks a lower limit for the decoherence mulj
-knorr . In self-assembled QDs it is indeed the elastic phonon scattering
(pure dephasing) which dominates the loss of coherence on a picosecond time
scale at temperatures below $100$K borri . The effects of electron-phonon
interactions on strong exciton-photon coupling in cavity-QED has been
considered wilson . It has been shown vagov2 that the phonon-induced damping
of Rabi oscillations in a QD is a non-monotonic function of the laser-field
intensity that is increasing at low fields and decreasing at high fields.
QDs are also promising candidates for efficient, deterministic single photon
sources shield -santor . Then the QDs are important sources of non-classical
light. For this kind of application an understanding of the coherence
properties of its optical transitions is of great importance. Therefore, there
are two processes in optical manipulation with semiconductor QDs: coherent
control of the QD exciton state axt and measurement of quantum statistics of
emitted light with QD muller . A theoretical investigation of exciton dynamics
and the possibility of generation of non-classical light has been considered
without taking into account the phonon effects perea .
In this paper, we investigate the effects of electron-phonon interactions on
optical coherence and quantum statistics of light emitted by a pulse driven QD
interacting with a cavity mode. The photon statistics from a driven QD under
the influence of the phonon environment has been considered recently zahir .
On the other hand, influence of phonons on incoherent photon emission of a QD
in the presence of pulse excitation had been considered ahn . We use the most
widely studied model for phonon effects in QDs which accounts two electronic
levels coupled to a laser pulse and to non-interacting phonons machni . As
mentioned, phonon interaction provides a dephasing mechanism for optically
induced coherence on a time scale (a few picosecond) much shorter than for
radiative interaction and recombination borri1 . Due to the different
correlation time for a phonon reservoir (few picosecond) and for a radiative
reservoir (several ten nanosecond) we restrict our attention to the time
scales which dephasing effects due to the phonon system play an important role
(with the radiative reservoir we mean a reservoir for photon system. The
mentioned time scale relates to the decay time of cavity photons). Then we do
not consider any damping effect on cavity mode and spontaneous emission. In
our consideration the only damping effect is related to phonons.
The paper is organized as follows: In section II we describe the model
Hamiltonian and master equation that allows to calculate the evolution of
populations and coherence of the energy levels. In section III we present the
exciton dynamics and its coherence while driven with a laser pulse. The photon
statistics and exciton dynamics of pulse driven QD interacting with a cavity
mode is presented and discussed in section IV. Section V is devoted to a
summery and conclusion.
## II Theoretical model
We consider a single QD inside a semiconductor microcavity that is pumped with
a laser pulse and interacts with a cavity mode. It is assumed that the system
is initially prepared in its ground state. We consider a solid-state reservoir
for the exciton population and we focus on time scales which phonon effects
are important. We neglect other sources of damping in the system. We model the
QD by a two-level system with ground state $|g\rangle$ (the semiconductor
ground-state) and first excited state $|e\rangle$ (a single exciton),
separated by an energy $\hbar\omega_{ex}$. The phonon environment is modelled
by a bath of harmonic oscillators of frequencies $\omega_{k}$, with the
wavevector $k$. The Hamiltonian of the total system in the rotating wave
approximation is written as
$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\hbar\omega_{ex}\hat{\sigma}_{ee}+\hbar\omega_{c}\hat{a}^{{\dagger}}\hat{a}+\sum_{k}\hbar\omega_{k}\hat{b}^{{\dagger}}_{k}\hat{b}_{k}$
(1) $\displaystyle+$ $\displaystyle\hbar
g(\hat{\sigma}_{eg}\hat{a}+\hat{a}^{{\dagger}}\hat{\sigma}_{ge})+\hbar
f(t)(\hat{\sigma}_{eg}+\hat{\sigma}_{ge})$ $\displaystyle+$
$\displaystyle\hat{\sigma}_{ee}\sum_{k}\lambda_{k}(\hat{b}_{k}+\hat{b}^{\dagger}_{k}),$
where $\hat{\sigma}_{ij}=|i\rangle\langle j|$, $\hat{a}\;(\hat{a}^{\dagger})$
and $\hat{b}_{k}\;(\hat{b}^{\dagger}_{k})$ are the annihilation (creation)
operators for cavity mode, and $k$th phonon mode, respectively. The parameter
$g$ is the coupling constant of the exciton and cavity mode, and $f(t)$ is a
real envelope function of the driving pulse. The last term in the Hamiltonian
describes the exciton-phonon interaction. In this term, $\lambda_{k}$ is the
corresponding coupling constant. The coupling of the confined exciton to the
acoustic phonons by means of the deformation potential tends to dominant the
dephasing dynamics, over the piezoelectric interaction or coupling to optical
phonons krumm . In this case, the coupling constant is given by
$\lambda_{k}=kD(k)\sqrt{2n\omega_{k}V}$ mahan where $n$ is the sample density
and $V$ is the unit cell volume. $D(k)$ is the form factor of the confined
electron and hole in the ground state of the QD. The Hamiltonian in the
interaction picture can be written as
$\hat{H}_{int}=\hat{H}_{0}+\hat{H}_{R},$ (2)
where we decompose the coherent-field part and environment part as follow
$\displaystyle\hat{H}_{0}$ $\displaystyle=$ $\displaystyle\hbar
g(\hat{\sigma}_{eg}\hat{a}e^{i\Delta
t}+\hat{a}^{{\dagger}}\hat{\sigma}_{ge}e^{-i\Delta t})$ $\displaystyle+$
$\displaystyle\hbar
f(t)(\hat{\sigma}_{eg}e^{i\omega_{ex}t}+\hat{\sigma}_{ge}e^{-i\omega_{ex}t}),$
$\displaystyle\hat{H}_{R}$ $\displaystyle=$
$\displaystyle\hat{\sigma}_{ee}\sum_{k}\lambda_{k}(\hat{b}_{k}e^{-i\omega_{k}t}+\hat{b}^{{\dagger}}e^{i\omega_{k}t}).$
(3)
In this equation $\Delta=\omega_{ex}-\omega_{c}$ is detuning between the
exciton excitation energy in the QD and cavity field energy.
Now we consider the Liouville equation of density matrix in the interaction
picture
$\frac{d\hat{\rho}_{t}}{dt}=\frac{i}{\hbar}[\hat{\rho}_{t},\hat{H}_{int}].$
(4)
We define the reduced density matrix $\hat{\rho}$ for the exciton-photon
system by tracing out the phonon degrees of freedom in the total density
matrix, $\hat{\rho}=Tr_{ph}(\hat{\rho}_{t})$. Now we consider the master
equation in the Born approximation qo1 -qo2 in the case of the phonon
interaction while we consider the gain and pump parts exactly. Phonons are one
of the slowest process and this kind of reservoir has a correlation time of
the order of a few picosecond krumm and this reservoir is naturally non-
Markovian. To consider non-Markovian dynamics we have used time
convolutionless projection operator method bruer , up to second order of
expansion. We assume an uncorrelated state for initial state of the exciton-
photon system and phonon reservoir. At the initial time $t=0$ the phonon
system is assumed to be in a thermal equilibrium at temperature $T$. Then the
density operator of the exciton-photon system satisfies the following
dynamical equation
$\displaystyle\dot{\rho}(t)$ $\displaystyle=$
$\displaystyle\frac{i}{\hbar}[\rho(t),\hat{H}_{0}]-\int_{0}^{t}([\hat{\sigma}_{ee},\hat{\sigma}_{ee}\rho(t)]K(t-t^{\prime})$
(5) $\displaystyle-$
$\displaystyle[\hat{\sigma}_{ee},\rho(t)\hat{\sigma}_{ee}]K^{\ast}(t-t^{\prime}))dt^{\prime}.$
The first term describes the coherent evolution of the density matrix $\rho$
under the action of the Hamiltonian $\hat{H}_{0}$ of the dot-cavity-pulse
system. The kernel $K$ which is the correlation function of the environment is
written as
$K(t)=\frac{1}{\hbar^{2}}\int_{0}^{\infty}d\omega
j(\omega)\left[coth(\frac{\hbar\omega}{2k_{B}T})cos(\omega t)-isin(\omega
t)\right],$ (6)
with Boltzmann constant $k_{B}$. $j(\omega)$ is the spectral density of the
phonons which completely describes the interaction of exciton and phonons
weiss . Here, we introduce the following spectral density
$j(\omega)=\sum_{k}\lambda_{k}^{2}\delta(\omega-\omega_{k}).$ (7)
The density matrix dynamics is obtained under the Born-Markov approximation
for exciton-phonon interaction and the strong exciton-photon interaction and
pump effects are described exactly. We can extract exciton dynamics and photon
statistics from this equation.
## III Exciton dynamics under a driving pulse
In this section we consider the optical coherence of a driven QD under a pump
pulse. Here we neglect the cavity mode and we consider optical coherence and
exciton population dynamics under pulse excitation and effects of physical
parameters such as pulse duration on these physical quantities. Then the
density matrix of the excitonic system satisfies the following equation of
motion
$\displaystyle\dot{\rho}_{ex}(t)$ $\displaystyle=$
$\displaystyle\frac{i}{\hbar}[\rho_{ex}(t),\hbar(\hat{\sigma}_{eg}\alpha(t)+\hat{\sigma}_{ge}\alpha^{\ast}(t))]$
$\displaystyle-$
$\displaystyle\int_{0}^{t}([\hat{\sigma}_{ee},\hat{\sigma}_{ee}\rho(t)]K(t-t^{\prime})$
$\displaystyle-$
$\displaystyle[\hat{\sigma}_{ee},\rho(t)\hat{\sigma}_{ee}]K^{\ast}(t-t^{\prime}))dt^{\prime},$
(9)
where $\alpha(t)=f(t)e^{i\omega_{ex}t}$. Exciton population and optical
induced coherence in the QD system are defined through the different matrix
elements of the density matrix. Exciton population and optical coherence are
defined with the following set of equations, respectively
$\displaystyle\dot{P}(t)$ $\displaystyle=$ $\displaystyle
i\alpha(t)(2N_{e}(t)-1)-P(t)\int_{0}^{t}K(t-t^{\prime})dt^{\prime},$
$\displaystyle\dot{N}_{e}(t)$ $\displaystyle=$ $\displaystyle
2iIm(\alpha^{\ast}(t)P(t)),$ (10)
where $P(t)=\langle e|\hat{\rho}_{ex}(t)|g\rangle$ and $N_{e}(t)=\langle
e|\hat{\rho}_{ex}(t)|e\rangle$. We assume at $t=0$ the QD be in its ground
state and at this time it is excited with a Gaussian pulse excitation with
envelop function $f(t)=\frac{A}{\sqrt{2\pi}a}e^{-\frac{t^{2}}{a^{2}}}$ where
$a$ is the pulse width and $A$ is a measure of pulse amplitude. For numerical
integration of this set of equations, we shall take a GaAs QD with a spherical
shape. In this case the spectral density is given by
$j(\omega)=\frac{(\sigma_{e}-\sigma_{h})^{2}}{4\pi^{2}\rho
c^{5}}\omega^{3}e^{-\frac{3l^{2}}{2c^{2}}\omega^{2}},$ (11)
where $\sigma_{e}$ and $\sigma_{h}$ are the bulk deformation potential
constants for electron and hole, $c$ is the sound velocity in the sample and
$l$ is the electron and hole ground-state localization length (we assume a
spherically symmetric harmonic confinement potential for the QD and electron
and hole in the ground state). We use the following numerical values
$\sigma_{e}-\sigma_{h}=9eV$, $\rho=5350\frac{kg}{m^{3}}$, $c=5150\frac{m}{s}$
and $l=4.5nm$ (these material parameters are approximately acquired from amk
). Figure (1) shows plots of the time evolution of the exciton inversion for
two values of pulse duration. In first picoseconds of dynamics the time
evolution shows a strong decrease of exciton inversion due to the phonon
effects and then we see a stable oscillation in inversion behavior during the
pulse duration. It is clear from the figure that the phonon effects can
prevent exciton generation. On the other hand, we see the complex behavior on
the same timescales of initial dynamics for each pulse duration and after that
small oscillations will continue at the end of pulse duration. Then we
conclude that in the first steps of dynamics the influence of phonons is a
very important damping effect. Figure(2) shows plots of $ImP(t)$ to consider
the time evolution of optical coherence. As in the case of exciton population,
optical coherence experiences a very rapid decrease during some first
picoseconds. After this strong decrease we see a very small stable
oscillations in optical coherence. Therefore, we concloud phonon effects are
very important on timescales smaller than the spontaneous decay time and we
can consider phonon reservoir as dominant damping source during the first
steps of dynamics.
## IV Interaction of QD with cavity mode
In this section we consider the interaction of the QD embedded in a
microcavity with cavity mode. In this case, the density matrix for the system
satisfies Eq.(5). By using Eq.(5) one can get a set of differential equations
that describe the evolution of the populations and coherence of the cavity-QD
system. In the basis of product states between the QD states and Fock states
of the cavity mode ($|en\rangle$, $|gn\rangle$) we calculate the matrix
elements of the exciton-photon density matrix. By taking the matrix elements
in Eq.(5) we get the following set of linear differential equations for the
populations and coherence in the QD-photon system (we have used the notation
$\rho_{in,jm}=\langle in|\rho|jm\rangle$ in which $i$ and $j$ refer to QD
states)
$\displaystyle\dot{\rho}_{en-1,en-1}(t)=ig\sqrt{n}(\rho_{en-1,gn}(t)e^{-i\Delta
t}-\rho_{gn,en-1}(t)e^{i\Delta t})$
$\displaystyle+if(t)(\rho_{en-1,gn-1}(t)e^{-i\omega_{ex}t}-\rho_{gn-1,en-1}(t)e^{i\omega_{ex}t}),$
(12a)
$\displaystyle\dot{\rho}_{gn,gn}(t)=ig\sqrt{n}(\rho_{gn,en-1}(t)e^{i\Delta
t}-\rho_{en-1,gn}(t)e^{-i\Delta t})$
$\displaystyle+if(t)(\rho_{gn,en}(t)e^{i\omega_{ex}t}-\rho_{en,gn}(t)e^{-i\omega_{ex}t}),$
(12b)
$\displaystyle\dot{\rho}_{en-1,gn}(t)=ig\sqrt{n}(\rho_{en-1,en-1}(t)e^{i\Delta
t}$ $\displaystyle-\rho_{gn,gn}(t)e^{i\Delta
t})-\rho_{en-1,gn}(t)\int_{0}^{t}K(t-t^{\prime})dt^{\prime},$ (12c)
$\displaystyle\dot{\rho}_{en-1,gn-1}(t)=if(t)(\rho_{en-1,en-1}(t)e^{i\omega_{ex}t}$
$\displaystyle-\rho_{gn-1,gn-1}(t)e^{i\omega_{ex}t})-\rho_{en-1,gn-1}(t)\int_{0}^{t}K(t-t^{\prime})dt^{\prime}.$
(12d)
In the absence of pulse excitation, the matrix elements $\rho_{en-1,en-1}(t)$,
$\rho_{gn,gn}(t)$, $\rho_{en-1,gn}(t)$ and $\rho_{gn,en-1}(t)$, for a given
photon number, satisfy a closed set of differential equations. However, the
excitation pulse couples the different terms to each other and an infinite set
of equations has to be solved. In the process of obtaining the above set of
equations we neglect the terms like $\rho_{gn,gn-1}(t)$ and
$\rho_{en,en-1}(t)$ because these terms do not have physical meaning related
to the conditions under consideration. These terms show a coherence in photon
field while the QD remains in its state. This could be related to photon
damping which we have neglected such kind of terms. On the other hand, we
maintain terms like $\rho_{en,gn}(t)$ which describe coherence in QD system
while photon number is constant. As is clear from (12) these terms cam be
generated during the dynamics by the pump pulse.
As initial condition we take at $t=0$ the QD in its ground state and cavity
field in the vacuum state $\rho_{g0,g0}(0)=1$, and all other elements of the
density matrix equal to zero. For the numerical integration, the set of
equations can be truncated at a given value, which we take it equal to 90
(this value is choose with this condition that the results not change with
increasing the number of equation).
Photon statistics and material characteristics such as inversion population
and optical coherence can be obtained from (12). At first we consider Mandel
parameter of the cavity field which is defined as mandel
$M=\frac{\langle\hat{n}^{2}\rangle-\langle\hat{n}\rangle^{2}}{\langle\hat{n}\rangle}-1.$
(13)
This parameter vanishes for the Poisson distribution, is positive for the
super-Poisson distribution (photon bunching effect), and is negative for the
sub-Poisson distribution (photon anti-bunching effect). The mean number of
photons in the cavity is (other moments of $\hat{n}$ can be calculated in the
same manner)
$\langle\hat{n}\rangle=\sum_{n}n\left[\rho_{en,en}(t)+\rho_{gn,gn}(t)\right].$
(14)
Mandel parameter for the case of resonant interaction ($\Delta=0$) and in the
presence of detuning is plotted, respectively, in figures (3) and (4) for two
different values of pulse duration. As is seen, the cavity field mode exhibits
non-classical (sub-Poissonian statistics) in the course of time evolution.
Another important feature of this plot is the oscillatory behavior of Mandel
parameter for time scales approximately two times of pulse duration.
Therefore, the emitted photons to cavity mode by QD in the course of the
excitation duration can be reabsorbed by QD and re-excite the QD then after
the end of pulse duration we have oscillations in photon statistics. On the
other hand, it is clear that with increasing the detuning feature the
amplitude of oscillations in Mandel parameter decrease.
Another important quantity in photon statistics is second order coherence
function, $g^{(2)}(t,\tau)$ qo1 ,mandel which is a two-time correlation
function. Here we consider this quantity for the case of zero time delay,
$g^{(2)}(t,\tau=0)$. This quantity can be used as an indication of the
possible coherence of the state of the photon system. For the single mode
cavity field $g^{(2)}(t,\tau=0)$ has the following definition
$\displaystyle g^{(2)}(t,\tau=0)$ $\displaystyle=$ $\displaystyle\frac{\langle
a^{\dagger}a^{\dagger}aa\rangle}{\langle a^{\dagger}a\rangle^{2}}$
$\displaystyle=$
$\displaystyle\frac{\sum_{n}n(n-1)\left[\rho_{en,en}(t)+\rho_{gn,gn}(t)\right]}{\left(\sum_{n}n\left[\rho_{en,en}(t)+\rho_{gn,gn}(t)\right]\right)^{2}}.$
In the case of resonant interaction and off-resonant interaction, the plots of
this quantity are shown in figures(5) and (6), respectively. The figures show
non-classical nature of emitted photons (photon anti-bunching). This quantity
shows similar oscillatory behavior to the Mandel parameter and its oscillatory
behavior continue up to times twice the pulse duration. According to these
plots the detuning effects on $g^{(2)}(t,\tau=0)$ are similar to its effects
on the Mandel parameter and cause the amplitude of oscillation be reduced.
Therefore, in this conditions without any restriction on physical parameters
(damping coefficients and coupling constant) it is possible that QD emits
anti-bunched photons with sub-Poissonian statistics. The possibility of
emitting anti-bunched photons with sub-Poissonian statistics by a single QD
has been considered experimentally becher .
The time evolution of the QD coherence in the process of one photon
interaction $P(t)=\langle e0|\rho(t)|g1\rangle$ is shown in figures(7) and (8)
for different values of pulse duration and detuning. In these figures we plot
imaginary part of $P(t)$. These figures indicating occurrence of decoherence
(damping of the imaginary part of polarization) in the system. The main source
of this decoherence is phonon interaction. In the case of pulse with long
duration we see an irregular oscillation in some time periods. It is clear
that detuning prevents the coherence in this system. However the detuning is
increased the imaginary part of coherence $P(t)$ and increasing of detuning
leads to the regular oscillatory behavior and causes damping will decrease. In
turn, because of the detuning, which weakens the dynamics, the pumping should
be increased. Hence these two parameter can be considered as some experimental
parameters for controling the decoherence in the QD systems on the timescales
under consideration. On the other hand, by comparison Fig.(2) with Fig.(7) we
can conclude that while the QD interacts with a cavity mode its optical
coherence between energy levels has a longer life time. Then this can be
considered as another experimental condition for controlling of optical
coherence.
## V Conclusion
In this paper we have considered phonon effects (dephasing effects) on optical
properties of a pulse driven QD. We have shown that these effects strongly
affect the Rabi oscillations and optical coherence. In the time scales which
spontaneous emission and non-radiative recombination do not play an important
role in the dynamics (characteristic times of these effects are much longer
than the characteristic time of phonon reservoir) the phonons strongly affect
optical properties of QD. In the case of the interaction of system under
consideration with cavity mode we have shown that emitted photons are anti-
bunched and obey the sub-Poissonian statistics. Then in the microcavity with
high quality factor which contains a single QD it is possible to generate non-
classical light in the first some ten picoseconds. Here, we have considered a
Gaussian pulse as a pump. We have shown that with the ending of pump,
oscillations in the photon statistics continue until times twice the pulse
duration. This relates to cavity photon which remains in the cavity and after
ending of pump re-excites the QD. On the other hand, we have considered the
detuning effect on the optical coherence of QD and we have seen that detuning
can prevent decoherence effects. Hence, detuning can be considered as a
controlling parameter of optical coherence. While QD interacts resonantly with
the cavity mode, we have found that its optical coherence has a longer life
time in comparison with its interaction with classical pulse. Then by putting
the QD in the cavity it can maintain its coherence between energy levels.
Therefore, the off-resonant interaction of a QD with cavity mode can be
considered as an experimental tool for suppressing decoherence effects on the
exciton.
Acknowledgment The authors wish to thank the Office of Graduate Studies of the
University of Isfahan and Iranian Nanotechnology initiative for their support.
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* (19) B. Krummheuer, V. M. Axt, and T. Kuhn, Phys. Rev. B 65, 195313 (2002).
E. Pazy, Semicond. Sci. Technol. 17, 1172 (2002).
* (20) H.-P. Breuer, and F. Petruccione, The Theory of Open Quantum Systems(Oxford University Press, 2002).
* (21) G. Mahan, Many-Body Physics (Kluwer, New York, 2000).
* (22) U. Weiss, Quantum Dissipative Systems (WOrld Scientific, Singapore, 1999).
* (23) V. M. Axt, P. Machnikowski, T. Kuhn, Phys. Rev. B 71, 155305 (2005).
* (24) L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press 1995).
* (25) C. Becher, A. Kiraz, P. Michler, A. Imamoglu, W. V. Schoenfeld, P. M. Petroff, L. Zhang, and E. Hu, Phys. Rev. B 63, 121312(R) (2001).
Figure 1: Plots of exciton inversion versus time for two different values of
pulse duration: (a) $a=10ps$, (b) $a=40ps$. Material parameters are pointed
out in the text and $T=30K$. Figure 2: Plots of imaginary part of optical
polarization versus time for two different values of pulse duration: (a)
$a=10ps$, (b) $a=40ps$. Material parameters are pointed out in the text and
$T=30K$. Figure 3: Mandel parameter versus time for pulse duration $a=10ps$
and $T=30K$ for two different values of detuning $\Delta=0.0,\;\Delta=1.0$.
Figure 4: Mandel parameter versus time for pulse duration $a=40ps$ and $T=30K$
for two different values of detuning $\Delta=0.0,\;\Delta=1.0$. Figure 5:
$g^{(2)}(t,\tau=0)$ as a function of time for pulse duration $a=10ps$ and
$T=30K$ for two different values of detuning $\Delta=0.0,\;\Delta=1.0$. Figure
6: $g^{(2)}(t,\tau=0)$ as a function of time for pulse duration $a=40ps$ and
$T=30K$ for two different values of detuning $\Delta=0.0,\;\Delta=1.0$. Figure
7: $ImP(t)$ as a function of time for three different values of detuning and
for pulse duration $a=10ps$. In this plot $T=30K$. Figure 8: $ImP(t)$ as a
function of time for three different values of detuning and for pulse duration
$a=40ps$. In this plot $T=30K$.
|
arxiv-papers
| 2011-12-26T11:41:17 |
2024-09-04T02:49:25.703260
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Bagheri Harouni, R. Roknizadeh and M. H. Naderi",
"submitter": "Malek Bagheri",
"url": "https://arxiv.org/abs/1112.5845"
}
|
1112.5902
|
# A note on the modified $q$-Genocchi numbers and polynomials with weight
$\left(\alpha,\beta\right)$ and their interpolation function at negative
integers
Serkan Aracı University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com , Mehmet Açıkgöz
University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY acikgoz@gantep.edu.tr , Feng Qi
Department of Mathematics, College of Science, Tianjin Polytechnic University,
Tianjin 300160, China qifeng618@gmail.com and Hassan Jolany School of
Mathematics, Statistics and Computer Science, University of Tehran, Iran
hassan.jolany@khayam.ut.ac.ir
(Date: December 12, 2011)
###### Abstract.
The purpose of this paper concerns to establish modified $q$-Genocchi numbers
and polynomials with weight ($\alpha$,$\beta$). In this paper we investigate
special generalized $q$-Genocchi polynomials and we apply the method of
generating function, which are exploited to derive further classes of
$q$-Genocchi polynomials and develop $q$-Genocchi numbers and polynomials. By
using the Laplace-Mellin transformation integral, we define $q$-Zeta function
with weight ($\alpha$,$\beta$) and by presenting a link between $q$-Zeta
function with weight ($\alpha$,$\beta$) and $q$-Genocchi numbers with weight
($\alpha$,$\beta$) we obtain an interpolation formula for the $q$-Genocchi
numbers and polynomials with weight ($\alpha$,$\beta$). Also we derive
distribution formula (Multiplication Theorem) and Witt’s type formula for
modified $q$-Genocchi numbers and polynomials with weight ($\alpha$,$\beta$)
which yields a deeper insight into the effectiveness of this type of
generalizations for $q$-Genocchi numbers and polynomials. Our new generating
function possess a number of interesting properties which we state in this
paper.
###### Key words and phrases:
Genocchi numbers and polynomials, $q$-Genocchi numbers and polynomials,
$q$-Genocchi numbers and polynomials with weight $\alpha$
###### 2000 Mathematics Subject Classification:
Primary 46A15, Secondary 41A65
## 1\. Introduction, Definitions and Notations
Recently, $q$-calculus has served as a bridge between mathematics and physics.
Therefore, there is a significant increase of activity in the area of the
$q$-calculus due to applications of the $q$-calculus in mathematics,
statistics and physics. The majority of scientists in the world who use
$q$-calculus today are physicists. $q$-Calculus is a generalization of many
subjects, like hypergeometric series, generating functions, complex analysis,
and particle physics. In short, $q$-calculus is quite a popular subject today.
One of Important Branch of $q$-calculus in number theory is $q$-type of
special generating functions, for instance $q$-Bernoulli numbers, $q$-Euler
numbers, and $q$-Genocchi numbers, here we introduce a new class of $q$-type
generating function. We introduce $q$-Genocchi numbers with weight
$\left(\alpha,\beta\right)$. When we define a new class of generating
functions like, $q$-Genocchi numbers with weight $\left(\alpha,\beta\right)$,
then we face to with this question that “can we define a new $q$-Zeta type
function in related of this new class of $q$-type generating function?”. We
give a positive answer for our new class of numbers and polynomials. More
precisely we show that our $q$-type generating function is generalization of
the Hurwitz Zeta function. Historically many authors have tried to give
$q$-analogues of the Riemann Zeta function $\zeta\left(s\right)$, and its
related functions. By just following the method of Kaneko et al. [M. Kaneko,
N. Kurokawa and M. Wakayama, A variation of Euler’s approach to the Riemann
Zeta function, Kyushu J. Math. 57 (2003), 175–192], who mainly used Euler-
Maclaurin summation formula to present and investigate a $q$-analogue of the
Riemann zeta function $\zeta\left(s\right)$, and gave a good and reasonable
explanation that their $q$-analogue may be a best choice. They also commented
that $q$-analogue of $\zeta\left(s\right)$ can be achieved by modifying their
method. Furthermore it is clear that $q$-Genocchi polynomials of weight
$\left(\alpha,\beta\right)$ are in a class of orthogonal polynomials and we
know that most such special functions that are orthogonal are satisfied in
multiplication theorem, so in this present paper we show this property is true
for $q$-Genocchi polynomials of weight $\left(\alpha,\beta\right)$. In this
introductory section, we present the definitions and notations (and some of
the Important properties and characteristics) of the various special
functions, polynomials and numbers, which are potentially useful in the
remainder of the paper.
Assume that $p$ be a fixed odd prime number. Throughout this paper we use the
following notations. By $\mathbb{Z}_{p}$ we denote the ring of $p$-adic
rational integers, $\mathbb{Q}$ denotes the field of rational numbers,
$\mathbb{Q}_{p}$ denotes the field of $p$-adic rational numbers, and
$\mathbb{C}_{p}$ denotes the completion of algebraic closure of
$\mathbb{Q}_{p}$. Let $\mathbb{N}$ be the set of natural numbers and
$\mathbb{Z}_{+}=\mathbb{N}\cup\left\\{0\right\\}.$ Let $v_{p}$ be the
normalized exponential valuation of $\mathbb{C}_{p}$ with
$\left|p\right|_{p}=p^{-v_{p}\left(p\right)}=p^{-1}.$ When one speaks of
$q$-extension, $q$ is considered in many ways such as an indeterminate, a
complex number $q\in\mathbb{C}$ or $p$-adic number $q\in\mathbb{C}_{p}.$ If
$q\in\mathbb{C}$ one normally assume that $\left|q\right|<1.$ If
$q\in\mathbb{C}_{p},$ we assume that $\left|1-q\right|_{p}<p^{-\frac{1}{p-1}}$
so that $q^{x}=\exp\left(x\log q\right)$ for $\left|x\right|_{p}\leq 1.$ We
use the following notation as follows:
$\left[x\right]_{q}=\frac{1-q^{x}}{1-q}\text{,
}\left[x\right]_{-q}=\frac{1-\left(-q\right)^{x}}{1+q}$
Note that $\lim_{q\rightarrow 1}\left[x\right]_{q}=x$; cf. [1-24].
For a fixed positive integer $d$ with $\left(d,f\right)=1,$ we set
$\displaystyle X$ $\displaystyle=$ $\displaystyle
X_{d}=\lim_{\overleftarrow{N}}\mathbb{Z}/dp^{N}\mathbb{Z},$ $\displaystyle
X^{\ast}$ $\displaystyle=$
$\displaystyle\underset{\underset{\left(a,p\right)=1}{0<a<dp}}{\cup}a+dp\mathbb{Z}_{p}$
and
$a+dp^{N}\mathbb{Z}_{p}=\left\\{x\in X\mid x\equiv
a\left(\mathop{\mathrm{m}od}dp^{N}\right)\right\\},$
where $a\in\mathbb{Z}$ satisfies the condition $0\leq a<dp^{N}.$
By use Koblitz [N. Koblitz, $p$-adic Numbers $p$-adic Analysis and Zeta
Functions, Springer-Verlag, New York Inc, 1977] notations, A $p$-adic
distribution $\mu$ on $X$ is a $\mathbb{Q}_{p}$-linear vector space
homomorphism from the $\mathbb{Q}_{p}$-vector space of locally constant
functions on $X$ to $\mathbb{Q}_{p}$. If $f:X\rightarrow\mathbb{Q}_{p}$ is
locally constant, instead of writing $\mu\left(f\right)$ for the value of
$\mu$ at $f$, we usually write $\int f\mu$. Also it is known that we can write
$\mu_{q}$ as follows:
$\mu_{q}\left(x+p^{N}\mathbb{Z}_{p}\right)=\frac{q^{x}}{\left[p^{N}\right]_{q}}$
is a distribution on $X$ for $q\in\mathbb{C}_{p}$ with
$\left|1-q\right|_{p}\leq 1.$ For
$f\in UD\left(\mathbb{Z}_{p}\right)=\left\\{f\mid
f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{p}\text{ is uniformly differentiable
function}\right\\},$
the fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ is defined by T. Kim
as follows:
$\displaystyle I_{-q}\left(f\right)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{-q}\left(x\right)=\lim_{N\rightarrow\infty}\sum_{x=0}^{p^{N}-1}f\left(x\right)\mu_{-q}\left(x+p^{N}\mathbb{Z}_{p}\right)$
$\displaystyle=$
$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{\left[p^{N}\right]_{-q}}\sum_{x=0}^{p^{N}-1}\left(-1\right)^{x}f\left(x\right)q^{x}$
Let $q\rightarrow 1,$ then we have fermionic integration on $\mathbb{Z}_{p}$
as follows:
$I_{-1}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{-1}\left(x\right)=\lim_{N\rightarrow\infty}\sum_{x=0}^{p^{N}-1}\left(-1\right)^{x}f\left(x\right),$
So by applying $f\left(x\right)=e^{xt},$ we get
(1.2)
$t\int_{\mathbb{Z}_{p}}e^{tx}d\mu_{-1}\left(x\right)=\frac{2t}{e^{t}+1}=\sum_{n=0}^{\infty}G_{n}\frac{t^{n}}{n!}$
Where $G_{n}$ are Genocchi numbers. By using (1.2), we have
$\int_{\mathbb{Z}_{p}}e^{xt}d\mu_{-1}\left(x\right)=\sum_{n=0}^{\infty}\frac{G_{n+1}}{n+1}\frac{t^{n}}{n!}$
so from above, we obtain
$\sum_{n=0}^{\infty}\left(\int_{\mathbb{Z}_{p}}x^{n}d\mu_{-1}\left(x\right)\right)\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\left(\frac{G_{n+1}}{n+1}\right)\frac{t^{n}}{n!}$
By comparing coefficients of $\frac{t^{n}}{n!}$ on both sides of the above
equation it is fairly straightforward to deduce,
$\frac{G_{n+1}}{n+1}=\int_{\mathbb{Z}_{p}}x^{n}d\mu_{-1}\left(x\right).$
The definition of modified $q$-Euler numbers are given by
(1.3) $\varepsilon_{0,q}=\frac{\left[2\right]_{q}}{2},\text{
}\left(q\varepsilon+1\right)^{k}-\varepsilon_{k,q}=\left\\{\QATOP{\left[2\right]_{q},\text{
}k=0}{0,\text{ }k>0}\right.$
with usual the convention about replacing $\varepsilon^{k}$ by
$\varepsilon_{k,q}$ cf. [15],[16]. It was known that the modified $q$-euler
numbers can be represented by $p$-adic $q$-integral on $\mathbb{Z}_{p}$ as
follows:
$\varepsilon_{n,q}=\int_{\mathbb{Z}_{p}}q^{-t}\left[t\right]_{q}^{n}d\mu_{-q}\left(t\right).$
In [3,14,15,17], $q$-Genocchi numbers are defined as follows:
(1.4) $G_{0,q}=0,\text{ and
}q\left(qG_{q}+1\right)^{n}+G_{n,q}=\left\\{\QATOPD..{\left[2\right]_{q},n=1}{0,\text{
\ \ \ }n>1}\right.$
with the usual convention of replacing $\left(G_{q}\right)^{n}$ by $G_{n,q}.$
In [6], $\left(h,q\right)$-Genocchi numbers are indicated as:
$G_{0,q}^{\left(h\right)}=0,\text{ and
}q^{h-2}\left(qG_{q}^{\left(h\right)}+1\right)^{n}+G_{n,q}^{\left(h\right)}=\left\\{\QATOP{\left[2\right]_{q},\text{
}n=1}{0,\text{ \ \ \ }n>1,}\right.$
with the usual convention about replacing
$\left(G_{q}^{\left(h\right)}\right)^{n}$ by $G_{n,q}^{\left(h\right)}.$
Recently, for $n\in\mathbb{Z}_{+},$ Araci et al. are considered weighted
$q$-Genocchi numbers by
(1.5) $\widetilde{G}_{0,q}^{\left(\alpha\right)}=0,\text{
}q^{1-\alpha}\left(q\widetilde{G}_{q}^{\left(\alpha\right)}+1\right)^{n}+\widetilde{G}_{n,q}^{\left(\alpha\right)}=\left\\{\QATOP{\left[2\right]_{q},\text{
}n=1}{0,\text{ \ \ \ \ \ }n\neq 1,}\right.$
with the usual convention about replacing $\left(\widetilde{G}_{q}\right)^{n}$
by $\widetilde{G}_{n,q}$ (for more information, see [1])
For $\alpha,n\in\mathbb{Z}_{+}$ and $h\in\mathbb{N},$ Araci et al. [2] defined
weighted $\left(h,q\right)$-Genocchi numbers as follows:
$\widetilde{G}_{n+1,q}^{\left(\alpha,h\right)}=\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)x}\left[x\right]_{q^{\alpha}}^{n}d\mu_{-q}\left(x\right).$
Taekyun Kim, by using $p$-adic $q$-integral on $\mathbb{Z}_{p}$, introduced a
new class of numbers and polynomials. He added a weight on $q$-Bernoulli
numbers and polynomials and defined $q$-Bernoulli numbers with weight
$\alpha$. He is given some interesting properties concerning $q$-Bernoulli
numbers and polynomials with weight $\alpha$. After, by using $p$-adic
$q$-integral on $\mathbb{Z}_{p},$ several mathematicians started to study on
this new branch of generating function theory and extended most of the
symmetric properties of $q$-Bernoulli numbers and polynomials to $q$-Bernoulli
numbers and polynomials with weight $\alpha$ (for more informations, see
[1],[2],[4],[5],[6],[7],[8],[9],[10],[11],[12]). With the same motivation, we
also introduce modified $q$-Genocchi numbers and polynomials with weight
$\left(\alpha,\beta\right).$ Also, we give some interesting properties this
type of polynomials. Furthermore, we derive the $q$-extensions of zeta type
functions with weight $\left(\alpha,\beta\right)$ from the Mellin
transformation to this generating function which interpolates the $q$-Genocchi
polynomials with weight $\left(\alpha,\beta\right)$ at negative integers.
## 2\. Modified $q$-Genocchi numbers and polynomials with weight
$\left(\alpha,\beta\right)$
In this section, we derive some interesting properties Modified $q$-Genocchi
numbers and polynomials with weight $\left(\alpha,\beta\right)$.
###### Lemma 1.
For $n\in\mathbb{Z}_{+},$we obtain
(2.1) $I_{-q}^{\left(\beta\right)}\left(q^{-\beta
x}f_{n}\right)+\left(-1\right)^{n-1}I_{-q}^{\left(\beta\right)}\left(q^{-\beta
x}f\right)=\left[2\right]_{q^{\beta}}\sum_{l=0}^{n-1}\left(-1\right)^{n-l-1}f\left(l\right),$
###### Proof.
Let be $f_{n}\left(x\right)=f\left(x+n\right)$ and
$I_{-q}^{\left(\beta\right)}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{-q^{\beta}}\left(x\right)$
by the (1), we easily get
(2.2) $\displaystyle-I_{-q}^{\left(\beta\right)}\left(q^{-\beta
x}f_{1}\right)$ $\displaystyle=$
$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{\left[p^{N}\right]_{-q^{\beta}}}\sum_{x=0}^{p^{N}-1}f\left(x+1\right)\left(-1\right)^{x}$
$\displaystyle=$
$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{\left[p^{N}\right]_{-q^{\beta}}}\sum_{x=0}^{p^{N}-1}f\left(x\right)\left(-1\right)^{x}-\left[2\right]_{q^{\beta}}\lim_{N\rightarrow\infty}\frac{f\left(p^{N}\right)+f\left(0\right)}{1+q^{\beta
p^{N}}}$ $\displaystyle=$ $\displaystyle
I_{-q}^{\left(\beta\right)}\left(q^{-\beta
x}f\right)-\left[2\right]_{q^{\beta}}f\left(0\right)$
and
$\displaystyle I_{-q}^{\left(\beta\right)}\left(q^{-\beta x}f_{2}\right)$
$\displaystyle=$ $\displaystyle\int_{\mathbb{Z}_{p}}q^{-\beta
x}f\left(x+2\right)d\mu_{-q^{\beta}}\left(x\right)=\lim_{N\rightarrow\infty}\frac{1}{\left[p^{N}\right]_{-q^{\beta}}}\sum_{x=0}^{p^{N}-1}f\left(x+2\right)\left(-1\right)^{x}$
$\displaystyle=$ $\displaystyle I_{-q}^{\left(\beta\right)}\left(q^{-\beta
x}f\right)+\left[2\right]_{q^{\beta}}\lim_{N\rightarrow\infty}\frac{-f\left(0\right)+f\left(1\right)-f\left(p^{N}\right)+f\left(p^{N}+1\right)}{1+q^{\beta
p^{N}}}$ $\displaystyle=$ $\displaystyle
I_{-q}^{\left(\beta\right)}\left(q^{-\beta
x}f\right)+\left[2\right]_{q^{\beta}}\left(f\left(1\right)-f\left(0\right)\right)$
Thus, we have
$I_{-q}^{\left(\beta\right)}\left(q^{-\beta
x}f_{2}\right)-I_{-q}^{\left(\beta\right)}\left(q^{-\beta
x}f\right)=\left[2\right]_{q^{\beta}}\sum_{l=0}^{1}\left(-1\right)^{1-l}f\left(l\right)$
By continuing this process, we arrive at the desired result.
###### Definition 1.
Let $\alpha,n,\beta\in\mathbb{Z}_{+}.$ We define modified $q$-Genocchi numbers
with weight $\left(\alpha,\beta\right)$ as follows:
(2.3)
$\frac{g_{n+1,q}^{\left(\alpha,\beta\right)}}{n+1}=\left[2\right]_{q^{\beta}}\sum_{m=0}^{\infty}\left(-1\right)^{m}\left[m\right]_{q^{\alpha}}^{n}$
###### Theorem 1.
For $\alpha,n,\beta\in\mathbb{Z}_{+},$ we get
(2.4)
$\frac{g_{n+1,q}^{\left(\alpha,\beta\right)}}{n+1}=\frac{\left[2\right]_{q^{\beta}}}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}\frac{1}{1+q^{\alpha
l}}$
###### Proof.
By (2.3), we develop as follows:
$\displaystyle\frac{g_{n+1,q}^{\left(\alpha,\beta\right)}}{n+1}$
$\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q^{\beta}}}{\left(1-q^{\alpha}\right)^{n}}\sum_{m=0}^{\infty}\left(-1\right)^{m}\left(1-q^{m\alpha}\right)^{n}$
$\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q^{\beta}}}{\left(1-q^{\alpha}\right)^{n}}\sum_{m=0}^{\infty}\left(-1\right)^{m}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}\left(q^{m\alpha}\right)^{l}$
$\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q^{\beta}}}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}\sum_{m=0}^{\infty}\left(-1\right)^{m}q^{m\alpha
l}$ $\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q^{\beta}}}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}\frac{1}{1+q^{\alpha
l}}.$
Thus, we complete the proof of Theorem.
By the following Theorem, we get Witt’s type formula of this type polynomials.
###### Theorem 2.
For $\beta,\alpha,n\in\mathbb{Z}_{+},$ we get
(2.5)
$\frac{g_{n+1,q}^{\left(\alpha,\beta\right)}}{n+1}=\int_{\mathbb{Z}_{p}}q^{-\beta
x}\left[x\right]_{q^{\alpha}}^{n}d\mu_{-q^{\beta}}\left(x\right).$
###### Proof.
By using $p$-adic $q$-integral on $\mathbb{Z}_{p}$, namely, replace $f(x)$ by
$q^{-\beta x}\left[x\right]_{q^{\alpha}}^{n}$ and
$\mu_{-q}\left(x+p^{N}\mathbb{Z}_{p}\right)$ by
$\mu_{-q^{\beta}}\left(x+p^{N}\mathbb{Z}_{p}\right)$ into (1), we get
$\displaystyle\int_{\mathbb{Z}_{p}}q^{-\beta
x}\left[x\right]_{q^{\alpha}}^{n}d\mu_{-q^{\beta}}\left(x\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}\int_{\mathbb{Z}_{p}}q^{\alpha
lx-\beta x}d\mu_{-q^{\beta}}\left(x\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}\lim_{N\rightarrow\infty}\frac{1}{\left[p^{N}\right]_{-q^{\beta}}}\sum_{x=0}^{p^{N}-1}\left(-q^{\alpha
l}\right)^{x}$ $\displaystyle=$
$\displaystyle\frac{1}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}\frac{\left[2\right]_{q^{\beta}}}{1+q^{\alpha
l}}\lim_{N\rightarrow\infty}\frac{1+\left(q^{\alpha
l}\right)^{p^{N}}}{1+q^{\beta p^{N}}}$ $\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q^{\beta}}}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}\frac{1}{1+q^{\alpha
l}}$
Use of (2.4) and (2), we arrive at the desired result.
The Witt’s type formula of modified $q$-Genocchi numbers with weight
$\left(\alpha,\beta\right)$ asserted by Theorem 2, do aid in translating the
various properties and results involving $q$-Genocchi numbers with weight
$\left(\alpha,\beta\right)$ which we state some of them in this section. We
put $\alpha\rightarrow 1$ and $\beta\rightarrow 1$ into (2.5), we readily see
$\frac{g_{n+1,q}^{\left(1,1\right)}}{n+1}=\varepsilon_{n,q}.$
###### Corollary 1.
Let $C_{q}^{\left(\alpha,\beta\right)}\left(t\right)=\sum_{n=0}^{\infty}$
$g_{n,q}^{\left(\alpha,\beta\right)}\frac{t^{n}}{n!}.$ Then we have
$C_{q}^{\left(\alpha,\beta\right)}\left(t\right)=\left[2\right]_{q^{\beta}}t\sum_{m=0}^{\infty}\left(-1\right)^{m}e^{t\left[m\right]_{q^{\alpha}}}.$
###### Proof.
From (2.3) we easily get,
(2.7) $\int_{\mathbb{Z}_{p}}q^{-\beta
x}e^{t\left[x\right]_{q^{\alpha}}}d\mu_{-q^{\beta}}\left(x\right)=\left[2\right]_{q^{\beta}}t\sum_{m=0}^{\infty}\left(-1\right)^{m}e^{t\left[m\right]_{q^{\alpha}}}$
By expression (2.7), we have
$\sum_{n=0}^{\infty}g_{n,q}^{\left(\alpha,\beta\right)}\frac{t^{n}}{n!}=\left[2\right]_{q^{\beta}}t\sum_{m=0}^{\infty}\left(-1\right)^{m}e^{t\left[m\right]_{q^{\alpha}}}$
Thus, we complete the proof of Theorem.
Now, we consider the modified $q$-Genocchi polynomials polynomials with weight
$\alpha$ as follows:
(2.8)
$\frac{g_{n+1,q}^{\left(\alpha,\beta\right)}(x)}{n+1}=\int_{\mathbb{Z}_{p}}q^{-\beta
t}\left[x+t\right]_{q^{\alpha}}^{n}d\mu_{-q^{\beta}}\left(t\right),\text{ \
}n\in\mathbb{N}\text{ and }\alpha\in\mathbb{Z}_{+}$
From expression (2.8), we see readily
(2.9) $\displaystyle\frac{g_{n+1,q}^{\left(\alpha,\beta\right)}(x)}{n+1}$
$\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q^{\beta}}}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}q^{\alpha
lx}\frac{1}{1+q^{\alpha l}}$ $\displaystyle=$
$\displaystyle\left[2\right]_{q^{\beta}}\sum_{m=0}^{\infty}\left(-1\right)^{m}\left[m+x\right]_{q^{\alpha}}^{n}$
Let
$C_{q}^{\left(\alpha,\beta\right)}\left(t,x\right)=\sum_{n=0}^{\infty}g_{n,q}^{\left(\alpha,\beta\right)}(x)\frac{t^{n}}{n!}.$
Then we have
(2.10) $\displaystyle C_{q}^{\left(\alpha,\beta\right)}\left(t,x\right)$
$\displaystyle=$
$\displaystyle\left[2\right]_{q^{\beta}}t\sum_{m=0}^{\infty}\left(-1\right)^{m}e^{t\left[m+x\right]_{q^{\alpha}}}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}g_{n,q}^{\left(\alpha,\beta\right)}\left(x\right)\frac{t^{n}}{n!}.$
By Lemma 1, we get the following Theorem:
###### Theorem 3.
For $m\in\mathbb{N},$and $\alpha,\beta,n\in\mathbb{Z}_{+},$ we get
$\frac{g_{m+1,q}^{\left(\alpha,\beta\right)}}{m+1}+\left(-1\right)^{n-1}\frac{g_{m+1,q}^{\left(\alpha,\beta\right)}\left(n\right)}{m+1}=\left[2\right]_{q^{\beta}}\sum_{l=0}^{n-1}\left(-1\right)^{n-l-1}\left[l\right]_{q^{\alpha}}^{m}$
###### Proof.
By applying Lemma 1 the methodology and techniques used above in getting some
identities for the generating functions of the modified $q$-Genocchi numbers
and polynomials with weight $\left(\alpha,\beta\right),$ we arrive at the
desired result.
###### Theorem 4.
The following identity holds:
$g_{0,q}^{\left(\alpha,\beta\right)}=0,\text{ and \
}g_{n,q}^{\left(\alpha,\beta\right)}\left(1\right)+g_{n,q}^{\left(\alpha,\beta\right)}=\left\\{\QATOP{\left[2\right]_{q^{\beta}},\text{if
}n=1,}{0,\text{ if }n>1.}\right.$
###### Proof.
In (2.2) it is known that
$I_{-q}^{\left(\beta\right)}\left(q^{-\beta
x}f_{1}\right)+I_{-q}^{\left(\beta\right)}\left(q^{-\beta
x}f\right)=\left[2\right]_{q^{\beta}}f\left(0\right)$
If we take $f(x)=e^{t\left[x\right]_{q^{\alpha}}},$ then we have
(2.11) $\displaystyle\left[2\right]_{q^{\beta}}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}q^{-\beta
x}e^{t\left[x+1\right]_{q^{\alpha}}}d\mu_{-q^{\beta}}\left(x\right)+\int_{\mathbb{Z}_{p}}q^{-\beta
x}e^{t\left[x\right]_{q^{\alpha}}}d\mu_{-q^{-\beta}}\left(x\right)$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(g_{n,q}^{\left(\alpha,\beta\right)}\left(1\right)+g_{n,q}^{\left(\alpha,\beta\right)}\right)\frac{t^{n-1}}{n!}$
Therefore, we get the Proof of Theorem.
###### Theorem 5.
For $d\equiv 1\left(\mathop{\mathrm{m}od}2\right)$,
$\alpha,\beta\in\mathbb{Z}_{+}$ and $n\in\mathbb{N},$ we get,
$g_{n,q}^{\left(\alpha,\beta\right)}\left(dx\right)=\frac{\left[d\right]_{q^{\alpha}}^{n-1}}{\left[d\right]_{-q^{\beta}}}\sum_{a=0}^{d-1}\left(-1\right)^{a}g_{n,q^{d}}^{\left(\alpha,\beta\right)}\left(x+\frac{a}{d}\right).$
###### Proof.
From (2.8), we can easily derive the following (2.12)
(2.12) $\displaystyle\int_{\mathbb{Z}_{p}}q^{-\beta
t}\left[x+t\right]_{q^{\alpha}}^{n}d\mu_{-q^{\beta}}\left(t\right)$
$\displaystyle=$
$\displaystyle\frac{\left[d\right]_{q^{\alpha}}^{n}}{\left[d\right]_{-q^{\beta}}}\sum_{a=0}^{d-1}\left(-1\right)^{a}\int_{\mathbb{Z}_{p}}q^{-\beta
t}\left[\frac{x+a}{d}+t\right]_{q^{d\alpha}}^{n}d\mu_{\left(-q^{d}\right)^{\beta}}\left(t\right)$
$\displaystyle=$
$\displaystyle\frac{\left[d\right]_{q^{\alpha}}^{n}}{\left[d\right]_{-q^{\beta}}}\sum_{a=0}^{d-1}\left(-1\right)^{a}\frac{g_{n+1,q^{d}}^{\left(\alpha,\beta\right)}\left(\frac{x+a}{d}\right)}{n+1}.$
So, by applying expression (2.12), we get at the desired result and proof is
complete.
## 3\. Interpolation function of the polynomials
$g_{n,q}^{\left(\alpha,\beta\right)}\left(x\right)$
In this section, we derive the interpolation function of the generating
functions of modified $q$-Genocchi polynomials with weight $\alpha$ and we
give the value of $q$-extension zeta function with weight
$\left(\alpha,\beta\right)$ at negative integers explicitly. For
$s\in\mathbb{C}$, by applying the Mellin transformation to (2.10), we obtain
$\displaystyle\xi^{\left(\alpha,\beta\right)}\left(s,x\mid q\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}t^{s-2}\left\\{-C_{q}^{\left(\alpha,\beta\right)}\left(-t,x\right)\right\\}dt$
$\displaystyle=$
$\displaystyle\left[2\right]_{q^{\beta}}\sum_{m=0}^{\infty}\left(-1\right)^{m}\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}t^{s-1}e^{-t\left[m+x\right]_{q^{\alpha}}}dt$
where $\Gamma\left(s\right)$ is Euler gamma function. We have
$\xi^{\left(\alpha,\beta\right)}\left(s,x\mid
q\right)=\left[2\right]_{q^{\beta}}\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}}{\left[m+x\right]_{q^{\alpha}}^{s}}$
So, we define $q$-extension zeta function with weight
$\left(\alpha,\beta\right)$ as follows:
###### Definition 2.
For $s\in\mathbb{C}$ and $\alpha,\beta\in\mathbb{N},$ we have
(3.1) $\xi^{\left(\alpha,\beta\right)}\left(s,x\mid
q\right)=\left[2\right]_{q^{\beta}}\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}}{\left[m+x\right]_{q^{\alpha}}^{s}}$
$\xi^{\left(\alpha,\beta\right)}\left(s,x\mid q\right)$ can be continued
analytically to an entire function.
Observe that, if $q\rightarrow 1,$ then
$\xi^{\left(\alpha,\beta\right)}\left(s,x\mid 1\right)=\zeta\left(s,x\right)$
which is the Hurwitz- Euler zeta functions. Relation between
$\xi^{\left(\alpha,\beta\right)}\left(s,x\mid q\right)$ and
$g_{n,q}^{\left(\alpha,\beta\right)}\left(x\right)$ are given by the following
theorem:
###### Theorem 6.
For $\alpha,\beta\in\mathbb{N}$ and $n\in\mathbb{N},$ we get
$\xi^{\left(\alpha,\beta\right)}\left(-n,x\mid
q\right)=\frac{g_{n+1,q}^{\left(\alpha,\beta\right)}\left(x\right)}{n+1}.$
###### Proof.
By substituting $s=-n$ into (3.1), we arrive at the desired result.
## References
* [1] Araci, S., Erdal, D., and Seo, J-J., 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.
* [2] Araci, S., Seo, J-J., and Erdal, D., New construction weighted $\left(h,q\right)$-Genocchi numbers and polynomials related to zeta type functions, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 487490, 7 pages, doi:10.1155/2011/487490.
* [3] Araci, S., Erdal, D.,and Kang, D-J., Some new properties on the $q$-Genocchi numbers and polynomials associated with $q$-Bernstein polynomials, Honam Mathematical J. 33 (2011), no. 2, pp. 261-270.
* [4] Araci, S., Aslan, N., and Seo, J-J., A Note on the weighted Twisted Dirichlet’s type $q$-Euler numbers and polynomials, Honam Mathematical J. 33 (2011), no. 3, pp. 311-320.
* [5] Araci, S., Açıkgöz, M., and Seo, J-J., A study on the weighted $q$-Genocchi numbers and polynomials with their Interpolation Function, Accepted in Honam Mathematical Journal.
* [6] Araci, S., and Açıkgöz, M., and Park, K-H., Some identities concerning the $\left(h,q\right)$-Genocchi numbers and polynomials via the $p$-adic $q$-integral on $\mathbb{Z}_{p}$ and $q$-Bernstein polynomials,(submitted).
* [7] Kim, T., On the weighted $q$-Bernoulli numbers and polynomials, Advanced Studies in Contemporary Mathematics 21(2011), no.2, p. 207-215, http://arxiv.org/abs/1011.5305.
* [8] Kim, T., Lee, S. H., Dolgy, D. V., Ryoo, C. S., A note on the generalized $q$-Bernoulli measures with weight $\alpha$, Abstract and Applied Analysis, Article ID 867217, 9 pages, doi:10.1155/2011/867217.
* [9] Kim, T., Choi, J., Kim, Y. H., Ryoo, C. S., A note on the weighted $p$-adic $q$-Euler measure on $\mathbb{Z}_{p}$, Advn. Stud. Contemp. Math. 21 (2011), 35-40.
* [10] Hwang, K-W., Dolgy, D-V., Lee, S. H., and Kim, T., On the Higher-Order $q$-Euler numbers and polynomials with weight $\alpha,$ Discrete Dynamics in Nature and Society, Volume (2011), Article ID 354329, 12 pages.
* [11] Kim, T., and Choi, J., On the $q$-Bernoulli numbers and polynomials with weight $\alpha$, Abstract and Applied Analysis, Volume 2011 (2011), Article ID 392025, 14 pages.
* [12] Kim, T., Dolgy, D-V., Lee, B., and Rim S-H., Identities on the weighted $q$-Euler numbers of higher order, Discrete Dynamics in Nature and Society, Volume 2011 (2011), Article ID 918364, 6 pages.
* [13] Kim, T., On the $q$-extension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007) 1458-1465.
* [14] Kim, T., On the multiple $q$-Genocchi and Euler numbers, Russian J. Math. Phys. 15 (4) (2008) 481-486. arXiv:0801.0978v1 [math.NT]
* [15] Kim, T., The modified $q$-Euler numbers and polynomials, Advn. Stud. Contemp. Math. 16 (2008), 161-170.
* [16] Ozden, H., Simsek, Y., Rim, S-H., Cangul, I-N., A note on $p$-adic $q$-euler measure, Adv. Stud. Contemp. Math. 14 (2007), 233-239.
* [17] Kim, T., $q$-Volkenborn integration, Russ. J. Math. phys. 9$\left(2002\right),$ 288-299.
* [18] Kim, T., $q$-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients, Russ. J. Math. Phys. 15$\left(2008\right),$ 51-57.
* [19] Kim, T., An invariant $p$-adic $q$-integrals on $\mathbb{Z}_{p}$, Applied Mathematics Letters, vol. 21, pp. 105-108,2008.
* [20] Kim, T., A Note on the $q$-Genocchi Numbers and Polynomials, Journal of Inequalities and Applications, Article ID 71452, 8 pages, doi:10.1155/2007/71452.
* [21] Kim, T. Choi, J. Kim, Y. H. and Jang, L. C., On $p$-Adic Analogue of $q$-Bernstein Polynomials and Related Integrals, Discrete Dynamics in Nature and Society, Article ID 179430, 9 pages, doi:10.1155/2010/179430.
* [22] Kim, T., $q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals, J. Nonlinear Math. Phys., 14 (2007), no. 1, 15–27.
* [23] Kim, T., New approach to $q$-Euler polynomials of higher order, Russ. J. Math. Phys., 17 (2010), no. 2, 218–225.
* [24] Kim, T., Some identities on the $q$-Euler polynomials of higher order and $q$-Stirling numbers by the fermionic $p$-adic integral on $\mathbb{Z}_{p}$, Russ. J. Math. Phys., 16 (2009), no.4,484–491.
|
arxiv-papers
| 2011-12-26T23:52:29 |
2024-09-04T02:49:25.711418
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Mehmet A\\c{c}ikg\\\"oz, Feng Qi, and Hassan Jolany",
"submitter": "Hassan Jolany",
"url": "https://arxiv.org/abs/1112.5902"
}
|
1112.5909
|
# Geodesic Structure of Test Particle in Bardeen Spacetime
Sheng Zhou Juhua Chen jhchen@hunnu.edu.cn Yongjiu Wang College of Physics
and Information Science, Hunan Normal University, Changsha, Hunan 410081, P.
R. China
###### Abstract
The Bardeen model describes a regular space-time, i.e. a singularity-free
black hole space-time. In this paper, by analyzing the behavior of the
effective potential for the particles and photons, we investigate the time-
like and null geodesic structures in the space-time of Bardeen model. At the
same time, all kinds of orbits, which are allowed according to the energy
level corresponding to the effective potentials, are numerically simulated in
detail. We find many-world bound orbits, two-world escape orbits and escape
orbits in this spacetime. We also find that bound orbits precession directions
are opposite and their precession velocities are different, the inner bound
orbits shift along counter-clockwise with high velocity while the exterior
bound orbits shift along clockwise with low velocity.
###### pacs:
04.20.Jb, 02.30.Hq,04.70.-s
## I Introduction
The Bardeen model describes a regular space-time Bardeen ; Borde using the
energy-momentum tensor of nonlinear electrodynamics as the source of the field
equations and it is also known as a regular black-hole solution which obeys
the weak energy condition. This global regularity of black hole solutions is
quite important to understand the final state of gravitational collapse of
initially regular configurations. When ratio of mass to charge is
$27g^{2}\leqslant 16m^{2}$, the Bardeen model represents a black hole and a
singularity-free structure Eloy1 . When $27g^{2}=16m^{2}$, the horizons shrink
into a single one, which corresponds to an extreme black hole such as the
extreme Reissner-Nordström solution. The physically reasonable source for
regular black hole solution to Einstein equations has been reported around
1998 Eloy2 ; Eloy3 ; Eloy4 ; Magli . In the Bardeen model, the parameter $g$
representing the magnetic charge of the nonlinear self-gravitating
monopoleEloy1 , was studied later on.
It is well known that many effects, such as bending of light, gravitational
time-delay, gravitational red-shift and precession of planetary orbits, were
predicted by General Relativity. Because these gravitational effects are very
important for theories and observations, many theoretical physics and
astrophysics are interested in investigating them for different gravitational
systems. The geodesic structure with a positive cosmological constant was
investigated by Jaklitsch et al.Jaklitsch , the corresponding effective
potential was analyzed in detail. The analysis of the effective potential for
null geodesics in the Reissner-Nordström-de Sitter and Kerr-de Sitter space-
time was carried out in Refs. Stuchlik and Jiao . All possible geodesic
motions in the extreme Schwarzschild-de Sitter space-time were investigated by
Podolsky Podolsky . Lake investigated light deflection in the Schwarzschild-de
Sitter space-timeLake . Exact solutions in closed analytic form for the
geodesic motion in the Kottler space-time were considered by Kraniotis et al
Kraniotis1 . Kraniotis Kraniotis2 investigated the geodesic motion of a
massive particle in the Kerr and Kerr (anti)de Sitter gravitational field by
solving the Hamilton acobi partial differential equation. Cruz et al.Cruz
studied the geodesic structure of the Schwarzschild anti-de Sitter black hole.
Chen and Wang chen1 ; chen2 ; chen3 ; chen4 have investigated the orbital
dynamics of a test particle in gravitational fields with an electric dipole
and a mass quadrupole, and in the extreme Reissner-Nordström black hole
spacetime. The motion of test particle in Ho$\breve{r}$ava-Lifshitz black hole
space-times was studied using numerical techniques Enolskii .
To find all of the possible orbits which are allowed by the energy levels for
time-like and null geodesic in Bardeen spacetime, we analysis the effective
potentials in detail. To describe the trajectories of massive and null
particles, we have a direct visualization of the allowed motions. This paper
is organized as follows: In Section II, we give a brief review on the Bardeen
spacetime. In Section III, we give out the motion equations, and define the
effective potential. In Section IV and V, we discuss the time-like and null
geodesic structure of the Bardeen spacetime in detail. A conclusion is given
in the last section.
## II The Bardeen spacetime
The line element representing the Bardeen spacetime is given byBorde
$\displaystyle ds^{2}=$ $\displaystyle-$
$\displaystyle[1-\frac{2mr^{2}}{(r^{2}+g^{2})^{\frac{3}{2}}}]dt^{2}+[1-\frac{2mr^{2}}{(r^{2}+g^{2})^{\frac{3}{2}}}]^{-1}dr^{2}$
(1) $\displaystyle+$ $\displaystyle r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}),$
where the parameter g represents the magnetic charge of the nonlinear self-
gravitating monopoleEloy1 . The corresponding lapse function is
$\displaystyle f(r)=1-\frac{2mr^{2}}{(r^{2}+g^{2})^{\frac{3}{2}}}.$ (2)
The Bardeen model describes a regular space-time for the following inequality:
$\displaystyle g^{2}\leqslant\frac{16}{27}m^{2}.$ (3)
When $g^{2}<\frac{16}{27}m^{2}$, there are two horizons in Bardeen spacetime.
For the equality $g^{2}=\frac{16}{27}m^{2}$, the horizons shrink into a single
one, which are showed in Fig.1 in detail.
Figure 1: Horizons of the Bardeen spacetime.
## III Geodesics equation
It is well known that the Euler-Lagrange equations for the variational problem
associated to spacetime metric describes the geodesics. So we set up the
corresponding Lagrangian according to Eq.(1)
$\displaystyle\mathcal{L}=-f(r)\dot{t}^{2}+f(r)^{-1}\dot{r}^{2}+r^{2}(\dot{\theta}^{2}+sin^{2}\theta\dot{\phi}^{2}),$
(4)
in which the dots denote the derivative with respect to the affine parameter
$\tau$. The Hamiltonian motion equations are
$\dot{\Pi}_{q}-\frac{\partial{\mathcal{L}}}{\partial q}=0,$ (5)
where $\Pi_{q}=\partial{\mathcal{L}}/\partial\dot{q}$ is the momentum to
coordinate $q$. Since the Lagrangian is independent of $(t,\phi)$, the
corresponding conjugate momentums are conserved, therefore
$\Pi_{t}=-(1-\frac{2mr^{2}}{(r^{2}+g^{2})^{\frac{3}{2}}})\dot{t}=-E,$ (6)
$\Pi_{\phi}=r^{2}sin^{2}\theta\dot{\phi}=L,$ (7)
where $E$ and $L$ are motion constants.
From the motion equation for $\theta$
$\displaystyle\dot{\Pi}_{\theta}-\frac{\partial{\L}}{\partial\theta}=0,$ (8)
we obtain
$\frac{d(r^{2}\dot{\theta})}{d\tau}=r^{2}sin\theta cos\theta\dot{\phi}^{2}.$
(9)
If we simplify the above equation by choosing the initial conditions
$\theta=\pi/2$, $\dot{\theta}=0$ and $\ddot{\theta}=0$, the Eq.(7) becomes
$\Pi_{\phi}=r^{2}\dot{\phi}=L,$ (10)
from Eqs.(6, 7), the Lagrangian (4) can be written in the following form
$2{\mathcal{L}}\equiv
h=\frac{E^{2}}{1-\frac{2mr^{2}}{(r^{2}+g^{2})^{\frac{3}{2}}}}-\frac{\dot{r}^{2}}{1-\frac{2mr^{2}}{(r^{2}+g^{2})^{\frac{3}{2}}}}-\frac{L^{2}}{r^{2}}.$
(11)
Now we solve the above equation for $\dot{r}^{2}$ in order to obtain the
radial equation, which allows us to characterize possible moments of test
particles and explicit solutions of the motion equation of test particles in
the invariant plane
$\dot{r}^{2}=E^{2}-(1-\frac{2mr^{2}}{(r^{2}+g^{2})^{\frac{3}{2}}})(h+\frac{L^{2}}{r^{2}}),$
(12)
It is useful to rewrite the above motion equation as a one-dimensional problem
$\dot{r}^{2}=E^{2}-V_{eff}^{2},$ (13)
where $V_{eff}^{2}$ is defined as an effective potential
$V_{eff}^{2}=(1-\frac{2mr^{2}}{(r^{2}+g^{2})^{\frac{3}{2}}})(h+\frac{L^{2}}{r^{2}}).$
(14)
## IV Time-like Geodesic Structure
For time-like geodesic $h=1$, the corresponding effective potential becomes
$V_{eff}^{2}=(1-\frac{2mr^{2}}{(r^{2}+g^{2})^{\frac{3}{2}}})(1+\frac{L^{2}}{r^{2}}).$
(15)
Figure 2: The behavior of the effective potential of non-radial particle for
fixed $L=3.5m$(left) and fixed $g=0.87m$(right).
and the orbit equation for massive particle is
$\dot{r}^{2}=E^{2}-(1-\frac{2mr^{2}}{(r^{2}+g^{2})^{\frac{3}{2}}})(1+\frac{L^{2}}{r^{2}}).$
(16)
By using Eq.(10) and making the change of variable $u^{-1}=r$, we can obtain
orbit equation for massive particle
$(\frac{du}{d\phi})^{2}=\frac{E^{2}-1}{L^{2}}-u^{2}+\frac{2mu+2mu^{3}L^{2}}{L^{2}(1+u^{2}g^{2})^{\frac{3}{2}}},$
(17)
Differentiating (17), we have its second order motion equation
$\frac{d^{2}u}{d\phi^{2}}+u=\frac{3mu^{2}L^{2}+3m}{L^{2}(1+u^{2}g^{2})^{\frac{5}{2}}}-\frac{2m}{L^{2}(1+u^{2}g^{2})^{\frac{3}{2}}},$
(18)
We solved (17) and (18) numerically to find all types of geodesics and examine
how the parameters influence on the timelike geodesics in the space-time of
Bardeen model in detail.
From the effective potential curve (see Fig.3), we can identify 3 classes of
orbits: i.e. planetary orbits, escape orbits and circular orbits when the
energy of particle $E$ satisfies two critical values $E_{C_{1}}$ and
$E_{C_{2}}$.
Figure 3: The behavior of the effective potential in Bardeen space-time with
$g=0.70m$, $L=3.5m$, $m=1$ and energy levels $E_{C_{2}}^{2}=0.90$ and
$E_{C_{1}}^{2}=0.92$.
### IV.1 Time-like bound geodesics
Figure 4: The behavior of the effective potential of time-like bound geodesics
for $E^{2}=0.91,g=0.7m,L=3.5m,m=1$
In Fig.4 the dashed line denotes the value of the energy $E^{2}=0.91$, i.e.
$E_{C_{1}}^{2}<E^{2}<E_{C_{2}}^{2}$. From potential curve, we can find two
kinds of bound orbits for this energy level:
I) The particle orbits on a many-world bound orbit between the range
$r_{A}<r<r_{B}$, which is near the singularity and can cross the two event
horizons. The $r_{A}$ and $r_{B}$ are the perihelion and aphelion distance of
the planetary orbits, respectively. We also can find the clockwise precession
of planetary orbits which it is a well-known gravitational effect in general
relativity theory.
II) The particle orbit is on a two-world bound orbit in the range
$r_{D}<r<r_{E}$, where the $r_{D}$ and $r_{E}$ are the perihelion and aphelion
distance, which are larger than the orbit of Case I. The two-world bound orbit
is outside the event horizon. However, we also can find that the precession
direction of the planetary orbit is counter-clockwise, and the precession
velocity is slower than the orbit of Case I.
These two kinds of two-world bound orbits are simulated in Fig.6.
Figure 5: Examples of the two-world bound orbit in the Bardeen space-time with
$E^{2}=0.91$, $g=0.7m$, $L=3.5m$ and $m=1$.
### IV.2 Time-like circle geodesics
From Fig.6 and 7, we can see that there are two different circular orbits. One
is a unstable circular orbit, the other one is a stable circular orbit.
I) When the energy of particle $E$ equals the peak value $E_{C_{2}}$ of the
effective potential, the particle can orbit on a unstable circular orbit at
$r=r_{C_{2}}$. Any perturbation would make such unstable orbit recede from
$r=r_{C_{2}}$ to $r=r_{A}$, then reflect at $r=r_{A}$, or move from
$r=r_{C_{2}}$ to $r=r_{B}$, then reflect at $r=r_{B}$. The particle will move
between $r=r_{A}$ and $r=r_{B}$ and will make a unstable choice on a movement
direction due to the perturbation. Figure 6 shows two cases numerically.
II) When the energy of particle $E$ equals the bottom value $E_{C_{1}}$ of the
effective potential, the particle can orbit on a stable circular orbit at
$r=r_{C_{1}}$. Or the particle orbits on a many-world bound orbit in the range
$r_{A}<r<r_{D}$, where the $r_{A}$ and $r_{D}$ are the perihelion and aphelion
distance, respectively. Figure 7 shows two cases numerically.
Figure 6: Examples of the unstable time-like circle orbit in the Bardeen
space-time with $E_{C_{2}}^{2}=0.92$, $g=0.7m$, $L=3.5m$ and $m=1$.
Figure 7: Examples of the stable time-like circle orbit in the Bardeen space-
time with $E^{2}=0.90$, $g=0.7m$, $L=3.5m$ and $m=1$.
### IV.3 Time-like escape geodesics
When the particle energy is above the critical value (i.e. the peak value
$E_{C_{2}}$ of the effective potential), the particle can orbit on a two-world
escape orbit with a curly structure and cross the two horizons which is showed
in Fig.8a. When the energy of particle is much higher than the critical value,
the escape orbit straightly deflects without curls, which is showed in Fig.8b.
This means that the test particle coming from infinite would be reflected at a
value of $r$ and would not be able to reach $r=0$, due to the infinite
potential barrier at $r=0$. on the other words the particle approaches the
black hole from an asymptotically flat region, crosses the horizons twice and
moves away into another asymptotically region.
Figure 8: Examples of the two-world escape orbit in the Bardeen space-time
with $E_{1}^{2}=1.2$, $g=0.7m$, $L=3.5m$ and $m=1$ (top) and the time-like
escape orbit in the Bardeen space-time with $E_{2}^{2}=7$, $g=0.7m$, $L=3.5m$
and $m=1$(bottom).
## V Null Geodesics
For the null geodesic $h=0$, we get the corresponding effective potential from
Eq.(14)
$V_{eff}^{2}=(1-\frac{2mr^{2}}{(r^{2}+g^{2})^{\frac{3}{2}}})\frac{L^{2}}{r^{2}}.$
(19)
Figure 9: The behavior of the effective potential of the null geodesics for
fixed $L=3.5m,m=1(left)$ and for fixed $g=0.8m,m=1(right)$.
The behavior of the effective potential depends on the parameters $g$, $L$ and
the corresponding orbit equation is
$\dot{r}^{2}=E^{2}-(1-\frac{2mr^{2}}{(r^{2}+g^{2})^{\frac{3}{2}}})\frac{L^{2}}{r^{2}}.$
(20)
By using Eq.(20) and making the change of variable $u^{-1}=r$, we obtain the
orbit equation for massive particle
$(\frac{du}{d\phi})^{2}=\frac{E^{2}}{L^{2}}-u^{2}(1-\frac{2mu}{(1+u^{2}g^{2})^{\frac{3}{2}}}).$
(21)
By differentiating the Eq.(21), we have
$\frac{d^{2}u}{d\phi^{2}}+u=\frac{3mu^{2}}{(1+u^{2}g^{2})^{\frac{5}{2}}}.$
(22)
We must solve the geodesic equations (21) and (22) numerically to investigate
the null geodesics structure and how the space-time parameters influence on
the null geodesics structures in the Bardeen spacetime. We continue to follow
the similar process of the Section IV.
### V.1 Null bound geodesics
From the effective potential curve for photons in Fig.10, we can see that
there are two different types of orbit when the energy $E$ belows the peak
energy value $E_{C}$. When the initial position is between $r_{A}$ and
$r_{B}$, the particle will move on a many-world bound orbit with the range of
radius from $r_{A}$ to $r_{B}$. When the particle initial position is on the
right hand side of the potential barrier, the particle approaches $r_{D}$ from
an asymptotically flat region, then will be reflected to move away into
another asymptotically region. These two kinds of orbits corresponding to the
energy level are plotted on the right side of Fig.10, respectively.
Figure 10: Examples of of the many-world null bound and escape geodesics in
Bardeen spacetime with $E^{2}=0.4$, $g=0.6m$, $L=3.5m$ and $m=1$.
### V.2 Null circle geodesics
When the energy $E=E_{C}$, The photon can orbit on a unstable circular orbit
at $r=r_{C}$, and the photo on such orbit will more likely recede from $r_{C}$
to $r_{A}$ crossing the horizons and will be reflected at $r_{A}$ or escape to
the infinity on the other side of the potential barrier due the initial
conditions and outside perturbation. Examples of such two kinds of orbits are
shown in Fig.11.
Figure 11: Examples of of two kinds of unstable null circle geodesics in
Bardeen spacetime with $E^{2}=0.52$, $g=0.6m$, $L=3.5m$ and $m=1$.
### V.3 Null escape geodesics
When the energy $E>E_{C}$, The photon will be on the three different kinds of
the escape geodesics which are shown in Fig.12. We can see that when the
energy level becomes larger from 0.6 to 7, the corresponding orbit changes
from the two-world escape orbit to the escape orbit without intersection
point. this means the particle approaches the black hole from an
asymptotically flat region, crosses the horizons and then moves away into
another asymptotically region.
Figure 12: Examples of three kinds of null escape geodesics in Bardeen
spacetime with $g=0.6m$, $L=3.5m$, $m=1$ and two energy levels $E_{1}^{2}=0.6$
and $E_{2}^{2}=0.8$.
## VI conclusions
By analyzing the effective potential of massless and massive test particles in
the Bardeen space-time, which describes a regular space-time and also
represents a singularity free black hole for $g^{2}<(16/27)m^{2}$, where the
parameter $g$ represents the magnetic charge of the nonlinear self-gravitating
monopole, and numerically simulating all possible orbits corresponding to all
kinds of energy levels, we have found that there exist two kinds of bound
orbits, one is close to the center of the black hole and crosses the two
horizons, the other is outside the exterior horizon. The interesting result is
that the planetary orbital precession direction is opposite and heir
precession velocities are different, the inner bound orbit shifts along
counter-clockwise with higher velocity while the exterior bound orbit shifts
along clockwise with low velocity, as shown in Fig.5. We have also found two
kinds of circular orbits, the inside one which closes to the exterior horizon
is unstable and the outside one is a stable circular orbits, and two kinds of
escape orbits. For the photon particle, there only exist one many-world bound
orbit which can cross the inner- and out-horizons, one unstable circular orbit
and three kinds of escape orbits.
## VII Acknowledgments
This project is supported by the National Natural Science Foundation of China
under Grant No.10873004, the State Key Development Program for Basic Research
Program of China under Grant No.2010CB832803 and the Program for Changjiang
Scholars and Innovative Research Team in University, No. IRT0964.
## References
* (1) J. Bardeen, presented at GR5, Tiflis, U.S.S.R., and published in the conference proceedings in the U.S.R. (1968).
* (2) A. Borde, Phys. Rev. D. 55, 7615 (1997).
* (3) E. Ayón-Beato and A. García, Phys. Lett. B. 493, 149 (2000).
* (4) E. Ayón-Beato and A. García, Phys. Rev. Lett. 80, 5056 (1998).
* (5) E. Ayón-Beato and A. García, Gen. Rel. Gravit. 31, 629 (1999).
* (6) E. Ayón-Beato and A. García, Phys. Lett. B. 464, 25 (1999).
* (7) G. Magli, Rept. Math. Phys. 44, 407 (1999).
* (8) M. J. Jaklitsch, C. Hellaby and D. R. Matravers, Gen. Rel. Grav. 21, 94 (1989).
* (9) Z. Stuchlik and M. Calvani, Gen. Rel. Grav. 23, 507 (1991).
* (10) J. Podolsky, Gen. Rel. Grav. 31, 1703 (1999).
* (11) Z. Y. Jiao and Y. C. Li, Chin. Phys. 11, 467 (2002).
* (12) K. Lake, Phys. Rev. D 65, 087301 (2002).
* (13) J. H. Chen and Y. J. Wang, Class. Quantum. Grav. 20, 3897 (2003).
* (14) G. V. Kraniotis and S. B. Whitehouse, Class. Quantum. Grav. 20, 4817 (2003).
* (15) G. V. Kraniotis, Class. Quantum Grav. 21, 4743 (2004)
* (16) N. Cruz, M. Olivares and J. R. Villanueva, Class. Quant. Grav. 22, 1167 (2005).
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|
arxiv-papers
| 2011-12-27T02:51:27 |
2024-09-04T02:49:25.718321
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sheng Zhou, Juhua Chen and Yongjiu Wang",
"submitter": "Juhua Chen",
"url": "https://arxiv.org/abs/1112.5909"
}
|
1112.5938
|
# Estimates for eigenvalues of $\mathfrak{L}$ operator
on Self-Shrinkers*
Qing-Ming Cheng and Yejuan Peng Qing-Ming Cheng
Department of Applied Mathematics, Faculty of Sciences,
Fukuoka University, Fukuoka 814-0180, Japan. cheng@fukuoka-u.ac.jp Yejuan
Peng
Department of Mathematics, Graduate School of Science and Engineering
Saga University, Saga 840-8502, Japan. yejuan666@gmail.com
###### Abstract.
In this paper, we study eigenvalues of the closed eigenvalue problem of the
differential operator $\mathfrak{L}$, which is introduced by Colding and
Minicozzi in [4], on an $n$-dimensional compact self-shrinker in
$\mathbf{R}^{n+p}$. Estimates for eigenvalues of the differential operator
$\mathfrak{L}$ are obtained. Our estimates for eigenvalues of the differential
operator $\mathfrak{L}$ are sharp. Furthermore, we also study the Dirichlet
eigenvalue problem of the differential operator $\mathfrak{L}$ on a bounded
domain with a piecewise smooth boundary in an $n$-dimensional complete self-
shrinker in $\mathbf{R}^{n+p}$. For Euclidean space $\mathbf{R}^{n}$, the
differential operator $\mathfrak{L}$ becomes the Ornstein-Uhlenbeck operator
in stochastic analysis. Hence, we also give estimates for eigenvalues of the
Ornstein-Uhlenbeck operator.
††footnotetext: Key words and phrases: mean curvature flows, self-shrinkers,
spheres, the differential operator $\mathfrak{L}$ and
eigenvalues††footnotetext: 2010 Mathematics Subject Classification: 58G25,
53C40.††footnotetext: * Research partially Supported by JSPS Grant-in-Aid for
Scientific Research (B) No. 24340013.
## 1\. introduction
Let $X:M^{n}\to\mathbf{R}^{n+p}$ be an isometric immersion from an
$n$-dimensional Riemannian manifold $M^{n}$ into a Euclidean space
$\mathbf{R}^{n+p}$. One considers a smooth one-parameter family of immersions:
$F(\cdot,t):M^{n}\to\mathbf{R}^{n+p}$
satisfying $F(\cdot,0)=X(\cdot)$ and
(1.1) $\bigl{(}\dfrac{\partial F(p,t)}{\partial
t}\bigl{)}^{N}=H(p,t),\quad(p,t)\in M\times[0,T),$
where $H(p,t)$ denotes the mean curvature vector of submanifold
$M_{t}=F(M^{n},t)$ at point $F(p,t)$. The equation (1.1) is called the mean
curvature flow equation. A submanifold $X:M^{n}\to\mathbf{R}^{n+p}$ is said to
be a self-shrinker in $\mathbf{R}^{n+p}$ if it satisfies
(1.2) $H=-X^{N},$
where $X^{N}$ denotes the orthogonal projection into the normal bundle of
$M^{n}$ (cf. Ecker-Huisken [10]). Self-shrinkers play an important role in the
study of the mean curvature flow since they are not only solutions of the mean
curvature flow equation, but they also describe all possible blow ups at a
given singularity of a mean curvature flow. Huisken [11] proved that the
sphere of radius $\sqrt{n}$ is the only closed embedded self-shrinker
hypersurfaces with non-zero mean curvature. For classifications of complete
non-compact embedded self-shrinker hypersurfaces, Huisken [12] and Colding and
Minicozzi [4] proved that an $n$-dimensional complete embedded self-shrinker
hypersurface with non-negative mean curvature and polynomial volume growth in
$\mathbf{R}^{n+1}$ is a Riemannian product $S^{k}\times\mathbf{R}^{n-k}$,
$0\leq k<n$. Smoczyk [14] has obtained several results for complete self-
shrinkers with higher co-dimensions.
For study of the rigidity problem for self-shrinkers, Le and Sesum [13] and
Cao and Li [1] have classified $n$-dimensional complete embedded self-
shrinkers in $\mathbf{R}^{n+p}$ with polynomial volume growth if the squared
norm $|A|^{2}$ of the second fundamental form satisfies $|A|^{2}\leq 1$. For a
further study, see Colding and Minicozzi [5, 6], Ding and Wang [7], Ding and
Xin [8, 9], Wang [15] and so on. In [4], Colding and Minicozzi introduced a
differential operator $\mathfrak{L}$ and used it to study self-shrinkers. The
differential operator $\mathfrak{L}$ is defined by
(1.3) $\mathfrak{L}f=\Delta f-\langle X,\nabla f\rangle$
for a smooth function $f$, where $\Delta$ and $\nabla$ denote the Laplacian
and the gradient operator on the self-shrinker, respectively and
$\langle\cdot,\cdot\rangle$ denotes the standard inner product of
$\mathbf{R}^{n+p}$. We should notice that the differential operator
$\mathfrak{L}$ plays a very important role in studying of $n$-dimensional
complete embedded self-shrinkers in $\mathbf{R}^{n+p}$ with polynomial volume
growth in order to guarantee integration by part holds as in [4]. The purpose
of this paper is to study eigenvalues of the closed eigenvalue problem for the
differential operator $\mathfrak{L}$ on compact self-shrinkers in
$\mathbf{R}^{n+p}$ in sections 3 and 4 and eigenvalues of the Dirichlet
eigenvalue problem of the differential operator $\mathfrak{L}$ on a bounded
domain with a piecewise smooth boundary in complete self-shrinkers in
$\mathbf{R}^{n+p}$ in section 5. I shall adapt the idea of Cheng and Yang in
[2] for studying eigenvalues of the Dirichlet eigenvalue problem of the
Laplacian $\Delta$ to the differential operator $\mathfrak{L}$ by constructing
appropriated trial functions for the differential operator $\mathfrak{L}$.
Since the differential operator $\mathfrak{L}$ is self-adjoint with respect to
measure $e^{-\frac{|X|^{2}}{2}}dv$, where $dv$ is the volume element of
$M^{n}$ and $|X|^{2}=\langle X,X\rangle$, we know that the closed eigenvalue
problem:
(1.4) $\mathfrak{L}u=-\lambda u\quad\text{on}\ M^{n}$
for the differential operator $\mathfrak{L}$ on compact self-shrinkers in
$\mathbf{R}^{n+p}$ has a real and discrete spectrum:
$0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{k}\leq\cdots\longrightarrow\infty,$
where each eigenvalue is repeated according to its multiplicity. We shall
prove the following:
###### Theorem 1.1.
Let $M^{n}$ be an $n$-dimensional compact self-shrinker in $\mathbf{R}^{n+p}$.
Then, eigenvalues of the closed eigenvalue problem ${\rm(1.4)}$ satisfy
(1.5)
$\sum_{i=0}^{k}(\lambda_{k+1}-\lambda_{i})^{2}\leq\frac{4}{n}\sum_{i=0}^{k}(\lambda_{k+1}-\lambda_{i})(\lambda_{i}+\frac{2n-\min_{M^{n}}{|X|^{2}}}{4}).$
###### Remark 1.1.
The sphere $S^{n}(\sqrt{n})$ of radius $\sqrt{n}$ is a compact self-shrinker
in $\mathbf{R}^{n+p}$. For $S^{n}(\sqrt{n})$ and for any $k$, the inequality
${\rm(1.5)}$ for eigenvalues of the closed eigenvalue problem ${\rm(1.4)}$
becomes equality. Hence our results in theorem 1.1 are sharp.
Furthermore, from the recursion formula of Cheng and Yang [3], we can obtain
an upper bound for eigenvalue $\lambda_{k}$:
###### Theorem 1.2.
Let $M^{n}$ be an $n$-dimensional compact self-shrinker in $\mathbf{R}^{n+p}$.
Then, eigenvalues of the closed eigenvalue problem ${\rm(1.4)}$ satisfy, for
any $k\geq 1$,
$\lambda_{k}+\frac{2n-\min_{M^{n}}{|X|^{2}}}{4}\leq(1+\frac{a(min\\{n,k-1\\})}{n})(\frac{2n-\min_{M^{n}}{|X|^{2}}}{4})k^{2/n},$
where the bound of $a(m)$ can be formulated as:
$\left\\{\begin{aligned} a(0)&\leq 4,\\\ a(1)&\leq 2.64,\\\ a(m)&\leq
2.2-4\log(1+\frac{1}{50}(m-3)),\qquad\mbox{for}\quad m\geq
2.\end{aligned}\right.$
In particular, for $n\geq 41$ and $k\geq 41$, we have
$\lambda_{k}+\frac{2n-\min_{M^{n}}{|X|^{2}}}{4}\leq(\frac{2n-\min_{M^{n}}{|X|^{2}}}{4})k^{2/n}.$
Results for eigenvalues of the Dirichlet eigenvalue problem of the
differential operator $\mathfrak{L}$ are given in section 5.
Acknowledgements. We would like to express our gratitude to the referee for
valuable suggestions and comments.
## 2\. Preliminaries
Suppose $X:M^{n}\longrightarrow\mathbf{R}^{n+p}$ is an isometric immersion
from Riemannian manifold $M^{n}$ into the (n+p)-dimensional Euclidean space
$\mathbf{R}^{n+p}$. Let $\\{E_{A}\\}_{A=1}^{n+p}$ be the standard basis of
$\mathbf{R}^{n+p}$. The position vector can be written by
$X=(x_{1},x_{2},\cdots,x_{n+p})$. We choose a local orthonormal frame field
$\\{e_{1},e_{2},\cdots,e_{n},e_{n+1},\cdots,e_{n+p}\\}$ and the dual coframe
field
$\\{\omega_{1},\omega_{2},\cdots,\omega_{n},\omega_{n+1},\cdots,\omega_{n+p}\\}$
along $M^{n}$ of $\mathbf{R}^{n+p}$ such that $\\{e_{1},e_{2},\cdots,e_{n}\\}$
is a local orthonormal basis on $M^{n}$. Thus, we have
$\omega_{\alpha}=0,\quad n+1\leq\alpha\leq n+p$
on $M^{n}$. From the Cartan’s lemma, we have
$\omega_{i\alpha}=\sum_{j=1}^{n}h^{\alpha}_{ij}\omega_{j},\quad
h^{\alpha}_{ij}=h^{\alpha}_{ji}.$
The second fundamental form ${\bf h}$ of $M^{n}$ and the mean curvature vector
$H$ are defined, respectively, by
${\bf
h}=\sum_{\alpha=n+1}^{n+p}\sum_{i,j=1}^{n}h^{\alpha}_{ij}\omega_{i}\otimes\omega_{j}e_{\alpha}\quad
H=\sum_{\alpha=n+1}^{n+p}\sum_{i=1}^{n}h^{\alpha}_{ii}e_{\alpha}.$
One considers the mean curvature flow for a submanifold
$X:M^{n}\to\mathbf{R}^{n+p}$. Namely, we consider a one-parameter family of
immersions:
$F(\cdot,t):M^{n}\to\mathbf{R}^{n+p}$
satisfying $F(\cdot,0)=X(\cdot)$ and
(2.1) $\bigl{(}\dfrac{\partial F(p,t)}{\partial
t}\bigl{)}^{N}=H(p,t),\quad(p,t)\in M\times[0,T),$
where $H(p,t)$ denotes the mean curvature vector of submanifold
$M_{t}=F(M^{n},t)$ at point $F(p,t)$. An important class of solutions to the
mean curvature flow equation (2.1) are self-similar shrinkers, which profiles,
self-shrinkers, satisfy
$H=-X^{N},$
which is a system of quasi-linear elliptic partial differential equations of
the second order. Here $X^{N}$ denotes the orthogonal projection of $X$ into
the normal bundle of $M^{n}$.
In [4], Colding and Minicozzi introduced a differential operator
$\mathfrak{L}$ and used it to study self-shrinkers. The differential operator
$\mathfrak{L}$ is defined by
(2.2) $\mathfrak{L}f=\Delta f-\langle X,\nabla f\rangle$
for a smooth function $f$, where $\Delta$ and $\nabla$ denote the Laplacian
and the gradient operator on the self-shrinker, respectively. For a compact
self-shrinker $M^{n}$ without boundary, we have
$\displaystyle\int_{M^{n}}f\mathfrak{L}u\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=\int_{M^{n}}f(\Delta u-\langle X,\nabla u\rangle)\
e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=\int_{M^{n}}f\text{div}(e^{-\frac{|X|^{2}}{2}}\nabla u)dv$
$\displaystyle=\int_{M^{n}}u\mathfrak{L}f\ e^{-\frac{|X|^{2}}{2}}dv,$
that is,
(2.3) $\displaystyle\int_{M^{n}}f\mathfrak{L}u\
e^{-\frac{|X|^{2}}{2}}dv=\int_{M^{n}}u\mathfrak{L}f\
e^{-\frac{|X|^{2}}{2}}dv,$
for any smooth functions $u,f$. Hence, the differential operator
$\mathfrak{L}$ is self-adjoint with respect to the measure
$e^{-\frac{|X|^{2}}{2}}dv$. Therefore, we know that the closed eigenvalue
problem:
(2.4) $\mathfrak{L}u=-\lambda u\quad\text{on}\ M^{n}$
has a real and discrete spectrum:
$0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{k}\leq\cdots\longrightarrow\infty.$
Furthermore, we have
(2.5) $\mathfrak{L}x_{A}=-x_{A}.$
In fact,
$\displaystyle\mathfrak{L}x_{A}=\Delta\langle X,E_{A}\rangle-\langle X,\nabla
x_{A}\rangle$ $\displaystyle=\langle\Delta X,E_{A}\rangle-\langle
X,E_{A}^{T}\rangle$ $\displaystyle=\langle H,E_{A}\rangle-\langle
X,E_{A}^{T}\rangle$ $\displaystyle=-\langle X^{N},E_{A}\rangle-\langle
X,E_{A}^{T}\rangle=-x_{A}.$
Denote the induced metric by $g$ and define $\nabla u\cdot\nabla v=g(\nabla
u,\nabla v)$ for functions $u,v$. We get, from (2.5),
(2.6)
$\mathfrak{L}|X|^{2}=\sum_{A=1}^{n+p}\bigl{(}2x_{A}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla x_{A}\bigl{)}=2(n-|X|^{2}).$
Here we have used
$\sum_{A=1}^{n+p}\nabla x_{A}\cdot\nabla x_{A}=n.$
###### Proposition 2.1.
For an $n$-dimensional compact self-shrinker $M^{n}$ without boundary in
$\mathbf{R}^{n+p}$, we have
$\min_{M^{n}}|X|^{2}\leq n=\dfrac{\int_{M^{n}}|X|^{2}\
e^{-\frac{|X|^{2}}{2}}dv}{\int_{M^{n}}e^{-\frac{|X|^{2}}{2}}dv}\leq\max_{M^{n}}|X^{N}|^{2}.$
###### Proof.
Since $\mathfrak{L}$ is self-adjoint with respect to the measure
$e^{-\frac{|X|^{2}}{2}}dv$, from (2.6), we have
$n\int_{M^{n}}e^{-\frac{|X|^{2}}{2}}dv=\int_{M^{n}}|X|^{2}\
e^{-\frac{|X|^{2}}{2}}dv\geq\min_{M^{n}}|X|^{2}\int_{M^{n}}e^{-\frac{|X|^{2}}{2}}dv.$
Furthermore, since
(2.7) $\Delta|X|^{2}=2(n+\langle X,H\rangle)=2(n-|X^{N}|^{2}),$
we have
$n\leq\max_{M^{n}}|X^{N}|^{2}.$
It completes the proof of this proposition. ∎
## 3\. Universal estimates for eigenvalues
In this section, we give proof of the theorem 1.1. In order to prove our
theorem 1.1, we need to construct trial functions. Thank to
$\mathfrak{L}X=-X$. We can use coordinate functions of the position vector $X$
of the self-shrinker $M^{n}$ to construct trial functions.
Proof of Theorem 1.1. For an $n$-dimensional compact self-shrinker $M^{n}$ in
$\mathbf{R}^{n+p}$, the closed eigenvalue problem:
(3.1) $\mathfrak{L}u=-\lambda u\quad\text{on}\ M^{n}$
for the differential operator $\mathfrak{L}$ has a discrete spectrum. For any
integer $j\geq 0$, let $u_{j}$ be an eigenfunction corresponding to the
eigenvalue $\lambda_{j}$ such that
(3.2) $\begin{cases}\begin{aligned}
&\mathfrak{L}u_{j}=-\lambda_{j}u_{j}\quad\text{on}\ M^{n}\\\
&\int_{M^{n}}u_{i}u_{j}\ e^{-\frac{|X|^{2}}{2}}dv=\delta_{ij},\ \text{for
any}\ i,j.\end{aligned}\end{cases}$
From the Rayleigh-Ritz inequality, we have
(3.3)
$\lambda_{k+1}\leq\frac{\displaystyle{-\int_{M^{n}}}\varphi\mathfrak{L}\varphi\
e^{-\frac{|X|^{2}}{2}}dv}{\displaystyle{\int_{M^{n}}}\varphi^{2}\
e^{-\frac{|X|^{2}}{2}}dv},$
for any function $\varphi$ satisfies $\int_{M^{n}}\varphi u_{j}\
e^{-\frac{|X|^{2}}{2}}dv$, $0\leq j\leq k$. Since $X:M^{n}\to\mathbf{R}^{n+p}$
is a self-shrinker in $\mathbf{R}^{n+p}$, we have
(3.4) $H=-X^{N}.$
Letting $x_{A}$, $A=1,2,\cdots,n+p$, denote components of the position vector
$X$, we define, for $0\leq i\leq k$,
(3.5) $\varphi_{i}^{A}:=x_{A}u_{i}-\sum_{j=0}^{k}a_{ij}^{A}u_{j},\quad
a_{ij}^{A}=\int_{M^{n}}x_{A}u_{i}u_{j}\ e^{-\frac{|X|^{2}}{2}}dv.$
By a simple calculation, we obtain
(3.6)
$\displaystyle{\int_{M^{n}}u_{j}\varphi_{i}^{A}e^{-\frac{|X|^{2}}{2}}dv}=0,\hskip
14.22636pti,j=0,1,\cdots,k.$
From the Rayleigh-Ritz inequality, we have
(3.7)
$\lambda_{k+1}\leq\frac{\displaystyle{-\int_{M^{n}}}\varphi_{i}^{A}\mathfrak{L}\varphi_{i}^{A}\
e^{-\frac{|X|^{2}}{2}}dv}{\displaystyle{\int_{M^{n}}}(\varphi_{i}^{A})^{2}\
e^{-\frac{|X|^{2}}{2}}dv}.$
Since
(3.8) $\displaystyle\mathfrak{L}\varphi_{i}^{A}=\Delta\varphi_{i}^{A}-\langle
X,\nabla\varphi_{i}^{A}\rangle$
$\displaystyle=\Delta(x_{A}u_{i}-\sum_{j=0}^{k}a_{ij}^{A}u_{j})-\langle
X,\nabla(x_{A}u_{i}-\sum_{j=0}^{k}a_{ij}^{A}u_{j})\rangle$
$\displaystyle=x_{A}\Delta u_{i}+u_{i}\Delta x_{A}+2\nabla x_{A}\cdot\nabla
u_{i}-\langle X,x_{A}\nabla u_{i}+u_{i}\nabla x_{A}\rangle$
$\displaystyle-\sum_{j=0}^{k}a_{ij}^{A}\Delta u_{j}+\langle
X,\sum_{j=0}^{k}a_{ij}^{A}\nabla u_{j})$
$\displaystyle=-\lambda_{i}x_{A}u_{i}+u_{i}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla u_{i}+\sum_{j=0}^{k}a_{ij}^{A}\lambda_{j}u_{j},$
we have, from (3.7) and (3.8),
(3.9) $\begin{split}&(\lambda_{k+1}-\lambda_{i})||\varphi_{i}^{A}||^{2}\\\
&\leq-\int_{M^{n}}\varphi_{i}^{A}(u_{i}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla u_{i})\ e^{-\frac{|X|^{2}}{2}}dv:=W_{i}^{A},\end{split}$
where
$||\varphi_{i}^{A}||^{2}=\int_{M^{n}}(\varphi_{i}^{A})^{2}\
e^{-\frac{|X|^{2}}{2}}dv.$
On the other hand, defining
$b_{ij}^{A}=-\int_{M^{n}}(u_{j}\mathfrak{L}x_{A}+2\nabla x_{A}\cdot\nabla
u_{j})u_{i}\ e^{-\frac{|X|^{2}}{2}}dv$
we obtain
(3.10) $b_{ij}^{A}=(\lambda_{i}-\lambda_{j})a_{ij}^{A}.$
In fact,
$\displaystyle\lambda_{i}a_{ij}^{A}$
$\displaystyle=\int_{M^{n}}\lambda_{i}u_{i}u_{j}x_{A}e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=-\int_{M^{n}}u_{j}x_{A}\mathfrak{L}u_{i}\
e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=-\int_{M^{n}}u_{i}\mathfrak{L}(u_{j}x_{A})\
e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=-\int_{M^{n}}u_{i}(x_{A}\mathfrak{L}u_{j}+u_{j}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla u_{j})\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=\lambda_{j}a_{ij}^{A}+b_{ij}^{A},$
that is,
$b_{ij}^{A}=(\lambda_{i}-\lambda_{j})a_{ij}^{A}.$
Hence, we have
(3.11) $b_{ij}^{A}=-b_{ji}^{A}.$
From (3.6), (3.9) and the Cauchy-Schwarz inequality, we infer
(3.12) $\displaystyle W_{i}^{A}$
$\displaystyle=-\int_{M^{n}}\varphi_{i}^{A}\bigl{(}u_{i}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla u_{i}\bigl{)}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=-\int_{M^{n}}\varphi_{i}^{A}\bigl{(}u_{i}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla u_{i}-\sum_{j=0}^{k}b_{ij}^{A}u_{j}\bigl{)}\
e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle\leq\|\varphi_{i}^{A}\|\|u_{i}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla u_{i}-\sum_{j=0}^{k}b_{ij}^{A}u_{j}\|.$
Hence, we have, from (3.9) and (3.12),
$\begin{split}&(\lambda_{k+1}-\lambda_{i})(W_{i}^{A})^{2}\\\
&=(\lambda_{k+1}-\lambda_{i})\|\varphi_{i}^{A}\|^{2}\|u_{i}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla u_{i}-\sum_{j=0}^{k}b_{ij}^{A}u_{j}\|^{2}\\\ &\leq
W_{i}^{A}\|u_{i}\mathfrak{L}x_{A}+2\nabla x_{A}\cdot\nabla
u_{i}-\sum_{j=0}^{k}b_{ij}^{A}u_{j}\|^{2}.\\\ \end{split}$
Therefore, we obtain
(3.13)
$(\lambda_{k+1}-\lambda_{i})^{2}W_{i}^{A}\leq(\lambda_{k+1}-\lambda_{i})\|u_{i}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla u_{i}-\sum_{j=0}^{k}b_{ij}^{A}u_{j}\|^{2}.$
Summing on $i$ from $0$ to $k$ for (3.13), we have
(3.14)
$\sum_{i=0}^{k}(\lambda_{k+1}-\lambda_{i})^{2}W_{i}^{A}\leq\sum_{i=0}^{k}(\lambda_{k+1}-\lambda_{i})\|u_{i}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla u_{i}-\sum_{j=0}^{k}b_{ij}^{A}u_{j}\|^{2}.$
By the definition of $b_{ij}^{A}$ and (3.10), we have
(3.15) $\displaystyle\|u_{i}\mathfrak{L}x_{A}+2\nabla x_{A}\cdot\nabla
u_{i}-\sum_{j=0}^{k}b_{ij}^{A}u_{j}\|^{2}$
$\displaystyle=\|u_{i}\mathfrak{L}x_{A}+2\nabla x_{A}\cdot\nabla u_{i}\|^{2}$
$\displaystyle-2\sum_{j=0}^{k}b_{ij}^{A}\int_{M^{n}}(u_{i}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla u_{i})u_{j}\
e^{-\frac{|X|^{2}}{2}}dv+\sum_{j=0}^{k}(b_{ij}^{A})^{2}$
$\displaystyle=\|u_{i}\mathfrak{L}x_{A}+2\nabla x_{A}\cdot\nabla
u_{i}\|^{2}-\sum_{j=0}^{k}(b_{ij}^{A})^{2}$
$\displaystyle=\|u_{i}\mathfrak{L}x_{A}+2\nabla x_{A}\cdot\nabla
u_{i}\|^{2}-\sum_{j=0}^{k}(\lambda_{i}-\lambda_{j})^{2}(a_{ij}^{A})^{2}.$
Furthermore, according to the definitions of $W_{i}^{A}$ and
$\varphi_{i}^{A}$, we have from (3.10)
(3.16) $\displaystyle W_{i}^{A}$
$\displaystyle=-\int_{M^{n}}\varphi_{i}^{A}\bigl{(}u_{i}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla u_{i}\bigl{)}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=-\int_{M^{n}}(x_{A}u_{i}-\sum_{j=0}^{k}a_{ij}^{A}u_{j})\bigl{(}u_{i}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla u_{i}\bigl{)}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=-\int_{M^{n}}(x_{A}u_{i}^{2}\mathfrak{L}x_{A}+2x_{A}u_{i}\nabla
x_{A}\cdot\nabla u_{i}\bigl{)}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle+\sum_{j=0}^{k}a_{ij}^{A}\int_{M^{n}}u_{j}(u_{i}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla u_{i}\bigl{)}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=-\int_{M^{n}}\bigl{(}x_{A}\mathfrak{L}x_{A}-\frac{1}{2}\mathfrak{L}(x_{A})^{2}\bigl{)}u_{i}^{2}\
e^{-\frac{|X|^{2}}{2}}dv+\sum_{j=0}^{k}a_{ij}^{A}b_{ij}^{A}$
$\displaystyle=\int_{M^{n}}\nabla x_{A}\cdot\nabla x_{A}u_{i}^{2}\
e^{-\frac{|X|^{2}}{2}}dv+\sum_{j=0}^{k}(\lambda_{i}-\lambda_{j})(a_{ij}^{A})^{2}.$
Since
(3.17) $\displaystyle
2\sum_{i,j=0}^{k}(\lambda_{k+1}-\lambda_{i})^{2}(\lambda_{i}-\lambda_{j})(a_{ij}^{A})^{2}$
$\displaystyle=\sum_{i,j=0}^{k}(\lambda_{k+1}-\lambda_{i})^{2}(\lambda_{i}-\lambda_{j})(a_{ij}^{A})^{2}-\sum_{i,j=0}^{k}(\lambda_{k+1}-\lambda_{j})^{2}(\lambda_{i}-\lambda_{j})(a_{ij}^{A})^{2}$
$\displaystyle=-\sum_{i,j=0}^{k}(\lambda_{k+1}-\lambda_{i}+\lambda_{k+1}-\lambda_{j})(\lambda_{i}-\lambda_{j})^{2}(a_{ij}^{A})^{2}$
$\displaystyle=-2\sum_{i,j=0}^{k}(\lambda_{k+1}-\lambda_{i})(\lambda_{i}-\lambda_{j})^{2}(a_{ij}^{A})^{2},$
from (3.14), (3.15), (3.16) and (3.17), we obtain, for any $A$,
$A=1,2,\cdots,n+p$,
(3.18)
$\displaystyle\sum_{i=0}^{k}(\lambda_{k+1}-\lambda_{i})^{2}\int_{M^{n}}\nabla
x_{A}\cdot\nabla x_{A}u_{i}^{2}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle\leq\sum_{i=0}^{k}(\lambda_{k+1}-\lambda_{i})\|u_{i}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla u_{i}\|^{2}.$
On the other hand, since
$\mathfrak{L}x_{A}=-x_{A},\quad\sum_{A=1}^{n+p}(\nabla x_{A}\cdot\nabla
u_{i})^{2}=\nabla u_{i}\cdot\nabla u_{i},$
we infer, from (2.6),
(3.19) $\displaystyle\sum_{A=1}^{n+p}\|u_{i}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla u_{i}\|^{2}$
$\displaystyle=\sum_{A=1}^{n+p}\int_{M^{n}}\bigl{(}u_{i}\mathfrak{L}x_{A}+2\nabla
x_{A}\cdot\nabla u_{i}\bigl{)}^{2}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=\sum_{A=1}^{n+p}\int_{M^{n}}\bigl{(}u_{i}^{2}(x_{A})^{2}-4u_{i}x_{A}\nabla
x_{A}\cdot\nabla u_{i}+4(\nabla x_{A}\cdot\nabla u_{i})^{2}\bigl{)}\
e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=\sum_{A=1}^{n+p}\int_{M^{n}}\bigl{(}u_{i}^{2}(x_{A})^{2}-\nabla(x_{A})^{2}\cdot\nabla
u_{i}^{2}\bigl{)}\ e^{-\frac{|X|^{2}}{2}}dv+4\int_{M^{n}}\nabla
u_{i}\cdot\nabla u_{i}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=\int_{M^{n}}(\mathfrak{L}|X|^{2}+|X|^{2})u_{i}^{2}\
e^{-\frac{|X|^{2}}{2}}dv+4\lambda_{i}$
$\displaystyle=\int_{M^{n}}(2n-|X|^{2})u_{i}^{2}\
e^{-\frac{|X|^{2}}{2}}dv+4\lambda_{i}$
$\displaystyle\leq(2n-\min_{M^{n}}{|X|^{2}})+4\lambda_{i}.$
Furthermore, because of
(3.20) $\sum_{A=1}^{n+p}\nabla x_{A}\cdot\nabla x_{A}=n,$
taking summation on $A$ from $1$ to $n+p$ for (3.18) and using (3.19) and
(3.20), we get
$\displaystyle\sum_{i=0}^{k}(\lambda_{k+1}-\lambda_{i})^{2}\leq\frac{4}{n}\sum_{i=0}^{k}(\lambda_{k+1}-\lambda_{i})(\lambda_{i}+\frac{2n-\min_{M^{n}}{|X|^{2}}}{4}).$
It finished the proof of the theorem 1.1.
$\square$
## 4\. Upper bounds for eigenvalues
The following recursion formula of Cheng and Yang [3] plays a very important
role in order to prove the theorem 1.2.
A recursion formula of Cheng and Yang. Let
$\mu_{1}\leq\mu_{2}\leq\dots,\leq\mu_{k+1}$ be any positive real numbers
satisfying
$\sum_{i=1}^{k}(\mu_{k+1}-\mu_{i})^{2}\leq\frac{4}{n}\sum_{i=1}^{k}\mu_{i}(\mu_{k+1}-\mu_{i}).$
Define
$\Lambda_{k}=\frac{1}{k}\sum_{i=1}^{k}\mu_{i},\qquad
T_{k}=\frac{1}{k}\sum_{i=1}^{k}\mu_{i}^{2},\ \ \
F_{k}=\left(1+\frac{2}{n}\right)\Lambda_{k}^{2}-T_{k}.$
Then, we have
(4.1) $F_{k+1}\leq C(n,k)\left(\frac{k+1}{k}\right)^{\frac{4}{n}}F_{k},$
where
$C(n,k)=1-\frac{1}{3n}\left(\frac{k}{k+1}\right)^{\frac{4}{n}}\frac{\left(1+\frac{2}{n}\right)\left(1+\frac{4}{n}\right)}{(k+1)^{3}}<1.$
Proof of Theorem 1.2. From the proposition 2.1, we know
$\mu_{i+1}=\lambda_{i}+\dfrac{2n-\min_{M^{n}}|X|^{2}}{4}>0,$
for any $i=0,1,2,\cdots$. Then, we obtain from (1.5)
(4.2)
$\sum_{i=1}^{k}(\mu_{k+1}-\mu_{i})^{2}\leq\frac{4}{n}\sum_{i=1}^{k}(\mu_{k+1}-\mu_{i})\mu_{i}.$
Thus, we know that $\mu_{i}$’s satisfy the condition of the above recursion
formula of Cheng and Yang [3]. Furthermore, since
$\mathfrak{L}x_{A}=-x_{A}\ \text{\rm and}\ \int_{M^{n}}x_{A}\
e^{-\frac{|X|^{2}}{2}}dv=0,\quad\text{\rm for}\ A=1,2,\cdots,n+p,$
$\lambda=1$ is an eigenvalue of $\mathfrak{L}$ with multiplicity at least
$n+p$. Thus,
$\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n+1}\leq 1.$
Hence, we have
(4.3) $\sum_{j=1}^{n}(\mu_{j+1}-\mu_{1})=\sum_{j=1}^{n}\lambda_{j}\leq n\leq
2n-\min_{M^{n}}|X|^{2}=4\mu_{1}$
because of $\min_{M^{n}}|X|^{2}\leq n$ according to the proposition 2.1.
Hence, we can prove the theorem 1.2 as in Cheng and Yang [3] almost word by
word. For the convenience of readers, we shall give a self contained proof.
First of all, according to the above recursion formula of Cheng and Yang, we
have
$F_{k}\leq C(n,k-1)\left(\frac{k}{k-1}\right)^{\frac{4}{n}}F_{k-1}\leq
k^{\frac{4}{n}}F_{1}=\frac{2}{n}k^{\frac{4}{n}}\mu_{1}^{2}.$
Furthermore, we infer, from (4.2)
$\left[\mu_{k+1}-\left(1+\frac{2}{n}\right)\Lambda_{k}\right]^{2}\leq\left(1+\frac{4}{n}\right)F_{k}-\frac{2}{n}\left(1+\frac{2}{n}\right)\Lambda_{k}^{2}.$
Hence, we have
$\frac{\frac{2}{n}}{\left(1+\frac{4}{n}\right)}\mu_{k+1}^{2}+\frac{1+\frac{2}{n}}{1+\frac{4}{n}}\left(\mu_{k+1}-\left(1+\frac{4}{n}\right)\Lambda_{k}\right)^{2}\leq\left(1+\frac{4}{n}\right)F_{k}.$
Thus, we derive
(4.4)
$\mu_{k+1}\leq\left(1+\frac{4}{n}\right)\sqrt{\frac{n}{2}F_{k}}\leq\left(1+\frac{4}{n}\right)k^{\frac{2}{n}}\mu_{1}.$
Define
$\displaystyle
a_{1}(n)=\frac{n(1+\frac{4}{n})\left(1+\frac{8}{n+1}+\frac{8}{(n+1)^{2}}\right)^{\frac{1}{2}}}{(n+1)^{\frac{2}{n}}}-n,$
$\displaystyle
a_{2}(k,n)=\frac{n}{k^{\frac{2}{n}}}\left(1+\frac{4(n+k+4)}{n^{2}+5n-4(k-1)}\right)-n,$
$\displaystyle a_{2}(k)=\max\\{a(n,k),k\leq n\leq 400\\},$ $\displaystyle
a_{3}(k)=\dfrac{4}{1-\frac{k}{400}}-2\log k,$ $\displaystyle
a(k)=\max\\{a_{1}(k),a_{2}(k+1)),a_{3}(k+1)\\}.$
The case 1. For $k\geq n+1$, we have
(4.5) $\displaystyle\mu_{k+1}\leq$
$\displaystyle\frac{\left(1+\frac{4}{n}\right)\left(1+\frac{8}{n+1}+\frac{8}{(n+1)^{2}}\right)^{\frac{1}{2}}}{(n+1)^{\frac{2}{n}}}k^{\frac{2}{n}}\mu_{1}$
$\displaystyle=\left(1+\frac{a_{1}(n)}{n}\right)k^{\frac{2}{n}}\mu_{1},$
where $a_{1}(n)\leq 2.31$. In fact, since $\mu_{k+1}$ satisfies (4.2), we
have, from (4.1),
(4.6)
$\mu_{k+1}^{2}\leq\frac{n}{2}\left(1+\frac{4}{n}\right)^{2}F_{k}\leq\frac{n}{2}\left(1+\frac{4}{n}\right)^{2}\left(\frac{k}{n+1}\right)^{\frac{4}{n}}F_{n+1}.$
On the other hand,
(4.7) $\displaystyle F_{n+1}=$
$\displaystyle\frac{2}{n}\Lambda_{n+1}^{2}-\sum_{i=1}^{n+1}\frac{(\mu_{i}-\Lambda_{n+1})^{2}}{n+1}$
$\displaystyle\leq$
$\displaystyle\frac{2}{n}\Lambda_{n+1}^{2}-\frac{(\mu_{1}-\Lambda_{n+1})^{2}+\frac{1}{n}(\mu_{1}-\Lambda_{n+1})^{2}}{n+1}$
$\displaystyle=$
$\displaystyle\frac{2}{n}\left(\Lambda_{n+1}^{2}-\frac{(\mu_{1}-\Lambda_{n+1})^{2}}{2}\right).$
It is obvious that $\Lambda_{n+1}^{2}-\dfrac{(\mu_{1}-\Lambda_{n+1})^{2}}{2}$
is an increasing function of $\Lambda_{n+1}$. From (4.3), we have
(4.8) $\mu_{n+1}+\dots+\mu_{2}\leq(n+4)\mu_{1}.$
Thus, we derive
(4.9) $\Lambda_{n+1}\leq(1+\frac{4}{n+1})\mu_{1}.$
Hence, we have
(4.10)
$\frac{n}{2}F_{n+1}\leq\left(1+\frac{8}{n+1}+\frac{8}{(n+1)^{2}}\right)\mu_{1}^{2}.$
From (4.6) and (4.10), we complete the proof of (4.5). The case 2. For $k\geq
55$ and $n\geq 54$, we have
(4.11) $\displaystyle\mu_{k+1}\leq k^{\frac{2}{n}}\mu_{1}.$
If $k\geq n+1$, from the case 1, we have
$\mu_{k+1}\leq\frac{1}{(n+1)^{\frac{2}{n}}}\left(1+\frac{4}{n}\right)^{2}k^{\frac{2}{n}}\mu_{1}.$
Since
(4.12)
$\displaystyle(n+1)^{\frac{2}{n}}=\exp\left(\frac{2}{n}\log(n+1)\right)$
$\displaystyle\geq 1+\frac{2}{n}\log(n+1)+\frac{2}{n^{2}}(\log(n+1))^{2}$
$\displaystyle\geq\left(1+\frac{1}{n}\log(n+1)\right)^{2},$
we have
(4.13)
$\mu_{k+1}\leq\left(\frac{1+\frac{4}{n}}{1+\frac{1}{n}\log(n+1)}\right)^{2}k^{\frac{2}{n}}\mu_{1}.$
Then, when $n\geq 54$, $\log(n+1)\geq 4$, we have
$\mu_{k+1}\leq k^{\frac{2}{n}}\mu_{1}.$
On the other hand, if $k\leq n$, then $\Lambda_{k}\leq\Lambda_{n+1}$. Since
$\displaystyle\frac{n}{2}F_{k}$
$\displaystyle=\Lambda_{k}^{2}-\frac{n}{2}\frac{\sum_{i=1}^{k}(\mu_{i}-\Lambda_{k})^{2}}{k}$
$\displaystyle\leq\Lambda_{k}^{2}-\frac{n}{2}\frac{(\mu_{1}-\Lambda_{k})^{2}+\dfrac{\left\\{\sum_{i=2}^{k}(\mu_{i}-\Lambda_{k})\right\\}^{2}}{k-1}}{k}$
$\displaystyle\leq\Lambda_{k}^{2}-\frac{(\mu_{1}-\Lambda_{k})^{2}}{2}$
$\displaystyle\leq\Lambda_{n+1}^{2}-\frac{(\mu_{1}-\Lambda_{n+1})^{2}}{2}\leq(1+\frac{4}{n})^{2}\mu_{1}^{2},$
we have
$\displaystyle\mu_{k+1}\leq$
$\displaystyle\left(1+\frac{4}{n}\right)\sqrt{\frac{n}{2}F_{k}}\leq\frac{1}{k^{\frac{2}{n}}}\left(1+\frac{4}{n}\right)^{2}k^{\frac{2}{n}}\mu_{1}\leq\left(\frac{1+\frac{4}{n}}{1+\frac{\log
k}{n}}\right)^{2}k^{\frac{2}{n}}\mu_{1}.$
Here we used $k^{\frac{2}{n}}\geq(1+\frac{\log k}{n})^{2}$. By the same
assertion as above, when $k\geq 55$, we also have
$\mu_{k+1}\leq k^{\frac{2}{n}}\mu_{1}.$
The case 3. For $k\leq 54$ and $k\leq n$, we have
$\displaystyle\mu_{k+1}\leq(1+\frac{\max\\{a_{2}(k),a_{3}(k)\\}}{n})k^{\frac{2}{n}}\mu_{1}.$
Because of $k\leq n$ and $k\leq 54$, from (4.3), we derive,
(4.14) $\mu_{k+1}\leq\frac{1}{n-k+1}\\{(n+5)\mu_{1}-k\Lambda_{k}\\}.$
Since the formula (4.2) is a quadratic inequality for $\mu_{k+1}$, we have
(4.15) $\mu_{k+1}\leq\left(1+\frac{4}{n}\right)\Lambda_{k}.$
Since the right hand side of (4.14) is a decreasing function of $\Lambda_{k}$
and the right hand side of (4.15) is an increasing function of $\Lambda_{k}$,
for
$\dfrac{1}{n-k+1}\\{(n+5)\mu_{1}-k\Lambda_{k}\\}=\left(1+\frac{4}{n}\right)\Lambda_{k}$,
we infer
(4.16) $\displaystyle\mu_{k+1}\leq$
$\displaystyle\frac{1}{k^{\frac{2}{n}}}\left(1+\frac{4(n+k+4)}{n^{2}+5n-4(k-1)}\right)k^{\frac{2}{n}}\mu_{1}$
$\displaystyle=\left(1+\frac{a_{2}(k,n)}{n}\right)k^{\frac{2}{n}}\mu_{1}.$
From the definition of $a_{2}(k)=\max\\{a(n,k),k\leq n\leq 400\\}$, when
$n\leq 400$, we obtain
(4.17) $\mu_{k+1}\leq\left(1+\frac{a_{2}(k)}{n}\right)k^{\frac{2}{n}}\mu_{1}.$
When $n>400$ holds, from (4.4), we have
$\mu_{k+1}\leq\left(1+\frac{4}{n-k}\right)\mu_{1}.$
Since $n>400$ and $k\leq 54$, we know $\frac{2}{n}\log k<\frac{1}{50}$. Hence,
we have
$\displaystyle k^{-\frac{2}{n}}=e^{-\frac{2}{n}\log k}$
$\displaystyle=1-\frac{2}{n}\log k+\frac{1}{2}(\frac{2}{n}\log k)^{2}-\cdots$
$\displaystyle\leq 1-\frac{2}{n}\log k+\frac{1}{2}(\frac{2}{n}\log k)^{2}.$
Therefore, we obtain
$\displaystyle(1+\frac{4}{n-k})k^{-\frac{2}{n}}$
$\displaystyle\leq(1+\frac{4}{n-k})\left(1-\frac{2}{n}\log
k+\frac{1}{2}(\frac{2}{n}\log k)^{2}\right)$ $\displaystyle\leq
1+\frac{\left(4/(1-\frac{k}{400})-2\log k\right)}{n}.$
Hence, we infer
(4.18) $\displaystyle\mu_{k+1}\leq$
$\displaystyle\left(1+\frac{4}{n-k}\right)k^{-\frac{2}{n}}k^{\frac{2}{n}}\mu_{1}$
$\displaystyle\leq$
$\displaystyle\left(1+\frac{\left(4/(1-\frac{k}{400})-2\log
k\right)}{n}\right)k^{\frac{2}{n}}\mu_{1}$ $\displaystyle=$
$\displaystyle\left(1+\frac{a_{3}(k)}{n}\right)k^{\frac{2}{n}}\mu_{1}.$
By Table 1 of the values of $a_{1}(k)$, $a_{2}(k+1)$ and $a_{3}(k+1)$ which
are calculated by using Mathematica and are listed up in the next page, we
have $a_{1}(1)\leq a_{2}(2)\leq a_{3}(2)=a(1)\leq 2.64$ and, for $k\geq 2$,
$a_{3}(k+1)\leq a_{2}(k+1)\leq a_{1}(k).$
Hence, $a(k)=a_{1}(k)$ for $k\geq 2$. Further, for $k\geq 41$, we know
$a(k)<0$. Hence, for $k\geq 2$, we derive
$\mu_{k+1}\leq(1+\frac{a(\min\\{n,k-1\\})}{n})k^{\frac{2}{n}}\mu_{1}$
and for $n\geq 41$ and $k\geq 41$, we have
$\mu_{k+1}\leq k^{\frac{2}{n}}\mu_{1}.$
When $k=1$, $a(0)=4$ from (4.4). It is easy to check that, when $k\geq 3$, by
a simple calculation,
$a(k)\leq 2.2-4\log(1+\frac{k-3}{50}).$
This completes the proof of the theorem 1.2.
$\square$
Table 1: The values of $a_{1}(k)$, $a_{2}(k+1)$ and $a_{3}(k+1)$
$k$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|---|---
$a_{1}(k)\leq$ | 2.31 | 2.27 | 2.2 | 2.12 | 2.03 | 1.94 | 1.86 | 1.77 | 1.69 | 1.61 | |
$a_{2}(k+1)\leq$ | 2.62 | 2.05 | 2.00 | 1.96 | 1.90 | 1.84 | 1.77 | 1.70 | 1.63 | 1.56 | |
$a_{3}(k+1)\leq$ | 2.64 | 1.84 | 1.27 | 0.84 | 0.48 | 0.18 | -0.07 | -0.30 | -0.50 | -0.68 | |
$k$ | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | |
$a_{1}(k)\leq$ | 1.53 | 1.46 | 1.39 | 1.32 | 1.25 | 1.18 | 1.12 | 1.06 | 1.00 | 0.94 | |
$a_{2}(k+1)\leq$ | 1.49 | 1.42 | 1.35 | 1.29 | 1.22 | 1.16 | 1.10 | 1.04 | 0.98 | 0.92 | |
$a_{3}(k+1)\leq$ | -0.84 | -0.99 | -1.13 | -1.26 | -1.37 | -1.48 | -1.59 | -1.68 | -1.78 | -1.86 | |
$k$ | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | |
$a_{1}(k)\leq$ | 0.89 | 0.83 | 0.78 | 0.72 | 0.67 | 0.62 | 0.58 | 0.53 | 0.48 | 0.44 | |
$a_{2}(k+1)\leq$ | 0.87 | 0.82 | 0.76 | 0.71 | 0.66 | 0.61 | 0.57 | 0.52 | 0.47 | 0.43 | |
$a_{3}(k+1)\leq$ | -1.94 | -2.02 | -2.10 | -2.17 | -2.23 | -2.30 | -2.36 | -2.42 | -2.47 | -2.53 | |
$k$ | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 |
$a_{1}(k)\leq$ | 0.39 | 0.35 | 0.31 | 0.27 | 0.23 | 0.19 | 0.15 | 0.11 | 0.07 | 0.03 | -0.00 |
$a_{2}(k+1)\leq$ | 0.38 | 0.34 | 0.30 | 0.26 | 0.22 | 0.18 | 0.14 | 0.10 | 0.07 | 0.03 | -0.01 |
$a_{3}(k+1)\leq$ | -2.58 | -2.63 | -2.68 | -2.72 | -2.77 | -2.81 | -2.85 | -2.89 | -2.93 | -2.97 | -3.00 |
## 5\. The Dirichlet eigenvalue problem
For a bounded domain $\Omega$ with a piecewise smooth boundary
$\partial\Omega$ in an $n$-dimensional complete self-shrinker in
$\mathbf{R}^{n+p}$, we consider the following Dirichlet eigenvalue problem of
the differential operator $\mathfrak{L}$:
(5.1) $\begin{cases}\mathfrak{L}u=-\lambda u&\text{in $\Omega$},\\\
u=0&\text{on $\partial\Omega$}.\end{cases}$
This eigenvalue problem has a real and discrete spectrum:
$0<\lambda_{1}<\lambda_{2}\leq\cdots\leq\lambda_{k}\leq\cdots\longrightarrow\infty,$
where each eigenvalue is repeated according to its multiplicity. We have
following estimates for eigenvalues of the Dirichlet eigenvalue problem
$(5.1)$.
###### Theorem 5.1.
Let $\Omega$ be a bounded domain with a piecewise smooth boundary
$\partial\Omega$ in an $n$-dimensional complete self-shrinker $M^{n}$ in
$\mathbf{R}^{n+p}$. Then, eigenvalues of the Dirichlet eigenvalue problem
$(5.1)$ satisfy
$\sum_{i=1}^{k}(\lambda_{k+1}-\lambda_{i})^{2}\leq\frac{4}{n}\sum_{i=1}^{k}(\lambda_{k+1}-\lambda_{i})(\lambda_{i}+\frac{2n-\inf_{\Omega}{|X|^{2}}}{4}).$
###### Proof.
. By making use of the same proof as in the proof of the theorem 1.1, we can
prove the theorem 5.1 if one notices to count the number of eigenvalues from
$1$. ∎
From the recursion formula of Cheng and Yang [3], we can give an upper bound
for eigenvalue $\lambda_{k+1}$:
###### Theorem 5.2.
Let $\Omega$ be a bounded domain with a piecewise smooth boundary
$\partial\Omega$ in an $n$-dimensional complete self-shrinker $M^{n}$ in
$\mathbf{R}^{n+p}$. Then, eigenvalues of the Dirichlet eigenvalue problem
$(5.1)$ satisfy, for any $k\geq 1$,
$\lambda_{k+1}+\frac{2n-\inf_{\Omega}{|X|^{2}}}{4}\leq(1+\frac{a(min\\{n,k-1\\})}{n})(\lambda_{1}+\frac{2n-\inf_{\Omega}{|X|^{2}}}{4})k^{2/n},$
where the bound of $a(m)$ can be formulated as:
$\left\\{\begin{aligned} a(0)&\leq 4,\\\ a(1)&\leq 2.64,\\\ a(m)&\leq
2.2-4\log(1+\frac{1}{50}(m-3)),\qquad\mbox{for}\quad m\geq
2.\end{aligned}\right.$
In particular, for $n\geq 41$ and $k\geq 41$, we have
$\lambda_{k+1}+\frac{2n-\inf_{\Omega}{|X|^{2}}}{4}\leq(\lambda_{1}+\frac{2n-\inf_{\Omega}{|X|^{2}}}{4})k^{2/n}.$
###### Remark 5.1.
For the Euclidean space $\mathbf{R}^{n}$, the differential operator
$\mathfrak{L}$ is called Ornstein-Uhlenbeck operator in stochastic analysis.
Since the Euclidean space $\mathbf{R}^{n}$ is a complete self-shrinker in
$\mathbf{R}^{n+1}$, our theorems also give estimates for eigenvalues of the
Dirichlet eigenvalue problem of the Ornstein-Uhlenbeck operator.
For the Dirichlet eigenvalue problem $(5.1)$, components $x_{A}$’s of the
position vector $X$ are not eigenfunctions corresponding to the eigenvalue $1$
because they do not satisfy the boundary condition. In order to prove the
theorem 5.2, we need to obtain the following estimates for lower order
eigenvalues.
###### Proposition 5.1.
Let $\Omega$ be a bounded domain with a piecewise smooth boundary
$\partial\Omega$ in an $n$-dimensional complete self-shrinker $M^{n}$ in
$\mathbf{R}^{n+p}$. Then, eigenvalues of the Dirichlet eigenvalue problem
$(5.1)$ satisfy
$\sum_{j=1}^{n}(\lambda_{j+1}-\lambda_{1})\leq(2n-\inf_{\Omega}{|X|^{2}})+4\lambda_{1}.$
###### Proof.
Let $u_{j}$ be an eigenfunction corresponding to the eigenvalue $\lambda_{j}$
such that
(5.2) $\begin{cases}\begin{aligned}
&\mathfrak{L}u_{j}=-\lambda_{j}u_{j}\quad\text{in}\ {\Omega}\\\ &u_{j}=0,\ \
\text{on $\partial\Omega$}\\\ &\int_{\Omega}u_{i}u_{j}\
e^{-\frac{|X|^{2}}{2}}dv=\delta_{ij},\ \text{for any}\
i,j=1,2,\cdots.\end{aligned}\end{cases}$
We consider an $(n+p)\times(n+p)$-matrix $B=(b_{AB})$ defined by
$b_{AB}=\int_{\Omega}x_{A}u_{1}u_{B+1}e^{-\frac{|X|^{2}}{2}}dv.$
From the orthogonalization of Gram and Schmidt, there exist an upper triangle
matrix $R=(R_{AB})$ and an orthogonal matrix $Q=(q_{AB})$ such that $R=QB$.
Thus,
(5.3)
$R_{AB}=\sum_{C=1}^{n+p}q_{AC}b_{CB}=\int_{\Omega}\sum_{C=1}^{n+p}q_{AC}x_{C}u_{1}u_{B+1}=0,\
\text{for}\ 1\leq B<A\leq n+p.$
Defining $y_{A}=\sum_{C=1}^{n+p}q_{AC}x_{C}$, we have
(5.4)
$\int_{\Omega}y_{A}u_{1}u_{B+1}=\int_{\Omega}\sum_{C=1}^{n+p}q_{AC}x_{C}u_{1}u_{B+1}=0,\
\text{for}\ 1\leq B<A\leq n+p.$
Therefore, the functions $\varphi_{A}$ defined by
$\varphi_{A}=(y_{A}-a_{A})u_{1},\ a_{A}=\int_{\Omega}y_{A}u_{1}^{2}\
e^{-\frac{|X|^{2}}{2}}dv,\ \text{for}\ \ 1\leq A\leq n+p$
satisfy
$\int_{\Omega}\varphi_{A}u_{B}=0,\qquad\mbox{for}\ 1\leq B\leq A\leq n+p.$
Therefore, $\varphi_{A}$ is a trial function. From the Rayleigh-Ritz
inequality, we have, for $1\leq A\leq n+p$,
(5.5)
$\lambda_{A+1}\leq\frac{\displaystyle{-\int_{\Omega}}\varphi_{A}\mathfrak{L}\varphi_{A}\
e^{-\frac{|X|^{2}}{2}}dv}{\displaystyle{\int_{\Omega}}(\varphi_{A})^{2}\
e^{-\frac{|X|^{2}}{2}}dv}.$
From the definition of $\varphi_{A}$, we derive
$\displaystyle\mathfrak{L}\varphi_{A}=\Delta\varphi_{A}-\langle
X,\nabla\varphi_{A}\rangle$
$\displaystyle=\Delta\\{(y_{A}-a_{A})u_{1}\\}-\langle
X,\nabla\\{(y_{A}-a_{A})u_{1}\\}\rangle$
$\displaystyle=y_{A}\mathfrak{L}u_{1}+u_{1}\mathfrak{L}y_{A}+2\nabla
y_{A}\cdot\nabla u_{1}-a_{A}\mathfrak{L}u_{1}$
$\displaystyle=-\lambda_{1}y_{A}u_{1}-u_{1}y_{A}+2\nabla y_{A}\cdot\nabla
u_{1}+a_{A}\lambda_{1}u_{1}.$
Thus, (5.5) can be written as
(5.6)
$(\lambda_{A+1}-\lambda_{1})\|\varphi_{A}\|^{2}\leq\int_{\Omega}\bigl{(}y_{A}u_{1}-2\nabla
y_{A}\cdot\nabla u_{1}\bigl{)}\varphi_{A}\ e^{-\frac{|X|^{2}}{2}}dv.$
From the Cauchy-Schwarz inequality, we obtain
$\left(\int_{\Omega}\bigl{(}y_{A}u_{1}-2\nabla y_{A}\cdot\nabla
u_{1}\bigl{)}\varphi_{A}\
e^{-\frac{|X|^{2}}{2}}dv\right)^{2}\leq\|\varphi_{A}\|^{2}\|y_{A}u_{1}-2\nabla
y_{A}\cdot\nabla u_{1}\|^{2}.$
Multiplying the above inequality by $(\lambda_{A+1}-\lambda_{1})$, we infer,
from (5.6),
(5.7)
$\displaystyle(\lambda_{A+1}-\lambda_{1})\left(\int_{\Omega}\bigl{(}y_{A}u_{1}-2\nabla
y_{A}\cdot\nabla u_{1}\bigl{)}\varphi_{A}\
e^{-\frac{|X|^{2}}{2}}dv\right)^{2}$
$\displaystyle\leq(\lambda_{A+1}-\lambda_{1})\|\varphi_{A}\|^{2}\|y_{A}u_{1}-2\nabla
y_{A}\cdot\nabla u_{1}\|^{2}$
$\displaystyle\leq\Big{(}\int_{\Omega}\bigl{(}y_{A}u_{1}-2\nabla
y_{A}\cdot\nabla u_{1}\bigl{)}\varphi_{A}\
e^{-\frac{|X|^{2}}{2}}dv\Big{)}\|y_{A}u_{1}-2\nabla y_{A}\cdot\nabla
u_{1}\|^{2}$
Hence, we derive
(5.8)
$\displaystyle(\lambda_{A+1}-\lambda_{1})\int_{\Omega}\bigl{(}y_{A}u_{1}-2\nabla
y_{A}\cdot\nabla u_{1}\bigl{)}\varphi_{A}\
e^{-\frac{|X|^{2}}{2}}dv\leq\|y_{A}u_{1}-2\nabla y_{A}\cdot\nabla u_{1}\|^{2}$
Since
$\sum_{A=1}^{n+p}y_{A}^{2}=\sum_{A=1}^{n+p}x_{A}^{2}=|X|^{2},$
we infer
(5.9) $\displaystyle\sum_{A=1}^{n+p}\|y_{A}u_{1}-2\nabla y_{A}\cdot\nabla
u_{1}\|^{2}$
$\displaystyle=\sum_{A=1}^{n+p}\int_{\Omega}\bigl{(}y_{A}^{2}u_{1}^{2}-4y_{A}u_{1}\nabla
y_{A}\cdot\nabla u_{1}+4(\nabla y_{A}\cdot\nabla u_{1})^{2}\bigl{)}\
e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=\int_{\Omega}\bigl{(}|X|^{2}u_{1}^{2}-\nabla|X|^{2}\cdot\nabla
u_{1}^{2}+4\nabla u_{1}\cdot\nabla u_{1}\bigl{)}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=\int_{\Omega}\bigl{(}|X|^{2}u_{1}^{2}+\mathfrak{L}|X|^{2}u_{1}^{2}+4\nabla
u_{1}\cdot\nabla u_{1}\bigl{)}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=\int_{\Omega}(2n-|X|^{2})u_{1}^{2}\
e^{-\frac{|X|^{2}}{2}}dv+4\lambda_{1}\leq(2n-\inf_{\Omega}|X|^{2})+4\lambda_{1}.$
On the other hand, from the definition of $\varphi_{A}$, we have
(5.10) $\displaystyle\int_{\Omega}\bigl{(}y_{A}u_{1}-2\nabla y_{A}\cdot\nabla
u_{1}\bigl{)}\varphi_{A}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=\int_{\Omega}\bigl{(}y_{A}^{2}u_{1}^{2}-a_{A}y_{A}u_{1}^{2}+2a_{A}u_{1}\nabla
y_{A}\cdot\nabla u_{1}-2y_{A}u_{1}\nabla y_{A}\cdot\nabla u_{1}\bigl{)}\
e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=\int_{\Omega}\bigl{(}y_{A}^{2}u_{1}^{2}-a_{A}y_{A}u_{1}^{2}-a_{A}\mathfrak{L}y_{A}u_{1}^{2}+\frac{1}{2}\mathfrak{L}y_{A}^{2}u_{1}^{2}\bigl{)}\
e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=\int_{\Omega}\bigl{(}y_{A}^{2}u_{1}^{2}+\frac{1}{2}\mathfrak{L}y_{A}^{2}u_{1}^{2}\bigl{)}\
e^{-\frac{|X|^{2}}{2}}dv$ $\displaystyle=\int_{\Omega}\nabla y_{A}\cdot\nabla
y_{A}u_{1}^{2}\ e^{-\frac{|X|^{2}}{2}}dv.$
For any point $p$, we choose a new coordinate system
$\bar{X}=(\bar{x}_{1},\cdots,\bar{x}_{n+p})$ of $\mathbf{R}^{n+p}$ given by
$X-X(p)=\bar{X}O$ such that
$(\frac{\partial}{\partial\bar{x}_{1}})_{p},\cdots,(\frac{\partial}{\partial\bar{x}_{n}})_{p}$
span $T_{p}M^{n}$ and at $p$,
$g\Big{(}\frac{\partial}{\partial\bar{x}_{i}},\frac{\partial}{\partial\bar{x}_{j}}\Big{)}=\delta_{ij}$,
where $O=(o_{AB})\in O(n+p)$ is an $(n+p)\times(n+p)$ orthogonal matrix.
$\displaystyle\nabla y_{A}\cdot\nabla y_{A}=g(\nabla y_{A},\nabla
y_{A})=\sum_{B,C=1}^{n+p}q_{AB}q_{AC}g(\nabla x_{B},\nabla x_{C})$
$\displaystyle=\sum_{B,C=1}^{n+p}q_{AB}q_{AC}g(\sum_{D=1}^{n+p}o_{DB}\nabla\bar{x}_{D},\sum_{E=1}^{n+p}o_{EC}\nabla\bar{x}_{E})$
$\displaystyle=\sum_{B,C,D,E=1}^{n+p}q_{AB}o_{DB}q_{AC}o_{EC}g(\nabla\bar{x}_{D},\nabla\bar{x}_{E})$
$\displaystyle=\sum_{j=1}^{n}\bigl{(}\sum_{B=1}^{n+p}q_{AB}o_{jB}\bigl{)}^{2}\leq
1$
since $OQ$ is an orthogonal matrix if $Q$ and $O$ are orthogonal matrices,
that is, we have
(5.11) $\displaystyle\nabla y_{A}\cdot\nabla y_{A}\leq 1.$
Thus, we obtain, from (5.10) and (5.11),
(5.12)
$\displaystyle\sum_{A=1}^{n+p}(\lambda_{A+1}-\lambda_{1})\int_{\Omega}\bigl{(}y_{A}u_{1}-2\nabla
y_{A}\cdot\nabla u_{1}\bigl{)}\varphi_{A}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=\sum_{A=1}^{n+p}(\lambda_{A+1}-\lambda_{1})\int_{\Omega}\nabla
y_{A}\cdot\nabla y_{A}u_{1}^{2}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=\sum_{j=1}^{n}(\lambda_{j+1}-\lambda_{1})\int_{\Omega}\nabla
y_{j}\cdot\nabla y_{j}u_{1}^{2}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle+\sum_{A=n+1}^{n+p}(\lambda_{A+1}-\lambda_{1})\int_{\Omega}\nabla
y_{A}\cdot\nabla y_{A}u_{1}^{2}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle\geq\sum_{j=1}^{n}(\lambda_{j+1}-\lambda_{1})\int_{\Omega}\nabla
y_{j}\cdot\nabla y_{j}u_{1}^{2}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle+\sum_{A=n+1}^{n+p}(\lambda_{n+1}-\lambda_{1})\int_{\Omega}\nabla
y_{A}\cdot\nabla y_{A}u_{1}^{2}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=\sum_{j=1}^{n}(\lambda_{j+1}-\lambda_{1})\int_{\Omega}\nabla
y_{j}\cdot\nabla y_{j}u_{1}^{2}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle+(\lambda_{n+1}-\lambda_{1})\int_{\Omega}(n-\sum_{j=1}^{n}\nabla
y_{j}\cdot\nabla y_{j})u_{1}^{2}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle=\sum_{j=1}^{n}(\lambda_{j+1}-\lambda_{1})\int_{\Omega}\nabla
y_{j}\cdot\nabla y_{j}u_{1}^{2}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle+(\lambda_{n+1}-\lambda_{1})\int_{\Omega}\sum_{j=1}^{n}(1-\nabla
y_{j}\cdot\nabla y_{j})u_{1}^{2}\ e^{-\frac{|X|^{2}}{2}}dv$
$\displaystyle\geq\sum_{j=1}^{n}(\lambda_{j+1}-\lambda_{1}).$
According to (5.8), (5.9) and (5.12), we obtain
$\sum_{j=1}^{n}(\lambda_{j+1}-\lambda_{1})\leq(2n-\inf_{\Omega}|X|^{2})+4\lambda_{1}.$
This completes the proof of the proposition 5.1. ∎
Proof of Theorem 5.2. By making use of the proposition 5.1 and the same proof
as in the proof of the theorem 1.2, we can prove the theorem 5.2 if one
notices to count the number of eigenvalues from $1$.
$\square$
## References
* [1] H.-D. Cao and H. Li, A Gap Theorem for Self-shrinkers of the Mean Curvature Flow in Arbitrary Codimension, arXiv:1101.0516, 2011, to appear in Calc. Var. Partial Differential Equations.
* [2] Q. -M. Cheng and H. C. Yang, Estimates on eigenvalues of Laplacian, Math. Ann., 331 (2005), 445-460.
* [3] Q. -M. Cheng and H. C. Yang, Bounds on eigenvalues of Dirichlet Laplacian, Math. Ann., 337 (2007), 159-175.
* [4] Tobias H. Colding and William P. Minicozzi II, Generic Mean Curvature Flow I; Generic Singularities, Ann. of Math., 175 (2012), 755-833.
* [5] Tobias H. Colding and William P. Minicozzi II, Smooth Compactness of Self-shrinkers, arXiv:0907.2594, 2009.
* [6] Tobias H. Colding and William P. Minicozzi II, Minimal Surfaces and Mean Curvature Flow, arXiv:1102.1411, 2011.
* [7] Q. Ding and Z. Wang, On the self-shrinking systems in arbitrary codimension spaces, arXiv:1012.0429, 2010.
* [8] Q. Ding, Y. L. Xin, Volume growth, eigenvalue and compactness for self-shrinkers, arXiv:1101.1411, 2011.
* [9] Q. Ding, Y. L. Xin, the rigidity theorems of self-shrnkers, arXiv:1105. 4962, 2011.
* [10] K. Ecker and G. Huisken, Mean Curvature Evolution of Entire Graphs, Ann. of Math., 130 (1989), 453-471.
* [11] G. Huisken, Asymptotic Behavior for Singularities of the Mean Curvature Flow, J. Differential Geom., 31 (1990), 285-299.
* [12] G. Huisken, Local and global behaviour of hypersurfaces moving by mean curvature. Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., 54, Part 1, Amer. Math. Soc., Providence, RI, (1993), 175-191.
* [13] Nam Q. Le and N. Sesum, Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers, arXiv:1011.5245, 2010.
* [14] K. Smoczyk, Self-Shrinkers of the Mean Curvature Flow in Arbitrary Codimension, International Mathematics Research Notices, 48 (2005), 2983-3004.
* [15] L. Wang, A Bernstein Type Theorem for Self-similar Shrinkers, to appear in Geom. Dedicata.
|
arxiv-papers
| 2011-12-27T08:29:56 |
2024-09-04T02:49:25.725241
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qing-Ming Cheng and Yejuan Peng",
"submitter": "Qing-Ming Cheng",
"url": "https://arxiv.org/abs/1112.5938"
}
|
1112.6047
|
# Characterization of $2^{n}$-periodic binary sequences with fixed 3-error or
4-error linear complexity
Jianqin Zhou1,2, Jun Liu1 Telecommunication School, Hangzhou Dianzi
University, Hangzhou, 310018 China
Computer Science School, Anhui Univ. of Technology, Ma’anshan, 243002 China
zhou9@yahoo.com
Wanquan Liu
Department of Computing, Curtin University, Perth, WA 6102 Australia
w.liu@curtin.edu.au
###### Abstract
The linear complexity and the $k$-error linear complexity of a sequence have
been used as important security measures for key stream sequence strength in
linear feedback shift register design. By using the sieve method of
combinatorics, the $k$-error linear complexity distribution of
$2^{n}$-periodic binary sequences is investigated based on Games-Chan
algorithm. First, for $k=2,3$, the complete counting functions on the
$k$-error linear complexity of $2^{n}$-periodic binary sequences with linear
complexity less than $2^{n}$ are characterized. Second, for $k=3,4$, the
complete counting functions on the $k$-error linear complexity of
$2^{n}$-periodic binary sequences with linear complexity $2^{n}$ are
presented. Third, for $k=4,5$, the complete counting functions on the
$k$-error linear complexity of $2^{n}$-periodic binary sequences with linear
complexity less than $2^{n}$ are derived. As a consequence of these results,
the counting functions for the number of $2^{n}$-periodic binary sequences
with the $3$-error linear complexity are obtained, and the complete counting
functions on the $4$-error linear complexity of $2^{n}$-periodic binary
sequences are obvious.
Keywords: Periodic sequence; linear complexity; $k$-error linear complexity;
$k$-error linear complexity distribution
MSC2000: 94A55, 94A60, 11B50
## I Introduction
The linear complexity of a sequence is defined as the length of the shortest
linear feedback shift register (LFSR) that can generate the sequence. The
concept of linear complexity is very useful in the study of security of stream
ciphers for cryptographic applications and it has attracted many attentions in
cryptographic community [1, 15]. In fact, a necessary condition for the
security of a key stream generator in LFSR is that it produces a sequence with
high linear complexity. However, high linear complexity can not necessarily
guarantee that the sequence is safe since the linear complexity of some
sequences is unstable. For example, if a small number of changes to a sequence
greatly reduce its linear complexity, then the resulting key stream is
cryptographically weak. Ding, Xiao and Shan noticed this problem first in
their book [1], and proposed the weight complexity and sphere complexity.
Stamp and Martin [15] introduced $k$-error linear complexity, which is similar
to the sphere complexity, and put forward the concept of $k$-error linear
complexity profile. Specifically, suppose that $s$ is a sequence with period
$N$. For any $k(0\leq k\leq N)$, $k$-error linear complexity of $s$, denoted
as $L_{k}(s)$, is defined as the smallest linear complexity that can be
obtained when any $k$ or fewer bits of the sequence are changed within one
period.
One important result, proved by Kurosawa et al. [10] is that the minimum
number $k$ for which the $k$-error linear complexity of a $2^{n}$-periodic
binary sequence $s$ is strictly less than a linear complexity $L(s)$ of $s$ is
determined by $k_{\min}=2^{W_{H}(2^{n}-L(s))}$, where $W_{H}(a)$ denotes the
Hamming weight of the binary representation of an integer $a$. Also Rueppel
[14] derived the number $N(L)$ of $2^{n}$-periodic binary sequences with given
linear complexity $L,0\leq L\leq 2^{n}$.
For $k=1,2$, Meidl [13] characterized the complete counting functions on the
$k$-error linear complexity of $2^{n}$-periodic binary sequences having
maximal possible linear complexity $2^{n}$. For $k=2,3$, Zhu and Qi [19]
further showed the complete counting functions on the $k$-error linear
complexity of $2^{n}$-periodic binary sequences with linear complexity
$2^{n}-1$. By using algebraic and combinatorial methods, Fu et al. [4] studied
the linear complexity and the $1$-error linear complexity for $2^{n}$-periodic
binary sequences, and characterized the $2^{n}$-periodic binary sequences with
given 1-error linear complexity and derived the counting function for the
1-error linear complexity for $2^{n}$-periodic binary sequences.
By investigating sequences with linear complexity $2^{n}$ or linear complexity
less than $2^{n}$ together, Kavuluru [7, 8] characterized $2^{n}$-periodic
binary sequences with fixed 2-error or 3-error linear complexity, and obtained
the counting functions for the number of $2^{n}$-periodic binary sequences
with given $k$-error linear complexity for $k=2$ and 3. These results are
important progress on the $k$-error linear complexity. Unfortunately, the
results in [7, 8] on the $3$-error linear complexity are not completely
correct, as pointed out in [18].
In current literature [13, 19, 7, 8], sequences $s$ with $L_{k}(s)=c$ are
directly investigated. In contrast with that, we will study the $k$-error
linear complexity by proposing a new approach. Let
$S=\\{s|L(s)=c\\},E=\\{e|W_{H}(e)\leq k\\},SE=\\{s+e|s\in S,e\in E\\}$, where
$s$ is a sequence with linear complexity $c$, and $e$ is an error sequence
with $W_{H}(e)\leq k$. With the sieve method of combinatorics, we sieve
sequences $s+e$ with $L_{k}(s+e)=c$ in $SE$.
First we investigate sequences with linear complexity $2^{n}$, and sequences
with linear complexity less than $2^{n}$, separately. It is observed that for
sequences with linear complexity $2^{n}$, the $k$-error linear complexity is
equal to $(k+1)$-error linear complexity, when $k$ is odd. For sequences with
linear complexity less than $2^{n}$, the $k$-error linear complexity is equal
to $(k+1)$-error linear complexity, when $k$ is even. Then we investigate the
$3$-error linear complexity in two cases and this reduces the complexity of
this problem. Finally, by combining the results of two cases, we obtain the
complete counting functions for the number of $2^{n}$-periodic binary
sequences with $3$-error linear complexity.
The contribution of this paper can be summarized as follows. i) As the results
in [7, 8] on the $3$-error linear complexity are not completely correct, the
correct results are given here. ii) A new approach is proposed for the
$k$-error linear complexity problem, which can decompose this problem into two
sub problems with less complexity. iii) Generally, the complete counting
functions for the number of $2^{n}$-periodic binary sequences with given
$k$-error linear complexity for $k>4$ can be obtained using a similar
approach.
## II Preliminaries
In this section we give some preliminary results which will be used in the
sequel.
We will consider sequences over $GF(q)$, which is the finite field of order
$q$. Let $x=(x_{1},x_{2},\cdots,x_{n})$ and $y=(y_{1},y_{2},\cdots,y_{n})$ be
vectors over $GF(q)$. Then define
$x+y=(x_{1}+y_{1},x_{2}+y_{2},\cdots,x_{n}+y_{n})$.
When $n=2m$, we define $Left(x)=(x_{1},x_{2},\cdots,x_{m})$ and
$Right(x)=(x_{m+1},x_{m+2},\cdots,x_{2m})$.
The Hamming weight of an $N$-periodic sequence $s$ is defined as the number of
nonzero elements in per period of $s$, denoted by $W(s)$. Let $s^{N}$ be one
period of $s$. If $N=2^{n}$, $s^{N}$ is also denoted as $s^{(n)}$. Obviously,
$W(s^{(n)})=W(s^{N})=W(s)$. $supp(s)$ is defined as a set of the positions
with nonzero elements in per period of $s$.
The generating function of a sequence
$s=\\{s_{0},s_{1},s_{2},s_{3},\cdots,\\}$ is defined by
$s(x)=s_{0}+s_{1}x+s_{2}x^{2}+s_{3}x^{3}+\cdots=\sum\limits^{\infty}_{i=0}s_{i}x^{i}$
The generating function of a finite sequence
$s^{N}=\\{s_{0},s_{1},s_{2},\cdots,s_{N-1},\\}$ is defined by
$s^{N}(x)=s_{0}+s_{1}x+s_{2}x^{2}+\cdots+s_{N-1}x^{N-1}$. If $s$ is a periodic
sequence with the first period $s^{N}$, then,
$\displaystyle s(x)$ $\displaystyle=$ $\displaystyle
s^{N}(x)(1+x^{N}+x^{2N}+\cdots)=\frac{s^{N}(x)}{1-x^{N}}$ (1) $\displaystyle=$
$\displaystyle\frac{s^{N}(x)/\gcd(s^{N}(x),1-x^{N})}{(1-x^{N})/\gcd(s^{N}(x),1-x^{N})}$
$\displaystyle=$ $\displaystyle\frac{g(x)}{f_{s}(x)}$
where
$f_{s}(x)=(1-x^{N})/\gcd(s^{N}(x),1-x^{N}),g(x)=s^{N}(x)/\gcd(s^{N}(x),1-x^{N})$.
Obviously, $\gcd(g(x),f_{s}(x))=1,\deg(g(x)<\deg(f_{s}(x)))$. $f_{s}(x)$ is
called the minimal polynomial of $s$, and the degree of $f_{s}(x)$ is called
the linear complexity of $s$, that is $\deg(f_{s}(x))=L(s)$.
Suppose that $N=2^{n}$ and $GF(q)=GF(2)$, then
$1-x^{N}=1-x^{2^{n}}=(1-x)^{2^{n}}=(1-x)^{N}$. Thus for binary sequences with
period $2^{n}$, its linear complexity is equal to the degree of factor $(1-x)$
in $s^{N}(x)$.
The following three lemmas are well known results on $2^{n}$-periodic binary
sequences.
Lemma 2.1 Suppose that s is a binary sequence with period $N=2^{n}$, then
$L(s)=N$ if and only if the Hamming weight of a period of the sequence is odd.
If an element one is removed from a sequence whose Hamming weight is odd, the
Hamming weight of the sequence will be changed to even, so the main concern
hereinafter is about sequences whose Hamming weight are even.
Lemma 2.2 Let $s_{1}$ and $s_{2}$ be two binary sequences with period
$N=2^{n}$. If $L(s_{1})\neq L(s_{2})$, then
$L(s_{1}+s_{2})=\max\\{L(s_{1}),L(s_{2})\\}$; otherwise if
$L(s_{1})=L(s_{2})$, then $L(s_{1}+s_{2})<L(s_{1})$.
Suppose that the linear complexity of s can decline when at least $k$ elements
of s are changed. By Lemma 2.2, the linear complexity of the binary sequence,
in which elements at exactly those $k$ positions are all nonzero, must be
L(s). Therefore, for the computation of $k$-error linear complexity, we only
need to find the binary sequence whose Hamming weight is minimum and its
linear complexity is L(s).
Lemma 2.3 Let $E_{i}$ be a $2^{n}$-periodic sequence with one nonzero element
at position $i$ and 0 elsewhere in each period, $0\leq i<2^{n}$. If
$j-i=2^{r}(1+2a),a\geq 0,0\leq i<j<2^{n},r\geq 0$, then
$L(E_{i}+E_{j})=2^{n}-2^{r}$.
## III Counting functions with the $k$-error linear complexity
For $2^{n}$-periodic binary sequences with linear complexity less than
$2^{n}$, the change of one bit per period results in a sequence with odd
number of nonzero bits per period, which has again linear complexity $2^{n}$.
In this section, we thus first focus on the $2$-error linear complexity.
Further more, in order to derive the counting functions on the $3$-error
linear complexity of $2^{n}$-periodic binary sequences with linear complexity
less than $2^{n}$, we only need to investigate the $2$-error linear complexity
of $2^{n}$-periodic binary sequences with linear complexity less than $2^{n}$.
Second, for $k=3,4$, the complete counting functions on the $k$-error linear
complexity of $2^{n}$-periodic binary sequences with linear complexity $2^{n}$
are presented. Third, for $k=4,5$, the complete counting functions on the
$k$-error linear complexity of $2^{n}$-periodic binary sequences with linear
complexity less than $2^{n}$ are derived.
Given a $2^{n}$-periodic binary sequence s, its linear complexity L(s) can be
determined by the Games-Chan algorithm [3]. Based on Games-Chan algorithm, the
following Lemma 3.1 is given in [13].
Lemma 3.1 Suppose that s is a binary sequence with first period
$s^{(n)}=\\{s_{0},s_{1},s_{2},\cdots,s_{2^{n}-1}\\}$, a mapping $\varphi_{n}$
from $F^{2^{n}}_{2}$ to $F^{2^{n-1}}_{2}$ is given by
$\displaystyle\varphi_{n}(s^{(n)})$ $\displaystyle=$
$\displaystyle\varphi_{n}((s_{0},s_{1},s_{2},\cdots,s_{2^{n}-1}))$
$\displaystyle=$
$\displaystyle(s_{0}+s_{2^{n-1}},s_{1}+s_{2^{n-1}+1},\cdots,s_{2^{n-1}-1}+s_{2^{n}-1})$
Let $W(\mathbf{\upsilon})$ denote the Hamming weight of a vector
$\mathbf{\upsilon}$. Then mapping $\varphi_{n}$ has the following properties
1) $W(\varphi_{n}(s^{(n)}))\leq W(s^{(n)})$;
2) If $n\geq 2$ then $W(\varphi_{n}(s^{(n)}))$ and $W(s^{(n)})$ are either
both odd or both even;
3) The set
$\varphi^{-1}_{n+1}(s^{(n)})=\\{v\in
F^{2^{n+1}}_{2}|\varphi_{n+1}(v)=s^{(n)}\\}$
of the preimage of $s^{(n)}$ has cardinality $2^{2^{n}}$.
Rueppel [14] presented the following.
Lemma 3.2 The number $N(L)$ of $2^{n}$-periodic binary sequences with given
linear complexity $L,0\leq L\leq 2^{n}$, is given by
$N(L)=\left\\{\begin{array}[]{l}1,\ \ \ \ \ L=0\\\ 2^{L-1},\ 1\leq L\leq
2^{n}\end{array}\right.$
It is known that the computation of $k$-error linear complexity can be
converted to finding error sequences with minimal Hamming weight. Hence
2-error linear complexity of $s^{(n)}$ is the smallest linear complexity that
can be obtained when any $u^{(n)}$ with $W(u^{(n)})=0$ or 2 is added to
$s^{(n)}$. So, the main approach of this section and next section is as
follows. Let $s^{(n)}$ be a binary sequence with linear complexity $c$,
$u^{(n)}$ a binary sequence with $W(u^{(n)})\leq k$. We derive the counting
functions on the $k$-error linear complexity of $2^{n}$-periodic binary
sequences by investigating $s^{(n)}+u^{(n)}$. Based on this idea, we first
prove the following lemmas
Lemma 3.3 1). If $s^{(n)}$ is a binary sequence with linear complexity
$c,1\leq c\leq 2^{n-1}-3$, $c\neq 2^{n-1}-2^{m},2\leq m<n-1$, $u^{(n)}$ is a
binary sequence, and $W(u^{(n)})=0$ or 2. Then the 2-error linear complexity
of $s^{(n)}+u^{(n)}$ is still $c$.
2). If $s^{(n)}$ is a binary sequence with linear complexity
$c=2^{n-1}-2^{m},0\leq m<n-1$, then there exists a binary sequence $u^{(n)}$
with $W(u^{(n)})=2$, such that the 2-error linear complexity of
$s^{(n)}+u^{(n)}$ is less than $c$.
###### Proof:
Without loss of generality, we suppose that $v^{(n)}\neq u^{(n)}$, and
$W(v^{(n)})=0$ or 2.
1). As $c\leq 2^{n-1}-3$, we only need to consider the case
$L(u^{(n)}+v^{(n)})<2^{n-1}$. Thus
$Left(u^{(n)}+v^{(n)})=Right(u^{(n)}+v^{(n)})$ and
$W(Left(u^{(n)}+v^{(n)}))=2$.
By Lemma 2.3, $L(u^{(n)}+v^{(n)})=2^{n-1}-2^{m}$, $0\leq m<n-1$.
Thus $L(s^{(n)}+u^{(n)}+v^{(n)})\geq L(s^{(n)})$, so the 2-error linear
complexity of $s^{(n)}+u^{(n)}$ is $c$.
2). As $s^{(n)}$ is a binary sequence with linear complexity
$c=2^{n-1}-2^{m},0\leq m<n-1$, so the 2-error linear complexity of
$s^{(n)}+u^{(n)}$ must be less than $c$ when $L(u^{(n)}+v^{(n)})=c$. ∎
Lemma 3.4 Suppose that $s^{(n)}$ and $t^{(n)}$ are two different binary
sequences with linear complexity $c,1\leq c\leq 2^{n-2}$, and $u^{(n)}$ and
$v^{(n)}$ are two different binary sequences, and $W(u^{(n)})=0$ or 2, and
$W(v^{(n)})=0$ or 2. Then $s^{(n)}+u^{(n)}\neq t^{(n)}+v^{(n)}$.
###### Proof:
The following is obvious
$s^{(n)}+u^{(n)}\neq t^{(n)}+v^{(n)}$
$\Leftrightarrow$ $s^{(n)}+u^{(n)}+v^{(n)}\neq t^{(n)}$
$\Leftrightarrow$ $u^{(n)}+v^{(n)}\neq s^{(n)}+t^{(n)}$
Note that $s^{(n)}$ and $t^{(n)}$ are two different binary sequences with
linear complexity $c,1\leq c\leq 2^{n-2}$, so the linear complexity of
$s^{(n)}+t^{(n)}$ is less than $2^{n-2}$, hence one period of
$s^{(n)}+t^{(n)}$ can be divided into 4 equal parts.
Suppose that $u^{(n)}+v^{(n)}=s^{(n)}+t^{(n)}$, then one period of
$u^{(n)}+v^{(n)}$ can be divided into 4 equal parts. It follows that the
linear complexity of $u^{(n)}+v^{(n)}$ is $2^{n-2}$, which contradicts the
fact that the linear complexity of $s^{(n)}+t^{(n)}$ is less than $2^{n-2}$. ∎
Next we divide the 2-error linear complexity into three categories. First
consider the category of $2^{n-1}-2^{n-m}$.
Lemma 3.5 Let $N_{2}(2^{n-1}-2^{n-m})$ be the number of $2^{n}$-periodic
binary sequences with linear complexity less than $2^{n}$ and given 2-error
linear complexity $2^{n-1}-2^{n-m},n\geq 2,1<m\leq n$. Then
$N_{2}(2^{n-1}-2^{n-m})=(1+\left(\begin{array}[]{c}2^{n}\\\
2\end{array}\right)-3\times 2^{n+m-3})2^{2^{n-1}-2^{n-m}-1}$
###### Proof:
Suppose that $s^{(n)}$ is a binary sequence with linear complexity
$2^{n-1}-2^{n-m}$, and $u^{(n)}$ is a binary sequence with $W(u^{(n)})=2$. By
Lemma 3.3, there exists a binary sequence $v^{(n)}$ with $W(v^{(n)})=2$, such
that $L(u^{(n)}+v^{(n)})=2^{n-1}-2^{n-m}$. So the 2-error linear complexity of
$u^{(n)}+s^{(n)}$ is less than $2^{n-1}-2^{n-m}$.
Suppose that $u^{(n)}$ is a binary sequence with linear complexity $2^{n}$ and
$W(u^{(n)})=2$, and there exist 2 nonzero elements whose distance is
$2^{n-m}(2k+1)$ or $2^{n-1}$, with $k$ being an integer. It is easy to verify
that there exists a binary sequence $v^{(n)}$ with $W(v^{(n)})=2$, such that
$L(u^{(n)}+v^{(n)})=2^{n-1}-2^{n-m}$. So the 2-error linear complexity of
$u^{(n)}+s^{(n)}$ is less than $2^{n-1}-2^{n-m}$.
Let us divide one period of $u^{(n)}$ into $2^{n-m}$ subsequences of form
$\\{a,a+2^{n-m},a+2^{n-m+1},\cdots,a+(2^{m}-1)\times 2^{n-m}\\}$.
If 2 nonzero elements of $u^{(n)}$ are in the same subsequence, then the
number of these $u^{(n)}$ can be given by
$C1=2^{n-m}\times\left(\begin{array}[]{c}2^{m}\\\ 2\end{array}\right)\times
2^{m}.$
Suppose that 2 nonzero elements of $u^{(n)}$ are in the same subsequence, and
the distance of the 2 nonzero elements is not $2^{n-m}(2k+1)$, then the number
of these $u^{(n)}$ can be given by
$2^{n-m+1}\times\left(\begin{array}[]{c}2^{m-1}\\\ 2\end{array}\right).$ Of
these $u^{(n)}$, there are $2^{n-m}\times 2^{m-1}=2^{n-1}$ sequences, in each
sequence the distance of the 2 nonzero elements is $2^{n-1}$.
So, if 2 nonzero elements of $u^{(n)}$ are in the same subsequence, and the
distance of the 2 nonzero elements is neither $2^{n-m}(2k+1)$ nor $2^{n-1}$,
then the number of these $u^{(n)}$ can be given by
$C2=2^{n-m+1}\times\left(\begin{array}[]{c}2^{m-1}\\\
2\end{array}\right)-2^{n-1}.$
Suppose that $u^{(n)}$ is a binary sequence with $W(u^{(n)})=2$, and there
exist 2 nonzero elements whose distance is a multiple of $2^{n-m+1}$. Then
there exists one binary sequence $v^{(n)}$ with $W(v^{(n)})=2$, such that
$L(u^{(n)}+v^{(n)})=2^{n-1}-2^{n-r},1<r<m$. Let
$t^{(n)}=s^{(n)}+u^{(n)}+v^{(n)}$. Then
$L(t^{(n)})=L(s^{(n)})=2^{n-1}-2^{n-m}$ and $s^{(n)}+u^{(n)}=t^{(n)}+v^{(n)}$.
By Lemma 3.2, the number of $2^{n}$-periodic binary sequences with given
linear complexity $2^{n-1}-2^{n-m}$ is $2^{2^{n-1}-2^{n-m}-1}$. This leads to
the following,
$\displaystyle N_{2}(2^{n-1}-2^{n-m})$ $\displaystyle=$
$\displaystyle[1+\left(\begin{array}[]{c}2^{n}\\\
2\end{array}\right)-(C1-C2)-C2/2]2^{2^{n-1}-2^{n-m}-1}$ $\displaystyle=$
$\displaystyle[1+\left(\begin{array}[]{c}2^{n}\\\
2\end{array}\right)-2^{n-m}\left(\begin{array}[]{c}2^{m}\\\
2\end{array}\right)+2^{n-m}\left(\begin{array}[]{c}2^{m-1}\\\
2\end{array}\right)-2^{n-2}]$ $\displaystyle\times 2^{2^{n-1}-2^{n-m}-1}$
$\displaystyle=$ $\displaystyle(1+\left(\begin{array}[]{c}2^{n}\\\
2\end{array}\right)-3\times 2^{n+m-3})2^{2^{n-1}-2^{n-m}-1}$
∎
Next we consider the category of $2^{n-1}-2^{n-m}+x$.
Lemma 3.6 Let $N_{2}(2^{n-1}-2^{n-m}+x)$ be the number of $2^{n}$-periodic
binary sequences with linear complexity less than $2^{n}$ and given 2-error
linear complexity $2^{n-1}-2^{n-m}+x,n>3,1<m<n-1,0<x<2^{n-m-1}$. Then
$\displaystyle N_{2}(2^{n-1}-2^{n-m}+x)$ $\displaystyle=$
$\displaystyle[1+\left(\begin{array}[]{c}2^{n}\\\
2\end{array}\right)+2^{n-m}-2^{n+m-2}]2^{2^{n-1}-2^{n-m}+x-1}$
###### Proof:
Suppose that $s^{(n)}$ is a binary sequence with linear complexity
$2^{n-1}-2^{n-m}+x$, and $u^{(n)}$ is a binary sequence with $W(u^{(n)})=2$.
By Lemma 3.3, the 2-error linear complexity of $u^{(n)}+s^{(n)}$ is still
$2^{n-1}-2^{n-m}+x$. The number of these $u^{(n)}$ can be given by
$\left(\begin{array}[]{c}2^{n}\\\ 2\end{array}\right).$
Suppose that $u^{(n)}$ is a binary sequence with $W(u^{(n)})=2$, and there
exist 2 nonzero elements whose distance is $2^{n-r}(1+2a),1<r\leq m,a\geq 0$.
Then there exists one binary sequence $v^{(n)}$ with $W(v^{(n)})=2$, such that
$L(u^{(n)}+v^{(n)})=2^{n-1}-2^{n-r}$. Let $t^{(n)}=s^{(n)}+u^{(n)}+v^{(n)}$.
Then $L(t^{(n)})=L(s^{(n)})=2^{n-1}-2^{n-m}+x$ and
$s^{(n)}+u^{(n)}=t^{(n)}+v^{(n)}$.
Let us divide one period of $u^{(n)}$ into $2^{n-m}$ subsequences of form
$\\{a,a+2^{n-m},a+2^{n-m+1},\cdots,a+(2^{m}-1)\times 2^{n-m}\\}$.
If 2 nonzero elements of $u^{(n)}$ are in the same subsequence, and their
distance is $2^{n-1}$, then there exist $2^{m-1}-1$ binary sequences $v^{(n)}$
with $W(v^{(n)})=2$, such that $L(u^{(n)}+v^{(n)})=2^{n-1}-2^{n-r},1<r\leq m$.
Let $t^{(n)}=s^{(n)}+u^{(n)}+v^{(n)}$. Then $s^{(n)}+u^{(n)}=t^{(n)}+v^{(n)}$.
The number of these $u^{(n)}$ can be given by $D1=2^{n-m}\times
2^{m-1}=2^{n-1}.$
Suppose that 2 nonzero elements of $u^{(n)}$ are in the same subsequence, and
their distance is not $2^{n-1}$. Then there exist one binary sequence
$v^{(n)}$, with $W(v^{(n)})=2$, such that
$L(u^{(n)}+v^{(n)})=2^{n-1}-2^{n-r},1<r\leq m$. The number of these $u^{(n)}$
can be given by
$D2=2^{n-m}[\left(\begin{array}[]{c}2^{m}\\\ 2\end{array}\right)-2^{m-1}]$
By Lemma 3.2, the number of $2^{n}$-periodic binary sequences with given
linear complexity $2^{n-1}-2^{n-m}+x$ is $2^{2^{n-1}-2^{n-m}+x-1}$. This will
derive the following,
$\displaystyle N_{2}(2^{n-1}-2^{n-m}+x)$ $\displaystyle=$
$\displaystyle[1+\left(\begin{array}[]{c}2^{n}\\\
2\end{array}\right)-\frac{2^{m-1}-1}{2^{m-1}}\times D1-\frac{1}{2}\times D2]$
$\displaystyle\ \ \ \ 2^{2^{n-1}-2^{n-m}+x-1}$ $\displaystyle=$
$\displaystyle\\{1+\left(\begin{array}[]{c}2^{n}\\\
2\end{array}\right)-\frac{2^{m-1}-1}{2^{m-1}}\times 2^{n-1}$
$\displaystyle-2^{n-m-1}[\left(\begin{array}[]{c}2^{m}\\\
2\end{array}\right)-2^{m-1}]\\}2^{2^{n-1}-2^{n-m}+x-1}$ $\displaystyle=$
$\displaystyle\\{1+\left(\begin{array}[]{c}2^{n}\\\
2\end{array}\right)-(2^{m-1}-1)\times 2^{n-m}$
$\displaystyle-2^{n-m-1}[\left(\begin{array}[]{c}2^{m}\\\
2\end{array}\right)-2^{m-1}]\\}2^{2^{n-1}-2^{n-m}+x-1}$ $\displaystyle=$
$\displaystyle[1+\left(\begin{array}[]{c}2^{n}\\\
2\end{array}\right)+2^{n-m}-2^{n+m-2}]2^{2^{n-1}-2^{n-m}+x-1}$
∎
Finally we consider the simplest category, that is $1\leq c\leq 2^{r-2}-1$.
Lemma 3.7 Let $L(r,c)=2^{n}-2^{r}+c,3\leq r\leq n,1\leq c\leq 2^{r-2}-1$, and
$N_{2}(L(r,c))$ be the number of $2^{n}$-periodic binary sequences with linear
complexity less than $2^{n}$ and given 2-error linear complexity $L(r,c)$.
Then
$N_{2}(L)=\left\\{\begin{array}[]{l}1+\left(\begin{array}[]{c}2^{n}\\\
2\end{array}\right),\ \ \ \ \ \ \ \ L=0\\\
2^{L-1}(1+\left(\begin{array}[]{c}2^{r}\\\ 2\end{array}\right)),\
L=L(r,c)\end{array}\right.$
###### Proof:
Suppose that $s$ is a binary sequence with first period
$s^{(n)}=\\{s_{0},s_{1},s_{2},\cdots,s_{2^{n}-1}\\}$, and $L(s)<2^{n}$. By
Games-Chan algorithm, $Left(s^{(t)})\neq Right(s^{(t)}),1\leq t\leq n$, where
$s^{(t)}=\varphi_{t+1}\cdots\varphi_{n}(s^{(n)})$.
First consider the case of $W(s^{(n)})=0$. There is only one binary sequence
of this kind.
Consider the case of $W(s^{(n)})=2$. There is 2 nonzero bits in
$\\{s_{0},s_{1},\cdots,s_{2^{n}-1}\\}$, thus there are
$\left(\begin{array}[]{c}2^{n}\\\ 2\end{array}\right)$ binary sequences of
this kind.
So $N_{2}(0)=1+\left(\begin{array}[]{c}2^{n}\\\ 2\end{array}\right)$.
Consider $L(r,c)=2^{n}-2^{r}+c$, $3\leq r\leq n,1\leq c\leq 2^{r-2}-1$.
Suppose that $s^{(n)}$ is a binary sequence with $L(s^{(n)})=L(r,c)$. Note
that $L(r,c)=2^{n}-2^{r}+c=2^{n-1}+\cdots+2^{r}+c$. By Games-Chan algorithm,
$Left(s^{(r)})=Right(s^{(r)})$, and $L(s^{(r)})=c$.
It is known that the number of binary sequences $t^{(r)}$ with $W(t^{(r)})=0$
or 2 is $1+\left(\begin{array}[]{c}2^{r}\\\ 2\end{array}\right)$.
By Lemma 3.3, the 2-error linear complexity of $s^{(r)}+t^{(r)}$ is $c$.
By Lemma 3.2 and Lemma 3.4, the number of binary sequences $s^{(r)}+t^{(r)}$
is $2^{c-1}\times(1+\left(\begin{array}[]{c}2^{r}\\\ 2\end{array}\right))$
By Lemma 3.1, there are $2^{2^{n-1}+\cdots+2^{r}}=2^{2^{n}-2^{r}}$ binary
sequences $s^{(n)}+t^{(n)}$, such that
$s^{(r)}+t^{(r)}=\varphi_{r+1}\cdots\varphi_{n}(s^{(n)}+t^{(n)})$,
$t^{(r)}=\varphi_{r+1}\cdots\varphi_{n}(t^{(n)})$ and $W(t^{(n)})=W(t^{(r)})$.
Thus the 2-error linear complexity of $s^{(n)}+t^{(n)}$ is
$2^{n-1}+\cdots+2^{r}+L_{2}(s^{(r)}+t^{(r)})=2^{n}-2^{r}+c=L(r,c).$
Therefore, $N_{2}(L(r,c))=2^{2^{n}-2^{r}}\times
2^{c-1}\times(1+\left(\begin{array}[]{c}2^{r}\\\
2\end{array}\right))=2^{L(r,c)-1}(1+\left(\begin{array}[]{c}2^{r}\\\
2\end{array}\right))$ ∎
Based on the results above, we have the following theorem.
Theorem 3.1 Let $L(r,c)=2^{n}-2^{r}+c$, $2\leq r\leq n,1\leq c\leq 2^{r-1}-1$,
and $N_{2}(L(r,c))$ be the number of $2^{n}$-periodic binary sequences with
linear complexity less than $2^{n}$ and given 2-error linear complexity
$L(r,c)$. Then
$N_{2}(L)=\left\\{\begin{array}[]{l}\left(\begin{array}[]{c}2^{n}\\\
2\end{array}\right)+1,\ \ \ \ \ \ \ \ \ \ L=0\\\
2^{L-1}(\left(\begin{array}[]{c}2^{r}\\\ 2\end{array}\right)+1),\
L=L(r,c),1\leq c\leq 2^{r-2}-1,r>2\\\ 2^{L-1}(\left(\begin{array}[]{c}2^{r}\\\
2\end{array}\right)+1-3\times 2^{r+m-3}),\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
L=L(r,c),c=2^{r-1}-2^{r-m},1<m\leq r,r\geq 2\\\
2^{L-1}(\left(\begin{array}[]{c}2^{r}\\\
2\end{array}\right)+1+2^{r-m}-2^{r+m-2}),\\\ \ \ \ \ \ \ \ \ \ \ \ \ \
L=L(r,c),c=2^{r-1}-2^{r-m}+x,\\\ \ \ \ \ \ \ \ \ \ \ \ \ \
1<m<r-1,0<x<2^{r-m-1},r>3\\\ 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \mbox{others}\end{array}\right.$
###### Proof:
By Lemma 3.7, we now only need to consider the case of $3\leq r\leq
n,2^{r-2}\leq c\leq 2^{r-1}-1$.
By Lemma 3.1 and Lemma 3.5,
$N_{2}(L(r,c))=2^{L(r,c)-1}(\left(\begin{array}[]{c}2^{r}\\\
2\end{array}\right)+1-3\times 2^{r+m-3})$
for $3\leq r\leq n,c=2^{r-1}-2^{r-m},1<m\leq r$
By Lemma 3.1 and Lemma 3.6,
$N_{2}(L(r,c))=2^{L(r,c)-1}(\left(\begin{array}[]{c}2^{r}\\\
2\end{array}\right)+1+2^{r-m}-2^{r+m-2})$
for $4\leq r\leq n,c=2^{r-1}-2^{r-m}+x,1<m<r-1,0<x<2^{r-m-1}$
This completes the proof. ∎
Now we give an example to illustrate Theorem 3.1.
For $n=4$, the number of $2^{n}$-periodic binary sequences with linear
complexity less than $2^{n}$ is $count=2^{2^{4}-1}=2^{15}$.
$N_{2}(L(2,1))=2^{12}$
$N_{2}(L(3,1))=2^{8}[\left(\begin{array}[]{c}2^{3}\\\
2\end{array}\right)+1]=count\times\frac{29}{128}$.
$N_{2}(L(3,2))=count\times\frac{17}{64}$.
$N_{2}(L(3,3))=count\times\frac{5}{32}$.
$N_{2}(0)=N_{2}(L(4,1))=\left(\begin{array}[]{c}2^{4}\\\
2\end{array}\right)+1=121$.
$N_{2}(L(4,2))=2\times 121=242$.
$N_{2}(L(4,3))=4\times 121=484$.
$N_{2}(L(4,4))=776$. $N_{2}(L(4,5))=1744$.
$N_{2}(L(4,6))=2336$. $N_{2}(L(4,7))=1600$.
It is easy to verify that the number of all these sequences is $2^{15}$. These
results are also checked by computer.
Notice that for $2^{n}$-periodic binary sequences with linear complexity less
than $2^{n}$, the change of three bits per period results in a sequence with
odd number of nonzero bits per period, which has again linear complexity
$2^{n}$. So from Theorem 3.1, we also know the counting functions on the
$3$-error linear complexity for $2^{n}$-periodic binary sequences with linear
complexity less than $2^{n}$.
Similarly, we can have the following theorem.
Theorem 3.2 Let $L(r,c)=2^{n}-2^{r}+c$, or $2^{n}-2^{3}+1$, $4\leq r\leq
n,1\leq c\leq 2^{r-1}-1$, and $N_{3}(L(r,c))$ be the number of
$2^{n}$-periodic binary sequences with linear complexity $2^{n}$ and given
3-error linear complexity $L(r,c)$. Let
$\displaystyle f(n,m)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}2^{n}\\\
3\end{array}\right)-2^{n-m}\left(\begin{array}[]{c}2^{m}\\\
3\end{array}\right)-\left(\begin{array}[]{c}2^{n-m}\\\
2\end{array}\right)\left(\begin{array}[]{c}2^{m}\\\
2\end{array}\right)2^{m+1}$ $\displaystyle+\left(\begin{array}[]{c}2^{n-m}\\\
2\end{array}\right)\times
2^{2m}(2^{m-2}-1)+2^{n-m-1}\times\left(\begin{array}[]{c}2^{m-1}\\\
3\end{array}\right)$ $\displaystyle-2^{n-2}\times(2^{m-2}-1)$ $\displaystyle
g(n,m)$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}2^{n}\\\
3\end{array}\right)-(2^{m-2}-1)\times 2^{n+1}$
$\displaystyle-(2^{m-1}-1)\times\left(\begin{array}[]{c}2^{n-m}\\\
2\end{array}\right)\times 2^{m+1}$ $\displaystyle-3\times
2^{n-m-2}[\left(\begin{array}[]{c}2^{m}\\\
3\end{array}\right)-4\left(\begin{array}[]{c}2^{m-1}\\\ 2\end{array}\right)]$
$\displaystyle-\left(\begin{array}[]{c}2^{n-m}\\\
2\end{array}\right)\times[\left(\begin{array}[]{c}2^{m}\\\
2\end{array}\right)-2^{m-1}]\times 2^{m}$
Then
$N_{3}(L)=\left\\{\begin{array}[]{l}\left(\begin{array}[]{c}2^{n}\\\
3\end{array}\right)+2^{n},\ \ \ \ L=0\\\
2^{L(r,c)-1}(\left(\begin{array}[]{c}2^{r}\\\ 3\end{array}\right)+2^{r}),\\\ \
\ \ \ \ \ \ \ L=L(r,c),1\leq c\leq 2^{r-2}-1,r>2\\\ 2^{L(r,c)-1}f(r,m),\\\ \ \
\ \ \ \ \ \ L=L(r,c),c=2^{r-1}-2^{r-m},1<m\leq r,r>3\\\ 2^{L(r,c)-1}g(r,m),\\\
\ \ \ \ \ \ \ \ L=L(r,c),c=2^{r-1}-2^{r-m}+x,\\\ \ \ \ \ \ \ \ \
1<m<r-1,0<x<2^{r-m-1},r>3\\\ 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\mbox{others}\end{array}\right.$
Based on Theorem 3.1 and Theorem 3.2, the counting functions for the number of
$2^{n}$-periodic binary sequences with fixed 3-error linear complexity can be
easily derived as follows.
Theorem 3.3 Let $L(r,c)=2^{n}-2^{r}+c$, $4\leq r\leq n,1\leq c\leq 2^{r-1}-1$,
and $N_{3}(L(r,c))$ be the number of $2^{n}$-periodic binary sequences with
3-error linear complexity $L(r,c)$. Then
$N_{3}(L)=\left\\{\begin{array}[]{l}\left(\begin{array}[]{c}2^{n}\\\
3\end{array}\right)+\left(\begin{array}[]{c}2^{n}\\\
2\end{array}\right)+2^{n}+1,\ \ \ \ \ \ \ L=0\\\
2^{L-1}(\left(\begin{array}[]{c}2^{r}\\\
3\end{array}\right)+\left(\begin{array}[]{c}2^{r}\\\
2\end{array}\right)+2^{r}+1),\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ L=L(r,c),1\leq
c\leq 2^{r-2}-1,r>3\\\ 2^{L-1}(\left(\begin{array}[]{c}2^{r}\\\
2\end{array}\right)+1-3\times 2^{r+m-3}+f(r,m)),\\\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ L=L(r,c),c=2^{r-1}-2^{r-m},1<m\leq r,r>3\\\
2^{L-1}(\left(\begin{array}[]{c}2^{r}\\\
2\end{array}\right)+1+2^{r-m}-2^{r+m-2}+g(r,m)),\\\ \ \ \ \ \ \ \ \ \ \ \ \ \
L=L(r,c),c=2^{r-1}-2^{r-m}+x,\\\ \ \ \ \ \ \ \ \ \ \ \ \ \
1<m<r-1,0<x<2^{r-m-1},r>3\\\ 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \mbox{others}\end{array}\right.$ where $f(r,m)$ and $g(r,m)$ are
defined in Theorem 3.2.
Let $L(r,c)=2^{n}-2^{r}+c$, $3\leq r\leq n,1\leq c\leq 2^{r-1}-1$. By dividing
the 4-error linear complexity into six categories: $c=2^{r-2}-2^{r-m}$,
$c=2^{r-2}-2^{r-m}+x$, $c=2^{r-1}-2^{r-m}$, $c=2^{r-1}-(2^{r-m}+2^{r-j})$,
$c=2^{r-1}-(2^{r-m}+2^{r-j})+x$, and $1\leq c\leq 2^{r-3}-1$, we finally got
the counting functions for the number of $2^{n}$-periodic binary sequences
with linear complexity less than $2^{n}$ and fixed 4-error linear complexity.
As a consequence of the result, the complete counting functions on the
$4$-error linear complexity of $2^{n}$-periodic binary sequences (with linear
complexity $2^{n}$ or less than $2^{n}$) are obvious.
Here only the cases of $c=2^{r-1}-2^{r-m}$ and $c=2^{r-1}-(2^{r-m}+2^{r-j})$
are presented. The results about other cases are omitted.
Lemma 3.8 Let $N_{4}(2^{n-1}-2^{n-m})$ be the number of $2^{n}$-periodic
binary sequences with linear complexity less than $2^{n}$ and given 4-error
linear complexity $2^{n-1}-2^{n-m},2\leq m\leq n$. Then
$\displaystyle N_{4}(2^{n-1}-2^{n-m})$ $\displaystyle=$
$\displaystyle[\left(\begin{array}[]{c}2^{n}\\\
4\end{array}\right)-E1+E2/4-E3+E4/2-E5+E6/4-E7+E8/8]$ $\displaystyle\ \ \times
2^{2^{n-1}-2^{n-m}-1}$
where
$E1=\left(\begin{array}[]{c}2^{n-m}\\\
2\end{array}\right)\times\left(\begin{array}[]{c}2^{m}\\\
2\end{array}\right)\times\left(\begin{array}[]{c}2^{m}\\\ 2\end{array}\right)$
$\displaystyle E2=4\times\left(\begin{array}[]{c}2^{n-m}\\\
2\end{array}\right)\times\left(\begin{array}[]{c}2^{m-1}\\\
2\end{array}\right)\times\left(\begin{array}[]{c}2^{m-1}\\\
2\end{array}\right)$ $\displaystyle\ \ \ \ \ \ \
-\left(\begin{array}[]{c}2^{n-m}\\\
2\end{array}\right)\times[2^{2m-2}+2^{m+1}(\left(\begin{array}[]{c}2^{m-1}\\\
2\end{array}\right)$ $\displaystyle\ \ \ \ \ \ \ -2^{m-2})]$
$E3=\left(\begin{array}[]{c}2^{n-m}\\\
3\end{array}\right)\times\left(\begin{array}[]{c}3\\\
1\end{array}\right)\times\left(\begin{array}[]{c}2^{m}\\\
2\end{array}\right)\times 2^{m}\times 2^{m}$
$\displaystyle E4=\left(\begin{array}[]{c}2^{n-m}\\\
3\end{array}\right)\left(\begin{array}[]{c}3\\\
1\end{array}\right)\left(\begin{array}[]{c}2^{m-1}\\\
2\end{array}\right)\times 2^{2m+1}$ $\displaystyle\ \ \ \ \ \ \ \
-\left(\begin{array}[]{c}2^{n-m}\\\
3\end{array}\right)\left(\begin{array}[]{c}3\\\ 1\end{array}\right)\times
2^{3m-1}$
$E5=\left(\begin{array}[]{c}2^{n-m}\\\
2\end{array}\right)\left(\begin{array}[]{c}2\\\
1\end{array}\right)\left(\begin{array}[]{c}2^{m}\\\ 3\end{array}\right)\times
2^{m}$
$\displaystyle E6=2^{m+2}\times\left(\begin{array}[]{c}2^{n-m}\\\
2\end{array}\right)\times\left(\begin{array}[]{c}2^{m-1}\\\
3\end{array}\right)$ $\displaystyle\ \ \ \ \ \ \ \
-\left(\begin{array}[]{c}2^{n-m}\\\ 2\end{array}\right)\times(2^{m-1}-2)\times
2^{2m}$
$E7=2^{n-m}\times\left(\begin{array}[]{c}2^{m}\\\ 4\end{array}\right)$
$\displaystyle E8=2^{n-m+1}\times\left(\begin{array}[]{c}2^{m-1}\\\
4\end{array}\right)-[2^{n-m+1}\times 2^{m-2}$ $\displaystyle\ \ \ \ \ \ \ \
\times\left(\begin{array}[]{c}2^{m-1}-2\\\
2\end{array}\right)-2^{n-m+1}\times\left(\begin{array}[]{c}2^{m-2}\\\
2\end{array}\right)]$
By Lemma 3.8, for $n=5,m=5$, $N_{4}(15)=4587520$, which is checked by
computer.
Lemma 3.9 Let $N_{4}(2^{n-1}-(2^{n-m}+2^{n-j}))$ be the number of
$2^{n}$-periodic binary sequences with linear complexity less than $2^{n}$ and
given 4-error linear complexity $2^{n-1}-(2^{n-m}+2^{n-j}),n>3,2<m<j\leq n$.
Then
$\displaystyle N_{4}(2^{n-1}-(2^{n-m}+2^{n-j}))$ $\displaystyle=$
$\displaystyle[1+\left(\begin{array}[]{c}2^{n}\\\
2\end{array}\right)+\left(\begin{array}[]{c}2^{n}\\\ 4\end{array}\right)-F4$
$\displaystyle-\sum\limits_{k=m+1}^{j-1}(\frac{2^{2m-3}-1}{2^{2m-3}}F6+\frac{2^{m-1}-1}{2^{m-1}}F7+F8/2)$
$\displaystyle-\frac{2^{m-2}-1}{2^{m-2}}F10-F11/2-F13-\frac{3}{4}F14-\frac{2^{m-1}-1}{2^{m-1}}F17$
$\displaystyle-\frac{3}{4}F18-\frac{2^{2m-4}-1}{2^{2m-4}}F19-F22/2-\frac{2^{m-2}-1}{2^{m-2}}F23$
$\displaystyle-F25-F26-\frac{7}{8}F27]\times 2^{2^{n-1}-(2^{n-m}+2^{n-j})-1}$
where
$F4=2^{n+2m+j-6}+2^{n+m-4}+2^{n+j-4}+3\times 2^{n+m+j-4}$
$F6=2^{n+k-4},F7=3\times 2^{n+m+k-4},F8=2^{n+2m+k-6}$
$F10=2^{n-1},F11=2^{n-m+1}\left(\begin{array}[]{c}2^{m-1}\\\
2\end{array}\right)-2^{n-1}$ $F13=\left(\begin{array}[]{c}2^{n-m+1}\\\
2\end{array}\right)\times\left(\begin{array}[]{c}2\\\
1\end{array}\right)\times 2^{m-2}\times(2^{m-1}-2)\times 2^{m-1}$
$F14=\left(\begin{array}[]{c}2^{n-m+1}\\\
2\end{array}\right)\times\frac{2^{m-1}-4}{3}\times(2^{m-1}-2)\times 2^{2m-2}$
$F17=\left(\begin{array}[]{c}2^{n-m+1}\\\
2\end{array}\right)\times\left(\begin{array}[]{c}2\\\
1\end{array}\right)\times 2^{m-2}\times[\left(\begin{array}[]{c}2^{m-1}\\\
2\end{array}\right)-2^{m-2}]$ $F18=\left(\begin{array}[]{c}2^{n-m+1}\\\
2\end{array}\right)\times[\left(\begin{array}[]{c}2^{m-1}\\\
2\end{array}\right)-2^{m-2}]^{2}$
$F19=\left(\begin{array}[]{c}2^{n-m+1}\\\
2\end{array}\right)\times\left(\begin{array}[]{c}2^{m-1}\\\
2\end{array}\right)^{2}-2^{n+m-4}-\sum\limits_{k=m+1}^{j}2^{n+k-4}-F17-F18$
$F22=\left(\begin{array}[]{c}2^{n-m+1}\\\
3\end{array}\right)\left(\begin{array}[]{c}3\\\
1\end{array}\right)\times(\left(\begin{array}[]{c}2^{m-1}\\\
2\end{array}\right)-2^{m-2})\times(2^{m-1})^{2}$
$F23=\left(\begin{array}[]{c}2^{n-m+1}\\\
3\end{array}\right)\times\left(\begin{array}[]{c}3\\\
1\end{array}\right)\times\left(\begin{array}[]{c}2^{m-1}\\\
2\end{array}\right)\times(2^{m-1})^{2}-3\sum\limits_{k=m+1}^{j}2^{n+m+k-4}-F22$
$F25=2^{n-m+1}\times\left(\begin{array}[]{c}2^{m-2}\\\ 2\end{array}\right)$
$F26=2^{n-m+1}\times 2^{m-2}\times[\left(\begin{array}[]{c}2^{m-1}-2\\\
2\end{array}\right)-(2^{m-2}-1)]$
$F27=2^{n-m+1}\times\left(\begin{array}[]{c}2^{m-1}\\\
4\end{array}\right)-F25-F26$
By Lemma 3.9, for $n=5,m=4,j=5$, $N_{4}(13)=46845952$, which is checked by
computer.
## IV Conclusion
By using the sieve method of combinatorics, an approach to construct the
complete counting functions on the $k$-error linear complexity of
$2^{n}$-periodic binary sequences was developed. The complete counting
functions on the $k$-error linear complexity of $2^{n}$-periodic binary
sequences were obtained for $k=3$ and 4.
Using the approach proposed, we can deal with the $k$-error linear complexity
distribution of sequences over $GF(q)$ with period $p^{n}$ or $2p^{n}$, where
$p$ and $q$ are odd primes, and $q$ is a primitive root modulo $p^{2}$.
## Acknowledgment
The research was supported by Zhejiang Natural Science Foundation(No.Y1100318,
R1090138) and NSAF (No. 10776077).
## References
* [1] Ding,C.S., Xiao,G.Z. and Shan,W.J., The Stability Theory of Stream Ciphers[M]. Lecture Notes in Computer Science, Vol.561. Berlin/ Heidelberg, Germany: Springer-Verlag, 1991,85-88.
* [2] Etzion T., Kalouptsidis N., Kolokotronis N., Limniotis K. and Paterson K. G., Properties of the Error Linear Complexity Spectrum, IEEE Transactions on Information Theory, 2009, 55(10): 4681-4686.
* [3] Games, R.A., and Chan, A.H., A fast algorithm for determining the complexity of a binary sequence with period $2^{n}$. IEEE Trans on Information Theory, 1983, 29 (1):144-146.
* [4] Fu F, Niederreiter H., and Su M., The characterization of $2^{n}$-periodic binary sequences with fixed 1-error linear complexity, In: Gong G., Helleseth T., Song H.-Y., Yang K. (eds.) SETA 2006, LNCS, vol. 4086, 88-103. Springer (2006).
* [5] Han Y.K., Chung J.H., and Yang K., On the $k$-error linear complexity of $p^{m}$-periodic binary sequences. IEEE Transactions on Information Theory, 2007, 53(6): 2297-2304.
* [6] Kaida T., Uehara S., and Imamura K., An algorithm for the $k$-error linear complexity of sequences over GF($p^{m}$) with period $p^{n}$, $p$ a prime. Information and Computation, 1999,151(1):134 -147.
* [7] Kavuluru R., $2^{n}$-periodic binary sequences with fixed 2-error or 3-error linear complexity. In: Golomb S., Parker M., Pott A., Winterhof A. (eds.) SETA 2008. LNCS, vol. 5203, 252-265. Springer (2008).
* [8] Kavuluru R., Characterization of $2^{n}$-periodic binary sequences with fixed 2-error or 3-error linear complexity, Des. Codes Cryptogr. 2009,53: 75-97.
* [9] Kolokotronis N., Rizomiliotis P. and Kalouptsidis N., Minimum linear span approximation of binary sequences. IEEE Transactions on Information Theory, 2002, 48:2758-2764.
* [10] Kurosawa K., Sato F., Sakata T. and Kishimoto W., A relationship between linear complexity and $k$-error linear complexity. IEEE Transactions on Information Theory, 2000, 46(2): 694-698.
* [11] Lauder A. and Paterson K., Computing the error linear complexity spectrum of a binary sequence of period $2^{n}$. IEEE Transactions on Information Theory, 2003, 49(1):273-280.
* [12] Meidl W., How many bits have to be changed to decrease the linear complexity?, Des. Codes Cryptogr., 2004, 33:109-122.
* [13] Meidl W., On the stablity of $2^{n}$-periodic binary sequences. IEEE Transactions on Information Theory, 2005, 51(3): 1151-1155.
* [14] Rueppel R A. Analysis and Design of Stream Ciphers. Berlin: Springer-Verlag, 1986, chapter 4.
* [15] Stamp,M., and Martin,C. F., An algorithm for the $k$-error linear complexity of binary sequences with period $2^{n}$, IEEE Trans. Inform. Theory, 1993, 39:1398-1401.
* [16] Wei,S.M., Xiao,G.Z., and Chen,Z., A fast algorithm for determining the minimal polynomial of a sequence with period $2p^{n}$ over $GF(q)$, IEEE Trans on Information Theory, 2002, 48(10):2754-2758.
* [17] Xiao,G.Z., Wei,S.M., Lam K. Y., and Imamura K., A fast algorithm for determining the linear complexity of a sequence with period $p^{n}$ over $GF(q)$. IEEE Trans on Information Theory, 2000,46: 2203-2206.
* [18] Zhou,J.Q., A counterexample concerning the 3-error linear complexity of $2^{n}$-periodic binary sequences, Des. Codes Cryptogr., 2011, http://www.springerlink.com/content/7562p69561624154/
* [19] Zhu,F.X. and Qi,W.F., The 2-error linear complexity of $2^{n}$-periodic binary sequences with linear complexity $2^{n}$-1. Journal of Electronics (China), 2007,24(3): 390-395, http://www.springerlink.com/content /3200vt810p232769/
|
arxiv-papers
| 2011-12-28T02:57:58 |
2024-09-04T02:49:25.736283
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jianqin Zhou, Jun Liu and Wanquan Liu",
"submitter": "Jianqin Zhou",
"url": "https://arxiv.org/abs/1112.6047"
}
|
1112.6200
|
# Quantum Signatures of the Optomechanical Instability
Jiang Qian Arnold Sommerfeld Center for Theoretical Physics, Center for
NanoScience and Department of Physics, Ludwig-Maximilians-Universität München,
Theresienstrasse 37, 80333, München, Germany A. A. Clerk Department of
Physics, McGill University, Montreal, Quebec, Canada H3A 2T8 K. Hammerer
Institute for Theoretical Physics, Institute for Gravitational Physics,
Leibniz University Hanover, Callinstrasse 38, D-30167 Hanover, Germany
Florian Marquardt Institute for Theoretical Physics, Department of Physics,
Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstr. 7, 91058
Erlangen, Germany Max Planck Institute for the Science of Light, Günter-
Scharowsky-Straße 1/Bau 24, 91058 Erlangen, Germany
###### Abstract
In the past few years, coupling strengths between light and mechanical motion
in optomechanical setups have improved by orders of magnitude. Here we show
that, in the standard setup under continuous laser illumination, the steady
state of the mechanical oscillator can develop a non-classical, strongly
negative Wigner density if the optomechanical coupling is comparable to or
larger than the optical decay rate and the mechanical frequency. Because of
its robustness, such a Wigner density can be mapped using optical homodyne
tomography. This feature is observed near the onset of the instability towards
self-induced oscillations. We show that there are also distinct signatures in
the photon-photon correlation function $g^{(2)}(t)$ in that regime, including
oscillations decaying on a time scale not only much longer than the optical
cavity decay time, but even longer than the mechanical decay time.
By coupling optical and mechanical degrees of freedom, the emerging field of
optomechanics provides exciting new opportunities to study the quantum
mechanical behavior of macroscopic objects (for reviews see Marquardt2009 ;
Kippenberg2008 ). Recent optomechanical cooling experiments are successfully
bringing nanomechanical oscillators into their quantum mechanical ground state
TeufelCooling2011 ; PainterGround2011 . The same optomechanical coupling also
promises the possibility of single-quadrature measurements of the resulting
mechanical quantum states with the help of the light field Braginsky ;
ClerkMarquardtQND ; 2010_HertzbergQND_NaturePhysics . For a reproducible and
persistent quantum state, such measurements would result in an experimental
determination of its full Wigner density via tomography, similar to what has
been achieved in microscopic systems, for single ions or photons IonTrapWigner
; PhotonFockState . The recent advances in fabricating optomechanical devices
have drastically improved coupling parameters, _e.g._ for optomechanical
crystals PainterCrystal , in microwave setups TeufelCooling2011 , and other
devices like GaAs disks 2010_Favero_GaAsDisk or toroidal optical microcavity
Kippenberg2012 . It will likely be possible relatively soon to achieve
optomechanical coupling strengths $g_{0}$ at the single-photon level that are
comparable to the optical cavity decay rate $\kappa$, a feat that has already
been achieved in cold atom optomechanical systemsMurch2008 ;
EsslingerColdAtom2008 . This regime of strongly nonlinear quantum
optomechanics promises to pave the way towards generating and detecting novel
quantum states in optomechanical systems. It is currently only beginning to be
explored theoretically LudwigNJP2008 ; Rabl ; NunnenkampPRL , although very
early work already discussed quantum optomechanical effects in the
(unrealistic) absence of any dissipation Mancini1997 ; Bose1997 .
In the classical regime, nonlinear dynamics is known to occur when the system
is driven by a blue detuned laser. When the input laser power crosses a
certain threshold, the mechanical oscillator will undergo a Hopf bifurcation
and start self-induced mechanical oscillations, a phenomenon termed
“parametric instability” Braginsky1967 ; Kippenberg2005 ; Carmon2005 ;
MarquardtPRL2006 ; Ludwig2008 ; GrudininPhononLaser2010 . The quantum dynamics
of this regime has first been studied in LudwigNJP2008 , and there is
interesting synchronization behaviour for arrays of coupled oscillators of
this type OurPRL2011 .
In this paper, we show that, for strong optomechanical couplings $g_{0}$
comparable to or greater than the optical decay rate $\kappa$ and mechanical
frequency $\omega_{M}$ ($g_{0}/\kappa\gtrsim
1,g_{0}^{2}/(\kappa\cdot\omega_{M})\gtrsim 1$), a large laser driving and an
effectively zero temperature thermal bath, a non-classical state of the
mechanical oscillator with strongly negative Wigner density can be produced
around the onset of self-induced oscillations. Because the state is time-
independent, one may use single-quadrature homodyne tomography to
experimentally reconstruct its non-classical Wigner density.
In addition, we propose to use the two-point photon correlation function
$g^{(2)}(t)$ as an experimentally convenient probe for the peculiar quantum
dynamics near the bifurcation. We identify two distinct signatures that enable
experimentalists to reliably detect the onset and growth of the self-induced
oscillation. We provide an explanation of the non-classical decay of
$g^{(2)}(t)$ in both the red and blue-detuned regime.
| | |
---|---|---|---
Figure 1: Non-classical states in an optomechanical system. The laser input
$\alpha_{L}$ is held constant and the laser detuning $\Delta$ increases from
the steady state “A” to “D”. The mechanical Wigner densities of these states
are shown in (a)-(d). $x_{ZPF}$ and $p_{ZPF}$ are zero-point fluctuations of
the oscillator’s position and momentum, respectively. Plot (e) shows the start
of the self-induced oscillation, where the phonon number $n_{b}$ of the
oscillator rises quickly between state “B” and “C”. As the detuning further
increases to “D”, a non-classical mechanical quantum state with _negative_
mechanical Wigner density state appears, as shown in (d). In (f) the evolution
of the mechanical Fano factor $F$ as a function of $\Delta$ is shown. It dips
below the Poisson value $1$ (dashed line) in non-classical state shown here.
In plot (g) and (h), we show that the negative Wigner density states have more
sharply peaked phonon number distributions $p(n)$ compared with non-negative
states. In (g) the $p(n)$ of state “C” and “D” (plot (c),(d)) are compared. In
(h), where $g_{0}=0.6\omega_{M}$, the negative state (solid line) has two
clear peaks in $p(n)$, in contrast to a single smooth peak for the non-
negative state (dashed line). The Wigner density of these two states are shown
as insets. Finally, in (i) we show two regions in the parameter space of
detuning $\Delta$ and coupling $g_{0}$ where significant negative Wigner
density states exist. States “A”-“D” are indicated here. In all plots other
physical parameters are
$g_{0}=0.36\omega_{M},\kappa_{M}=0.3\omega_{M},\Gamma_{M}=0.00147\omega_{M},\alpha_{L}=0.311\omega_{M}$,
except for (h), where $g_{0}=0.6\omega_{M},\alpha_{L}=0.186\omega_{M}$. The
intra-cavity photon number is $n_{a}\approx 0.1$—$0.7$ when
$g_{0}=0.36\omega_{M},-\omega_{M}\leq\Delta\leq 0$.
Within the rotating wave approximation, an optomechanical system can be
described by the following standard Hamiltonian:
$\hat{H}=\hbar(-\Delta+g_{0}(\hat{b}^{\dagger}+\hat{b}))\hat{a}^{\dagger}\hat{a}+\hbar\omega_{M}\hat{b}^{\dagger}\hat{b}+\hbar\alpha_{L}(\hat{a}^{\dagger}+\hat{a})+{\hat{H}}_{\rm
diss}.$ (1)
Here ${\hat{a}}/{\hat{b}}$ are the operators for the photon/phonon modes,
$\omega_{M}$ is the mechanical frequency and $\alpha_{L}$ is the laser driving
amplitude. $\Delta=\omega_{L}-\omega_{C}$ is the detuning of the laser from
the cavity’s _unperturbed_ resonance (_i.e._ evaluated for _zero mechanical
displacement_). $g_{0}$ describes the strength of the optomechanical coupling
at the single-photon level.
When the dissipative terms in $H_{\rm diss}$ are taken into account, the
density matrix $\hat{\rho}$ of the combined photon-phonon system evolves
according to the quantum master equation:
$\frac{d\hat{\rho}}{dt}=\mathcal{L}[\hat{\rho}]=\frac{[\hat{H},\hat{\rho}]}{i\hbar}+\Gamma\mathcal{D}[\hat{b},\hat{\rho}]+\kappa\mathcal{D}[\hat{a},\hat{\rho}].$
(2)
Here $\mathcal{L}$ is the quantum Liouville operator describing the time
evolution of the density matrix $\hat{\rho}$, where we incorporate dissipation
in the photon and phonon subsystems with decay rates $\kappa$ and $\Gamma$,
respectively. The standard Lindblad term is given by
$\mathcal{D}[{\hat{O}},\hat{\rho}]=\hat{O}\hat{\rho}\hat{O}^{\dagger}-\frac{1}{2}(\hat{O}^{\dagger}\hat{O}\hat{\rho}+\hat{\rho}\hat{O}^{\dagger}\hat{O})$.
Note that we will assume zero bath temperature in our simulations, which will
be reachable to a good approximation when ${\rm GHz}$-frequency setups (_e.g._
optomechanical crystals) are deployed in dilution refrigerator settings. In
this paper, we are interested in the steady state solution of Eq. 2, where all
the transient dynamics has died out. This is obtained numerically by finding
the density matrix satisfying $\mathcal{L}[\hat{\rho}]=0$ using the standard
Arnoldi algorithm, as implemented in the ARPACK package. Due to its
persistence, this state is ideal for making homodyne measurements of its
mechanical Wigner density, in contrast to transient scenarios.
Specifically we are interested in the mechanical Wigner density $W_{\rm
M}(x,p)=\frac{1}{\pi\hbar}\int_{-\infty}^{\infty}\langle x-y|\hat{\rho}_{\rm
M}|x+y\rangle e^{2ipy/\hbar}dy$, where $\hat{\rho}_{\rm M}$ is the mechanical
density matrix, obtained by tracing out the optical degrees of freedom from
$\hat{\rho}$. The Wigner density is the quantum analog of the classical
Liouville phase space probability density. A negative Wigner density is a
strong signature of a non-classical state. Early investigations LudwigNJP2008
of the optomechanical instability in the regime around $g_{0}\sim\kappa$ did
not turned up nonclassical states.
In Fig 1, (a)-(e), we show the overall properties of the steady state
solutions. As we increase the laser detuning while keeping the input laser
power fixed (points $\textrm{A}\to\textrm{B}\to\textrm{C}$), the phonon number
in the mechanical oscillator rises sharply (plot (e)), signaling the onset of
the self-induced oscillations. This is also reflected in the mechanical Wigner
density $W_{\rm M}(x,p)$. Below the onset (point “A”), $W_{\rm M}(x,p)$ is a
simple Gaussian, which starts to broaden just below the threshold, as the
susceptibility of the system diverges and quantum fluctuations are strongly
amplified (point “B”). Above the threshold, we have a coherent state
undergoing circular motion in phase space, but with an undetermined phase,
which is the Wigner density observed at point “C” LudwigNJP2008 ;
NunnenkampPRL .
However, such a simple picture is inadequate for an optomechanical system with
$g_{0}\sim\kappa$, _i.e._ when one approaches the optomechanical instability
deep in the quantum regime 111Another necessary condition is that
$g_{0}^{2}/(\kappa\cdot\omega_{M})$ is not much smaller than one. In such a
system, we observe that for a range of detuning $\Delta$ and laser driving
$\alpha_{L}$, the mechanical self-induced oscillation produces strongly non-
classical states with large negative areas in the Wigner density. This can be
seen in the example of Fig. 1 (d). Negative rims, shown in brighter color,
develop at amplitudes slightly smaller than the average amplitude of
oscillation. Plots (f)-(h) in Fig. 1 analyze negative states more deeply. In
state “D”, (f) shows the mechanical Fano factor $F=\frac{\langle\Delta
n_{b}^{2}\rangle}{\langle n_{b}\rangle}$ dips below the coherent state value
$1$, and its phonon number distribution (g) has a reduced variance. At larger
coupling $g=0.6\omega_{M}$ (h), the negative state exhibits a sharp peak and a
smoother one, as opposed to a single broader peak of the non-negative state
222Note here, due to the two-peak structure, the Fano factor of the negative
Wigner density state remains above one. Overall, (f)-(h) show that the
negative states are closer to a _single_ Fock state _or_ a superposition of
_few_ Fock states as compared with a coherent state Rodrigues2010 . Note,
however, the origin of this non-classical state is _not_ the same as that in
the well-studied micromaser Filipowicz:86 ; Meystre:88 ; Krause87 ; Varcoe00 .
In the micromaser, the mechanism relies crucially on the swapping of a single
excitation between an excited atom and cavity over a fixed interaction time.
These features are absent in our system.
Fig. 1 (i) maps out the regions in parameter space where negative Wigner
densities occur. This ‘phase diagram’ is shown as a function of the “quantum
parameter” $\zeta=\frac{g_{0}}{\kappa}$ LudwigNJP2008 and of the laser
detuning $\Delta\omega_{M}$, at a fixed value of the laser driving strength
$\alpha_{L}$. It has been obtained by solving for the steady state of the
optomechanical system under constant illumination, and the Wigner density is
considered as nonclassical if a sufficiently large area turns out to be
negative. The threshold criterion is a negative area of at least 3% of the
positive area, and the minimum value being at least 5% in absolute value of
the maximum. The numerical results shown here indicate that, for the
parameters considered here, starting at $\frac{g_{0}}{\kappa}=0.8$, the
negative Wigner density states appear around detuning $\Delta/\omega_{M}=0$,
and a second negative Wigner density region opens up at
$\frac{g_{0}}{\kappa}=1.6$, around $\Delta/\omega_{M}=0.9$ at the first blue
sideband, where the instability is driven efficiently. The phonon number
distribution displays a pronounced narrowing, getting closer to a single or
few mechanical Fock states. However, we find that still many photon/phonon
levels are involved in the dynamics in the regime considered here, and there
seems to be no simple explanation involving only a few levels.
These _steady-state_ non-classical Wigner densities could be reconstructed via
optomechanical Quantum Non-demolition quadrature detection Braginsky ;
ClerkMarquardtQND and subsequent quantum state tomography
2009_Lvovsky_Review_QuantumStateTomography . This merely involves illumination
with another amplitude-modulated laser beam for read-out, as explained in
ClerkMarquardtQND . When observed, these would provide an accessible example
of non-classical states in a fabricated mesoscopic mechanical object. To date,
there has been no experimental observation of non-classical Wigner densities
in the domain of micro- or nanomechanical structures. The experiment that came
closest to that goal, and in the process did produce nonclassical mechanical
Fock states, employed a complex multi-layered superconducting circuit with
piezoelectric coupling to a superconducting qubit and ultrafast pulse
sequences ClelandPiezo2010 . Furthermore in their setup the resonator lifetime
is too short to permit the reconstruction of the full Wigner density. By
contrast, once optomechanical parameters can be improved to reach the single-
photon strong coupling regime, the scheme discussed here would be relatively
straightforward, being based on continuous laser illumination of an
optomechanical setup whose fabrication is much less complex as it involves
only one material. Recently a coupling $g_{0}/\kappa\approx 0.007$ has been
achieved in an optomechanical crystal system painter2012 and further
improvement is expected in that setup. In addition, there is the possibility
that the parameters required here may be reached in cold atom optomechanical
setups Murch2008 ; EsslingerColdAtom2008 .
The full mechanical state reconstruction in the nonlinear quantum regime is an
enticing and challenging goal. Nevertheless, it requires many experimental
runs. It will be helpful to have other means of optically probing the quantum
dynamics of the system around the onset of the instability. A very suitable
probe for the dynamics is provided by the two-point photon correlation
function:
$g^{(2)}(t)=\frac{\langle\hat{a}^{\dagger}_{\tau}\hat{a}^{\dagger}_{\tau+t}\hat{a}_{\tau+t}\hat{a}_{\tau}\rangle}{(\langle\hat{a}^{\dagger}_{\tau}\hat{a}_{\tau}\rangle)^{2}}.$
(3)
$\langle\cdots\rangle$ denotes the average over $\hat{\rho}$. Here we employ
the two-point correlator for the intra-cavity photon field, extractable from
our numerical simulations. However, we emphasize that it can be shown using
input-output theory (See Appendix A-1) that Eq. 3 also directly provides the
$g^{(2)}$ function for the fluctuations of the output optical field.
In steady state, $g^{(2)}$ does not depend on the initial time $\tau$. Photon
correlations are readily accessible in quantum optics experiments today with
single-photon detectors (_e.g._ using a Hanbury-Brown Twiss setup), and they
have been successfully employed to characterize the change of photonic
statistics upon transmission through nonlinear systems. The most important
example is photon anti-bunching in the resonance fluorescence of single photon
emitters, which has recently also been predicted to occur in optomechanical
systems for sufficiently strong coupling Rabl .
Figure 2: Time-dependence of photon-photon correlations near the regime of
quantum optomechanical oscillations. “A,B,C” labels the same states as in Fig.
1. These plots show that there is a remarkably slow long-term decay near the
onset of self-induced oscillations at point “B” (see main text). Inset also
shows the appearance of higher harmonics at point “C”. Figure 3: Quantifying
the slow approach of $g^{(2)}(t)\to 1$ near the onset of the self-induced
oscillations, as observed in Fig. 2. $g^{(2)}(t)-1$ obeys an exponential decay
$e^{-t/\tau_{g}}$ in the long-time limit $t\to\infty$. The inset shows the
decay time $\tau_{g}$ peaking toward very large values around point $B$,
_i.e._ $\Delta=0.6\omega_{M}$.
As can be seen in Fig. 2, there are clear signatures in the photon correlator
around the onset of parametric instability (point B). In particular,
$g^{(2)}(t)$ persists at some value above unity over a very long time (middle
panel, Fig. 2). It can be proven (see Appendix A-2) that as long as the steady
state of the system is not degenerate, we always have $g^{(2)}(t)\to
1+\alpha\exp{(-t/\tau_{g})}$ in the long-time limit $t\to\infty$. Here the
decay rate is $1/\tau_{g}=\mathrm{Re}{(\lambda_{1})}$, where $\lambda_{1}$ is
the eigenvalue of the Liouville operator $\mathcal{L}$ in Eq. 2 with the
largest non-zero real part, characterizing the slowest decay in the system.
This can be verified by plotting $\ln(g^{(2)}(t)-1)$ to extract $\tau_{g}$,
which indeed agrees with the $\lambda_{1}$ obtained from $\mathcal{L}$ (see
Fig. 3). As can be seen in the inset, $\tau_{g}$ rises strongly around the
start of the self-induced oscillation (point B). This is connected to the fact
that the overall mechanical damping rate goes to zero near the Hopf
bifurcation MarquardtPRL2006 .
The second signature in $g^{(2)}$ is the appearance of higher harmonics when
the self-induced oscillations are fully developed (see insets of Fig. 2). To
understand these in a semiclassical picture, we approximate the photon
correlator via the classical intensity correlator,
${\langle|\alpha(t+\tau)|^{2}|\alpha(\tau)|^{2}\rangle_{\tau}}$. The light
amplitude $\alpha(t)=e^{i\phi(t)}\sum\limits_{n}\alpha_{n}e^{in\omega_{M}t}$
is modulated harmonically by the mechanical oscillations, as detailed in
MarquardtPRL2006 . In Appendix A-3 we show that a fully developed mechanical
self-induced oscillation results in higher harmonics in $g^{(2)}$. To
understand the decay of the resulting oscillations in the $g^{(2)}$, we take
into account the mechanical phase diffusion induced by the radiation pressure
shot noise. VahalaPhaseNoise presented the first analysis of the quantum
contribution to phase diffusion in the parametric instability regime. Here we
follow a slightly modified approach. The phase fluctuates according to
$\delta{\phi}(t)=(m\omega_{M}A)^{-1}\int_{0}^{t}~{}dt^{\prime}~{}\delta
F(t^{\prime})~{}\cos(\omega_{M}t^{\prime})$, which yields:
$\mathrm{Var}(\delta\phi(t))=\frac{1}{(m\omega_{M}A)^{2}}\frac{t}{4}\left(S_{FF}(\omega_{M})+S_{FF}(-\omega_{M})\right),$
where $S_{FF}$ is the force noise spectrum (see MarquardtPRL2007 ). Thus:
${\left\langle|\alpha(t+\tau)|^{2}|\alpha(\tau)|^{2}\right\rangle_{\tau}}=\sum_{n=-\infty}^{+\infty}Z_{n}e^{in\omega_{M}t}e^{-n^{2}\left\langle\delta\phi(t)^{2}\right\rangle/2},$
where $Z_{n}=|\sum_{m=-\infty}^{\infty}\alpha_{m}\alpha^{*}_{m-n}|^{2}$. This
theory explains qualitatively the shape of the correlator even deep in the
quantum regime (Appendix A-4). Finally, we note that in the red detuned
regime, the photon correlator decay can be described by the optomechanical
cooling rate (see Appendix A-4).
To summarize, in this paper we investigated quantum signatures of light and
mechanics for an optomechanical system in the parametric instability regime.
We found that, at strong optomechanical coupling
($g_{0}\sim\kappa,g_{0}^{2}\sim(\kappa\cdot\omega_{M})$), for a range of
detuning and input power, the steady state mechanical Wigner density contains
strong negative parts, signaling stable non-classical states. Single-
quadrature homodyne measurements can be used to reconstruct the Wigner
density. In addition, the two-point photon correlator $g^{(2)}(t)$ displays
two clear signatures near the onset of parametric instability. Finally we
explained the slow long-time decay of the photon correlations as due to the
mechanical phase diffusion induced by photon shot noise. One should note that
experimental observation of some of these photon correlation features does not
require being in the nonlinear quantum regime and could succeed even in
existing setups.
F.M. acknowledges the DFG (Emmy-Noether) and an ERC Starting Grant. F.M. and
A.C. acknowledge the DARPA ORCHID program. J.Q. acknowledges the support of
DFG SFB 631 and NIM. K.H acknowledges the support through QUEST.
## Appendix A-1 A-1. Correlation Function: from Intra-cavity to Output Field
Here we summarize how the calculation of the $g^{(2)}$-function for the output
field $\hat{a}_{\mathrm{out}}$ can be traced back to calculating $g^{(2)}$ for
the intra-cavity field $\hat{a}$, following Gardiner . We are interested in
the normally ordered two-time correlation function of the output field
$g^{(2)}(t)=\frac{\langle\hat{a}^{\dagger}_{\mathrm{out}}(\tau)\hat{a}^{\dagger}_{\mathrm{out}}(t+\tau)\hat{a}_{\mathrm{out}}(t+\tau)\hat{a}_{\mathrm{out}}(\tau)\rangle}{\langle\hat{a}^{\dagger}_{\mathrm{out}}(\tau)\hat{a}_{\mathrm{out}}(\tau)\rangle^{2}}.$
(A-1)
We substitute the input-output relation
$\hat{a}_{\mathrm{out}}(t)=\hat{a}_{\mathrm{in}}(t)+\sqrt{\kappa}\hat{a}(t)$,
and use that $\hat{a}^{\dagger}(\tau)$ commutes with
$\hat{a}^{\dagger}_{\mathrm{in}}(t+\tau)$, and $\hat{a}(t+\tau)$ commutes with
$\hat{a}_{\mathrm{in}}(\tau)$ for $t\geq 0$ as a consequence of causality, see
Gardiner for details. This permits us to bring the two time correlation
function to a form where the $\hat{a}_{\mathrm{in}}^{\dagger}$ stand to the
left, and the $\hat{a}_{\mathrm{in}}$ to the right of all other operators.
Moreover, note that for vacuum input
$\hat{a}_{\mathrm{in}}\rho_{\mathrm{in}}=\rho_{\mathrm{in}}\hat{a}^{\dagger}_{\mathrm{in}}=0$.
This ultimately establishes the identity
$\displaystyle\langle\hat{a}^{\dagger}_{\mathrm{out}}(\tau)\hat{a}^{\dagger}_{\mathrm{out}}(t+\tau)\hat{a}_{\mathrm{out}}(t+\tau)\hat{a}_{\mathrm{out}}(\tau)\rangle$
$\displaystyle=\kappa^{2}\langle\hat{a}^{\dagger}(\tau)\hat{a}^{\dagger}(t+\tau)\hat{a}(t+\tau)\hat{a}(\tau)\rangle,$
(A-2)
such that the normalized correlation function for the output field is
_identical_ to the normalized correlation function of the intra-cavity field.
This is what we calculated in Eq. (3) of the main text.
## Appendix A-2 A-2. Proof Concerning the Longtime Limit of $g^{(2)}(t)$
In this section we give a proof that the $g^{(2)}(t)$ defined in Eq. 3 of the
main text, approaches one as $t\to\infty$.
We can rewrite the unnormalized correlation function (the numerator of Eq. 3
of the main text) as follows:
$\displaystyle g_{0}^{(2)}(t)$ $\displaystyle=$
$\displaystyle\mathrm{tr}[\hat{\rho}~{}\hat{a}^{\dagger}_{\tau}\hat{a}^{\dagger}_{t+\tau}\hat{a}_{t+\tau}\hat{a}_{\tau}]$
(A-3) $\displaystyle=$
$\displaystyle\mathrm{tr}[(\hat{a}_{\tau}\hat{\rho}\hat{a}^{\dagger}_{\tau})~{}\hat{a}^{\dagger}_{t+\tau}\hat{a}_{t+\tau}]$
$\displaystyle=$
$\displaystyle\mathrm{tr}[(\hat{a}_{\tau}\hat{\rho}\hat{a}^{\dagger}_{\tau})~{}e^{\frac{i\hat{H}t}{\hbar}}~{}\hat{a}^{\dagger}_{\tau}\hat{a}_{\tau}~{}e^{-\frac{i\hat{H}t}{\hbar}}]$
$\displaystyle=$
$\displaystyle\mathrm{tr}[\hat{a}^{\dagger}_{\tau}\hat{a}_{\tau}~{}e^{\frac{-i\hat{H}t}{\hbar}}(\hat{a}_{\tau}\hat{\rho}\hat{a}^{\dagger}_{\tau})e^{\frac{i\hat{H}t}{\hbar}}]$
$\displaystyle=$
$\displaystyle\mathrm{tr}[\hat{a}^{\dagger}_{\tau}\hat{a}_{\tau}~{}e^{\mathcal{L}t}\hat{\rho}^{\prime}].$
Here $\hat{\rho}^{\prime}=\hat{a}_{\tau}\hat{\rho}\hat{a}^{\dagger}_{\tau}$
and $e^{\mathcal{L}t}\hat{\rho}^{\prime}$ is its time evolution under the
quantum Liouville operator Eq. 2 of the main text. In the last step we use the
quantum regression approximation.
Let us now consider the right eigenvectors of $\mathcal{L}$:
$\mathcal{L}\hat{\rho}_{n}=\lambda_{n}\hat{\rho}_{n}.$ (A-4)
Here we rank $\hat{\rho}_{n}$ in descending order of $\rm{Re}~{}\lambda_{n}$.
Assuming the steady state $\lambda_{0}=0$ is not degenerate, we have
$\rm{Re}~{}\lambda_{n}<0$ for $n>1$. Since the trace is conserved in the time
evolution according to Eq. 2 of the main text, $\rm{tr}(\hat{\rho}_{n})=0$ for
$n>0$ and $\rm{tr}(\hat{\rho}_{0})=1$ (by normalization).
Expand the
$\hat{\rho}^{\prime}=\hat{a}_{\tau}\hat{\rho}\hat{a}^{\dagger}_{\tau}$ in
eigenvectors $\hat{\rho}_{n}$:
$\displaystyle\hat{\rho}^{\prime}$ $\displaystyle=$
$\displaystyle\sum_{n}c_{n}\hat{\rho}_{n},$ (A-5) $\displaystyle
e^{\mathcal{L}t}\hat{\rho}^{\prime}$ $\displaystyle=$
$\displaystyle\sum_{n}c_{n}\hat{\rho}_{n}~{}e^{\lambda_{n}}.$ (A-6)
we can then evaluate the correlator in Eq. A-3 as $t\to\infty$:
$\displaystyle g_{0}^{(2)}(t\to\infty)$ $\displaystyle=$
$\displaystyle\lim_{t\to\infty}\mathrm{tr}[\hat{a}^{\dagger}_{\tau}\hat{a}_{\tau}~{}e^{\mathcal{L}t}\hat{\rho}^{\prime}]$
(A-7) $\displaystyle=$
$\displaystyle\mathrm{tr}[\hat{a}^{\dagger}_{\tau}\hat{a}_{\tau}~{}c_{0}\hat{\rho}_{0}]$
$\displaystyle=$ $\displaystyle
c_{0}~{}\mathrm{tr}[\hat{a}^{\dagger}_{\tau}\hat{a}_{\tau}\hat{\rho}_{0}]$
$\displaystyle=$ $\displaystyle
c_{0}~{}\mathrm{tr}[\hat{a}^{\dagger}_{\tau}\hat{a}_{\tau}\hat{\rho}].$
In the last step we use the fact that at time $\tau$ the system is in a steady
state where the photon number no longer changes with time. Taking the trace of
both sides of Eq. A-5 and utilizing the properties of
$\rm{tr}(\hat{\rho}_{n})$ discussed above, we have:
$\displaystyle
c_{0}=\mathrm{tr}[\rho^{\prime}]=\mathrm{tr}[\hat{a}_{\tau}\hat{\rho}\hat{a}^{\dagger}_{\tau}]=\mathrm{tr}[\hat{a}^{\dagger}_{\tau}\hat{a}_{\tau}\hat{\rho}]=\langle\hat{a}^{\dagger}_{\tau}\hat{a}_{\tau}\rangle.$
(A-8)
Thus we arrive at:
$\displaystyle g^{(2)}(t\to\infty)$ $\displaystyle=$
$\displaystyle\frac{g_{0}^{(2)}(t\to\infty)}{(\langle\hat{a}^{\dagger}_{\tau}\hat{a}_{\tau}\rangle)^{2}}$
(A-9) $\displaystyle=$
$\displaystyle\frac{(\mathrm{tr}[\hat{a}^{\dagger}_{\tau}\hat{a}_{\tau}\hat{\rho}])^{2}}{(\langle\hat{a}^{\dagger}_{\tau}\hat{a}_{\tau}\rangle)^{2}}$
$\displaystyle=$ $\displaystyle 1.$
Finally, we point out that the leading term governing the asymptotic approach
of $g^{(2)}\to 1$ is $\lambda_{1}$ in Eq. A-4, since it has the slowest
exponential decay in Eq. A-6. This gives the asymptotic behavior of
$g^{(2)}(t)$ shown in the main text.
## Appendix A-3 A-3. Correlation Function: Semiclassical Picture
Under typical experimental conditions, when the classical self-induced
oscillation starts, the mechanical motion is to a good approximation harmonic
$x(t)\approx\bar{x}+A\cos(\omega_{M}t)$. The laser amplitude, influenced by
the mechanical oscillation, will contain higher harmonics MarquardtPRL2006
$\alpha(t)=e^{i\phi(t)}\sum\limits_{n}\alpha_{n}e^{in\omega_{M}t}$, where
$\alpha_{n}=\frac{\alpha_{L}J_{n}(-g_{0}A)}{-n\omega_{M}+(\Delta+g_{0}\bar{x})+i\kappa/2}$
(A-10)
and $\phi(t)=g_{0}A\sin(\omega_{M}t)$. Here we take the length unit to be the
mechanical zero point fluctuation $x_{\textrm{ZPF}}$ and frequency unit to be
$\omega_{M}$. $J_{n}(x)$ is the n-th order Bessel function. The oscillation
amplitude $A$ and equilibrium position $\bar{x}$ can be determined self-
consistently. We can express $g^{(2)}(t)$ in terms of the coefficients in Eq.
A-10:
$\displaystyle
Z_{n}=|\sum_{m=-\infty}^{\infty}\alpha_{m}\alpha^{*}_{m-n}|^{2}$ ,
$\displaystyle\langle|\alpha(\tau)|^{2}\rangle_{\tau}=\sum_{n=-\infty}^{\infty}|\alpha_{n}|^{2}=Z_{0}.$
$\displaystyle{\langle|\alpha(t+\tau)|^{2}|\alpha(\tau)|^{2}\rangle_{\tau}}$
$\displaystyle=$ $\displaystyle
Z_{0}+2\sum_{n=1}^{\infty}\cos(n\omega_{M}t)Z_{n}.$ (A-11)
In this paper we’re interested in the strongly quantum regime where
$g_{0}\approx\kappa$, thus in the sideband resolved regime we have
$g_{0}/\omega_{M}<1$. From Eq. A-10 we see that only when $A\gg
x_{\textrm{ZPF}}$ would there be significant contribution of higher harmonics
in the light amplitude $\alpha(t)$, which, as can be seen from Eq. A-3 is also
the condition of having higher harmonics in $g^{(2)}(t)$. This explains
qualitatively the appearance of higher harmonics in the insets of Fig. $2$ in
the main text when the quantum self-induced oscillation gains large amplitude.
## Appendix A-4 A-3. Correlation Function: Quantum Fluctuation
However, even when the self-induced oscillation has amplitude much larger than
the $x_{\textrm{ZPF}}$, there are important quantum effects that are not
accounted for by Eq. A-10 and Eq. A-3. As seen in Fig. A-1, the classical
solution (bottom) is fully periodic, as there is a balance between the optical
and mechanical dissipation and laser driving. However, over the period of 60
cycles, the amplitude of the quantum mechanical $g^{(2)}(t)$ decays
significantly (top three panels, blue curves). We can account for this decay
by calculating the effect of shot noise fluctuations in the radiation pressure
force $F(t)=(\hbar\omega_{R}/L)\hat{a}(t)^{\dagger}\hat{a}(t)$ on the phase
$\phi$ of the mechanical oscillations:
Figure A-1: Semiclassical approximation for photon correlations in the
parametric instability regime of optomechanics. By adding the mechanical
phase-diffusion to the classical light field dynamics (red), one can
qualitatively account for the slow long-time decay of the photon correlator
$g^{(2)}(t)$ in the full quantum simulation (blue). The lowest panel plots the
classical solution without phase diffusion (here $\Delta=\omega_{M}$). Figure
A-2: The decay of the correlation function $g^{(2)}(t)$ in the red-detuned
regime $\Delta=-0.8\omega_{M}$ can be qualitatively accounted for with
$\Gamma_{opt}$ obtained from optomechanical cooling.
$\displaystyle\delta{\phi}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{m\omega_{M}A}\int_{0}^{t}~{}dt^{\prime}~{}F(t^{\prime})~{}\cos(\phi(t^{\prime})),$
(A-12) $\displaystyle\mathrm{Var}(\delta\phi(t))$ $\displaystyle=$
$\displaystyle\frac{1}{(m\omega_{M}A)^{2}}\frac{t}{4}\left(S_{FF}(\omega_{M})+S_{FF}(-\omega_{M})\right).$
The noise spectrum of the radiation force is can be easily computed
MarquardtPRL2007 :
$S_{FF}(\omega)=\left(\frac{\hbar\omega_{R}}{L}\right)\bar{n}_{p}\frac{\kappa}{(\omega+\Delta)^{2}+(\kappa/2)^{2}}.$
(A-13)
The fluctuations of the mechanical oscillator’s phase feed back to the time-
dependence of optical amplitude. Thus, under the semi-classical assumption
where we take into account the photon shot noise but still treat the photon
amplitude classically, we can modify Eq. A-3 to be:
$\displaystyle{\langle|\alpha(t+\tau)|^{2}|\alpha(\tau)|^{2}\rangle_{\tau}}=\sum_{n=-\infty}^{\infty}Z_{n}~{}e^{in\omega_{M}t}\langle
e^{in(\delta\phi(t+\tau)-\delta\phi(t))}\rangle$
$\displaystyle=Z_{0}+2\sum_{n=1}^{\infty}Z_{n}\cos(n\omega_{M}t)e^{\frac{-n^{2}}{2}\mathrm{Var}(\delta\phi(t))}.$
Here we assume the phase fluctuation $\delta\phi$ is Gaussian. The result of
this semi-classical accounting of the photon shot noise can be seen in the red
curves in the top three panels of Fig. A-1. We can see that this simple
analysis can account qualitatively for the decay of the $g^{(2)}(t)$ in the
large amplitude self-induced oscillation regime.
Finally, as we see in Fig. A-2, there are also significant oscillation
structure and decay for the $g^{(2)}(t)$ in the so-called red detuned regime,
before the start of the self-induced oscillation. This regime cannot be
understood at all by the classical picture, since classically the system has
no dynamics there. The oscillation in $g^{(2)}(t)$ can be understood as the
dynamical response of the mechanical oscillator to the quantum fluctuation of
the photon field. The decay can then be modeled using the theory of the
optomechanical cooling of the mechanical oscillation in the red detuned
regime, giving a cooling rate
$\Gamma_{opt}=\frac{x_{ZPF}^{2}}{\hbar^{2}}\left[S_{FF}(\omega_{M})-S_{FF}(-\omega_{M})\right].$
(A-14)
Here $S_{FF}$ is the same noise spectrum given by Eq. A-13. As seen in Fig.
A-2, this rate gives a qualitative account for the decay rate of $g^{(2)}(t)$
in the red-detuned regime.
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|
arxiv-papers
| 2011-12-28T23:17:12 |
2024-09-04T02:49:25.746804
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jiang Qian, A. A. Clerk, K. Hammerer and Florian Marquardt",
"submitter": "Jiang Qian",
"url": "https://arxiv.org/abs/1112.6200"
}
|
1112.6275
|
This work is partially based on the paper <cit.>, which appeared in
In system design, model checking is a well-established formal method
that allows to automatically check for global system
correctness <cit.>.
In such a framework, in order to check whether a system satisfies a required
property, we describe its structure in a mathematical model (such as
Kripke structures <cit.> or labeled transition
systems <cit.>), specify the property with a formula of a temporal
logic (such as <cit.>, <cit.>, or <cit.>), and
check formally that the model satisfies the formula.
In the last decade, interest has arisen in analyzing the behavior of
individual components or sets of them in systems with several entities.
This interest has started in reactive systems, which are systems that interact
continually with their environments.
In module checking <cit.>, the system is modeled as a module that
interacts with its environment and correctness means that a desired property
holds with respect to all such interactions.
Starting from the study of module checking, researchers have looked for logics
focusing on the strategic behavior of agents in multi-agent
systems <cit.>.
One of the most important development in this field is Alternating-Time
Temporal Logic (, for short), introduced by Alur, Henzinger, and
Kupferman <cit.>.
allows reasoning about strategies of agents with temporal goals.
Formally, it is obtained as a generalization of in which the path
quantifiers, there exists “$\E$” and for all “$\A$”, are
replaced with strategic modalities of the form “$\EExs{\ASet}$” and
“$\AAll{\ASet}$”, where $\ASet$ is a set of agents (a.k.a.
Strategic modalities over agent sets are used to express cooperation and
competition among them in order to achieve certain goals.
In particular, these modalities express selective quantifications over those
paths that are the result of infinite games between a coalition and its
formulas are interpreted over concurrent game structures (,
for short) <cit.>, which model interacting processes.
Given a $\GName$ and a set $\ASet$ of agents, the formula
$\EExs{\ASet} \psi$ is satisfied at a state $\sElm$ of $\GName$ if there is a
set of strategies for agents in $\ASet$ such that, no matter strategies are
executed by agents not in $\ASet$, the resulting outcome of the interaction in
$\GName$ satisfies $\psi$ at $\sElm$.
Thus, can express properties related to the interaction among
components, while can only express property of the global system.
As an example, consider the property “processes $\alpha$ and $\beta$
cooperate to ensure that a system (having more than two processes) never
enters a failure state”.
This can be expressed by the formula $\EExs{\{ \alpha, \beta \}} \G
\neg \mathit{fail}$, where $\G$ is the classical temporal operators
, in contrast, cannot express this property <cit.>.
Indeed, it can only assert whether the set of all agents may or may not
prevent the system from entering a failure state.
The price that one has to pay for the greater expressiveness of is the
increased complexity of model checking.
Indeed, both its model-checking and satisfiability problems are
2 <cit.>.
Despite its powerful expressiveness, suffers from a strong limitation,
due to the fact that strategies are treated only implicitly, through
modalities that refer to games between competing coalitions.
To overcome this problem, Chatterjee, Henzinger, and Piterman introduced
Strategy Logic (, for short) <cit.>, a logic that treats
strategies in two-player turn-based games as explicit first-order
In , the formula $\EExs{\{ \alpha \}} \psi$, for a system modeled
by a with agents $\alpha$ and $\beta$, becomes $\exists \xSym. \forall
\ySym. \psi(\xSym, \ySym)$, i.e., “there exists a player-$\alpha$ strategy
$\xSym$ such that for all player-$\beta$ strategies $\ySym$, the unique
infinite path resulting from the two players following the strategies $\xSym$
and $\ySym$ satisfies the property $\psi$”.
The explicit treatment of strategies in this logic allows to state many
properties not expressible in .
In particular, it is shown in <cit.> that , in the restricted case
of two-agent turn-based games, corresponds to a proper one-alternation
fragment of .
The authors of that work have also shown that the model-checking problem for
is decidable, although only a non-elementary algorithm for it, both in
the size of system and formula, has been provided, leaving as open question
whether an algorithm with a better complexity exists or not.
The complementary question about the decidability of the satisfiability
problem for was also left open and, as far as we known, it is not
addressed in other papers apart our preliminary work <cit.>.
While the basic idea exploited in <cit.> to quantify over strategies and
then to commit agents explicitly to certain of these strategies turns to be
very powerful and useful <cit.>, still presents severe
Among the others, it needs to be extended to the more general concurrent
multi-agent setting.
Also, the specific syntax considered there allows only a weak kind of strategy
For example, does not allow different players to share the same
strategy, suggesting that strategies have yet to become first-class objects in
this logic.
Moreover, an agent cannot change his strategy during a play without forcing
the other to do the same.
These considerations, as well as all questions left open about decision
problems, led us to introduce and investigate a new Strategy
Logic, denoted , as a more general framework than , for explicit
reasoning about strategies in multi-agent concurrent games.
Syntactically, extends by means of two strategy quantifiers,
the existential $\EExs{\xElm}$ and the universal $\AAll{\xElm}$, as well as
agent binding $(\aElm, \xElm)$, where $\aElm$ is an agent and $\xElm$ a
Intuitively, these elements can be respectively read as “there exists a
strategy $\xElm$”, “for all strategies $\xElm$”, and “bind
agent $\aElm$ to the strategy associated with $\xElm$”.
For example, in a with the three agents $\alpha$, $\beta$, $\gamma$, the
previous formula $\EExs{\{ \alpha, \beta \}} \G \neg \mathit{fail}$ can
be translated in the formula $\EExs{\xSym} \EExs{\ySym} \AAll{\zSym}
(\alpha, \xSym) (\beta, \ySym) (\gamma, \zSym) (\G \neg \mathit{fail})$.
The variables $\xSym$ and $\ySym$ are used to select two strategies for the
agents $\alpha$ and $\beta$, respectively, while $\zSym$ is used to select one
for the agent $\gamma$ such that their composition, after the binding, results
in a play where $\mathit{fail}$ is never met.
Note that we can also require, by means of an appropriate choice of agent
bindings, that agents $\alpha$ and $\beta$ share the same strategy, using the
formula $\EExs{\xSym} \AAll{\zSym} (\alpha, \xSym) (\beta, \xSym) (\gamma,
\zSym) (\G \neg \mathit{fail})$.
Furthermore, we may vary the structure of the game by changing the way the
quantifiers alternate, as in the formula $\EExs{\xSym} \AAll{\zSym}
\EExs{\ySym} (\alpha, \xSym) (\beta, \ySym) (\alpha, \zSym) (\G \neg
\mathit{fail})$.
In this case, $\xSym$ remains uniform w.r.t. $\zSym$, but $\ySym$ becomes
dependent on it.
Finally, we can change the strategy that one agent uses during the play
without changing those of the other agents, by simply using nested bindings,
as in the formula $\EExs{\xSym} \EExs{\ySym} \AAll{\zSym} \EExs{\wSym}
(\alpha, \xSym) (\beta, \ySym) (\gamma, \zSym) (\G (\gamma, \wSym) \G \neg
\mathit{fail})$.
The last examples intuitively show that is a extension of both and
It is worth noting that the pattern of modal quantifications over strategies
and binding to agents can be extended to other linear-time temporal logics
than , such as the linear <cit.>.
In fact, the use of here is only a matter of simplicity in presenting
our framework, and changing the embedded temporal logic only involves few
side-changes in proofs and decision procedures.
As one of the main results in this paper about , we show that the
model-checking problem is non-elementarily decidable.
To gain this, we use an automata-theoretic approach <cit.>.
Precisely, we reduce the decision problem for our logic to the emptiness
problem of a suitable alternating parity tree automaton, which is an
alternating tree automaton (see <cit.>, for a survey) along with
a parity acceptance condition <cit.>.
Due to the operations of projection required by the elimination of
quantifications on strategies, which induce at any step an exponential
blow-up, the overall size of the required automaton is non-elementary in the
size of the formula, while it is only polynomial in the size of the model.
Thus, together with the complexity of the automata-nonemptiness calculation,
we obtain that the model checking problem is in , w.r.t. the size of the
model, and , w.r.t. the size of the specification.
Hence, the algorithm we propose is computationally not harder than the best
one known for and even a non-elementary improvement with respect to
the model.
This fact allows for practical applications of in the field of system
verification just as those done for the monadic second-order logic on infinite
objects <cit.>.
Moreover, we prove that our problem has a non-elementary lower bound.
Specifically, it is $k$- in the alternation number $k$ of
quantifications in the specification.
The contrast between the high complexity of the model-checking problem for our
logic and the elementary one for has spurred us to investigate
syntactic fragments of , strictly subsuming , with a better
In particular, by means of these sublogics, we would like to understand why
is computationally more difficult than .
The main fragments we study here are Nested-Goal, Boolean-Goal,
and One-Goal Strategy Logic, respectively denoted by , , and
Note that the last, differently from the first two, was introduced
in <cit.>.
They encompass formulas in a special prenex normal form having nested temporal
goals, Boolean combinations of goals, and a single goal at a time,
For goal we mean an formula of the type $\bpElm \psi$, where $\bpElm$ is
a binding prefix of the form $(\alpha_{1}, \xElm[1]), \ldots, (\alpha_{n},
\xElm[n])$ containing all the involved agents and $\psi$ is an agent-full
With more detail, the idea behind is that, when in $\psi$ there is a
quantification over a variable, then there are quantifications of all free
variables contained in the inner subformulas.
So, a subgoal of $\psi$ that has a variable quantified in $\psi$ itself cannot
use other variables quantified out of this formula.
Thus, goals can be only nested or combined with Boolean and temporal
and further restrict the use of goals.
In particular, in , each temporal formula $\psi$ is prefixed by a
quantification-binding prefix $\qpElm \bpElm$ that quantifies over a tuple of
strategies and binds them to all agents.
As main results about these fragments, we prove that the model-checking
problem for is 2, thus not harder than the one for .
On the contrary, for , it is both and and thus we
enforce the corresponding result for .
Finally, we observe that includes , while the relative
model-checking problem relies between 2 and .
To achieve all positive results about , we use a fundamental property of
the semantics of this logic, called elementariness, which allows us to
strongly simplify the reasoning about strategies by reducing it to a set of
reasonings about actions.
This intrinsic characteristic of , which unfortunately is not shared by
the other fragments, asserts that, in a determined history of the play, the
value of an existential quantified strategy depends only on the values of
strategies, from which the first depends, on the same history.
This means that, to choose an existential strategy, we do not need to know the
entire structure of universal strategies, as for , but only their values on
the histories of interest.
Technically, to describe this property, we make use of the machinery of
dependence map, which defines a Skolemization procedure for ,
inspired by the one in first order logic.
By means of elementariness, we can modify the model-checking procedure
via alternating tree automata in such a way that we avoid the projection
operations by using a dedicated automaton that makes an action quantification
for each node of the tree model.
Consequently, the resulting automaton is only exponential in the size of the
formula, independently from its alternation number.
Thus, together with the complexity of the automata-nonemptiness calculation,
we get that the model-checking procedure for is 2.
Clearly, the elementariness property also holds for , as it is included
in .
In particular, although it has not been explicitly stated, this property is
crucial for most of the results achieved in literature about by means
of automata (see <cit.>, as an example).
Moreover, we believe that our proof techniques are of independent interest and
applicable to other logics as well.
Related works.
Several works have focused on extensions of to incorporate more
powerful strategic constructs.
Among them, we recall Alternating-Time (, for
short) <cit.>, Game Logic (, for short) <cit.>,
Quantified Decision Modality (qD$\mu$, for
short) <cit.>, Coordination Logic (, for short) <cit.>,
and some extensions of considered in <cit.>.
and qD$\mu$ are intrinsically different from (as
well as from and ) as they are obtained by extending the
propositional $\mu$-calculus <cit.> with strategic modalities.
is similar to qD$\mu$ but with temporal operators
instead of explicit fixpoint constructors.
is strictly included in , in the case of two-player turn-based
games, but it does not use any explicit treatment of strategies, neither it
does the extensions of introduced in <cit.>.
In particular, the latter work consider restrictions on the memory for
strategy quantifiers.
Thus, all above logics are different from , which we recall it aims
to be a minimal but powerful logic to reason about strategic behavior in
multi-agent systems.
A very recent generalization of , which results to be expressive but
a proper sublogic of , is also proposed in <cit.>.
In this logic, a quantification over strategies does not reset the
strategies previously quantified but allows to maintain them in a particular
context in order to be reused.
This makes the logic much more expressive than .
On the other hand, as it does not allow agents to share the same strategy,
it is not comparable with the fragments we have considered in this paper.
Finally, we want to remark that our non-elementary hardness proof about the
model-checking problem is inspired by and improves a proof proposed for
their logic and communicated to us <cit.> by the authors
of <cit.>.
Note on <cit.>.
Preliminary results on appeared in <cit.>.
We presented there a 2 algorithm for the model-checking problem.
The described procedure applies only to the fragment, as model
checking for full is non-elementary.
The remaining part of this work is structured as follows.
In Section <ref>, we recall the semantic framework based on
concurrent game structures and introduce syntax and semantics of .
Then, in Section <ref>, we show the non-elementary lower bound
for the model-checking problem.
After this, in Section <ref>, we start the study of few syntactic
and semantic fragments and introduce the concepts of dependence map and
elementary satisfiability.
Finally, in Section <ref>, we describe the model-checking
automata-theoretic procedures for all fragments.
Note that, in the accompanying Appendix <ref>, we recall standard
mathematical notation and some basic definitions that are used in the paper.
However, for the sake of a simpler understanding of the technical part, we
make a reminder, by means of footnotes, for each first use of a non trivial
or immediate mathematical concept.
The paper is self contained.
All missing proofs in the main body of the work are reported in appendix.
Strategy Logic
In this section, we introduce Strategy Logic, an extension of the
classic linear-time temporal logic <cit.> along with the concepts
of strategy quantifications and agent binding.
Our aim is to define a formalism that allows to express strategic plans over
temporal goals in a way that separates the part related to the strategic
reasoning from that concerning the tactical one.
This distinctive feature is achieved by decoupling the instantiation of
strategies, done through the quantifications, from their application by means
of bindings.
Our proposal, on the line marked by its precursor <cit.>
and differently from classical temporal logics <cit.>, turns in a logic
that is not simply propositional but predicative, since we treat strategies as
a first order concept via the use of agents and variables as explicit
syntactic elements.
This fact let us to write Boolean combinations and nesting of complex
predicates, linked together by some common strategic choice, which may
represent each one a different temporal goal.
However, it is worth noting that the technical approach we follow here is
quite different from that used for the definition of , which is based,
on the syntactic side, on the formula framework <cit.> and,
The section is organized as follows.
In Subsection <ref>, we recall the definition of concurrent
game structure used to interpret Strategy Logic, whose syntax is introduced in
Subsection <ref>.
Then, in Subsection <ref>, we give, among the others, the
notions of strategy and play, which are finally used, in
Subsection <ref>, to define the semantics of the logic.
Underlying framework
As semantic framework for our logic language, we use a graph-based
model for multi-player games named concurrent game
structure <cit.>.
Intuitively, this mathematical formalism provides a generalization of
Kripke structures <cit.> and labeled transition
systems <cit.>, modeling multi-agent systems viewed as games,
in which players perform concurrent actions chosen strategically as a
function on the history of the play.
A concurrent game structure (, for short) is a tuple
$\GName \defeq \CGSStruct$, where $\APSet$ and $\AgSet$ are finite
non-empty sets of atomic propositions and agents, $\AcSet$
and $\StSet$ are enumerable non-empty sets of actions and
states, $\sElm[0] \in \StSet$ is a designated initial state,
and $\labFun : \StSet \to \pow{\APSet}$ is a labeling function that
maps each state to the set of atomic propositions true in that state.
Let $\DecSet \defeq \AcSet^{\AgSet}$ be the set of decisions, i.e.,
functions from $\AgSet$ to $\AcSet$ representing the choices of an action
for each agent. [In the following, we use both $\XSet \to \YSet$
and $\YSet^{\XSet}$ to denote the set of functions from the domain $\XSet$
to the codomain $\YSet$.]
Then, $\trnFun: \StSet \times \DecSet \to \StSet$ is a transition
function mapping a pair of a state and a decision to a state.
Observe that elements in $\StSet$ are not global states of the system, but
states of the environment in which the agents operate.
Thus, they can be viewed as states of the game, which do not include the
local states of the agents.
From a practical point of view, this means that all agents have perfect
information on the whole game, since local states are not taken into account
in the choice of actions <cit.>.
Observe also that, differently from other similar formalizations, each agent
has the same set of possible executable actions, independently of the
current state and of choices made by other agents.
However, as already reported in literature <cit.>, this simplifying
choice does not result in a limitation of our semantics framework and allow
us to give a simpler and clearer explanation of all formal definitions and
techniques we work on.
From now on, apart from the examples and if not differently stated, all s are defined on the same sets of atomic propositions $\APSet$ and agents
$\AgSet$, so, when we introduce a new structure in our reasonings, we do not
make explicit their definition anymore.
In addition, we use the italic letters $\pElm$, $\aElm$, $\cElm$, and
$\sElm$, possibly with indexes, as meta-variables on, respectively, the
atomic propositions $\pSym, \qSym, \ldots$ in $\APSet$, the agents $\alpha,
\beta, \gamma, \ldots$ in $\AgSet$, the actions $0, 1, \ldots$ in $\AcSet$,
and the states $\sSym, \ldots$ in $\StSet$.
Finally, we use the name of a as a subscript to extract the components
from its tuple-structure.
Accordingly, if $\GName = \CGSStruct$, we have that $\AcSet[\GName] =
\AcSet$, $\labFun[\GName] = \labFun$, $\sElm[0\GName] = \sSym[0]$, and so
Furthermore, we use the same notational concept to make explicit to which
the set $\DecSet$ of decisions is related to.
Note that,
Now, to get attitude to the introduced semantic framework, let us describe
two running examples of simple concurrent games.
In particular, we start by modeling the paper, rock, and scissor
[Paper, Rock, and Scissor]
Consider the classic two-player concurrent game paper, rock, and
scissor (PRS, for short) as represented in Figure fig:exm:prs,
where a play continues until one of the participants catches the move of
the other.
Vertexes are states of the game and labels on edges represent decisions of
agents or sets of them, where the symbol $*$ is used in place of every
possible action.
In this specific case, since there are only two agents, the pair of
symbols $**$ indicates the whole set $\DecSet$ of decisions.
The agents “Alice” and “Bob” in $\AgSet \defeq \{ \ASym, \BSym \}$
have as possible actions those in the set $\AcSet \defeq \{ \PSym, \RSym,
\SSym \}$, which stand for “paper”, “rock”, and “scissor”,
During the play, the game can stay in one of the three states in $\StSet
\defeq \{ \sSym[i], \sSym[\ASym], \sSym[\BSym] \}$, which represent,
respectively, the waiting moment, named idle, and the two winner
The latter ones are labeled with one of the atomic propositions in $\APSet
\defeq \{ \wSym[\ASym], \wSym[\BSym] \}$, in order to represent who is the
The catch of one action over another is described by the relation $\CRel
\defeq \{ (\PSym, \RSym), (\RSym, \SSym), (\SSym, \PSym) \} \subseteq
\AcSet \times \AcSet$.
We can now define the $\GName[P\!RS] \defeq \CGSStruct[ { \sSym[i] }
]$ for the PRS game, with the labeling given by $\labFun(\sSym[i]) \defeq
\emptyset$, $\labFun(\sSym[\ASym]) \defeq \{ \wSym[\ASym] \}$, and
$\labFun(\sSym[\BSym]) \defeq \{ \wSym[\BSym] \}$ and the transition
function set as follows, where $\DSet[\ASym] \defeq \set{ \decFun \in
\DecSet[ { \GName[P\!RS] } ] }{ (\decFun(\ASym), \decFun(\BSym)) \in \CRel
}$ and $\DSet[\BSym] \defeq \set{ \decFun \in \DecSet[ { \GName[P\!RS] } ]
}{ (\decFun(\BSym), \decFun(\ASym)) \in \CRel }$ are the sets of winning
decisions for the two agents: if $\sElm = \sSym[i]$ and $\decFun \in
\DSet[\ASym]$ then $\trnFun(\sElm, \decFun) \defeq \sSym[\ASym]$, else if
$\sElm = \sSym[i]$ and $\decFun \in \DSet[\BSym]$ then $\trnFun(\sElm,
\decFun) \defeq \sSym[\BSym]$, otherwise $\trnFun(\sElm, \decFun) \defeq
\sElm$.
Note that, when none of the two agents catches the action of the other,
i.e., the used decision is in $\DSet[i] \defeq \DecSet[ { \GName[P\!RS] }
] \setminus (\DSet[\ASym] \cup \DSet[\BSym])$, the play remains in the
idle state to allow another try, otherwise it is stuck in a winning
position forever.
We now describe a non-classic qualitative version of the well-known
prisoner's dilemma.
[Prisoner's Dilemma]
In the prisoner's dilemma (PD, for short), two accomplices are
interrogated in separated rooms by the police, which offers them the same
If one defects, i.e., testifies for the prosecution against the other,
while the other cooperates, i.e., remains silent, the defector goes free
and the silent accomplice goes to jail.
If both cooperate, they remain free, but will be surely interrogated in
the next future waiting for a defection.
On the other hand, if every one defects, both go to jail.
It is ensured that no one will know about the choice made by the other.
This tricky situation can be modeled by the $\GName[P\!D] \defeq
\CGSStruct[ {\sSym[i]} ]$ depicted in Figure fig:exm:pd, where the
agents “Accomplice-1” and “Accomplice-2” in $\AgSet \defeq \{
\ASym[1], \ASym[2] \}$ can chose an action in $\AcSet \defeq \{ \CSym,
\DSym \}$, which stand for “cooperation” and “defection”,
There are four states in $\StSet \defeq \{ \sSym[i], \sSym[ {\ASym[1]} ],
\sSym[ {\ASym[2]} ], \sSym[j] \}$.
In the idle state $\sSym[i]$ the agents are waiting for the interrogation,
while $\sSym[j]$ represents the jail for both of them.
The remaining states $\sSym[ {\ASym[1]} ]$ and $\sSym[ {\ASym[2]} ]$
indicate, instead, the situations in which only one of the agents become
definitely free.
To characterize the different meaning of these states, we use the atomic
propositions in $\APSet \defeq \{ \fSym[ {\ASym[1]} ], \fSym[ {\ASym[2]} ]
\}$, which denote who is “free”, by defining the following labeling:
$\labFun(\sSym[i]) \defeq \{ \fSym[ {\ASym[1]} ], \fSym[ {\ASym[2]} ] \}$,
$\labFun(\sSym[ {\ASym[1]} ]) \defeq \{ \fSym[ {\ASym[1]} ] \}$,
$\labFun(\sSym[ {\ASym[2]} ]) \defeq \{ \fSym[ {\ASym[2]} ] \}$, and
$\labFun(\sSym[j]) \defeq \emptyset$.
The transition function $\trnFun$ can be easily deduced by the figure.
Strategy Logic (, for short) syntactically extends by
means of two strategy quantifiers, the existential $\EExs{\xElm}$ and
the universal $\AAll{\xElm}$, and agent binding $(\aElm, \xElm)$,
where $\aElm$ is an agent and $\xElm$ a variable.
Intuitively, these new elements can be respectively read as “there
exists a strategy $\xElm$”, “for all strategies $\xElm$”, and
“bind agent $\aElm$ to the strategy associated with the variable
The formal syntax of follows.
formulas are built inductively from the sets of atomic
propositions $\APSet$, variables $\VarSet$, and agents $\AgSet$, by using
the following grammar, where $\pElm \in \APSet$, $\xElm \in \VarSet$, and
$\aElm \in \AgSet$:
$\varphi ::= \pElm \mid \neg \varphi \mid \varphi \wedge \varphi \mid
\varphi \vee \varphi \mid \X \varphi \mid \varphi \:\U \varphi \mid
\varphi \:\R \varphi \mid \EExs{\xElm} \varphi \mid \AAll{\xElm} \varphi
\mid (\aElm, \xElm) \varphi$.
denotes the infinite set of formulas generated by the above rules.
Observe that, by construction, is a proper syntactic fragment of ,
i.e., $\LTL \subset \SL$.
In order to abbreviate the writing of formulas, we use the boolean values
true $\Tt$ and false $\Ff$ and the well-known temporal operators future $\F
\varphi \defeq \Tt\: \U \varphi$ and globally $\G \varphi \defeq \Ff\: \R
\varphi$.
Moreover, we use the italic letters $\xElm, \yElm, \zElm, \ldots$, possibly
with indexes, as meta-variables on the variables $\xSym, \ySym, \zSym,
\ldots$ in $\VarSet$.
A first classic notation related to the syntax that we need to
introduce is that of subformula, i.e., a syntactic expression that is
part of an a priori given formula.
By $\mthfun{sub} : \SL \to \pow{\SL}$ we formally denote the function
returning the set of subformulas of an formula.
For instance, consider $\varphi = \EExs{\xSym} (\alpha, \xSym) (\F \pSym)$.
Then, it is immediate to see that $\sub{\varphi} = \{ \varphi, (\alpha,
\xSym) (\F \pSym), (\F \pSym), \pSym, \Tt \}$.
Normally, predicative logics need the concepts of free and bound
placeholders in order to formally define the meaning of their
The placeholders are used to represent particular positions in syntactic
expressions that have to be highlighted, since they have a crucial role in
the definition of the semantics.
In first order logic, for instance, there is only one type of placeholders,
which is represented by the variables.
In , instead, we have both agents and variables as placeholders, as it
can be noted by its syntax, in order to distinguish between the
quantification of a strategy and its application by an agent.
Consequently, we need a way to differentiate if an agent has an associated
strategy via a variable and if a variable is quantified.
To do this, we use the set of free agents/variables as the subset of
$\AgSet \cup \VarSet$ containing (i) all agents for which there is no
binding after the occurrence of a temporal operator and (ii) all
variables for which there is a binding but no quantifications.
The set of free agents/variables of an formula is given by the
function $\mthfun{free} : \SL \to \pow{\AgSet \cup \VarSet}$ defined as
$\free{\pElm} \defeq \emptyset$, where $\pElm \in \APSet$;
$\free{\neg \varphi} \defeq \free{\varphi}$;
$\free{\varphi_{1} \Opr \varphi_{2}} \defeq \free{\varphi_{1}} \cup
\free{\varphi_{2}}$, where $\Opr\! \in \{ \wedge, \vee \}$;
$\free{\X \varphi} \defeq \AgSet \cup \free{\varphi}$;
$\free{\varphi_{1} \Opr \varphi_{2}} \defeq \AgSet \cup
\free{\varphi_{1}} \cup \free{\varphi_{2}}$, where $\Opr\! \in \{
\U\!\!, \R\! \}$;
$\free{\Qnt \varphi} \defeq \free{\varphi} \setminus \{ \xElm \}$,
where $\Qnt\! \in \set{ \EExs{\xElm}, \AAll{\xElm} }{ \xElm \in
\VarSet }$;
$\free{(\aElm, \xElm) \varphi} \defeq \free{\varphi}$, if $\aElm
\not\in \free{\varphi}$, where $\aElm \in \AgSet$ and $\xElm \in
\VarSet$;
$\free{(\aElm, \xElm) \varphi} \defeq (\free{\varphi} \setminus \{
\aElm \}) \cup \{ \xElm \}$, if $\aElm \in \free{\varphi}$, where
$\aElm \in \AgSet$ and $\xElm \in \VarSet$.
A formula $\varphi$ without free agents (resp., variables), i.e., with
$\free{\varphi} \cap \AgSet = \emptyset$ (resp., $\free{\varphi} \cap
\VarSet = \emptyset$), is named agent-closed (resp.,
If $\varphi$ is both agent- and variable-closed, it is referred to as a
The function $\mthfun{snt} : \SL \to \pow{\SL}$ returns the set of
subsentences $\snt{\varphi} \defeq \set{ \phi \in \sub{\varphi} }{
\free{\phi} = \emptyset }$ for each formula $\varphi$.
Observe that, on one hand, free agents are introduced in
Items <ref> and <ref>
and removed in Item <ref>.
On the other hand, free variables are introduced in
Item <ref> and removed in
Item <ref>.
As an example, let $\varphi = \EExs{\xSym} (\alpha, \xSym) (\beta, \ySym)
(\F \pSym)$ be a formula on the agents $\AgSet = \{ \alpha, \beta, \gamma
\}$.
Then, we have $\free{\varphi} = \{ \gamma, \ySym \}$, since $\gamma$ is an
agent without any binding after $\F \pSym$ and $\ySym$ has no quantification
at all.
Consider also the formulas $(\alpha, \zSym) \varphi$ and $(\gamma, \zSym)
\varphi$, where the subformula $\varphi$ is the same as above.
Then, we have $\free{(\alpha, \zSym) \varphi} = \free{\varphi}$ and
$\free{(\gamma, \zSym) \varphi} = \{ \ySym, \zSym \}$, since $\alpha$ is not
free in $\varphi$ but $\gamma$ is, i.e., $\alpha \notin \free{\varphi}$ and
$\gamma \in \free{\varphi}$.
So, $(\gamma, \zSym) \varphi$ is agent-closed while $(\alpha, \zSym)
\varphi$ is not.
Similarly to the case of first order logic, another important concept that
characterizes the syntax of is that of the alternation number of
quantifiers, i.e., the maximum number of quantifier switches $\EExs{\cdot}
\AAll{\cdot}$, $\AAll{\cdot} \EExs{\cdot}$, $\EExs{\cdot} \neg
\EExs{\cdot}$, or $\AAll{\cdot} \neg \AAll{\cdot}$ that bind a variable in a
subformula that is not a sentence.
The constraint on the kind of subformulas that are considered here means
that, when we evaluate the number of such switches, we consider each
possible subsentence as an atomic proposition, hence, its quantifiers are
not taken into account.
Moreover, it is important to observe that vacuous quantifications, i.e.,
quantifications on variable that are not free in the immediate inner
subformula, need to be not considered at all in the counting of quantifier
This value is crucial when we want to analyze the complexity of the decision
problems of fragments of our logic, since higher alternation can usually
mean higher complexity.
By $\mthfun{alt} : \SL \to \SetN$ we formally denote the function returning
the alternation number of an formula.
Furthermore, the fragment $\SL[k-\text{alt}] \defeq \set{ \varphi \in \SL }{
\forall \varphi' \in \sub{\varphi} \:.\: \alt{\varphi'} \leq k }$ of ,
for $k \in \SetN$, denotes the subset of formulas having all subformulas
with alternation number bounded by $k$.
For instance, consider the sentence $\varphi = \AAll{\xSym} \EExs{\ySym}
(\alpha, \xSym) (\beta, \ySym) (\F \varphi')$ with $\varphi' = \AAll{\xSym}
\EExs{\ySym} (\alpha, \xSym) (\beta, \ySym) (\X \pSym)$, on the set of
agents $\AgSet = \{ \alpha, \beta \}$.
Then, the alternation number $\alt{\varphi}$ is $1$ and not $3$, as one can
think at a first glance, since $\varphi'$ is a sentence.
Moreover, it holds that $\alt{\varphi'} = 1$.
Hence, $\varphi \in \SL[1-\text{alt}]$.
On the other hand, if we substitute $\varphi'$ with $\varphi'' =
\AAll{\xSym} (\alpha, \xSym) (\X \pSym)$, we have that $\alt{\varphi} = 2$,
since $\varphi''$ is not a sentence.
Thus, it holds that $\varphi \not\in \SL[1-\text{alt}]$ but $\varphi \in
\SL[2-\text{alt}]$.
At this point, in order to practice with the syntax of our logic by
expressing game-theoretic concepts through formulas, we describe two
examples of important properties that are possible to write in , but
neither in <cit.> nor in .
This is clarified later in the paper.
The first concept we introduce is the well-known deterministic concurrent
multi-player Nash equilibrium for Boolean valued payoffs.
[Nash Equilibrium]
Consider the $n$ agents $\alpha_{1}, \ldots, \alpha_{n}$ of a game, each
of them having, respectively, a possibly different temporal goal described
by one of the formulas $\psi_{1}, \!\ldots\!, \psi_{n}$.
Then, we can express the existence of a strategy profile $(\xSym[1],
\ldots, \xSym[n])$ that is a Nash equilibrium (NE, for short) for
$\alpha_{1}, \ldots, \alpha_{n}$ w.r.t. $\psi_{1}, \ldots, \psi_{n}$ by
using the [$1$-alt] sentence $\varphi_{\!N\!\!E} \!\defeq\!
\EExs{\xSym[1]} (\alpha_{1}, \xSym[1]) \cdots \EExs{\xSym[n]} (\alpha_{n},
\xSym[n]) \: \psi_{\!N\!\!E}$, where $\psi_{\!N\!\!E} \!\defeq\!
\bigwedge_{i = 1}^{n} (\EExs{\ySym} (\alpha_{i}, \ySym) \psi_{i})
\rightarrow \psi_{i}$ is a variable-closed formula.
Informally, this asserts that every agent $\alpha_{i}$ has $\xSym[i]$ as
one of the best strategy w.r.t. the goal $\psi_{i}$, once all the other
strategies of the remaining agents $\alpha_{j}$, with $j \neq i$, have
been fixed to $\xSym[j]$.
Note that here we are only considering equilibria under deterministic
As in physics, also in game theory an equilibrium is not always stable.
Indeed, there are games like the PD of Example exm:pd having Nash
equilibria that are instable.
One of the simplest concepts of stability that is possible to think is
called stability profile.
[Stability Profile]
Think about the same situation of the above example on NE.
Then, a stability profile (SP, for short) is a strategy profile
$(\xSym[1], \ldots, \xSym[n])$ for $\alpha_{1}, \ldots, \alpha_{n}$
w.r.t. $\psi_{1}, \ldots, \psi_{n}$ such that there is no agent
$\alpha_{i}$ that can choose a different strategy from $\xSym[i]$ without
changing its own payoff and penalizing the payoff of another agent
$\alpha_{j}$, with $j \neq i$.
To represent the existence of such a profile, we can use the [$1$-alt]
sentence $\varphi_{\!S\!P} \defeq \EExs{\xSym[1]} (\alpha_{1}, \xSym[1])
\cdots \EExs{\xSym[n]} (\alpha_{n}, \xSym[n]) \: \psi_{\!S\!P}$, where
$\psi_{\!S\!P} \defeq \bigwedge_{i,j = 1, i \neq j}^{n} \psi_{j}
\rightarrow \AAll{\ySym} ((\psi_{i} \leftrightarrow (\alpha_{i}, \ySym)
\psi_{i}) \rightarrow (\alpha_{i}, \ySym) \psi_{j})$.
Informally, with the $\psi_{\!S\!P}$ subformula, we assert that, if
$\alpha_{j}$ is able to achieve his goal $\psi_{j}$, all strategies
$\ySym$ of $\alpha_{i}$ that left unchanged the payoff related to
$\psi_{i}$, also let $\alpha_{j}$ to maintain his achieved goal.
At this point, it is very easy to ensure the existence of an NE that is
also an SP, by using the [$1$-alt] sentence $\varphi_{\!S\!N\!\!E}
\defeq \EExs{\xSym[1]} (\alpha_{1}, \xSym[1]) \cdots \EExs{\xSym[n]}
(\alpha_{n}, \xSym[n]) \: \psi_{\!S\!P} \wedge \psi_{\!N\!\!E}$.
Basic concepts
Before continuing with the description of our logic, we have to introduce
some basic concepts, regarding a generic , that are at the base of the
semantics formalization.
Remind that a description of used mathematical notation is reported in
Appendix <ref>.
We start with the notions of track and path.
Intuitively, tracks and paths of a $\GName$ are legal sequences of
reachable states in $\GName$ that can be respectively seen as partial and
complete descriptions of possible outcomes of the game modeled by $\GName$
A track (resp., path) in a $\GName$ is a finite
(resp., an infinite) sequence of states $\trkElm \in \StSet^{*}$ (resp.,
$\pthElm \in \StSet^{\omega}$) such that, for all $i \in
\numco{0}{\card{\trkElm} - 1}$ (resp., $i \in \SetN$), there exists a
decision $\decFun \in \DecSet$ such that $(\trkElm)_{i + 1} =
\trnFun((\trkElm)_{i}, \decFun)$ (resp., $(\pthElm)_{i + 1} =
\trnFun((\pthElm)_{i}, \decFun)$). [The notation $(\wElm)_{i} \in
\Sigma$ indicates the element of index $i \in
\numco{0}{\card{\wElm}}$ of a non-empty sequence $\wElm \in
\Sigma^{\infty}$.]
A track $\trkElm$ is non-trivial if it has non-zero length, i.e.,
$\card{\trkElm} > 0$ that is $\trkElm \neq \epsilon$. [The
Greek letter $\epsilon$ stands for the empty sequence.]
The set $\TrkSet \subseteq \StSet^{+}$ (resp., $\PthSet \subseteq
\StSet^{\omega}$) contains all non-trivial tracks (resp., paths).
Moreover, $\TrkSet(\sElm) \defeq \set{ \trkElm \in \TrkSet }{
\fst{\trkElm} = \sElm }$ (resp., $\PthSet(\sElm) \defeq \set{ \pthElm \in
\PthSet }{ \fst{\pthElm} = \sElm }$) indicates the subsets of tracks
(resp., paths) starting at a state $\sElm \in \StSet$. [By
$\fst{\wElm} \defeq (\wElm)_{0}$ it is denoted the first element of
a non-empty sequence $\wElm \in \Sigma^{\infty}$.]
For instance, consider the PRS game of Example exm:prs.
Then, $\trkSym = \sSym[i] \cdot \sSym[\ASym] \in \StSet^{+}$ and $\pthSym =
\sSym[i]^{\omega} \in \StSet^{\omega}$ are, respectively, a track and a path
in the $\GName[P\!RS]$.
Moreover, it holds that $\TrkSet = \sSym[i]^{+} + \sSym[i]^{*} \cdot
(\sSym[\ASym]^{+} + \sSym[\BSym]^{+})$ and $\PthSet = \sSym[i]^{\omega} +
\sSym[i]^{*} \cdot (\sSym[\ASym]^{\omega} + \sSym[\BSym]^{\omega})$.
At this point, we can define the concept of strategy.
Intuitively, a strategy is a scheme for an agent that contains all choices
of actions as a function of the history of the current outcome.
However, observe that here we do not set an a priori connection between a
strategy and an agent, since the same strategy can be used by more than one
agent at the same time.
A strategy in a $\GName$ is a partial function $\strFun :
\TrkSet \pto \AcSet$ that maps each non-trivial track in its domain to an
For a state $\sElm \in \StSet$, a strategy $\strFun$ is said
$\sElm$-total if it is defined on all tracks starting in $\sElm$,
i.e., $\dom{\strFun} = \TrkSet(\sElm)$.
The set $\StrSet \defeq \TrkSet \pto \AcSet$ (resp., $\StrSet(\sElm)
\defeq \TrkSet(\sElm) \to \AcSet$) contains all (resp., $\sElm$-total)
An example of strategy in the $\GName[P\!RS]$ is the function
$\strFun[1] \in \StrSet(\sSym[i])$ that maps each track having length
multiple of $3$ to the action $\PSym$, the tracks whose remainder of length
modulo $3$ is $1$ to the action $\RSym$, and the remaining tracks to the
action $\SSym$.
A different strategy is given by the function $\strFun[2] \in
\StrSet(\sSym[i])$ that returns the action $\PSym$, if the tracks ends in
$\sSym[\ASym]$ or $\sSym[\BSym]$ or if its length is neither a second nor a
third power of a positive number, the action $\RSym$, if the length is a
square power, and the action $\SSym$, otherwise.
An important operation on strategies is that of translation along a
given track, which is used to determine which part of a strategy has yet to
be used in the game.
Let $\strFun \in \StrSet$ be a strategy and $\trkElm \in \dom{\strFun}$ a
track in its domain.
Then, $(\strFun)_{\trkElm} \in \StrSet$ denotes the translation of
$\strFun$ along $\trkElm$, i.e., the strategy with
$\dom{(\strFun)_{\trkElm}} \defeq \set{ \trkElm' \in
\TrkSet(\lst{\trkElm}) }{ \trkElm \cdot \trkElm'_{\geq 1} \in
\dom{\strFun} }$ such that $(\strFun)_{\trkElm}(\trkElm') \defeq
\strFun(\trkElm \cdot \trkElm'_{\geq 1})$, for all $\trkElm' \in
\dom{(\strFun)_{\trkElm}}$. [By $\lst{\wElm} \defeq
(\wElm)_{\card{\wElm} - 1}$ it is denoted the last element of a
finite non-empty sequence $\wElm \in \Sigma^{*}$.] [The notation
$(\wElm)_{\geq i} \in \Sigma^{\infty}$ indicates the suffix from
index $i \in \numcc{0}{\card{\wElm}}$ inwards of a non-empty sequence
$\wElm \in \Sigma^{\infty}$.]
Intuitively, the translation $(\strFun)_{\trkElm}$ is the update of the
strategy $\strFun$, once the history of the game becomes $\trkElm$.
It is important to observe that, if $\strFun$ is a $\fst{\trkElm}$-total
strategy then $(\strFun)_{\trkElm}$ is $\lst{\trkElm}$-total.
For instance, consider the two tracks $\trkElm[1] = \sSym[i]^{4} \in
\TrkSet(\sSym[i])$ and $\trkElm[2] = \sSym[i]^{4} \cdot \sSym[\ASym]^{2} \in
\TrkSet(\sSym[i])$ in the $\GName[P\!RS]$ and the strategy $\strFun[1]
\in \StrSet(\sSym[i])$ previously described.
Then, we have that $(\strFun[1])_{\trkElm[1]} = \strFun[1]$, while
$(\strFun[1])_{\trkElm[2]} \in \StrSet(\sSym[\ASym])$ maps each track having
length multiple of $3$ to the action $\SSym$, each track whose remainder of
length modulo $3$ is $1$ to the action $\PSym$, and the remaining tracks to
the action $\RSym$.
We now introduce the notion of assignment.
Intuitively, an assignment gives a valuation of variables with strategies,
where the latter are used to determine the behavior of agents in the game.
With more detail, as in the case of first order logic, we use this concept
as a technical tool to quantify over strategies associated with variables,
independently of agents to which they are related to.
So, assignments are
An assignment in a $\GName$ is a partial function $\asgFun :
\VarSet \cup \AgSet \pto \StrSet$ mapping variables and agents in its
domain to a strategy.
An assignment $\asgFun$ is complete if it is defined on all agents,
i.e., $\AgSet \subseteq \dom{\asgFun}$.
For a state $\sElm \in \StSet$, it is said that $\asgFun$ is
$\sElm$-total if all strategies $\asgFun(\lElm)$ are $\sElm$-total,
for $\lElm \in \dom{\asgFun}$.
The set $\AsgSet \defeq \VarSet \cup \AgSet \pto \StrSet$ (resp.,
$\AsgSet(\sElm) \defeq \VarSet \cup \AgSet \pto \StrSet(\sElm)$) contains
all (resp., $\sElm$-total) assignments.
Moreover, $\AsgSet(\XSet) \defeq \XSet \to \StrSet$ (resp.,
$\AsgSet(\XSet, \sElm) \defeq \XSet \to \StrSet(\sElm)$) indicates the
subset of $\XSet$-defined (resp., $\sElm$-total) assignments, i.e.,
(resp., $\sElm$-total) assignments defined on the set $\XSet \subseteq
\VarSet \cup \AgSet$.
As an example of assignment, let us consider the function $\asgFun[1]
\in \AsgSet$ in the $\GName[P\!RS]$, defined on the set $\{ \ASym,
\xSym \}$, whose values are $\strFun[1]$ on $\ASym$ and $\strFun[2]$ on
$\xSym$, where the strategies $\strFun[1], \strFun[2] \in \StrSet(\sSym[i])$
are those described above.
Another examples is given by the assignment $\asgFun[2] \in \AsgSet$,
defined on the set $\{ \ASym, \BSym \}$, such that $\asgFun[2](\ASym) =
\asgFun[1](\xSym)$ and $\asgFun[2](\BSym) = \asgFun[1](\ASym)$.
Note that both are $\sSym[i]$-total and the latter is also complete while
the former is not.
As in the case of strategies, it is useful to define the operation of
translation along a given track for assignments too.
For a given state $\sElm \in \StSet$, let $\asgFun \in \AsgSet(\sElm)$ be
an $\sElm$-total assignment and $\trkElm \in \TrkSet(\sElm)$ a track.
Then, $(\asgFun)_{\trkElm} \in \AsgSet(\lst{\trkElm})$ denotes the
translation of $\asgFun$ along $\trkElm$, i.e., the
$\lst{\trkElm}$-total assignment, with $\dom{(\asgFun)_{\trkElm}} \defeq
\dom{\asgFun}$, such that $(\asgFun)_{\trkElm}(\lElm) \defeq
(\asgFun(\lElm))_{\trkElm}$, for all $\lElm \in \dom{\asgFun}$.
Intuitively, the translation $(\asgFun)_{\trkElm}$ is the simultaneous
update of all strategies $\asgFun(\lElm)$ defined by the assignment
$\asgFun$, once the history of the game becomes $\trkElm$.
Given an assignment $\asgFun$, an agent or variable $\lElm$, and a strategy
$\strFun$, it is important to define a notation to represent the
redefinition of $\asgFun$, i.e., a new assignment equal to the first
on all elements of its domain but $\lElm$, on which it assumes the value
Let $\asgFun \in \AsgSet$ be an assignment, $\strFun \in \StrSet$ a
strategy and $\lElm \in \VarSet \cup \AgSet$ either an agent or a
Then, $\asgFun[][\lElm \mapsto \strFun] \in \AsgSet$ denotes the new
assignment defined on $\dom{\asgFun[][\lElm \mapsto \strFun]} \defeq
\dom{\asgFun} \cup \{ \lElm \}$ that returns $\strFun$ on $\lElm$ and is
equal to $\asgFun$ on the remaining part of its domain, i.e.,
$\asgFun[][\lElm \mapsto \strFun](\lElm) \defeq \strFun$ and
$\asgFun[][\lElm \mapsto \strFun](\lElm') \defeq \asgFun(\lElm')$, for all
$\lElm' \in \dom{\asgFun} \setminus \{ \lElm \}$.
Intuitively, if we have to add or update a strategy that needs to be bound
by an agent or variable, we can simply take the old assignment and redefine
it by using the above notation.
It is worth to observe that, if $\asgFun$ and $\strFun$ are $\sElm$-total
then $\asgFun[][\lElm \mapsto \strFun]$ is $\sElm$-total too.
Now, we can introduce the concept of play in a game.
Intuitively, a play is the unique outcome of the game determined by all
agent strategies participating to it.
A path $\playElm \in \PthSet(\sElm)$ starting at a state $\sElm \in
\StSet$ is a play w.r.t. a complete $\sElm$-total assignment
$\asgFun \in \AsgSet(\sElm)$ ($(\asgFun, \sElm)$-play, for short)
if, for all $i \in \SetN$, it holds that $(\playElm)_{i + 1} =
\trnFun((\playElm)_{i}, \decFun)$, where $\decFun(\aElm) \defeq
\asgFun(\aElm)((\playElm)_{\leq i})$, for each $\aElm \in
\AgSet$. [The notation $(\wElm)_{\leq i} \in \Sigma^{*}$
indicates the prefix up to index $i \in \numcc{0}{\card{\wElm}}$ of
a non-empty sequence $\wElm \in \Sigma^{\infty}$.]
The partial function $\playFun : \AsgSet \times \StSet \pto \PthSet$, with
$\dom{\playFun} \defeq \set{ (\asgFun, \sElm) }{ \AgSet \subseteq
\dom{\asgFun} \land \asgFun \in \AsgSet(\sElm) \land \sElm \in \StSet }$,
returns the $(\asgFun, \sElm)$-play $\playFun(\asgFun, \sElm) \in
\PthSet(\sElm)$, for all pairs $(\asgFun, \sElm)$ in its domain.
As a last example, consider again the complete $\sSym[i]$-total assignment
$\asgFun[2]$ previously described for the $\GName[P\!RS]$, which
returns the strategies $\strFun[2]$ and $\strFun[1]$ on the agents $\ASym$
and $\BSym$, respectively.
Then, we have that $\playFun(\asgFun[2], \sSym[i]) = \sSym[i]^{3} \cdot
\sSym[\BSym]^{\omega}$.
This means that the play is won by the agent $\BSym$.
Finally, we give the definition of global translation of a complete
assignment together with a related state, which is used to calculate, at a
certain step of the play, what is the current state and its updated
For a given state $\sElm \in \StSet$ and a complete $\sElm$-total
assignment $\asgFun \in \AsgSet(\sElm)$, the $i$-th global
translation of $(\asgFun, \sElm)$, with $i \in \SetN$, is the pair of a
complete assignment and a state $(\asgFun, \sElm)^{i} \defeq
((\asgFun)_{(\playElm)_{\leq i}}, (\playElm)_{i})$, where $\playElm =
\playFun(\asgFun, \sElm)$.
In order to avoid any ambiguity of interpretation of the described notions,
we may use the name of a as a subscript of the sets and functions just
introduced to clarify to which structure they are related to, as in the case
of components in the tuple-structure of the itself.
As already reported at the beginning of this section, just like and
differently from , the semantics of is defined w.r.t. concurrent
game structures.
For a $\GName$, one of its states $\sElm$, and an $\sElm$-total
assignment $\asgFun$ with $\free{\varphi} \subseteq \dom{\asgFun}$, we write
$\GName, \asgFun, \sElm \models \varphi$ to indicate that the formula
$\varphi$ holds at $\sElm$ in $\GName$ under $\asgFun$.
The semantics of formulas involving the atomic propositions, the
Boolean connectives $\neg$, $\wedge$, and $\vee$, as well as the temporal
operators $\X\!$, $\U\!$, and $\R\!$ is defined as usual in .
The novel part resides in the formalization of the meaning of strategy
quantifications $\EExs{\xElm}$ and $\AAll{\xElm}$ and agent binding $(\aElm,
\xElm)$.
Given a $\GName$, for all formulas $\varphi$, states $\sElm \in
\StSet$, and $\sElm$-total assignments $\asgFun \in \AsgSet(\sElm)$ with
$\free{\varphi} \subseteq \dom{\asgFun}$, the modeling relation $\GName,
\asgFun, \sElm \models \varphi$ is inductively defined as follows.
$\GName, \asgFun, \sElm \models \pElm$ if $\pElm \in \labFun(\sElm)$,
with $\pElm \in \APSet$.
For all formulas $\varphi$, $\varphi_{1}$, and $\varphi_{2}$, it holds
$\GName, \asgFun, \sElm \models \neg \varphi$ if not $\GName,
\asgFun, \sElm \models \varphi$, that is $\GName, \asgFun, \sElm
\not\models \varphi$;
$\GName, \asgFun, \sElm \models \varphi_{1} \wedge \varphi_{2}$ if
$\GName, \asgFun, \sElm \models \varphi_{1}$ and $\GName, \asgFun,
\sElm \models \varphi_{2}$;
$\GName, \asgFun, \sElm \models \varphi_{1} \vee \varphi_{2}$ if
$\GName, \asgFun, \sElm \models \varphi_{1}$ or $\GName, \asgFun,
\sElm \models \varphi_{2}$.
For a variable $\xElm \in \VarSet$ and a formula $\varphi$, it holds
$\GName, \asgFun, \sElm \models \EExs{\xElm} \varphi$ if there
exists an $\sElm$-total strategy $\strFun \in \StrSet(\sElm)$ such
that $\GName, \asgFun[][\xElm \mapsto \strFun], \sElm \models
\varphi$;
$\GName, \asgFun, \sElm \models \AAll{\xElm} \varphi$ if for all
$\sElm$-total strategies $\strFun \in \StrSet(\sElm)$ it holds
that $\GName, \asgFun[][\xElm \mapsto \strFun], \sElm \models
\varphi$.
For an agent $\aElm \in \AgSet$, a variable $\xElm \in \VarSet$, and a
formula $\varphi$, it holds that $\GName, \asgFun, \sElm \models
(\aElm, \xElm) \varphi$ if $\GName, \asgFun[][\aElm \mapsto
\asgFun(\xElm)], \sElm \models \varphi$.
Finally, if the assignment $\asgFun$ is also complete, for all
formulas $\varphi$, $\varphi_{1}$, and $\varphi_{2}$, it holds that:
$\GName, \asgFun, \sElm \models \X \varphi$ if $\GName, (\asgFun,
\sElm)^{1} \models \varphi$;
$\GName, \asgFun, \sElm \models \varphi_{1} \U \varphi_{2}$ if
there is an index $i \in \SetN$ with $k \leq i$ such that $\GName,
(\asgFun, \sElm)^{i} \models \varphi_{2}$ and, for all indexes $j
\in \SetN$ with $k \leq j < i$, it holds that $\GName, (\asgFun,
\sElm)^{j} \models \varphi_{1}$;
$\GName, \asgFun, \sElm \models \varphi_{1} \R \varphi_{2}$ if,
for all indexes $i \in \SetN$ with $k \leq i$, it holds that
$\GName, (\asgFun, \sElm)^{i} \models \varphi_{2}$ or there is an
index $j \in \SetN$ with $k \leq j < i$ such that $\GName,
(\asgFun, \sElm)^{j} \models \varphi_{1}$.
Intuitively, at Items <ref>
and <ref>, respectively, we evaluate the existential
$\EExs{\xElm}$ and universal $\AAll{\xElm}$ quantifiers over strategies, by
associating them to the variable $\xElm$.
Moreover, at Item <ref>, by means of an agent binding
$(\aElm, \xElm)$, we commit the agent $\aElm$ to a strategy associated with
the variable $\xElm$.
It is evident that, due to Items <ref>,
<ref>, and <ref>, the
semantics is simply embedded into the one.
In order to complete the description of the semantics, we now give the
classic notions of model and satisfiability of an sentence.
We say that a $\GName$ is a model of an sentence
$\varphi$, in symbols $\GName \models \varphi$, if $\GName, \emptyfun,
\sElm[0] \models \varphi$. [The symbol $\emptyfun$ stands for the
empty function.]
In general, we also say that $\GName$ is a model for $\varphi$ on
$\sElm \in \StSet$, in symbols $\GName, \sElm \models \varphi$, if
$\GName, \emptyfun, \sElm \models \varphi$.
An sentence $\varphi$ is satisfiable if there is a model for
It remains to introduce the concepts of implication and
equivalence between formulas, which are useful to describe
transformations preserving the meaning of a specification.
Given two formulas $\varphi_{1}$ and $\varphi_{2}$ with
$\free{\varphi_{1}} = \free{\varphi_{2}}$, we say that $\varphi_{1}$
implies $\varphi_{2}$, in symbols $\varphi_{1} \implies
\varphi_{2}$, if, for all s $\GName$, states $\sElm \in \StSet$, and
$\free{\varphi_{1}}$-defined $\sElm$-total assignments $\asgFun \in
\AsgSet(\free{\varphi_{1}}, \sElm)$, it holds that if $\GName, \asgFun,
\sElm \models \varphi_{1}$ then $\GName, \asgFun, \sElm \models
\varphi_{2}$.
Accordingly, we say that $\varphi_{1}$ is equivalent to
$\varphi_{2}$, in symbols $\varphi_{1} \equiv \varphi_{2}$, if both
$\varphi_{1} \implies \varphi_{2}$ and $\varphi_{2} \implies \varphi_{1}$
In the rest of the paper, especially when we describe a decision procedure,
we may consider formulas in existential normal form (, for
short) and positive normal form (, for short), i.e.,
formulas in which only existential quantifiers appear or in which the
negation is applied only to atomic propositions.
In fact, it is to this aim that we have considered in the syntax of both the Boolean connectives $\wedge$ and $\vee$, the temporal operators
$\U\!\!$, and $\R\!\!$, and the strategy quantifiers $\EExs{ \cdot }$ and
$\AAll{ \cdot }$.
Indeed, all formulas can be linearly translated in by using De
Morgan's laws together with the following equivalences, which directly
follow from the semantics of the logic: $\neg \X \varphi \equiv \X \neg
\varphi$, $\neg (\varphi_{1} \U \varphi_{2}) \equiv (\neg \varphi_{1}) \R
(\neg \varphi_{2})$, $\neg \EExs{x} \varphi \equiv \AAll{x} \neg \varphi$,
and $\neg (\aElm, \xElm) \varphi \equiv (\aElm, \xElm) \neg \varphi$.
At this point, in order to better understand the meaning of our logic, we
discuss two examples in which we describe the evaluation of the semantics of
some formula w.r.t. the a priori given s.
We start by explaining how a strategy can be shared by different agents.
[Shared Variable]
Consider the [$2$-alt] sentence $\varphi = \EExs{\xSym} \AAll{\ySym}
\EExs{\zSym} ((\alpha, \xSym) (\beta, \ySym) (\X \pSym) \wedge (\alpha,
\ySym) (\beta, \zSym) (\X \qSym))$.
It is immediate to note that both agents $\alpha$ and $\beta$ use the
strategy associated with $\ySym$ to achieve simultaneously the temporal goals $\X \pSym$ and $\X \qSym$.
A model for $\varphi$ is given by the $\GName[S\!V] \defeq \CGSTuple
{\{ \pSym, \qSym \}} {\{ \alpha, \beta\}} {\{ 0, 1 \}} {\{ \sSym[0],
\sSym[1], \sSym[2], \sSym[3] \}} {\labFun} {\trnFun} {\sSym[0]}$, where
$\labFun(\sSym[0]) \defeq \emptyset$, $\labFun(\sSym[1]) \defeq \{ \pSym
\}$, $\labFun(\sSym[2]) \defeq \{ \pSym, \qSym \}$, $\labFun(\sSym[3])
\defeq \{ \qSym \}$, $\trnFun(\sSym[0], (0, 0)) \defeq \sSym[1]$,
$\trnFun(\sSym[0], (0, 1)) \defeq \sSym[2]$, $\trnFun(\sSym[0], (1, 0))
\defeq \sSym[3]$, and all the remaining transitions (with any decision) go
to $\sSym[0]$.
In Figure fig:exm:sv, we report a graphical representation of the
Clearly, $\GName[S\!V] \models \varphi$ by letting, on $\sSym[0]$, the
variables $\xSym$ to chose action $0$ (the goal $(\alpha, \xSym) (\beta,
\ySym) (\X \pSym)$ is satisfied for any choice of $\ySym$, since we can
move from $\sSym[0]$ to either $\sSym[1]$ or $\sSym[2]$, both labeled with
$\pSym$) and $\zSym$ to choose action $1$ when $\ySym$ has action $0$ and,
vice versa, $0$ when $\ySym$ has $1$ (in both cases, the goal
$(\alpha, \ySym) (\beta, \zSym) (\X \qSym)$ is satisfied, since one can
move from $\sSym[0]$ to either $\sSym[2]$ or $\sSym[3]$, both labeled with
We now discuss an application of the concepts of Nash equilibrium and
stability profile to both the prisoner's dilemma and the paper, rock, and
scissor game.
[Equilibrium Profiles]
Let us first to consider the $\GName[P\!D]$ of the prisoner's
dilemma described in the Example exm:pd.
Intuitively, each of the two accomplices $\ASym[1]$ and $\ASym[2]$ want to
avoid the prison.
These goals can be, respectively, represented by the formulas
$\psi_{\ASym[1]} \defeq \G \fSym[ {\ASym[1]} ]$ and $\psi_{\ASym[2]}
\defeq \G \fSym[ {\ASym[2]} ]$.
The existence of a Nash equilibrium in $\GName[P\!D]$ for the two
accomplices w.r.t. the above goals can be written as $\phi_{\!N\!E}
\defeq \EExs{\xSym[1]} (\ASym[1], \xSym[1]) \EExs{\xSym[2]} (\ASym[2],
\xSym[2]) \: \psi_{\!N\!E}$, where $\psi_{\!N\!E} \defeq ((\EExs{\ySym}
(\ASym[1], \ySym) \psi_{\ASym[1]}) \rightarrow \psi_{\ASym[1]}) \wedge
((\EExs{\ySym} (\ASym[2], \ySym) \psi_{\ASym[2]}) \rightarrow
\psi_{\ASym[2]})$, which results to be an instantiation of the general
sentence $\varphi_{\!N\!E}$ of Example exm:ne.
In the same way, the existence of a stable Nash equilibrium can be
represented with the sentence $\phi_{\!S\!N\!\!E} \defeq \EExs{\xSym[1]}
(\ASym[1], \xSym[1]) \EExs{\xSym[2]} (\ASym[2], \xSym[2]) \: \psi_{\!N\!E}
\wedge \psi_{\!S\!P}$, where $\psi_{\!S\!P} \defeq (\psi_{1} \rightarrow
\AAll{\ySym} ((\psi_{2} \leftrightarrow (\ASym[2], \ySym) \psi_{2})
\rightarrow (\ASym[2], \ySym) \psi_{1})) \wedge (\psi_{2} \rightarrow
\AAll{\ySym} ((\psi_{1} \leftrightarrow (\ASym[1], \ySym) \psi_{1})
\rightarrow (\ASym[1], \ySym) \psi_{2}))$, which is a particular case of
the sentence $\varphi_{\!S\!N\!\!E}$ of Example exm:sp.
Now, it is easy to see that $\GName[P\!D] \models \phi_{\!S\!N\!E}$ and,
so, $\GName[P\!D] \models \phi_{\!N\!E}$.
Indeed, an assignment $\asgFun \in \AsgSet[ {\GName[P\!D]} ](\AgSet,
\sSym[i])$, for which $\asgFun(\ASym[1])(\sSym[i]) =
\asgFun(\ASym[2])(\sSym[i]) = \DSym$, is a stable equilibrium profile,
i.e., it is such that $\GName[P\!D], \asgFun, \sSym[i] \models
\psi_{\!N\!E} \wedge \psi_{\!S\!P}$.
This is due to the fact that, if an agent $\ASym[k]$, for $k \in \{ 1, 2
\}$, choses another strategy $\strFun \in \StrSet[ {\GName[P\!D]}
](\sSym[i])$, he is still unable to achieve his goal $\psi_{k}$, i.e.,
$\GName[P\!D], \asgFun[][\ASym[k] \mapsto \strFun], \sSym[i] \not\models
\psi_{k}$, so, he cannot improve his payoff.
Moreover, this equilibrium is stable, since the payoff of an agent cannot
be made worse by the changing of the strategy of the other agent.
However, it is interesting to note that there are instable equilibria too.
One of these is represented by the assignment $\asgFun' \in \AsgSet[
{\GName[P\!D]} ](\AgSet, \sSym[i])$, for which
$\asgFun'(\ASym[1])(\sSym[i]^{j}) = \asgFun'(\ASym[2])(\sSym[i]^{j}) =
\CSym$, for all $j \in \SetN$.
Indeed, we have that $\GName[P\!D], \asgFun', \sSym[i] \models
\psi_{\!N\!E}$, since $\GName[P\!D], \asgFun', \sSym[i] \models \psi_{1}$
and $\GName[P\!D], \asgFun', \sSym[i] \models \psi_{2}$, but
$\GName[P\!D], \asgFun', \sSym[i] \not\models \psi_{\!S\!P}$.
The latter property holds because, if one of the agents $\ASym[k]$, for $k
\in \{ 1, 2 \}$, choses a different strategy $\strFun' \in \StrSet[
{\GName[P\!D]} ](\sSym[i])$ for which there is a $j \in \SetN$ such that
$\strFun'(\sSym[i]^{j}) = \DSym$, he cannot improve his payoff but makes
surely worse the payoff of the other agent, i.e., $\GName[P\!D],
\asgFun'[\ASym[k] \mapsto \strFun'], \sSym[i] \models \psi_{k}$ but
$\GName[P\!D], \asgFun'[\ASym[k] \mapsto \strFun'], \sSym[i] \not\models
\psi_{3 - k}$.
Finally, consider the $\GName[P\!RS]$ of the paper, rock, and
scissor game described in the Example exm:prs together with the
associated formula for the Nash equilibrium $\phi_{\!N\!E} \defeq
\EExs{\xSym[1]} (\ASym, \xSym[1]) \EExs{\xSym[2]} (\BSym, \xSym[2]) \:
\psi_{\!N\!E}$, where $\psi_{\!N\!E} \defeq ((\EExs{\ySym} (\ASym, \ySym)
\psi_{\ASym}) \rightarrow \psi_{\ASym}) \wedge ((\EExs{\ySym} (\BSym,
\ySym) \psi_{\BSym}) \rightarrow \psi_{\BSym})$ with $\psi_{\ASym} \defeq
\F \wSym[\ASym]$ and $\psi_{\BSym} \defeq \F \wSym[\BSym]$ representing
the temporal goals for Alice and Bob, respectively.
Then, it is not hard to see that $\GName[P\!RS] \not\models
\phi_{\!N\!E}$, i.e., there are no Nash equilibria in this game, since
there is necessarily an agent that can improve his/her payoff by changing
his/her strategy.
Finally, we want to remark that our semantics framework, based on concurrent
game structures, is enough expressive to describe turn-based features in the
multi-agent case too.
This is possible by simply allowing the transition function to depend only
on the choice of actions of an a priori given agent for each state.
A $\GName$ is turn-based if there exists a function $\ownFun
: \StSet \to \AgSet$, named owner function, such that, for all
states $\sElm \in \StSet$ and decisions $\decFun[1], \decFun[2] \in
\DecSet$, it holds that if $\decFun[1](\ownFun(\sElm)) =
\decFun[2](\ownFun(\sElm))$ then $\trnFun(\sElm, \decFun[1]) =
\trnFun(\sElm, \decFun[2])$.
Intuitively, a is turn-based if it is possible to associate with each
state an agent, i.e., the owner of the state, which is responsible for the
choice of the successor of that state.
It is immediate to observe that $\ownFun$ introduces a partitioning of the
set of states into $\card{\rng{\ownFun}}$ components, each one ruled by a
single agent.
Moreover, observe that a having just one agent is trivially
turn-based, since this agent is the only possible owner of all states.
In the following, as one can expect, we also consider the case in which has its semantics defined on turn-based only.
In such an eventuality, we name the resulting semantic fragment
Turn-based Strategy Logic (, for short) and refer to the related
satisfiability concept as turn-based satisfiability.
Model-Checking Hardness
In this section, we show the non-elementary lower bound for the model-checking
problem of .
Precisely, we prove that, for sentences having alternation number $k$, this
problem is $k$-.
To this aim, in Subsection <ref>, we first recall syntax and
semantics of <cit.>.
Then, in Subsection <ref>, we give a reduction from the
satisfiability problem for this logic to the model-checking problem for .
Quantified propositional temporal logic
Quantified Propositional Temporal Logic (, for short)
syntactically extends the old-style temporal logic with the future
$\F\!$ and global $\G\!$ operators by means of two proposition
quantifiers, the existential $\exists \qElm .$ and the universal $\forall
\qElm .$, where $\qElm$ is an atomic proposition.
Intuitively, these elements can be respectively read as “there exists
an evaluation of $\qElm$” and “for all evaluations of $\qElm$”.
The formal syntax of follows.
formulas are built inductively from the sets of atomic
propositions $\APSet$, by using the following grammar, where $\pElm \in
\APSet$:
$\varphi ::= \pElm \mid \neg \varphi \mid \varphi \wedge \varphi \mid
\varphi \vee \varphi \mid \X \varphi \mid \F \varphi \mid \G \varphi
\mid \exists \pElm . \varphi \mid \forall \pElm . \varphi$.
denotes the infinite set of formulas generated by the above
Similarly to , we use the concepts of subformula, free atomic
proposition, sentence, and alternation number, together with the syntactic fragment of bounded alternation [$k$-alt], with $k \in
\SetN$.
In order to define the semantics of , we have first to introduce the
concepts of truth evaluations used to interpret the meaning of atomic
propositions at the passing of time.
A temporal truth evaluation is a function $\tteFun : \SetN \to \{
\Ff, \Tt \}$ that maps each natural number to a Boolean value.
Moreover, a propositional truth evaluation is a partial function
$\pteFun : \APSet \pto \TTESet$ mapping every atomic proposition in its
domain to a temporal truth evaluation.
The sets $\TTESet \defeq \SetN \to \{ \Ff, \Tt \}$ and $\PTESet \defeq
\APSet \pto \TTESet$ contain, respectively, all temporal and propositional
truth evaluations.
At this point, we have the tool to define the interpretation of formulas.
For a propositional truth evaluation $\pteFun$ with $\free{\varphi}
\subseteq \dom{\pteFun}$ and a number $k$, we write $\pteFun, k \models
\varphi$ to indicate that the formula $\varphi$ holds at the $k$-th position
of the $\pteFun$.
For all formulas $\varphi$, propositional truth evaluation $\pteFun
\in \PTESet$ with $\free{\varphi} \subseteq \dom{\pteFun}$, and numbers $k
\in \SetN$, the modeling relation $\pteFun, k \models \varphi$ is
inductively defined as follows.
$\pteFun, k \models \pElm$ iff $\pteFun(\pElm)(k) = \Tt$, with
$\pElm \in \APSet$.
For all formulas $\varphi$, $\varphi_{1}$, and $\varphi_{2}$, it holds
$\pteFun, k \models \neg \varphi$ iff not $\pteFun, k \models
\varphi$, that is $\pteFun, k \not\models \varphi$;
$\pteFun, k \models \varphi_{1} \wedge \varphi_{2}$ iff
$\pteFun, k \models \varphi_{1}$ and $\pteFun, k \models
\varphi_{2}$;
$\pteFun, k \models \varphi_{1} \vee \varphi_{2}$ iff $\pteFun,
k \models \varphi_{1}$ or $\pteFun, k \models \varphi_{2}$;
$\pteFun, k \models \X \varphi$ iff $\pteFun, k + 1 \models
\varphi$;
$\pteFun, k \models \F \varphi$ iff there is an index $i \in
\SetN$ with $k \leq i$ such that $\pteFun, i \models \varphi$;
$\pteFun, k \models \G \varphi$ iff, for all indexes $i \in \SetN$
with $k \leq i$, it holds that $\pteFun, i \models \varphi$.
For an atomic proposition $\qElm \in \APSet$ and a formula $\varphi$,
it holds that:
$\pteFun, k \models \exists \qElm . \varphi$ iff there exists a
temporal truth evaluation $\tteFun \in \TTESet$ such that
$\pteFun[][\qElm \mapsto \tteFun], k \models \varphi$;
$\pteFun, k \models \forall \qElm . \varphi$ iff for all temporal
truth evaluations $\tteFun \in \TTESet$ it holds that
$\pteFun[][\qElm \mapsto \tteFun], k \models \varphi$.
Obviously, a sentence $\varphi$ is satisfiable if $\emptyfun,
0 \models \varphi$.
Observe that the described semantics is slightly different but completely
equivalent to that proposed and used in <cit.> to prove the
non-elementary hardness result for the satisfiability problem.
Non-elementary lower-bound
We can show how the solution of satisfiability problem can be reduced
to that of the model-checking problem for , over a turn-based constant
size with a unique atomic proposition.
In order to do this, we first prove the following auxiliary lemma, which
actually represents the main step of the above mentioned reduction.
There is a one-agent $\GName[Rdc]$ such that, for each
[$k$-alt] sentence $\varphi$, with $k \in \SetN$, there exists an
[$k$-alt] variable-closed formula $\trn{\varphi}$ such that $\varphi$
is satisfiable iff $\GName[Rdc], \asgFun, \sElm[0] \models \trn{\varphi}$,
for all complete assignments $\asgFun \in \AsgSet(\AgSet, \sElm[0])$.
Consider the one-agent $\GName[Rdc] \defeq \CGSTuple { \{ \pSym \} }
{ \{ \alpha \} } { \{ \Ff, \Tt \} } { \{ \sSym[0], \sSym[1] \} } {\labFun}
{\trnFun} {\sSym[0]}$ depicted in Figure fig:lmm:qptl(rdc), where
the two actions are the Boolean values false and true and where the
labeling and transition functions $\labFun$ and $\trnFun$ are set as
follows: $\labFun(\sSym[0]) \defeq \emptyset$, $\labFun(\sSym[1]) \defeq
\{ \pSym \}$, and $\trnFun(\sElm, \decFun) = \sSym[0]$ iff
$\decFun(\alpha) = \Ff$, for all $\sElm \in \StSet$ and $\decFun \in
\DecSet$.
It is evident that $\GName[Rdc]$ is a turn-based .
Moreover, consider the transformation function $\trn{\cdot} : \QPTL \to
\SL$ inductively defined as follows:
* $\trn{\qElm} \defeq (\alpha, \xSym[\qElm]) \X \pSym$, for $\qElm \in
\APSet$;
* $\trn{\exists \qElm . \varphi} \defeq \EExs{\xSym[\qElm]}
\trn{\varphi}$;
* $\trn{\forall \qElm . \varphi} \defeq \AAll{\xSym[\qElm]}
\trn{\varphi}$;
* $\trn{\Opr \varphi} \defeq \Opr \trn{\varphi}$, where $\Opr \in \{
\neg, \X\!, \F\!, \G\! \}$;
* $\trn{\varphi_{1} \Opr \varphi_{2}} \defeq \trn{\varphi_{1}} \Opr
\trn{\varphi_{2}}$, where $\Opr \in \{ \wedge, \vee \}$.
It is not hard to see that a formula $\varphi$ is a sentence iff
$\trn{\varphi}$ is variable-closed.
Furthermore, we have that $\alt{\trn{\varphi}} = \alt{\varphi}$.
At this point, it remains to prove that, a sentence $\varphi$ is
satisfiable iff $\GName[Rdc], \asgFun, \sSym[0] \! \models \trn{\varphi}$,
for all total assignments $\asgFun \in \AsgSet(\{ \alpha \}, \sSym[0])$.
To do this by induction on the structure of $\varphi$, we actually show a
stronger result asserting that, for all subformulas $\psi \in
\sub{\varphi}$, propositional truth evaluations $\pteFun \in \PTESet$, and
$i \in \SetN$, it holds that $\pteFun, i \models \psi$ iff $\GName[Rdc],
(\asgFun, \sSym[0])^{i} \models \trn{\psi}$, for each total assignment
$\asgFun \in \AsgSet(\{ \alpha \} \cup \set{ \xSym[\qElm] \in \VarSet }{
\qElm \in \free{\psi} }, \sSym[0])$ such that
$\asgFun(\xSym[\qElm])((\playElm)_{\leq n}) = \pteFun(\qElm)(n)$, where
$\playElm \defeq \playFun(\asgFun, \sSym[0])$, for all $\qElm \in
\free{\psi}$ and $n \in \numco{i}{\omega}$.
Here, we only show the base case of atomic propositions and the two
inductive cases regarding the proposition quantifiers.
The remaining cases of Boolean connectives and temporal operators are
straightforward and left to the reader as a simple exercise.
* $\psi = \qElm$.
By Item <ref> of
Definition <ref> of semantics, we have that
$\pteFun, i \models \qElm$ iff $\pteFun(\qElm)(i) = \Tt$.
Thus, due to the above constraint on the assignment, it follows that
$\pteFun, i \models \qElm$ iff $\asgFun(\xSym[\qElm])((\playElm)_{\leq
i}) = \Tt$.
Now, by applying Items <ref>
and <ref> of Definition <ref>
of semantics, we have that $\GName[Rdc], (\asgFun, \sSym[0])^{i}
\models (\alpha, \xSym[\qElm]) \X \pSym$ iff $\GName[Rdc],
(\asgFun'[\alpha \mapsto \asgFun'(\xSym[\qElm])], \sElm')^{1} \models
\pSym$, where $(\asgFun', \sElm') = (\asgFun, \sSym[0])^{i}$.
At this point, due to the particular structure of the $\GName[Rdc]$, we have that $\GName[Rdc], (\asgFun'[\alpha \mapsto
\asgFun'(\xSym[\qElm])], \sElm')^{1} \models \pSym$ iff
$(\playElm')_{1} = \sSym[1]$, where $\playElm' \defeq
\playFun(\asgFun'[\alpha \mapsto \asgFun'(\xSym[\qElm])], \sElm')$,
which in turn is equivalent to
$\asgFun'(\xSym[\qElm])((\playElm')_{\leq 0}) = \Tt$.
So, $\GName[Rdc], (\asgFun, \sSym[0])^{i} \models (\alpha,
\xSym[\qElm]) \X \pSym$ iff $\asgFun'(\xSym[\qElm])((\playElm')_{\leq
0}) = \Tt$.
Now, by observing that $(\playElm')_{\leq 0} = (\playElm)_{i}$ and
using the above definition of $\asgFun'$, we obtain that
$\asgFun'(\xSym[\qElm])((\playElm')_{\leq 0}) =
\asgFun(\xSym[\qElm])((\playElm)_{\leq i})$.
Hence, $\pteFun, i \models \qElm$ iff $\pteFun(\qElm)(i) =
\asgFun(\xSym[\qElm])((\playElm)_{\leq i}) = \Tt =
\asgFun'(\xSym[\qElm])((\playElm')_{\leq 0})$ iff $\GName[Rdc],
(\asgFun, \sSym[0])^{i} \models (\alpha, \xSym[\qElm]) \X \pSym$.
* $\psi = \exists \qElm. \psi'$.
[Only if].
If $\pteFun, i \models \exists \qElm. \psi'$, by
Item <ref> of
Definition <ref>, there exists a temporal truth
evaluation $\tteFun \in \TTESet$ such that $\pteFun[][\qElm \mapsto
\tteFun], i \models \psi'$.
Now, consider a strategy $\strFun \in \StrSet(\sSym[0])$ such that
$\strFun((\playElm)_{\leq n}) = \tteFun(n)$, for all $n \in
\numco{i}{\omega}$.
Then, it is evident that $\asgFun[][\xSym[\qElm] \mapsto
\strFun](\xSym[\qElm'])((\playElm)_{\leq n}) = \pteFun[][\qElm \mapsto
\tteFun](\qElm')(n)$, for all $\qElm' \in \free{\psi}$ and $n \in
\numco{i}{\omega}$.
So, by the inductive hypothesis, it follows that $\GName[Rdc],
(\asgFun[][\xSym[\qElm] \mapsto \strFun], \sSym[0])^{i} \models
\trn{\psi'}$.
Thus, we have that $\GName[Rdc], (\asgFun, \sSym[0])^{i} \models
\EExs{\xSym[\qElm]} \trn{\psi'}$.
If $\GName[Rdc], (\asgFun, \sSym[0])^{i} \models \EExs{\xSym[\qElm]}
\trn{\psi'}$, there exists a strategy $\strFun \in \StrSet(\sSym[0])$
such that $\GName[Rdc], (\asgFun[][\xSym[\qElm] \mapsto \strFun],
\sSym[0])^{i} \models \trn{\psi'}$.
Now, consider a temporal truth evaluation $\tteFun \in \TTESet$ such
that $\tteFun(n) = \strFun((\playElm)_{\leq n})$, for all $n \in
\numco{i}{\omega}$.
Then, it is evident that $\asgFun[][\xSym[\qElm] \mapsto
\strFun](\xSym[\qElm'])((\playElm)_{\leq n}) = \pteFun[][\qElm \mapsto
\tteFun](\qElm')(n)$, for all $\qElm' \in \free{\psi}$ and $n \in
\numco{i}{\omega}$.
So, by the inductive hypothesis, it follows that $\pteFun[][\qElm
\mapsto \tteFun], i \models \psi'$.
Thus, by Item <ref> of
Definition <ref>, we have that $\pteFun, i \models
\exists \qElm. \psi'$.
* $\psi = \forall \qElm. \psi'$.
[Only if].
For each strategy $\strFun \in \StrSet(\sSym[0])$, consider a temporal
truth evaluation $\tteFun \in \TTESet$ such that $\tteFun(n) =
\strFun((\playElm)_{\leq n})$, for all $n \in \numco{i}{\omega}$.
It is evident that $\asgFun[][\xSym[\qElm] \mapsto
\strFun](\xSym[\qElm'])((\playElm)_{\leq n}) = \pteFun[][\qElm \mapsto
\tteFun](\qElm')(n)$, for all $\qElm' \in \free{\psi}$ and $n \in
\numco{i}{\omega}$.
Now, since $\pteFun, i \models \forall \qElm. \psi'$, by
Item <ref> of
Definition <ref>, it follows that $\pteFun[][\qElm
\mapsto \tteFun], i \models \psi'$.
So, by the inductive hypothesis, for each strategy $\strFun \in
\StrSet(\sSym[0])$, it holds that $\GName[Rdc],
(\asgFun[][\xSym[\qElm] \mapsto \strFun], \sSym[0])^{i} \models
\trn{\psi'}$.
Thus, we have that $\GName[Rdc], (\asgFun, \sSym[0])^{i} \models
\AAll{\xSym[\qElm]} \trn{\psi'}$.
For each temporal truth evaluation $\tteFun \in \TTESet$, consider a
strategy $\strFun \in \StrSet(\sSym[0])$ such that
$\strFun((\playElm)_{\leq n}) = \tteFun(n)$, for all $n \in
\numco{i}{\omega}$.
It is evident that $\asgFun[][\xSym[\qElm] \mapsto
\strFun](\xSym[\qElm'])((\playElm)_{\leq n}) =
\pteFun[][\qElm \mapsto \tteFun](\qElm')(n)$, for all $\qElm' \in
\free{\psi}$ and $n \in \numco{i}{\omega}$.
Now, since $\GName[Rdc], (\asgFun, \sSym[0])^{i} \models
\AAll{\xSym[\qElm]} \trn{\psi'}$, it follows that $\GName[Rdc],
(\asgFun[][\xSym[\qElm] \mapsto \strFun], \sSym[0])^{i} \models
\trn{\psi'}$.
So, by the inductive hypothesis, for each temporal truth evaluation
$\tteFun \in \TTESet$, it holds that $\pteFun[][\qElm \mapsto
\tteFun], i \models \psi'$.
Thus, by Item <ref> of
Definition <ref>, we have that $\pteFun, i \models
\forall \qElm. \psi'$.
Thus, we are done with the proof.
Now, we can show the full reduction that allows us to state the existence of
a non-elementary lower-bound for the model-checking problem of and,
thus, of .
The model-checking problem for [$k$-alt] is $k$-.
Let $\varphi$ be a [$k$-alt] sentence, $\trn{\varphi}$ the related
[$k$-alt] variable-closed formula, and $\GName[Rdc]$ the turn-based
of Lemma <ref> of reduction.
Then, by applying the previous mentioned lemma, it is easy to see that
$\varphi$ is satisfiable iff $\GName[Rdc] \models \AAll{\xSym} (\alpha,
\xSym) \trn{\varphi}$ iff $\GName[Rdc] \models \EExs{\xSym} (\alpha,
\xSym) \trn{\varphi}$.
Thus, the satisfiability problem for can be reduced to the
model-checking problem for .
Now, since the satisfiability problem for [$k$-alt] is
$k$- <cit.>, we have that the model-checking problem for
[$k$-alt] is $k$- as well.
The following corollary is an immediate consequence of the previous theorem.
The model-checking problem for [$k$-alt] is $k$-.
Strategy Quantifications
Since model checking for is non-elementary hard while the same problem
for is only 2, a question that naturally arises is whether
there are proper fragments of of practical interest, still strictly
subsuming , that reside in such a complexity gap.
In this section, we answer positively to this question and go even further.
Precisely, we enlighten a fundamental property that, if satisfied, allows to
retain a 2 model-checking problem.
We refer to such a property as elementariness.
To formally introduce this concept, we use the notion of dependence map
as a machinery.
The remaining part of this section is organized as follows.
In Subsection <ref>, we describe three syntactic fragments of
, named , , and , having the peculiarity to use strategy
quantifications grouped in atomic blocks.
Then, in Subsection <ref>, we define the notion of dependence
map, which is used, in Subsection <ref>, to introduce the
concept of elementariness.
Finally, in Subsection <ref>, we prove a fundamental result,
which is at the base of our elementary model-checking procedure for .
Syntactic fragments
In order to formalize the syntactic fragments of we want to
investigate, we first need to define the concepts of quantification
and binding prefixes.
A quantification prefix over a set $\VSet \subseteq \VarSet$ of
variables is a finite word $\qpElm \in \set{ \EExs{\xElm}, \AAll{\xElm} }{
\xElm \in \VSet }^{\card{\VSet}}$ of length $\card{\VSet}$ such that each
variable $\xElm \in \VSet$ occurs just once in $\qpElm$, i.e., there is
exactly one index $i \in \numco{0}{\card{\VSet}}$ such that $(\qpElm)_{i}
\in \{ \EExs{\xElm}, \AAll{\xElm} \}$.
A binding prefix over a set of variables $\VSet \subseteq \VarSet$
is a finite word $\bpElm \in \set{ (\aElm, \xElm) }{ \aElm \in \AgSet
\land \xElm \in \VSet }^{\card{\AgSet}}$ of length $\card{\AgSet}$ such
that each agent $\aElm \in \AgSet$ occurs just once in $\bpElm$, i.e.,
there is exactly one index $i \in \numco{0}{\card{\AgSet}}$ for which
$(\bpElm)_{i} \in \set{ (\aElm, \xElm) }{ \xElm \in \VSet }$.
Finally, $\QPSet(\VSet) \subseteq \set{ \EExs{\xElm}, \AAll{\xElm} }{
\xElm \in \VSet }^{\card{\VSet}}$ and $\BPSet(\VSet) \subseteq \set{
(\aElm, \xElm) }{ \aElm \in \AgSet \land \xElm \in \VSet
}^{\card{\AgSet}}$ denote, respectively, the sets of all quantification
and binding prefixes over variables in $\VSet$.
We now have all tools to define the syntactic fragments we want to analyze,
which we name, respectively, Nested-Goal, Boolean-Goal, and
One-Goal Strategy Logic (, , and , for short).
For goal we mean an agent-closed formula of the kind $\bpElm
\varphi$, with $\AgSet \subseteq \free{\varphi}$, being $\bpElm \in
\BndSet(\VarSet)$ a binding prefix.
The idea behind is that, when there is a quantification over a
variable used in a goal, we are forced to quantify over all free variables
of the inner subformula containing the goal itself, by using a
quantification prefix.
In this way, the subformula is build only by nesting and Boolean
combinations of goals.
In addition, with we avoid nested goals sharing the variables of a
same quantification prefix, but allow their Boolean combinations.
Finally, forces the use of a different quantification prefix for each
single goal in the formula.
The formal syntax of , , and follows.
formulas are built inductively from the sets of atomic propositions
$\APSet$, quantification prefixes $\QPSet(\VSet)$ for any $\VSet \subseteq
\VarSet$, and binding prefixes $\BPSet(\VarSet)$, by using the following
grammar, with $\pElm \in \APSet$, $\qpElm \in \cup_{\VSet \subseteq
\VarSet} \QPSet(\VSet)$, and $\bpElm \in \BPSet(\VarSet)$:
$\varphi ::= \pElm \mid \neg \varphi \mid \varphi \wedge \varphi \mid
\varphi \vee \varphi \mid \X \varphi \mid \varphi \:\U \varphi \mid
\varphi \:\R \varphi \mid \qpElm \varphi \mid \bpElm \varphi$,
where in the formation rule $\qpElm \varphi$ it is ensured that $\varphi$
is agent-closed and $\qpElm \in \QPSet(\free{\varphi})$.
In addition, formulas are determined by splitting the above
syntactic class in two different parts, of which the second is dedicated
to build the Boolean combinations of goals avoiding their nesting:
$\varphi ::= \pElm \mid \neg \varphi \mid \varphi \wedge \varphi \mid
\varphi \vee \varphi \mid \X \varphi \mid \varphi \:\U \varphi \mid
\varphi \:\R \varphi \mid \qpElm \psi$,
$\psi ::= \bpElm \varphi \mid \neg \psi \mid \psi \wedge \psi \mid \psi
\vee \psi$,
where in the formation rule $\qpElm \psi$ it is ensured that $\qpElm \in
\QPSet(\free{\psi})$.
Finally, the simpler formulas are obtained by forcing each goal to
be coupled with a quantification prefix:
$\varphi ::= \pElm \mid \neg \varphi \mid \varphi \wedge \varphi \mid
\varphi \vee \varphi \mid \X \varphi \mid \varphi \:\U \varphi \mid
\varphi \:\R \varphi \mid \qpElm \bpElm \varphi$,
where in the formation rule $\qpElm \bpElm \varphi$ it is ensured that
$\qpElm \in \QPSet(\free{\bpElm \varphi})$.
$\SL \supset \NGSL \supset \BGSL \supset \OGSL$ denotes the syntactic
chain of infinite sets of formulas generated by the respective grammars
with the associated constraints on free variables of goals.
Intuitively, in , , and , we force the writing of formulas to
use atomic blocks of quantifications and bindings, where the related free
variables are strictly coupled with those that are effectively quantified in
the prefix just before the binding.
In a nutshell, we can only write formulas by using sentences of the form
$\qpElm \psi$ belonging to a kind of prenex normal form in which the
quantifications contained into the matrix $\psi$ only belong to the
prefixes $\qpElm'$ of some inner subsentence $\qpElm' \psi' \in \snt{\qpElm
\psi}$.
An sentence $\phi$ is principal if it is of the form $\phi =
\qpElm \psi$, where $\psi$ is agent-closed and $\qpElm \in
\QPSet(\free{\psi})$.
By $\psnt{\varphi} \subseteq \snt{\varphi}$ we denote the set of all
principal subsentences of the formula $\varphi$.
We now introduce other two general restrictions in which the numbers
$\card{\AgSet}$ of agents and $\card{\VarSet}$ of variables that are used
to write a formula are fixed to the a priori values $n, m \in
\numco{1}{\omega}$, respectively.
Moreover, we can also forbid the sharing of variables, i.e., each variable
is binded to one agent only, so, we cannot force two agents to use the same
We name these three fragments [$n$-ag], [$m$-var], and [fvs],
Note that, in the one agent fragment, the restriction on the sharing of
variables between agents, naturally, does not act, i.e., $\SL[$1$-ag, fvs] =
\SL[$1$-ag]$.
To start to practice with the above fragments, consider again the sentence
$\varphi$ of Example exm:sv.
It is easy to see that it actually belongs to [$2$-ag, $3$-var,
$2$-alt], and so, to , but not to , since it is of the form
$\qpSym (\bpSym[1] \X \pSym \wedge \bpSym[2] \X \qSym)$, where the
quantification prefix is $\qpSym = \EExs{\xSym} \AAll{\ySym} \EExs{\zSym}$
and the binding prefixes of the two goals are $\bpSym[1] = (\alpha, \xSym)
(\beta, \ySym)$ and $\bpSym[2] = (\alpha, \ySym) (\beta, \zSym)$.
Along the paper, sometimes we assert that a given formula $\varphi$ belongs
to an syntactic fragment also if its syntax does not precisely
correspond to what is described by the relative grammar.
We do this in order to make easier the reading and interpretation of the
formula $\varphi$ itself and only in the case that it is simple to translate
it into an equivalent formula that effectively belongs to the intended
logic, by means of a simple generalization of classic rules used to put a
formula of first order logic in the prenex normal form.
For example, consider the sentence $\varphi_{\!N\!\!E}$ of
Example exm:ne representing the existence of a Nash equilibrium.
This formula is considered to belong to [$n$-ag, $2n$-var, fvs,
$1$-alt], since it can be easily translated in the form $\phi_{\!N\!\!E} =
\qpSym \bigwedge_{i = 1}^{n} \bpSym[i] \psi_{i} \rightarrow \bpSym
\psi_{i}$, where $\qpSym = \EExs{\xSym[1]} \cdots \EExs{\xSym[n]}
\AAll{\ySym[1]} \cdots \AAll{\ySym[n]}$, $\bpSym = (\alpha_{1}, \xSym[1])
\cdots (\alpha_{n}, \xSym[n])$, $\bpSym[i] = (\alpha_{1}, \xSym[1]) \cdots
(\alpha_{i - 1}, \xSym[i - 1]) (\alpha_{i}, \ySym[i]) (\alpha_{i + 1},
\xSym[i + 1]) \cdots (\alpha_{n}, \xSym[n])$, and $\free{\psi_{i}} =
\AgSet$.
As another example, consider the sentence $\varphi_{\!S\!P}$ of
Example exm:sp representing the existence of a stability profile.
Also this formula is considered to belong to [$n$-ag, $2n$-var, fvs,
$1$-alt], since it is equivalent to $\phi_{\!S\!P} = \qpSym \bigwedge_{i, j
= 1, i \neq j}^{n} \bpSym \psi_{j} \rightarrow ((\bpSym \psi_{i}
\leftrightarrow \bpSym[i] \psi_{i}) \rightarrow \bpSym[i] \psi_{j})$.
Note that both $\phi_{\!N\!\!E}$ and $\phi_{\!S\!P}$ are principal
Now, it is interesting to observe that and are exactly
equivalent to [fvs, $0$-alt] and [fvs, $1$-alt], respectively.
Moreover, <cit.> is the very simple fragment of [fvs,
$1$-alt] that forces all goals in a formula to have a common part containing
all variables quantified before the unique possible alternation of the
quantification prefix.
Finally, we have that is the [$2$-ag, fvs] fragment.
It is well-known that the non-elementary hardness result for the
satisfiability problem of <cit.> already holds for formulas in
prenex normal form.
Now, it is not hard to see that the transformation described in
Lemma <ref> of reduction puts [$k$-alt]
sentences $\varphi$ in prenex normal form into [$1$-ag, $k$-alt]
variable-closed formulas $\trn{\varphi} = \qpElm \psi$.
Moreover, the derived [$1$-ag, $k$-alt] sentence $\EExs{\xSym}
(\alpha, \xSym) \qpElm \psi$ used in Theorem <ref> of
model-checking hardness is equivalent to the [$1$-ag,
$k$-alt] principal sentence $\EExs{\xSym} \qpElm (\alpha, \xSym) \psi$,
since $\xSym$ is not used in the quantification prefix $\qpElm$.
Thus, the hardness result for the model-checking problem holds for
[$1$-ag, $k$-alt] and, consequently, for [$1$-ag, $k$-alt] as
However, it is important to observe that, unfortunately, it is not know
if such an hardness result holds for or and, in
particular, for .
We leave this problem open here.
At this point, we prove that is strictly less expressive than and, consequently, than and .
To do this, we show the existence of two structures that result to be
equivalent only w.r.t. sentences having alternation number bounded by $1$.
It can be interesting to note that, we use an ad-hoc technique based on a
brute-force check to verify that all formulas cannot distinguish
between the two structures.
A possible future line of research is to study variants of the
Ehrenfeucht-Fraïssé game <cit.> for , which allow to
determine whether two structures are or not equivalent w.r.t. a particular
There exists an [$3$-ag, fvs, $2$-alt] sentence having no equivalent.
Consider the two s $\GName[1] \defeq \CGSTuple {\{ \pSym \}} {\{
\alpha, \beta, \gamma \}} {\{ 0, 1 \}} {\{ \sSym[0], \sSym[1], \sSym[2]
\}} {\labFun} {\trnFun[1]} {\sSym[0]}$ and $\GName[2] \defeq \CGSTuple {\{
\pSym \}} {\{ \alpha, \beta, \gamma \}} {\{ 0, 1, 2 \}} {\{ \sSym[0],
\sSym[1], \sSym[2] \}} {\labFun} {\trnFun[2]} {\sSym[0]}$ depicted in
Figure fig:thm:ogslvsatls(exp), where $\labFun(\sSym[0]) =
\labFun(\sSym[2]) \defeq \emptyset$, $\labFun(\sSym[1]) \defeq \{ \pSym
\}$, $\DSet[1] \defeq \{ 00*, 11* \}$, and $\DSet[2] \defeq \{ 00*, 11*,
12*, 200, 202, 211 \}$.
Moreover, consider the [$3$-ag, fvs, $2$-alt] sentence $\varphi^{*}
\defeq \qpSym^{*} \bpSym^{*} \X \pSym$, where $\qpSym^{*} \defeq
\AAll{\xSym} \EExs{\ySym} \AAll{\zElm}$ and $\bpSym^{*} \defeq (\alpha,
\xSym) (\beta, \ySym) (\gamma, \zSym)$.
Then, it is easy to see that $\GName[1] \models \varphi^{*}$ but
$\GName[2] \not\models \varphi^{*}$.
Indeed, $\GName[1], \asgFun[1], \sSym[0] \models \bpSym^{*} \X \pSym$, for
all $\asgFun[1] \in \AsgSet[ {\GName[1]} ](\{ \xSym, \ySym, \zSym \},
\sSym[0])$ such that $\asgFun[1](\ySym)(\sSym[0]) =
\asgFun[1](\xSym)(\sSym[0])$, and $\GName[2], \asgFun[2], \sSym[0] \models
\bpSym^{*} \X \neg \pSym$, for all $\asgFun[2] \in \AsgSet[ {\GName[2]}
](\{ \xSym, \ySym, \zSym \}, \sSym[0])$ such that
$\asgFun[2](\xSym)(\sSym[0]) = 2$ and $\asgFun[2](\zSym)(\sSym[0]) =
(\asgFun[2](\ySym)(\sSym[0]) + 1) \bmod 3$.
Now, due to the particular structure of the s $\GName[i]$ under exam,
with $i \in \{ 1, 2 \}$, for each path $\pthElm \in \PthSet[ {\GName[i]}
](\sSym[0])$, we have that either $\labFun((\pthElm)_{j}) = \{ \pSym \}$
or $\labFun((\pthElm)_{j}) = \emptyset$, for all $j \in
\numco{1}{\omega}$, i.e., apart from the initial state, the path is
completely labeled either with $\{ \pSym \}$ or with $\emptyset$.
Thus, it is easy to see that, for each formula $\qpElm \bpElm
\psi$, there is a literal $\lElm[\psi] \in \{ \pSym, \neg \pSym \}$ such
that $\GName[i] \models \qpElm \bpElm \psi$ iff $\GName[i] \models \qpElm
\bpElm \X \!\lElm[\psi]$, for all $i \in \{ 1, 2 \}$.
W.l.o.g., we can suppose that $\bpElm = \bpSym^{*}$, since we are always
able to uniformly rename the variables of the quantification and binding
prefixes without changing the meaning of the sentence.
At this point, it is easy to see that there exists an index $k \in \{ 1,
2, 3 \}$ for which it holds that either $\qpSym[k] \bpSym^{*} \X
\!\lElm[\psi] \implies \qpElm \bpSym^{*} \X \!\lElm[\psi]$ or $\qpElm
\bpSym^{*} \X \!\lElm[\psi] \implies \dual{\qpElm[k]} \bpSym^{*} \X
\!\lElm[\psi]$, where $\qpSym[1] \defeq \AAll{\xSym} \AAll{\zElm}
\EExs{\ySym}$, $\qpSym[2] \defeq \EExs{\xSym} \EExs{\yElm} \AAll{\zSym}$,
and $\qpSym[3] \defeq \AAll{\ySym} \AAll{\zElm} \EExs{\xSym}$.
Thus, to prove that every formula cannot distinguish between
$\GName[1]$ and $\GName[2]$, we can simply show that the sentences
$\qpSym[k] \bpSym^{*} \X \!\lElm$, with $k \in \{ 1, 2, 3 \}$ and $\lElm
\in \{ \pSym, \neg \pSym \}$, do the same.
In fact, it holds that $\GName[i] \models \qpSym[k] \bpSym^{*} \X
\!\lElm$, for all $i \in \{ 1, 2 \}$, $k \in \{ 1, 2, 3 \}$, and $\lElm
\in \{ \pSym, \neg \pSym \}$.
Hence, the thesis holds.
The check of the latter fact is trivial and left to the reader as an
Dependence Maps
We now introduce the concept of dependence map of a quantification and show
how any quantification prefix contained into an formula can be
represented by an adequate choice of a dependence map over strategies.
The main idea here is inspired by what Skolem proposed for the first order
logic in order to eliminate each existential quantification over variables,
by substituting them with second order existential quantifications over
functions, whose choice is uniform w.r.t. the universal variables.
First, we introduce some notation regarding quantification prefixes.
Let $\qpElm \in \QPSet(\VSet)$ be a quantification prefix over a set
$\QPVSet(\qpElm) \defeq \VSet \subseteq \VarSet$ of variables.
By $\QPEVSet{\qpElm} \defeq \set{ \xElm \in \QPVSet(\qpElm) }{ \exists i \in
\numco{0}{\card{\qpElm}} .\: (\qpElm)_{i} = \EExs{\xElm} }$ and
$\QPAVSet{\qpElm} \defeq \QPVSet(\qpElm) \setminus \QPEVSet{\qpElm}$ we
denote, respectively, the sets of existential and universal
variables quantified in $\qpElm$.
For two variables $\xElm, \yElm \in \QPVSet(\qpElm)$, we say that $\xElm$
precedes $\yElm$ in $\qpElm$, in symbols $\xElm \qpordRel[\qpElm]
\yElm$, if $\xElm$ occurs before $\yElm$ in $\qpElm$, i.e., there are two
indexes $i, j \in \numco{0}{\card{\qpElm}}\!$, with $i < j$, such that
$(\qpElm)_{i} \in \{ \EExs{\xElm}, \AAll{\xElm} \}$ and $(\qpElm)_{j} \in \{
\EExs{\yElm}, \AAll{\yElm} \}$.
Moreover, we say that $\yElm$ is functional dependent on $\xElm$, in
symbols $\xElm \qpdepRel[\qpElm] \yElm$, if $\xElm \in \QPAVSet{\qpElm}$,
$\yElm \in \QPEVSet{\qpElm}$, and $\xElm \qpordRel_{\qpElm} \yElm$, i.e.,
$\yElm$ is existentially quantified after that $\xElm$ is universally
quantified, so, there may be a dependence between a value chosen by $\xElm$
and that chosen by $\yElm$.
This definition induces the set $\QPDepSet(\qpElm) \defeq \set{ (\xElm,
\yElm) \in \QPVSet(\qpElm) \times \QPVSet(\qpElm) }{ \xElm \qpdepRel[\qpElm]
\yElm }$ of dependence pairs and its derived version
$\QPDepSet(\qpElm, \yElm) \defeq \set{ \xElm \in \QPVSet(\qpElm) }{ \xElm
\qpdepRel[\qpElm] \yElm }$ containing all variables from which $\yElm$
Finally, we use $\dual{\qpElm} \in \QPSet(\QPVSet(\qpElm))$ to indicate the
quantification derived from $\qpElm$ by dualizing each quantifier
contained in it, i.e., for all indexes $i \in \numco{0}{\card{\qpElm}}\!$,
it holds that $(\dual{\qpElm})_{i} = \EExs{\xElm}$ iff $(\qpElm)_{i} =
\AAll{\xElm}$, with $\xElm \in \QPVSet(\qpElm)$.
It is evident that $\QPEVSet{\dual{\qpElm}} = \QPAVSet{\qpElm}$ and
$\QPAVSet{\dual{\qpElm}} = \QPEVSet{\qpElm}$.
As an example, let $\qpSym = \AAll{\xSym} \EExs{\ySym} \EExs{\zSym}
\AAll{\wSym} \EExs{\vSym}$.
Then, we have $\QPEVSet{\qpSym} = \{ \ySym, \zSym, \vSym \}$,
$\QPAVSet{\qpSym} = \{ \xSym, \wSym \}$, $\QPDepSet(\qpSym, \xSym) =
\QPDepSet(\qpSym, \wSym) = \emptyset$,
Finally, we define the notion of valuation of variables over a
generic set $\DSet$, called domain, i.e., a partial function $\valFun
: \VarSet \pto \DSet$ mapping every variable in its domain to an element in
By $\ValSet[\DSet](\VSet) \defeq \VSet \to \DSet$ we denote the set of all
valuation functions over $\DSet$ defined on $\VSet \subseteq \VarSet$.
At this point, we give a general high-level semantics for the quantification
prefixes by means of the following main definition of dependence map.
Let $\qpElm \in \QPSet(\VSet)$ be a quantification prefix over a set
$\VSet \subseteq \VarSet$ of variables, and $\DSet$ a set.
Then, a dependence map for $\qpElm$ over $\DSet$ is a function
$\spcFun : \ValSet[\DSet](\QPAVSet{\qpElm}) \to \ValSet[\DSet](\VSet)$
satisfying the following properties:
$\spcFun(\valFun)_{\rst \QPAVSet{\qpElm}} \!=\! \valFun$, for all
$\valFun \in \ValSet[\DSet](\QPAVSet{\qpElm})$; [By
$\gFun_{\rst \ZSet} : (\XSet \cap \ZSet) \to \YSet$ we denote the
restriction of a function $\gFun : \XSet \to \YSet$ to the
elements in the set $\ZSet$.]
$\spcFun(\valFun[1])(\xElm) \!=\! \spcFun(\valFun[2])(\xElm)$, for all
$\valFun[1], \valFun[2] \!\in\! \ValSet[\DSet](\QPAVSet{\qpElm})$ and
$\xElm \!\in\! \QPEVSet{\qpElm}$ such that $\valFun[1]_{\rst
\QPDepSet(\qpElm, \xElm)} \!=\! \valFun[2]_{\rst \QPDepSet(\qpElm,
\xElm)}$.
$\SpcSet[\DSet](\qpElm)$ denotes the set of all dependence maps for
$\qpElm$ over $\DSet$.
Intuitively, Item <ref> asserts that $\spcFun$ takes the
same values of its argument w.r.t. the universal variables in $\qpElm$ and
Item <ref> ensures that the value of $\spcFun$ w.r.t. an
existential variable $\xElm$ in $\qpElm$ does not depend on variables not in
$\QPDepSet(\qpElm, \xElm)$.
To get a better insight into this definition, a dependence map $\spcFun$ for
$\qpElm$ can be considered as a set of Skolem functions that, given a
value for each variable in $\QPVSet(\qpElm)$ that is universally quantified
in $\qpElm$, returns a possible value for all the existential variables in
$\qpElm$, in a way that is consistent w.r.t. the order of quantifications.
Observe that, each $\spcFun \in \SpcSet[\DSet](\qpElm)$ is injective, so,
$\card{\rng{\spcFun}} = \card{\dom{\spcFun}} =
\card{\DSet}^{\card{\QPAVSet{\qpElm}}}$.
Moreover, $\card{\SpcSet[\DSet](\qpElm)} = \prod_{\xElm \in
\QPEVSet{\qpElm}} \card{\DSet}^{\card{\DSet}^{\card{\QPDepSet(\qpElm,
\xElm)}}}$.
As an example, let $\DSet = \{ 0, 1 \}$ and $\qpSym = \AAll{\xSym}
\EExs{\ySym} \AAll{\zSym} \in \QPSet(\VSet)$ be a quantification prefix over
$\VSet = \{ \xSym, \ySym, \zSym \}$.
Then, we have that $\card{\SpcSet[\DSet](\qpSym)} = 4$ and
$\card{\SpcSet[\DSet](\dual{\qpSym})} = 8$.
Moreover, the dependence maps $\spcFun[i] \in \SpcSet[\DSet](\qpSym)$ with
$i \in \numcc{0}{3}$ and $\dual[i]{\spcFun} \in
\SpcSet[\DSet](\dual{\qpSym})$ with $i \in \numcc{0}{7}$, for a particular
fixed order, are such that $\spcFun[0](\valFun)(\ySym) = 0$,
$\spcFun[1](\valFun)(\ySym) = \valFun(\xSym)$, $\spcFun[2](\valFun)(\ySym) =
1 - \valFun(\xSym)$, and $\spcFun[3](\valFun)(\ySym) = 1$, for all $\valFun
\in \ValSet[\DSet](\QPAVSet{\qpSym})$, and
$\dual[i]{\spcFun}(\dual{\valFun})(\xSym) = 0$ with $i \in \numcc{0}{3}$,
$\dual[i]{\spcFun}(\dual{\valFun})(\xSym) = 1$ with $i \in \numcc{4}{7}$,
$\dual[0]{\spcFun}(\dual{\valFun})(\zSym) =
\dual[4]{\spcFun}(\dual{\valFun})(\zSym) = 0$,
$\dual[1]{\spcFun}(\dual{\valFun})(\zSym) =
\dual[5]{\spcFun}(\dual{\valFun})(\zSym) = \dual{\valFun}(\ySym)$,
$\dual[2]{\spcFun}(\dual{\valFun})(\zSym) =
\dual[6]{\spcFun}(\dual{\valFun})(\zSym) = 1 - \dual{\valFun}(\ySym)$, and
$\dual[3]{\spcFun}(\dual{\valFun})(\zSym) =
\dual[7]{\spcFun}(\dual{\valFun})(\zSym) = 1$, for all $\dual{\valFun} \in
\ValSet[\DSet](\QPAVSet{\dual{\qpSym}})$.
We now prove the following fundamental theorem that describes how to
eliminate the strategy quantifications of an formula via a choice of a
suitable dependence map over strategies.
This procedure can be seen as the equivalent of Skolemization in
first order logic (see <cit.>, for more details).
Let $\GName$ be a and $\varphi = \qpElm \psi$ an formula, being
$\qpElm \in \QPSet(\VSet)$ a quantification prefix over a set $\VSet
\subseteq \free{\psi} \cap \VarSet$ of variables.
Then, for all assignments $\asgFun \in \AsgSet(\free{\varphi}, \sElm[0])$,
the following holds: $\GName, \asgFun, \sElm[0] \models \varphi$ iff there
exists a dependence map $\spcFun \in \SpcSet[ {\StrSet(\sElm[0])}
](\qpElm)$ such that $\GName, \asgFun \umrg \spcFun(\asgFun'), \sElm[0]
\models \psi$, for all $\asgFun' \in \AsgSet(\QPAVSet{\qpElm},
\sElm[0])$. [By $\gFun[1] \umrg \gFun[2] : (\XSet[1] \cup
\XSet[2]) \to (\YSet[1] \cup \YSet[2])$ we denote the operation of
union of two functions $\gFun[1] : \XSet[1] \to \YSet[1]$ and
$\gFun[2] : \XSet[2] \to \YSet[2]$ defined on disjoint domains, i.e.,
$\XSet[1] \cap \XSet[2] = \emptyset$.]
The proof proceeds by induction on the length of the quantification prefix
For the base case $\card{\qpElm} = 0$, the thesis immediately follows,
since $\QPAVSet{\qpElm} = \emptyset$ and, consequently, both
$\SpcSet[\StrSet( {\sElm[0]} )](\qpElm)$ and $\AsgSet(\QPAVSet{\qpElm},
\sElm[0])$ contain the empty function only (we are assuming, by
convention, that $\emptyfun(\emptyfun) \defeq \emptyfun$).
We now prove, separately, the two directions of the inductive case.
[Only if].
Suppose that $\GName, \asgFun, \sElm[0] \models \varphi$, where $\qpElm =
\Qnt \cdot \qpElm'$.
Then, two possible cases arise: either $\Qnt = \EExs{\xElm}$ or $\Qnt =
\AAll{\xElm}$.
* $\Qnt = \EExs{\xElm}$.
By Item <ref> of
Definition <ref> of semantics, there is a
strategy $\strFun \in \StrSet(\sElm[0])$ such that $\GName,
\asgFun[][\xElm \mapsto \strFun], \sElm[0] \models \qpElm' \psi$.
Note that $\QPAVSet{\qpElm} = \QPAVSet{\qpElm'}$.
By the inductive hypothesis, we have that there exists a dependence
map $\spcFun \in \SpcSet[\StrSet( {\sElm[0]} )](\qpElm')$ such that
$\GName, \asgFun[][\xElm \mapsto \strFun] \umrg \spcFun(\asgFun'),
\sElm[0] \models \psi$, for all $\asgFun' \in
\AsgSet(\QPAVSet{\qpElm'}, \sElm[0])$.
Now, consider the function $\spcFun' : \AsgSet(\QPAVSet{\qpElm},
\sElm[0]) \to \AsgSet(\VSet, \sElm[0])$ defined by $\spcFun'(\asgFun')
\defeq \spcFun(\asgFun')[\xElm \mapsto \strFun]$, for all $\asgFun'
\in \AsgSet(\QPAVSet{\qpElm}, \sElm[0])$.
It is easy to check that $\spcFun'$ is a dependence map for $\qpElm$
over $\StrSet(\sElm[0])$, i.e., $\spcFun' \in \SpcSet[\StrSet(
{\sElm[0]} )](\qpElm)$.
Moreover, $\asgFun[][\xElm \mapsto \strFun] \umrg \spcFun(\asgFun') =
\asgFun \umrg \spcFun(\asgFun')[\xElm \mapsto \strFun] = \asgFun \umrg
\spcFun'(\asgFun')$, for $\asgFun' \in \AsgSet(\QPAVSet{\qpElm},
\sElm[0])$.
Hence, $\GName, \asgFun \umrg \spcFun'(\asgFun'), \sElm[0] \models
\psi$, for all $\asgFun' \in \AsgSet(\QPAVSet{\qpElm}, \sElm[0])$.
* $\Qnt = \AAll{\xElm}$.
By Item <ref> of
Definition <ref>, we have that, for all strategies
$\strFun \in \StrSet(\sElm[0])$, it holds that $\GName,
\asgFun[][\xElm \mapsto \strFun], \sElm[0] \models \qpElm' \psi$.
Note that $\QPAVSet{\qpElm} = \QPAVSet{\qpElm'} \cup \{ \xElm \}$.
By the inductive hypothesis, we derive that, for each $\strFun \in
\StrSet(\sElm[0])$, there exists a dependence map $\spcFun[\strFun]
\in \SpcSet[\StrSet( {\sElm[0]} )](\qpElm')$ such that $\GName,
\asgFun[][\xElm \mapsto \strFun] \umrg \spcFun[\strFun](\asgFun'),
\sElm[0] \models \psi$, for all $\asgFun' \in
\AsgSet(\QPAVSet{\qpElm'}, \sElm[0])$.
Now, consider the function $\spcFun' : \AsgSet(\QPAVSet{\qpElm},
\sElm[0]) \to \AsgSet(\VSet, \sElm[0])$ defined by $\spcFun'(\asgFun')
\defeq \spcFun[\asgFun'(\xElm)](\asgFun[|']_{\rst
\QPAVSet{\qpElm'}})[\xElm \mapsto \asgFun'(\xElm)]$, for all $\asgFun'
\in \AsgSet(\QPAVSet{\qpElm}, \sElm[0])$.
It is evident that $\spcFun'$ is a dependence map for $\qpElm$ over
$\StrSet(\sElm[0])$, i.e., $\spcFun' \in \SpcSet[\StrSet( {\sElm[0]}
Moreover, $\asgFun[][\xElm \mapsto \strFun] \umrg
\spcFun[\strFun](\asgFun') = \asgFun \umrg
\spcFun[\strFun](\asgFun')[\xElm \mapsto \strFun] = \asgFun \umrg
\spcFun'(\asgFun'[\xElm \mapsto \strFun])$, for $\strFun \in
\StrSet(\sElm[0])$ and $\asgFun' \in \AsgSet(\QPAVSet{\qpElm'},
\sElm[0])$.
Hence, $\GName, \asgFun \umrg \spcFun'(\asgFun'), \sElm[0] \models
\psi$, for all $\asgFun' \in \AsgSet(\QPAVSet{\qpElm}, \sElm[0])$.
Suppose that there exists a dependence map $\spcFun \in \SpcSet[\StrSet(
{\sElm[0]} )](\qpElm)$ such that $\GName, \asgFun \umrg \spcFun(\asgFun'),
\sElm[0] \models \psi$, for all $\asgFun' \in \AsgSet(\QPAVSet{\qpElm},
\sElm[0])$, where $\qpElm = \Qnt \cdot \qpElm'$.
Then, two possible cases arise: either $\Qnt = \EExs{\xElm}$ or $\Qnt =
\AAll{\xElm}$.
* $\Qnt = \EExs{\xElm}$.
There is a strategy $\strFun \in \StrSet(\sElm[0])$ such that $\strFun
= \spcFun(\asgFun')(\xElm)$, for all $\asgFun' \in
\AsgSet(\QPAVSet{\qpElm}, \sElm[0])$.
Note that $\QPAVSet{\qpElm} = \QPAVSet{\qpElm'}$.
Consider the function $\spcFun' : \AsgSet(\QPAVSet{\qpElm'}, \sElm[0])
\to \AsgSet(\VSet \setminus \{ \xElm \}, \sElm[0])$ defined by
$\spcFun'(\asgFun') \defeq \spcFun(\asgFun')_{\rst (\VSet \setminus \{
\xElm \})}$, for all $\asgFun' \in \AsgSet(\QPAVSet{\qpElm'},
\sElm[0])$.
It is easy to check that $\spcFun'$ is a dependence map for $\qpElm'$
over $\StrSet(\sElm[0])$, i.e., $\spcFun' \in \SpcSet[\StrSet(
{\sElm[0]} )](\qpElm')$.
Moreover, $\asgFun \umrg \spcFun(\asgFun') = \asgFun \umrg
\spcFun'(\asgFun')[\xElm \mapsto \strFun] = \asgFun[][\xElm \mapsto
\strFun] \umrg \spcFun'(\asgFun')$, for $\asgFun' \in
\AsgSet(\QPAVSet{\qpElm'}, \sElm[0])$.
Then, it is evident that $\GName, \asgFun[][\xElm \to \strFun] \umrg
\spcFun'(\asgFun'), \sElm[0] \models \psi$, for all $\asgFun' \in
\AsgSet(\QPAVSet{\qpElm'}, \sElm[0])$.
By the inductive hypothesis, we derive that $\GName, \asgFun[][\xElm
\mapsto \strFun], \sElm[0] \models \qpElm' \psi$, which means that
$\GName, \asgFun, \sElm[0] \models \varphi$, by
Item <ref> of
Definition <ref> of semantics.
* $\Qnt = \AAll{\xElm}$.
First note that $\QPAVSet{\qpElm} = \QPAVSet{\qpElm'} \cup \{ \xElm
\}$.
Also, consider the functions $\spcFun[\strFun|'] :
\AsgSet(\QPAVSet{\qpElm'}, \sElm[0]) \to \AsgSet(\VSet \setminus \{
\xElm \}, \sElm[0])$ defined by $\spcFun[\strFun|'](\asgFun') \defeq
\spcFun(\asgFun'[\xElm \mapsto \strFun])_{\rst (\VSet \setminus \{
\xElm \})}$, for each $\strFun \in \StrSet(\sElm[0])$ and $\asgFun'
\in \AsgSet(\QPAVSet{\qpElm'}, \sElm[0])$.
It is easy to see that every $\spcFun[\strFun|']$ is a dependence map
for $\qpElm'$ over $\StrSet(\sElm[0])$, i.e.,
$\spcFun[\strFun|'] \in \SpcSet[\StrSet( {\sElm[0]} )](\qpElm')$.
Moreover, $\asgFun \umrg \spcFun(\asgFun') = \asgFun \umrg
\spcFun[\asgFun'(\xElm)|'](\asgFun'_{\rst \QPAVSet{\qpElm'}})[\xElm
\mapsto \asgFun'(\xElm)] = \asgFun[][\xElm \mapsto \asgFun'(\xElm)]
\umrg \spcFun[\asgFun'(\xElm)|'](\asgFun'_{\rst \QPAVSet{\qpElm'}})$,
for $\asgFun' \in \AsgSet(\QPAVSet{\qpElm}, \sElm[0])$.
Then, it is evident that $\GName, \asgFun[][\xElm \to \strFun] \umrg
\spcFun[\strFun|'](\asgFun'), \sElm[0] \models \psi$, for all $\strFun
\in \StrSet(\sElm[0])$ and $\asgFun' \in \AsgSet(\QPAVSet{\qpElm'},
\sElm[0])$.
By the inductive hypothesis, we derive that $\GName, \asgFun[][\xElm
\mapsto \strFun], \sElm[0] \models \qpElm' \psi$, for all $\strFun \in
\StrSet(\sElm[0])$, which means that $\GName, \asgFun, \sElm[0]
\models \varphi$, by Item <ref> of
Definition <ref>.
Thus, the thesis of the theorem holds.
As an immediate consequence of the previous result, we derive the following
Let $\GName$ be a and $\varphi = \qpElm \psi$ an sentence,
where $\psi$ is agent-closed and $\qpElm \in \QPSet(\free{\psi})$.
Then, $\GName \models \varphi$ iff there exists a dependence map $\spcFun
\in \SpcSet[ {\StrSet(\sElm[0])} ](\qpElm)$ such that $\GName,
\spcFun(\asgFun), \sElm[0] \models \psi$, for all $\asgFun \in
\AsgSet(\QPAVSet{\qpElm}, \sElm[0])$.
Elementary quantifications
We now have all tools we need to introduce the property of elementariness
for a particular class of dependence maps.
Intuitively, a dependence map over functions from a set $\TSet$ to a set
$\DSet$ is elementary if it can be split into a set of dependence maps over
$\DSet$, one for each element of $\TSet$.
This idea allows us to enormously simplify the reasoning about strategy
quantifications, since we can reduce them to a set of quantifications over
actions, one for each track in their domains.
This means that, under certain conditions, we can transform a dependence map
$\spcFun \in \SpcSet[\StrSet(\sElm)](\qpElm)$ over strategies in a function
$\adj{\spcFun} : \TrkSet(\sElm) \to \SpcSet[\AcSet](\qpElm)$ that associates
with each track a dependence map over actions.
To formally develop the above idea, we have first to introduce the generic
concept of adjoint function and state an auxiliary lemma.
Let $\DSet$, $\TSet$, $\USet$, and $\VSet$ be four sets, and $\mFun :
(\TSet \to \DSet)^{\USet} \to (\TSet \to \DSet)^{\VSet}$ and $\adj{\mFun}
: \TSet \to (\DSet^{\USet} \to \DSet^{\VSet})$ two functions.
Then, $\adj{\mFun}$ is the adjoint of $\mFun$ if
$\adj{\mFun}(\tElm)(\flip{\gFun}(\tElm))(\xElm) =
\mFun(\gFun)(\xElm)(\tElm)$, for all $\gFun \in (\TSet \to
\DSet)^{\USet}$, $\xElm \in \VSet$, and $\tElm \in \TSet$ [By
$\flip{\gFun} : \YSet \to \XSet \to \ZSet$ we denote the operation of
flipping of a function $\gFun : \XSet \to \YSet \to \ZSet$.]
Intuitively, $\adj{\mFun}$ is the adjoint of $\mFun$ if the dependence from
the set $\TSet$ in both domain and codomain of the latter can be extracted
and put as a common factor of the functor given by the former.
This means also that, for every pair of functions $\gFun[1], \gFun[2] \in
(\TSet \to \DSet)^{\USet}$ such that $\flip{\gFun[1]}(\tElm) =
\flip{\gFun[2]}(\tElm)$ for some $\tElm \in \TSet$, it holds that
$\mFun(\gFun[1])(\xElm)(\tElm) = \mFun(\gFun[2])(\xElm)(\tElm)$, for all
$\xElm \in \VSet$.
It is immediate to observe that if a function has an adjoint then this
adjoint is unique.
At the same way, if one has an adjoint function then it is possible to
determine the original function without any ambiguity.
Thus, it is established a one-to-one correspondence between functions
admitting an adjoint and the adjoint itself.
Next lemma formally states the property briefly described above, i.e., that
each dependence map over a set $\TSet \to \DSet$, admitting an adjoint
function, can be represented as a function, with $\TSet$ as domain, which
returns dependence maps over $\DSet$ as values.
Let $\qpElm \in \QPSet(\VSet)$ be a quantification prefix over a set
$\VSet \subseteq \VarSet$ of variables, $\DSet$ and $\TSet$ two sets, and
$\spcFun : \ValSet[\TSet \to \DSet](\QPAVSet{\qpElm}) \to \ValSet[\TSet
\to \DSet](\VSet)$ and $\adj{\spcFun} : \TSet \to
(\ValSet[\DSet](\QPAVSet{\qpElm}) \to \ValSet[\DSet](\VSet))$ two
functions such that $\adj{\spcFun}$ is the adjoint of $\spcFun$.
Then, $\spcFun \in \SpcSet[\TSet \to \DSet](\qpElm)$ iff, for all $t \in
\TSet$, it holds that $\adj{\spcFun}(t) \in \SpcSet[\DSet](\qpElm)$.
We now define the formal meaning of the elementariness of a dependence map
over functions.
Let $\qpElm \in \QPSet(\VSet)$ be a quantification prefix over a set
$\VSet \subseteq \VarSet$ of variables, $\DSet$ and $\TSet$ two sets, and
$\spcFun \in \SpcSet[\TSet \to \DSet](\qpElm)$ a dependence map for
$\qpElm$ over $\TSet \to \DSet$.
Then, $\spcFun$ is elementary if it admits an adjoint function.
$\ESpcSet[\TSet \to \DSet](\qpElm)$
It is important to observe that, unfortunately, there are dependence maps
that are not elementary.
To easily understand why this is actually the case, it is enough to count
both the number of dependence maps $\SpcSet[\TSet \to \DSet](\qpElm)$ and of
adjoint functions $\TSet \to \SpcSet[\DSet](\qpElm)$, where $\card{\DSet} >
1$, $\card{\TSet} > 1$ and $\qpElm$ is such that there is an $\xElm \in
\QPEVSet{\qpElm}$ for which $\QPDepSet(\qpElm, \xElm) \neq \emptyset$.
Indeed, it holds that $\card{\SpcSet[\TSet \to \DSet](\qpElm)} =
\prod_{\xElm \in \QPEVSet{\qpElm}} \card{\DSet}^{\card{\TSet} \cdot
\card{\DSet}^{\card{\TSet} \cdot \card{\QPDepSet(\qpElm, \xElm)}}} >
\prod_{\xElm \in \QPEVSet{\qpElm}} \card{\DSet}^{\card{\TSet} \cdot
\card{\DSet}^{\card{\QPDepSet(\qpElm, \xElm)}}} = \card{\TSet \to
\SpcSet[\DSet](\qpElm)}$.
So, there are much more dependence maps, a number double exponential in
$\card{\TSet}$, than possible adjoint functions, whose number is only
exponential in this value.
Furthermore, observe that the simple set
$\QPSet[\exists^{*}\forall^{*}](\VSet) \defeq \set{ \qpElm \in \QPSet(\VSet)
}{ \exists i \in \numcc{0}{\card{\qpElm}} \:.\: \QPAVSet{(\qpElm)_{< i}} =
\emptyset \wedge \QPEVSet{(\qpElm)_{\geq i}} = \emptyset }$, for $\VSet
\subseteq \VarSet$, is the maximal class of quantification prefixes that
admits only elementary dependence maps over $\TSet \to \DSet$, i.e., it is
such that each $\spcFun \in \SpcSet[\TSet \to \DSet](\qpElm)$ is elementary,
for all $\qpElm \in \QPSet[\exists^{*}\forall^{*}](\VSet)$.
This is due to the fact that there are no functional dependences between
variables, i.e., for each $\xElm \in \QPEVSet{\qpElm}$, it holds that
$\QPDepSet(\qpElm, \xElm) = \emptyset$.
Finally, we can introduce a new very important semantics for syntactic
fragments, which is based on the concept of elementary dependence map over
strategies, and we refer to the related satisfiability concept as
elementary satisfiability, in symbols $\emodels$.
Intuitively, such a semantics has the peculiarity that a strategy, used in
an existential quantification in order to satisfy a formula, is only chosen
between those that are elementary w.r.t. the universal quantifications.
In this way, when we have to decide what is its value $\cElm$ on a given
track $\trkElm$, we do it only in dependence of the values on the same track
of the strategies so far quantified, but not on their whole structure, as it
is the case instead of the classic semantics.
This means that $\cElm$ does not depend on the values of the other
strategies on tracks $\trkElm'$ that extend $\trkElm$, i.e., it does not
depend on future choices made on $\trkElm'$.
In addition, we have that $\cElm$ does not depend on values on parallel
tracks $\trkElm'$ that only share a prefix with $\trkElm$, i.e., it is
independent on choices made on the possibly alternative futures $\trkElm'$.
The elementary semantics of formulas involving atomic propositions,
Boolean connectives, temporal operators, and agent bindings is defined as
for the classic one, where the modeling relation $\models$ is substituted
with $\emodels$, and we omit to report it here.
In the following definition, we only describe the part concerning the
quantification prefixes.
Let $\GName$ be a , $\sElm \in \StSet$ one of its states, and $\qpElm
\psi$ an formula, where $\psi$ is agent-closed and $\qpElm \in
\QPSet(\free{\psi})$.
Then $\GName, \emptyfun, \sElm \emodels \qpElm \psi$ if there is an
elementary dependence map $\spcFun \in \ESpcSet[\StrSet(\sElm)](\qpElm)$
for $\qpElm$ over $\StrSet(\sElm)$ such that $\GName, \spcFun(\asgFun),
\sElm \emodels \psi$, for all $\asgFun \in \AsgSet(\QPAVSet{\qpElm},
\sElm)$.
It is immediate to see a strong similarity between the statement of
Corollary <ref> of strategy quantification and the
previous definition.
The only crucial difference resides in the choice of the kind of dependence
Moreover, observe that, differently from the classic semantics, the
quantifications in the prefix are not treated individually but as an atomic
This is due to the necessity of having a strict correlation between the
point-wise structure of the quantified strategies.
It can be interesting to know that we do not define an elementary
semantics for the whole , since we are not able, at the moment, to
easily use the concept of elementary dependence map, when the
quantifications are not necessarily grouped in prefixes, i.e., when the
formula is not in prenex normal form.
In fact, this may represent a challenging problem, whose solution is left
to future works.
Due to the new semantics of , we have to redefine the related concepts
of model and satisfiability, in order to differentiate between the classic
relation $\models$ and the elementary one $\emodels$.
Indeed, as we show later, there are sentences that are satisfiable but not
elementary satisfiable and vice versa.
We say that a $\GName$ is an elementary model of an sentence $\varphi$, in symbols $\GName \emodels \varphi$, if $\GName,
\emptyfun, \sElm[0] \emodels \varphi$.
In general, we also say that $\GName$ is a elementary model for
$\varphi$ on $\sElm \in \StSet$, in symbols $\GName, \sElm \emodels
\varphi$, if $\GName, \emptyfun, \sElm \emodels \varphi$.
An sentence $\varphi$ is elementarily satisfiable if there
is an elementary model for it.
We have to modify the concepts of implication and equivalence, as well.
Indeed, also in this case we can have pairs of equivalent formulas that are
not elementarily equivalent, and vice versa.
Thus, we have to be careful when we use natural transformation between
formulas, since it can be the case that they preserve the meaning only under
the classic semantics.
An example of this problem can arise when one want to put a formula in .
Given two formulas $\varphi_{1}$ and $\varphi_{2}$ with
$\free{\varphi_{1}} = \free{\varphi_{2}}$, we say that $\varphi_{1}$
elementarily implies $\varphi_{2}$, in symbols $\varphi_{1}
\eimplies \varphi_{2}$, if, for all s $\GName$, states $\sElm \in
\StSet$, and $\free{\varphi_{1}}$-defined $\sElm$-total assignments
$\asgFun \in \AsgSet(\free{\varphi_{1}}, \sElm)$, it holds that if
$\GName, \asgFun, \sElm \emodels \varphi_{1}$ then $\GName, \asgFun, \sElm
\emodels \varphi_{2}$.
Accordingly, we say that $\varphi_{1}$ is elementarily equivalent
to $\varphi_{2}$, in symbols $\varphi_{1} \!\eequiv\! \varphi_{2}$, if
both $\varphi_{1} \!\eimplies\! \varphi_{2}$ and $\varphi_{2}
\!\eimplies\! \varphi_{1}$ hold.
Elementariness and non-elementariness
Finally, we show that the introduced concept of elementary satisfiability is
relevant to the context of our logic, as its applicability represents a
demarcation line between “easy” and “hard” fragments of .
Moreover, we believe that it is because of this fundamental property that
several well-known temporal logics are so robustly decidable <cit.>.
It is interesting to observe that, for every $\GName$ and
[$0$-alt] sentence $\varphi$, it holds that $\GName \models \varphi$
iff $\GName \emodels \varphi$.
This is an immediate consequence of the fact that all quantification
prefixes $\qpElm$ used in $\varphi$ belong to
$\QPSet[\exists^{*}\forall^{*}](\VSet)$, for a given set $\VSet \subseteq
\VarSet$ of variables.
Thus, as already mentioned, the related dependence maps on strategies
$\spcFun \in \SpcSet[\StrSet( {\sElm[0]} )](\qpElm)$ are necessarily
By Corollary <ref> of strategy quantification, it is
easy to see that the following coherence property about the elementariness
of the satisfiability holds.
Intuitively, it asserts that every elementarily satisfiable sentence in
is satisfiable too.
Let $\GName$ be a , $\sElm \in \StSet$ one of its states, $\varphi$ an
formula in , and $\asgFun \in \AsgSet(\sElm)$ an $\sElm$-total
assignment with $\free{\varphi} \subseteq \dom{\asgFun}$.
Then, it holds that $\GName, \asgFun, \sElm \emodels \varphi$ implies
$\GName, \asgFun, \sElm \models \varphi$.
The proof proceeds by induction on the structure of the formula.
For the sake of succinctness, we only show the crucial case of principal
subsentences $\phi \in \psnt{\varphi}$, i.e., when $\phi$ is of the form
$\qpSym \psi$, where $\qpElm \in \QPSet(\free{\psi})$ is a quantification
prefix, and $\psi$ is an agent-closed formula.
Suppose that $\GName, \emptyfun, \sElm \emodels \qpSym \psi$.
Then, by Definition <ref> of elementary semantics, there is an elementary dependence map $\spcFun \in
\ESpcSet[ {\StrSet(\sElm)} ](\qpElm)$ such that, for all assignments
$\asgFun \in \AsgSet(\QPAVSet{\qpElm}, \sElm)$, it holds that $\GName,
\spcFun(\asgFun), \sElm \emodels \psi$.
Now, by the inductive hypothesis, there is a dependence map $\spcFun \in
\SpcSet[ {\StrSet(\sElm)} ](\qpElm)$ such that, for all assignments
$\asgFun \in \AsgSet(\QPAVSet{\qpElm}, \sElm)$, it holds that $\GName,
\spcFun(\asgFun), \sElm \models \psi$.
Hence, by Corollary <ref> of strategy quantification,
we have that $\GName, \emptyfun, \sElm \models \qpSym \psi$.
However, it is worth noting that the converse property may not hold, as we
show in the next theorem, i.e., there are sentences in that are
satisfiable but not elementarily satisfiable.
Note that the following results already holds for .
There exists a satisfiable [$1$-ag, $2$-var, $1$-alt] sentence in
that is not elementarily satisfiable.
Consider the [$1$-ag, $2$-var, $1$-alt] sentence $\varphi \defeq
\varphi_{1} \wedge \varphi_{2}$ in where $\varphi_{1} \defeq \qpSym
(\psi_{1} \wedge \psi_{2})$, with $\qpSym \defeq \AAll{\xSym}
\EExs{\ySym}$, $\psi_{1} \defeq (\alpha, \xSym) \X \pSym \leftrightarrow
(\alpha,\ySym) \X \neg \pSym$, and $\psi_{2} \defeq (\alpha, \xSym) \X \X
\pSym \leftrightarrow (\alpha, \ySym) \X \X \pSym$, and $\varphi_{2}
\defeq \AAll{\xSym} (\alpha, \xSym) \X ((\EExs{\xSym} (\alpha, \xSym) \X
\pSym) \wedge (\EExs{\xSym} (\alpha, \xSym) \X \neg \pSym))$.
Moreover, note that the [$1$-ag, $1$-var, $0$-alt] sentence
$\varphi_{2}$ is equivalent to the formula $\A \X ((\E \X \pSym)
\wedge (\E \X \neg \pSym))$.
Then, it is easy to see that the turn-based $\GName[Rdc]$ of
Figure fig:lmm:qptl(rdc) satisfies $\varphi$.
Indeed, $\GName[Rdc], \spcFun(\asgFun), \sSym[0] \models \psi_{1} \wedge
\psi_{2}$, for all assignments $\asgFun \in \AsgSet(\{ \xSym \},
\sSym[0])$, where the non-elementary dependence map $\spcFun \in \SpcSet[
{\StrSet(\sSym[0])} ](\qpSym)$ is such that
$\spcFun(\asgFun)(\ySym)(\sSym[0]) = \neg \asgFun(\xSym)(\sSym[0])$ and
$\spcFun(\asgFun)(\ySym)(\sSym[0] \cdot \sElm[i]) =
\asgFun(\xSym)(\sSym[0] \cdot \sElm[1 - i])$, for all $i \in \{ 0, 1 \}$.
Now, let $\GName$ be a generic .
If $\GName \not\models \varphi$, by Theorem <ref> of
elementary coherence, it holds that $\GName \not\emodels \varphi$.
Otherwise, we have that $\GName \models \varphi$ and, in particular,
$\GName \models \varphi_{1}$, which means that $\GName \models \qpSym
(\psi_{1} \wedge \psi_{2})$.
At this point, to prove that $\GName \not\emodels \varphi$, we show that,
for all elementary dependence maps $\spcFun \in \ESpcSet[
{\StrSet(\sElm[0])} ](\qpSym)$, there exists an assignment $\asgFun \in
\AsgSet(\{ \xSym \}, \sElm[0])$ such that $\GName, \spcFun(\asgFun),
\sElm[0] \not\emodels \psi_{1} \wedge \psi_{2}$.
To do this, let us fix an elementary dependence map $\spcFun$ and an
assignment $\asgFun$.
Also, assume $\sElm[1] \defeq \trnFun(\sElm[0], \allowbreak
\emptyset[\alpha \mapsto \asgFun(\xSym)(\sElm[0])])$ and
* $\pSym \in \labFun(\sElm[1])$ iff $\pSym \in \labFun(\sElm[2])$.
In this case, we can easily observe that $\GName, \spcFun(\asgFun),
\sElm[0] \not\models \psi_{1}$ and consequently, by
Theorem <ref>, it holds that $\GName,
\spcFun(\asgFun), \sElm[0] \not\emodels \psi_{1} \wedge \psi_{2}$.
So, we are done.
* $\pSym \in \labFun(\sElm[1])$ iff $\pSym \not\in \labFun(\sElm[2])$.
If $\GName, \spcFun(\asgFun), \sElm[0] \not\models \psi_{2}$ then, by
Theorem <ref>, it holds that $\GName,
\spcFun(\asgFun), \sElm[0] \not\emodels \psi_{1} \wedge \psi_{2}$.
So, we are done.
Otherwise, let $\sElm[3] \defeq \trnFun(\sElm[1], \emptyset[\alpha
\mapsto \asgFun(\xSym)(\sElm[0] \cdot \sElm[1])])$ and $\sElm[4]
\defeq \trnFun(\sElm[2], \emptyset[\alpha \mapsto
\spcFun(\asgFun)(\ySym)(\sElm[0] \cdot \sElm[2])])$.
Then, it holds that $\pSym \in \labFun(\sElm[3])$ iff $\pSym \in
\labFun(\sElm[4])$.
Now, consider a new assignment $\asgFun' \in \AsgSet(\{ \xSym \},
\sElm[0])$ such that $\asgFun'(\xElm)(\sElm[0] \cdot \sElm[2]) =
\asgFun(\xElm)(\sElm[0] \cdot \sElm[2])$ and $\pSym \in
\labFun(\sElm[3]')$ iff $\pSym \not\in \labFun(\sElm[4])$, where
$\sElm[3]' \defeq \trnFun(\sElm[1], \emptyset[\alpha \mapsto
\asgFun'(\xSym)(\sElm[0] \cdot \sElm[1])])$.
Observe that the existence of such an assignment, with particular
reference to the second condition, is ensured by the fact that $\GName
\models \varphi_{2}$.
At this point, due to the elementariness of the dependence map
$\spcFun$, we have that $\spcFun(\asgFun')(\ySym)(\sElm[0] \cdot
\sElm[2]) = \spcFun(\asgFun)(\ySym)(\sElm[0] \cdot \sElm[2])$.
Consequently, it holds that $\sElm[4] = \trnFun(\sElm[2],
\emptyset[\alpha \mapsto \spcFun(\asgFun')(\ySym)(\sElm[0] \cdot
\sElm[2])])$.
Thus, $\GName, \spcFun(\asgFun'), \sElm[0] \not\models \psi_{2}$,
which implies, by Theorem <ref>, that $\GName,
\spcFun(\asgFun'), \sElm[0] \not\emodels \psi_{1} \wedge \psi_{2}$.
So, we are done.
Thus, the thesis of the theorem holds.
The following corollary is an immediate consequence of the previous theorem.
It is interesting to note that, at the moment, we do not know if such a
result can be extended to the simpler fragment.
There exists a satisfiable [$1$-ag, $2$-var, $1$-alt] sentence in
that is not elementarily satisfiable.
It is worth remarking that the kind of non-elementariness of the
sentence $\varphi$ shown in the above theorem can be called
essential, i.e., it cannot be eliminated, due to the fact that
$\varphi$ is satisfiable but not elementarily satisfiable.
However, there are different sentences, such as the conjunct
$\varphi_{1}$ in $\varphi$, having both models on which they are
elementarily satisfiable and models, like the $\GName[Rdc]$, on
which they are only non-elementarily satisfiable.
Such a kind of non-elementariness can be called non-essential,
since it can be eliminated by an opportune choice of the underlying model.
Note that a similar reasoning can be done for the dual concept of
elementariness, which we call essential
Before continuing, we want to show the reason why we have redefined the
concepts of implication and equivalence in the context of elementary
Consider the [$1$-ag, $2$-var, $1$-alt] sentence $\varphi_{1}$ used in
Theorem <ref> of non-elementariness.
It is not hard to see that it is equivalent to the [$1$-ag, $1$-var,
$0$-alt] $\varphi' \defeq (\EExs{\xSym} (\alpha, \xSym) \psi_{1}
\leftrightarrow \EExs{\xSym} (\alpha, \xSym) \psi_{2}) \wedge (\EExs{\xSym}
(\alpha, \xSym) \psi_{3} \leftrightarrow \EExs{\xSym} (\alpha, \xSym)
\psi_{4})$, where $\psi_{1} \defeq \X (\pSym \wedge \X \pSym)$, $\psi_{2}
\defeq \X (\neg \pSym \wedge \X \pSym)$, $\psi_{3} \defeq \X (\pSym \wedge
\X \neg \pSym)$, and $\psi_{4} \defeq \X (\neg \pSym \wedge \X \neg \pSym)$.
Note that $\varphi'$ is in turn equivalent to the formula $(\E
\psi_{1} \leftrightarrow \E \psi_{2}) \wedge (\E \psi_{3} \leftrightarrow \E
\psi_{4})$.
However, $\varphi_{1}$ and $\varphi'$ are not elementarily equivalent, since
we have that $\GName[Rdc] \not\emodels \varphi_{1}$ but $\GName[Rdc]
\emodels \varphi'$, where $\GName[Rdc]$ is the of
Figure fig:lmm:qptl(rdc).
At this point, we can proceed with the proof of the elementariness of
satisfiability for .
It is important to note that there is no gap, in our knowledge, between the
logics that are elementarily satisfiable and those that are not, since the
fragment [$1$-ag, $2$-var, $1$-alt] used in the previous theorem
cannot be further reduced, due to the fact that otherwise it collapses into
Before starting, we have to describe some notation regarding classic
two-player games on infinite words <cit.>, which are used here as a
technical tool.
Note that we introduce the names of scheme and match in place of the more
usual strategy and play, in order to avoid confusion between the concepts
related to a and those related to the tool.
A two-player arena (, for short) is a tuple $\AName \defeq
\TPAStruct$, where $\NdESet$ and $\NdOSet$ are non-empty non-intersecting
sets of nodes for player even and odd, respectively,
$\EdgRel \defeq \EdgERel \cup \EdgORel$, with $\EdgERel \subseteq \NdESet
\times \NdOSet$ and $\EdgORel \subseteq \NdOSet \times \NdESet$, is the
edge relation between nodes, and $\nElm[0] \in \NdOSet$ is a
designated initial node.
An even position in $\AName$ is a finite non-empty sequence of nodes
$\posElm \in \NdESet^{+}$ such that $(\posElm)_{0} = \nElm[0]$ and, for all
$i \in \numco{0}{\card{\posElm} - 1}\!$, there exists a node $\nElm \in
\NdOSet$ for which $((\posElm)_{i}, \nElm) \in \EdgERel$ and $(\nElm,
(\posElm)_{i + 1}) \in \EdgORel$ hold.
In addition, an odd position in $\AName$ is a finite non-empty
sequence of nodes $\posElm = \posElm' \cdot \nElm \in \NdESet^{+} \cdot
\NdOSet$, with $\nElm \in \NdOSet$, such that $\posElm'$ is an even position
and $(\lst{\posElm'}, \nElm) \in \EdgERel$.
An even (resp., odd) scheme in $\AName$ is a function
$\scheFun : \PosESet \to \NdOSet$ (resp., $\schoFun : \PosOSet \to
\NdESet$) that maps each even (resp., odd) position to an odd (resp., even)
node in a way that is compatible with the edge relation $\EdgERel$ (resp.,
$\EdgORel$), i.e., for all $\posElm \in \PosESet$ (resp., $\posElm \in
\PosOSet$), it holds that $(\lst{\posElm}, \scheFun(\posElm)) \in \EdgERel$
(resp., $(\lst{\posElm}, \schoFun(\posElm)) \in \EdgORel$).
By $\SchESet$ (resp., $\SchOSet$) we indicate the sets of even (resp., odd)
A match in $\AName$ is an infinite sequence of nodes $\mtcElm \in
\NdESet^{\omega}$ such that $(\mtcElm)_{0} = \nElm[0]$ and, for all $i \in
\SetN$, there exists a node $\nElm \in \NdOSet$ such that $((\mtcElm)_{i},
\nElm) \in \EdgERel$ and $(\nElm, (\mtcElm)_{i + 1}) \in \EdgORel$.
By $\MtcSet$ we denote the set of all matches.
A match map $\mtcFun : \SchESet \times \SchOSet \to \MtcSet$ is a
function that, given two schemes $\scheFun \in \SchESet$ and $\schoFun \in
\SchOSet$, returns the unique match $\mtcElm = \mtcFun(\scheFun, \schoFun)$
such that, for all $i \in \SetN$, it holds that $(\mtcElm)_{i + 1} =
\schoFun((\mtcElm)_{\leq i} \cdot \scheFun((\mtcElm)_{\leq i}))$.
A two-player game (, for short) is a tuple $\HName \defeq
\TPGStruct$, where $\AName$ is a and $\WinSet \subseteq \MtcSet$.
On one hand, we say that player even wins $\HName$ if there exists an even
scheme $\scheFun \in \SchESet$ such that, for all odd schemes $\schoFun \in
\SchOSet$, it holds that $\mtcFun(\scheFun, \schoFun) \in \WinSet$.
On the other hand, we say that player odd wins $\HName$ if there exists an
odd scheme $\schoFun \in \SchOSet$ such that, for all even schemes $\scheFun
\in \SchESet$, it holds that $\mtcFun(\scheFun, \schoFun) \not\in \WinSet$.
In the following, for a given binding prefix $\bpElm \in \BndSet(\VSet)$
with $\VSet \subseteq \VarSet$, we denote by $\bndFun[\bpElm] : \AgSet \to
\VSet$ the function associating with each agent the related variable in
$\bpElm$, i.e., for all $\aElm \in \AgSet$, there is $i \in
\numco{0}{\card{\bpElm}}$ such that $(\bpElm)_{i} = (\aElm,
\bndFun[\bpElm](\aElm))$.
As first step towards the proof of the elementariness of , we have to
give a construction of a two-player game, based on an a priori chosen ,
in which the players are explicitly viewed one as a dependence map and the
other as a valuation, both over actions.
This construction results to be a deep technical evolution of the proof
method used for the dualization of alternating automata on infinite
objects <cit.>.
Let $\GName$ be a , $\sElm \in \StSet$ one of its states, $\PSet
\subseteq \PthSet(\sElm)$ a set of paths, $\qpElm \in \QPSet(\VSet)$ a
quantification prefix over a set $\VSet \subseteq \VarSet$ of variables,
and $\bpElm \in \BndSet(\VSet)$ a binding.
Then, the dependence-vs-valuation game for $\GName$ in $\sElm$ over
$\PSet$ w.r.t. $\qpElm$ and $\bpElm$ is the $\HName(\GName, \sElm,
\PSet, \qpElm, \bpElm) \defeq \TPGTuple {\AName(\GName, \sElm, \qpElm,
\bpElm)} {\PSet}$, where the $\AName(\GName, \sElm, \qpElm, \bpElm)
\defeq \TPATuple {\StSet} {\StSet \times \SpcSet[\AcSet](\qpElm)}
{\EdgRel} {\sElm}$ has the edge relations defined as follows:
* $\EdgERel \defeq \set{ (\tElm, (\tElm, \spcFun)) }{ \tElm \in \StSet
\land \spcFun \in \SpcSet[\AcSet](\qpElm) }$;
* $\EdgORel \defeq \set{ ((\tElm, \spcFun), \trnFun(\tElm,
\spcFun(\valFun) \cmp \bndFun[\bpElm])) }{ \tElm \in \StSet \land
\spcFun \in \SpcSet[\AcSet](\qpElm) \land \valFun \in
\ValSet[\AcSet](\QPAVSet{\qpElm}) }$ [By $\gFun[2] \cmp
\gFun[1] : \XSet \to \ZSet$ we denote the operation of
composition of two functions $\gFun[1] : \XSet \to \YSet[1]$
and $\gFun[2] : \YSet[2] \to \ZSet$ with $\YSet[1] \subseteq
\YSet[2]$.].
In the next lemma we state a fundamental relationship between
dependence-vs-valuation games and their duals.
Basically, we prove that if a player wins the game then the opposite player
can win the dual game, and vice versa.
Let $\GName$ be a , $\sElm \in \StSet$ one of its states, $\PSet
\subseteq \PthSet(\sElm)$ a set of paths, $\qpElm \in \QPSet(\VSet)$ a
quantification prefix over a set $\VSet \subseteq \VarSet$ of variables,
and $\bpElm \in \BndSet(\VSet)$ a binding.
Then, player even wins the $\HName(\GName, \sElm, \PSet, \qpElm,
\bpElm)$ iff player odd wins the dual $\HName(\GName, \sElm,
\PthSet(\sElm) \setminus \PSet, \dual{\qpElm}, \bpElm)$.
Now, we are going to give the definition of the important concept of
Informally, an encasement is a particular subset of paths in a given that “works to encase” an elementary dependence map on strategies, in the
sense that it contains all plays obtainable by complete assignments derived
from the evaluation of the above mentioned dependence map.
In our context, this concept is used to summarize all
Let $\GName$ be a , $\sElm \in \StSet$ one of its states, $\PSet
\subseteq \PthSet(\sElm)$ a set of paths, $\qpElm \in \QPSet(\VSet)$ a
quantification prefix over a set $\VSet \subseteq \VarSet$ of variables,
and $\bpElm \in \BndSet(\VSet)$ a binding.
Then, $\PSet$ is an encasement w.r.t. $\qpElm$ and $\bpElm$ if
there exists an elementary dependence map $\spcFun \!\in\! \ESpcSet[
{\StrSet(\sElm)} ](\qpElm)$ such that, for all assignments $\asgFun
\!\in\! \AsgSet(\QPAVSet{\qpElm}, \sElm)$, it holds that
$\playFun(\spcFun(\asgFun) \cmp \bndFun[\bpElm], \sElm) \!\in\! \PSet$.
In the next lemma, we give the second of the two crucial steps in our
elementariness proof.
In particular, we are able to show a one-to-one relationship between the
wining in the dependence-vs-valuation game of player even and the
verification of the encasement property of the associated winning set.
Moreover, in the case that the latter is a Borelian set, by using Martin's
Determinacy Theorem <cit.>, we obtain a complete characterization of
the winning concept by means of that of encasements.
Let $\GName$ be a , $\sElm \in \StSet$ one of its states, $\PSet
\subseteq \PthSet(\sElm)$ a set of paths, $\qpElm \in \QPSet(\VSet)$ a
quantification prefix over a set $\VSet \subseteq \VarSet$ of variables,
and $\bpElm \in \BndSet(\VSet)$ a binding.
Then, the following hold:
player even wins $\HName(\GName, \sElm, \PSet, \qpElm, \bpElm)$ iff
$\PSet$ is an encasement w.r.t. $\qpElm$ and $\bpElm$;
if player odd wins $\HName(\GName, \sElm, \PSet, \qpElm, \bpElm)$ then
$\PSet$ is not an encasement w.r.t. $\qpElm$ and $\bpElm$;
if $\PSet$ is a Borelian set and it is not an encasement w.r.t. $\qpElm$ and $\bpElm$ then player odd wins $\HName(\GName, \sElm,
\PSet, \qpElm, \bpElm)$.
Finally, we have all technical tools useful to prove the elementariness of
the satisfiability for .
Intuitively, we describe a bidirectional reduction of the problem of
interest to the verification of the winning in the dependence-vs-valuation
The idea behind this construction resides in the strong similarity between
the statement of Corollary <ref> of strategy
quantification and the definition of the winning condition in a two-player
Indeed, on one hand, we say that a sentence is satisfiable iff “there
exists a dependence map such that, for all all assignments, it holds that
On the other hand, we say that player even wins a game iff “there exists
an even scheme such that, for all odd schemes, it holds that ...”.
In particular, for the fragment, we can resolve the gap between these
two formulations, by using the concept of elementary quantification.
Let $\GName$ be a , $\varphi$ an formula, $\sElm \in \StSet$ a
state, and $\asgFun \in \AsgSet(\sElm)$ an $\sElm$-total assignment with
$\free{\varphi} \subseteq \dom{\asgFun}$.
Then, it holds that $\GName, \asgFun, \sElm \models \varphi$ iff $\GName,
\asgFun, \sElm \emodels \varphi$.
The proof proceeds by induction on the structure of the formula.
For the sake of succinctness, we only show the most important inductive
case of principal subsentences $\phi \in \psnt{\varphi}$, i.e., when
$\phi$ is of the form $\qpSym \bpElm \psi$, where $\qpElm \in
\QPSet(\VSet)$ and $\bpElm \in \BPSet(\VSet)$ are, respectively, a
quantification and binding prefix over a set $\VSet \subseteq \VarSet$ of
variables, and $\psi$ is a variable-closed formula.
The proof of this direction is practically the same of the one used in
Theorem <ref> of elementary coherence.
So, we omit to report it here.
[Only if].
Assume that $\GName, \emptyfun, \sElm \models \qpSym \bpElm \psi$.
Then, it is easy to see that, for all elementary dependence maps $\spcFun
\in \ESpcSet[ {\StrSet(\sElm)} ](\dual{\qpElm})$, there is an assignment
$\asgFun \in \AsgSet(\QPAVSet{\dual{\qpElm}}, \sElm)$ such that $\GName,
\spcFun(\asgFun) \cmp \bndFun[\bpElm], \sElm \models \psi$.
Indeed, suppose by contradiction that there exists an elementary
dependence map $\spcFun \in \ESpcSet[ {\StrSet(\sElm)} ](\dual{\qpElm})$
such that, for all assignments $\asgFun \in
\AsgSet(\QPAVSet{\dual{\qpElm}}, \sElm)$, it holds that $\GName,
\spcFun(\asgFun) \cmp \bndFun[\bpElm], \sElm \not\models \psi$, i.e.,
$\GName, \spcFun(\asgFun) \cmp \bndFun[\bpElm], \sElm \models \neg \psi$,
and so $\GName, \spcFun(\asgFun), \sElm \models \bpElm \neg \psi$.
Then, by Corollary <ref> of strategy quantification,
we have that $\GName, \emptyfun, \sElm \models \dual{\qpElm} \bpElm \neg
\psi$, i.e., $\GName, \emptyfun, \sElm \models \neg \qpElm \bpElm \psi$,
and so $\GName, \emptyfun, \sElm \not\models \qpElm \bpElm \psi$, which is
Now, let $\PSet \defeq \set{ \playFun(\asgFun, \sElm) \in \PthSet(\sElm)
}{ \asgFun \in \AsgSet(\AgSet, \sElm) \land \GName, \asgFun, \sElm
\not\models \psi}$.
Then, it is evident that, for all elementary dependence maps $\spcFun \in
\ESpcSet[ {\StrSet(\sElm)} ](\dual{\qpElm})$, there is an assignment
$\asgFun \in \AsgSet(\QPAVSet{\dual{\qpElm}}, \sElm)$ such that
$\playFun(\spcFun(\asgFun) \cmp \bndFun[\bpElm], \sElm) \not\in \PSet$.
At this point, by Definition <ref> of encasement, it is
clear that $\PSet$ is not an encasement w.r.t. $\dual{\qpElm}$ and
Moreover, since $\psi$ describes a regular language, the derived set
$\PSet$ is Borelian <cit.>.
Consequently, by Item <ref> of
Lemma <ref> of encasement characterization, we have that
player odd wins the $\HName(\GName, \sElm, \PSet, \dual{\qpElm},
\bpElm)$.
Thus, by Lemma <ref> of dependence-vs-valuation duality,
player even wins the dual $\HName(\GName, \sElm, \PthSet(\sElm)
\setminus \PSet, \qpElm, \bpElm)$.
Hence, by Item <ref> of Lemma <ref>,
we have that $\PthSet(\sElm) \setminus \PSet$ is an encasement w.r.t. $\qpElm$ and $\bpElm$.
Finally, again by Definition <ref>, there exists an
elementary dependence map $\spcFun \in \ESpcSet[ {\StrSet(\sElm)}
](\qpElm)$ such that, for all assignments $\asgFun \in
\AsgSet(\QPAVSet{\qpElm}, \sElm)$, it holds that
$\playFun(\spcFun(\asgFun) \cmp \bndFun[\bpElm], \sElm) \in \PthSet(\sElm)
\setminus \PSet$.
Now, it is immediate to observe that $\PthSet(\sElm) \setminus \PSet =
\set{ \playFun(\asgFun, \sElm) \in \PthSet(\sElm) }{ \asgFun \in
\AsgSet(\AgSet, \sElm) \land \GName, \asgFun, \sElm \models \psi}$.
So, by the inductive hypothesis, we have that $\PthSet(\sElm) \setminus
\PSet = \set{ \playFun(\asgFun, \sElm) \in \PthSet(\sElm) }{ \asgFun \in
\AsgSet(\AgSet, \sElm) \land \GName, \asgFun, \sElm \emodels \psi}$, from
which we derive that there exists an elementary dependence map $\spcFun
\in \ESpcSet[ {\StrSet(\sElm)} ](\qpElm)$ such that, for all assignments
$\asgFun \in \AsgSet(\QPAVSet{\qpElm}, \sElm)$, it holds that $\GName,
\spcFun(\asgFun) \cmp \bndFun[\bpElm], \sElm \emodels \psi$.
Consequently, by Definition <ref> of elementary semantics, we have that $\GName, \emptyfun, \sElm \emodels
\qpSym \bpElm \psi$.
As an immediate consequence of the previous theorem, we derive the following
fundamental corollary.
Let $\GName$ be a and $\varphi$ an sentence.
Then, $\GName \models \varphi$ iff $\GName \emodels \varphi$.
It is worth to observe that the elementariness property for is a
crucial difference w.r.t. , which allows us to obtain an elementary
decision procedure for the related model-checking problem, as described in
the last part of the next section.
Model-Checking Procedures
In this section, we study the model-checking problem for and show that,
in general, it is non-elementarily decidable, while, in the particular case of
sentences, it is just 2, as for .
For the algorithmic procedures, we follow an automata-theoretic
approach <cit.>, reducing the decision problem for the logics to the
emptiness problem of an automaton.
In particular, we use a bottom-up technique through which we recursively
label each state of the of interest by all principal subsentences of the
specification that are satisfied on it, starting from the innermost
subsentences and terminating with the sentence under exam.
In this way, at a given step of the recursion, since the satisfaction of all
subsentences of the given principal sentence has already been determined, we
can assume that the matrix of the latter is only composed by Boolean
combinations and nesting of goals whose temporal part is simply .
The procedure we propose here extends that used for in <cit.> by
means of a richer structure of the automata involved in.
The rest of this section is organized as follows.
In Subsection <ref>, we recall the definition of alternating
parity tree automata.
Then, in Subsection <ref>, we build an automaton accepting a
tree encoding of a iff this satisfies the formula of interest, which is
used to prove the main result about and model checking.
Finally, in Subsection <ref>, we refine the previous result
to obtain an elementary decision procedure for .
Alternating tree automata
Nondeterministic tree automata are a generalization to infinite trees
of the classical nondeterministic word automata on infinite words
(see <cit.>, for an introduction).
Alternating tree automata are a further generalization of
nondeterministic tree automata <cit.>.
Intuitively, on visiting a node of the input tree, while the latter sends
exactly one copy of itself to each of the successors of the node, the former
can send several own copies to the same successor.
Here we use, in particular, alternating parity tree automata, which
are alternating tree automata along with a parity acceptance
condition (see <cit.>, for a survey).
We now give the formal definition of alternating tree automata.
An alternating tree automaton (, for short) is a tuple
$\AName \defeq \ATAStruct$, where $\LabSet$, $\DirSet$, and $\QSet$ are,
respectively, non-empty finite sets of input symbols,
directions, and states, $\qElm[0] \in \QSet$ is an
initial state, $\aleph$ is an acceptance condition to be
defined later, and $\delta : \QSet \times \LabSet \to \PBoolSet(\DirSet
\times \QSet)$ is an alternating transition function that maps each
pair of states and input symbols to a positive Boolean combination on the
set of propositions of the form $(\dElm, \qElm) \in \DirSet \times \QSet$,
a.k.a. moves.
On one side, a nondeterministic tree automaton (, for
short) is a special case of in which each conjunction in the
transition function $\delta$ has exactly one move $(\dElm, \qElm)$
associated with each direction $\dElm$.
This means that, for all states $\qElm \in \QSet$ and symbols $\sigma \in
\LabSet$, we have that $\atFun(\qElm, \sigma)$ is equivalent to a Boolean
formula of the form $\bigvee_{i} \bigwedge_{\dElm \in \DirSet} (\dElm,
\qElm[i, \dElm])$.
On the other side, a universal tree automaton (, for
short) is a special case of in which all the Boolean combinations that
appear in $\delta$ are conjunctions of moves.
Thus, we have that $\atFun(\qElm, \sigma) = \bigwedge_{i} (\dElm[i],
\qElm[i])$, for all states $\qElm \in \QSet$ and symbols $\sigma \in
\LabSet$.
The semantics of the s is given through the following concept of run.
A run of an $\AName = \ATAStruct$ on a $\LabSet$-labeled
$\DirSet$-tree $\TName = \LTStruct$ is a $(\DirSet \times \QSet)$-tree
$\RSet$ such that, for all nodes $\xElm \in \RSet$, where $\xElm =
\prod_{i = 1}^{n} (\dElm[i], \qElm[i])$ and $\yElm \defeq \prod_{i =
1}^{n} \dElm[i]$ with $n \in \numco{0}{\omega}$, it holds that (i)
$\yElm \in \TSet$ and (ii), there is a set of moves $\SSet
\subseteq \DirSet \times \QSet$ with $\SSet \models \delta(\qElm[n],
\vFun(\yElm))$ such that $\xElm \cdot (\dElm, \qElm) \in \RSet$, for all
$(\dElm, \qElm) \in \SSet$.
In the following, we consider s along with the parity acceptance
condition (, for short) $\aleph \defeq (\FSet_{1}, \ldots,
\FSet_{k}) \in (\pow{\QSet})^{+}$ with $\FSet_{1} \subseteq \ldots \subseteq
\FSet_{k} = \QSet$ (see <cit.>, for more).
The number $k$ of sets in the tuple $\aleph$ is called the index of
the automaton.
We also consider s with the co-Büchi acceptance condition
(, for short) that is the special parity condition with index
Let $\RSet$ be a run of an $\AName$ on a tree $\TName$ and $\wElm$ one
of its branches.
Then, by $\infFun(\wElm) \defeq \set{ \qElm \in \QSet }{ \card{\set{ i \in
\SetN }{ \exists \dElm \in \DirSet . (w)_{i} = (\dElm, \qElm) }} = \omega }$
we denote the set of states that occur infinitely often as the second
component of the letters along the branch $w$.
Moreover, we say that $w$ satisfies the parity acceptance condition $\aleph
\!=\! (\FSet_{1}, \ldots, \FSet_{k})$ if the least index $i \!\in\!
\numcc{1}{k}$ for which $\infFun(w) \cap \FSet_{i} \neq \emptyset$ is even.
At this point, we can define the concept of language accepted by an .
An $\AName = \ATAStruct$ accepts a $\LabSet$-labeled
$\DirSet$-tree $\TName$ iff is there exists a run $\RSet$ of $\AName$ on
$\TName$ such that all its infinite branches satisfy the acceptance
condition $\aleph$.
By $\LangSet(\AName)$ we denote the language accepted by the $\AName$,
i.e., the set of trees $\TName$ accepted by $\AName$.
Moreover, $\AName$ is said to be empty if $\LangSet(\AName) =
\emptyset$.
The emptiness problem for $\AName$ is to decide whether
$\LangSet(\AName) = \emptyset$.
We finally show a simple but useful result about the direction
To do this, we first need to introduce an extra notation.
Let $\fElm \in \BoolSet(\PSet)$ be a Boolean formula on a set of
propositions $\PSet$.
Then, by $\fElm[][\pElm / \qElm \:\!|\:\! \pElm \in \PSet']$ we denote the
formula in which all occurrences of the propositions $\pElm \in \PSet'
\subseteq \PSet$ in $\fElm$ are replaced by the proposition $\qElm$
belonging to a possibly different set.
Let $\AName \defeq \TATuple {\LabSet \times \DirSet} {\DirSet} {\QSet}
{\atFun} {\qElm[0]} {\aleph}$ be an over a set of $m$ directions
with $n$ states and index $k$.
Moreover, let $\dElm[0] \in \DirSet$ be a distinguished direction.
Then, there exists an $\NName^{\dElm[0]} \defeq \TATuple {\LabSet}
{\DirSet} {\QSet'} {\atFun'} {\qElm[0 | ']} {\aleph'}$ with $m \cdot
2^{\AOmicron{k \cdot n \cdot \log n}}$ states and index $\AOmicron{k \cdot
n \cdot \log n}$ such that, for all $\LabSet$-labeled $\DirSet$-tree
$\TName \defeq \LTStruct$, it holds that $\TName \in
\LangSet(\NName^{\dElm[0]})$ iff $\TName' \in \LangSet(\AName)$, where
$\TName'$ is the $(\LabSet \times \DirSet)$-labeled $\DirSet$-tree
$\LTTuple{}{}{\TSet}{\vFun'}$ such that $\vFun'(\tElm) \defeq (\vFun(t),
\lst{\dElm[0] \cdot t})$, for all $\tElm \in \TSet$.
As first step, we use the well-known nondeterminization procedure for
s <cit.> in order to transform the $\AName$ into an
equivalent $\NName = \TATuple {\LabSet \times \DirSet} {\DirSet}
{\QSet''} {\atFun''} {\qElm[0 | '']} {\aleph''}$ with $2^{\AOmicron{k
\cdot n \cdot \log n}}$ states and index $k' = \AOmicron{k \cdot n \cdot
\log n}$.
Then, we transform the latter into the new $\NName^{\dElm[0]} \defeq
\TATuple {\LabSet} {\DirSet} {\QSet'} {\atFun'} {\qElm[0 | ']} {\aleph'}$
with $m \cdot 2^{\AOmicron{k \cdot n \cdot \log n}}$ states and same index
$k'$, where $\QSet' \defeq \QSet'' \times \DirSet$, $\qElm[0 | '] \defeq
(\qElm[0 | ''], \dElm[0])$, $\aleph' \defeq (\FSet[1] \times \DirSet,
\ldots, \FSet[k'] \times \DirSet)$ with $\aleph'' \defeq (\FSet[1],
\ldots, \FSet[k'])$, and $\atFun'((\qElm, \dElm), \sigma) \defeq
\atFun''(\qElm, (\sigma, \dElm)) [(\dElm', \qElm') / (\dElm', (\qElm',
\dElm')) \:\!|\:\! (\dElm', \qElm') \in \DirSet \times \QSet'']$, for all
$(\qElm, \dElm) \in \QSet'$ and $\sigma \in \LabSet$.
Now, it easy to see that $\NName^{\dElm[0]}$ satisfies the declared
Model Checking
A first step towards our construction of an algorithmic procedure for the
solution of the model-checking problem is to define, for each possible
formula $\varphi$, an alternating parity tree automaton $\AName[\varphi |
^{\GName}]$ that recognizes a tree encoding $\TName$ of a $\GName$, containing the information on an assignment $\asgFun$ on the free
variables/agents of $\varphi$, iff $\GName$ is a model of $\varphi$ under
The high-level idea at the base of this construction is an evolution and
merging of those behind the translations of and , respectively,
To proceed with the formal description of the model-checking procedure, we
have to introduce a concept of encoding for the assignments of a .
Let $\GName$ be a , $\sElm \in \StSet[\GName]$ one of its states, and
$\asgFun \in \AsgSet[\GName](\VSet, \sElm)$ an assignment defined on the
set $\VSet \subseteq \VarSet \cup \AgSet$.
Then, a $(\ValSet[ {\AcSet[\GName]} ](\VSet) \times
\StSet[\GName])$-labeled $\StSet[\GName]$-tree $\TName \defeq
\LTTuple{}{}{\TSet}{\uFun}$, where $\TSet \defeq \set{ \trkElm_{\geq 1} }{
\trkElm \in \TrkSet[\GName](\sElm) }$, is an assignment-state
encoding for $\asgFun$ if it holds that $\uFun(\tElm) \defeq
(\flip{\asgFun}(\sElm \cdot \tElm), \lst{\sElm \cdot \tElm})$, for all
$\tElm \in \TSet$.
Observe that there is a unique assignment-state encoding for each given
In the next lemma, we prove the existence of an for each and
formula that is able to recognize all the assignment-state encodings of
an a priori given assignment, made the assumption that the formula is
satisfied on the under this assignment.
Let $\GName$ be a and $\varphi$ an formula.
Then, there exists an $\AName[\varphi | ^{\GName}] \defeq \TATuple
{\ValSet[ {\AcSet[\GName]} ](\free{\varphi}) \times \StSet[\GName]}
{\StSet[\GName]} {\QSet[\varphi]} {\atFun[\varphi]} {\qElm[0\varphi]}
{\aleph_{\varphi}}$ such that, for all states $\sElm \in \StSet[\GName]$
and assignments $\asgFun \in \AsgSet[\GName](\free{\varphi}, \sElm)$, it
holds that $\GName, \asgFun, \sElm \models \varphi$ iff $\TName \in
\LangSet(\AName[\varphi | ^{\GName}])$, where $\TName$ is the
assignment-state encoding for $\asgFun$.
The construction of the $\AName[\varphi | ^{\GName}]$ is done
recursively on the structure of the formula $\varphi$, which w.l.o.g. is supposed to be in , by using a variation of the transformation,
via alternating tree automata, of the S$1$S and S$k$S logics into
nondeterministic Büchi word and tree automata recognizing all models of
the formula of interest <cit.>.
The detailed construction of $\AName[\varphi | ^{\GName}]$, by a case
analysis on $\varphi$, follows.
* If $\varphi \in \APSet$, the automaton has to verify if the atomic
proposition is locally satisfied or not.
To do this, we set $\AName[\varphi | ^{\GName}] \defeq \TATuple
{\ValSet[ {\AcSet[\GName]} ](\emptyset) \times \StSet[\GName]}
{\StSet[\GName]} {\{ \varphi \}} {\atFun[\varphi]} {\varphi} {(\{
\varphi \})}$, where $\atFun[\varphi](\varphi, (\valFun, \sElm))
\defeq \Tt$, if $\varphi \in \labFun[\GName](\sElm)$, and
$\atFun[\varphi](\varphi, (\valFun, \sElm)) \defeq \Ff$, otherwise.
Intuitively, $\AName[\varphi | ^{\GName}]$ only verifies that the
state $\sElm$ in the labeling of the root of the assignment-state
encoding of the empty assignment $\emptyfun$ satisfies $\varphi$.
* The boolean case $\varphi = \neg \varphi'$ is treated in the classical
way, by simply dualizing the automaton $\AName[\varphi' | ^{\GName}] =
\TATuple {\ValSet[ {\AcSet[\GName]} ](\free{\varphi'}) \times
\StSet[\GName]} {\StSet[\GName]} {\QSet[\varphi']} {\atFun[\varphi']}
{\qElm[0\varphi']} {\aleph_{\varphi'}}$ <cit.>.
* The boolean cases $\varphi = \varphi_{1} \Opr \varphi_{2}$, with $\Opr
\in \{ \wedge, \vee \}$, are treated in a way that is similar to the
classical one, by simply merging the two automata $\AName[\varphi_{1}
| ^{\GName}] = \TATuple {\ValSet[ {\AcSet[\GName]}
](\free{\varphi_{1}}) \times \StSet[\GName]} {\StSet[\GName]}
{\QSet[\varphi_{1}]} {\atFun[\varphi_{1}]} {\qElm[0\varphi_{1}]}
{\aleph_{\varphi_{1}}}$ and $\AName[\varphi_{2} | ^{\GName}] =
\TATuple {\ValSet[ {\AcSet[\GName]} ](\free{\varphi_{2}}) \times
\StSet[\GName]} {\StSet[\GName]} {\QSet[\varphi_{2}]}
{\atFun[\varphi_{2}]} {\qElm[0\varphi_{2}]} {\aleph_{\varphi_{2}}}$
into the automaton $\AName[\varphi | ^{\GName}] \defeq \TATuple
{\ValSet[ {\AcSet[\GName]} ](\free{\varphi}) \times \StSet[\GName]}
{\StSet[\GName]} {\QSet[\varphi]} {\atFun[\varphi]} {\qElm[0\varphi]}
{\aleph_{\varphi}}$, where the following hold:
* $\QSet[\varphi] \defeq \{ \qElm[0\varphi] \} \cup
\QSet[\varphi_{1}] \cup \QSet[\varphi_{2}]$, with $\qElm[0\varphi]
\not\in \QSet[\varphi_{1}] \cup \QSet[\varphi_{2}]$;
* $\atFun[\varphi](\qElm[0\varphi], (\valFun, \sElm)) \defeq
\atFun[\varphi_{1}](\qElm[0\varphi_{1}], (\valFun_{\rst
\free{\varphi_{1}}}, \sElm)) \:\Opr
\atFun[\varphi_{2}](\qElm[0\varphi_{2}], (\valFun_{\rst
\free{\varphi_{2}}}, \sElm))$, for all $(\valFun, \sElm) \in
\ValSet[ {\AcSet[\GName]} ](\free{\varphi}) \times
\StSet[\GName]$;
* $\atFun[\varphi](\qElm, (\valFun, \sElm)) \defeq
\atFun[\varphi_{1}](\qElm, (\valFun_{\rst \free{\varphi_{1}}},
\sElm))$, if $\qElm \in \QSet[\varphi_{1}]$, and
$\atFun[\varphi](\qElm, (\valFun, \sElm)) \defeq
\atFun[\varphi_{2}](\qElm, (\valFun_{\rst \free{\varphi_{2}}},
\sElm))$, otherwise, for all $\qElm \in \QSet[\varphi_{1}] \cup
\QSet[\varphi_{2}]$ and $(\valFun, \sElm) \in \ValSet[
{\AcSet[\GName]} ](\free{\varphi}) \times \StSet[\GName]$;
* $\aleph_{\varphi} \defeq (\FSet[1\varphi], \ldots,
\FSet[k\varphi])$, where (i) $\aleph_{\varphi_{1}} \defeq
(\FSet[1\varphi_{1}], \ldots, \FSet[k_{1}\varphi_{1}])$ and
$\aleph_{\varphi_{2}} \defeq (\FSet[1\varphi_{2}], \ldots,
\FSet[k_{2}\varphi_{2}])$, (ii) $h = \min \{ k_{1}, k_{2}
\}$ and $k = \max \{ k_{1}, k_{2} \}$, (iii)
$\FSet[i\varphi] \defeq \FSet[i\varphi_{1}] \cup
\FSet[i\varphi_{2}]$, for $i \in \numcc{1}{h}$, (iv)
$\FSet[i\varphi] \defeq \FSet[i\varphi_{j}]$, for $i \in
\numcc{h + 1}{k - 1}$ with $k_{j} = k$, and (v)
$\FSet[k\varphi] \defeq \QSet[\varphi]$.
* The case $\varphi = \X \varphi'$ is solved by running the automaton
$\AName[\varphi' | ^{\GName}] = \TATuple {\ValSet[ {\AcSet[\GName]}
](\free{\varphi'}) \times \StSet[\GName]} {\StSet[\GName]}
{\QSet[\varphi']} {\atFun[\varphi']} {\qElm[0\varphi']}
{\aleph_{\varphi'}}$ on the successor node of the root of the
assignment-state encoding in the direction individuated by the
assignment itself.
To do this, we use the automaton $\AName[\varphi | ^{\GName}] \defeq
\TATuple {\ValSet[ {\AcSet[\GName]} ](\free{\varphi}) \times
\StSet[\GName]} {\StSet[\GName]} {\QSet[\varphi]} {\atFun[\varphi]}
{\qElm[0\varphi]} {\aleph_{\varphi}}$, where the following hold:
* $\QSet[\varphi] \defeq \{ \qElm[0\varphi] \} \cup
\QSet[\varphi']$, with $\qElm[0\varphi] \not\in \QSet[\varphi']$;
* $\atFun[\varphi](\qElm[0\varphi], (\valFun, \sElm)) \defeq
(\trnFun[\GName](\sElm, \valFun_{\rst \AgSet}),
\qElm[0\varphi'])$, for all $(\valFun, \sElm) \in \ValSet[
{\AcSet[\GName]} ](\free{\varphi}) \times \StSet[\GName]$;
* $\atFun[\varphi](\qElm, (\valFun, \sElm)) \defeq
\atFun[\varphi'](\qElm, (\valFun_{\rst \free{\varphi'}}, \sElm))$,
for all $\qElm \in \QSet[\varphi']$ and $(\valFun, \sElm) \in
\ValSet[ {\AcSet[\GName]} ](\free{\varphi}) \times
\StSet[\GName]$;
* $\aleph_{\varphi} \defeq (\FSet[1\varphi'], \ldots,
\FSet[k\varphi'] \cup \{ \qElm[0\varphi] \})$, where
$\aleph_{\varphi'} \defeq (\FSet[1\varphi'], \ldots,
\FSet[k\varphi'])$.
* To handle the case $\varphi = \varphi_{1} \U \varphi_{2}$, we use
the automaton $\AName[\varphi | ^{\GName}] \defeq \TATuple {\ValSet[
{\AcSet[\GName]} ](\free{\varphi}) \times \StSet[\GName]}
{\StSet[\GName]} {\QSet[\varphi]} {\atFun[\varphi]} {\qElm[0\varphi]}
{\aleph_{\varphi}}$ that verifies the truth of the until operator
using its one-step unfolding equivalence $\varphi_{1} \U \varphi_{2}
\equiv \varphi_{2} \vee \varphi_{1} \wedge \X \varphi_{1} \U
\varphi_{2}$, by appropriately running the two automata
$\AName[\varphi_{1} | ^{\GName}] = \TATuple {\ValSet[ {\AcSet[\GName]}
](\free{\varphi_{1}}) \times \StSet[\GName]} {\StSet[\GName]}
{\QSet[\varphi_{1}]} {\atFun[\varphi_{1}]} {\qElm[0\varphi_{1}]}
{\aleph_{\varphi_{1}}}$ and $\AName[\varphi_{2} | ^{\GName}] =
\TATuple {\ValSet[ {\AcSet[\GName]} ](\free{\varphi_{2}}) \times
\StSet[\GName]} {\StSet[\GName]} {\QSet[\varphi_{2}]}
{\atFun[\varphi_{2}]} {\qElm[0\varphi_{2}]} {\aleph_{\varphi_{2}}}$
for the inner formulas $\varphi_{1}$ and $\varphi_{2}$.
* $\QSet[\varphi] \defeq \{ \qElm[0\varphi] \} \cup
\QSet[\varphi_{1}] \cup \QSet[\varphi_{2}]$, with $\qElm[0\varphi]
\not\in \QSet[\varphi_{1}] \cup \QSet[\varphi_{2}]$;
* $\atFun[\varphi](\qElm[0\varphi], (\valFun, \sElm)) \defeq
\atFun[\varphi_{2}](\qElm[0\varphi_{2}], (\valFun_{\rst
\free{\varphi_{2}}}, \sElm)) \vee
\atFun[\varphi_{1}](\qElm[0\varphi_{1}], (\valFun_{\rst
\free{\varphi_{1}}}, \sElm)) \wedge (\trnFun[\GName](\sElm,
\valFun_{\rst \AgSet}), \qElm[0\varphi])$, for all $(\valFun,
\sElm) \in \ValSet[ {\AcSet[\GName]} ](\free{\varphi}) \times
\StSet[\GName]$;
* $\atFun[\varphi](\qElm, (\valFun, \sElm)) \defeq
\atFun[\varphi_{1}](\qElm, (\valFun_{\rst \free{\varphi_{1}}},
\sElm))$, if $\qElm \in \QSet[\varphi_{1}]$, and
$\atFun[\varphi](\qElm, (\valFun, \sElm)) \defeq
\atFun[\varphi_{2}](\qElm, (\valFun_{\rst \free{\varphi_{2}}},
\sElm))$, otherwise, for all $\qElm \in \QSet[\varphi_{1}] \cup
\QSet[\varphi_{2}]$ and $(\valFun, \sElm) \in \ValSet[
{\AcSet[\GName]} ](\free{\varphi}) \times \StSet[\GName]$;
* $\aleph_{\varphi} \defeq (\FSet[1\varphi], \ldots,
\FSet[k\varphi])$, where (i) $\aleph_{\varphi_{1}} \defeq
(\FSet[1\varphi_{1}], \ldots, \FSet[k_{1}\varphi_{1}])$ and
$\aleph_{\varphi_{2}} \defeq (\FSet[1\varphi_{2}], \ldots,
\FSet[k_{2}\varphi_{2}])$, (ii) $h = \min \{ k_{1}, k_{2}
\}$ and $k = \max \{ k_{1}, k_{2} \}$, (iii)
$\FSet[i\varphi] \defeq \{ \qElm[0\varphi] \} \cup
\FSet[i\varphi_{1}] \cup \FSet[i\varphi_{2}]$, for $i \in
\numcc{1}{h}$, (iv) $\FSet[i\varphi] \defeq \{
\qElm[0\varphi] \} \cup \FSet[i\varphi_{j}]$, for $i \in \numcc{h
+ 1}{k - 1}$ with $k_{j} = k$, and (v) $\FSet[k\varphi]
\defeq \QSet[\varphi]$.
It is important to observe that the initial state $\qElm[0\varphi]$ is
included in all sets of the parity acceptance condition, in particular
in $\FSet[1\varphi]$, in order to avoid its regeneration for an
infinite number of times.
* To handle the case $\varphi = \varphi_{1} \R \varphi_{2}$, we use
the automaton $\AName[\varphi | ^{\GName}] \defeq \TATuple {\ValSet[
{\AcSet[\GName]} ](\free{\varphi}) \times \StSet[\GName]}
{\StSet[\GName]} {\QSet[\varphi]} {\atFun[\varphi]} {\qElm[0\varphi]}
{\aleph_{\varphi}}$ that verifies the truth of the release operator
using its one-step unfolding equivalence $\varphi_{1} \R \varphi_{2}
\equiv \varphi_{2} \wedge (\varphi_{1} \vee \X \varphi_{1} \R
\varphi_{2})$, by appropriately running the two automata
$\AName[\varphi_{1} | ^{\GName}] = \TATuple {\ValSet[ {\AcSet[\GName]}
](\free{\varphi_{1}}) \times \StSet[\GName]} {\StSet[\GName]}
{\QSet[\varphi_{1}]} {\atFun[\varphi_{1}]} {\qElm[0\varphi_{1}]}
{\aleph_{\varphi_{1}}}$ and $\AName[\varphi_{2} | ^{\GName}] =
\TATuple {\ValSet[ {\AcSet[\GName]} ](\free{\varphi_{2}}) \times
\StSet[\GName]} {\StSet[\GName]} {\QSet[\varphi_{2}]}
{\atFun[\varphi_{2}]} {\qElm[0\varphi_{2}]} {\aleph_{\varphi_{2}}}$
for the inner formulas $\varphi_{1}$ and $\varphi_{2}$.
* $\QSet[\varphi] \defeq \{ \qElm[0\varphi] \} \cup
\QSet[\varphi_{1}] \cup \QSet[\varphi_{2}]$, with $\qElm[0\varphi]
\not\in \QSet[\varphi_{1}] \cup \QSet[\varphi_{2}]$;
* $\atFun[\varphi](\qElm[0\varphi], (\valFun, \sElm)) \defeq
\atFun[\varphi_{2}](\qElm[0\varphi_{2}], (\valFun_{\rst
\free{\varphi_{2}}}, \sElm)) \wedge
(\atFun[\varphi_{1}](\qElm[0\varphi_{1}], (\valFun_{\rst
\free{\varphi_{1}}}, \sElm)) \vee (\trnFun[\GName](\sElm,
\valFun_{\rst \AgSet}), \qElm[0\varphi]))$, for all $(\valFun,
\sElm) \in \ValSet[ {\AcSet[\GName]} ](\free{\varphi}) \times
\StSet[\GName]$;
* $\atFun[\varphi](\qElm, (\valFun, \sElm)) \defeq
\atFun[\varphi_{1}](\qElm, (\valFun_{\rst \free{\varphi_{1}}},
\sElm))$, if $\qElm \in \QSet[\varphi_{1}]$, and
$\atFun[\varphi](\qElm, (\valFun, \sElm)) \defeq
\atFun[\varphi_{2}](\qElm, (\valFun_{\rst \free{\varphi_{2}}},
\sElm))$, otherwise, for all $\qElm \in \QSet[\varphi_{1}] \cup
\QSet[\varphi_{2}]$ and $(\valFun, \sElm) \in \ValSet[
{\AcSet[\GName]} ](\free{\varphi}) \times \StSet[\GName]$;
* $\aleph_{\varphi} \defeq (\FSet[1\varphi], \ldots,
\FSet[k\varphi])$, where (i) $\aleph_{\varphi_{1}} \defeq
(\FSet[1\varphi_{1}], \ldots, \FSet[k_{1}\varphi_{1}])$ and
$\aleph_{\varphi_{2}} \defeq (\FSet[1\varphi_{2}], \ldots,
\FSet[k_{2}\varphi_{2}])$, (ii) $h = \min \{ k_{1}, k_{2}
\}$ and $k = \max \{ k_{1}, k_{2} \}$, (iii)
$\FSet[1\varphi] \defeq \FSet[1\varphi_{1}] \cup
\FSet[1\varphi_{2}]$, (iv) $\FSet[i\varphi] \defeq \{
\qElm[0\varphi] \} \cup \FSet[i\varphi_{1}] \cup
\FSet[i\varphi_{2}]$, for $i \!\in\! \numcc{2}{h}$, (iv)
$\FSet[i\varphi] \!\defeq\! \{ \qElm[0\varphi] \} \cup
\FSet[i\varphi_{j}]$, for $i \!\in\! \numcc{h + 1}{k - 1}$ with
$k_{j} \!=\! k$, and (v) $\FSet[k\varphi] \!\defeq\!
\QSet[\varphi]$.
It is important to observe that, differently from the case of the
until operator, the initial state $\qElm[0\varphi]$ is included in all
sets of the parity acceptance condition but $\FSet[1\varphi]$, in
order to allow its regeneration for an infinite number of time.
* The case $\varphi = (\aElm, \xElm) \varphi'$ is solved by simply
transforming the transition function of the automaton $\AName[\varphi'
| ^{\GName}] = \TATuple {\ValSet[ {\AcSet[\GName]} ](\free{\varphi'})
\times \StSet[\GName]} {\StSet[\GName]} {\QSet[\varphi']}
{\atFun[\varphi']} {\qElm[0\varphi']} {\aleph_{\varphi'}}$, by setting
the value of the valuations in input w.r.t. the agent $\aElm$ to the
value of the same valuation w.r.t. the variable $\xElm$.
The definitions of the transition function for $\AName[\varphi |
^{\GName}] \defeq \TATuple {\ValSet[ {\AcSet[\GName]}
](\free{\varphi}) \times \StSet[\GName]} {\StSet[\GName]}
{\QSet[\varphi']} {\atFun[\varphi]} {\qElm[0\varphi']}
{\aleph_{\varphi'}}$ follows: $\atFun[\varphi](\qElm, (\valFun,
\sElm)) \defeq \atFun[\varphi'](\qElm, (\valFun', \sElm))$, where
$\valFun' = \valFun[][\aElm \mapsto \valFun(\xElm)]_{\rst
\free{\varphi'}}$, if $\aElm \in \free{\varphi'}$, and $\valFun' =
\valFun$, otherwise, for all $\qElm \in \QSet[\varphi']$ and
$(\valFun, \sElm) \in \ValSet[ {\AcSet[\GName]} ](\free{\varphi})
\times \StSet[\GName]$.
* To handle the case $\varphi = \EExs{\xElm} \varphi'$, assuming that
$\xElm \in \free{\varphi'}$, we use the operation of existential
projection for nondeterministic tree automata.
To do this, we have first to nondeterminize the $\AName[\varphi'
| ^{\GName}]$, by applying the classic transformation <cit.>.
In this way, we obtain an equivalent $\NName[\varphi' |
^{\GName}] = \TATuple {\ValSet[ {\AcSet[\GName]} ](\free{\varphi'})
\times \StSet[\GName]} {\StSet[\GName]} {\QSet[\varphi']}
{\atFun[\varphi']} {\qElm[0\varphi']} {\aleph_{\varphi'}}$.
Now, we make the projection, by defining the new $\AName[\varphi
| ^{\GName}] \defeq \TATuple {\ValSet[ {\AcSet[\GName]}
](\free{\varphi}) \times \StSet[\GName]} {\StSet[\GName]}
{\QSet[\varphi']} {\atFun[\varphi]} {\qElm[0\varphi']}
{\aleph_{\varphi'}}$ where $\atFun[\varphi](\qElm, (\valFun, \sElm))
\defeq \bigvee_{\cElm \in \AcSet[\GName]} \atFun[\varphi'](\qElm,
(\valFun[][\xElm \mapsto \cElm], \sElm))$, for all $\qElm \in
\QSet[\varphi']$ and $(\valFun, \sElm) \in \ValSet[ {\AcSet[\GName]}
](\free{\varphi}) \times \StSet[\GName]$.
At this point, it only remains to prove that, for all states $\sElm \in
\StSet[\GName]$ and assignments $\asgFun \in
\AsgSet[\GName](\free{\varphi}, \sElm)$, it holds that $\GName, \asgFun,
\sElm \models \varphi$ iff $\TName \in \LangSet(\AName[\varphi |
^{\GName}])$, where $\TName$ is the assignment-state encoding for
The proof can be developed by a simple induction on the structure of the
formula $\varphi$ and is left to the reader as a simple exercise.
We now have the tools to describe the recursive model-checking procedure on
nested subsentences for and its fragments under the general semantics.
To proceed, we have first to prove the following theorem that represents the
core of our automata-theoretic approach.
Let $\GName$ be a , $\sElm \in \StSet[\GName]$ one of its states, and
$\varphi$ an sentence.
Then, there exists an $\NName[\varphi | ^{\GName, \sElm}]$ such that
$\GName, \emptyfun, \sElm \models \varphi$ iff $\LangSet(\NName[\varphi |
^{\GName, \sElm}]) \neq \emptyset$.
To construct the $\NName[\varphi | ^{\GName, \sElm}]$ we apply
Theorem <ref> of direction projection with
distinguished direction $\sElm$ to the $\AName[\varphi | ^{\GName}]$
derived by Lemma <ref> of formula automaton.
In this way, we can ensure that the state labeling of nodes of the
assignment-state encoding is coherent with the node itself.
Observe that, since $\varphi$ is a sentence, we have that $\free{\varphi}
= \emptyset$, and so, the unique assignment-state encoding of interest is
that related to the empty assignment $\emptyfun$.
[Only if].
Suppose that $\GName, \emptyfun, \sElm \models \varphi$.
Then, by Lemma <ref>, we have that $\TName \in
\LangSet(\AName[\varphi | ^{\GName}])$, where $\TName$ is the elementary
dependence-state encoding for $\emptyfun$.
Hence, by Theorem <ref>, it holds that
$\LangSet(\NName[\varphi | ^{\GName, \sElm}]) \neq \emptyset$.
Suppose that $\LangSet(\NName[\varphi | ^{\GName, \sElm}]) \neq
\emptyset$.
Then, by Theorem <ref>, there exists an $( \{ \emptyfun \}
\times \StSet[\GName])$-labeled $\StSet[\GName]$-tree $\TName$ such that
$\TName \in \LangSet(\AName[\varphi | ^{\GName}])$.
Now, it is immediate to see that $\TName$ is the assignment-state encoding
for $\emptyfun$.
Hence, by Lemma <ref>, we have that $\GName, \emptyfun,
\sElm \models \varphi$.
Before continuing, we define the length $\lng{\varphi}$ of an formula
$\varphi$ as the number $\card{\sub{\varphi}}$ of its subformulas.
We also introduce a generalization of the Knuth's double arrow notation in
order to represents a tower of exponentials: $a \uparrow\uparrow_{b} 0
\defeq b$ and $a \uparrow\uparrow_{b} (c + 1) \defeq a^{a
\uparrow\uparrow_{b} c}$, for all $a, b, c \in \SetN$.
At this point, we prove the main theorem about the non-elementary complexity
of model-checking problem.
The model-checking problem for is w.r.t. the size of the
model and w.r.t. the size of the specification.
By Theorem <ref> of sentence automaton, to verify that
$\GName, \emptyfun, \sElm \models \varphi$, we simply calculate the
emptiness of the $\NName[\varphi | ^{\GName, \sElm}]$ having
$\card{\StSet[\GName]} \cdot (2 \uparrow\uparrow_{m} m)$ states and index
$2 \uparrow\uparrow_{m} m$, where $m = \AOmicron{\lng{\varphi} \cdot \log
\lng{\varphi}}$.
It is well-known that the emptiness problem for such a kind of automaton
with $n$ states and index $h$ is solvable in time
$\AOmicron{n^{h}}$ <cit.>.
Thus, we get that the time complexity of checking whether $\GName,
\emptyfun, \sElm \models \varphi$ is $\card{\StSet[\GName]}^{2
\uparrow\uparrow_{m} m}$.
Hence, the membership of the model-checking problem for in w.r.t. the size of the model and w.r.t. the size of the
specification directly follows.
Finally, by getting the relative lower bound on the model from the same
problem for <cit.>, the thesis is proved.
Finally, we show a refinement of the previous result, when we consider
sentences of the fragment.
The model-checking problem for is w.r.t. the size of the
model and $(k + 1)$- w.r.t. the maximum alternation $k$ of the
By Theorem <ref> of sentence automaton, to verify that
$\GName, \emptyfun, \sElm \models \qpElm \psi$, where $\qpElm \psi$ is an
principal sentence without proper subsentences, we can simply
calculate the emptiness of the $\NName[\qpElm \psi | ^{\GName,
\sElm}]$ having $\card{\StSet[\GName]} \cdot (2 \uparrow\uparrow_{m} k)$
states and index $2 \uparrow\uparrow_{m} k$, where $m =
\AOmicron{\lng{\psi} \cdot \log \lng{\psi}}$ and $k = \alt{\qpElm \psi}$.
Thus, we get that the time complexity of checking whether $\GName,
\emptyfun, \sElm \models \qpElm \psi$ is $\card{\StSet[\GName]}^{2
\uparrow\uparrow_{m} k}$.
At this point, since we have to do this verification for each possible
state $\sElm \in \StSet[\GName]$ and principal subsentence $\qpElm \psi
\in \psnt{\varphi}$ of the whole specification $\varphi$, we derive
that the bottom-up model-checking procedure requires time
$\card{\StSet[\GName]}^{2 \uparrow\uparrow_{\lng{\varphi}} k}$, where $k =
\max \set{ \alt{\qpElm \psi} }{ \qpElm \psi \in \psnt{\varphi} }$.
Hence, the membership of the model-checking problem for in w.r.t. the size of the model and $(k + 1)$- w.r.t. the maximum
alternation $k$ of the specification directly follows.
Finally, by getting the relative lower bound on the model from the same
Model Checking
We now show how the concept of elementariness of dependence maps over
strategies can be used to enormously reduce the complexity of the
model-checking procedure for the fragment.
The idea behind our approach is to avoid the use of projections used to
handle the strategy quantifications, by reducing them to action
quantifications, which can be managed locally on each state of the model
without a tower of exponential blow-ups.
To start with the description of the ad-hoc procedure for , we first
prove the existence of a for each and goal $\bpElm \psi$
that recognizes all the assignment-state encodings of an a
priori given assignment,
Let $\GName$ be a and $\bpElm \psi$ an goal without principal
Then, there exists an $\UName[\bpElm \psi | ^{\GName}] \defeq
\TATuple {\ValSet[ {\AcSet[\GName]} ](\free{\bpElm \psi}) \times
\StSet[\GName]} {\StSet[\GName]} {\QSet[\bpElm \psi]} {\atFun[\bpElm
\psi]} {\qElm[\bpElm \psi]} {\aleph_{\bpElm \psi}}$ such that, for all
states $\sElm \in \StSet[\GName]$ and assignments $\asgFun \in
\AsgSet[\GName](\free{\bpElm \psi}, \sElm)$, it holds that $\GName,
\asgFun, \sElm \models \bpElm \psi$ iff $\TName \in \LangSet(\UName[\bpElm
\psi | ^{\GName}])$, where $\TName$ is the assignment-state encoding for
A first step in the construction of the $\UName[\bpElm \psi |
^{\GName}]$ is to consider the $\UName[\psi] \defeq \WATuple
{\pow{\APSet}} {\QSet[\psi]} {\atFun[\psi]} {\QSet[0\psi]}
{\aleph_{\psi}}$ obtained by dualizing the resulting from the
application of the classic Vardi-Wolper construction to the formula
$\neg \psi$ <cit.>.
Observe that $\LangSet(\UName[\psi]) = \LangSet(\psi)$, i.e.,
$\UName[\psi]$ recognizes all infinite words on the alphabet
$\pow{\APSet}$ that satisfy the formula $\psi$.
Then, define the components of $\UName[\bpElm \psi | ^{\GName}] \defeq
\TATuple {\ValSet[ {\AcSet[\GName]} ](\free{\bpElm \psi}) \times
\StSet[\GName]} {\StSet[\GName]} {\QSet[\bpElm \psi]} {\atFun[\bpElm
\psi]} {\qElm[0\bpElm\psi]} {\aleph_{\bpElm \psi}}$ as follows:
* $\QSet[\bpElm \psi] \defeq \{ \qElm[0\bpElm\psi] \} \cup \QSet[\psi]$,
with $\qElm[0\bpElm\psi] \not\in \QSet[\psi]$;
* $\atFun[\bpElm \psi](\qElm[0\bpElm\psi], (\valFun, \sElm)) \defeq
\bigwedge_{\qElm \in \QSet[0\psi]} \atFun[\bpElm \psi](\qElm,
(\valFun, \sElm))$, for all $(\valFun, \sElm) \in \ValSet[
{\AcSet[\GName]} ](\free{\bpElm \psi}) \times \StSet[\GName]$;
* $\atFun[\bpElm \psi](\qElm, (\valFun, \sElm)) \!\defeq\!
\bigwedge_{\qElm' \!\in\! \atFun[\psi](\qElm, \labFun[\GName](\sElm))}
(\trnFun[\GName](\sElm, \valFun \cmp \bndFun[\bpElm]), \qElm')$, for
all $\qElm \!\in\! \QSet[\psi]$ and $(\valFun, \sElm) \!\in\! \ValSet[
{\AcSet[\GName]} ](\free{\bpElm \psi}) \times \StSet[\GName]$;
* $\aleph_{\bpElm \psi} \defeq \aleph_{\psi}$.
Intuitively, the $\UName[\bpElm \psi | ^{\GName}]$ simply runs the
$\UName[\psi]$ on the branch of the encoding individuated by the
assignment in input.
Thus, it is easy to see that, for all states $\sElm \in \StSet[\GName]$
and assignments $\asgFun \in \AsgSet[\GName](\free{\bpElm \psi}, \sElm)$,
it holds that $\GName, \asgFun, \sElm \models \bpElm \psi$ iff $\TName \in
\LangSet(\UName[\bpElm \psi | ^{\GName}])$, where $\TName$ is the
assignment-state encoding for $\asgFun$.
Now, to describe our modified technique, we introduce a new concept of
encoding regarding the elementary dependence maps over strategies.
Let $\GName$ be a , $\sElm \in \StSet[\GName]$ one of its states, and
$\spcFun \in \ESpcSet[ {\StrSet[\GName](\sElm)} ](\qpElm)$ an elementary
dependence map over strategies for a quantification prefix $\qpElm \in
\QPSet(\VSet)$ over the set $\VSet \subseteq \VarSet$.
Then, a $(\SpcSet[ {\AcSet[\GName]} ](\qpElm) \times
\StSet[\GName])$-labeled $\StSet[\GName]$-tree $\TName \defeq
\LTTuple{}{}{\TSet}{\uFun}$, where $\TSet \defeq \set{ \trkElm_{\geq 1} }{
\trkElm \in \TrkSet[\GName](\sElm) }$, is an elementary
dependence-state encoding for $\spcFun$ if it holds that $\uFun(\tElm)
\defeq (\adj{\spcFun}(\sElm \cdot \tElm), \lst{\sElm \cdot \tElm})$, for
all $\tElm \in \TSet$.
Observe that there exists a unique elementary dependence-state encoding for
each elementary dependence map over strategies.
In the next lemma, we show how to handle locally the strategy
quantifications on each state of the model, by simply using a quantification
over actions, which is modeled by the choice of an action dependence map.
Intuitively, we guess in the labeling what is the right part of the
dependence map over strategies for each node of the tree and then verify
that, for all assignments of universal variables, the corresponding complete
assignment satisfies the inner formula.
Let $\GName$ be a and $\qpElm \bpElm \psi$ an principal
sentence without principal subsentences.
Then, there exists a $\UName[\qpElm \bpElm \psi | ^{\GName}] \defeq
\TATuple {\SpcSet[ {\AcSet[\GName]} ](\qpElm) \times \StSet[\GName]}
{\StSet[\GName]} {\QSet[\qpElm \bpElm \psi]} {\atFun[\qpElm \bpElm \psi]}
{\qElm[0\qpElm\bpElm\psi]} {\aleph_{\qpElm \bpElm \psi}}$ such that, for
all states $\sElm \in \StSet[\GName]$ and elementary dependence maps over
strategies $\spcFun \in \ESpcSet[ {\StrSet[\GName](\sElm)} ](\qpElm)$, it
holds that $\GName, \spcFun(\asgFun), \sElm \emodels \bpElm \psi$, for all
$\asgFun \in \AsgSet[\GName](\QPAVSet{\qpElm}, \sElm)$, iff $\TName \in
\LangSet(\UName[\qpElm \bpElm \psi | ^{\GName}])$, where $\TName$ is the
elementary dependence-state encoding for $\spcFun$.
By Lemma <ref> of goal automaton, there is an
$\UName[\bpElm \psi | ^{\GName}] = \TATuple {\ValSet[
{\AcSet[\GName]} ](\free{\bpElm \psi}) \times \StSet[\GName]}
{\StSet[\GName]} {\QSet[\bpElm \psi]} {\atFun[\bpElm \psi]}
{\qElm[0\bpElm\psi]} {\aleph_{\bpElm \psi}}$ such that, for all states
$\sElm \!\in\! \StSet[\GName]$ and assignments $\asgFun \!\in\!
\AsgSet[\GName](\free{\bpElm \psi}, \sElm)$, it holds that $\GName,
\asgFun, \sElm \models \bpElm \psi$ iff $\TName \in \LangSet(\UName[\bpElm
\psi | ^{\GName}])$, where $\TName$ is the assignment-state encoding for
Now, transform $\UName[\bpElm \psi | ^{\GName}]$ into the new $\UName[\qpElm \bpElm \psi | ^{\GName}] \defeq \TATuple {\SpcSet[
{\AcSet[\GName]} ](\qpElm) \times \StSet[\GName]} {\StSet[\GName]}
{\QSet[\qpElm \bpElm \psi]} {\atFun[\qpElm \bpElm \psi]}
{\qElm[0\qpElm\bpElm\psi]} {\aleph_{\qpElm \bpElm \psi}}$, with
$\QSet[\qpElm \bpElm \psi] \defeq \QSet[\bpElm \psi]$,
$\qElm[0\qpElm\bpElm\psi] \defeq \qElm[0\bpElm\psi]$, and
$\aleph_{\qpElm \bpElm \psi} \defeq \aleph_{\bpElm \psi}$, which is used
to handle the quantification prefix $\qpElm$ atomically, where the
transition function is defined as follows: $\atFun[\qpElm \bpElm
\psi](\qElm, (\spcFun, \sElm)) \defeq \bigwedge_{\valFun \in \ValSet[
{\AcSet[\GName]} ](\QPAVSet{\qpElm})} \atFun[\bpElm \psi](\qElm,
(\spcFun(\valFun), \sElm))$, for all $\qElm \in \QSet[\qpElm \bpElm \psi]$
and $(\spcFun, \sElm) \in \SpcSet[ {\AcSet[\GName]} ](\qpElm) \times
\StSet[\GName]$.
Intuitively, $\UName[\qpElm \bpElm \psi | ^{\GName}]$ reads an action
dependence map $\spcFun$ on each node of the input tree $\TName$ labeled
with a state $s$ of $\GName$ and simulates the execution of the transition
function $\atFun[\bpElm \psi](\qElm, (\valFun, \sElm))$ of $\UName[\bpElm
\psi | ^{\GName}]$, for each possible valuation $\valFun =
\spcFun(\valFun')$ on $\free{\bpElm \psi}$ obtained from $\spcFun$ by a
universal valuation $\valFun' \in \ValSet[ {\AcSet[\GName]}
It is important to observe that we cannot move the component set $\SpcSet[
{\AcSet[\GName]} ](\qpElm)$ from the input alphabet to the states of
$\UName[\qpElm \bpElm \psi | ^{\GName}]$, by making a related guessing of
the dependence map $\spcFun$ in the transition function, since we have to
ensure that all states in a given node of the tree $\TName$, i.e., in each
track of the original model $\GName$, make the same choice for $\spcFun$.
Finally, it remains to prove that, for all states $\sElm \in
\StSet[\GName]$ and elementary dependence map over strategies $\spcFun \in
\ESpcSet[ {\StrSet[\GName](\sElm)} ](\qpElm)$, it holds that $\GName,
\spcFun(\asgFun), \sElm \emodels \bpElm \psi$, for all $\asgFun \in
\AsgSet[\GName](\QPAVSet{\qpElm}, \sElm)$, iff $\TName \in
\LangSet(\UName[\qpElm \bpElm \psi | ^{\GName}])$, where $\TName$ is the
elementary dependence-state encoding for $\spcFun$.
[Only if].
Suppose that $\GName, \spcFun(\asgFun), \sElm \emodels \bpElm \psi$, for
all $\asgFun \in \AsgSet[\GName](\QPAVSet{\qpElm}, \sElm)$.
Since $\psi$ does not contain principal subsentences, we have that
$\GName, \spcFun(\asgFun), \sElm \models \bpElm \psi$.
So, due to the property of $\UName[\bpElm \psi | ^{\GName}]$, it follows
that there exists an assignment-state encoding $\TName[\asgFun] \in
\LangSet(\UName[\bpElm \psi | ^{\GName}])$, which implies the existence of
an $(\StSet[\GName] \times \QSet[\bpElm \psi])$-tree $\RSet[\asgFun]$ that
is an accepting run for $\UName[\bpElm \psi | ^{\GName}]$ on
At this point, let $\RSet \defeq \bigcup_{\asgFun \in
\AsgSet[\GName](\QPAVSet{\qpElm}, \sElm)} \RSet[\asgFun]$ be the union of
all runs.
Then, due to the particular definition of the transition function of
$\UName[\qpElm \bpElm \psi | ^{\GName}]$, it is not hard to see that
$\RSet$ is an accepting run for $\UName[\qpElm \bpElm \psi | ^{\GName}]$
on $\TName$.
Hence, $\TName \in \LangSet(\UName[\qpElm \bpElm \psi | ^{\GName}])$.
Suppose that $\TName \in \LangSet(\UName[\qpElm \bpElm \psi |
Then, there exists an $(\StSet[\GName] \times \QSet[\qpElm \bpElm
\psi])$-tree $\RSet$ that is an accepting run for $\UName[\qpElm \bpElm
\psi | ^{\GName}]$ on $\TName$.
Now, for each $\asgFun \in \AsgSet[\GName](\QPAVSet{\qpElm}, \sElm)$, let
$\RSet[\asgFun]$ be the run for $\UName[\bpElm \psi | ^{\GName}]$ on the
assignment-state encoding $\TName[\asgFun]$ for $\spcFun(\asgFun)$.
Due to the particular definition of the transition function of
$\UName[\qpElm \bpElm \psi | ^{\GName}]$, it is easy to see that
$\RSet[\asgFun] \subseteq \RSet$.
Thus, since $\RSet$ is accepting, we have that $\RSet[\asgFun]$ is
accepting as well.
So, $\TName[\asgFun] \in \LangSet(\UName[\bpElm \psi | ^{\GName}])$.
At this point, due to the property of $\UName[\bpElm \psi | ^{\GName}]$,
it follows that $\GName, \spcFun(\asgFun), \sElm \models \bpElm \psi$.
Now, since $\psi$ does not contain principal subsentences, we have that
$\GName, \spcFun(\asgFun), \sElm \emodels \bpElm \psi$, for all $\asgFun
\in \AsgSet[\GName](\QPAVSet{\qpElm}, \sElm)$.
At this point, we can prove the following theorem that is at the base of the
elementary model-checking procedure for .
Let $\GName$ be a , $\sElm \in \StSet[\GName]$ one of its states, and
$\qpElm \bpElm \psi$ an principal sentence without principal
Then, there exists an $\NName[\qpElm \bpElm \psi | ^{\GName,
\sElm}]$ such that $\GName, \emptyfun, \sElm \models \qpElm \bpElm \psi$
iff $\LangSet(\NName[\qpElm \bpElm \psi | ^{\GName, \sElm}]) \neq
\emptyset$.
As in the general case of sentence automaton, we have to ensure that
the state labeling of nodes of the elementary dependence-state encoding is
coherent with the node itself.
To do this, we apply Theorem <ref> of direction
projection with distinguished direction $\sElm$ to the $\UName[\qpElm \bpElm \psi | ^{\GName}]$ derived by
Lemma <ref> of the sentence automaton, thus
obtaining the required $\NName[\qpElm \bpElm \psi | ^{\GName,
\sElm}]$.
[Only if].
Suppose that $\GName, \emptyfun, \sElm \models \qpElm \bpElm \psi$.
By Corollary <ref> of elementariness, it means that
$\GName, \emptyfun, \sElm \emodels \qpElm \bpElm \psi$.
Then, by Definition <ref> of elementary semantics, there exists an elementary dependence map $\spcFun
\in \ESpcSet[ {\StrSet[\GName](\sElm)} ](\qpElm)$ such that $\GName,
\spcFun(\asgFun), \sElm \emodels \bpElm \psi$, for all $\asgFun \in
\AsgSet[\GName](\QPAVSet{\qpElm}, \sElm)$.
Thus, by Lemma <ref>, we have that $\TName \in
\LangSet(\UName[\qpElm \bpElm \psi | ^{\GName}])$, where $\TName$ is the
elementary dependence-state encoding for $\spcFun$.
Hence, by Theorem <ref>, it holds that
$\LangSet(\NName[\qpElm \bpElm \psi | ^{\GName, \sElm}]) \neq \emptyset$.
Suppose that $\LangSet(\NName[\qpElm \bpElm \psi | ^{\GName, \sElm}]) \neq
\emptyset$.
Then, by Theorem <ref>, there exists an $(\SpcSet[
{\AcSet[\GName]} ](\qpElm) \times \StSet[\GName])$-labeled
$\StSet[\GName]$-tree $\TName$ such that $\TName \in
\LangSet(\UName[\qpElm \bpElm \psi | ^{\GName}])$.
Now, it is immediate to see that there is an elementary dependence map
$\spcFun \in \ESpcSet[ {\StrSet[\GName](\sElm)} ](\qpElm)$ for which
$\TName$ is the elementary dependence-state encoding.
Thus, by Lemma <ref>, we have that $\GName,
\spcFun(\asgFun), \sElm \emodels \bpElm \psi$, for all $\asgFun \in
\AsgSet[\GName](\QPAVSet{\qpElm}, \sElm)$.
By Definition <ref> of elementary
semantics, it holds that $\GName, \emptyfun, \sElm \emodels \qpElm \bpElm
\psi$.
Hence, by Corollary <ref> of elementariness, it means
that $\GName, \emptyfun, \sElm \models \qpElm \bpElm \psi$.
Finally, we show in the next fundamental theorem the precise complexity of
the model-checking for .
The model-checking problem for is w.r.t. the size of the
model and 2 w.r.t. the size of the specification.
By Theorem <ref> of sentence automaton, to verify
that $\GName, \emptyfun, \sElm \models \qpElm \bpElm \psi$, we simply
calculate the emptiness of the $\NName[\qpElm \bpElm \psi |
^{\GName, \sElm}]$.
This automaton is obtained by the operation of direction projection on the
$\UName[\qpElm \bpElm \psi | ^{\GName}]$, which is in turn derived
by the $\UName[\bpElm \psi | ^{\GName}]$.
Now, it is easy to see that the number of states of $\UName[\bpElm \psi |
^{\GName}]$, and consequently of $\UName[\qpElm \bpElm \psi | ^{\GName}]$,
is $2^{\AOmicron{\lng{\psi}}}$.
So, $\NName[\qpElm \bpElm \psi | ^{\GName, \sElm}]$ has
$\card{\StSet[\GName]} \cdot 2^{2^{\AOmicron{\lng{\psi}}}}$ states and
index $2^{\AOmicron{\lng{\psi}}}$.
The emptiness problem for such a kind of automaton with $n$ states and
index $h$ is solvable in time $\AOmicron{n^{h}}$ <cit.>.
Thus, we get that the time complexity of checking whether $\GName,
\emptyfun, \sElm \models \qpElm \bpElm \psi$ is
At this point, since we have to do this verification for each possible
state $\sElm \in \StSet[\GName]$ and principal subsentence $\qpElm \bpElm
\psi \in \psnt{\varphi}$ of the whole specification $\varphi$, we
derive that the whole bottom-up model-checking procedure requires time
Hence, the membership of the model-checking problem for in w.r.t. the size of the model and 2 w.r.t. the size of the
specification directly follows.
Finally the thesis is proved, by getting the relative lower bounds from
the same problem for <cit.>.
In this paper, we introduced and studied as a very powerful logic
formalism to reasoning about strategic behaviors of multi-agent concurrent
In particular, we proved that it subsumes the classical temporal and game
logics not using explicit fix-points.
As one of the main results about , we shown that the relative
model-checking problem is decidable but non-elementary hard.
As further and interesting practical results, we investigated several of its
syntactic fragments.
The most appealing one is , which is obtained by restricting to deal
with one temporal goal at a time.
Interestingly, strictly extends , while maintaining all its
positive properties.
In fact, the model-checking problem is 2, hence not harder than the
one for .
Moreover, although for the sake of space it is not reported in this paper, we
shown that it is invariant under bisimulation and decision-unwinding, and
consequently, it has the decision-tree model property.
The main reason why has all these positive properties is that it
satisfies a special model property, which we name “elementariness”.
Informally, this property asserts that all strategy quantifications in a
sentence can be reduced to a set of quantifications over actions, which turn
out to be easier to handle.
We remark that among all fragments we investigated, is the only
one that satisfies this property.
As far as we know, is the first significant proper extension of having an elementary model-checking problem, and even more, with the same
computational complexity.
All these positive aspects make us strongly believe that is a valid
alternative to to be used in the field of formal verification for
multi-agent concurrent systems.
As another interesting fragment we investigated in this paper, we recall
This logic allows us to express important game-theoretic properties, such as
Nash equilibrium, which cannot be defined in .
Unfortunately, we do not have an elementary model-checking procedure for it,
neither we can exclude it.
We leave to investigate this as future work.
Last but not least, from a theoretical point of view, we are convinced that
our framework can be used as a unifying basis for logic reasonings about
strategic behaviors in multi-agent scenarios and their relationships.
In particular, it can be used to study variations and extensions of in
a way similar as it has been done in the literature for .
For example, it could be interesting to investigate memoryful , by
inheriting and extending the “memoryful” concept used for and and investigated in <cit.> and <cit.>, respectively.
Also, we recall that this concept implicitly allows to deal with backwards
temporal modalities.
As another example, it would be interesting to investigate the graded
extension of , in a way similar as it has been done
in <cit.> and <cit.> for and ,
We recall that graded quantifiers in branching-time temporal logics allow to
count how many equivalent classes of paths satisfy a given property.
This concept in would further allow the counting of strategies and so
to succinctly check the existence of more than one nonequivalent winning
strategy for a given agent, in one shot.
We hope to lift to graded questions left open about graded
branching-time temporal logic, such as the precise satisfiability complexity
of graded full computation tree logic <cit.>.
Mathematical Notation
In this short reference appendix, we report the classical mathematical
notation and some common definitions that are used along the whole work.
*Classic objects
We consider $\SetN$ as the set of natural numbers and $\numcc{m}{n}
\defeq \set{ k \in \SetN }{ m \leq k \leq n }$, $\numco{m}{n} \defeq \set{ k
\in \SetN }{ m \leq k < n }$, $\numoc{m}{n} \defeq \set{ k \in \SetN }{ m <
k \leq n }$, and $\numoo{m}{n} \defeq \set{ k \in \SetN }{ m < k < n }$ as
its interval subsets, with $m \in \SetN$ and $n \in \SetNI \defeq
\SetN \cup \{ \omega \}$, where $\omega$ is the numerable infinity,
i.e., the least infinite ordinal.
Given a set $\XSet$ of objects, we denote by $\card{\XSet} \in
\SetNI \cup \{ \infty \}$ the cardinality of $\XSet$, i.e., the
number of its elements, where $\infty$ represents a more than
countable cardinality, and by $\pow{\XSet} \defeq \set{ \YSet }{ \YSet
\subseteq \XSet }$ the powerset of $\XSet$, i.e., the set of all its
By $\RRel \subseteq \XSet \times \YSet$ we denote a relation between
the domain $\dom{\RRel} \defeq \XSet$ and codomain
$\cod{\RRel} \defeq \YSet$, whose range is indicated by $\rng{\RRel}
\defeq \set{ \yElm \in \YSet }{ \exists \xElm \in \XSet .\: (\xElm, \yElm)
\in \RRel }$.
We use $\RRel^{-1} \defeq \set{ (\yElm, \xElm) \in \YSet \times \XSet }{
(\xElm, \yElm) \in \RRel }$ to represent the inverse of $\RRel$
Moreover, by $\SRel \cmp \RRel$, with $\RRel \subseteq \XSet \times \YSet$
and $\SRel \subseteq \YSet \times \ZSet$, we denote the composition
of $\RRel$ with $\SRel$, i.e., the relation $\SRel \cmp \RRel \defeq \set{
(\xElm, \zElm) \in \XSet \times \ZSet }{ \exists \yElm \in \YSet .\: (\xElm,
\yElm) \in \RRel \land (\yElm, \zElm) \in \SRel }$.
We also use $\RRel^{n} \defeq \RRel^{n - 1} \cmp \RRel$, with $n \in
\numco{1}{\omega}\!$, to indicate the $n$-iteration of $\RRel
\subseteq \XSet \times \YSet$, where $\YSet \subseteq \XSet$ and $\RRel^{0}
\defeq \set{ (\yElm, \yElm) }{ \yElm \in \YSet }$ is the identity on
With $\RRel^{+} \defeq \bigcup_{n = 1}^{< \omega} \RRel^{n}$ and $\RRel^{*}
\defeq \RRel^{+} \cup \RRel^{0}$ we denote, respectively, the
transitive and reflexive-transitive closure of $\RRel$.
Finally, for an equivalence relation $\RRel \subseteq \XSet \times
\XSet$ on $\XSet$, we represent with $\class{ \XSet }{ \:\RRel } \defeq
\set{ [\xElm]_{\RRel} }{ \xElm \in \XSet }$, where $[\xElm]_{\RRel} \defeq
\set{ \xElm' \in \XSet }{ (\xElm, \xElm') \in \RRel }$, the quotient
set of $\XSet$ w.r.t. $\RRel$, i.e., the set of all related equivalence
classes $[\cdot]_{\RRel}$.
We use the symbol $\YSet^{\XSet} \subseteq \pow{\XSet \times \YSet}$ to
denote the set of total functions $\fFun$ from $\XSet$ to $\YSet$,
i.e., the relations $\fFun \subseteq \XSet \times \YSet$ such that for all
$\xElm \in \dom{\fFun}$ there is exactly one element $\yElm \in \cod{\fFun}$
such that $(\xElm, \yElm) \in \fFun$.
Often, we write $\fFun : \XSet \to \YSet$ and $\fFun : \XSet \pto \YSet$ to
indicate, respectively, $\fFun \in \YSet^{\XSet}$ and $\fFun \in
\bigcup_{\XSet' \subseteq \XSet} \YSet^{\XSet'}$.
Regarding the latter, note that we consider $\fFun$ as a partial
function from $\XSet$ to $\YSet$, where $\dom{\fFun} \subseteq \XSet$
contains all and only the elements for which $\fFun$ is defined.
Given a set $\ZSet$, by $\fFun[\rst \ZSet] \defeq \fFun \cap (\ZSet \times
\YSet)$ we denote the restriction of $\fFun$ to the set $\XSet \cap
\ZSet$, i.e., the function $\fFun[\rst \ZSet] : \XSet \cap \ZSet \pto \YSet$
such that, for all $\xElm \in \dom{\fFun} \cap \ZSet$, it holds that
$\fFun[\rst \ZSet](\xElm) = \fFun(\xElm)$.
Moreover, with $\emptyfun$ we indicate a generic empty function,
i.e., a function with empty domain.
Note that $\XSet \cap \ZSet = \emptyset$ implies $\fFun[\rst \ZSet] =
\emptyfun$.
Finally, for two partial functions $\fFun, \gFun : \XSet \pto \YSet$, we use
$\fFun \Cup \gFun$ and $\fFun \Cap \gFun$ to represent, respectively, the
union and intersection of these functions defined as follows:
$\dom{\fFun \Cup \gFun} \defeq \dom{\fFun} \cup \dom{\gFun} \setminus \set{
\xElm \in \dom{\fFun} \cap \dom{\gFun} }{ \fFun(\xElm) \neq \gFun(\xElm) }$,
$\dom{\fFun \Cap \gFun} \defeq \set{ \xElm \in \dom{\fFun} \cap \dom{\gFun}
}{ \fFun(\xElm) = \gFun(\xElm) }$, $(\fFun \Cup \gFun)(\xElm) =
\fFun(\xElm)$ for $\xElm \in \dom{\fFun \Cup \gFun} \cap \dom{\fFun}$,
$(\fFun \Cup \gFun)(\xElm) = \gFun(\xElm)$ for $\xElm \in \dom{\fFun \Cup
\gFun} \cap \dom{\gFun}$, and $(\fFun \Cap \gFun)(\xElm) = \fFun(\xElm)$ for
$\xElm \in \dom{\fFun \Cap \gFun}$.
By $\XSet^{n}$, with $n \in \SetN$, we denote the set of all
$n$-tuples of elements from $\XSet$, by $\XSet^{*} \defeq \bigcup_{n
= 0}^{< \omega} \XSet^{n}$ the set of finite words on the
alphabet $\XSet$, by $\XSet^{+} \defeq \XSet^{*} \setminus \{
\epsilon \}$ the set of non-empty words, and by $\XSet^{\omega}$ the
set of infinite words, where, as usual, $\epsilon \in \XSet^{*}$ is
the empty word.
The length of a word $\wElm \in \XSet^{\infty} \defeq \XSet^{*} \cup
\XSet^{\omega}$ is represented with $\card{\wElm} \in \SetNI$.
By $(\wElm)_{i}$ we indicate the $i$-th letter of the finite word
$\wElm \in \XSet^{+}$, with $i \in \numco{0}{\card{\wElm}}$.
Furthermore, by $\fst{\wElm} \defeq (\wElm)_{0}$ (resp., $\lst{\wElm} \defeq
(\wElm)_{\card{\wElm} - 1}$), we denote the first (resp.,
last) letter of $\wElm$.
In addition, by $(\wElm)_{\leq i}$ (resp., $(\wElm)_{> i}$), we indicate the
prefix up to (resp., suffix after) the letter of index $i$ of
$\wElm$, i.e., the finite word built by the first $i + 1$ (resp., last
$\card{\wElm} - i - 1$) letters $(\wElm)_{0}, \ldots, (\wElm)_{i}$ (resp.,
$(\wElm)_{i + 1}, \ldots, (\wElm)_{\card{\wElm} - 1}$).
We also set, $(\wElm)_{< 0} \defeq \epsilon$, $(\wElm)_{< i} \defeq
(\wElm)_{\leq i - 1}$, $(\wElm)_{\geq 0} \defeq \wElm$, and $(\wElm)_{\geq
i} \defeq (\wElm)_{> i - 1}$, for $i \in \numco{1}{\card{\wElm}}$.
Mutatis mutandis, the notations of $i$-th letter, first, prefix, and suffix
apply to infinite words too.
Finally, by $\pfx{\wElm[1], \wElm[2]} \in \XSet^{\infty}$ we denote the
maximal common prefix of two different words $\wElm[1], \wElm[2] \in
\XSet^{\infty}$, i.e., the finite word $\wElm \in \XSet^{*}$ for which
there are two words $\wElm[1|'], \wElm[2|'] \in \XSet^{\infty}$ such that
$\wElm[1] = \wElm \cdot \wElm[1|']$, $\wElm[2] = \wElm \cdot \wElm[2|']$,
and $\fst{\wElm[1|']} \neq \fst{\wElm[2|']}$.
By convention, we set $\pfx{\wElm, \wElm} \defeq \wElm$.
For a set $\DirSet$ of objects, called directions, a
$\DirSet$-tree is a set $\TSet \subseteq \DirSet^{*}$ closed under
prefix, i.e., if $\tElm \cdot \dElm \in \TSet$, with $\dElm \in \DirSet$,
then also $\tElm \in \TSet$.
We say that it is complete if it holds that $\tElm \cdot \dElm' \in
\TSet$ whenever $\tElm \cdot \dElm \in \TSet$, for all $\dElm' < \dElm$,
where $< \: \subseteq \DirSet \times \DirSet$ is an a priori fixed strict
total order on the set of directions that is clear from the context.
Moreover, it is full if $\TSet = \DirSet^{*}$.
The elements of $\TSet$ are called nodes and the empty word
$\epsilon$ is the root of $\TSet$.
For every $\tElm \in \TSet$ and $\dElm \in \DirSet$, the node $\tElm \cdot
\dElm \in \TSet$ is a successor of $\tElm$ in $\TSet$.
The tree is $b$-bounded if the maximal number $b$ of its successor
nodes is finite, i.e., $b = \max_{\tElm \in \TSet} \card{\set{ \tElm \cdot
\dElm \in \TSet }{ \dElm \in \DirSet }} < \omega$.
A branch of the tree is an infinite word $\wElm \in \DirSet^{\omega}$
such that $(\wElm)_{\leq i} \in \TSet$, for all $i \in \SetN$.
For a finite set $\LabSet$ of objects, called symbols, a
$\LabSet$-labeled $\DirSet$-tree is a quadruple
$\LTDef{\LabSet}{\DirSet}$, where $\TSet$ is a $\DirSet$-tree and $\vFun :
\TSet \to \LabSet$ is a labeling function.
When $\DirSet$ and $\LabSet$ are clear from the context, we call $\LTStruct$
simply a (labeled) tree.
Proofs of Section <ref>
In this appendix, we report the proofs of lemmas needed to prove the
elementariness of .
Before this, we describe two relevant properties that link together dependence
maps of a given quantification prefix with those of the dual one.
These properties report, in the dependence maps framework, what is known to
hold, in an equivalent way, for first and second order logic.
In particular, they result to be two key points towards a complete
understanding of the strategy quantifications of our logic.
The first of these properties enlighten the fact that two arbitrary dual
dependence maps $\spcFun$ and $\dual{\spcFun}$ always share a common valuation
To better understand this concept, consider for instance the functions
$\spcFun[1]$ and $\dual[6]{\spcFun}$ of the examples illustrated just after
Definition <ref> of dependence maps.
Then, it is easy to see that the valuation $\valFun \in
\ValSet[\DSet](\VSet)$ with $\valFun(\xSym) = \valFun(\ySym) = 1$ and
$\valFun(\zSym) = 0$ resides in both the ranges of $\spcFun[1]$ and
$\dual[6]{\spcFun}$, i.e., $\valFun \in \rng{\spcFun[1]} \cap
\rng{\dual[6]{\spcFun}}$.
Let $\qpElm \in \QPSet(\VSet)$ be a quantification prefix over a set of
variables $\VSet \subseteq \VarSet$ and $\DSet$ a generic set.
Moreover, let $\spcFun \in \SpcSet[\DSet](\qpElm)$ and $\dual{\spcFun} \in
\SpcSet[\DSet](\dual{\qpElm})$ be two dependence maps.
Then, there exists a valuation $\valFun \in \ValSet[\DSet](\VSet)$ such
that $\valFun = \spcFun(\valFun_{\rst \QPAVSet{\qpElm}}) =
\dual{\spcFun}(\valFun_{\rst \QPAVSet{\dual{\qpElm}}})$.
W.l.o.g., suppose that $\qpElm$ starts with an existential quantifier.
If this is not the case, the dual prefix $\dual{\qpElm}$ necessarily
satisfies the above requirement, so, we can simply shift our reasoning on
The whole proof proceeds by induction on the alternation number
$\alt{\qpElm}$ of $\qpElm$.
As base case, if $\alt{\qpElm} = 0$, we define $\valFun \defeq
\spcFun(\emptyfun)$, since $\QPAVSet{\qpElm} = \emptyset$.
Obviously, it holds that $\valFun = \spcFun(\valFun_{\rst
\QPAVSet{\qpElm}}) = \dual{\spcFun}(\valFun_{\rst
\QPAVSet{\dual{\qpElm}}})$, due to the fact that $\valFun_{\rst
\QPAVSet{\qpElm}} = \emptyfun$ and $\valFun_{\rst
\QPAVSet{\dual{\qpElm}}} = \valFun$.
Now, as inductive case, suppose that the statement is true for all
prefixes $\qpElm' \in \QPSet(\VSet')$ with $\alt{\qpElm'} = n$, where
$\VSet' \subset \VSet$.
Then, we prove that it is true for all prefixes $\qpElm \in \QPSet(\VSet)$
with $\alt{\qpElm} = n + 1$ too.
To do this, we have to uniquely split $\qpElm = \qpElm' \cdot \qpElm''$
into the two prefixes $\qpElm' \in \QPSet(\VSet')$ and $\qpElm'' \in
\QPSet(\VSet \setminus \VSet')$ such that $\alt{\qpElm'} = n$ and
$\alt{\qpElm''} = 0$.
At this point, the following two cases can arise.
* If $n$ is even, it is immediate to see that $\QPEVSet{\qpElm''} =
\emptyset$.
So, consider the dependence maps $\spcFun' \in \SpcSet[\DSet](\qpElm')$
and $\dual{\spcFun'} \in \SpcSet[\DSet](\dual{\qpElm'})$ such that
$\spcFun'(\valFun_{\rst \QPAVSet{\qpElm'}}) = \spcFun(\valFun)_{\rst
\VSet'}$ and $\dual{\spcFun'}(\dual{\valFun}) =
\dual{\spcFun}(\dual{\valFun})_{\rst \VSet'}$, for all valuations
$\valFun \in \ValSet[\DSet](\QPAVSet{\qpElm})$ and $\dual{\valFun} \in
\ValSet[\DSet](\QPAVSet{\dual{\qpElm}}) =
\ValSet[\DSet](\QPAVSet{\dual{\qpElm'}})$.
By the inductive hypothesis, there exists a valuation $\valFun' \in
\ValSet[\DSet](\VSet')$ such that $\valFun' = \spcFun'(\valFun'_{\rst
\QPAVSet{\qpElm'}}) = \dual{\spcFun'}(\valFun'_{\rst
\QPAVSet{\dual{\qpElm'}}})$.
So, set $\valFun \defeq \dual{\spcFun}(\valFun'_{\rst
\QPAVSet{\dual{\qpElm}}})$.
* If $n$ is odd, it is immediate to see that $\QPAVSet{\qpElm''} =
\emptyset$.
So, consider the dependence maps $\spcFun' \in \SpcSet[\DSet](\qpElm')$
and $\dual{\spcFun'} \in \SpcSet[\DSet](\dual{\qpElm'})$ such that
$\spcFun'(\valFun) = \spcFun(\valFun)_{\rst \VSet'}$ and
$\dual{\spcFun'}(\dual{\valFun}_{\rst \QPAVSet{\dual{\qpElm'}}}) =
\dual{\spcFun}(\dual{\valFun})_{\rst \VSet'}$, for all valuations
$\valFun \in \ValSet[\DSet](\QPAVSet{\qpElm}) =
\ValSet[\DSet](\QPAVSet{\qpElm'})$ and $\dual{\valFun} \in
\ValSet[\DSet](\QPAVSet{\dual{\qpElm}})$.
By the inductive hypothesis, there exists a valuation $\valFun' \in
\ValSet[\DSet](\VSet')$ such that $\valFun' = \spcFun'(\valFun'_{\rst
\QPAVSet{\qpElm'}}) = \dual{\spcFun'}(\valFun'_{\rst
\QPAVSet{\dual{\qpElm'}}})$.
So, set $\valFun \defeq \spcFun(\valFun'_{\rst
\QPAVSet{\qpElm}})$.
Now, it is easy to see that in both cases the valuation $\valFun$
satisfies the thesis, i.e., $\valFun = \spcFun(\valFun_{\rst
\QPAVSet{\qpElm}}) = \dual{\spcFun}(\valFun_{\rst
\QPAVSet{\dual{\qpElm}}})$.
The second property we are going to prove describes the fact that, if all
dependence maps $\spcFun$ of a given prefix $\qpElm$, for a dependent specific
universal valuation $\valFun$, share a given property then there is a dual
dependence maps $\dual{\spcFun}$ that has the same property, for all universal
valuations $\dual{\valFun}$.
To have a better understanding of this idea, consider again the examples
reported just after Definition <ref> and let $\PSet \defeq \{ (0,
0, 1), (0, 1, 0) \} \subset \ValSet[\DSet](\VSet)$, where the triple $(\lElm,
\mElm, \nElm)$ stands for the valuation that assigns $\lElm$ to $\xSym$,
$\mElm$ to $\ySym$, and $\nElm$ to $\zSym$.
Then, it is easy to see that all ranges of the dependence maps $\spcFun[i]$
for $\qpSym$ intersect $\PSet$, i.e., for all $i \in \numcc{0}{3}$, there is
$\valFun \in \ValSet[\DSet](\QPAVSet{\qpSym})$ such that $\spcFun[i](\valFun)
\in \PSet$.
Moreover, consider the dual dependence maps $\dual[2]{\spcFun}$ for
Then, it is not hard to see that $\dual[2]{\spcFun}(\dual{\valFun}) \in
\PSet$, for all $\dual{\valFun} \in
\ValSet[\DSet](\QPAVSet{\dual{\qpSym}})$.
Let $\qpElm \in \QPSet(\VSet)$ be a quantification prefix over a set of
variables $\VSet \subseteq \VarSet$, $\DSet$ a generic set, and $\PSet
\subseteq \ValSet[\DSet](\VSet)$ a set of valuations of $\VSet$ over
Moreover, suppose that, for all dependence maps $\spcFun \in
\SpcSet[\DSet](\qpElm)$, there is a valuation $\valFun \in
\ValSet[\DSet](\QPAVSet{\qpElm})$ such that $\spcFun(\valFun) \in \PSet$.
Then, there exists a dependence map $\dual{\spcFun} \in
\SpcSet[\DSet](\dual{\qpElm})$ such that, for all valuations
$\dual{\valFun} \in \ValSet[\DSet](\QPAVSet{\dual{\qpElm}})$, it holds
that $\dual{\spcFun}(\dual{\valFun}) \in \PSet$.
The proof easily proceeds by induction on the length of the prefix
As base case, when $\card{\qpElm} = 0$, we have that
$\SpcSet[\DSet](\qpElm) = \SpcSet[\DSet](\dual{\qpElm}) = \{ \emptyfun
\}$, i.e., the only possible dependence maps is the empty function, which
means that the statement is vacuously verified.
As inductive case, we have to distinguish between two cases, as follows.
* $\qpElm = \EExs{\xElm} \cdot \qpElm'$.
As first thing, note that $\QPAVSet{\qpElm} = \QPAVSet{\qpElm'}$ and,
for all elements $\eElm \in \DSet$, consider the projection
$\PSet[\eElm] \defeq \set{ \valFun' \in
\ValSet[\DSet](\QPVSet(\qpElm'))}{ \valFun'[\xElm \mapsto \eElm] \in
\PSet }$ of $\PSet$ on the variable $\xElm$ with value $\eElm$.
Then, by hypothesis, we can derive that, for all $\eElm \in \DSet$ and
$\spcFun' \in \SpcSet[\DSet](\qpElm')$, there exists $\valFun' \in
\ValSet[\DSet](\QPAVSet{\qpElm'})$ such that $\spcFun'(\valFun') \in
\PSet[\eElm]$.
Indeed, let $\eElm \in \DSet$ and $\spcFun' \in
\SpcSet[\DSet](\qpElm')$, and build the function $\spcFun :
\ValSet[\DSet](\QPAVSet{\qpElm}) \to \ValSet[\DSet](\VSet)$ given by
$\spcFun(\valFun') \defeq \spcFun'(\valFun')[\xElm \mapsto \eElm]$,
for all $\valFun' \in \ValSet[\DSet](\QPAVSet{\qpElm}) =
\ValSet[\DSet](\QPAVSet{\qpElm'})$.
It is immediate to see that $\spcFun \in \SpcSet[\DSet](\qpElm)$.
So, by the hypothesis, there is $\valFun' \in
\ValSet[\DSet](\QPAVSet{\qpElm})$ such that $\spcFun(\valFun') \in
\PSet$, which implies $\spcFun'(\valFun')[\xElm \mapsto \eElm] \in
\PSet$, and so, $\spcFun'(\valFun') \in \PSet[\eElm]$.
Now, by the inductive hypothesis, for all elements $\eElm \in \DSet$,
there exists $\dual[\eElm]{\spcFun'} \in
\SpcSet[\DSet](\dual{\qpElm'})$ such that, for all $\dual{\valFun'}
\in \ValSet[\DSet](\QPAVSet{\dual{\qpElm'}})$, it holds that
$\dual[\eElm]{\spcFun'}(\dual{\valFun'}) \in \PSet[\eElm]$, i.e.,
$\dual[\eElm]{\spcFun'}(\dual{\valFun'})[\xElm \mapsto \eElm] \in
\PSet$.
At this point, consider the function $\dual{\spcFun} :
\ValSet[\DSet](\QPAVSet{\dual{\qpElm}}) \to \ValSet[\DSet](\VSet)$
given by $\dual{\spcFun}(\dual{\valFun}) \defeq
\dual[{\dual{\valFun}(\xElm)}]{\spcFun'}(\dual{\valFun}_{\rst
\QPAVSet{\dual{\qpElm'}}})[\xElm \mapsto \dual{\valFun}(\xElm)]$,
for all $\dual{\valFun} \in
\ValSet[\DSet](\QPAVSet{\dual{\qpElm}})$.
Then, it is possible to verify that $\dual{\spcFun} \in
\SpcSet[\DSet](\dual{\qpElm})$.
Indeed, for each $\yElm \in \QPAVSet{\dual{\qpElm}}$ and
$\dual{\valFun} \in \ValSet[\DSet](\QPAVSet{\dual{\qpElm}})$, we
have that $\dual{\spcFun}(\dual{\valFun})(\yElm) =
\dual[{\dual{\valFun}(\xElm)}]{\spcFun'}(\dual{\valFun}_{\rst
\QPAVSet{\dual{\qpElm'}}})[\xElm \mapsto
\dual{\valFun}(\xElm)](\yElm)$.
Now, if $\yElm = \xElm$ then $\dual{\spcFun}(\dual{\valFun})(\yElm)
= \dual{\valFun}(\yElm)$.
Otherwise, since $\dual[\dual{\valFun}(\xElm)]{\spcFun'}$ is a
dependence map, it holds that $\dual{\spcFun}(\dual{\valFun})(\yElm) =
\dual[{\dual{\valFun}(\xElm)}]{\spcFun'}(\dual{\valFun}_{\rst
\QPAVSet{\dual{\qpElm'}}})(\yElm) = \dual{\valFun}_{\rst
\QPAVSet{\dual{\qpElm'}}}(\yElm) = \dual{\valFun}(\yElm)$.
So, Item <ref> of Definition <ref> of
dependence maps is verified.
It only remains to prove Item <ref>.
Let $\yElm \in \QPEVSet{\dual{\qpElm}}$ and $\dual{\valFun[1]},
\dual{\valFun[2]} \in \ValSet[\DSet](\QPAVSet{\dual{\qpElm}})$, with
$\dual{\valFun[1]}_{\rst \QPDepSet(\dual{\qpElm}, \yElm)} =
\dual{\valFun[2]}_{\rst \QPDepSet(\dual{\qpElm}, \yElm)}$.
It is immediate to see that $\xElm \in \QPDepSet(\dual{\qpElm},
\yElm)$, so, $\dual{\valFun[1]}(\xElm) = \dual{\valFun[2]}(\xElm)$,
which implies that $\dual[\dual{\valFun[1]}(\xElm)]{\spcFun'} =
\dual[\dual{\valFun[2]}(\xElm)]{\spcFun'}$.
At this point, again for the fact that
$\dual[\dual{\valFun}(\xElm)]{\spcFun'}$ is a dependence map, for each
$\dual{\valFun} \in \ValSet[\DSet](\QPAVSet{\dual{\qpElm}})$, we have
that $\dual[\dual{\valFun[1]}(\xElm)]{\spcFun'}(\dual{\valFun[1]}_{\rst
\QPAVSet{\dual{\qpElm'}}})(\yElm) =
\dual[\dual{\valFun[2]}(\xElm)]{\spcFun'}(\dual{\valFun[2]}_{\rst
\QPAVSet{\dual{\qpElm'}}})(\yElm)$.
Thus, it holds that $\dual{\spcFun}(\dual{\valFun[1]})(\yElm) \!=\!
\dual[\dual{\valFun[1]}(\xElm)]{\spcFun'}(\dual{\valFun[1]}_{\rst
\QPAVSet{\dual{\qpElm'}}})[\xElm \mapsto
\dual{\valFun[1]}(\xElm)](\yElm) =
\dual[\dual{\valFun[2]}(\xElm)]{\spcFun'}(\dual{\valFun[2]}_{\rst
\QPAVSet{\dual{\qpElm'}}})[\xElm \mapsto
\dual{\valFun[2]}(\xElm)](\yElm) =
\dual{\spcFun}(\dual{\valFun[2]})(\yElm)$.
Finally, it is enough to observe that, by construction,
$\dual{\spcFun}(\dual{\valFun}) \in \PSet$, for all $\dual{\valFun} \in
\ValSet[\DSet](\QPAVSet{\dual{\qpElm}})$, since
\QPAVSet{\dual{\qpElm'}}}) \in \PSet[{\dual{\valFun}(\xElm)}]$.
Thus, the thesis holds for this case.
* $\qpElm = \AAll{\xElm} \cdot \qpElm'$.
We first show that there exists $\eElm \in \DSet$ such that, for all
$\spcFun' \in \SpcSet[\DSet](\qpElm')$, there is $\valFun' \in
\ValSet[\DSet](\QPAVSet{\qpElm'})$ for which $\spcFun'(\valFun')
\in \PSet[\eElm]$ holds, where the set $\PSet[\eElm]$ is defined as
To do this, suppose by contradiction that, for all $\eElm \in \DSet$,
there is a $\spcFun[\eElm|'] \in \SpcSet[\DSet](\qpElm')$ such that,
for all $\valFun' \in \ValSet[\DSet](\QPAVSet{\qpElm'})$, it holds
that $\spcFun[\eElm|'](\valFun') \not\in \PSet[\eElm]$.
Also, consider the function $\spcFun :
\ValSet[\DSet](\QPAVSet{\qpElm}) \to \ValSet[\DSet](\VSet)$ given by
$\spcFun(\valFun) \defeq \spcFun[\valFun(\xElm)|'](\valFun_{\rst
\QPAVSet{\qpElm'}})[\xElm \mapsto \valFun(\xElm)]$, for all $\valFun
\in \ValSet[\DSet](\QPAVSet{\qpElm})$.
Then, is possible to verify that $\spcFun \in \SpcSet[\DSet](\qpElm)$.
Indeed, for each $\yElm \in \QPAVSet{\qpElm}$ and $\valFun \in
\ValSet[\DSet](\QPAVSet{\qpElm})$, we have that
$\spcFun(\valFun)(\yElm) = \spcFun[\valFun(\xElm)|'](\valFun_{\rst
\QPAVSet{\qpElm'}})[\xElm \mapsto \valFun(\xElm)](\yElm)$.
Now, if $\yElm = \xElm$ then $\spcFun(\valFun)(\yElm) =
\valFun(\yElm)$.
Otherwise, since $\spcFun[\valFun(\xElm)|']$ is a dependence map, it
holds that $\spcFun(\valFun)(\yElm) =
\spcFun[\valFun(\xElm)|'](\valFun_{\rst \QPAVSet{\qpElm'}})(\yElm) =
\valFun_{\rst \QPAVSet{\qpElm'}}(\yElm) = \valFun(\yElm)$.
So, Item <ref> of Definition <ref> of
dependence maps is verified.
It only remains to prove Item <ref>.
Let $\yElm \in \QPEVSet{\qpElm}$ and $\valFun[1], \valFun[2] \in
\ValSet[\DSet](\QPAVSet{\qpElm})$, with $\valFun[1]_{\rst
\QPDepSet(\qpElm, \yElm)} = \valFun[2]_{\rst \QPDepSet(\qpElm,
\yElm)}$.
It is immediate to see that $\xElm \in \QPDepSet(\qpElm, \yElm)$, so,
$\valFun[1](\xElm) = \valFun[2](\xElm)$, which implies that
$\spcFun[{\valFun[1](\xElm)}|'] = \spcFun[{\valFun[2](\xElm)}|']$.
At this point, again for the fact that $\spcFun[\valFun(\xElm)|']$ is a
dependence map, for each $\valFun \in \ValSet[\DSet](\QPAVSet{\qpElm})$,
we have that $\spcFun[{\valFun[1](\xElm)}|'](\valFun[1]_{\rst
\QPAVSet{\qpElm'}})(\yElm) =
\spcFun[{\valFun[2](\xElm)}|'](\valFun[2]_{\rst
\QPAVSet{\qpElm'}})(\yElm)$.
Thus, it holds that $\spcFun(\valFun[1])(\yElm) =
\spcFun[{\valFun[1](\xElm)}|'](\valFun[1]_{\rst
\QPAVSet{\qpElm'}})[\xElm \mapsto \valFun[1](\xElm)](\yElm) =
\spcFun[{\valFun[2](\xElm)}|'](\valFun[2]_{\rst
\QPAVSet{\qpElm'}})[\xElm \mapsto \valFun[2](\xElm)](\yElm) =
\spcFun(\valFun[2])(\yElm)$.
Now, by the contradiction hypothesis, we have that $\spcFun(\valFun)
\not\in \PSet$, for all $\valFun \in \ValSet(\QPAVSet{\qpElm})$, since
$\spcFun[\valFun(\xElm)|'](\valFun_{\rst \QPAVSet{\qpElm'}}) \not\in
\PSet[\valFun(\xElm)]$, which is in evident contradiction with the
At this point, by the inductive hypothesis, there exists
$\dual{\spcFun'} \in \SpcSet[\DSet](\dual{\qpElm'})$ such that, for
all $\dual{\valFun'} \in \ValSet[\DSet](\QPAVSet{\dual{\qpElm'}})$, it
holds that $\dual{\spcFun'}(\dual{\valFun'}) \in \PSet[\eElm]$, i.e.,
$\dual{\spcFun'}(\dual{\valFun'})[\xElm \mapsto \eElm] \in \PSet$.
Finally, build the function $\dual{\spcFun} :
\ValSet[\DSet](\QPAVSet{\dual{\qpElm}}) \to \ValSet[\DSet](\VSet)$
given by $\dual{\spcFun}(\dual{\valFun}) \defeq
\dual{\spcFun'}(\dual{\valFun})[\xElm \mapsto \eElm]$, for all
$\dual{\valFun} \in \ValSet[\DSet](\QPAVSet{\dual{\qpElm}}) =
\ValSet[\DSet](\QPAVSet{\dual{\qpElm'}})$.
It is immediate to see that $\dual{\spcFun} \in
\SpcSet[\DSet](\dual{\qpElm})$.
Moreover, for all valuations $\dual{\valFun} \in
\ValSet[\DSet](\QPAVSet{\dual{\qpElm}})$, it holds that
$\dual{\spcFun}(\dual{\valFun}) \in \PSet$.
Thus, the thesis holds for this case too.
Hence, we have done with the proof of the lemma.
At this point, we are able to give the proofs of Lemma <ref> of
adjoint dependence maps, Lemma <ref> of dependence-vs-valuation
duality, and Lemma <ref> of encasement characterization.
Let $\qpElm \in \QPSet(\VSet)$ be a quantification prefix over a set of
variables $\VSet \subseteq \VarSet$, $\DSet$ and $\TSet$ two generic sets,
and $\spcFun : \ValSet[\TSet \to \DSet](\QPAVSet{\qpElm}) \to
\ValSet[\TSet \to \DSet](\VSet)$ and $\adj{\spcFun} : \TSet \to
(\ValSet[\DSet](\QPAVSet{\qpElm}) \to \ValSet[\DSet](\VSet))$ two
functions such that $\adj{\spcFun}$ is the adjoint of $\spcFun$.
Then, $\spcFun \in \SpcSet[\TSet \to \DSet](\qpElm)$ iff, for all $t \in
\TSet$, it holds that $\adj{\spcFun}(t) \in \SpcSet[\DSet](\qpElm)$.
To prove the statement, it is enough to show, separately, that
Items <ref> and <ref> of
Definition <ref> of dependence maps hold for $\spcFun$ if the
$\adj{\spcFun}(\tElm)$ satisfies the same items, for all $\tElm \in \TSet$,
and vice versa.
[Item <ref>, if].
Assume that $\adj{\spcFun}(\tElm)$ satisfies Item <ref>,
for each $\tElm \in \TSet$, i.e., $\adj{\spcFun}(\tElm)(\valFun)_{\rst
\QPAVSet{\qpElm}} = \valFun$, for all $\valFun \in
\ValSet[\DSet](\QPAVSet{\qpElm})$.
Then, we have that $\adj{\spcFun}(\tElm)(\flip{\gFun}(\tElm)) =
\flip{\gFun}(\tElm)$, so,
$\adj{\spcFun}(\tElm)(\flip{\gFun}(\tElm))(\xElm) =
\flip{\gFun}(\tElm)(\xElm)$, for all $\gFun \in \ValSet[\TSet \to
\DSet](\QPAVSet{\qpElm})$ and $\xElm \in \QPAVSet{\qpElm}$.
By hypothesis, we have that $\spcFun(\gFun)(\xElm)(\tElm) =
\adj{\spcFun}(\tElm)(\flip{\gFun}(\tElm))(\xElm)$, thus
$\spcFun(\gFun)(\xElm)(\tElm) = \flip{\gFun}(\tElm)(\xElm) =
\gFun(\xElm)(\tElm)$, which means that $\spcFun(\gFun)_{\rst
\QPAVSet{\qpElm}} = \gFun$, for all $\gFun \in \ValSet[\TSet \to
\DSet](\QPAVSet{\qpElm})$.
[Item <ref>, only if].
Assume now that $\spcFun$ satisfies Item <ref>,
i.e., $\spcFun(\gFun)_{\rst \QPAVSet{\qpElm}} = \gFun$, for all $\gFun \in
\ValSet[\TSet \to \DSet](\QPAVSet{\qpElm})$.
Then, we have that $\spcFun(\gFun)(\xElm)(\tElm) = \gFun(\xElm)(\tElm)$,
for all $\xElm \in \QPAVSet{\qpElm}$ and $\tElm \in \TSet$.
By hypothesis, we have that
$\adj{\spcFun}(\tElm)(\flip{\gFun}(\tElm))(\xElm) =
\spcFun(\gFun)(\xElm)(\tElm)$, so,
$\adj{\spcFun}(\tElm)(\flip{\gFun}(\tElm))(\xElm) = \gFun(\xElm)(\tElm) =
\flip{\gFun}(\tElm)(\xElm)$, which means that
$\adj{\spcFun}(\tElm)(\flip{\gFun}(\tElm))_{\rst \QPAVSet{\qpElm}} =
\flip{\gFun}(\tElm)$.
Now, since for each $\valFun \in \ValSet[\DSet](\QPAVSet{\qpElm})$, there
is an $\gFun \in \ValSet[\TSet \to \DSet](\QPAVSet{\qpElm})$ such that
$\flip{\gFun}(\tElm) = \valFun$, we obtain that
$\adj{\spcFun}(\tElm)(\valFun)_{\rst \QPAVSet{\qpElm}} = \valFun$, for all
$\valFun \in \ValSet[\DSet](\QPAVSet{\qpElm})$ and $\tElm \in \TSet$.
[Item <ref>, if].
Assume that $\adj{\spcFun}(\tElm)$ satisfies Item <ref>,
for each $\tElm \in \TSet$, i.e., $\adj{\spcFun}(\tElm)(\valFun[1])(\xElm)
= \adj{\spcFun}(\tElm)(\valFun[2])(\xElm)$, for all $\valFun[1],
\valFun[2] \in \ValSet[\DSet](\QPAVSet{\qpElm})$ and $\xElm \in
\QPEVSet{\qpElm}$ such that $\valFun[1]_{\rst \QPDepSet(\qpElm, \xElm)} =
\valFun[2]_{\rst \QPDepSet(\qpElm, \xElm)}$.
Then, we have that $\adj{\spcFun}(\tElm)(\flip{\gFun[1]}(\tElm))(\xElm) =
\adj{\spcFun}(\tElm)(\flip{\gFun[2]}(\tElm))(\xElm)$, for all $\gFun[1],
\gFun[2] \in \ValSet[\TSet \to \DSet](\QPAVSet{\qpElm})$ such that
$\gFun[1]_{\rst \QPDepSet(\qpElm, \xElm)} = \gFun[2]_{\rst
\QPDepSet(\qpElm, \xElm)}$.
By hypothesis, we have that $\spcFun(\gFun[1])(\xElm)(\tElm) =
\adj{\spcFun}(\tElm)(\flip{\gFun[1]}(\tElm))(\xElm)$ and
$\adj{\spcFun}(\tElm)(\flip{\gFun[2]}(\tElm))(\xElm) =
\spcFun(\gFun[2])(\xElm)(\tElm)$, thus $\spcFun(\gFun[1])(\xElm)(\tElm) =
\spcFun(\gFun[2])(\xElm)(\tElm)$.
Hence, $\spcFun(\gFun[1])(\xElm) \!=\! \spcFun(\gFun[2])(\xElm)$, for all
$\gFun[1], \gFun[2] \in \ValSet[\TSet \to \DSet](\QPAVSet{\qpElm})$ and
$\xElm \in \QPEVSet{\qpElm}$ such that $\gFun[1]_{\rst \QPDepSet(\qpElm,
\xElm)} = \gFun[2]_{\rst \QPDepSet(\qpElm, \xElm)}$.
[Item <ref>, only if].
Assume that $\spcFun$ satisfies Item <ref>, i.e.,
$\spcFun(\gFun[1])(\xElm) = \spcFun(\gFun[2])(\xElm)$, for all $\gFun[1],
\gFun[2] \in \ValSet[\TSet \to \DSet](\QPAVSet{\qpElm})$ and $\xElm \in
\QPEVSet{\qpElm}$ such that $\gFun[1]_{\rst \QPDepSet(\qpElm, \xElm)} =
\gFun[2]_{\rst \QPDepSet(\qpElm, \xElm)}$.
Then, we have that $\spcFun(\gFun[1])(\xElm)(\tElm) =
\spcFun(\gFun[2])(\xElm)(\tElm)$, for all $\tElm \in \TSet$.
By hypothesis, we have that
$\adj{\spcFun}(\tElm)(\flip{\gFun[1]}(\tElm))(\xElm) \allowbreak =
\spcFun(\gFun[1])(\xElm)(\tElm)$ and $\spcFun(\gFun[2])(\xElm)(\tElm) =
\adj{\spcFun}(\tElm)(\flip{\gFun[2]}(\tElm))(\xElm)$, hence
$\adj{\spcFun}(\tElm)(\flip{\gFun[1]}(\tElm))(\xElm) =
\adj{\spcFun}(\tElm)(\flip{\gFun[2]}(\tElm))(\xElm)$.
Now, since for each $\valFun[1], \valFun[2] \in
\ValSet[\DSet](\QPAVSet{\qpElm})$, with $\valFun[1]_{\rst
\QPDepSet(\qpElm, \xElm)} = \valFun[2]_{\rst \QPDepSet(\qpElm, \xElm)}$,
there are $\gFun[1], \gFun[2] \in \ValSet[\TSet \to
\DSet](\QPAVSet{\qpElm})$ such that $\flip{\gFun[1]}(\tElm) = \valFun[1]$
and $\flip{\gFun[2]}(\tElm) = \valFun[2]$, with $\gFun[1]_{\rst
\QPDepSet(\qpElm, \xElm)} = \gFun[2]_{\rst \QPDepSet(\qpElm, \xElm)}$, we
obtain that $\adj{\spcFun}(\tElm)(\valFun[1])(\xElm) =
\adj{\spcFun}(\tElm)(\valFun[2])(\xElm)$, for all $\valFun[1], \valFun[2]
\in \ValSet[\DSet](\QPAVSet{\qpElm})$ and $\xElm \in \QPEVSet{\qpElm}$
such that $\valFun[1]_{\rst \QPDepSet(\qpElm, \xElm)} = \valFun[2]_{\rst
\QPDepSet(\qpElm, \xElm)}$.
Let $\GName$ be a , $\sElm \in \StSet$ one of its states, $\PSet
\subseteq \PthSet(\sElm)$ a set of paths, $\qpElm \in \QPSet(\VSet)$ a
quantification prefix over a set of variables $\VSet \subseteq \VarSet$,
and $\bpElm \in \BndSet(\VSet)$ a binding.
Then, player even wins the $\HName(\GName, \sElm, \PSet, \qpElm,
\bpElm)$ iff player odd wins the dual $\HName(\GName, \sElm,
\PthSet(\sElm) \setminus \PSet, \dual{\qpElm}, \bpElm)$.
Let $\AName$ and $\dual{\AName}$ be, respectively, the two s
$\AName(\GName, \sElm, \qpElm, \bpElm)$ and $\AName(\GName, \sElm,
\dual{\qpElm}, \bpElm)$.
It is easy to observe that $\PosESet[\AName] = \PosESet[\dual{\AName}] =
\TrkSet(\sElm)$.
Moreover, it holds that $\PosOSet[\AName] = \set{ \trkElm \cdot
(\lst{\trkElm}, \spcFun) }{ \trkElm \in \TrkSet(\sElm) \land \spcFun \in
\SpcSet[\AcSet](\qpElm) }$ and $\PosOSet[\dual{\AName}] = \set{ \trkElm
\cdot (\lst{\trkElm}, \dual{\spcFun}) }{ \trkElm \in \TrkSet(\sElm) \land
\dual{\spcFun} \in \SpcSet[\AcSet](\dual{\qpElm}) }$.
We now prove, separately, the two directions of the statement.
[Only if].
Suppose that player even wins the $\HName(\GName, \sElm, \PSet,
\qpElm, \bpElm)$.
Then, there exists an even scheme $\scheFun \in \SchESet[\AName]$ such
that, for all odd schemes $\schoFun \in \SchOSet[\AName]$, it holds that
$\mtcFun[\AName](\scheFun, \schoFun) \in \PSet$.
Now, to prove that odd wins the dual $\HName(\GName, \sElm,
\PthSet(\sElm) \setminus \PSet, \dual{\qpElm}, \bpElm)$, we have to show
that there exists an odd scheme $\dual{\schoFun} \in
\SchOSet[\dual{\AName}]$ such that, for all even schemes $\dual{\scheFun}
\in \SchESet[\dual{\AName}]$, it holds that
$\mtcFun[\dual{\AName}](\dual{\scheFun}, \dual{\schoFun}) \in \PSet$.
To do this, let us first consider a function $\zFun :
\SpcSet[\AcSet](\qpElm) \times \SpcSet[\AcSet](\dual{\qpElm}) \to
\ValSet[\AcSet](\VSet)$ such that $\zFun(\spcFun, \dual{\spcFun}) =
\spcFun(\zFun(\spcFun, \dual{\spcFun})_{\rst \QPAVSet{\qpElm}}) =
\dual{\spcFun}(\zFun(\spcFun, \dual{\spcFun})_{\rst
\QPAVSet{\dual{\qpElm}}})$, for all $\spcFun \in \SpcSet[\AcSet](\qpElm)$
and $\dual{\spcFun} \in \SpcSet[\AcSet](\dual{\qpElm})$.
The existence of such a function is ensured by Lemma <ref> on
the dependence incidence.
Now, define the odd scheme $\dual{\schoFun} \in \SchOSet[\dual{\AName}]$
in $\dual{\AName}$ as follows: $\dual{\schoFun}(\trkElm \cdot
(\lst{\trkElm}, \dual{\spcFun})) \defeq \trnFun(\lst{\trkElm}, \allowbreak
\zFun(\spcFun, \dual{\spcFun}) \cmp \bndFun[\bpElm])$, for all $\trkElm
\in \TrkSet(\sElm)$ and $\dual{\spcFun} \in
\SpcSet[\AcSet](\dual{\qpElm})$, where $\spcFun \in
\SpcSet[\AcSet](\qpElm)$ is such that $\scheFun(\trkElm) = (\lst{\trkElm},
\spcFun)$.
Moreover, let $\dual{\scheFun} \in \SchESet[\dual{\AName}]$ be a generic
even scheme in $\dual{\AName}$ and consider the derived odd scheme
$\schoFun \in \SchOSet[\AName]$ in $\AName$ defined as follows:
$\schoFun(\trkElm \cdot (\lst{\trkElm}, \spcFun)) \defeq
\trnFun(\lst{\trkElm}, \zFun(\spcFun, \dual{\spcFun}) \cmp
\bndFun[\bpElm])$, for all $\trkElm \in \TrkSet(\sElm)$ and $\spcFun \in
\SpcSet[\AcSet](\qpElm)$, where $\dual{\spcFun} \in
\SpcSet[\AcSet](\dual{\qpElm})$ is such that $\dual{\scheFun}(\trkElm) =
(\lst{\trkElm}, \dual{\spcFun})$.
At this point, it remains only to prove that $\mtcElm = \dual{\mtcElm}$,
where $\mtcElm \defeq \mtcFun[\AName](\scheFun, \schoFun)$ and
$\dual{\mtcElm} \defeq \mtcFun[\dual{\AName}](\dual{\scheFun},
\dual{\schoFun})$.
To do this, we proceed by induction on the prefixes of the matches, i.e.,
we show that $(\mtcElm)_{\leq i} = (\dual{\mtcElm})_{\leq i}$, for all $i
\in \SetN$.
The base case is immediate by definition of match, since we have that
$(\mtcElm)_{\leq 0} = \sElm = (\dual{\mtcElm})_{\leq 0}$.
Now, as inductive case, suppose that $(\mtcElm)_{\leq i} =
(\dual{\mtcElm})_{\leq i}$, for $i \in \SetN$.
By the definition of match, we have that $(\mtcElm)_{i + 1} =
\schoFun((\mtcElm)_{\leq i} \cdot \scheFun((\mtcElm)_{\leq i}))$ and
$(\dual{\mtcElm})_{i + 1} = \dual{\schoFun}((\dual{\mtcElm})_{\leq i}
\cdot \dual{\scheFun}((\dual{\mtcElm})_{\leq i}))$.
Moreover, by the inductive hypothesis, it follows that
$\schoFun((\mtcElm)_{\leq i} \cdot \scheFun((\mtcElm)_{\leq i})) =
\schoFun((\dual{\mtcElm})_{\leq i} \cdot \scheFun((\dual{\mtcElm})_{\leq
At this point, let $\spcFun \in \SpcSet[\AcSet](\qpElm)$ and
$\dual{\spcFun} \in \SpcSet[\AcSet](\dual{\qpElm})$ be two quantification
dependence maps such that $\scheFun((\dual{\mtcElm})_{\leq i}) =
((\dual{\mtcElm})_{i}, \spcFun)$ and
$\dual{\scheFun}((\dual{\mtcElm})_{\leq i}) = ((\dual{\mtcElm})_{i},
\dual{\spcFun})$.
Consequently, by substituting the values of the even schemes $\scheFun$
and $\dual{\scheFun}$, it holds that $(\mtcElm)_{i + 1} =
\schoFun((\dual{\mtcElm})_{\leq i} \cdot ((\dual{\mtcElm})_{i}, \spcFun))$
and $(\dual{\mtcElm})_{i + 1} = \dual{\schoFun}((\dual{\mtcElm})_{\leq i}
\cdot ((\dual{\mtcElm})_{i}, \dual{\spcFun}))$.
Furthermore, by the definition of the odd schemes $\schoFun$ and
$\dual{\schoFun}$, it follows that $\schoFun((\dual{\mtcElm})_{\leq i}
\cdot ((\dual{\mtcElm})_{i}, \spcFun)) = \trnFun((\dual{\mtcElm})_{i},
\zFun(\spcFun, \dual{\spcFun}) \cmp \bpFun[\bpElm]) =
\dual{\schoFun}((\dual{\mtcElm})_{\leq i} \cdot ((\dual{\mtcElm})_{i},
\dual{\spcFun}))$.
Thus, we have that $(\mtcElm)_{i + 1} = (\dual{\mtcElm})_{i + 1}$, which
implies $(\mtcElm)_{\leq i + 1} = (\dual{\mtcElm})_{\leq i + 1}$.
Suppose that player odd wins the dual $\HName(\GName, \sElm,
\PthSet(\sElm) \setminus \PSet, \dual{\qpElm}, \bpElm)$.
Then, there exists an odd scheme $\dual{\schoFun} \in
\SchOSet[\dual{\AName}]$ such that, for all even schemes $\dual{\scheFun}
\in \SchESet[\dual{\AName}]$, it holds that
$\mtcFun[\dual{\AName}](\dual{\scheFun}, \dual{\schoFun}) \in \PSet$.
Now, to prove that even wins the $\HName(\GName, \sElm, \PSet,
\qpElm, \bpElm)$, we have to show that there exists an even scheme
$\scheFun \in \SchESet[\AName]$ such that, for all odd schemes $\schoFun
\in \SchOSet[\AName]$, it holds that $\mtcFun[\AName](\scheFun, \schoFun)
\in \PSet$.
To do this, let us first consider the two functions $\gFun :
\TrkSet(\sElm) \to \pow{\ValSet[\AcSet](\VSet)}$ and $\hFun :
\TrkSet(\sElm) \to \pow{\StSet}$ such that $\gFun(\trkElm) \defeq \set{
\dual{\spcFun}(\dual{\valFun}) }{ \dual{\spcFun} \in
\SpcSet[\AcSet](\dual{\qpElm}) \land \dual{\valFun} \in
\ValSet[\AcSet](\QPAVSet{\dual{\qpElm}}) \land \dual{\schoFun}(\trkElm
\cdot (\lst{\trkElm}, \dual{\spcFun})) = \trnFun(\lst{\trkElm},
\dual{\spcFun}(\dual{\valFun}) \cmp \bndFun[\bpElm]) }$ and
$\hFun(\trkElm) \defeq \set{ \dual{\schoFun}(\trkElm \cdot (\lst{\trkElm},
\dual{\spcFun})) }{ \dual{\spcFun} \in \SpcSet[\AcSet](\dual{\qpElm}) }$,
for all $\trkElm \in \TrkSet(\sElm)$.
Now, it is easy to see that, for each $\trkElm \in \TrkSet(\sElm)$ and
$\dual{\spcFun} \in \SpcSet[\AcSet](\dual{\qpElm})$, there is
$\dual{\valFun} \in \ValSet[\AcSet](\QPAVSet{\dual{\qpElm}})$ such that
$\dual{\spcFun}(\dual{\valFun}) \in \gFun(\trkElm)$.
Consequently, by Lemma <ref> on dependence dualization, for all
$\trkElm \in \TrkSet(\sElm)$, there is $\spcFun[\trkElm] \in
\SpcSet[\AcSet](\qpElm)$ such that, for each $\valFun \in
\ValSet[\AcSet](\QPAVSet{\qpElm})$, it holds that
$\spcFun[\trkElm](\valFun) \in \gFun(\trkElm)$, and so,
$\trnFun(\lst{\trkElm}, \spcFun[\trkElm](\valFun) \cmp \bndFun[\bpElm])
\in \hFun(\trkElm)$.
Now, define the even scheme $\scheFun \in \SchESet[\AName]$ in $\AName$ as
follows: $\scheFun(\trkElm) \defeq (\lst{\trkElm}, \spcFun[\trkElm])$, for
all $\trkElm \in \TrkSet(\sElm)$.
Moreover, let $\schoFun \in \SchESet[\AName]$ be a generic odd scheme in
$\AName$ and consider the derived even scheme $\dual{\scheFun} \in
\SchESet[\dual{\AName}]$ in $\dual{\AName}$ defined as follows:
$\dual{\scheFun}(\trkElm) \defeq (\lst{\trkElm},
\dual[\trkElm]{\spcFun})$, for all $\trkElm \in \TrkSet(\sElm)$, where
$\dual[\trkElm]{\spcFun} \in \SpcSet[\AcSet](\dual{\qpElm})$ is such that
$\schoFun(\trkElm \cdot (\lst{\trkElm}, \spcFun[\trkElm])) =
\dual{\schoFun}(\trkElm \cdot (\lst{\trkElm}, \dual[\trkElm]{\spcFun}))$.
The existence of such a dependence map is ensure by the previous membership
of the successor of $\lst{\trkElm}$ in $\hFun(\trkElm)$.
At this point, it remains only to prove that $\mtcElm = \dual{\mtcElm}$,
where $\mtcElm \defeq \mtcFun[\AName](\scheFun, \schoFun)$ and
$\dual{\mtcElm} \defeq \mtcFun[\dual{\AName}](\dual{\scheFun},
\dual{\schoFun})$.
To do this, we proceed by induction on the prefixes of the matches, i.e.,
we show that $(\mtcElm)_{\leq i} = (\dual{\mtcElm})_{\leq i}$, for all $i
\in \SetN$.
The base case is immediate by definition of match, since we have that
$(\mtcElm)_{\leq 0} = \sElm = (\dual{\mtcElm})_{\leq 0}$.
Now, as inductive case, suppose that $(\mtcElm)_{\leq i} =
(\dual{\mtcElm})_{\leq i}$, for $i \in \SetN$.
By the definition of match, we have that $(\mtcElm)_{i + 1} =
\schoFun((\mtcElm)_{\leq i} \cdot \scheFun((\mtcElm)_{\leq i}))$ and
$(\dual{\mtcElm})_{i + 1} = \dual{\schoFun}((\dual{\mtcElm})_{\leq i}
\cdot \dual{\scheFun}((\dual{\mtcElm})_{\leq i}))$.
Moreover, by the inductive hypothesis, it follows that
$\schoFun((\mtcElm)_{\leq i} \cdot \scheFun((\mtcElm)_{\leq i})) =
\schoFun((\dual{\mtcElm})_{\leq i} \cdot \scheFun((\dual{\mtcElm})_{\leq
Now, by substituting the values of the even schemes $\scheFun$ and
$\dual{\scheFun}$, we have that $(\mtcElm)_{i + 1} =
\schoFun((\dual{\mtcElm})_{\leq i} \cdot ((\dual{\mtcElm})_{i},
\spcFun_{(\dual{\mtcElm})_{\leq i}}))$ and $(\dual{\mtcElm})_{i + 1} =
\dual{\schoFun}((\dual{\mtcElm})_{\leq i} \cdot ((\dual{\mtcElm})_{i},
\dual{\spcFun}_{\dual{\mtcElm}_{\leq i}}))$.
At this point, due to the choice of the dependence map
$\dual{\spcFun}_{(\dual{\mtcElm})_{\leq i}}$, it holds that
$\schoFun((\dual{\mtcElm})_{\leq i} \cdot ((\dual{\mtcElm})_{i},
\spcFun_{(\dual{\mtcElm})_{\leq i}})) =
\dual{\schoFun}((\dual{\mtcElm})_{\leq i} \cdot ((\dual{\mtcElm})_{i},
\dual{\spcFun}_{(\dual{\mtcElm})_{\leq i}}))$.
Thus, we have that $(\mtcElm)_{i + 1} = (\dual{\mtcElm})_{i + 1}$, which
implies $(\mtcElm)_{\leq i + 1} = (\dual{\mtcElm})_{\leq i + 1}$.
Let $\GName$ be a , $\sElm \in \StSet$ one of its states, $\PSet
\subseteq \PthSet(\sElm)$ a set of paths, $\qpElm \in \QPSet(\VSet)$ a
quantification prefix over a set of variables $\VSet \subseteq \VarSet$,
and $\bpElm \in \BndSet(\VSet)$ a binding.
Then, the following hold:
* player even wins $\HName(\GName, \sElm, \PSet, \qpElm, \bpElm)$ iff
$\PSet$ is an encasement w.r.t. $\qpElm$ and $\bpElm$;
* if player odd wins $\HName(\GName, \sElm, \PSet, \qpElm, \bpElm)$ then
$\PSet$ is not an encasement w.r.t. $\qpElm$ and $\bpElm$;
* if $\PSet$ is a Borelian set and it is not an encasement w.r.t. $\qpElm$ and $\bpElm$ then player odd wins $\HName(\GName, \sElm,
\PSet, \qpElm, \bpElm)$.
[Item <ref>, only if].
Suppose that player even wins the $\HName(\GName, \sElm, \PSet,
\qpElm, \bpElm)$.
Then, there exists an even scheme $\scheFun \in \SchESet$ such that, for
all odd schemes $\schoFun \in \SchOSet$, it holds that $\mtcFun(\scheFun,
\schoFun) \in \PSet$.
Now, to prove the statement, we have to show that there exists an elementary
dependence map $\spcFun \in \ESpcSet[ {\StrSet(\sElm)} ](\qpElm)$ such that,
for all assignments $\asgFun \in \AsgSet(\QPAVSet{\qpElm}, \sElm)$, it holds
that $\playFun(\spcFun(\asgFun) \cmp \bndFun[\bpElm], \sElm) \in \PSet$.
To do this, consider the function $\wFun : \TrkSet(\sElm) \to
\SpcSet[\AcSet](\qpElm)$ constituting the projection of $\scheFun$ on the
second component of its codomain, i.e., for all $\trkElm \in
\TrkSet(\sElm)$, it holds that $\scheFun(\trkElm) = (\lst{\trkElm},
\wFun(\trkElm))$.
By Lemma <ref> on adjoint dependence maps, there exists an
elementary dependence map $\spcFun \in \ESpcSet[ {\StrSet(\sElm)} ](\qpElm)$
for which $\wFun$ is the adjoint, i.e., $\wFun = \adj{\spcFun}$.
Moreover, let $\asgFun \in \AsgSet(\QPAVSet{\qpElm}, \sElm)$ be a generic
assignment and consider the derived odd scheme $\schoFun \in \SchOSet$
defined ad follows: $\schoFun(\trkElm \cdot (\lst{\trkElm},
\spcFun')) = \trnFun(\lst{\trkElm}, \spcFun'(\flip{\asgFun}(\trkElm)) \cmp
\bndFun[\bpElm])$, for all $\trkElm \in \TrkSet(\sElm)$ and $\spcFun' \in
\SpcSet[\AcSet](\qpElm)$.
At this point, it remains only to prove that $\playElm = \mtcElm$, where
$\playElm \defeq \playFun(\spcFun(\asgFun) \cmp \bndFun[\bpElm], \sElm)$
and $\mtcElm \defeq \mtcFun(\scheFun, \schoFun)$.
To do this, we proceed by induction on the prefixes of both the play and
the match, i.e., we show that $(\playElm)_{\leq i} = (\mtcElm)_{\leq i}$,
for all $i \in \SetN$.
The base case is immediate by definition, since we have that
$(\playElm)_{\leq 0} = \sElm = (\mtcElm)_{\leq 0}$.
Now, as inductive case, suppose that $(\playElm)_{\leq i} =
(\mtcElm)_{\leq i}$, for $i \in \SetN$.
On one hand, by the definition of match, we have that $(\mtcElm)_{i + 1} =
\schoFun((\mtcElm)_{\leq i} \cdot \scheFun((\mtcElm)_{\leq i}))$, from
which, by substituting the value of the even scheme $\scheFun$, we derive
$(\mtcElm)_{i + 1} = \schoFun((\mtcElm)_{\leq i} \cdot ((\mtcElm)_{i},
\adj{\spcFun}((\mtcElm)_{\leq i})))$.
On the other hand, by the definition of play, we have that $(\pthElm)_{i +
1} = \trnFun((\pthElm)_{i}, \adj{\spcFun}((\pthElm)_{\leq
i})(\flip{\asgFun}((\pthElm)_{\leq i}))\cmp \bpFun[\bpElm])$, from which,
by using the definition of the odd scheme $\schoFun$, we derive
$(\pthElm)_{i + 1} = \schoFun((\pthElm)_{\leq i} \cdot ((\pthElm)_{i},
\adj{\spcFun}((\pthElm)_{\leq i})))$.
Then, by the inductive hypothesis, we have that $(\mtcElm)_{i + 1} =
\schoFun((\mtcElm)_{\leq i} \cdot ((\mtcElm)_{i},
\adj{\spcFun}((\mtcElm)_{\leq i}))) = \schoFun((\pthElm)_{\leq i} \cdot
((\pthElm)_{i}, \adj{\spcFun}((\pthElm)_{\leq i}))) = (\pthElm)_{i + 1}$,
which implies $(\mtcElm)_{\leq i + 1} = (\pthElm)_{\leq i + 1}$.
[Item <ref>, if].
Suppose that $\PSet$ is an encasement w.r.t. $\qpElm$ and $\bpElm$.
Then, there exists an elementary dependence map $\spcFun \in \ESpcSet[
{\StrSet(\sElm)} ](\qpElm)$ such that, for all assignments $\asgFun \in
\AsgSet(\QPAVSet{\qpElm}, \sElm)$, it holds that $\playFun(\spcFun(\asgFun)
\cmp \bndFun[\bpElm], \sElm) \in \PSet$.
Now, to prove the statement, we have to show that there exists an even
scheme $\scheFun \in \SchESet$ such that, for all odd schemes $\schoFun
\in \SchOSet$, it holds that $\mtcFun(\scheFun, \schoFun) \in \PSet$.
To do this, consider the even scheme $\scheFun \in \SchESet$ defined as
follows: $\scheFun(\trkElm) \!\defeq\! (\lst{\trkElm},
\adj{\spcFun}(\trkElm))$, for all $\trkElm \in \TrkSet(\sElm)$.
Observe that, by Lemma <ref> on adjoint dependence maps, the
definition is well-formed.
Moreover, let $\schoFun \in \SchOSet$ be a generic odd scheme and consider
a derived assignment $\asgFun \in \AsgSet(\QPAVSet{\qpElm}, \sElm)$
satisfying the following property: $\flip{\asgFun}(\trkElm) \in \set{
\valFun \in \ValSet[\AcSet](\QPAVSet{\qpElm}) }{ \schoFun(\trkElm \cdot
(\lst{\trkElm}, \adj{\spcFun}(\trkElm))) = \trnFun(\lst{\trkElm},
\adj{\spcFun}(\valFun) \cmp \bndFun[\bpElm]) }$, for all $\trkElm \in
\TrkSet(\sElm)$.
At this point, it remains only to prove that $\playElm = \mtcElm$, where
$\playElm \defeq \playFun(\spcFun(\asgFun) \cmp \bndFun[\bpElm], \sElm)$
and $\mtcElm \defeq \mtcFun(\scheFun, \schoFun)$.
To do this, we proceed by induction on the prefixes of both the play and
the match, i.e., we show that $(\playElm)_{\leq i} = (\mtcElm)_{\leq i}$,
for all $i \in \SetN$.
The base case is immediate by definition, since we have that
$(\playElm)_{\leq 0} = \sElm = (\mtcElm)_{\leq 0}$.
Now, as inductive case, suppose that $(\playElm)_{\leq i} =
(\mtcElm)_{\leq i}$, for $i \in \SetN$.
On one hand, by the definition of match, we have that $(\mtcElm)_{i + 1} =
\schoFun((\mtcElm)_{\leq i} \cdot \scheFun((\mtcElm)_{\leq i}))$, from
which, by the definition of the even scheme $\scheFun$, we derive
$(\mtcElm)_{i + 1} = \schoFun((\mtcElm)_{\leq i} \cdot ((\mtcElm)_{i},
\adj{\spcFun}((\mtcElm)_{\leq i})))$.
On the other hand, by the definition of play, we have that $(\pthElm)_{i +
1} = \trnFun((\pthElm)_{i}, \adj{\spcFun}((\pthElm)_{\leq
i})(\flip{\asgFun}((\pthElm)_{\leq i}))\cmp \bpFun[\bpElm])$, from which,
by the choice of the assignment $\asgFun$, we derive $(\pthElm)_{i + 1} =
\schoFun((\pthElm)_{\leq i} \cdot ((\pthElm)_{i},
\adj{\spcFun}((\pthElm)_{\leq i})))$.
Then, by the inductive hypothesis, we have that $(\mtcElm)_{i + 1} =
\schoFun((\mtcElm)_{\leq i} \cdot ((\mtcElm)_{i},
\adj{\spcFun}((\mtcElm)_{\leq i}))) = \schoFun((\pthElm)_{\leq i} \cdot
((\pthElm)_{i}, \adj{\spcFun}((\pthElm)_{\leq i}))) = (\pthElm)_{i + 1}$,
which implies $(\mtcElm)_{\leq i + 1} = (\pthElm)_{\leq i + 1}$.
[Item <ref>].
If player odd wins the $\HName(\GName, \sElm, \PSet, \qpElm,
\bpElm)$, we have that player even does not win the same game.
Consequently, by Item <ref>, it holds that $\PSet$ is
not an encasement w.r.t. $\qpElm$ and $\bpElm$.
[Item <ref>].
If $\PSet$ is not an encasement w.r.t. $\qpElm$ and $\bpElm$, by
Item <ref>, we have that player even does not win the
$\HName(\GName, \sElm, \PSet, \qpElm, \bpElm)$.
Now, since $\PSet$ is Borelian, by the determinacy
theorem <cit.>, it holds that player odd wins the same game.
We wish to thank the authors of <cit.> for their helpful comments and
discussions on a preliminary version of the paper.
? 20?? 20?? 20?
|
arxiv-papers
| 2011-12-29T11:12:19 |
2024-09-04T02:49:25.760354
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fabio Mogavero, Aniello Murano, Giuseppe Perelli, and Moshe Y. Vardi",
"submitter": "Fabio Mogavero PhD",
"url": "https://arxiv.org/abs/1112.6275"
}
|
1112.6279
|
# Complex Effective Path: A Semi-Classical Probe of Quantum Effects
Suprit Singh111suprit@iucaa.ernet.in and T.
Padmanabhan222nabhan@iucaa.ernet.in
IUCAA, Pune University Campus, Ganeshkhind,
Pune 411007, INDIA.
###### Abstract
We discuss the notion of an effective, average, quantum mechanical path which
is a solution of the dynamical equations obtained by extremizing the quantum
effective action. Since the effective action can, in general, be complex, the
effective path will also, in general, be complex. The imaginary part of the
effective action is known to be related to the probability of particle
creation by an external source and hence we expect the imaginary part of the
effective path also to contain information about particle creation. We try to
identify such features using simple examples including that of effective path
through the black hole horizon leading to thermal radiation. Implications of
this approach are discussed.
## 1 Introduction
The study of a quantum mechanical system interacting with an externally
specified classical background is of importance in several physical contexts.
Such an external classical source will, in general, lead to vacuum
polarization and particle production. Well known examples of these phenomena
occur in the study of Schwinger effect [1, 2, 3], particle creation in
expanding universe [2, 4] and black hole evaporation [2, 5]. A powerful
technique to study such external source problems is that of the effective
action which captures the quantum effects through a c-numbered effective
action functional, $S_{\rm eff}\equiv\Gamma$ of the dynamical variables [2,
3]. In general, the effective action will be a complex quantity with its real
and imaginary parts being related to vacuum polarization and particle
production respectively. Conventionally, one writes down the effective
dynamical equations for the system by varying only the real part of the
effective action thereby identifying the quantum corrections to the classical
equations. For example, in the case of electromagnetic field, such an approach
will lead to the Euler-Heisenberg effective action which can provide quantum
corrections to classical Maxwell’s equations [3, 6]. The imaginary part of the
effective action is not usually considered in such a variational principle
since in many applications the effect of vacuum polarization dominates over
that due to particle production.
It is interesting to ask whether one can extend the above formalism to include
the effects of imaginary part of the effective action as well since it could,
potentially, provide a formal procedure for handling the back reaction due to
particle production. The obvious procedure would be to look for the solutions
of $\delta\Gamma=0$ where both the real and imaginary part of $\Gamma$ are
retained. These equations will, in general, be complex rendering the solutions
also to be complex. For example, in the elementary context of non-relativistic
quantum mechanics, such a solution is the effective average path
$X(t;x_{2},t_{2};x_{1},t_{1})$ obeying the appropriate boundary conditions at
the end points. This function will, in general, be complex and one would
presume that its imaginary part will contain some information about the
particle production due to the external source. The primary aim of this paper
is to investigate the properties of this function.
It might seem that, since the effective path $X(t)$ is a solution to the
effective field equation $\delta\Gamma=0$, it can be determined only after
$\Gamma$ is explicitly obtained which in turn would depend on the system under
consideration. We shall see, however, that there is a simple way of
characterizing $X(t)$ as a path integral average of all paths so that it can
be expressed as an integral involving the standard path integral kernel. (This
idea was first developed in [7] but we could not find any follow up of this
idea in the literature, hence we shall provide fair amount of details of the
approach in this paper.) This is the approach we shall use to investigate the
properties of $X(t)$ in this paper.
In the above discussion, we have made a correspondence between the imaginary
part of the action with the existence of phenomena like particle production or
vacuum instability. This is indeed the case for the specific examples which we
will be concerned with in this paper. However, it should be mentioned that one
can have situations in which imaginary terms arise in the Euclidean action due
to other reasons, which are usually topological. One key example of this is in
the context of terms in the Minkowski action which are odd under time
reversal. When analytically continued to the Euclidean sector such terms can
give rise to an imaginary part in the Euclidean action. Examples of this
include topological terms, Wess-Zumino term, Chern-Simons term etc. (see for
e.g. ref. [8]). We will not be concerned with actions containing such terms in
this paper.
The plan of the paper is as follows. In Section 2 we briefly review the
concept of the effective path as a solution to the effective action equations
and its connection with the path integral. We evaluate the effective path in
the case of a harmonic oscillator interacting with an external source in
Section 3 We show that the effective path for this case is complex and its
modulus square can be related to the total energy input into the system by the
external source due to the production of particles. (Interestingly, the
effective path in this case is similar to a complex quantity constructed by
Landau and Lifshitz in [9] to solve the problem of a forced oscillator in
classical mechanics.) We next consider (Section 4.1) the effective path for a
non-quadratic system with potential $-1/x^{2}$ and evaluate the modulus square
in a suitable approximation. We find that this quantity has a rather curious
form in that it contains a ‘Planck spectrum’. We know, however, from previous
work [10] that the problem of thermal radiation from a horizon can be mapped
to that of quantum mechanics in an inverse square potential. We study (Section
4.3) the properties of the effective path in this context and show that its
modulus square can be related to the Hawking temperature (except for a factor
of 2, the origin of which has been discussed extensively in the literature
[11]). In Appendix B we also extend the results of [7] to a more general class
singular potentials with the hope that it will be of future use.
## 2 Effective action and the concept of effective path
We shall begin by introducing the notion of effective path and its relation to
the standard effective action. We shall work in the context of point quantum
mechanics because it is adequate for our purposes; the generalization to a
field theoretic context is conceptually similar though mathematically more
involved. In the context of point quantum mechanics, the path integral kernel
describing the system is given by the Feynman path integral
$\displaystyle K(x_{2},t_{2}|x_{1},t_{1})$ $\displaystyle=\langle
x_{2},t_{2}|x_{1},t_{1}\rangle$
$\displaystyle=\int\mathcal{D}x(t)\,\exp\frac{i}{\hbar}S[x(t)]$ (1)
where the sum is over all the paths satisfying the indicated boundary
conditions. This suggests a very natural definition of an effective average
path using the path integral average:
$X(t)\equiv\frac{\int\mathcal{D}x\,x\,\exp[iS/\hbar]}{\int\mathcal{D}x\,\exp[iS/\hbar]}=\frac{\langle
x_{2},t_{2}|\hat{x}(t)|x_{1},t_{1}\rangle}{\langle
x_{2},t_{2}|x_{1},t_{1}\rangle}.$ (2)
In terms of the path integral kernel, the effective path can be expressed as
$\displaystyle X(t)$ $\displaystyle=\frac{\langle
x_{2},t_{2}|\hat{x}(t)|x_{1},t_{1}\rangle}{\langle
x_{2},t_{2}|x_{1},t_{1}\rangle}$
$\displaystyle=\frac{\int_{-\infty}^{\infty}\mathrm{d}x\,x\,K(x_{2},t_{2}|x,t)\,K(x,t|x_{1},t_{1})}{K(x_{2},t_{2}|x_{1},t_{1})}.$
(3)
We can evaluate this function once we know the path integral kernel for the
system. While the path integral average in Eq. (2) appears to be a natural
quantity to define, it should be noted that — being a transition matrix
element rather than the expectation value of an operator — it is in general a
complex quantity (which is probably why it has not received any attention in
the literature; we could not find any published study of this quantity except
in ref. [7]). But what makes $X(t)$ important is that it is a solution to the
effective action equations $\delta\Gamma=0$ including the imaginary part of
the effective action. We shall now provide a short proof of this claim for the
sake of completeness.
The standard procedure for defining the effective action is as follows. We
introduce an external source $J(t)$ and define
$\displaystyle\exp\frac{i}{\hbar}W[J(t)]$ $\displaystyle=\langle
x_{2},t_{2}|x_{1},t_{1}\rangle_{J}$
$\displaystyle=\int\mathcal{D}x(t)\,\exp\frac{i}{\hbar}\left(S[x(t)]+\int
dtJ(t)x(t)\right)$ (4)
where $W[J]$ is the generating functional for Green functions. Functional
differentiation of the generating function with respect to $J$ then
immediately leads to something very similar to the quantity in Eq. (2), and,
of course, is used in the literature:
$X[J]\equiv\frac{\delta W[J]}{\delta J}=\frac{\langle
x_{2},t_{2}|\hat{x}(t)|x_{1},t_{1}\rangle_{J}}{\langle
x_{2},t_{2}|x_{1},t_{1}\rangle_{J}}$ (5)
which is the effective average path of the system for the specified boundary
conditions but in the presence of the external source. This relation can be
inverted to get $J=J[X]$ and hence allows us to naturally define a functional
of $X$, $\Gamma[X]$, as the Legendre transform of $W[J]$ with respect to $J$
as
$\Gamma[X]\equiv W[J]-\int J(t)X(t)dt$ (6)
where $J$ is now considered a functional of $X$. It is easy to see that the
functional derivative of $\Gamma[X]$ is given by
$\frac{\delta\Gamma[X]}{\delta X}=-J.$ (7)
Thus, the extremum condition for effective action, giving the effective,
quantum corrected, dynamical equation, $\delta_{X}\Gamma=0$, implies $J=0$.
Therefore its solution is just $X[J]$ evaluated at $J=0$ which is
$X[0]\equiv\left.\frac{\delta W[J]}{\delta
J}\right\arrowvert_{J=0}=\frac{\langle
x_{2},t_{2}|\hat{x}(t)|x_{1},t_{1}\rangle}{\langle
x_{2},t_{2}|x_{1},t_{1}\rangle},$ (8)
the effective path given by Eq. (2) in the absence of the source. Since the
effective action can, in general, be complex, it follows that the complex
nature of $X(t)$ contains information about the complex nature of effective
action. It is this aspect of the effective path which we will focus our
attention on using simple examples.
## 3 Effective path for forced harmonic oscillator
We begin by considering the case of a harmonic oscillator coupled linearly to
an external source, $J(t)$. We will assume that $J(t)$ was switched on and
switched off sufficiently fast when $t\to\pm\infty$. The oscillator evolves
from the initial vacuum state in the asymptotic past to the final vacuum state
in the asymptotic future. The in-out vacuum-to-vacuum amplitude can be
calculated [12] to be
$\langle
0_{\mathrm{out}}|0_{\mathrm{in}}\rangle=\exp\left(-\frac{1}{4\hbar\omega}|\tilde{J}(\omega)|^{2}\right)$
(9)
where $\tilde{J}(\omega)$ is the Fourier mode of $J(t)$ at the oscillator’s
natural frequency, $\omega$. Since the oscillator can only absorb quanta at
its natural frequency $\omega$, we see that only the fourier mode of $J(t)$ at
the natural frequency of the oscillator is relevant for particle production.
The calculation of the effective action for this system proceeds in a
straightforward manner. By definition,
$\exp[iW[J(t)]/\hbar]=\int\mathcal{D}x(t)\exp iS[J(t),x(t)]/\hbar.$ (10)
where the action is given by
$S[x,J]=-\int\left(\frac{1}{2}x\hat{D}x-J(t)\,x\right)dt$ (11)
with $\hat{D}$ as the standard harmonic oscillator differential operator. The
path integral for the system can be computed by elementary procedures to give
$\exp[iW[J(t)]/\hbar]=(\det
D)^{-\frac{1}{2}}\exp\frac{i}{2\hbar}\int\mathrm{d}t\int\mathrm{d}t^{\prime}\,J(t)G_{F}(t,t^{\prime})J(t^{\prime}).$
(12)
where $G_{F}$ is the Feynman Green function for the harmonic oscillator. The
corresponding generating function is given by
$\displaystyle W[J(t)]$ $\displaystyle=\frac{1}{2}\int
J(t)J(t^{\prime})G_{F}(t,t^{\prime})\mathrm{d}t\mathrm{d}t^{\prime}$
$\displaystyle=\frac{i}{4\omega}|\tilde{J}(\omega)|^{2}+\int
J(t)J(t^{\prime})\frac{\sin\omega|t-t^{\prime}|}{4\omega}\mathrm{d}t\mathrm{d}t^{\prime}$
(13)
apart from a $J$-independent part from the $(\det D)^{-1/2}$ which is
irrelevant for our purpose. Using the definition
$|\tilde{J}(\omega)|^{2}=\int\mathrm{d}t\mathrm{d}t^{\prime}e^{i\omega(t-t^{\prime})}J(t)J(t^{\prime})$
(14)
we see that the imaginary part of $\Gamma[J]$ is precisely the in-out matrix
element (which, of course, can be evaluated directly in this simple case):
$\langle
0_{\mathrm{out}}|0_{\mathrm{in}}\rangle=\exp\left(-\frac{1}{4\hbar\omega}|\tilde{J}(\omega)|^{2}\right).$
(15)
The transition probability is the modulus squared of the amplitude
$|\langle
0_{\mathrm{out}}|0_{\mathrm{in}}\rangle|^{2}=\exp\left(-\frac{1}{\hbar\omega}\frac{|\tilde{J}(\omega)|^{2}}{2}\right)$
(16)
from which we can read off the energy transferred to the oscillator by the
external source to be
$\mathcal{E}=\frac{1}{2}|\tilde{J}(\omega)|^{2}.$ (17)
Thus, a time-dependent source with non-zero $\tilde{J}(\omega)$ driving a
harmonic oscillator does produce transitions of eigenstates so that the ‘in’
and the ‘out’ states are different with the amplitude given by imaginary part
of the effective action.
All this is fairly standard and we shall now introduce the effective path for
the system as a solution to the effective dynamical equations obtained by
extremizing the effective action. It is obvious that while the equation of
motion is the same as the classical one,
$\ddot{x}+\omega^{2}x=J(t)$ (18)
its solution should be now obtained in terms of the Feynman Green function
(rather than the standard retarded Green function) which makes the effective
path complex:
$X(t)=\int\mathrm{d}t^{\prime}G_{F}(t,t^{\prime})J(t^{\prime})+x^{H}(t).$ (19)
Here $x^{H}(t)$ is the solution to the homogenous equation of motion without
the external source. The oscillator in the absence of external force evolves
as
$x^{H}_{cl}(t)=x_{1}\frac{\sin\omega(t_{2}-t)}{\sin\omega(t_{2}-t_{1})}+x_{2}\frac{\sin\omega(t-t_{1})}{\sin\omega(t_{2}-t_{1})}$
(20)
between the boundary points $x_{1}(t_{1})$ and $x_{2}(t_{2})$. Letting
$t_{2}=-t_{1}=T$ and taking the limit $iT\rightarrow\infty$, we see that
$x^{H}_{cl}$ vanishes in our case when we consider sufficiently large time
intervals. This gives the effective path to be
$\displaystyle X(t)$
$\displaystyle=\int\mathrm{d}t^{\prime}G_{F}(t,t^{\prime})J(t^{\prime})=\int\mathrm{d}t^{\prime}J(t^{\prime})\frac{e^{-i\omega|t-t^{\prime}|}}{2\omega}$
$\displaystyle=\int\mathrm{d}t^{\prime}J(t^{\prime})\frac{i}{2\omega}\left[e^{-i\omega(t-t^{\prime})}\theta(t-t^{\prime})+e^{i\omega(t-t^{\prime})}\theta(t^{\prime}-t)\right]$
$\displaystyle=\int_{-\infty}^{t}\mathrm{d}t^{\prime}J(t^{\prime})\frac{i}{2\omega}e^{-i\omega(t-t^{\prime})}+\int_{t}^{\infty}\mathrm{d}t^{\prime}J(t^{\prime})\frac{i}{2\omega}e^{i\omega(t-t^{\prime})}$
(21)
with the real and imaginary parts
$\displaystyle\mathrm{Re}X(t)$
$\displaystyle=\int_{-\infty}^{t}\mathrm{d}t^{\prime}J(t^{\prime})\frac{\sin\omega(t-t^{\prime})}{\omega}-\frac{1}{2}\int_{-\infty}^{\infty}\mathrm{d}t^{\prime}J(t^{\prime})\frac{\sin\omega(t-t^{\prime})}{\omega}$
$\displaystyle=x_{cl}(t)-\frac{1}{2}\int_{-\infty}^{\infty}\mathrm{d}t^{\prime}J(t^{\prime})\frac{\sin\omega(t-t^{\prime})}{\omega}$
(22) $\displaystyle\mathrm{Im}X(t)$
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}\mathrm{d}t^{\prime}J(t^{\prime})\frac{\cos\omega(t-t^{\prime})}{\omega}$
(23)
where $x_{cl}(t)$ is the classical solution to the driven oscillator evaluated
with retarded boundary conditions.
$\displaystyle x_{cl}(t)$
$\displaystyle=\int\mathrm{d}t^{\prime}G_{R}(t,t^{\prime})J(t^{\prime})=\int\mathrm{d}t^{\prime}J(t^{\prime})\frac{\sin\omega(t-t^{\prime})}{\omega}\theta(t-t^{\prime})$
$\displaystyle=\int_{-\infty}^{t}\mathrm{d}t^{\prime}J(t^{\prime})\frac{\sin\omega(t-t^{\prime})}{\omega}.$
(24)
It is obvious that the net effect of the source is to introduce an imaginary
part to $X(t)$ and modify the real part by an extra term.
Since we have already shown that the effective path $X(t)$ is a solution to
the effective action equations, one can also compute the effective action for
our system by evaluating it for the effective complex path given above. An
elementary calculation shows that the result is given by
$\Gamma[X_{\mathrm{eff}}]=-\frac{1}{2}\int\mathrm{d}t\,\left(X_{\mathrm{eff}}J-2JX_{\mathrm{eff}}\right)=\frac{1}{2}\int\mathrm{d}t\,JX_{\mathrm{eff}}$
(25)
so that
$\displaystyle\mathrm{Im}\,\Gamma[X_{\mathrm{eff}}]$
$\displaystyle=\frac{1}{2}\int\mathrm{d}t\,J\,\mathrm{Im}X_{\mathrm{eff}}$
(26)
$\displaystyle=\int\mathrm{d}t\mathrm{d}t^{\prime}\,\frac{\cos\omega(t-t^{\prime})}{4\omega}\,J(t)J(t^{\prime})=\frac{1}{4\omega}|\tilde{J}(\omega)|^{2}$
(27)
which agrees with the result obtained in Eq. (15).
We will now highlight the above aspects with an explicit example. Consider the
source $J(t)=|t|e^{-\lambda|t|}$, which is chosen specifically to distinguish
the cases in which the particle production occurs from those in which it does
not. We have seen that the energy that goes into the system from external
source is proportional to the modulus square of fourier mode of the source
evaluated at $\omega$, natural frequency of the oscillator. For our choice of
$J(t)=|t|e^{-\lambda|t|}$ we have:
$|\tilde{J}(\omega)|^{2}=\frac{(\lambda^{2}-\omega^{2})^{2}}{(\omega^{2}+\lambda^{2})^{4}}$
(28)
which vanishes for the parameter $\lambda=\omega$ and hence there is no
particle production in that case. We have tabulated the results for the two
cases, one with a general $\lambda$ and the other with $\lambda=\omega$:
$J(t)$ | $|t|e^{-\lambda|t|}$ | $|t|e^{-\omega|t|}$
---|---|---
$x_{cl}(t)$ | $\left(\frac{2(\lambda^{2}-\omega^{2})\sin\omega t}{\omega(\omega^{2}+\lambda^{2})^{2}}+\frac{e^{-\lambda t}(\lambda(2+\lambda t)+\omega^{2}t)}{(\omega^{2}+\lambda^{2})^{2}}\right)\theta(t)-\left(\frac{e^{\lambda t}(\lambda(-2+\lambda t)+\omega^{2}t)}{(\omega^{2}+\lambda^{2})^{2}}\right)\theta(-t)$ | $\frac{e^{-\omega t}(\omega(2+\omega t)+\omega^{2}t)}{4\omega^{4}}\theta(t)-\frac{e^{\omega t}(\omega(-2+\omega t)+\omega^{2}t)}{4\omega^{4}}\theta(-t)$
$\mathrm{Re}X(t)$ | $\left(\frac{(\lambda^{2}-\omega^{2})\sin\omega t}{\omega(\omega^{2}+\lambda^{2})^{2}}+\frac{e^{-\lambda t}(\lambda(2+\lambda t)+\omega^{2}t)}{(\omega^{2}+\lambda^{2})^{2}}\right)\theta(t)-\left(\frac{(\lambda^{2}-\omega^{2})\sin\omega t}{\omega(\omega^{2}+\lambda^{2})^{2}}+\frac{e^{\lambda t}(\lambda(-2+\lambda t)+\omega^{2}t)}{(\omega^{2}+\lambda^{2})^{2}}\right)\theta(-t)$ | $x_{cl}|_{\lambda=\omega}$
$\mathrm{Im}X(t)$ | $\frac{(\lambda^{2}-\omega^{2})\cos\omega t}{\omega(\omega^{2}+\lambda^{2})^{2}}$ | $0$
$|X|^{2}_{t=\infty}$ | $\frac{(\lambda^{2}-\omega^{2})^{2}}{\omega^{2}(\omega^{2}+\lambda^{2})^{4}}$ | $0$
$|\tilde{J}(\omega)|^{2}=2\mathrm{Im}\,W$ | $\frac{(\lambda^{2}-\omega^{2})^{2}}{(\omega^{2}+\lambda^{2})^{4}}$ | $0$
It is obvious that the imaginary part of the effective path is related to the
particle production and vanishes when there is no particle production.
Further, when $\lambda t\to\infty$ we can approximate the real and imaginary
parts of $X(t)$ by
$\mathrm{Re}X(t)\approx\frac{(\lambda^{2}-\omega^{2})\sin\omega
t}{\omega(\omega^{2}+\lambda^{2})^{2}};\quad\mathrm{Im}X(t)=\frac{(\lambda^{2}-\omega^{2})\cos\omega
t}{\omega(\omega^{2}+\lambda^{2})^{2}}.$ (29)
It follows that
$\mathcal{E}=\frac{\omega^{2}|X|^{2}_{t\to\infty}}{2}=\frac{|\tilde{J}(\omega)|^{2}}{2}=\mathrm{Im}\,W$
(30)
giving a direct relation between the particle production rate and the squared
modulus of the effective path. It is also worth mentioning here that the
effective path which we get as the solution of effective action equation of
motion, interestingly, gives an interpretation to the complex quantity,
$\xi(t)=\dot{x}+i\,\omega x$ (31)
constructed in [9] purely as a mathematical trick for solving the problem of
forced harmonic oscillator. The energy input into the system in terms of $\xi$
is
$\mathcal{E}=\frac{|\xi(\infty)|^{2}}{2}.$ (32)
We can identify the corresponding real and imaginary parts in $X(t)$ and
$\xi(t)$ apart from a factor of $\omega$. This elementary illustration shows
that even in the context of such a simple system the concept of effective path
can be related to a tangible result.
## 4 Inverse square potential in quantum mechanics and applications to
horizon thermodynamics
The results in the above case are rather simple because the coupling was
linear. We next investigate the complex path formalism in the case of a
nontrivial example, involving one-dimensional inverse square potential. The
primary motivation for this arises from the fact that the problem of a scalar
field in Schwarzschild background — and, more generally, in any spacetime in
which the near horizon geometry can be approximated as Rindler — can be
reduced to dynamics of a particle in an inverse square potential across the
singularity. We explore the nature of the effective path in this potential and
show that it has some curious features which find application to the problem
of black hole evaporation.
### 4.1 Complex effective path for the inverse square potential
We will consider an inverse square potential of the form
$V(x)=-\frac{\hbar^{2}}{2m}\left(a^{2}+\frac{1}{4}\right)\frac{1}{x^{2}}=-\frac{\tilde{\alpha}}{x^{2}}.$
(33)
where $a,\tilde{\alpha}$ are constants. Since $a$ is real,
$\tilde{\alpha}>\hbar^{2}/8m$. To calculate the effective path in this case,
we will use the path integral average. The kernel for a particle to propagate
from points $(x_{1},t_{1})$ to $(x_{2},t_{2})$ in an inverse square potential,
$V=-\tilde{\alpha}x^{-2}$ is given by (see Appendix A.1 for details),
$K(t_{2},x_{2}|t_{1},x_{1})=e^{-\frac{1}{2}i\pi(\gamma+1)}\left(\frac{m}{2\hbar(t_{2}-t_{1})}\right)(x_{1}x_{2})^{1/2}\exp\left[\frac{im(x_{1}^{2}+x_{2}^{2})}{2\hbar(t_{2}-t_{1})}\right]H_{\gamma}^{(2)}\left(\frac{mx_{1}x_{2}}{\hbar(t_{2}-t_{1})}\right)$
(34)
where $H_{\gamma}^{(2)}(z)$ is the Hankel function of the second kind of order
$\gamma=\sqrt{\frac{1}{4}-\frac{2m\tilde{\alpha}}{\hbar^{2}}}=ia.$ (35)
which is a dimensionless constant and we have substituted for $\tilde{\alpha}$
from Eq. (33). The effective path defined in Eq. (2) is given by the integral,
$X(t)=\frac{\langle x_{2},t_{2}|\hat{x}(t)|x_{1},t_{1}\rangle}{\langle
x_{2},t_{2}|x_{1},t_{1}\rangle}=\frac{1}{K(x_{2},t_{2}|x_{1},t_{1})}\int_{-\infty}^{\infty}\mathrm{d}x\,x\,K(x_{2},t_{2}|x,t)\,K(x,t|x_{1},t_{1}).$
(36)
Substituting the kernel from Eq. (34), we get,
$\displaystyle X(t)=$
$\displaystyle\lambda\>\exp\left[\frac{-i\pi}{2}(ia+1)\right]\exp\left[\frac{im}{2\hbar}\left(\frac{x_{2}^{2}}{t_{2}-t}+\frac{x_{1}^{2}}{t-t_{1}}-\frac{(x_{1}^{2}+x_{2}^{2})}{t_{2}-t_{1}}\right)\right]\left[H_{ia}^{(2)}\left(\frac{mx_{1}x_{2}}{\hbar(t_{2}-t_{1})}\right)\right]^{-1}$
$\displaystyle\int_{-\infty}^{\infty}\,\mathrm{d}x\,x^{2}e^{i\lambda
x^{2}}H_{ia}^{(2)}(px)H_{ia}^{(2)}(qx)$ (37)
where we have defined,
$\lambda\equiv\frac{m(t_{2}-t_{1})}{2\hbar(t_{2}-t)(t-t_{1})}\hskip
2.0pt\hbox{,}\hskip 14.0ptp\equiv\frac{mx_{1}}{\hbar(t-t_{1})}\hskip
7.0pt\hbox{and}\hskip 7.5ptq\equiv\frac{mx_{2}}{\hbar(t_{2}-t)}.$ (38)
Note that $\lambda$ has the dimension of inverse length squared while $p$ and
$q$ both have dimensions of inverse length. Since the interesting physics
takes place when a particle crosses the singularity at the origin, $x=0$, we
will take $x_{1}=-\epsilon$ at $t_{1}=0$ and $x_{2}=\epsilon$ as
$t_{2}\rightarrow\infty$ with limit $\epsilon\rightarrow 0^{+}$ taken
eventually so that the particle has to cross from left to right in the late-
time limit. To begin with it is convenient to keep $t_{1}$ and $t_{2}$
arbitrary and take the limit at the end of the calculation. Under these
conditions, the effective path becomes
$\displaystyle X(t)$ $\displaystyle=\lambda
e^{-i\frac{\pi}{2}(ia+1)}\left[H_{ia}^{(2)}\left(\frac{-m\epsilon^{2}}{\hbar(t_{2}-t_{1})}\right)\right]^{-1}\int_{-\infty}^{\infty}\mathrm{d}x\,x^{2}e^{i\lambda
x^{2}}H_{ia}^{(2)}\left(\frac{-m\epsilon
x}{\hbar(t-t_{1})}\right)H_{ia}^{(2)}\left(\frac{m\epsilon
x}{\hbar(t_{2}-t)}\right)$
Unfortunately, the integral in the above expression cannot be evaluated
exactly in closed from but we can calculate it under the limit
$\epsilon\rightarrow 0^{+}$ as follows. We first express the Hankel functions
in the integrand in terms of the Bessel functions which reduces the integral
to the form,
$\displaystyle I=$
$\displaystyle\int_{-\infty}^{\infty}\mathrm{d}x\,x^{2}e^{i\lambda
x^{2}}H_{ia}^{(2)}\left(\frac{-m\epsilon
x}{\hbar(t-t_{1})}\right)H_{ia}^{(2)}\left(\frac{m\epsilon
x}{\hbar(t_{2}-t)}\right)$ $\displaystyle=$ $\displaystyle\left(1-\coth\pi
a\right)^{2}\int_{-\infty}^{\infty}\mathrm{d}x\;x^{2}e^{i\lambda
x^{2}}J_{ia}(px)J_{ia}(qx)+\frac{1}{\sinh^{2}\pi
a}\int_{-\infty}^{\infty}\mathrm{d}x\;x^{2}e^{i\lambda
x^{2}}J_{-ia}(px)J_{-ia}(qx)$ $\displaystyle+\frac{\left(1-\coth\pi
a\right)}{\sinh\pi a}\int_{-\infty}^{\infty}\mathrm{d}x\;x^{2}e^{i\lambda
x^{2}}\left(J_{ia}(px)J_{-ia}(qx)+J_{-ia}(px)J_{ia}(qx)\right).$ (40)
Now we can use the following identity (see [13]),
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\;x^{\lambda+1}e^{-\alpha
x^{2}}J_{\mu}(\beta x)J_{\nu}(\gamma x)$
$\displaystyle=\frac{\beta^{\mu}\gamma^{\nu}\alpha^{-(\mu+\nu+\lambda+2)/2}}{2^{\nu+\mu+1}\Gamma(\nu+1)}\sum_{m=0}^{\infty}\frac{\Gamma(m+\frac{1}{2}(\nu+\mu+\lambda+2))}{\Gamma(m+\mu+1)\Gamma(m+1)}\left(\frac{-\beta^{2}}{4\alpha}\right)^{m}$
$\displaystyle\hskip 15.5ptF(-m,-\mu-m;\nu+1;\frac{\gamma^{2}}{\beta^{2}})$
(41)
and evaluate the integral in the limit of $\epsilon\to 0^{+}$ (see Appendix
A.2 for details). In the same limit the Hankel function in the denominator can
be approximated by:
$H_{ia}^{(2)}(z)\approx\frac{i2^{-ia}e^{-\pi
a}\Gamma(-ia)z^{ia}}{\pi}+\frac{i2^{ia}\Gamma(ia)z^{-ia}}{\pi}$ (42)
With these manipulations the effective path can be expressed as,
$X(t)=-i\lambda\,e^{\pi a/2}\,\frac{I}{D}$ (43)
where
$\displaystyle I$ $\displaystyle=\left\\{-\frac{e^{\pi
a/2}}{2\pi^{2}}2^{-ia}\left[\Gamma(-ia)\right]^{2}\left(\frac{m\epsilon^{2}}{\hbar(t_{2}-t_{1})}\right)^{ia}\left(1+e^{-2\pi
a}\right)(-i\lambda)^{-3/2}(ia+1/2)\Gamma(ia+1/2)\right.$
$\displaystyle-\left.\frac{e^{-\pi
a/2}\,2^{ia}\left[\Gamma(ia)\right]^{2}}{\pi^{2}}\left(\frac{m\epsilon^{2}}{\hbar(t_{2}-t_{1})}\right)^{-ia}\left(1+e^{-2\pi
a}\right)(-i\lambda)^{-3/2}(-ia+1/2)\Gamma(-ia+1/2)\right.$
$\displaystyle\left.-\frac{e^{\pi a}\sqrt{\pi}}{2\pi a\sinh\pi
a}(-i\lambda)^{-3/2}\left[e^{\pi
a}\left(\frac{t_{2}-t}{t-t_{1}}\right)^{ia}+e^{-\pi
a}\left(\frac{t_{2}-t}{t-t_{1}}\right)^{-ia}\right]\right\\}$ (44)
and
$D=\left[\frac{i2^{-ia}\Gamma(-ia)}{\pi}\left(\frac{m\epsilon^{2}}{\hbar(t_{2}-t_{1})}\right)^{ia}+\frac{i2^{ia}\Gamma(ia)}{\pi}\left(\frac{m\epsilon^{2}}{\hbar(t_{2}-t_{1})}\right)^{-ia}e^{-\pi
a}\right].$ (45)
Based on our previous analysis of forced harmonic oscillator in Section 3, we
would suspect $|X|^{2}$ to contain information about the analogue of particle
creation in a quantum theory. It is obvious that $|X|^{2}$ arising from the
above expression will be quite complicated partially due to the fact that it
is evaluated for finite time and space interval. To understand the physical
significance of this quantity it is again useful to take the limit of
$t_{2}\rightarrow\infty$ with $t_{1}=0$ and $\epsilon\rightarrow 0^{+}$. In
this limit, one can ignore transient terms which oscillate rapidly and obtain
a simpler expression for $|X|^{2}$. Somewhat tedious but straight forward
algebra (see Appendix A.3) yields an interesting final result: We find that
$|X|^{2}$ increases linearly with time allowing us to define a constant,
finite, rate which itself takes a very suggestive form as:
$\frac{\mathrm{d}|X(t)|^{2}}{\mathrm{d}t}=\left(\frac{4\hbar}{ma}\right)\left(a^{2}+\frac{1}{4}\right)\left[N+\frac{1}{2}\right]$
(46)
where
$N=\frac{1}{e^{2\pi a}-1}$ (47)
has the form of a Planckian spectrum of particles. If one thinks of
$d|X|^{2}/dt$ as the rate of production of particles, then it is rather
curious that we have a thermal radiation term related to a parameter in the
potential, $a$. Obviously, in this particular quantum mechanical example, this
result has no physical interpretation but we will next show how this result
connects up with radiation from a horizon.
### 4.2 Quantum mechanics of the scalar field near the horizon
It turns out that the problem of a scalar field near a black hole spacetime
(more generally in any spacetime with a horizon when we consider the Rindler
limit of the horizon) can be reduced to that of a quantum mechanical particle
in an inverse square potential. In that context, the $d|X|^{2}/dt$ can be
thought of as rate of production of particles by the horizon and the
mathematical result obtained above acquires a physical meaning.
We shall first briefly sketch how the problem of a scalar field near a horizon
can be mapped to a quantum mechanical problem of a particle in an inverse-
square potential [10]. Consider a scalar field in a 1+1 spacetime with the
metric
$ds^{2}=B(r)dt^{2}-B^{-1}(r)dr^{2}$ (48)
where $B(r)$ has a simple zero at $r=r_{0}$ with $B^{\prime}(r)=dB/dr$ being
finite and nonzero at $r_{0}$. (We will work with (1+1) dimensional system
since it captures all the essential physics.) The vanishing of $B(r)$ at point
$r=r_{0}$ indicates the presence of a horizon. Near the horizon, we can expand
$B(r)$ as
$B(r)=B^{\prime}(r_{0})(r-r_{0})+\mathcal{O}[(r-r_{0})^{2}]\approx
B^{\prime}(r_{0})(r-r_{0}).$ (49)
Note that in the Schwarzschild case, $B^{\prime}(r_{0})=r_{0}^{-2}$ with
$r_{0}=2M$ as the Schwarzschild radius. The field equation for the scalar
field $\Phi(t,r)$,
$\left(\Box+\frac{m_{0}^{2}c^{2}}{\hbar^{2}}\right)\Phi=0$ (50)
when written for the metric in Eq. (48) becomes
$c^{-2}B(r)^{-1}\,\partial^{2}_{t}\Phi-\partial_{r}\left(B(r)\partial_{r}\Phi\right)=-m_{0}^{2}c^{2}\hbar^{-2}\,\Phi.$
(51)
We substitute the following ansatz for $\Phi$ in the above equation,
$\Phi(r,t)=e^{-i\omega t}\frac{\psi(r)}{\sqrt{B(r)}}$ (52)
and find that $\psi(r)$ satisfies the equation
$-\frac{\hbar^{2}}{2}\,\frac{d^{2}\psi(r)}{dr^{2}}-\frac{\alpha}{(r-r_{0})^{2}}\,\psi(r)=0$
(53)
where $\alpha=\hbar^{2}\omega^{2}/2c^{2}[B^{\prime}(r_{0})]^{2}$ near the
horizon (Note that in the near-horizon limit, the term with $m_{0}$ does not
contribute in the leading order). For the Schwarzschild metric,
$\alpha=\hbar^{2}\omega^{2}r_{0}^{2}/2c^{2}$ hence we see that $\alpha$ has
dimensions of $\hbar^{2}$, as it should. With $x=(r-r_{0})$, and mass, $m$ put
in, this equation is same as the Schr$\ddot{\mathrm{o}}$dinger equation for a
particle in an inverse square potential, $-\tilde{\alpha}/x^{2}$,
$-\frac{\hbar^{2}}{2m}\,\frac{d^{2}\psi(x)}{dx^{2}}-\frac{\tilde{\alpha}}{x^{2}}\,\psi(x)=\mathcal{E}\psi(x)$
(54)
where $\tilde{\alpha}=\alpha/m$ and we take the energy eigenvalue
$\mathcal{E}\,\rightarrow\,0$ at the end of the calculations. Thus the problem
of scalar field in Schwarzschild background is equivalent to quantum mechanics
of a particle in an inverse square potential near the origin.
### 4.3 Horizon thermodynamics
With the problem of a scalar field in the Schwarzschild background reduced to
an effective quantum mechanical problem in an inverse square potential, we can
identify the parameters of potential in the two situations.
$V(x)=\frac{\hbar^{2}}{2m}\left(a^{2}+\frac{1}{4}\right)\frac{1}{x^{2}}=-\frac{\hbar^{2}\omega^{2}r_{0}^{2}}{2mc^{2}}\frac{1}{x^{2}}$
(55)
which gives for $a$,
$a=\left(\frac{\omega^{2}r_{0}^{2}}{c^{2}}-\frac{1}{4}\right)^{1/2}\approx\frac{\omega
r_{0}}{c}.$ (56)
in the high-frequency limit. In this case, substituting for $a$ in our
expression for $d|X|^{2}/dt$ obtained earlier gives,
$\frac{\mathrm{d}|X(t)|^{2}}{\mathrm{d}t}=\frac{8GM}{mc^{3}}\left[\hbar\omega\left(N+\frac{1}{2}\right)\right]$
(57)
where
$N=\frac{1}{e^{2\pi a}-1}=\frac{1}{e^{\hbar\omega/k_{B}T}-1}$ (58)
with the temperature being given by
$T=\frac{\hbar c^{3}}{4\pi GMk_{B}}=\frac{1}{4\pi M}$ (59)
where the second result is valid in natural units. (The time asymmetry in our
boundary conditions, $t_{1}=0,t_{2}\to\infty$ makes it meaningful to treat
$d|X|^{2}/dt$ as a rate.) The expression $N$, of course, represents a
Planckian spectrum of particles at temperature $T$ and the part $N+(1/2)$
correctly describes the Planckian energy density of a cavity at temperature
$T$ along with the zero point contribution. The temperature $T=1/4\pi M$
however is twice the usual temperature associated with black holes. This
feature is well-known in the tunneling derivation of black hole temperature
and has been extensively discussed in the literature [11]. While the topic is
still somewhat controversial, the origin of this extra factor is attributed to
using singular coordinates at horizon [14]. Since we have started with
Schwarzschild coordinates (which are ill-defined at the horizon) it is
probably natural that we get this result.
Thus the effective path approach seems to capture the Planck spectrum (with
the temperature off by factor 2 which occurs in some other tunneling
computations as well) along with zero-point energy. So the squared modulus of
$X(t)$ does contain information related to the production of particles, this
time in a fairly non-trivial setting.
## 5 Conclusions
The concept of effective action is a well-known technique that is used in the
literature to study various aspects of quantum field theories in classical
backgrounds. The effective action is in general complex and its real and
imaginary parts contain information about the vacuum polarization and particle
production. In using the effective action to describe the back reaction
effects, one usually uses the real part of the effective action and discards
the imaginary part in order to obtain real equations of motion.
On the other hand if we retain both real and imaginary parts of the effective
action and obtain the equations of motion, then the solutions will be — in
general — complex. Because the imaginary part of the effective action contains
information about the particle production, it seems likely that the solutions
to the complex effective action will give us a handle to explore particle
production. This motivates us to study a quantum effective path $X(t)$ (in the
context of QM) which is a solution to $\delta\Gamma[X]=0$. Fortunately, this
$X(t)$ can be expressed as an integral over the path integral kernel and hence
can be evaluated, in principle, if the kernel is known.
In practice, the calculation turns out to be quite complicated. To gather a
preliminary insight we studied two important examples in this paper. First one
is the case of a forced harmonic oscillator in which we could directly link
the complex effective path to particle production in the asymptotic limit. The
imaginary part of the effective path is generated solely by the non-zero
Fourier mode of the external source at the natural frequency of the
oscillator. The modulus square of the complex effective path gave the particle
production rate in the system. (We also found that the complex effective path
obtained in this case also provided a nice interpretation to a quantity which
was purely a mathematical construction by Landau used in the case of forced
harmonic oscillator.)
The second case we studied was that of an attractive, inverse-square
potential. It was known from previous work that the problem of a scalar field
in a spacetime with a horizon (in which the near-horizon geometry can be
approximated as Rindler geometry) can be mapped to the
Schr$\mathrm{\ddot{o}}$dinger problem in an inverse square potential. We
expect the emission of particles by the black hole to get mapped to
propagation of particle through the singularity at the origin in the
equivalent Schr$\mathrm{\ddot{o}}$dinger problem, even though there are no
time-dependent sources. In this case the modulus square of the effective path
can be interpreted as a rate of emission of particles. This expression
correctly gives the Planckian distribution along with the zero point
contribution for the Hawking radiation. The temperature of the Planckian
distribution turns out to be $T=\hbar/4\pi M$ which is twice the standard
value for Hawking temperature. This factor of two discrepency has been noticed
in the literature previously and arises when one uses coordinates which are
singular at the horizon and hence is probably understandable.
Finally we would like to make some comments regarding the nature of the
potential ($V(x)\propto-x^{-2}$) considered in the last section and the
existence of imaginary part in the trajectory. The first example we studied in
section 3 has an explicitly time-dependant external source and hence it is not
surprising that we encounter particle production and an imaginary part to the
classical trajectory. In the case of an inverse square potential there is no
explicit time-dependance but we still obtain an imaginary part to the
trajectory. In fact, the wave equation for a scalar field in the black hole
spacetime — which contains the physics of black hole evaporation — does get
mapped to such a static potential. It is, however, known from previous work
that particle creation can occur even in the absence of explicit time-
dependance. A well studied example of this is the Schwinger effect in which
one obtains a steady particle production in the presence of static electric
field. In this particular case the wave equation can be mapped to an inverted
harmonic oscillator [10] for which the Hamiltonian is unbounded. In a way, it
is this singular behavior of the Hamiltonian which leads to the particle
production. (In contrast, the wave equation in the presence of a constant
magnetic field gets mapped to a normal harmonic oscillator with a bounded
Hamiltonian and — as expected — one does not have any particle creation in a
constant magnetic field.) The current situation is very similar: The wave
equation in the black hole spacetime gets mapped to an inverse square
potential and it is well-known that this potential leads to a Hamiltonian
which is not hermitian. In the previous work [10] which connects up particle
production in blackhole spacetime with the inverse square potential one
crucially used the singular structure of the potential (and an integration
around a singularity in complex plane) to obtain the result. This work has
established the essential connection between the singular nature of the
Hamiltonian in these potentials (both in the context of black hole spacetimes
as well as in the context of constant electric field) and the production of
particles. We, therefore, believe that path integral formalism studied in this
paper leads to complex trajectories for essentially the same reason viz. that
the Hamiltonian is non-Hermitian. It would be interesting to investigate this
question further and see whether one can provide a direct and rigorous proof
for the the existence of complex paths for certain class of Hamiltonians which
are unbounded or non-Hermitian.
## Acknowledgements
SS is supported by a fellowship from the Council of Scientific and Industrial
Research (CSIR), India. TP’s research is partially supported by the J.C.Bose
Research Grant of DST, India. We thank the referee for useful comments.
## Appendix A Detailed calculation of some results
### A.1 Path integral kernel for an inverse square potential
The path integral kernel is defined by,
$K(x_{2},T|x_{1},0)=\int\mathcal{D}x(t)\,e^{\frac{i}{\hbar}[\int_{0}^{t}\mathrm{d}t\,(\frac{1}{2}m\dot{x}^{2}-\alpha
x^{-2})]}$ (60)
To evaluate the kernel, we use the perturbative series expansion, which gives
$\displaystyle K(x_{2},T|x_{1},0)$ $\displaystyle=$ $\displaystyle
K_{0}(x_{2},t_{2}|x_{1},t_{1})+\sum_{n\,=\,1}^{\infty}\left(\frac{-i\alpha}{\hbar}\right)^{n}\int_{0}^{T}\mathrm{d}t_{n}\int_{0}^{t_{n}}\mathrm{d}t_{n-1}\cdot$
(61)
$\displaystyle\cdot\cdot\int_{0}^{t_{2}}\mathrm{d}t_{1}\int\prod_{j=1}^{n}\frac{\mathrm{d}x_{j}}{x_{j}^{2}}\left(\prod_{j=0}^{n}K_{0}(x_{j+1},t_{j+1}|x_{j},t_{j})\right)$
where $K_{0}(x_{2},t_{2}|x_{1},t_{1})$ is the free particle kernel.
Introducing
$G(x_{2},x_{1};E)\equiv\int_{0}^{\infty}\mathrm{d}Te^{-\frac{i}{\hbar}ET}K(x_{2},T|x_{1},0)$
(62)
we have,
$\displaystyle G(x_{2},x_{1};E)$ $\displaystyle=$ $\displaystyle
G_{0}(x_{2},x_{1};E)+\sum_{n\,=\,1}^{\infty}\left(\frac{-i\alpha}{\hbar}\right)^{n}\int\prod_{j=1}^{n}\frac{\mathrm{d}x_{j}}{x_{j}^{2}}\left(\prod_{j=0}^{n}G_{0}(x_{j+1},x_{j};E)\right)$
(63) $\displaystyle=$ $\displaystyle
G_{0}(x_{2},x_{1};E)+\sum_{n\,=\,1}^{\infty}\left(\frac{-i\alpha}{\hbar}\right)^{n}G_{n}$
We need to sum up the above series so as to get the closed form for the
Kernel. To do this we employ a trick [17] in which we first express the free
particle propagator, $G_{0}(x_{2},x_{1};E)$ in terms of the Hankel functions
and then use their orthogonality relation to evaluate the $n^{th}$ order
product, $G_{n}$.
$\displaystyle G_{0}(x_{2},x_{1};E)$
$\displaystyle\equiv\frac{1}{i}\left(\frac{m}{2E}\right)^{1/2}e^{ik|x_{2}-x_{1}|}$
$\displaystyle=\left(\frac{m\pi}{i2\hbar}\right)(x_{1}x_{2})^{1/2}\,H_{1/2}^{(1)}(kx_{>})H_{1/2}^{(2)}(kx_{<})$
$\displaystyle=\left(\frac{m}{i\hbar}\right)(x_{1}x_{2})^{1/2}\int_{0}^{\infty}\mathrm{d}\nu\,\frac{\nu\sinh(\nu\pi)}{\nu^{2}+1/4}H_{i\nu}^{(1)}(kx_{2})H_{i\nu}^{(1)*}(kx_{1})$
(64)
where $k=\sqrt{\frac{2mE}{\hbar^{2}}}$. Upon inserting the expression for
$G_{0}$, the n-fold integrations can be performed using the orthogonality
relation
$\int_{0}^{\infty}\frac{\mathrm{d}x}{x}\,H_{i\nu}^{(1)*}(kx)H_{i\nu^{\prime}}^{(1)}(kx)=\frac{2\delta(\nu-\nu^{\prime})}{\nu\sinh\,\nu\pi}$
and we obtain
$G_{n}(x_{2},x_{1};E)=\left(\frac{2m}{i\hbar}\right)^{n+1}\frac{(x_{1}x_{2})^{1/2}}{2}\int_{0}^{\infty}\mathrm{d}\nu\,\frac{\nu\sinh(\nu\pi)}{\nu^{2}+1/4}H_{i\nu}^{(1)}(kx_{2})H_{i\nu}^{(1)*}(kx_{1})$
(65)
We substitute this expression for $G_{n}$ and sum the resulting geometric
series to get
$G(x_{2},x_{1};E)=\left(\frac{m}{i\hbar}\right)(x_{1}x_{2})^{1/2}\int_{0}^{\infty}\mathrm{d}\nu\,\frac{\nu\sinh(\nu\pi)}{\nu^{2}+1/4+\frac{2m\alpha}{\hbar^{2}}}H_{i\nu}^{(1)}(kx_{2})H_{i\nu}^{(1)*}(kx_{1})$
(66)
which is the exact expression and is similar to the free Green function in Eq.
(A.1) with an addition to the denominator of the integrand. Noting that the
the free particle Kernel can be written as,
$\displaystyle K_{0}$ $\displaystyle=$ $\displaystyle\left(\frac{m}{2\pi
i\hbar T}\right)^{1/2}\exp\left[\frac{im}{2\hbar T}(x_{2}-x_{1})^{2}\right]$
(67) $\displaystyle=$ $\displaystyle e^{-\frac{i3\pi}{4}}\left(\frac{m}{2\hbar
T}\right)(x_{1}x_{2})^{1/2}\exp\left[\frac{im(x_{1}^{2}+x_{2}^{2})}{2\hbar
T}\right]H_{1/2}^{(2)}\left(\frac{mx_{1}x_{2}}{\hbar T}\right)$
We can obtain the Kernel for the problem by suitable replacements in the free
particle Kernel due to modified denominator in (59) from (57) as
$K(t_{2},x_{2}|t_{1},x_{1})=e^{-\frac{1}{2}i\pi(\gamma+1)}\left(\frac{m}{2\hbar
T}\right)(x_{1}x_{2})^{1/2}\exp\left[\frac{im(x_{1}^{2}+x_{2}^{2})}{2\hbar
T}\right]H_{\gamma}^{(2)}\left(\frac{mx_{1}x_{2}}{\hbar T}\right)$ (68)
where
$\gamma=\sqrt{\frac{1}{4}+\frac{2m\alpha}{\hbar^{2}}}$ (69)
When $\alpha=0$, we have $\gamma=1/2$ and the expression reduces to the free-
particle Kernel.
### A.2 Evaluation of the integral in Eq. (4.1)
We have,
$\displaystyle I=$
$\displaystyle\int_{-\infty}^{\infty}\mathrm{d}x\,x^{2}e^{i\lambda
x^{2}}H_{ia}^{(2)}\left(\frac{-m\epsilon
x}{\hbar(t-t_{1})}\right)H_{ia}^{(2)}\left(\frac{m\epsilon
x}{\hbar(t_{2}-t)}\right)$ $\displaystyle=$ $\displaystyle\left(1-\coth\pi
a\right)^{2}\int_{-\infty}^{\infty}\mathrm{d}x\;x^{2}e^{i\lambda
x^{2}}J_{ia}(px)J_{ia}(qx)+\frac{1}{\sinh^{2}\pi
a}\int_{-\infty}^{\infty}\mathrm{d}x\;x^{2}e^{i\lambda
x^{2}}J_{-ia}(px)J_{-ia}(qx)$ $\displaystyle+\frac{\left(1-\coth\pi
a\right)}{\sinh\pi a}\int_{-\infty}^{\infty}\mathrm{d}x\;x^{2}e^{i\lambda
x^{2}}\left(J_{ia}(px)J_{-ia}(qx)+J_{-ia}(px)J_{ia}(qx)\right).$ (70)
Using the following identity(see [13]),
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\;x^{\lambda+1}e^{-\alpha
x^{2}}J_{\mu}(\beta x)J_{\nu}(\gamma x)$
$\displaystyle=\frac{\beta^{\mu}\gamma^{\nu}\alpha^{-(\mu+\nu+\lambda+2)/2}}{2^{\nu+\mu+1}\Gamma(\nu+1)}\sum_{m=0}^{\infty}\frac{\Gamma(m+\frac{1}{2}(\nu+\mu+\lambda+2))}{\Gamma(m+\mu+1)\Gamma(m+1)}\left(\frac{-\beta^{2}}{4\alpha}\right)^{m}$
$\displaystyle\hskip 15.5ptF(-m,-\mu-m;\nu+1;\frac{\gamma^{2}}{\beta^{2}})$
(71)
the three integrals in $I$ can be evaluated in the following manner.
$\displaystyle I_{1}$ $\displaystyle=\left(1-\coth\pi
a\right)^{2}\int_{-\infty}^{\infty}\mathrm{d}x\;x^{2}e^{i\lambda
x^{2}}J_{ia}(px)J_{ia}(qx)$ $\displaystyle=\frac{e^{-2\pi a}}{(\sinh\pi
a)^{2}}\left(1+e^{2\pi a}\right)\int_{0}^{\infty}\mathrm{d}x\;x^{2}e^{i\lambda
x^{2}}J_{ia}(px)J_{ia}(qx)$ $\displaystyle=\frac{\left(1+e^{-2\pi
a}\right)}{(\sinh\pi
a)^{2}}\frac{(pq)^{ia}(-i\lambda)^{-ia-3/2}}{2^{2ia+1}\Gamma(ia+1)}\sum_{n=0}^{\infty}\frac{\Gamma(n+ia+3/2)}{n!\Gamma(n+ia+1)}\left(\frac{p^{2}}{4i\lambda}\right)^{n}F\left(-n,-ia-n;ia+1;\frac{q^{2}}{p^{2}}\right)$
$\displaystyle=\frac{e^{\pi a/2}\left(1+e^{-2\pi a}\right)}{2(\sinh\pi
a)^{2}}\left(\frac{m\epsilon^{2}}{\hbar(t_{2}-t_{1})}\right)^{ia}\frac{(-i\lambda)^{-3/2}2^{-ia}}{\Gamma(ia+1)}\sum_{n=0}^{\infty}\frac{\Gamma(n+ia+3/2)}{n!\Gamma(n+ia+1)}\left(\frac{(t_{2}-t_{1})m\epsilon^{2}}{2\hbar
i(t-t_{1})(t_{2}-t)}\right)^{n}$ $\displaystyle\hskip
25.0ptF\left(-n,-ia-n;ia+1;\left(\frac{t-t_{1}}{t_{2}-t}\right)^{2}\right).$
This expression cannot be simplified further in the general case. However, we
are interested in the $\epsilon\rightarrow 0$ limit when only the $n=0$ term
contributes and the expression reduces to:
$\displaystyle I_{1}$ $\displaystyle=\frac{e^{\pi a/2}\left(1+e^{-2\pi
a}\right)}{2(\sinh\pi
a)^{2}}\frac{(-i\lambda)^{-3/2}2^{-ia}}{[\Gamma(ia+1)]^{2}}\Gamma(ia+3/2)\left(\frac{m\epsilon^{2}}{\hbar(t_{2}-t_{1})}\right)^{ia}$
$\displaystyle=-\frac{e^{\pi a/2}\left(1+e^{-2\pi
a}\right)}{2\pi^{2}}(-i\lambda)^{-3/2}2^{-ia}[\Gamma(-ia)]^{2}\Gamma(ia+3/2)\left(\frac{m\epsilon^{2}}{\hbar(t_{2}-t_{1})}\right)^{ia}$
(72)
The integral $I_{2}$ is same as $I_{1}$ with $a\rightarrow-a$. Therefore,
$\displaystyle I_{2}$ $\displaystyle=\frac{1}{(\sinh\pi
a)^{2}}\int_{-\infty}^{\infty}\mathrm{d}x\;x^{2}e^{i\lambda
x^{2}}J_{-ia}(px)J_{-ia}(qx)$ $\displaystyle=e^{-2\pi a}I_{1}(a\rightarrow-a)$
$\displaystyle=-e^{-2\pi a}\frac{e^{-\pi a/2}\left(1+e^{2\pi
a}\right)}{2\pi^{2}}(-i\lambda)^{-3/2}2^{ia}[\Gamma(ia)]^{2}\Gamma(-ia+3/2)\left(\frac{m\epsilon^{2}}{\hbar(t_{2}-t_{1})}\right)^{-ia}$
$\displaystyle=-\frac{e^{-\pi a/2}\left(1+e^{-2\pi
a}\right)}{2\pi^{2}}(-i\lambda)^{-3/2}2^{ia}[\Gamma(ia)]^{2}\Gamma(-ia+3/2)\left(\frac{m\epsilon^{2}}{\hbar(t_{2}-t_{1})}\right)^{-ia}.$
(73)
Similarly, we can evaluate the third integral as well,
$\displaystyle I_{3}$ $\displaystyle=\frac{(1-\coth\pi a)}{\sinh\pi
a}\int_{-\infty}^{\infty}\mathrm{d}x\;x^{2}e^{i\lambda
x^{2}}[J_{ia}(px)J_{-ia}(qx)+J_{ia}(qx)J_{-ia}(px)]$
$\displaystyle=-\frac{2e^{-\pi a}}{(\sinh\pi
a)^{2}}\int_{0}^{\infty}\mathrm{d}x\;x^{2}e^{i\lambda
x^{2}}[J_{ia}(px)J_{-ia}(qx)+J_{ia}(qx)J_{-ia}(px)]$
$\displaystyle=\frac{-2e^{-\pi a}}{(\sinh\pi
a)^{2}}\left[\frac{p^{ia}q^{-ia}(-i\lambda)^{-\frac{3}{2}}}{2\Gamma(1-ia)}\sum_{n=0}^{\infty}\frac{\Gamma(n+\frac{3}{2})}{n!\Gamma(n+ia+1)}\left(\frac{p^{2}}{4i\lambda}\right)^{n}F\left(-n,-ia-n;-ia+1;\frac{q^{2}}{p^{2}}\right)+(a\rightarrow-a)\right]$
$\displaystyle=-\frac{e^{-\pi a}\sqrt{\pi}}{2\pi a\sinh\pi
a}(-i\lambda)^{-3/2}\left\\{e^{\pi
a}\left(\frac{t_{2}-t}{t-t_{1}}\right)^{ia}+e^{-\pi
a}\left(\frac{t_{2}-t}{t-t_{1}}\right)^{-ia}\right\\}.$ (74)
Combining the results,
$\displaystyle I$ $\displaystyle=I_{1}+I_{2}+I_{3}$
$\displaystyle=\left\\{-\frac{e^{\pi
a/2}}{2\pi^{2}}2^{-ia}\left[\Gamma(-ia)\right]^{2}\left(\frac{m\epsilon^{2}}{\hbar(t_{2}-t_{1})}\right)^{ia}\left(1+e^{-2\pi
a}\right)(-i\lambda)^{-3/2}(ia+1/2)\Gamma(ia+1/2)\right.$
$\displaystyle-\left.\frac{e^{-\pi
a/2}\,2^{ia}\left[\Gamma(ia)\right]^{2}}{\pi^{2}}\left(\frac{m\epsilon^{2}}{\hbar(t_{2}-t_{1})}\right)^{-ia}\left(1+e^{-2\pi
a}\right)(-i\lambda)^{-3/2}(-ia+1/2)\Gamma(-ia+1/2)\right.$
$\displaystyle\left.-\frac{e^{-\pi a}\sqrt{\pi}}{2\pi a\sinh\pi
a}(-i\lambda)^{-3/2}\left[e^{\pi
a}\left(\frac{t_{2}-t}{t-t_{1}}\right)^{ia}+e^{-\pi
a}\left(\frac{t_{2}-t}{t-t_{1}}\right)^{-ia}\right]\right\\}.$ (75)
### A.3 Evaluation of $|X|^{2}$ in the Eq. (46)
We have the effective path,
$X(t)=-i\lambda\,e^{\pi a/2}\frac{I}{D}$ (76)
where $I$ is given by Eq. (A.2) above and
$D=\left[\frac{i2^{-ia}\Gamma(-ia)}{\pi}\left(\frac{m\epsilon^{2}}{\hbar(t_{2}-t_{1})}\right)^{ia}+\frac{i2^{ia}\Gamma(ia)}{\pi}\left(\frac{m\epsilon^{2}}{\hbar(t_{2}-t_{1})}\right)^{-ia}e^{-\pi
a}\right].$ (77)
In general, $|X|^{2}$ arising from the above expression will be quite
complicated. But, working in the limit of $t_{2}\rightarrow\infty$ with
$t_{1}=0$ and $\epsilon\rightarrow 0^{+}$, we will be able to extract a
meaningful result. To see this, first note that $I_{1}$, $I_{2}$ and $D$ can
be written as
$\displaystyle I_{1}$ $\displaystyle=-\frac{e^{\pi a/2}(1+e^{-2\pi a})}{2\pi
a(-i\lambda)^{3/2}\sinh\pi a}\sqrt{\frac{\pi}{\cosh\pi
a}}\sqrt{a^{2}+\frac{1}{4}}\,\exp\left[i(2\theta+\psi+\phi)+ia\ln\frac{m\epsilon^{2}}{2\hbar(t_{2}-t_{1})}\right]$
$\displaystyle I_{2}$ $\displaystyle=-\frac{e^{-\pi a/2}(1+e^{-2\pi a})}{2\pi
a(-i\lambda)^{3/2}\sinh\pi a}\sqrt{\frac{\pi}{\cosh\pi
a}}\sqrt{a^{2}+\frac{1}{4}}\,\exp\left[-i(2\theta+\psi+\phi)-ia\ln\frac{m\epsilon^{2}}{2\hbar(t_{2}-t_{1})}\right]$
$\displaystyle D$ $\displaystyle=\frac{i}{\pi}\sqrt{\frac{\pi}{a\sinh\pi
a}}\left\\{\exp\left[i\theta+ia\ln\frac{m\epsilon^{2}}{2\hbar(t_{2}-t_{1})}\right]+e^{-\pi
a}\exp\left[-i\theta-
ia\ln\frac{m\epsilon^{2}}{2\hbar(t_{2}-t_{1})}\right]\right\\}$ (78)
where,
$\displaystyle\theta=\arg[\Gamma(-ia)],\hskip
5.0pt\phi=\arg[\Gamma(ia+1/2)]\hskip 5.0pt\mathrm{and}\hskip
2.0pt\psi=\arg[ia+1/2]$ (79)
Then, one sees immediately, that
$\displaystyle\frac{I_{1}+I_{2}}{D}=$ $\displaystyle\frac{ie^{\pi
a/2}(1+e^{-2\pi a})}{2a(-i\lambda)^{3/2}\sinh\pi a}\sqrt{\frac{a\sinh\pi
a}{\cosh\pi a}}\sqrt{a^{2}+\frac{1}{4}}\hskip 3.0pte^{i(\theta+\phi+\psi)}$
$\displaystyle\hskip 5.0pt\frac{1+e^{-\pi
a}\exp\left[-i(4\theta+2\phi+2\psi)-2ia\ln\left(m\epsilon^{2}/2\hbar(t_{2}-t_{1})\right)\right]}{1+e^{-\pi
a}\exp\left[-i2\theta-2ia\ln\left(m\epsilon^{2}/2\hbar(t_{2}-t_{1})\right)\right]}$
(80)
Similarly,
$\frac{I_{3}}{D}=\frac{ie^{-\pi a}}{2(-i\lambda)^{-3/2}\sqrt{a\sinh\pi
a}}\frac{e^{\pi a}\exp\left[ia\ln\frac{(t_{2}-t)}{(t-t_{1})}\right]+e^{-\pi
a}\exp\left[-ia\ln\frac{(t_{2}-t)}{(t-t_{1})}\right]}{\exp\left[i\theta+ia\ln\frac{m\epsilon^{2}}{2\hbar(t_{2}-t_{1})}\right]+e^{-\pi
a}\exp\left[-i\theta-ia\ln\frac{m\epsilon^{2}}{2\hbar(t_{2}-t_{1})}\right]}$
(81)
Imposing the late time condition $t_{2}\rightarrow\infty$ with $t_{1}=0$ and
$\epsilon\rightarrow 0$, and ignoring the oscillatory terms which do not
contribute on the average we can simplify this expression. In this limit, we
can neglect the contribution from $I_{3}/D$ term altogether while pre-factor
in the $(I_{1}+I_{2})/D$ term gives
$\displaystyle|X(t)|^{2}$ $\displaystyle=\frac{\lambda^{2}e^{2\pi
a}(1+e^{-2\pi a})^{2}}{4a^{2}\lambda^{3}\sinh^{2}\pi a}\frac{a\sinh\pi
a}{\cosh\pi a}\left(a^{2}+\frac{1}{4}\right)$
$\displaystyle=\left(\frac{4\hbar
t}{ma}\right)\left[N+\frac{1}{2}\right]\left(a^{2}+1/4\right)$ (82)
or
$\frac{\mathrm{d}|X(t)|^{2}}{\mathrm{d}t}=\left(\frac{4\hbar}{ma}\right)\left[N+\frac{1}{2}\right](a^{2}+1/4)$
(83)
where
$N=\frac{1}{e^{2\pi a}-1}$ (84)
It is worth mentioning here that if we include the leading transient terms,
then the above expression gets modified by an extra term:
$\frac{\mathrm{d}|X(t)|^{2}}{\mathrm{d}t}=\left(\frac{4\hbar}{ma}\right)\left[N+\frac{1}{2}+\sqrt{N(N+1)}\cos\xi\right](a^{2}+1/4)$
(85)
where
$\xi=2a\left(\frac{m\epsilon^{2}}{2\hbar(t_{2}-t_{1})}\right)+4\theta+2\phi+2\psi$
(86)
The factor $\sqrt{N(N+1)}$ has the physical meaning of the root-mean-square
fluctuation of the photons in Planck spectrum (see e.g. [15]). Given the large
phase in the cosine term (when $\xi\gg 1$), one may say that the relevant term
varies rapidly between $-\sqrt{N(N+1)}$ and $\sqrt{N(N+1)}$, matching the
magnitude of thermal fluctuations of photons in a bath. What is probably
remarkable is that a similar result was obtained years back [16] in a
completely different context. In [16], the authors showed that the Fourier
transform of a classical plane wave with respect to the Rindler time
coordinates leads to a very similar expression with exactly the three terms.
It is not obvious why the effective path method should lead to such a result
and this similarity is worth investigating. We hope to do this in a future
publication.
## Appendix B Effective path for a class of inverse square potentials
In general, evaluation of the effective path requires the knowledge of the
path integral kernel and tractability of the integral which appears in Eq.
(2). In many cases of interest, algebraic difficulties prevent the analysis of
the effective path in an explicit form. Given the fact that it could be a
useful tool in probing particle production, we present in this appendix some
specific cases in which such a calculation can be performed. We also provide
the calculational details for $X(t)$ including the case considered in [7]
since we could not find these details in the literature.
The simplest context in which the relevant equations are tractable occurs for
a special class of inverse square potentials having the form
$V(x)=l(l+1)\hbar^{2}(2m)^{-1}x^{-2}$ where $l$ is an integer. Note that this
potential has $\alpha>0$ unlike the case in the previous section and we would
like to probe the nature of effective path across the singularity in order to
display the tunneling feature via a complex path. For such a case,
$\gamma=(l+1/2)$, and we can using the property of Hankel functions of half-
integral orders [13],
$H_{n+\frac{1}{2}}^{(2)}(z)=i^{n}H_{\frac{1}{2}}^{(2)}(z)\sum_{k=0}^{n}\frac{(n+k)!}{k!(n-k)!}\frac{1}{(2iz)^{k}},$
(87)
write the generic kernel as
$K_{\gamma}(x_{2},t_{2}|x_{1},t_{1})=K_{\frac{1}{2}}(x_{2},t_{2}|x_{1},t_{1})\sum_{k=0}^{\gamma-1/2}\frac{(\gamma+k-1/2)!}{(\gamma-k-1/2)!k!}\left(\frac{\hbar(t_{2}-t_{1})}{2imx_{1}x_{2}}\right)^{k}$
(88)
where $K_{1/2}$ is the free particle kernel. The result is a finite series for
any particular choice (half-integral) of $\gamma$ and can, in principle, be
used to evaluate the effective path for any given value of $\gamma$.
As an example of the use of this result we will consider the nature of the
effective path near the origin along the lines studied in [7] for a more
general case. For this purpose, we will see that it suffices to look at two
starting simple non-zero values, $l=1$ and $2$. For the first case,
$\alpha=\hbar^{2}m^{-1}$ and $\gamma=3/2$, so that we have result which is
obtained earlier in [7], viz.
$K_{3/2}(x_{2},t_{2}|x_{1},t_{1})=\left\\{1-\frac{i\hbar(t_{2}-t_{1})}{mx_{1}x_{2}}\right\\}K_{1/2}(x_{2},t_{2}|x_{1},t_{1}).$
(89)
The effective trajectory for this case is
$\displaystyle X_{3/2}(t)$
$\displaystyle=\frac{1}{K_{3/2}(2|1)}\int\mathrm{d}xK_{3/2}(2|x,t)\,x\,K_{3/2}(x,t|1)$
$\displaystyle=\frac{1}{K_{3/2}(2|1)}\int\mathrm{d}xK_{1/2}(2|x)\,x\,K_{1/2}(x|1)\left\\{1-\frac{i\hbar(t_{2}-t_{1})}{mx_{1}x_{2}x}\bar{x}-\frac{\hbar^{2}}{m^{2}}\frac{(t_{2}-t)(t-t_{1})}{x_{1}x_{2}x^{2}}\right\\}$
$\displaystyle=\bar{x}+\frac{i\pi\hbar^{2}\sqrt{m(t_{2}-t)(t-t_{1})(t_{2}-t_{1})}}{(2\pi
i\hbar)^{1/2}m(mx_{1}x_{2}-i\hbar(t_{2}-t_{1}))}\exp(i\lambda\bar{x}^{2})\Phi(\bar{x}(i\lambda)^{1/2})$
(90)
where
$\bar{x}=\frac{x_{2}(t-t_{1})+x_{1}(t_{2}-t)}{(t_{2}-t_{1})}\\\ $
and $\Phi(x)$ is the probability integral. To study the small $\hbar$
behavior, that is, $\epsilon=\hbar(t_{2}-t_{1})/(mx_{1}x_{2})\ll 1$, we use
the properties of $\Phi(x)$ [13], and get
$X_{3/2}(t)=\bar{x}(t)-\hbar^{2}\frac{(t_{2}-t)(t-t_{1})}{m^{2}x_{1}x_{2}}[\bar{x}^{-1}-\sqrt{(1/2)i\pi\lambda}\exp(i\lambda\bar{x}^{2})]+\cdot\cdot\cdot.$
(91)
To the same order in $\epsilon$ the classical trajectory is given by
$x_{cl}(t)=\bar{x}-\frac{\hbar^{2}(t_{2}-t)(t-t_{1})}{m^{2}x_{1}x_{2}\bar{x}}+\mathrm{O}(\epsilon^{3}).$
(92)
In the limit $\hbar\,\rightarrow\,0$, effective trajectory becomes,
$X_{3/2}(t)=\bar{x}-\frac{\hbar^{2}(t_{2}-t)(t-t_{1})}{m^{2}x_{1}x_{2}}[\bar{x}^{-1}-i\pi\delta(\bar{x})].$
(93)
Using
$\lim_{\eta\,\rightarrow\,0}\,(\bar{x}+i\eta)^{-1}=\bar{x}^{-1}-i\pi\delta(\bar{x})$
(94)
we can rewrite this as
$X_{3/2}(t)=\bar{x}-\frac{\hbar^{2}(t_{2}-t)(t-t_{1})}{m^{2}x_{1}x_{2}(\bar{x}+i\eta)}$
(95)
We shall now evaluate the effective path for $l=2$ (for which $\gamma=5/2$) to
the same order. In this case the kernel is a series with three terms,
$K_{5/2}(x_{2},t_{2}|x_{1},t_{1})=\left\\{1-\frac{i\hbar(t_{2}-t_{1})}{mx_{1}x_{2}}-\frac{3\hbar^{2}(t_{2}-t_{1})^{2}}{m^{2}x_{1}^{2}x_{2}^{2}}\right\\}K_{1/2}(x_{2},t_{2}|x_{1},t_{1}).$
(96)
The calculation for effective path proceeds in the same way although the
algebra becomes tedious. Working out the effective path to the same order in
$\epsilon$ again shows similar pattern as Eq. (95):
$X_{5/2}(t)=\bar{x}+\frac{9\hbar^{2}(t_{2}-t_{1})^{2}}{m^{2}x_{1}^{2}x_{2}^{2}}\bar{x}-\frac{5\hbar^{2}}{m^{2}}\frac{(t_{2}-t)(t-t_{1})}{x_{1}x_{2}(\bar{x}+i\eta)}$
(97)
Note that classically the particle cannot cross the origin and in fact the
classical trajectory in Eq. (92) has a singularity at $\bar{x}=0$. However,
the complex effective trajectories in Eqs. (95, 97) are non-singular at
$\bar{x}=0$ since it can move over to the imaginary axis. The trend persists
for higher values of $\gamma$ as well, displaying the excursion into the
complex plane near the origin. This can be shown quickly in symbolic terms.
For any $\gamma>5/2$, we will have,
$K_{\gamma}(2|1)=K_{1/2}(2|1)\left\\{1+\frac{C_{1}}{x_{1}x_{2}}+\frac{C_{2}}{(x_{1}x_{2})^{2}}+\cdot\cdot\cdot+\frac{C_{\gamma-1/2}}{(x_{1}x_{2})^{\gamma-1/2}}\right\\}$
(98)
Now, the effective path is
$\displaystyle
X_{\gamma}(t)=\frac{1}{K_{\gamma}(2|1)}\left[\int\mathrm{d}x\,xK_{1/2}(2|x,t)K_{1/2}(x,t|1)\left(1+\frac{A_{1}}{xx_{2}}+\frac{A_{2}}{(xx_{2})^{2}}+\cdot\cdot\cdot+\frac{A_{\gamma-1/2}}{(xx_{2})^{\gamma-1/2}}\right)\right.$
$\displaystyle\left.\left(1+\frac{B_{1}}{x_{1}x}+\frac{B_{2}}{(x_{1}x)^{2}}+\cdot\cdot\cdot+\frac{B_{\gamma-1/2}}{(x_{1}x)^{\gamma-1/2}}\right)\right]$
For the first few terms we have,
$\displaystyle
X_{\gamma}(t)=\frac{1}{K_{1/2}(2|1)}\left\\{1-\frac{C_{1}}{x_{1}x_{2}}-\frac{C_{2}}{(x_{1}x_{2})^{2}}-\cdot\cdot\cdot\right\\}\left[\bar{x}K_{1/2}(2|1)+(A_{1}/x_{2}+B_{1}/x_{1})K_{1/2}(2|1)+\right.$
$\displaystyle\left.f(A_{1},A_{2},B_{1},B_{2},x_{1},x_{2})\int\frac{\mathrm{d}x}{x}\,K_{1/2}(2|x,t)K_{1/2}(x,t|1)+\cdot\cdot\cdot\right]$
(100)
Then, we can easily see that in our limit of $\epsilon\ll 1$,
$X_{\gamma}(t)=\mathrm{Re}X_{\gamma}+i\,\mathrm{Im}X_{\gamma}$ (101)
where the imaginary part essentially comes from the integral in the Eq. (B)
which is the probability integral. Thus the result obtained for $\gamma=3/2$
in [7] turns out to be true for a much wider class of potentials.
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|
arxiv-papers
| 2011-12-29T11:32:09 |
2024-09-04T02:49:25.790643
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Suprit Singh and T. Padmanabhan",
"submitter": "Suprit Singh",
"url": "https://arxiv.org/abs/1112.6279"
}
|
1201.0081
|
# Resource Allocation with Subcarrier Pairing in OFDMA Two-Way Relay Networks
Hao Zhang, Yuan Liu, , and Meixia Tao Manuscript received November 16, 2011.
The associate editor approving it for publication was Dr. Harish
Viswanathan.The authors are with the Department of Electronic Engineering at
Shanghai Jiao Tong University, Shanghai, 200240, P. R. China (e-mail:
{gavinzhanghao, yuanliu, mxtao}@sjtu.edu.cn).This work was supported by the
NSF of China (60902019) and the Innovation Program of Shanghai Municipal
Education Commission (11ZZ19).
###### Abstract
This study considers an orthogonal frequency-division multiple-access
(OFDMA)-based multi-user two-way relay network where multiple mobile stations
(MSs) communicate with a common base station (BS) via multiple relay stations
(RSs). We study the joint optimization problem of subcarrier-pairing based
relay-power allocation, relay selection, and subcarrier assignment. The
problem is formulated as a mixed integer programming problem. By using the
dual method, we propose an efficient algorithm to solve the problem in an
_asymptotically_ optimal manner. Simulation results show that the proposed
method can improve system performance significantly over the conventional
methods.
###### Index Terms:
Two-way relaying, subcarrier pairing, resource allocation, orthogonal
frequency-division multiple-access.
## I Introduction
An important property of orthogonal frequency-division multiplexing
(OFDM)-based relaying is that the frequency diversity can be exploited by
_subcarrier pairing_ , which matches the incoming and outgoing subcarriers at
the relay based on channel dynamics and hence improves system performance. In
multi-user environments with orthogonal frequency-division multiple-access
(OFDMA), subcarriers should not only be carefully paired at the relay but also
be assigned adaptively for different users. If with multiple relays, it
further complicates the problem because relay selection tightly couples with
subcarrier pairing and assignment. Thus, subcarrier-pairing based resource
allocation in multi-user multi-relay OFDMA networks is highly challenging.
Subcarrier-pairing based resource allocation has been originally investigated
for single-user single-relay one-way communications (e.g., [1, 2]). In
particular, it is proved in [1] that the _ordered pairing_ is optimal for
amplify-and-forward (AF) protocol. Authors in [3] investigated _separated_
power allocation and subcarrier pairing in two-way communication using single
relay, where the power allocation is first employed by water-filling and then
subcarriers are paired at the relay by a heuristic method. In [4], the
subcarrier-pairing based joint optimization of power allocation, relay
selection and subcarrier assignment for single-user multi-relay systems was
studied. The subcarrier-pairing based joint optimization of power allocation
and subcarrier-user assignment for multi-user single-relay scenario was
studied in [5]. In [6], the authors studied relay-assisted bidirectional OFDMA
cellular networks, wherein the subcarrier-pairing based joint optimization of
bidirectional transmission mode selection, relay selection, and subcarrier
assignment was investigated by a graph approach. Authors in [7] investigated
the jointly optimal channel and relay assignment for multi-user multi-relay
two-way relay networks. These works [6, 7], however, did not consider power
allocation.
In this work, we consider an OFDMA two-way relay network with a common base
station (BS), multiple mobile stations (MSs) and multiple relay stations
(RSs). The downlink and uplink traffic for each MS is multiplexed through
analog network coding at the RSs. We formulate a joint optimization problem of
subcarrier-pairing based relay-power allocation, relay selection, and
subcarrier assignment. The problem is a mixed integer programming problem and
we solve it efficiently in dual domain with polynomial complexity.
Figure 1: System model.
## II System Model and Problem formulation
We consider a single-cell OFDMA two-way relay network, as shown in Fig. 1,
with one BS, multiple MSs and RSs. All the MSs are assumed to be cell-edge
users so that both the downlink and uplink traffic of each user needs to be
relayed through one or more RSs. This assumption is commonly used for cellular
relay networks in the literature (e.g., [5, 8, 9]). Each RS operates in a
half-duplex mode and relays the bi-directional traffic using AF protocol,
known as analog network coding. In specific, the AF two-way relay protocol
takes place in two phases [10]. In the first phase, also known as multiple-
access (MAC) phase, all the MSs and the BS concurrently transmit signals while
al the RSs listen. In the second phase, known as broadcast (BC) phase, the RSs
amplify the received signals and then forward them to the MSs and the BS. To
avoid multi-user interference, each MS and RS operate in non-overlapping
subcarriers in the first and second phases, respectively. The downlink-uplink
interference within each user is eliminated by self-interference cancelation.
Furthermore, the channel is assumed to be slowly time-varying and all the
channel state information can be perfectly estimated and known at the BS for
centralized processing.
Let $\mathcal{N}=\\{1,2,\cdot\cdot\cdot,N\\}$ denote the set of subcarriers,
$\mathcal{K}=\\{1,2,\cdot\cdot\cdot,K\\}$ denote the set of RSs, and
$\mathcal{U}=\\{1,2,\cdot\cdot\cdot,M\\}$ denote the set of MSs. The channel
coefficients from BS $b$ and MS $u$ to RS $k$ on subcarrier $i$ in the MAC
phase are denoted as $h_{b,k,i}$ and $f_{u,k,i}$, respectively, $\forall
u\in\mathcal{U},k\in\mathcal{K},i\in\mathcal{N}$. In the BC phase, the channel
coefficients from RS $k$ to BS $b$ and MS $u$ on subcarrier $j$ are denoted as
$h_{k,b,j}$ and $f_{k,u,j}$, respectively, $\forall
u\in\mathcal{U},k\in\mathcal{K},j\in\mathcal{N}$. Here channel reciprocity is
used, which is valid in TDD (time-division duplex) mode. Along with the paths,
we further denote $p_{b,k,i}$ and $p_{u,k,i}$ as the transmitted power of BS
$b$ and MS $u$ respectively, and $p_{k,u,j}$ as the transmitted power of RS
$k$. Then, the sum-rate of uplink-downlink transmission of MS $u$ over
subcarrier pair $(i,j)$ with the assistance of RS $k$ can be expressed as [10,
3]
$\begin{split}R_{u,k,i,j}=&\frac{1}{2}\log_{2}\left(1+\frac{p_{u,k,i}|f_{u,k,i}|^{2}p_{k,u,j}|h_{k,b,j}|^{2}}{p_{k,u,j}|h_{k,b,j}|^{2}+m_{u,k,i}}\right)\\\
+&\frac{1}{2}\log_{2}\left(1+\frac{p_{b,k,i}|h_{b,k,i}|^{2}p_{k,u,j}|f_{k,u,j}|^{2}}{p_{k,u,j}|f_{k,u,j}|^{2}+m_{u,k,i}}\right),\end{split}$
(1)
in which $m_{u,k,i}=1+p_{b,k,i}|h_{b,k,i}|^{2}+p_{u,k,i}|f_{u,k,i}|^{2}$. It
can be proved that the sum-rate $R_{u,k,i,j}$ is concave in the relay power
$p_{k,u,j}$.
We then introduce a set of binary variables $\rho_{u,k,i,j}\in\\{0,1\\}$ for
all $u$, $k$, $i$, $j$. When $\rho_{u,k,i,j}=1$, it means that subcarrier $i$
in the MAC phase is paired with subcarrier $j$ in the BC phase and they are
used by RS $k$ to relay the signals of MS $u$. Otherwise, we have
$\rho_{u,k,i,j}=0$. These binary variables must satisfy the following
constraints, due to the exclusive subcarrier assignment,
$\displaystyle\sum_{u=1}^{M}\sum_{k=1}^{K}\sum_{j=1}^{N}\rho_{u,k,i,j}$
$\displaystyle\leq$ $\displaystyle 1,~{}\forall i,$ (2)
$\displaystyle\sum_{u=1}^{M}\sum_{k=1}^{K}\sum_{i=1}^{N}\rho_{u,k,i,j}$
$\displaystyle\leq$ $\displaystyle 1,~{}\forall j.$ (3)
For simplicity, we study relay-power allocation and let the transmit power of
the BS and MSs be fixed. Each RS is subject to its own peak power constraint.
This can be expressed as:
$\displaystyle\sum_{u=1}^{M}\sum_{j=1}^{N}p_{k,u,j}\leq P_{k},~{}\forall k,$
(4)
where $P_{k}$ is the peak power constraint of RS $k$.
Our objective is to maximize the system total weighted throughput by jointly
optimizing the assignment variables $\\{\rho_{u,k,i,j}\\}$ and the relay power
variables $\\{p_{k,u,j}\\}$. Mathematically, this can be formulated as:
$\displaystyle\max_{\\{\boldsymbol{p},\boldsymbol{\rho}\\}}\sum_{u=1}^{M}w_{u}\sum_{k=1}^{K}\sum_{i=1}^{N}\sum_{j=1}^{N}\rho_{u,k,i,j}R_{u,k,i,j}(p_{k,u,j})$
(5)
$\displaystyle\textit{s.t.}~{}~{}(\ref{eqn:t1}),(\ref{eqn:t2}),(\ref{eqn:p}),$
where $w_{u}$ is the weight that represents the priority of MS $u$,
$\boldsymbol{p}\in\mathbb{R}_{+}^{K\times M\times N}$ and
$\boldsymbol{\rho}\in\\{0,1\\}^{M\times K\times N\times N}$ are matrices with
entries $p_{k,u,j}$ and $\rho_{u,k,i,j}$, respectively.
## III Dual Based Algorithm
We first define $\mathcal{T}$ as the set of all possible $\boldsymbol{\rho}$
satisfying (2) and (3), $\mathcal{P}$ as the set of all possible power
allocations $\boldsymbol{p}$ for the given $\boldsymbol{\rho}$ that satisfy
$p_{k,u,j}\geq 0$ for $\rho_{u,k,i,j}=1$ and $p_{k,u,j}=0$ for
$\rho_{u,k,i,j}=0$. Denote
$\boldsymbol{\lambda}=(\lambda_{1},\lambda_{2},...,\lambda_{K})\succeq 0$ as
the dual variables associated with the peak power constraints of the RSs. Then
the dual function of the problem in (5) can be defined as
$\displaystyle
g(\boldsymbol{\lambda})\triangleq\max_{\begin{subarray}{\boldsymbol{missing}}{p}\in\mathcal{P}(\boldsymbol{\rho})\\\
\boldsymbol{\rho}\in\mathcal{T}\end{subarray}}L(\boldsymbol{p},\boldsymbol{\rho},\boldsymbol{\lambda}),$
(6)
where the Lagrangian is
$\displaystyle L(\boldsymbol{p},\boldsymbol{\rho},\boldsymbol{\lambda})$
$\displaystyle=$
$\displaystyle\sum_{u=1}^{M}w_{u}\sum_{k=1}^{K}\sum_{i=1}^{N}\sum_{j=1}^{N}R_{u,k,i,j}(p_{k,u,j})$
(7)
$\displaystyle+\sum_{k=1}^{K}\lambda_{k}\left(P_{k}-\sum_{u=1}^{M}\sum_{j=1}^{N}p_{k,u,j}\right).$
Computing the dual function $g(\boldsymbol{\lambda})$ requires us to determine
the optimal $({\boldsymbol{p},\boldsymbol{\rho}})$ at the given dual vector
$\boldsymbol{\lambda}$. In the following we present the derivations in detail.
### III-A Optimizing the Primal Variables
$(\boldsymbol{p},\boldsymbol{\rho})$ for Given $\boldsymbol{\lambda}$
We first find the optimal power variables $\boldsymbol{p}$ by fixing the
binary assignment variables $\boldsymbol{\rho}$. Then we search the optimal
$\boldsymbol{\rho}$ by eliminating $\boldsymbol{p}$ in the objective function.
Such a method has been commonly used in the literature (e.g., [2, 4, 5, 11]).
Let us rewrite $L(\boldsymbol{p},\boldsymbol{\rho},\boldsymbol{\lambda})$ as
$L(\boldsymbol{p},\boldsymbol{\rho},\boldsymbol{\lambda})=\sum_{k=1}^{K}\sum_{u=1}^{U}\sum_{j=1}^{N}L_{k,u,j}(p_{k,u,j})+\sum_{k=1}^{K}\lambda_{k}P_{k},$
(8)
where
$\begin{split}L_{u,k,j}(p_{u,k,j})=w_{u}\sum_{i=1}^{N}R_{u,k,i,j}(p_{k,u,j})-\lambda_{k}p_{k,u,j}.\end{split}$
(9)
Suppose $\rho_{u,k,i,j}=1$ for a certain $(u,k,i,j)$. It is easy to verify
that $L_{u,k,j}(p_{u,k,j})$ is concave in $p_{k,u,j}$ and thus the optimal
$p_{k,u,j}^{*}(\lambda_{k})$ can be obtained by applying the Karush-Kuhn-
Tucker (KKT) conditions. More specifically, $p_{k,u,j}^{*}(\lambda_{k})$ is
the non-negative real root of the following quartic function
$\displaystyle ap_{k,u,j}^{4}+bp_{k,u,j}^{3}+cp_{k,u,j}^{2}+dp_{k,u,j}+e=0,$
(10)
where $a,b,c,d,e$ are coefficients determined by the dual variables, MSs’
weights, and channel gains as defined at the top of the next page.
$\displaystyle a$ $\displaystyle=2\ln
2\lambda_{k}|h_{b,k,j}|^{4}|f_{u,k,j}|^{4}/m_{u,k,i},$ $\displaystyle b$
$\displaystyle=4\ln
2\lambda_{k}|h_{b,k,j}|^{2}|f_{u,k,j}|^{2}(|f_{u,k,j}|^{2}+|h_{b,k,j}|^{2}),$
$\displaystyle c$ $\displaystyle=2m_{u,k,i}\ln
2\lambda_{k}(|h_{b,k,j}|^{4}+|f_{u,k,j}|^{4}+4|h_{b,k,j}|^{2}|f_{u,k,j}|^{2})$
$\displaystyle-
w_{u}|h_{b,k,j}|^{2}|f_{u,k,j}|^{2}(p_{u,k,i}|f_{u,k,i}|^{2}|f_{u,k,j}|^{2}+p_{b,k,i}|h_{b,k,i}|^{2}|h_{b,k,j}|^{2}),$
$\displaystyle d$ $\displaystyle=4m_{u,k,i}^{2}\ln
2\lambda_{k}(|f_{u,k,j}|^{2}+|h_{b,k,j}|^{2})-2w_{u}m_{u,k,i}|h_{b,k,j}|^{2}|f_{u,k,j}|^{2}((p_{u,k,i}|f_{u,k,i}|^{2}+p_{b,k,i}|h_{b,k,i}|^{2}),$
$\displaystyle e$ $\displaystyle=2m_{u,k,i}^{3}\ln
2\lambda_{k}-w_{u}m_{u,k,i}^{2}(p_{u,k,i}|f_{u,k,i}|^{2}|h_{b,k,j}|^{2}+p_{b,k,i}|h_{b,k,i}|^{2}|f_{u,k,j}|^{2}).$
Substituting the optimal power allocations
$\boldsymbol{p}^{*}(\boldsymbol{\lambda})$ into (6) to eliminate the power
variables, the dual function can be rewritten as
$\displaystyle
g(\boldsymbol{\lambda})=\max_{\boldsymbol{\rho}\in\mathcal{T}}\sum_{u=1}^{M}\sum_{k=1}^{K}\sum_{i=1}^{N}\sum_{j=1}^{N}\rho_{u,k,i,j}X_{u,k,i,j}+\sum_{k=1}^{K}\lambda_{k}P_{k},$
where
$X_{u,k,i,j}=w_{u}R_{u,k,i,j}(p_{k,u,j}^{*}(\lambda_{k}))-\lambda_{k}p_{k,u,j}^{*}(\lambda_{k}).$
(11)
Now we are ready to find the optimal $\boldsymbol{\rho}$. In the following, we
show that $X_{u,k,i,j}$ defined in (11) plays an important role in user and
relay selection for occupying a subcarrier pair $(i,j)$.
Noting the constraints (2) and (3), we conclude that there is at most one non-
zero element for a given subcarrier pair $(i,j)$. This suggests that at most
one MS and one RS can occupy the subcarrier pair $(i,j)$. Based on the
observation, we define
$\mathcal{X}_{i,j}=\max_{k\in\mathcal{K},u\in\mathcal{U}}X_{u,k,i,j},$ (12)
$(u^{*},k^{*})_{i,j}=\arg\max_{k\in\mathcal{K},u\in\mathcal{U}}X_{u,k,i,j}.$
(13)
Then the dual function can be further written as
$\displaystyle
g(\boldsymbol{\lambda})=\max_{\boldsymbol{\rho}\in\mathcal{T}}\sum_{i=1}^{N}\sum_{j=1}^{N}\rho_{u*^{,}k^{*},i,j}\mathcal{X}_{i,j}+\sum_{k=1}^{K}\lambda_{k}P_{k}.$
(14)
From (14) it can be seen that if subcarrier $i$ in the MAC phase is paired
with subcarrier $j$ in the BC phase, then the pair should be used by MS
$u^{*}$ with the help of RS $k^{*}$, i.e., the MS and RS with the maximum
$X_{u,k,i,j}$ as defined in (11). This can be interpreted from an economic
perspective. Suppose each dual variable $\lambda_{k}$ represents the power
price of RS $k$. Then $X_{u,k,i,j}$ can be regarded as the profit of letting
MS $u$ transmitting over the subcarrier pair $(i,j)$ with the help of RS $k$,
where the profit is defined as the throughput revenue $w_{u}R_{u,k,i,j}$ minus
the power cost $\lambda_{k}p^{*}_{k,u,j}$. Clearly, to maximize the system
total profit, each potential subcarrier pair $(i,j)$ should be assigned to the
MS and RS that can generate the maximum sub-profit.
The remaining problem is then to identify the optimal subcarrier pairings
$\rho_{u^{*},k^{*},i,j}$. This is a standard _two-dimensional assignment
problem_. The classical Hungarian method can be applied to find the optimal
$\boldsymbol{\rho}^{*}(\boldsymbol{\lambda})$ in polynomial time.
### III-B Optimizing the Dual Vector $\boldsymbol{\lambda}$
After computing $g(\boldsymbol{\lambda})$, we now solve the standard dual
optimization problem which is
$\displaystyle\min_{\boldsymbol{\lambda}}~{}g(\boldsymbol{\lambda})$ (15)
$\displaystyle s.t.~{}~{}\boldsymbol{\lambda}\succeq 0.$
Since a dual function is always convex, subgradient-based methods can be used
to minimize $g(\boldsymbol{\lambda})$ with global convergence with the fact
that
$\Delta\lambda_{k}=P_{k}-\sum_{u=1}^{M}\sum_{j=1}^{N}p_{k,u,j}^{*}(\lambda_{k})$
(16)
is the subgradient at $\lambda_{k},\forall k$. In specific, denote
$\Delta\boldsymbol{\lambda}^{(l)}=(\Delta\lambda_{1}^{(l)},\Delta\lambda_{2}^{(l)},...,\Delta\lambda_{K}^{(l)})$,
then we can update the dual variables as
$\boldsymbol{\lambda}^{(l+1)}=\boldsymbol{\lambda}^{(l)}+\omega^{(l)}\Delta\boldsymbol{\lambda}^{(l)}$.
Here, $\omega^{(l)}$ is the diminishing step size at the $l$th iteration to
guarantee the convergence of the subgradient method.
### III-C Refinement of Power Allocation
Having the dual point $\boldsymbol{\lambda}^{*}$, we now need to determine the
optimal solution to the primal problem (5). Due to the non-zero duality gap,
the optimal $\boldsymbol{\rho}^{*}(\boldsymbol{\lambda}^{*})$ and
$\boldsymbol{p}^{*}(\boldsymbol{\lambda}^{*})$ may not satisfy all the
constraints (2), (3), and (4) in the original problem. To overcome this
problem, we first determine the optimal assignment $\boldsymbol{\rho}^{*}$ in
dual domain, and then make a refinement of the power allocation to meet the
power constraints in the primal problem. More specifically, denote
$\mathcal{A}_{u,k}$ as the set of active subcarrier pairs assigned to MS $u$
and RS $k$ obtained from the dual problem. The problem can be written as:
$\displaystyle\max_{\boldsymbol{p}}\sum_{u=1}^{M}w_{u}\sum_{k=1}^{K}\sum_{(i,j)\in\mathcal{A}_{u,k}}R_{u,k,i,j}(p_{k,u,j})$
(17) $\displaystyle
s.t.~{}~{}\sum_{u=1}^{M}\sum_{(i,j)\in\mathcal{A}_{u,k}}p_{k,u,j}\leq
P_{k},~{}\forall k.$ (18)
Clearly, this is a convex problem. By applying KKT conditions, we can obtain
the optimal $p_{k,u,j}^{*}$ which has the same expression as that in the dual
domain.
Finally we summarize the overall procedure of the proposed dual-based solution
in Algorithm 1. This algorithm is asymptotically optimal when $N$ is
sufficiently large [12].
Algorithm 1 Proposed algorithm for problem (5)
1: initialize $\boldsymbol{\lambda}^{(0)}$ as a random non-negative vector,
$l=0$.
2: repeat
3: Compute $X_{u,k,i,j}$ using (11) for all $(u,k,i,j)$ with $p^{*}_{k,u,j}$
being the non-negative real root of (10).
4: Obtain $\mathcal{X}_{i,j}$ and $(u^{*},k^{*})$ using (12) and (13)
respectively for all $(i,j)$, then obtain optimal
$\boldsymbol{\rho}^{*}(\boldsymbol{\lambda}^{(l)})$ by solving (14).
5: Update $\boldsymbol{\lambda}^{(l)}$ using the subgradients
$\Delta\boldsymbol{\lambda}^{(l)}$ in (16); Let $l\leftarrow l+1$.
6: until $\boldsymbol{\lambda}$ converges.
7: Set the final $\boldsymbol{\rho}$ as $\boldsymbol{\rho}^{*}$ obtained in
the dual domain and refine the power parameter $\boldsymbol{p}^{*}$ by solving
(17) at the given $\boldsymbol{\rho}^{*}$.
### III-D Discussion on Complexity and Proportional Fairness
The complexity of updating the dual variables $\boldsymbol{\lambda}$ is
$\mathcal{O}(K^{q})$ (e.g., if the ellipsoid method is used, $q=2$). The
complexity in (12) and the Hungarian method are $\mathcal{O}(MK)$ and
$\mathcal{O}(N^{3})$, respectively. Combining all, the total complexity of the
proposed method is $\mathcal{O}((MK+N^{3})K^{q})$, which is polynomial.
If consider long-term fairness among the MSs, the weight of MS $u$ at time $t$
can be updated by $w_{u}^{(t)}=1/T_{u}^{(t)}$, $\forall u\in\mathcal{U}$,
where $T_{u}^{(t)}$ as the accumulated rate of MS $u$ at time $t$. Note that
we can let $w_{u}=1$ for every MS for pure throughput maximization.
## IV Simulation Results
We consider a cell with 2 km radius. The RSs are uniformly located on a circle
centered at the BS and with radius of 1 km. The MSs are randomly but uniformly
distributed in the outer circle as in Fig. 1. The path loss exponent is $4$
and the standard deviation of log-normal shadowing is $5.8$ dB. The small-
scale fading is modeled by multi-path Rayleigh fading process. A total of
$3000$ independent channel realizations were used. Each channel realization is
associated with a different set of node locations. We set $M=4$, $K=3$, and
$N=32$. All MS and the BS have the same maximum power constraints, so do all
RSs. We set the BS and MS power to be $10$ dB per-node and uniformly
distributed among all subcarriers.
As the benchmarks, the performance of Equal Power Assignment (EPA) based
resource allocation and Random Resource Allocation (RRA) schemes are also
presented. Specifically, EPA lets $\boldsymbol{p}$ be uniformly distributed
among all the subcarriers on each relay station and finds optimal
$\boldsymbol{\rho}^{*}$ as in Section III-A proposed algorithm. In RRA, the
power is uniformly distributed and the subcarrier pairs and relays are
randomly assigned. The complexity of the EPA and RRA schemes are
$\mathcal{O}(MK+N^{3})$ and $\mathcal{O}(N)$, respectively, which are lower
than that of the proposed algorithm.
Figure 2: Average sum-rate versus RS-power per-node.
Fig. 2 compares the average sum-rate achieved by different schemes. We first
observe that the proposed dual-based algorithm approaches the upper bound (the
optimal dual) closely. This verifies the effectiveness of the dual method at
large number of subcarriers. One also observes that the proposed algorithm
outperforms the two benchmarks by a significant margin. In particular, the
proposed algorithm obtains more than $30\%$ and $200\%$ throughput
improvements over the EPA and RRA schemes, respectively. This tremendous
improvement demonstrates the superiority of our proposed algorithm.
## V Conclusion
In this work, we have studied the subcarrier-pairing based resource allocation
in OFDMA-based two-way relay networks. By using the dual method, an efficient
algorithm for joint optimization of subcarrier-pairing based relay-power
allocation, relay selection, and subcarrier assignment was proposed.
Simulation results show that the proposed algorithm can significantly improve
the system performance compared with the conventional schemes.
## References
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* [5] M. Hajiaghayi, M. Dong, and B. Liang, “Optimal channel assignment and power allocation for dual-hop multi-channel multi-user relaying,” in _Proc. IEEE INFOCOM_ , Apr. 2011.
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|
arxiv-papers
| 2011-12-30T08:49:17 |
2024-09-04T02:49:25.810869
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hao Zhang, Yuan Liu, and Meixia Tao",
"submitter": "Yuan Liu Yuan Liu",
"url": "https://arxiv.org/abs/1201.0081"
}
|
1201.0162
|
# Uncovering the Birth of a Coronal Mass Ejection from Two-Viewpoint SECCHI
Observations
A. Vourlidas1P. Syntelis2,4K. Tsinganos3,4 1 Space Sciences Division, Naval
Research Laboratory, Washington DC, USA
2 Research Center for Astronomy and Applied Mathematics, Academy of Athens,
Athens, Greece
3 National Observatory of Athens, Athens, Greece
4 Section of Astrophysics, Astronomy and Mechanics, Department of Physics,
University of Athens, Athens, Greece
###### Abstract
We investigate the initiation and formation of Coronal Mass Ejections (CMEs)
via detailed two-viewpoint analysis of low corona observations of a relatively
fast CME acquired by the SECCHI instruments aboard the STEREO mission. The
event which occurred on January 2, 2008, was chosen because of several unique
characteristics. It shows upward motions for at least four hours before the
flare peak. Its speed and acceleration profiles exhibit a number of
inflections which seem to have a direct counterpart in the GOES light curves.
We detect and measure, in 3D, loops that collapse toward the erupting channel
while the CME is increasing in size and accelerates. We suggest that these
collapsing loops are our first evidence of magnetic evacuation behind the
forming CME flux rope. We report the detection of a hot structure which
becomes the core of the white light CME. We observe and measure unidirectional
flows along the erupting filament channel which may be associated with the
eruption process. Finally, we compare these observations to the predictions
from the standard flare-CME model and find a very satisfactory agreement. We
conclude that the standard flare-CME concept is a reliable representation of
the initial stages of CMEs and that multi-viewpoint, high cadence EUV
observations can be extremely useful in understanding the formation of CMEs.
###### keywords:
Coronal Mass Ejections, Low Coronal Signatures; Coronal Mass Ejections,
Initiation and Propagation; Magnetic Reconnection, Observational Signatures
## 1 Introduction
sec:Introduction
CMEs have been observed for more than 40 years now. They are one of the most
energetic phenomena in our solar system and the main driver of disturbances in
the terrestrial space environment. Despite observations of tens of thousands
of CMEs, the physical processes behind their formation and propagation have
not yet been understood completely [Klimchuk (2001), Forbes et al. (2006),
Chen (2011)].
To make progress, we need to select the model (or models) that best describe
the phenomenon. To accomplish this, it is necessary to test the theoretical
predictions of the various models against the observations as was discussed by
2001AGUGM.125..143K. Here, we concentrate on the ’standard’ flare-CME model,
also known as the CSHKP model [Švestka and Cliver (1992)]. This is not
actually a fully-fledged model derived from the solution of a set of
Magnetohydrodynamic (MHD) equations but it is rather a two-dimensional (2D)
cartoon representation of the erupting process. However it captures the key
ingredients of many MHD models (i.e., the three-part CME, the ejection of a
flux rope, post-CME flaring loops, etc) and demonstrates, in a straightforward
way, the possible connection between the erupting and flaring processes. For
our discussion, we use the detailed model representation in 2004ApJ…602..422L
(their Figure 1) but many more variations can be found in the literature.
Even as a cartoon, the CSHKP model makes several predictions that can be
tested against the observations. First, it predicts the eruption of a core
surrounded by a cavity (or bubble) that forms during the initiation process.
High temperatures are expected in both the cavity and the core as result of
magnetic reconnection [Chen (2011)]. Second, the reconnection behind the
erupting system creates a magnetic void which draws adjacent lines toward the
current sheet thereby creating an inflow of material from the surrounding flux
systems. Third, through the reconnection processes in the post-CME current
sheet, magnetic energy is transformed into thermal energy that powers the
flare and kinetic energy that powers the CME. Therefore, we expect a close
correspondence between the SXR light curve and the CME acceleration profile as
has been found in the past (e.g., 2004ApJ…604..420Z). A delay between the two
processes is also likely depending on the magnetic fields and reconnection
rates involved [Reeves (2006)]. Fourth, there are many candidates for the role
of the eruption trigger. Flux emergence, tether-cutting or even mass unloading
from the prominence channel, are all capable of driving the system out of its
equilibrium state to set off the eruption (see discussion and references in
2011LRSP….8….1C). Can the trigger be identified in the observations?
Many, if not all, of these predictions relate to the very first stages of the
CME; namely, its initiation and formation. However, the initiation and
formation stages of CMEs present some serious observational challenges. The
CME formation and initial evolution take place low in the corona which is
accessible only to imagers in the Extreme Ultraviolet (EUV) or (less often)
Soft X-Ray (SXR) wavelengths. These instruments observe in a relatively narrow
passband and hence are sensitive to only a narrow range of temperatures, at a
time. CME triggers, such as plasma instabilities occur within Alfvenic
temporal and spatial scales (of the order of tens of seconds or hundreds of km
for an active region). The subsequent energy release also occurs in similar
scales and the eruption is usually accompanied by other phenomena such as
flares, jets and lateral plasma motions that may have nothing to do with the
erupting structure but they complicate the interpretation of the EUV
observations.
Therefore, we need observations of the formation stages of a CME taken with
high cadence and spatial resolution but with minimal line-of-sight confusion.
The unique stereoscopic viewing and instrument complement provided by the Sun-
Earth Connection Coronal and Heliospheric Investigation (SECCHI;
2008SSRv..136…67H) on-board the Solar TErrestial RElations Observatory
(STEREO) [Kaiser et al. (2008)] fulfills these requirements nicely.
To demonstrate this, we undertake a CME initiation study for an event which
took place on January 2, 2008. The eruption in the low corona was observed
very well by both SECCHI Extreme Ultraviolet Imagers (EUVI). We are able to
examine in detail the various stages of the initiation of a CME and relate
them to the usual phenomena that accompany these eruptions, such as flares and
filament ejections. In addition, we capture the transition of loop arcades
into the forming flux rope and report the first three-dimensional observations
of loop ’implosion’. Taken together, these observations reveal many of the key
components of CME initiation and provide strong constraints for CME models.
The paper is organized as follows. In Section 2, we present the time history
of the CME and discuss in detail several key observations including the close
correspondence between the acceleration profile and the GOES SXR light curve,
the novel observation and 3D measurements of collapsing loops, the detection
of a hot CME core, and the observation of outflows along the filament channel.
We conclude in Section 3.
## 2 Stereoscopic Observations of the January 2, 2008 CME
The event under study erupted from active region NOAA 10980 located at S05E65
(Figure fig:context). The region has an alpha magnetic configuration with a
single leading negative polarity sunspot. The sunpot disappeared within a
couple of days leaving only an extended area of plage fields. The eruption
occurred along a filament channel (thick white line in Figure fig:context)
overlying a neutral line extending from the center of the region to its
periphery. The CME was accompanied by a GOES C1.2 flare starting at 06:51 UT,
peaking at 10:00 UT, and ending at 11:21 UT. It is therefore a long duration
Soft X-ray (SXR) event but the gradual rise of the light curve is also
indicative of a partially occulted event. Indeed, upward motions at the
location of the subsequent CME can be detected much earlier than the flare
peak as we shall see later. The event was observed by the SECCHI/EUVI imagers
on the STEREO-A and STEREO-B spacecraft which were located $21^{\circ}$ West
and $23^{\circ}$ East from the Sun-Earth line, respectively. Therefore, it was
a limb event for EUVI-A ($\sim 88^{\circ}$) and an eastern event for EUVI-B
($\sim 42^{\circ}$). The 3D kinematics of the CME in the SECCHI coronagraph
fields of view have been discussed in detail by [Zhao et al. (2010)]. Here, we
focus on the initiation of the CME in the low corona as witnessed in the EUVI
fields of view up to about 1.7 Rsun. We use mainly the 171Å images because of
their high cadence (150 sec) but we discuss the observations in the other
wavelengths as well. The images have been processed by the 2008ApJ…674.1201S
wavelet-based algorithm to enhance the visibility of the off-limb structures
by removing the instrumental stray light.
Figure 1.: Top panels: EUVI-A and -B 171Å full disk images on January 2, 2008
at 09:01 UT. The box marks the FOV around AR 10980 used in the subsequent
analysis. Bottom panels: MDI magnetograms of AR10980 on January 2 and 4
showing the magnetic field configuration for the event. The thick white line
marks the filament channel involved in the eruption. The arrow mark the
approximate location of collapsing loops discussed in sec:collapsing. The
magnetogram images are courtesy of SolarMonitor.org. fig:context
### 2.1 The time history of the CME formation in the low corona
Because of the unusual duration of the eruption, we have to find a reliable
marker for the start of the event. We use the time of the first unambiguous
detection of upward motion of EUV loops at the location of the subsequent CME.
This occurs at 06:13:30 UT (online movie and Figure fig:detail). We rely on
the EUVI-A images to describe the upward evolution since the CME is
propagating along the sky plane of STEREO-A and therefore the images are least
affected by projection effects.
The motion in EUVI-A originates in a high-lying loop system which appears to
encompass a cavity as evidenced by the lack of 171Å emission (Figure
fig:detail). Inside this cavity (in projection) we detect a single bright loop
(L1) that begins to collapse as the rest of the loop system expands slowly.
The loop is visible from 05:33:30 UT to 07:21:00 UT. The behavior of this
collapsing loop is almost immediately imitated by a larger loop arcade (L2).
Their collapse starts at around 08:18:30 UT. The CME front leaves the edge of
the EUVI-A field of view at 09:18:30 UT. The first evidence of a CME core, in
the traditional sense of a 3-part CME, becomes apparent at 09:15:30 UT while
the L2 system continues to collapse. At 10:03:30 UT, the loop arcade
disappears, the CME continues to accelerate and the usual post-eruptive arcade
forms. An EUV wave is launched by the expanding CME at around 10:33:30 UT.
Material continues to flow outward from the active region while the post-
eruptive arcade continues to grow until about 13:03:30 UT. We take this time
as the end of the eruption since it marks the end of the material outflow and
the growth of the flaring arcade.
The low-lying activity in the source region is not visible from EUVI-A but it
is clearly visible in EUVI-B. The images show that all the action takes place
along the filament channel running roughly east-west through the center of the
active region. The start of the event occurs at the easternmost edge of the
filament channel, closest to the leading sunspot of 10890. The post-eruption
loop system expands from that location toward the east. The collapsing loops
follow the same path as they collapse (see online movie). The time history is
summarized in Table tbl:history.
Table 1.: Time history of the CME eruption as marked by several key events.tbl:history Event | Time | Elapsed time
---|---|---
| (UT) | (min)
Upward motion (event starts) | 06:13:30 | 0
Single loop (L1) collapses | 06:36:00 | 22.5
SXR flare starts | 06:51:00 | 37.5
Single loop (L1) disappears | 07:21:00 | 67.5
Loop arcade (L2) collapses | 08:18:30 | 125.0
Core appears | 09:15:00 | 181.5
Flaring Arcade (FL1) appears | 09:21:00 | 207.5
SXR Flare peaks | 10:00:00 | 246.5
Loop Arcade (L2) disappears | 10:03:30 | 230.0
CME acceleration peaks | 10:23:00 | 249.5
EUV Wave appears | 10:33:30 | 260.0
End of SXR flare | 11:21:00 | 307.5
End of outflows (event ends) | 13:03:30 | 410.0
### 2.2 Height-Time Evolution of CME in the Low Corona
sec:height
Since the beginning of the day, the overlying loop system seems to be in a
steady state without noticeable motions other than the effect of the solar
rotation (the AR is rotating over the eastern limb as seen from EUVI-A).
Starting at around 6:13 UT, we can see upward motions within the loop system
and the whole system begins to expand after 6:31:30 UT. We choose to follow
the top of the loops for our height-time (ht) measurements. For the first two
hours, however, the motion is very slow and can be best appreciated by
examining the accompanying movie. Because of the slow rise, we use a running
cadence of 10 min (every four 171Å frames) for the ht measurements to make the
motion easier to see. Consequently, the height-time measurements were taken
with the full available cadence of 2.5 min.
EUVI-A is able to follow the loop top until 09:18 UT, when the CME exits the
telescope’s field of view (Figure fig:detail). We then turn to the COR1-A
images to obtain a complete set of ht measurements during the rise time of the
SXR flare. The measurements are presented in Figure fig:CME_height. The first
COR1-A ht point is plotted right next to the line labeled ’Core Appears’. We
did not attempt to triangulate the CME front positions in EUVI-B because the
front is visible only between 8:51 - 9:08 UT and is quite extended and
diffuse. The projection, however, does not affect our EUVI-A measurements
because it is clear that the CME lies very close to the EUVI-A plane of the
sky. To derive velocity and acceleration profiles from our sparse ht points,
it is always better to smooth the ht points first. We use the same smoothing
method as in 2010ApJ…724L.188P. Namely, we minimize the $\chi^{2}$ between the
data and a cubic spline plus a penalty function equal to the second derivative
of the spline multiplied by a weighting factor, $spar$, provided by the user.
In this case, $spar=0.6$ offers the best balance between noisy and overly
smooth acceleration profiles. The results are shown in Figure fig:CME_height
where the velocity is plotted in the top panel and the acceleration in the
bottom panel.
Figure 2.: Snapshots of the eruption as seen in simultaneous images from
SECCHI/EUVI-A (right) and EUVI-B (left). The frames are taken from the online
movie and the labeled features are discussed in Sections sec:height -
sec:collapsing. The times correspond to the EUVI-A observation time.
fig:detail
Figure 3.: The development of the eruption as seen through height-time and
velocity-time diagrams (top panel) and the Soft X-ray light curve and its
derivative (bottom panel). The heights correspond to the top of the CME
structure and the speed is derived using a smoothing procedure (Section 2.2).
Key events, such as collapsing loops, are also marked on the figure and are
discussed in Sections 2.3 and 2.4. fig:CME˙height
The last height measurement was taken in COR1-A at 10:50:18 UT but we show
results only until 10:30 UT. At that point the CME has reached a height of
$3R_{\odot}$ with velocity of $420$ km s-1. Both our speed and acceleration
results are consistent with the Zhao10 results which were based on ht
measurements after 10:00:00 UT and on a different technique.
In the bottom panel of Figure fig:CME_height, we compare the CME acceleration
to the 1-min GOES SXR light curve (1-8Å channel) and its time derivative which
is considered a proxy to energy release episodes. Both SXR curves are
normalized to their respective peaks.
First, we see that the CME acceleration profile follows closely the SXR rise
as seen before [Zhang et al. (2004), Temmer et al. (2008), Temmer et al.
(2010)], albeit with some time delay. This delay is consistent with the
gradual character of this CME. Generally speaking, impulsive CMEs tend to have
acceleration profiles leading the SXR flux profile [Patsourakos, Vourlidas,
and Stenborg (2010)] since it takes some time to heat the chromosphere and to
fill in the coronal loops with the hot plasma. In our case, the CME
acceleration peaks sharply after at about 10:23 UT when the flux rope core and
the post-CME flaring arcades appear. We return to this point in Section
sec:flux-rope.
Second, the impulsive phase of flare is a bit unusual because the rise of the
SXR flux is marked by two interim inflections (one at $\sim$8:25-8:50 and the
second at 9:20 UT) before the SXR peak at 10:00 UT. Remarkably, the CME
acceleration profile changes at almost the same times. We can discern
inflection points at approximately 8:30, 8:55, 9:10, 9:20, 9:55, and 10:25 UT
in the bottom panel of Figure fig:CME_height. These points bracket intensity
changes in the SXR light curve and coincide with peaks in the SXR derivative
(and hence energy release episodes). The correlations are positive
(acceleration) with the exception of the SXR derivative peak at 9:10 UT which
occurs during a decelerating phase of the CME. The time offsets between the
SXR and CME acceleration peaks are within 5 min of each other. There is even
indication for an earlier acceleration jump associated with a small step in
the SXR flux at around 7:00 UT. Since flaring and hence changes in the SXR
profile result from energy release in the low corona, it is tempting to
interpret the changes in the CME acceleration profile as a result of the same
energy release. For example, the CME speed increases from about 5 km s-1 to
almost 80 km s-1 during the first flaring episode, between 8:20 and 9:00 UT.
To investigate whether the correspondence between the SXR and CME acceleration
profiles is based on a causal relationship we look into the various phases of
the event in detail in the following.
### 2.3 Collapsing Loops
sec:collapsing
The observation of the two collapsing loop systems, L1 and L2, represents a
unique aspect of this event and drew our attention to it. The first system,
L1, appears to be a single loop which stands out because it is projected
against an area of reduced 171Å emission, possibly a cavity, as viewed from
EUVI-A. The loop appears to collapse starting at around 6:36 UT and disappears
at 7:21 UT. The loops do not appear to simply contract as has been seen in
other occasions (see 2011SSRv..158….5H and references therein) but it rather
seems to incline toward the cavity. At the same time, the cavity is slowly
rising and expanding. This behavior, especially the disappearance of the loop,
is seen for the first time and is suggestive of a magnetic relationship
between the loop and the cavity. But before we discuss this further, we have
to understand the 3D topology of the loop.
The loop is quite tall (0.15 R⊙ or $1.04\times 10^{5}$ km). However, it is
very hard to discern from the EUVI-B perspective because it is narrow (small
footpoint distance) and is oriented toward EUVI-B (Figure fig:detail, middle
panels). Nevertheless, its 3-dimensional (3D) orientation can be established
because it becomes visible in EUVI-B once it starts collapsing. We use
standard SECCHI software (the scc_measure routine) to derive its 3D parameters
as a function of time for the period 6:36-7:08 UT. Briefly, the algorithm
requires the user to select a point in the loop in one view. This selection
corresponds to a line (the epipolar line) in the other view. The successful
triangulation is achieved by identifying the location where the epipolar line
intersects the projection of the original point in the loop. In our case, the
obvious candidate is the bright loop-top in EUVI-A. Unfortunately it does not
have a clearly identifiable counterpart in EUVI-B because we view the loop-top
face-on. After careful examination of the movies, we decided to use a
relatively bright edge in EUVI-B as the starting point because it was easier
to find the intersection of the epipolar line with the loop in the EUVI-A
images. The intersection was located a few pixels below the bright loop apex
along the loop leg farthest from the EUVI-A observer. Here, we are primarily
interested in the temporal behavior of the loop height. The ht measurements
are shown in the Figure fig:collapsing-loops-ht. There is an obvious downward
trend despite some scatter in the measurements around 7 UT. The scatter arises
from inaccurate identification of the same part of the structure in the two
images. We repeated the measurements three times but we were not able to
improve the scatter in time. Although the scatter in the three measurements
(at the same time) was very small, we decided to adopt a conservative error
estimate equal to the standard deviation of all measurements in order to
account for the scatter in time. Given the scatter, we fit the ht data points
with a first order polynomial, assuming therefore, a constant speed. We
obtained a speed of $3$ km s-1.
Figure 4.: Height-time measurements of the two sets of collapsing loops
observed during this CME event. The heights are true radial distances obtained
via triangulation of the structures in the EUVI-A and -B 171Å images. The
solid lines represent linear fits to the ht points and result in speeds of 3
km s-1 and 2 km s-1 for the L1 and L2 systems, respectively.fig:collapsing-
loops-ht
Just an hour later, at 8:18 UT, a larger loop system (L2) begins to collapse
following an almost identical path to L1 (Figure fig:infalling-loops). The L2
system is located just a few pixels southeast of L1 and reaches almost the
same height, 0.15 R⊙. L2 is more discernible in the EUVI-B images but it could
easily be overlooked if it was not for the EUVI-A observations. This is a very
important point and explains the lack of such observations in the past. How
many times have we missed such inclining, collapsing loops in the past because
we had only one viewpoint available? Thanks to the two EUVI views, we can
derive the 3D orientation of L2 as we did for L1. The resulting ht points in
Figure fig:collapsing-loops-ht show a rather sharp drop in the first 15 mins
followed by a gradual contraction. We chose to fit again a first order
polynomial to describe the long-term evolution of the loop apex. In this case,
we derived a slight slower speed of $2$ km s-1. The L2 system collapses toward
the bottom of the erupting structure and the cavity is clearly rising while
the loop system is collapsing. The loops disappear similarly to L1, at a
height of 0.12 R⊙. We note that the CME clearly took off while the L2 system
was still collapsing and that the disappearance of the L2 loops coincides with
the flare peak. It is also worth noting (Figure fig:detail) that the first set
of bright flaring loops (in 171Å) appears at the location of the L2
footpoints.
Figure 5.: The collapsing loops toward the expanding CME cavity as seen from
EUVI-A (top right) and EUVI-B (top left). The arrows point to the direction of
the collapse. A flaring loop system with peculiar connectivity is also marked
(FL1). The bottom panels show snapshots at the time of the disappearance of
the L2 system and the appearance of bright flaring loops at their
footpoints.fig:infalling-loops
The coincidence of the collapsing loops to the rise and growth of the erupting
structure is very suggestive of a magnetic connection between the two and is
expected according to the standard CME models. Specifically, the models show
that as the flux rope rises and a current sheet forms behind it, the resulting
reconnection attracts nearby magnetic lines. The result is the creation of a
void which field lines further afield would rush to fill. The void, and
subsequent inflow, would occur across the erupting channel. Because most
models are essentially two-dimensional, the reconnection is symmetric and
proceeds from the center of the neutral line (or filament channel) outwards
and across the channel. In this situation, the inflows are depicted on either
side of the post-CME current sheet (e.g., 2004ApJ…602..422L). However, this
does not have to be, and most likely it is not the situation with the actual
observations. Erupting prominences (a usual proxy for the CME core) are often
seen rising asymmetrically and the majority of H$\alpha$ ribbons brighten
progressively both across and along the channel. If the eruption were to start
at one end of the filament channel then the ribbons would move from that end
of the channel to the opposite instead from starting at the middle and
propagate outwards along the channel as the symmetric picture would suggest
[Li and Zhang (2009)]. In that case, the void would form on end of the channel
and any likely inflows would occur there. Such an asymmetric eruption was
discussed by Patsourakos_Vourlidas_Kliem_2010. Therefore, we expect the
following: (i) inflows toward and behind the erupting structure, (ii) the
inflows would occur where the flux rope rises first, and (iii) the inflows and
flux rope growth would be correlated. The analysis of the collapsing loops
meets all three of these expectations and hence we claim that they constitute
the first direct evidence of the process of flux rope formation (or growth)
though the incorporation of neighboring flux systems into the erupting
structure.
### 2.4 The Detection of the Hot Flux Rope Core
sec:flux-rope
Figure 6.: Overlays of quasi-simultaneous EUVI-A observations at 284Å (green)
and 171Å (red) during the appearance of the CME core. The degree of color
dominance (green or red) at a given location can be used as a proxy for the
temperature of the material at that location. For example, the CME core
appears fully green at 9:26 UT which implies that most of the core material is
emitting at 284Å or about 1.8 MK, at that time.fig:fluxrope
The CME has a clear 3-part structure in the COR1 and COR2 observations [Zhao
et al. (2010)] and both the front and following cavity are easily discernible
in the 171Å observations. The counterpart for the core is not easy to identify
until 9:15 UT when a rather diffuse blob-like structure appears in the 195Å
images. No erupting prominence is detected in the 304Å observations. The core
is clearly visible in the 284Å image taken at 9:26:30 UT but it is very hard
to detect in the almost simultaneous 171Å image at 9:26 UT (Figure
fig:fluxrope). The dominant contribution in the 284Å bandpass comes from the
FeXV line which forms at around 1.8 MK. Therefore, the lack of 171Å emission
and the bright 284Å emission suggest that the majority of the core plasma
comes from hot temperatures. This is exactly what the models predict and
recent Solar Dynamics Observatory (SDO) observations show [Cheng et al.
(2011)]. Therefore, we conclude that the CME core in our event is hot and
comes at the tail end of the cavity within the erupting structure. Once the
core is identified in the 284Å and 194Å images, it is relatively
straightforward to follow in the 171Å as well although it remains quite faint
(see online movie).
### 2.5 Flows along the Filament Channel
Throughout the event, one can observe flows along the filament channel (FC).
They become more obvious along a bend of the FC at its eastern end. The
filament itself is observed as a collection of dark threads in the 171Å
channel due to the absorption from the cool material. It is anchored in the AR
on its western end and in the quiet sun at its eastern end. The flows seem to
evolve in two phases. In the first one, which lasts until 8:28 UT, the flows
are brighter. In the second phase, which lasts until 10:06 UT, the flowing
material acquires a blob-like character. Some of those blobs are depicted in
Figure fig:siphon. The symbols in this figure (cross, box, circle) indicate
the position the blobs we identified and measured at different time frames. In
Figure fig:blobs, the area of interest has been rotated to make the blob
movement more obvious. The position of each blob in this sequence of images is
connected with a line.
Figure 7.: Flows along the erupting filament channel. The symbols indicate a
particular blob tracked at different times in each of these EUVI-B 171Å
images. fig:siphon
Figure 8.: Demonstration of our tracking of the blobs in the EUVI-B images.
The area was rotated to make easier the display of lines connecting the blobs.
In the top panels, the upper line traces the blob marked with an X in Figure
fig:siphon, and the lower line traces the blob marked with a box. In the
bottom panels, the first two frames are repeats from the last two frames of
the top panel. The upper line is the continuation of the trace for the blob
marked with the box symbol, and the lower line traces the blob marked with the
circle. fig:blobs
After tracing the blobs, we measured their velocities. When the size of blobs
was small (e.g. at 09:23:00 UT), their position was assumed to be their
coordinates in the image. When the blobs became more extended (e.g. at
09:38:30 UT), we took the middle point as their average position, and their
length was taken as the error uncertainty.
Because the blobs were located very low in the corona, they were not visible
from EUVI-A. Because we know the angular distance of EUVI-A and the location
of the flows from EUVI-B, we can derive an upper limit for the height of the
channel of 0.015 R⊙ or $10.5\times 10^{4}$ km. Since they move parallel to the
surface and over a limited spatial extension, there was no need to correct for
spherical geometry. The effect is less than 4% for the full $30^{\circ}$
length of the filament which we did not use in our measurements. However, the
projection effect due to the proximity of the channel to the limb needs to be
taken into account. The flows are measured at about $65^{\circ}$ east
longitude so the correction factor is $\sim 1/cos(65^{\circ})\sim 2.36$. The
average deprojected velocities of the blobs are given in Table tab:siphon.
Each blob is named after the symbol we used to mark them in Figure fig:siphon.
The relation of the flows to the eruption is not immediately clear. First,
they appear to correspond to material flowing out of the AR into the quiet sun
because they propagate only in one direction, from the center of the AR toward
the quiet sun. Such behavior has been very common since the beginning of EUVI
observations and is always related to AR filaments that extend into the quiet
sun. Examples can be seen in the eruptions of 1, 16, and 19 May 2009, 5 and 9
April 2008, 14 and 18 August 2010. The event on 3 April 2010 has been analyzed
in the detail by 2011ApJ…727L..10S who connect such flows to off-loading of
cool plasma that may contributed to the subsequent CME eruption. Second, the
nature of the blobs changes at around 8:28 UT from thick elongated flows to
smaller blob-like features suggesting that the amount of the flowing material
has been reduced or the plasma has cooled down. It is interesting to note that
the CME underwent its first acceleration jump during that time. This apparent
correlation seems to support the 2011ApJ…727L..10S interpretation of the flows
as off-loading material and suggests that gravity may affect the early
acceleration profile of CMEs.
Table 2.: Average velocities for each of the three blobs. The names correspond
to the symbols used to mark the blobs in Figure fig:siphon.
Blob | Velocity $(km\,s^{-1})$ | Error
---|---|---
X | 125 | 5.3
Box | 116 | 4.9
Circle | 130 | 5.4
tab:siphon
There is an alternative explanation, however. The flows apparently trace
closed field lines along the filament. The movement of the blobs is directed
away from the site of the emerging fluxrope where energy input is taking place
leading to higher plasma pressures in its vicinity. Therefore, the observed
flows could be siphon flow imposed by a pressure difference between the two
footpoints of the filament [Cargill and Priest (1980), Cargill and Priest
(1982)].
## 3 Discussion and Conclusions
We investigate in detail the initiation and formation of a CME on January 2,
2008 using two-viewpoint EUV observations in the lower corona. The images are
obtained in the 171Å (150 sec cadence) and 284Å (20 min cadence) channels of
the EUVI instruments aboard the STEREO mission. The event evolves slowly for
several hours but it then quickly accelerates around the time of the
accompanying SXR flare. This allows us to study in detail both its evolution
toward the eruption, the subsequent formation of a CME, and its connection to
the flare energy release profile. Our main results can be summarized as
follows:
* •
The acceleration profile of the CME is quite variable with peaks and valleys.
The acceleration changes are similar, in time of appearance and duration, with
corresponding changes in the GOES SXR light curve.
* •
The CME acceleration peaks at 10:30 UT which is 30 mins after the peak of the
SXR flare.
* •
The upward motions of the (eventually) erupting structure started at 6:13 UT,
about 1 hour before a small SXR flux increase and 2 hours before a significant
increase of SXR flux occurred (Figure fig:CME_height).
* •
We detect, for the first time, two sets of collapsing loops. The two viewpoint
EUVI observations allow us to measure their 3D evolution. They shrink very
little (compared to past observations of shrinking loops) so most of their
collapse is due to their inclining toward the erupting channel, beneath the
rising cavity. They appear in all EUVI channels and they disappear in all of
them at a height of 0.12 R⊙. The post-CME arcades appear after the
disappearance of the collapsing loops and at the same location. The CME cavity
is clearly growing while the second loop system (L2) is collapsing. These
observations lead us to conclude that the two loop systems are likely drawn
behind the expanding magnetic cavity surrounding the CME core. This appears to
be the first detection of this process predicted by CME initiation models.
* •
We detect the core of the CME mostly in the hot EUVI channel at 284Å (1.8 MK)
and the 195Å channel. This observation provides further support that the CME
cavity contains hot plasma as recent AIA observations have shown [Cheng et al.
(2011)].
* •
We detect significant and long duration ($\sim 3$ hours) plasma flows along
the filament channel before its eruption. Their nature changes abruptly at
around 8:30 UT coincident with a sudden change in the rising speed of the
cavity. This coincidence suggests that mass unloading is perhaps playing a
role in the early CME kinematics.
* •
The direction of the flows, from the western to the eastern part of the active
region, is also in agreement with the temporal evolution of the flaring
ribbons and post-eruptive flaring arcades, and the direction of the collapsing
loops. Clearly, the eruption starts at the center of the active region and
propagates to the east along the filament channel and toward the quiet sun
footpoints of that channel.
* •
Despite the large number of novel observations and detailed measurements we
cannot tell with certainty whether the erupted flux rope was pre-existing or
was formed during the eruption. However, we are fairly certain that additional
flux was introduced in the erupting flux rope during its ascent. This is the
second event we reach this conclusion [Patsourakos, Vourlidas, and Kliem
(2010)] and is the expected outcome of several models [Lin, Raymond, and van
Ballegooijen (2004), Forbes et al. (2006), Chen (2011)]. It is, therefore,
important to take this effect into account in the estimation of magnetic flux
entrained in CMEs.
All these observations confirm corresponding expectations of the standard
flare-CME models and suggest that such models are likely reliable
representations of the eruption process in the corona. Our analysis
demonstrates the power of two-viewpoint observations of the low corona and the
importance of extended fields of view for EUV instruments so that the
acceleration profile of the CME and the relationships among the various
erupting structures can be measured consistently.
#### Acknowledgements
We thank the referee for the very useful comments and G. Stenborg for
providing the wavelet-enhanced EUVI images and S. Patsourakos for fruitful
discussions. The work of AV is supported by NASA contract S-136361-Y to the
Naval Research Laboratory. The SECCHI data are produced by an international
consortium of the NRL, LMSAL and NASA GSFC (USA), RAL and Univ. Bham (UK), MPS
(Germany), CSL (Belgium), IOTA and IAS (France).
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|
arxiv-papers
| 2011-12-30T18:11:49 |
2024-09-04T02:49:25.818662
|
{
"license": "Public Domain",
"authors": "A. Vourlidas, P. Syntelis, K. Tsinganos",
"submitter": "Angelos Vourlidas",
"url": "https://arxiv.org/abs/1201.0162"
}
|
1201.0247
|
# Nonclassical properties of a particle in a finite range trap: the
$f$-deformed quantum oscillator approach
M. Davoudi Darareh m.davoudi@sci.ui.ac.ir M. Bagheri Harouni
m-bagheri@phys.ui.ac.ir Department of Physics, Faculty of Science, University
of Isfahan, Hezar Jerib, Isfahan, 81746-73441, Iran
###### Abstract
A particle bounded in a potential with finite range is described by using an
$f$-deformed quantum oscillator approach. Finite range of this potential can
be considered as a controllable deformation parameter. The non-classical
quantum statistical properties of this deformed oscillator can be manipulated
by nonlinearities associated to the finite range.
###### keywords:
Modified Pöschl-Teller like coherent state , nonclassical property ,
$f$-deformed quantum oscillator
###### PACS:
03.65.Fd, 03.65.Ge, 42.50.Dv, 42.50.Ar
## 1 Introduction
The quantum harmonic oscillator, its associated coherent states and their
generalizations [1] play an important role in various theoretical and
experimental fields of modern physics, including quantum optics and atom
optics. Motivations for these generalizations have arisen from symmetry
considerations [2], dynamics [3] and algebraic aspects [4, 5].
The quantum groups approach [4] for generalizing the notion of quantum
harmonic oscillator and its realizations in physical systems, by providing an
algebraic method, has given the possibility of extending the creation and
annihilation operators of the usual quantum oscillator to introduce the
deformed oscillator. In a very general important case, the associated algebra
of this deformed oscillator may be viewed as a deformation of classical Lie
algebra by a generic function $f$, the so-called $f$-deformation function,
depending nonlinearly on the number of excitation quanta and some deformation
parameters. The corresponding oscillator is called an $f$-deformed oscillator
[6]. In contrast to the usual quantum harmonic oscillator, $f$-deformed
oscillators do not have equally-spaced energy spectrum. Furthermore, it has
been known that the most of nonlinear generalizations of some physical models,
such as considered in [7], are only particular cases of $f$-deformed models.
Thus, it is reasonable that $f$-deformed oscillators exhibit strongly various
nonclassical properties [6, 8, 9], such as the sub-Poissonian statistics,
squeezing and the quantum interference effects, displaying the striking
consequences of the superposition principle of quantum mechanics. In addition,
$f$-deformed models depend on one or more deformation parameters which should
permit more flexibility and more ability for manipulating the model [10, 11].
An important question in the $f$-deformed model is the physical meaning of its
deformation parameters. The $q$-deformed oscillator [5], as a special kind of
$f$-deformed oscillators with only one deformation parameter $q$, has been
extensively applied in describing physical models, such as vibrational and
rotational spectra of molecules [12]. The appearance of various nonclassical
features induced by a $q$-deformation relevant to some specific nonlinearity
is also studied [13].
Based on the above-mentioned considerations, $f$-deformed quantum oscillators
and their associated coherent states, such as $f$-coherent states [6] or
nonlinear coherent states [9], can be appropriately established in attempting
to describe certain physical phenomena where their effects could be modelled
through a deformation on their dynamical algebra with respect to conventional
or usual counterparts. This approach has been accomplished, for instance, in
the study of the stationary states of the center-of-mass motion of an ion in
the harmonic trap [9] and under effects associated with the curvature of
physical space [14], the influence of the spatial confinement on the center-
of-mass motion of an exciton in a quantum dot [15], the influence of atomic
collisions and the finite number of atoms in a Bose-Einstein condensate on
controlled manipulation of the nonclassical properties of radiation field
[10], some nonlinear processes in high intensity photon beam [6], intensity-
dependent atom-field interaction in absence and in presence of nonlinear
quantum dissipation in a micromaser [16] and finally, incorporating the
effects of interactions among the particles in the framework of the
$q$-deformed algebra [17].
It is shown that the trapped systems provide a powerful tool for preparation
and manipulation of nonclassical states [18], quantum computations [19] and
quantum communications [20]. Improved experimental techniques have caused
precise measurements on realistic trapping systems, for example, trapped ion-
laser systems [21], trapped gas of atoms [22] and electron-hole carriers
confined in a quantum well and quantum dot [23]. A study of confined quantum
systems using the Wood-Saxon potential [24] and the $q$-analogue harmonic
oscillator trap [25], are some efforts which can be used to explain some
experimentally observed deviations from the results predicted by calculations
based on the harmonic oscillator model.
A realistic case in any experimental setup is that the dimension of the trap
is finite and the realistic trapping potential is not the harmonic oscillator
potential extending to infinity. Thus, the realistic confining potential
becomes flat near the edges of the trap and can be simulated by the tanh-
shaped potential $V(x)=D\,tanh^{2}(x/\delta)$, so-called the modified (or
hyperbolic) Pöschl-Teller(MPT) potential [26]. The MPT potential presents
discrete (or bound) and continuum (or scattering) states. The dynamical
symmetry algebra associated with the bound part of the spectrum is $su(2)$
algebra [27] while for the complete spectra is $su(1,1)$ algebra [28]. The MPT
potential has been used very widely in many branches of physics, such as, atom
optics [29], molecular physics [30] and nanostructure physics [23].
Constructing coherent states for systems with discrete and continuous spectrum
[31] and for various kinds of confining potentials [32] have become a very
important tool in the study of some quantum systems. The Pöschl-Teller(PT)
potentials, including trigonometric PT(TPT) and MPT potentials with discrete
infinite and finite dimensional bound states respectively, because of their
relations to several other trapping potentials are of crucial importance. Some
types of the coherent states for the MPT potential have been constructed. The
minimum-uncertainty coherent states formalism [33], the Klauder-Perelomov
approach [1] by realization of lowering and raising operators in terms of the
physical variable $u=tanh(x/\delta)$ by means of factorizations [34] and
applying one kind generalized deformed oscillator algebra with a selected
deformed commutation relation [35], are some attempts for this purpose.
In the present paper, we intend to investigate the nonlinear effects appeared
due to finite dimension of the trapping potential on producing new
nonclassical quantum statistical properties using the $f$-deformed quantum
oscillator approach. For this aim, it will be shown that the finite range of
the trapping potential leads to the $f$-deformation of the usual harmonic
potential with the well depth $D$ as a controllable physical deformation
parameter. Then, the $f$-deformed bound coherent states [36] for the above-
mentioned MPT quantum oscillator are introduced and their nonclassical
properties are examined. We think that by this $f$-deformed quantum oscillator
approach the problem of trapped ion-laser system and trapped gas of atoms,
such as a Bose-Einstein condensate, in a realistic trap can be studied
analytically.
The paper is organized as follows. In section 2, we introduce the $f$-deformed
quantum oscillator equivalent to the MPT oscillator and obtain the associated
ladder operators. In section 3, we construct the $f$-deformed bound coherent
states of the MPT quantum oscillator and examine its resolution of identity.
Section 4 devoted to the study of the influence of the finite range potential
on producing and manipulating the nonclassical properties, including the sub-
Poissonian statistics and squeezing character. Finally, the summary and
conclusions are presented in section 5.
## 2 MPT Hamiltonian as an $f$-deformed quantum oscillator
In this section, we will consider a bounded particle inside the MPT potential,
called the MPT oscillator, and we will associate to this system an
$f$-deformed quantum oscillator. By using this mathematical model, we try to
investigate physical deformation parameters in the model, to manipulate the
nonlinearities related to the finite range effects on this system. For this
purpose, we first give the bound energy eigenvalues for the MPT potential.
Then, by comparing it with the energy spectrum of the general $f$-deformed
quantum oscillator, we will obtain the deformed annihilation and creation
operators.
Let us consider the MPT potential energy
$V(x)=D\,tanh^{2}(\frac{x}{\delta}),$ (1)
where $D$ is the depth of the well, $\delta$ determines the range of the
potential and $x$ gives the relative distance from the equilibrium position.
The well depth, D, can be defined as $D=\frac{1}{2}m\omega^{2}\delta^{2}$,
with mass of the particle $m$ and angular frequency $\omega$ of the harmonic
oscillator, so that, in the limiting case $D\rightarrow\infty$(or
$\delta\rightarrow\infty)$, but keeping the product $m\omega^{2}$ finite, the
MPT potential energy reduces to harmonic potential energy,
$\lim_{D\rightarrow\infty}V(x)=\frac{1}{2}m\omega^{2}x^{2}$. Figure 1 depicts
the MPT potential for three different values of the well depth $D$. Harmonic
potential limit by increasing $D$ is clear from this figure. Solving the
Schrödinger equation, the energy eigenvalues for the MPT potential are
obtained as [37]
$E_{n}=D-\frac{\hbar^{2}\omega^{2}}{4D}(s-n)^{2},\quad\quad
n=0,1,2,\cdots,[s]$ (2)
in which $s=(\sqrt{1+(\frac{4D}{\hbar\omega})^{2}}-1)/2$, and $[s]$ stands for
the closest integer to $s$ that is smaller than $s$. The MPT oscillator
quantum number $n$ can not be larger than the maximum number of bound states
$[s]$, because of the dissociation condition $s-n\geq 0$. Consequently, the
total number of bound states is $[s]+1$. We should note that for integer $s$,
the final bound state and the total number of bound states will be $s-1$ and
$s$, respectively. Also, for every small value of the well depth $D$, we
always have at least one bound state for the MPT oscillator, i.e., the ground
state. By introducing a dimensionless parameter
$N=\frac{4D}{\hbar\omega}=\frac{2m\omega\delta^{2}}{\hbar}$, the total number
of bound states will obtain from $[(\sqrt{1+(N)^{2}}-1)/2]+1$. For integer
$s$, a simple relation $N=2\sqrt{s(s+1)}$ will connect $N$ to the total number
of bound states, i.e., $s$. The bound energy spectrum in equation (2) can be
rewritten as
$E_{n}=\hbar\omega[-\frac{n^{2}}{N}+(\sqrt{1+\frac{1}{N^{2}}}-\frac{1}{N})n+\frac{1}{2}(\sqrt{1+\frac{1}{N^{2}}}-\frac{1}{N})].$
(3)
The relation (3) shows a nonlinear dependence on the quantum number $n$, so
that, different energy levels are not equally spaced. It is clear that, in the
limit $D\rightarrow\infty$ (or $N\rightarrow\infty$), the energy spectrum for
the quantum harmonic oscillator will be obtained, i.e.,
$E_{n}=\hbar\omega(n+\frac{1}{2})$. In contrast with some confined systems
such as a particle bounded in an infinite and finite square well potentials,
by decreasing the size of the confinement parameter, i.e., the finite range
$\delta$ of the MPT oscillator, energy eigenvalues decreases.
A quantity that has a close connection to experimental information is the
energy level spacing, $E_{n+1}-E_{n}$, where it corresponds to the transition
frequency between two adjacent energy levels. Furthermore, by this quantity
one can theoretically explore an algebraic representation for the quantum
mechanical potentials with discrete spectrum [38]. Based upon above
considerations, a useful illustration for the effects of the deformation
parameter $D$ on the nonlinear behavior of the deformed oscillator, can be
investigated by introducing the delta parameter $\Delta_{n}$ as
$\Delta_{n}=\frac{E_{n+1}-E_{n}}{\hbar\omega}-1$ (4)
which measures the amount of deviation of the adjacent energy level spacing of
the deformed oscillator with respect to the non-deformed or harmonic
oscillator. Substituting from equation (3) in equation (4) we can ontain the
delta parameter $\delta$ for the MPT potential
$\Delta_{n}=-\frac{2}{N}n+\sqrt{1+\frac{1}{N^{2}}}-\frac{2}{N}-1$ (5)
On the other hand, the $f$-deformed quantum oscillator [6], as a nonlinear
oscillator with a specific kind of nonlinearity, is characterized by the
following deformed dynamical variables $\hat{A}$ and $\hat{A}^{\dagger}$
$\displaystyle\hat{A}$ $\displaystyle=$
$\displaystyle\hat{a}f(\hat{n})=f(\hat{n}+1)\hat{a},$
$\displaystyle\hat{A}^{\dagger}$ $\displaystyle=$ $\displaystyle
f(\hat{n})\hat{a}^{\dagger}=\hat{a}^{\dagger}f(\hat{n}+1),\quad\quad\hat{n}=\hat{a}^{\dagger}\hat{a},$
(6)
where $\hat{a}$ and $\hat{a}^{\dagger}$ are usual boson annihilation and
creation operators $([\hat{a},\hat{a}^{\dagger}]=1)$, respectively. The real
deformation function $f(\hat{n})$ is a nonlinear operator-valued function of
the harmonic number operator $\hat{n}$, where it introduces some
nonlinearities to the system. From equation (2), it follows that the
$f$-deformed operators $\hat{A}$, $\hat{A}^{\dagger}$ and $\hat{n}$ satisfy
the following closed algebra
$\displaystyle[\hat{A},\hat{A}^{\dagger}]=$
$\displaystyle(\hat{n}+1)f^{2}(\hat{n}+1)-\hat{n}f^{2}(\hat{n}),$ (7)
$\displaystyle[\hat{n},\hat{A}]=$
$\displaystyle-\hat{A},\quad\quad[\hat{n},\hat{A}^{{\dagger}}]=\hat{A}^{{\dagger}}.$
The above-mentioned algebra, represents a deformed Heisenberg-Weyl algebra
whose nature depends on the nonlinear deformation function $f(\hat{n})$. An
$f$-deformed oscillator is a nonlinear system characterized by a Hamiltonian
of the harmonic oscillator form
$\hat{H}=\frac{\hbar\omega}{2}(\hat{A}^{\dagger}\hat{A}+\hat{A}\hat{A}^{\dagger}).$
(8)
Using equation (2) and the number state representation
$\hat{n}|n\rangle=n|n\rangle$, the eigenvalues of the Hamiltonian (8) can be
written as
$E_{n}=\frac{\hbar\omega}{2}[(n+1)f^{2}(n+1)+nf^{2}(n)].$ (9)
It is worth noting that in the limiting case $f(n)\rightarrow 1$, the deformed
algebra (7) and the deformed energy eigenvalues (9) will reduce to the
conventional Heisenberg-Weyl algebra and the harmonic oscillator spectrum,
respectively.
Comparing the bound energy spectrum of the MPT oscillator, equation (3), and
the energy spectrum of an $f$-deformed oscillator, equation (9), we obtain the
corresponding deformation function for the MPT oscillator as
$f^{2}(\hat{n})=\sqrt{1+\frac{1}{N^{2}}}-\frac{\hat{n}}{N}.$ (10)
Furthermore, the ladder operators of the bound eigenstates of the MPT
Hamiltonian can be written in terms of the conventional operators $\hat{a}$
and $\hat{a}^{\dagger}$ as follows
$\hat{A}=\hat{a}\sqrt{\sqrt{1+\frac{1}{N^{2}}}-\frac{\hat{n}}{N}},\quad\quad\hat{A}^{\dagger}=\sqrt{\sqrt{1+\frac{1}{N^{2}}}-\frac{\hat{n}}{N}}\hat{a}^{\dagger}.$
(11)
These two operators satisfy the deformed Heisenberg-Weyl commutation relation
$[\hat{A},\hat{A}^{\dagger}]=\sqrt{1+\frac{1}{N^{2}}}-\frac{2\hat{n}+1}{N},$
(12)
and they act upon the quantum number states $|n\rangle$, corresponding to the
energy eigenvalues $E_{n}$ given in equation (3), as
$\displaystyle\hat{A}|n\rangle$ $\displaystyle=$ $\displaystyle
f(n)\sqrt{n}|n-1\rangle,$ $\displaystyle\hat{A}^{{\dagger}}|n\rangle$
$\displaystyle=$ $\displaystyle f(n+1)\sqrt{n+1}|n+1\rangle.$ (13)
The commutation relation (12), can be identified with the usual $su(2)$
commutation relations by introducing the set of transformations
$\hat{A}\rightarrow\frac{\hat{J}_{+}}{\sqrt{N}},\quad\hat{A}^{{\dagger}}\rightarrow\frac{\hat{J}_{-}}{\sqrt{N}},\quad\hat{n}\rightarrow\frac{\sqrt{1+N^{2}}-1}{2}-\hat{J}_{0},$
(14)
where $\hat{J}_{\mu}$ satisfy the usual angular momentum relations [39]. The
$f$-deformed commutation relation (12) in a special case of large but finite
value of $N$, which corresponds to the small deformation, can lead to a maths-
type $q$-deformed commutation relation [40], i.e.,
$\hat{A}\hat{A}^{{\dagger}}-q\hat{A}^{{\dagger}}\hat{A}=1$, with
$q=1-\frac{2}{N}=1-\frac{\hbar\omega}{2D}$. The harmonic oscillator limit
corresponds to $D\rightarrow\infty$ then $q\rightarrow 1$. This result
confirms a correspondence between the $q$-deformed oscillators and finite
range potentials, which is studied elsewhere [41].
It is evident that, herein, we have focused our attention on the quantum
states of the MPT Hamiltonian which exhibit bound oscillations with finite
range. The remaining states, i.e., the scattering states or energy continuum
eigenstates, have non-evident boundary conditions. From physical point of
view, it means that the excitation energies of this confined system in the MPT
potential energy are small compared with the well depth potential energy $D$,
such that, only the vibrational modes dominated and the scattering or
continuum states should be neglected. Some important physical systems with
such circumstances are vibrational excitations of molecular systems [42],
trapped ions or atoms [43] and the electron-hole carriers confined in a
quantum well [23].
## 3 $f$-Deformed bound coherent states
In the context of the $f$-deformed quantum oscillator approach, we introduce
the $f$-deformed bound coherent states $|\alpha,f\rangle$ for the MPT
oscillator as a coherent superposition of all bound energy eigenstates of the
MPT Hamiltonian as below
$|\alpha,f\rangle=C_{f}\sum_{n=0}^{[s]}\frac{\alpha^{n}}{\sqrt{n!}f(n)!}|n\rangle,\quad
C_{f}=\left(\sum_{n=0}^{[s]}\frac{|\alpha|^{2n}}{n!(f(n)!)^{2}}\right)^{-1/2},$
(15)
so that $\hat{n}|n\rangle=n|n\rangle$, and $f(n)!=f(n)f(n-1)\cdots f(0)$,
where $f(n)$ is obtained in equation (10). Since the sum in the equation (15)
is finite, the states $|\alpha,f\rangle$, similar to the Klauder-Perelomov
coherent states [1], are not an eigenstate of the annihilation operator
$\hat{A}$. From equations (2) and (15), we arrive at
$\hat{A}|\alpha,f\rangle=\alpha|\alpha,f\rangle-\frac{C_{f}\alpha^{[s]+1}}{\sqrt{[s]!}f([s])!}|[s]\rangle.$
(16)
As is clear from this equation, these states can not be considered as a right-
hand eigenstate of annihilation operator $\hat{A}$. This property is common
character of all coherent states that are defined in a finite- dimensional
basis [36, 44].
The ensemble of the $f$-deformed bound coherent states $|\alpha,f\rangle$
labelled by the complex number $\alpha$ form an overcomplete set with the
resolution of the identity
$\int d^{2}\alpha|\alpha,f\rangle
m_{f}(|\alpha|)\langle\alpha,f|=\sum_{n=0}^{[s]}|n\rangle\langle
n|=\hat{\textbf{1}},$ (17)
where $m_{f}(|\alpha|)$ is the proper measure for this family of the bound
coherent states. Substituting from equation (15) in equation (17) and using
integral relation
$\int_{0}^{\infty}K_{\nu}(t)t^{\mu-1}dt=2^{\mu-2}\Gamma(\frac{\mu-\nu}{2})\Gamma(\frac{\mu+\nu}{2})$
for the modified Bessel function $K_{\nu}(t)$ of the second kind and of the
order $\nu$, we obtain the suitable choice for the measure function as
$m_{f}(|\alpha|)=\frac{K_{\nu}(|\alpha|)}{2^{l}\pi|\alpha|^{\nu}C_{f}^{2}(|\alpha|)},$
(18)
where $\nu=(1+\gamma)n-\eta$, $l=(1-\gamma)n+\eta+1$ and $\gamma=\frac{1}{N}$,
$\eta=\sqrt{1+\frac{1}{N^{2}}}$.
In contrast to the Gazeau-Klauder coherent states [31], the $f$-deformed
coherent states, such as introduced in equation (15), do not generally have
the temporal stability [6]. But it is possible to introduce a notion of
temporally stable $f$-deformed coherent states [45].
## 4 Quantum statistical properties of the MPT oscillator
### 4.1 Sub-Poissonian statistics
In order to determine the quantum statistics of the MPT quantum oscillator, we
consider Mandel parameter $Q$ defined by [46]
$Q=\frac{\langle\hat{n}^{2}\rangle-\langle\hat{n}\rangle^{2}}{\langle\hat{n}\rangle}-1.$
(19)
The sub-Poissonian statistics (antibunching effect), as an important
nonclassical property, exists whenever $Q<0$. When $Q>0$, the state of the
system is called super-Poissonian (bunching effect). The state with $Q=0$ is
called Poissonian. Calculating the Mandel parameter $Q$ in equation (19) over
the $f$-deformed bound coherent states $|\alpha,f\rangle$ defined in equation
(15), it can be described the finite range dependence of the Mandel parameter.
Figure 2 shows the parameter $Q$ for four different values of $|\alpha|$,
i.e., $|\alpha|=3,\,4,\,5,\,7$. As is seen, for every one of the values of
$|\alpha|$, the Mandel parameter $Q$ exhibits the sub-Poissonian statistics at
certain range of $D$ or the dimensionless parameter
$N=\frac{4D}{\hbar\omega}$, where this range is determined by the value of
$|\alpha|$. The bigger parameter $|\alpha|$ is, the more late the Mandel
parameter tends to the Poissonian statistics. As expected, with further
increasing values of $D$ or $N$, the Mandel parameter $Q$ finally stabilized
at an asymptotical zero value, corresponding to the Poissonian statistics
associated to the canonical harmonic oscillator coherent states. For the limit
$N\rightarrow 0$(or $D\rightarrow 0$) and for every values of $|\alpha|$, the
Mandel parameter becomes $Q=-1$, where it is reasonable, because in this
limit, only the ground state supports by the potential.
### 4.2 Quadrature squeezing
As another important nonclassical property, we examine the quadrature
squeezing of the MPT quantum oscillator. For this purpose, we consider
quadrature operators $\hat{q}_{\varphi}$ and $\hat{p}_{\varphi}$ defined as
[47]
$\hat{q}_{\varphi}=\frac{1}{\sqrt{2}}(\hat{a}e^{-i\varphi}+\hat{a}^{\dagger}e^{i\varphi}),\quad\quad\hat{p}_{\varphi}=\frac{i}{\sqrt{2}}(\hat{a}^{{\dagger}}e^{i\varphi}-\hat{a}e^{-i\varphi}),$
(20)
satisfying the commutation relation $[\hat{q}_{\varphi},\hat{p}_{\varphi}]=i$.
One can define the invariant squeezing coefficient $S$ as the difference
between the minimal value (with respect to the phase $\varphi$) of the
variances of each quadratures and the mean value $1/2$ of these variances in
the coherent or vacuum state. Simple calculations result in the formula
$S=\langle\hat{a}^{{\dagger}}\hat{a}\rangle-|\langle\hat{a}\rangle|^{2}-|\langle\hat{a}^{2}\rangle-\langle\hat{a}\rangle^{2}|,$
(21)
so that the condition of squeezing is $S<0$. Calculating the squeezing
parameter $S$ over the $f$-deformed bound coherent states in equation (15), we
examine the squeezed character of these states. In figure 3, we have plotted
the parameter $S$ with respect to the dimensionless deformation parameter
$N=\frac{4D}{\hbar\omega}$ for three different values of $|\alpha|$, namely
$|\alpha|=0.5,\,1,\,1.3$. As is seen, the states $|\alpha,f\rangle$ exhibit
squeezing for certain values of $|\alpha|$. Furthermore, the squeezing
character of the states $|\alpha,f\rangle$ tend to zero as $N$ or the well
depth $D$ of the MPT potential approaches to infinity, according to the
coherent states of the quantum harmonic oscillator. In the limit $N\rightarrow
0$(or $D\rightarrow 0$), this plot shows the quadrature squeezing $S=0$, where
it is in agreement with the only ground state supported by the potential in
this limit.
## 5 Conlusions
In this paper, we have introduced an algebraic approach based on the
$f$-deformed quantum oscillator for considering a particle in the real
confining potential which has finite trap dimension, in contrast to the
harmonic oscillator potential extending to infinity. Proposed confining model
potential is the modified Pöschl-Teller potential. We have shown that the
effects of the finite trap dimension in this model potential can be considered
as a natural deformation in the quantum harmonic oscillator algebra. This
quantum deformation approach makes possible analytical study of a wide
category of realistic bound quantum systems algebraically. It is shown that
the nonlinear behavior resulted from this finite range effects can lead to
generate and manipulate some important nonclassical properties for this
deformed quantum oscillator. We have obtained that the presented $f$-deformed
bound coherent states of the modified Pöschl-Teller potential can exhibit the
sub-Poissonian statistics and quadrature squeezing in definite domain of the
trap dimension or well depth $D$ of this potential. In the large but finite
value for the well depth $D$, i.e., small deformation, a $q$-deformed
oscillator with $q=1-\hbar\omega/(2D)$ will result. In the limit
$D\rightarrow\infty$, the harmonic oscillator counterpart is obtained.
Based on the approach in this paper, we can obtain exact solutions for
realistic confined physical systems such as, trapped ion-laser system (in
progress), Bose-Einstein condensate and confined carriers in nano-structures.
Acknowledgments
The authors wish to thank The Office of Graduate Studies and Research Vice
President of The University of Isfahan for their support.
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Figure captions
Fig. 1. Plots of the MPT potential for three different values of the well
depth $D$, $D=1$(solid curve), $D=2$(dashed curve) and
$D\rightarrow\infty$(dotted curve).
Fig. 2. Plots of the Mandel parameter $Q$ versus the dimensionless deformation
parameter $N$ for $|\alpha|=3$(solid curve), $|\alpha|=4$(dashed curve),
$|\alpha|=5$(dotted curve) and $|\alpha|=7$(dash-dotted curve).
Fig. 3. Plots of the invariant squeezing coefficient $S$ versus the
dimensionless deformation parameter $N$ for $|\alpha|=0.5$(solid curve),
$|\alpha|=1$(dashed curve) and $|\alpha|=1.3$(dotted curve).
|
arxiv-papers
| 2011-12-31T11:02:12 |
2024-09-04T02:49:25.831516
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Davoudi Darareh, M. Bagheri Harouni",
"submitter": "Malek Bagheri",
"url": "https://arxiv.org/abs/1201.0247"
}
|
1201.0319
|
# Theory of dissipative chaotic atomic transport in an optical lattice
V.Yu. Argonov and S.V. Prants Laboratory of Nonlinear Dynamical Systems,
Pacific Oceanological Institute of the Russian Academy of Sciences,
43 Baltiiskaya st., 690041 Vladivostok, Russia
###### Abstract
We study dissipative transport of spontaneously emitting atoms in a 1D
standing-wave laser field in the regimes where the underlying deterministic
Hamiltonian dynamics is regular and chaotic. A Monte Carlo stochastic
wavefunction method is applied to simulate semiclassically the atomic dynamics
with coupled internal and translational degrees of freedom. It is shown in
numerical experiments and confirmed analytically that chaotic atomic transport
can take the form either of ballistic motion or a random walking with specific
statistical properties. The character of spatial and momentum diffusion in the
ballistic atomic transport is shown to change abruptly in the atom-laser
detuning regime where the Hamiltonian dynamics is irregular in the sense of
dynamical chaos. We find a clear correlation between the behavior of the
momentum diffusion coefficient and Hamiltonian chaos probability which is a
manifestation of chaoticity of the fundamental atom-light interaction in the
diffusive-like dissipative atomic transport. We propose to measure a linear
extent of atomic clouds in a 1D optical lattice and predict that, beginning
with those values of the mean cloud’s momentum for which the probability of
Hamiltonian chaos is close to 1, the linear extent of the corresponding clouds
should increase sharply. A sensitive dependence of statistical characteristics
of dissipative transport on the values of the detuning allows to manipulate
the atomic transport by changing the laser frequency.
###### pacs:
37.10.Vz, 05.45.Mt, 05.45.-a
## I Introduction
An atom placed in a laser standing wave is acted upon by two radiation forces,
deterministic dipole and stochastic dissipative ones Minogin ; Kaz ; Meystre .
The mechanical action of light upon neutral atoms is at the heart of laser
cooling, trapping, and Bose-Einstein condensation. Numerous applications of
the mechanical action of light include isotope separation, atomic litography
and epitaxy, atomic-beam deflection and splitting, manipulating translational
and internal atomic states, measurement of atomic positions, and many others.
Atoms and ions in an optical lattice, formed by a laser standing wave, are
perspective objects for implementation of quantum information processing and
quantum computing. Advances in cooling and trapping of atoms, tailoring
optical potentials of a desired form and dimension (including one-dimensional
optical lattices), controlling the level of dissipation and noise are now
enabling the direct experiments with single atoms to study fundamental
principles of quantum physics, quantum chaos, decoherence, and quantum-
classical correspondence (for recent reviews on cold atoms in optical lattices
see Ref. GR01 ; MO06 ).
Experimental study of quantum chaos has been carried out with ultracold atoms
interacting in $\delta$-kicked optical lattices MR94 ; RB95 ; DM95 ; Amm98 ;
Rin00 ; From00 ; Hens03 ; JS04 . To suppress spontaneous emission (SE) and
provide a coherent quantum dynamics atoms in those experiments were detuned
far from the optical resonance. Adiabatic elimination of the excited state
amplitude leads to an effective Hamiltonian for the center-of-mass (CM) motion
GSZ92 , whose 3/2 degree-of-freedom classical analogue has a mixed phase space
with regular islands embedded in a chaotic sea. De Brogile waves of
$\delta$-kicked ultracold atoms have been shown to demonstrate under
appropriate conditions the effect of dynamical localization in momentum
distributions which means the quantum suppression of chaotic diffusion MR94 ;
RB95 ; DM95 ; Amm98 ; Rin00 ; From00 ; Hens03 ; JS04 ; GSZ92 . Decoherence due
to SE or noise tends to suppress this quantum effect and restore classical-
like dynamics KO98 ; Ball99 ; SM00 ; Arcy01 . Another important quantum
chaotic phenomenon with cold atoms in far detuned optical lattices is a chaos
assisted tunneling. In experiments Steck01 ; HH01 ; Steck02 ultracold atoms
have been demonstrated to oscillate coherently between two regular regions in
mixed phase space even though the classical transport between these regions is
forbidden by a constant of motion (other than energy).
The transport of cold atoms in optical lattices has been observed to take the
form of ballistic motion, oscillations in wells of the optical potential,
Brownian motion Chu85 , anomalous diffusion and Lévy flights BB02 ; BB94 ;
JD96 ; KS97 ; ME96 . The Lévy flights have been found in the context of
subrecoil laser cooling BB02 ; BB94 in the distributions of escape times for
ultracold atoms trapped in the potential wells with momentum states close to
the dark state. In those experiments the variance and the mean time for atoms
to leave the trap have been shown to be infinite.
A new arena of quantum nonlinear dynamics with atoms in optical lattices is
opened if we work near the optical resonance and take the dynamics of internal
atomic states into account. A single atom in a standing-wave laser field may
be treated as a nonlinear dynamical system with coupled internal
(electronical) and external (mechanical) degrees of freedom PRA01 ; JETPL01 ;
JETPL02 . In the semiclassical and Hamiltonian limits (when one treats atoms
as point-like particles and neglects SE and other losses of energy), a number
of nonlinear dynamical effects have been analytically and numerically
demonstrated with this system: chaotic Rabi oscillations PRA01 ; JETPL01 ;
JETPL02 , Hamiltonian chaotic atomic transport and dynamical fractals JETP03 ;
PLA03 ; pra07 , Lévy flights and anomalous diffusion PRA02 ; JETPL02 ; JRLR06
. These effects are caused by local instability of the CM motion in a laser
field. A set of atomic trajectories under certain conditions becomes
exponentially sensitive to small variations in initial quantum internal and
classical external states or/and in the control parameters, mainly, the atom-
laser detuning. Hamiltonian evolution is a smooth process that is well
described in a semiclassical approximation by the coupled Hamilton-Schrödinger
equations. A detailed theory of Hamiltonian chaotic transport of atoms in a
laser standing wave has been developed in our recent paper pra07 .
The aim of the present paper is to study dissipative chaotic transport of
atoms in a one-dimensional optical lattice in the presence of SE events which
interrupt coherent Hamiltonian evolution at random time instants. Generally
speaking, deterministic (dynamical) chaos is practically indistinguishable in
some manifestations from a random (stochastic) process. The problem becomes
much more complicated when noise acts on a dynamical system which is chaotic
in the absence of noise. Such systems are of a great practical interest.
Comparatively weak noise may be treated as a small perturbation to
deterministic equations of motion, and one can study in which way the noise
modifies deterministic evolution on different time scales. However, SE is a
specific shot quantum noise that cannot be treated as a weak one because the
internal state may change significantly after the emission of a spontaneous
photon. Special methods are needed to describe correctly the dynamics of a
spontaneously emitting single atom in an optical lattice. The purpose of this
paper is twofold. Our first goal is to give a description of possible regimes
of dissipative atomic transport in the presence of SE and to quantify their
statistical properties. Our secondary intent is a search for manifestations of
the fundamental dynamical instability and Hamiltonian atomic chaos in the
diffusive-like CM motion of spontaneously emitting atoms in a laser standing
wave which can be observed in real experiments.
The paper is organized as follows. In Sec. II we formulate a Monte Carlo
stochastic wavefunction approach to solving semiclassical Hamilton-Schrödinger
equations of motion for a two-level atom in a one-dimensional monochromatic
standing light wave. This approach allows to get the most probabilistic
outcome that can be compared directly with corresponding experimental output
with single atoms. In Sec. III we review briefly our previous results on
Hamiltonian chaotic CM motion which are necessary to quantify and interpret
dissipative dynamics. Sec. IV is devoted to description of possible regimes of
dissipative CM motion of spontaneously emitting atoms in a standing wave.
Monte Carlo simulations of the well-known effects of acceleration,
deceleration, and velocity grouping, and of a novel effect of dissipative
chaotic walking of atoms are presented in this section. Anomalous statistical
properties of dissipative chaotic walking are quantified and discussed in Sec.
V. Whereas Secs. IV and V are devoted to solving the first task of this paper,
in Sec. VI we consider the problem of manifestations of dynamical instability
and Hamiltonian chaos in dissipative atomic transport. We demonstrate
analytically and numerically that character of diffusion of spontaneously
emitting atoms changes qualitatively in the detuning regime where the
underlying Hamiltonian dynamics is chaotic. To observe this effect in a real
experiment with cold atoms in a one-dimensional optical lattice we propose to
measure the linear extent of atomic clouds with different values of their mean
momentum and predict that the extent should increase significantly with those
values of the mean momentum for which the underlying Hamiltonian evolution is
chaotic.
## II Monte Carlo wavefunction modeling of the atomic dynamics
In the frame rotating with the laser frequency $\omega_{f}$, the standard
Hamiltonian of a two-level atom in a strong standing-wave 1D laser field has
the form
$\displaystyle\hat{H}=$
$\displaystyle\frac{\hat{P}^{2}}{2m_{a}}+\frac{1}{2}\hbar(\omega_{a}-\omega_{f})\hat{\sigma}_{z}-$
(1)
$\displaystyle-\hbar\Omega\left(\hat{\sigma}_{-}+\hat{\sigma}_{+}\right)\cos{k_{f}\hat{X}}-i\hbar\frac{\Gamma}{2}\hat{\sigma}_{+}\hat{\sigma}_{-},$
where $\hat{\sigma}_{\pm,z}$ are the Pauli operators for the internal atomic
degrees of freedom, $\hat{X}$ and $\hat{P}$ are the atomic position and
momentum operators, $\omega_{a}$, $\omega_{f}$, and $\Omega$ are the atomic
transition, laser, and Rabi frequencies, respectively, and $\Gamma$ is the
spontaneous decay rate. Internal atomic states are described by the
wavefunction ${|\Psi(t)\closeket}=a(t){|2\closeket}+b(t){|1\closeket}$, with
$a$ and $b$ being the complex-valued probability amplitudes to find an atom in
the excited ${|2\closeket}$ and ground ${|1\closeket}$ states. Note that the
norm of the wavefunction, $|a|^{2}+|b|^{2}$, is not conserved due to non-
Hermitean term in the Hamiltonian.
We use the standard Monte Carlo wavefunction technique Carmichael to simulate
the atomic dynamics with the coupled internal and external degrees of freedom
in an optical lattice. The evolution of an atomic state ${|\Psi(t)\closeket}$
consists of two parts: (i) jumps to the ground state ($a=0$, $|b|^{2}=1$) each
of which is accompanied by the emission of an observable photon at random time
moments with the mean time $2/\Gamma$ (actually, the probability of SE depends
on the atomic population inversion) and (ii) coherent evolution with
continuously decaying norm of the atomic state vector without the emission of
an observable photon. The decaying norm of the state vector is equal to the
probability of spontaneous emission of the next photon. It is convenient to
introduce the new real-valued variables (normalized all the time) instead of
the amplitudes $a$ and $b$ (renormalized after SE events only)
$\displaystyle
u\equiv\frac{2\operatorname{Re}\left(ab^{*}\right)}{\left|a\right|^{2}+\left|b\right|^{2}},\quad
v\equiv\frac{-2\operatorname{Im}\left(ab^{*}\right)}{\left|a\right|^{2}+\left|b\right|^{2}},\quad
z\equiv\frac{\left|a\right|^{2}-\left|b\right|^{2}}{\left|a\right|^{2}+\left|b\right|^{2}},$
(2)
which have the meaning of synphase and quadrature components of the atomic
electric dipole moment and the population inversion, respectively. We stress
that the length of the Bloch vector, $u^{2}+v^{2}+z^{2}=1$, is conserved.
Since we study manifestation of quantum nonlinear effects in ballistic
transport of atoms, when the average atomic momentum is very large as compared
with the photon momentum $\hbar k_{f}$, the translational motion is described
classically by Hamilton equations. The whole atomic dynamics is governed by
the following Hamilton-Schrödinger equations Acta06 ; epl08
$\displaystyle\dot{x}$ $\displaystyle=\omega_{r}p,\quad\dot{p}=-u\sin
x+\sum\limits_{j=1}^{\infty}\rho_{j}\delta(\tau-\tau_{j}),$ (3)
$\displaystyle\dot{u}$ $\displaystyle=\Delta
v+\frac{\gamma}{2}uz-u\sum\limits_{j=1}^{\infty}\delta(\tau-\tau_{j}),$
$\displaystyle\dot{v}$ $\displaystyle=-\Delta u+2z\cos
x+\frac{\gamma}{2}vz-v\sum\limits_{j=1}^{\infty}\delta(\tau-\tau_{j}),$
$\displaystyle\dot{z}$ $\displaystyle=-2v\cos
x-\frac{\gamma}{2}(u^{2}+v^{2})-(z+1)\sum\limits_{j=1}^{\infty}\delta(\tau-\tau_{j}),$
where $x\equiv k_{f}{\lang\hat{X}\rang}$ and $p\equiv{\lang\hat{P}\rang}/\hbar
k_{f}$ are normalized atomic CM position and momentum. The dot denotes
differentiation with respect to the normalized time $\tau\equiv\Omega t$.
Throughout the paper we fix the values of the normalized decay rate
$\gamma\equiv\Gamma/\Omega$ and the recoil frequency $\omega_{r}\equiv\hbar
k_{f}^{2}/m_{a}\Omega$ to be $\gamma=3.3\cdot 10^{-3}$ and
$\omega_{r}=10^{-5}$. This values are similar to those used in experiments
with Na MR94 ; RB95 , Cs Amm98 ; Hood00 and Rb Hens03 cold atoms in a
standing-wave laser field with the maximal Rabi frequency of the order of
$1\div 5$ GHz. So, the normalized detuning between the field and atomic
frequencies, $\Delta\equiv(\omega_{f}-\omega_{a})/\Omega$, is a single
variable parameter. Also we fix the initial conditions as follows:
$x_{0}=v_{0}=u_{0}=0,z_{0}=-1$, and vary the initial momentum $p_{0}$ only. In
Eqs. (3) $\tau_{j}$ are random time moments of SE events and $\rho_{j}$ are
random recoil momenta with the values between $\pm 1$ (1D case). In terms of
the normalized time $\tau$ the rate of occurrence of SE events is
$\gamma(z+1)/2$. At $\tau=\tau_{j}$, the atomic variables change as follows:
$\tau=\tau_{j}\Rightarrow u\to 0,\ v\to 0,\ z\to-1,\ p\to p+\rho_{j},\ -1\leq
p_{j}\leq 1.$ (4)
## III A brief review of Hamiltonian atomic dynamics
In this section we review briefly our main results on Hamiltonian atomic
dynamics (see Refs. PRA01 ; JETPL01 ; JETPL02 ; JETP03 ; JRLR06 ; pra07 )
which will be used in the next sections. In the absence of any losses
($\gamma=0$) the total atomic energy is conserved,
$H\equiv\frac{\omega_{r}p^{2}}{2}+U,\quad U\equiv-u\cos x-\frac{\Delta}{2}z.$
(5)
The corresponding lossless equations of motion with two independent integrals
of motion, the energy $H$ and the length of the Bloch vector, have been shown
JETPL01 ; PRA01 to be chaotic in the sense of an exponential sensitivity to
small variations in initial conditions and/or the control parameters. The CM
motion is governed by the simple equation for a nonlinear physical pendulum
with the frequency modulation JETPLP02
$\ddot{x}+\omega_{r}u(\tau)\sin x=0,$ (6)
where the synchronized component of the atomic dipole $u$ is a function of all
the other atomic variables including the translational ones. Besides the
regular CM motion, namely, oscillations in a well of the optical potential and
a ballistic motion over its hills, we have found and quantified chaotic CM
motion JETPL01 ; PRA01 ; JETPLP02 . On the exact atom-laser resonance with
$\Delta=0$, $u$ is a constant, and the CM performs either regular
oscillations, if $H<|u|$, or moves ballistically, if $H>|u|$.
At $\Delta\neq 0$, the depth of the potential wells changes in course of time,
and atoms may wander in a rigid optical lattice (without any modulations of
its parameters) in a chaotic way with alternating trappings in the wells and
flights of different lengths and directions over the hills. At small
detunings, $|\Delta|\ll 1$, the second term of the potential energy $U$ in Eq.
(5) is small, and $U$ can be approximated by a function of only one internal
variable $u$. In this case we have approximate solutions for $v$ and $z$
$\displaystyle v(\tau)=\pm\sqrt{1-u^{2}}\
\cos\left(2\int\limits_{0}^{\tau}\cos xd\tau^{\prime}+\chi_{0}\right),$ (7)
$\displaystyle z(\tau)=\mp\sqrt{1-u^{2}}\
\sin\left(2\int\limits_{0}^{\tau}\cos xd\tau^{\prime}+\chi_{0}\right),$
where $\chi_{0}$ is an integration constant which is a function of initial
values of $z$ and $u$. Using these solutions one can prove that at
$|\Delta|\ll 1$, $u$ performs shallow oscillations when the atom moves between
the nodes (recall that $u=const$ at $\Delta=0$). These oscillations are
synchronized with the oscillations of $z$, and when an atom approaches any
node with $\cos x=0$, where the strength of the laser field changes the sign,
they slow down (see Eq. (7)). The swing of oscillations of $u$ gradually
increases, and exactly at the node $u$ changes abruptly its value (see Fig.
1). Thus, $u$ is practically a constant between the nodes and it performs a
sudden jump at every node.
In the Raman-Nath approximation, where $x=\omega_{r}p\tau$ and $p=const$, we
have managed to derive the deterministic mapping allowing to compute the value
$u_{m}$ just after crossing the $m$th node
$\displaystyle
u_{m}\simeq\sin\left\\{\frac{\Delta}{\sqrt{1-u^{2}_{m-1}}}\left[\sqrt{\frac{\pi}{\omega_{r}p}}\left(v_{0}\cos\left(\frac{2}{\omega_{r}p}-\frac{\pi}{4}\right)+\right.\right.\right.$
(8)
$\displaystyle\left.\left.\left.+(-1)^{m}z_{0}\sin\left(\frac{2}{\omega_{r}p}-\frac{\pi}{4}\right)\right)+(-1)^{m}z_{0}\right]+\arcsin
u_{m-1}\right\\},$
where $v_{0}$ and $z_{0}$ are the values of $v$ and $z$ at the antinodes of
the standing wave at $x=\pi k$, $k=0,1,2,...$. They are the same at all the
antinodes because in the Raman-Nath approximation $v$ and $z$ are periodic
functions of $x$ (see solution (7)). Formula (8) describes the series of jumps
of two alternating magnitudes (for odd and even $m$). Strictly speaking, (8)
is valid with fast ballistic atoms and not on a very long time scale.
Deviation of the analytic calculations with Eq. (8) from the exact numerical
results is demonstrated in Fig. 1a where we plot the function $u(\tau)$ for a
fast atom with $p_{0}=1900$. It is obvious that the signal is rather regular
but the magnitude of the jumps changes slowly because the Bloch components $v$
and $z$ are not strictly periodic functions of time.
Figure 1b plots $u(\tau)$ in the regime of Hamiltonian chaotic walking. To
quantify chaotic jumps of $u$ we proposed in Ref. pra07 the stochastic map
Figure 1: Time evolution ($\tau$ is in units of $\Omega^{-1}$) of the synphase
component of the electric dipole moment $u$ and the atomic energy $H$ (in
units of $\hbar\Omega$). (a) Regular Hamiltonian dynamics ($p_{0}=1900$,
$\gamma=0$), (b) chaotic Hamiltonian dynamics ($p_{0}=700$, $\gamma=0$), (c)
regular dissipative dynamics ($p_{0}=1900$, $\gamma=0.0033$), (d) chaotic
dissipative dynamics ($p_{0}=700$, $\gamma=0.0033$). In all the panels,
$\Delta=-0.0005$. The initial part of (a) agrees with approximate solution (8)
with $v_{0}=0$, $z_{0}=-1$.
$\displaystyle
u_{m}\simeq\sin\left(|\Delta|\sqrt{\frac{\pi}{\omega_{r}p_{\rm{node}}}}\sin\phi_{m}+\arcsin
u_{m-1}\right),$ (9)
which was derived from the deterministic map (8) by introducing random phases
$\phi_{m}$ $(0\leq\phi\leq 2\pi)$ instead of arguments of trigonometric
functions which may differ significantly from node to node due to strong
variations in the atomic momentum beyond the Raman-Nath approximation. Note
that the value of the momentum at the instant when the atom crosses a node,
$p_{\rm node}=\sqrt{2H/\omega_{r}}$, is approximately the same for all nodes.
The map (9) describes a random Markov process in the $u$ space with $u_{m}$
varying in the range $-1\leq u_{m}\leq 1$. This quantity may be smaller or
larger than the atomic energy $H$ (which is a constant in the Hamiltonian
limit). Since the values of $u$ define the atomic potential energy, its random
walking governs a random walking of atoms in the lattice. The possible regimes
of the Hamiltonian CM motion can be summarized as follows pra07 : At $|u|>H$,
an atom oscillates in one of the potential wells, at $|u|<H$, it moves
ballistically. It can walk chaotically if $0<H<1$. In the process of
Hamiltonian chaotic walking the atom wanders in an optical lattice with
alternating trappings in wells of the optical potential and flights over its
hills changing the direction of motion many times. “A flight” is an event when
the atom passes, at least, three nodes. CM oscillations in a well of the
optical potential is called “a trapping”. The number of node crossings $l$
during a single flight or a single trapping event is a discrete measure of the
length and durations of those events. We have derived in Ref. pra07 the
following formulas for the probability density functions (PDFs) for the flight
and trapping events in the diffusive approximation:
$\displaystyle P_{fl}(l)\simeq\frac{Q(D_{u})}{\arcsin^{3}H}$
$\displaystyle\sum\limits_{j=0}^{\infty}(j+1/2)^{2}*$ (10)
$\displaystyle*\exp\frac{-(j+1/2)^{2}\pi^{2}D_{u}l}{\arcsin^{2}H},$
$\displaystyle P_{tr}(l)\simeq\frac{Q(D_{u})}{\arccos^{3}H}$
$\displaystyle\sum\limits_{j=0}^{\infty}(j+1/2)^{2}*$
$\displaystyle*\exp\frac{-(j+1/2)^{2}\pi^{2}D_{u}l}{\arccos^{2}H}.$
Here $Q$ is a constant, $D_{u}=\Delta^{2}\pi/4\omega_{r}p_{\rm{node}}$ is a
diffusion coefficient in the $u$ space. For comparatively small values of $l$
(i. e., with short flights and trappings), we get from Eq. (10) the power
decay
$P_{fl}\propto P_{tr}\propto l^{-1.5},$ (11)
whereas for large $l$ the decay is exponential. Numerical simulation of the
Hamiltonian equations of motion agrees well with the analytical results (10)
in different ranges of the detunings. A typical PDF for the flight and
trapping events decays initially algebraically and has an exponential tail.
The length of the initial power-law segment is inversely proportional to the
value of the detuning $\Delta$ and can be rather large.
In which way SE changes the statistical properties of the Hamiltonian motion?
Can we find fingerprints of Hamiltonian instability and chaos in the motion of
spontaneously emitting atoms or SE totally suppresses any manifestations of
coherent (but chaotic!) Hamiltonian dynamics? These questions will be
addressed in the next sections.
## IV Dissipative atomic transport in a laser standing wave
The emission of a photon into the continuum of modes of the electromagnetic
field is accompanied by an atomic recoil. The dissipative (friction) force
$F\equiv{\lang\dot{p}\rang}$ (which does not exist in the Hamiltonian system)
depends on the atomic momentum $p$ and the sign of the detuning in a
complicated way Kaz ; PRA05 . The effects of acceleration, deceleration, and
velocity grouping (at $\Delta<0$) are well-known in the literature Kaz ;
Meystre . A novel effect we report in this section is dissipative chaotic
walking. It appears under appropriate conditions that are different from those
specified for Hamiltonian chaotic walking in the preceding section.
To illustrate the possible regimes of dissipative atomic transport in a
standing wave we integrate by the Monte Carlo method dissipative equations of
motion (3) with 2000 atoms whose positions and momenta are distributed in a
quasi-Gaussian manner (Fig. 2a). In Fig. 2b we demonstrate the
Figure 2: Atomic momentum and position distribution illustrating the effects
of atomic acceleration, deceleration, and the velocity grouping: (a) $\tau=0$,
(b) $\tau=10^{5}$, $\Delta=-0.2$, (c) $\tau=10^{5}$, $\Delta=0.1$, (d)
$\tau=10^{5}$, $\Delta=-0.05$. Momentum $p$ is given in units of $\hbar
k_{f}$, the position in units of $k_{f}^{-1}$.
velocity grouping effect at $\Delta=-0.2$ and $\tau=10^{5}$. A large number of
atoms is grouped around two values of the capture momentum $p_{\rm
cap}\simeq\pm 1300$ because of acceleration of slow atoms and deceleration of
the fast ones in the initial ensemble. The slower the atoms are the longer is
the process of the velocity grouping. Note that atoms with $|p|\lesssim 100$,
trapped initially in a well of the optical potential, could not quit the well
up to $\tau=10^{5}$. Contrary to that, at positive values of the detuning fast
atoms are accelerated and slow ones are decelerated. As a result, we observe a
pronounced peak around $x\simeq p\simeq 0$ shown in Fig. 2c at $\Delta=0.1$
and $\tau=10^{5}$.
Dependence of the friction force $F$ on the current atomic momentum $p$ is
shown in Fig. 3 at $\Delta=-0.2$. It has been computed with our main equations
(3) when averaging over seven thousands atoms with different initial momentum.
The function $F(p)$ resembles the behavior of the friction force computed with
another methods
Figure 3: Dependence of the friction force $F\equiv{\lang\dot{p}\rang}$ on the
current atomic momentum $p$ at the detuning $\Delta=-0.2$.
(see Kaz and Fig. 1a in PRA05 ). The friction force decreases up to zero
value and then begin to oscillate with increasing $p$. It has a number of
zeroes (the detailed view is shown in Fig. 3b) like the corresponding
functions in Refs. Kaz ; PRA05 . Zero values of $F$ correspond to
quasistationary values of the momentum which depend on $\Delta$. Some of them
are attractors and atoms with close values of the momentum tend to $p_{\rm
cap}$, another ones are repellors. The attractors and repellors are not
deterministic because of a random nature of SE. Thus, an atoms walks randomly
in the momentum space between different values of the capture momentum $p_{\rm
cap}$. When it reaches a certain value of the capture momentum the atom does
not stop in the momentum space and goes on to fluctuate because of both the
Hamiltonian instability and SE effect.
In the preceding section we described the Hamiltonian chaotic walking that may
occur in the absence of any losses. Dissipation causes additional strong
fluctuations of the momentum. If $\Delta>0$ or if it is negative but
comparatively large, nothing principally new happens to atoms as compared with
the Hamiltonian limit. However, at negative small values of $\Delta$, a
characteristic capture momentum becomes smaller than a typical range of
momentum fluctuations due to atomic recoils. As a result, atoms may change
their direction of motion in an irregular way. Such a dissipative chaotic
atomic walking is illustrated in Fig. 2d at $\Delta=-0.05$ and $\tau=10^{5}$
with the atoms distributed widely in the phase plane. Typical atomic
trajectories are shown in Fig. 4 in the momentum space. Figures 4a and b
illustrate how the friction force near the resonance ($\Delta=-0.001$ and
$\Delta=-0.01$) decelerates atoms with large values of the initial momentum
down to so small values of the capture momentum when the dissipative chaotic
walking becomes possible. With increasing the absolute value of the negative
$\Delta$, the capture momentum increases and the atom changes rarely the
direction of motion (Fig. 4c with $\Delta=-0.1$). Panels d and e in Fig. 4
illustrate the velocity grouping effect at $\Delta=-0.15$ with different
values of the initial momentum.
Figure 4: Typical atomic trajectories in the momentum space: (a)-(c)
dissipative chaotic walking with different statistics of atomic flights,
(d)-(e) the effect of velocity grouping. Note that atoms with very different
initial momentum acquire a close value of the capture momentum.
## V Statistical properties of dissipative chaotic walking
Statistics of Hamiltonian chaotic walking is quantified by the flight and
trapping PDFs (10) with algebraically decaying head segments and exponential
tails whose lengths strongly depend on $\Delta$. We will show in this section
that PDFs for dissipative chaotic walking are even more sensitive to
variations in $\Delta$. Figures 4a and b clearly demonstrate that at very
small detuning $\Delta=-0.001$ long flights dominate, whereas there appears a
number of short flights with larger value of $\Delta=-0.01$. In Fig. 5 we plot
the PDFs $P_{fl}$ for the duration of atomic flights $T$ with different values
of the detuning $\Delta$. At small detunings (Fig. 5a), the length of the
power-law segments depends on $\Delta$ in a similar way as in the Hamiltonian
case (compare this figure with Figs. 5a, 6a, and 7a in Ref. pra07 ). However,
the slope slightly differs from the Hamiltonian slope which is equal to
$-1.5$. The difference in the statistics of dissipative and Hamiltonian
walkings is more evident with larger values of the detuning (Fig. 5b). The
length of the power-law segments increases drastically with increasing
$\Delta$. This effect is absent in Hamiltonian dynamics. The corresponding
slope $\alpha$ decreases with changing the detuning from $\Delta=-0.09$ to
$\Delta=-0.12$ because of the corresponding increase in the length of atomic
flights (see Figs. 4a, b, and c). In Fig. 6 we plot the dependencies of the
mean duration of atomic flights ${\lang T\rang}$ and the slope of the PDF
powerlaw fragments $\alpha$ on the detuning $\Delta$ in the range of its
medium values $-0.12\leq\Delta\leq-0.06$. Both the quantities correlate well
with each other. It means that, changing the value of $\Delta$, one can
manipulate statistical properties of dissipative atomic transport in an
optical lattice. This control is nonlinear in the sense that slightly changing
$\Delta$, say, from $-0.08$ to $-0.12$, we increase the mean duration of
flights in a few orders of magnitude. This effect may be qualitatively
explained as follows. When increasing the absolute value of the negative
detuning, the capture momentum increases but fluctuations of the current
momentum $p$ decrease providing long atomic flights epl08 .
To explain the statistical properties of the dissipative chaotic walking let
us consider the behavior of the quasienergy
Figure 5: The PDFs $P_{\rm fl}$ for the duration of atomic flights $T$ with
(a) small detunings (crosses $\Delta=-0.01$, stars $\Delta=-0.001$, circles
$\Delta=-0.0001$, squares $\Delta=-0.00001$) and (b) medium detunings (stars
$\Delta=-0.09$, $\alpha=-0.77$; circles $\Delta=-0.1$, $\alpha=-0.27$; squares
$\Delta=-0.12$, $\alpha=-0.05$). Straight lines show slopes $\alpha$ of the
power-law fragments of the PDFs in log-log scale. Figure 6: Dependencies of
the logarithms of the mean duration of atomic flights $\langle T\rangle$
(solid line) and of the slope $\alpha$ of the PDF power-law fragments
(squares) on the detuning $\Delta$ (in units of $\Omega$).
$\displaystyle\tilde{H}_{j}$
$\displaystyle\equiv\frac{\omega_{r}}{2}p^{2}-u\cos
x-\frac{\Delta}{2}z-\frac{\Delta\gamma}{4}{\lang
1-z^{2}\rang}(\tau-\tau_{j})=$ (12)
$\displaystyle=H-\frac{\Delta\gamma}{4}{\lang
1-z^{2}\rang}(\tau-\tau_{j}),\quad\tau_{j}<\tau<\tau_{j-1},$
which is equal to the total atomic energy (5) in the absence of relaxation.
Near the resonance, $|\Delta|\ll 1$, $\tilde{H}_{j}$ is almost conserved
between SE events, i. e., in the interval $\tau_{j}<\tau<\tau_{j-1}$. The real
energy $H$ (see Fig. 1d) decreases a little in between in a linear way. The
rate of this decrease is defined by the coefficients of spontaneous emission
$\gamma$, the detuning $\Delta$, and the average probability to find the atom
in the excited state. Both the quantities, $H$ and $\tilde{H}$, changes
abruptly just after SE (because of the corresponding changes in the atomic
variables (4)). Just after emitting a $j$th spontaneous photon at
$\tau=\tau_{j}$, they have the same values. So, we will model the evolution of
the energy as a map $H_{j}\equiv H(\tau_{j}^{+})$ taken at the moments
$\tau_{j}^{+}$ just after SE
$\displaystyle
H_{j}=\tilde{H}_{j}-\tilde{H}_{j-1}+H_{j-1}=H_{j-1}+\omega_{r}p(\tau_{j}^{-})\rho_{j}+$
(13)
$\displaystyle+\frac{\omega_{r}}{2}\rho_{j}^{2}+\frac{\Delta}{2}+u(\tau_{j}^{-})\cos
x(\tau_{j})+$
$\displaystyle+\frac{\Delta}{2}z(\tau_{j}^{-})+\frac{\Delta\gamma}{4}{\lang
1-z^{2}\rang}(\tau_{j}-\tau_{j-1}),$
where the values of the atomic variables $p(\tau_{j}^{-}),u(\tau_{j}^{-})$,
and $z(\tau_{j}^{-})$ are taken at the moments $\tau_{j}^{-}$ just before SE.
They are, in turn, determined by the coherent evolution between SE.
The stochastic map for the atomic energy (13) provides an important
information about the CM motion. It has been shown in Ref. pra07 that atoms
move ballistically, if the atomic energy satisfies to the condition
$H\gtrsim|u|$, whereas at $H\lesssim|u|$ they may change the direction of
motion. The dissipative chaotic walking takes place when the atomic energy $H$
alternatively takes the values larger and smaller than a critical value
$H=|u|$. In the Hamiltonian limit, where the energy is conserved, the problem
of the CM chaotic walking has been reduced to the task of random walking of
the Bloch component $u$ (see Sec. III and Ref. pra07 ). The energy is not
conserved in the presence of relaxation, but the values of $u$ are always
small (see Appendix and Fig. 1c and d). Thus, atoms oscillate in the wells of
the optical potential if $H\lesssim 0$ and move ballistically if $H\gtrsim 0$.
On a time scale exceeding the mean time between SE events $2/\gamma$, the
evolution of energy can be treated as a diffusion process with a drift in the
energetic space. The probability to have the energy $H$ at time $\tau$ is
governed by the Fokker-Planck equation
$\dot{P}(H,\tau)=-2c_{H}\frac{\partial P}{\partial
H}+D_{H}\frac{\partial^{2}P}{\partial H^{2}},$ (14)
where $D_{H}$ is an energy diffusion coefficient and $c_{H}$ is an energy
drift coefficient which can be estimated with the help of Eq. (13) as follows:
$c_{H}\equiv\frac{{\lang
H_{j}-H_{j-1}\rang}}{{\lang\tau_{j}-\tau_{j-}\rang}}={\lang\dot{H}\rang}\simeq\frac{\omega_{r}\gamma}{12}+\frac{\Delta\gamma}{2}.$
(15)
In deriving this formula we adopt the average value of the squared recoil
momentum ${\lang\rho_{j}^{2}\rang}=1/3$ (the projections of the recoil momenta
on the standing-wave axis $x$, $\rho_{j}$, are assumed to be distributed in
the range $\pm 1$ with the same probability), the average value of the
population inversion just before a SE event to be $z(\tau_{j}^{-})=1/2$, the
average value of $z$ over the whole time scale and its mean squared deviation
from 1 to be ${\lang z\rang}=0$ and ${\lang 1-z^{2}\rang}=1/2$, respectively
(see the solution (33) in Appendix). Moreover, neglecting the correlation, we
put ${\lang u\cos x\rang}\simeq{\lang u\rang}{\lang\cos x\rang}=0$, which is
valid if $|\omega_{r}p|\gtrsim\gamma/2$, i. e., when $p\gtrsim 100$ with our
choice of the parameters. Since the first term in (15) is small and may be
neglected, the drift velocity of an atom in the energetic space is
approximately proportional to the detuning $\Delta$, and, therefore in average
atoms accelerate and decelerate at $\Delta>0$ and $\Delta<0$, respectively, as
it should be for $|\Delta|\ll 1$. In the weak Raman-Nath approximation, (32)
and (34), the drift coefficient in the energetic space is simply related to
the friction force $F$ acting upon atoms
$F\equiv{\lang\dot{p}\rang}\simeq\frac{{\lang\dot{H}\rang}}{\omega_{r}p}.$
(16)
The friction force plays the role of a drift coefficient in the momentum
space. Strictly speaking, the weak Raman-Nath approximation is not valid near
the turning points when the atomic velocity is comparatively small. However,
most of the flight time it is valid.
The diffusion coefficient in the energetic space is given by the formula
$\displaystyle\ D_{H}\equiv\frac{{\lang(H_{j}-H_{j-1})^{2}\rang}-{\lang
H_{j}-H_{j-1}\rang}^{2}}{2{\lang\tau_{j}-\tau_{j-1}\rang}},$ (17)
which can be rewritten with the help of (13) as follows:
$\displaystyle
D_{H}\simeq\frac{\gamma\omega_{r}^{2}p^{2}(\tau_{j}^{-})}{{12}}+\frac{{\lang
u^{2}(\tau_{j}^{-})\rang}\gamma}{8}.$ (18)
Using weak Raman-Nath approximation, (32) and (34), the first term can be
replaced by $\gamma\omega_{r}H/6$. Using the estimation (38) for ${\lang
u^{2}(\tau_{j}^{-})\rang}$ in the irregular CM motion regime (see Appendix),
we get the following expression for the energy diffusion coefficient:
$D_{H}^{ch}\simeq\frac{\gamma\omega_{r}H}{{6}}+\frac{\Delta^{2}}{8}.$ (19)
This expression is valid for moderately small momentums ($p\lesssim 1000$)
when the strong Raman-Nath approximation cannot be applied. In the process of
dissipative chaotic walking, the probability to get higher values of the
momentums is almost zero.
Now we will try to derive analytically a distribution of the durations $T$ of
atomic flights in the process of dissipative chaotic walking. In fact, it is a
problem of the first passage time for the atomic energy $H$ to return to its
zero value. Recall that at small detunings we have $H\simeq 0$ in the very
beginning of every flight. In course of time $H$ can reach rather large
values, and it returns to zero at the end of the flight. If the random jumps
of the energy would be symmetric ($c_{H}=0$), the probability to find a flight
with duration $T$ would be proportional to $T^{-1.5}$, where the exponent
$-1.5$ does not depend on the diffusion coefficient. This conjecture follows
from the known theorem in probability theory. More general result (see chapter
XIV in Ref. Feller ) proves that in the case of an asymmetric random walking
in the energetic space ($c_{H}\neq 0$) the PDF for the flight durations in
configuration space is
$P_{\rm{fl}}\propto e^{-c_{H}^{2}T/D_{H}}T^{-1.5},$ (20)
if the drift and diffusion coefficients in the Fokker-Planck equation for the
random walking are assumed to be constants. This formula gives a distribution
of the flight durations with a power-law fragment followed by an exponential
tail and agrees qualitatively with the exact numerical computations of
$P_{fl}$ shown in Fig. 5a for a few values of the detuning $\Delta$. The main
disadvantage of this formula is that (20) does not depend on $\Delta$ as the
exact PDFs in Fig. 5a. At very small $\Delta=-10^{-5}$, the PDF, shown by
squares in Fig. 5a, decays mostly algebraically, whereas at larger values of
the detuning the power-law fragments are much shorter. A discrepancy between
the analytical and numerical PDFs arises because we assumed in deriving (20)
that $D_{H}$ and $c_{H}$ do not depend on the energy $H$. In fact, it is not
the case for small values of the momentum $p$, and a more accurate formula for
$P_{fl}(T)$ is required.
The PDFs for Hamiltonian (10) and dissipative (20) transport are similar in
the sense that both $P_{fl}$ contain power-law fragments followed by
exponential tails, but the origin of each statistics is different. In the
Hamiltonian limit the statistics is governed by the behavior of $u$, not the
energy, as in the dissipative system. A turnover from a power law to an
exponential decay in the Hamiltonian case is explained by a boundedness of
$|u|\lesssim 1$, whereas in the dissipative system it is explained by a
negative drift of the energy $H$. Each of the factors prevents the
corresponding randomly walking quantity to go far away from its critical value
(at which the atoms can change the directions of motion) decreasing the
probability of long flights in at exponential way.
## VI Manifestation of Hamiltonian chaos in dissipative atomic transport
In the absence of SE the atomic dynamics can be regular or chaotic depending
on the initial conditions and/or the detuning. In experiments one measures
statistical characteristics of spontaneously emitting atoms. Is there a
correlation between those characteristics and the underlying Hamiltonian
dynamics? Can we find any manifestations of Hamiltonian instability, chaos,
and order, in the diffusive-like dissipative atomic transport? These questions
will be addressed in the present section.
The common quantitative criterion of deterministic chaos, the maximal Lyapunov
exponent $\lambda$, is a measure of a divergence of two trajectories in the
phase space with close initial conditions LL83 . To quantify probability of
chaos in the mixed Hamiltonian dynamics, when $\lambda=0$ with some values of
$p_{0}$ and $\lambda>0$ with another values of $p_{0}$, we introduce a
probabilistic measure of Hamiltonian chaos
$\Lambda\equiv{\lang 2\Theta(\lambda)-1\rang},$ (21)
where $\Theta(\lambda)$ is a Heaviside function ($\Theta=0$ if $\lambda<0$,
$\Theta=1/2$ if $\lambda=0$, and $\Theta=1$ if $\lambda>0$). The probability
of Hamiltonian chaos $\Lambda$ is computed by averaging over a large number of
atomic trajectories with different values of $p_{0}$. If all the trajectories
in the set turn out to be stable, one gets $\Lambda=0$, and if all they are
exponentially unstable, then $\Lambda=1$. One gets $0<\Lambda<1$, if some
trajectories in the set are stable but the other ones are not. The magnitude
of $\Lambda$ is proportional to the fraction of trajectories with positive
$\lambda$s.
To examine manifestations of the underlying Hamiltonian dynamics in
dissipative transport in is convenient to consider atomic diffusion not in the
energetic but in the momentum space. The momentum diffusion coefficient, which
is a measure of momentum fluctuations, can be written with the help of (17)
and (32) as follows:
Figure 7: Correlation between the average momentum diffusion coefficient in a
log-log scale $D_{p}$ (in units of $\hbar^{2}k^{2}_{f}\Omega$) and probability
of Hamiltonian chaos $\Lambda$ in their dependencies on current atomic
momentum $p$ (in units of $\hbar k_{f}$) at $\Delta=-0.01$ and
$\Delta=-0.0005$. Dashed lines with the slopes $~{}p^{-2}+{\rm const}$ and
$~{}p^{-1}+{\rm const}$ are theoretical curves (23) and (24) valid in the
regimes of Hamiltonian chaos and order, respectively. Solid line is a
theoretical curve (25) derived to fit the exact numerical results. An abrupt
change in the decay laws for $D_{p}$ occurs for those values of $p$ where the
transition from order to chaos takes place in the underlying Hamiltonian
dynamics.
$\displaystyle
D_{p}\simeq\frac{D_{H}}{\omega_{r}^{2}p^{2}}\simeq\frac{\gamma}{{12}}+\frac{{\lang
u^{2}(\tau_{j}^{-})\rang}\gamma}{8\omega_{r}^{2}p^{2}}.$ (22)
Using the formula (19), we get $D_{p}$ in the regime of chaotic oscillations
of the Bloch component $u$
$D_{p}^{ch}\simeq\frac{\gamma}{{12}}+\frac{\Delta^{2}}{8\omega_{r}^{2}p^{2}}.$
(23)
The momentum diffusion coefficient $D_{p}$ is computed with the main equations
(3) in the following way. The range of all possible values of the atomic
energy $H$ (5) is partitioned in a large number of bins. For a large number of
initial conditions (in fact, we change only the initial momentum $p_{0}$
keeping the other conditions to be fixed), after any $j$th SE event we compute
the difference $H_{j}-H_{j-1}$ and the squared difference
$(H_{j}-H_{j-1})^{2}$. They are random values, but their statistics depend on
the preceding energy value $H_{j-1}$. So we calculate the histograms of the
average values of ${\lang H_{j}-H_{j-1}\rang}$ and
${\lang(H_{j}-H_{j-1})^{2}\rang}$ as functions of energy $H_{j-1}$. After that
we can compute the energy diffusion coefficient $D_{H}$ (17) which, being
divided by $\omega_{r}^{2}p^{2}$, yields the momentum diffusion coefficient
$D_{p}$ which is better to present as a function of the current momentum
$p\simeq\sqrt{2H_{j-1}/\omega_{r}}$.
The main result in this section is illustrated with Fig. 7. In the upper left
and right panels the dependencies of the momentum diffusion coefficient
$D_{p}$ on the current momentum $p$ are plotted in a log-log scale for
$\Delta=-0.01$ and $\Delta=-0.0005$, respectively. In both the cases, we put
$\gamma=0.0033$. These plots should be compared with the corresponding lower
panels where the probability of the Hamiltonian chaos $\Lambda$ is plotted
against $p$ with $\gamma=0$ (i. e. in the Hamiltonian limit of Eqs. (3)). It
is evident that the character of the momentum diffusion changes abruptly at
those values of the current momentum where a transition from chaos to order
occurs in the underlying Hamiltonian dynamics. Such a turnover takes place in
a range of small negative detunings and is a manifestation of the
peculiarities of the underlying Hamiltonian evolution in the diffusive-like
dissipative transport of atoms in a standing-wave laser field. We may conclude
that in spite of random atomic recoils due to SE the chaotic (regular)
dynamics between the acts of SE clearly manifests itself in the behavior of
the measurable characteristic of the atomic transport, the momentum diffusion
coefficient $D_{p}$. The behavior of $D_{p}$ in the range of $p$, where the
underlying Hamiltonian evolution is chaotic, is well described by the formula
(23) with $D_{p}^{ch}\sim p^{2}+const$ (see both the upper panels in Fig. 7
where this dependence is shown by dashed lines).
However, the formula (23) does not work in the regimes when the underlying
Hamiltonian dynamics is mixed or regular because in deriving it we supposed
fully chaotic behavior of $u$. We have managed to estimate analytically
$D_{p}$ in the Hamiltonian regular regime at extremely small values of the
detuning $|\Delta|\lll 1$ and for atoms whose momentum is so large that we can
neglect its fluctuations between SE events (the exact Raman-Nath approximation
with $x=\omega_{r}p\tau$). Figure 1a illustrates the ladder-like behavior of
$u$ which is descibed by the deterministic mapping (8) on a comparatively
short timescale. To get $D_{p}^{RN}$ from Eq. (22) we use the expression (37)
for $u^{2}(\tau_{j}^{-})$ derived in Appendix
$D_{p}^{RN}\simeq\frac{\gamma}{{12}}+\frac{\Delta^{2}}{8\omega_{r}p\gamma\pi}.$
(24)
Thus, we derived the formulas for the momentum diffusion coefficient $D_{p}$
in the regimes of Hamiltonian chaos (23) with $\Lambda=1$ and Hamiltonian
order (24) with $\Lambda=0$. In a general case with $0\leq\Lambda\leq 1$, we
will assume a linear combination
$\displaystyle D_{p}\simeq(1-\Lambda)D_{p}^{RN}+\Lambda D_{p}^{ch}\simeq$ (25)
$\displaystyle\simeq\frac{\gamma}{{12}}+\frac{\Delta^{2}}{8\omega_{r}p}\left(\frac{1-\Lambda}{\gamma\pi}+\frac{\Lambda}{\omega_{r}p}\right).$
This function, shown by the solid line in the right upper panel in Fig. 7,
fits rather well exact numerical results.
We would like to end this section with the proposal of a simple experimental
scheme to observe our main theoretical and numerical result on an abrupt
change in the character of atomic diffusion in a laser standing wave under
conditions corresponding to two different regimes of the underlying
Hamiltonian evolution, chaotic and regular ones. Let us consider a small
atomic cloud moving in one direction with an average atomic momentum ${\lang
p_{c}\rang}$. Initial position and momentum distribution are assumed to be a
Gaussian with the standard deviations $\sigma^{2}_{x}\equiv{\lang(x-{\lang
x_{c}\rang})^{2}\rang}$ and $\sigma^{2}_{p}\equiv{\lang(p-{\lang
p_{c}\rang})^{2}\rang}$. The momentum diffusion coefficient is
$D_{p}=\frac{d(\sigma^{2}_{p})}{2d\tau}.$ (26)
The temperature of gas and its rate of heating in Kelvins per second are
$T\equiv\frac{2{\lang
E_{k}\rang}}{k_{B}}=\frac{\hbar^{2}k^{2}_{f}\sigma^{2}_{p}}{m_{a}k_{B}},\quad\frac{dT}{dt}=\frac{2\hbar^{2}k^{2}_{f}\Omega
D_{p}}{m_{a}k_{B}},$ (27)
where $E_{k}$ is a kinetic energy of atoms (in Joules) in the reference frame
moving with the CM of the cloud. It follows from (27) that the rate of heating
is proportional to $D_{p}$ whose behavior is different in the regimes of
regular and chaotic underlying Hamiltonian dynamics.
The linear extent of the cloud in meters is $L_{X}\equiv 2\sigma_{x}/k_{f}$.
On a comparatively short time scale, $\tau\ll|({\lang p_{c}\rang}/F)|$, and
low temperatures $\sigma_{p}\ll|{\lang p_{c}\rang}|$, $D_{p}$ is approximately
the same for all the atoms in the cloud because the CM velocity could not
change significantly under the action of the friction force $F$ during the
observation time. Using the first equation in the set (3) and the Eq.(26), we
obtain
$\sigma^{2}_{x}\simeq\sigma^{2}_{x}(0)+\frac{1}{2}\omega^{2}_{r}\sigma^{2}_{p}(0)\tau^{2}+\frac{2}{3}D_{p}\omega^{2}_{r}\tau^{3}.$
(28)
Figure 8: A log-log dependence of the cloud’s linear extent $L_{X}$ (in
microns) on the average initial momentum ${\lang p_{c}\rang}$ of atoms in a
cloud at two moments of time. The analytic dashed lines were computed with the
formula (28) with $D_{p}=D_{p}^{ch}$ valid in the chaotic regime on a short
time scale. Note an abrupt change in the decay of $L_{X}({\lang p_{c}\rang})$
in the range of ${\lang p_{c}\rang}\simeq 1200$ where a chaos-order transition
takes place in the underlying Hamiltonian motion. $\Delta=-0.01$,
$2\sigma_{x}(0)=0.5$, $2\sigma_{p}(0)=5$, wavelength
$\lambda_{f}=2\pi/k_{f}=850$ nm.
We have computed $L_{X}$ with that formula with $D_{p}$ given by (23) and
compare the result with numerical simulation of Eqs. (3) for a number of
atomic clouds with different initial values of ${\lang p_{c}\rang}$. In Fig. 8
the dependence $L_{X}({\lang p_{c}\rang})$ is plotted for $\Delta=-0.01$ at
two moments of time. The analytic dashed lines fit well the exact numerical
results in the range ${\lang p_{c}\rang}\lesssim 1200$ where the underlying
Hamiltonian dynamics is chaotic (see the left column in Fig. 7). Note an
abrupt change in the decay of $L_{X}({\lang p_{c}\rang})$ beginning with those
values of ${\lang p_{c}\rang}\simeq 1200$ where the Hamiltonian dynamics
becomes more regular. Since the linear extent of the atomic clouds changes
abruptly at the chaos-order border one may conclude that in real experiments
the value of $L_{X}$ should increase sharply with those values of the average
cloud momentum ${\lang p\rang}$ for which the underlying Hamiltonian evolution
is chaotic.
## VII Conclusion
Coherent evolution of the atomic state in a near-resonant standing-wave laser
field is interrupted by SE events at random moments of times. The Hamiltonian
evolution between these events has been shown previously (for a summary of
Hamiltonian theory for cold atoms in a 1D optical lattice see Ref. pra07 ) to
be regular or chaotic depending on the values of the detuning $\Delta$ and the
initial momentum $p_{0}$. We stress that dynamical chaos may happen without
any noise and any modulation of the lattice parameters. It is a specific kind
of dynamical instability in the fundamental interaction between the matter and
radiation.
In reality Hamiltonian chaos is masked by random events of SE. The behavior of
spontaneously emitting atoms in the detuning and momentum regimes where the
underlying Hamiltonian dynamics is chaotic may be called stochastic chaos. We
have specified and quantified two regimes of the stochastic chaos, namely,
random walking and dissipative ballistic transport. In the first regime, atoms
wander in an optical lattice in a random way performing flights in both the
directions with the PDFs strongly depending on the detuning (see Figs. 5 and
6). In the ballistic regime, atoms move in the same direction with momentum
fluctuations caused both by the Hamiltonian instability as well as SE events.
It has been shown in our numerical experiments and confirmed analytically that
the character of momentum diffusion changed abruptly in the regime where the
underlying Hamiltonian dynamics proved to be chaotic. A clear correlation
between the decay of the momentum diffusion coefficient $D_{p}$ and
probability of Hamiltonian chaos $\Lambda$ has been found (Fig. 7). In order
to observe the manifestation of Hamiltonian chaos in real experiments we
proposed to measure a linear extent of atomic clouds $L_{X}$ in a 1D optical
lattice and predicted a significant increase in $L_{X}$ for the atomic clouds
with $\Lambda\simeq 1$.
In conclusion we would like to discuss some possible applications of the
theory developed and the results obtained. A sensitive dependence of
statistical properties of dissipative chaotic walking and ballistic transport
on the values of the detuning $\Delta$ provides a possibility to manipulate
atomic CM motion by changing $\Delta$. For example, one can increase the mean
duration of atomic flights in three orders of magnitude by changing $\Delta$
only by thirty percents (see Fig. 6).
Cold atoms in optical lattices is an ideal system to study different phenomena
in statistical physics. Besides dynamical chaos, the phenomena of stochastic
resonance has been observed in a near-resonant optical lattice SC02 . Another
phenomenon of considerable current interest is cold atom ratchets ML99 ; SS03
; JG04 ; GB07 ; GP07 . A ratchet is a spatially periodic device which is able
to produce a directed transport of particles in the absence of a net bias (i.
e., when the time- and space-averaged forces are zero). In order to realize
the ratchet effect it is necessary to break time or/and spatial symmetries
which generate a countermoving partner to each trajectory FY00 . Different
classes of the ratchets have been experimentally realized with cold atoms in
optical lattices ML99 ; SS03 ; JG04 . The interrelation of Hamiltonian chaos
and SE noise, found in this paper, provides an additional possibility to
create and manipulate directed transport of atoms in rigid optical lattices.
## Appendix
We work in the regime of small detunings
$|\Delta|\ll 1,$ (29)
moderate mean atomic velocities
${\lang|\omega_{r}p|\rang}\sim\gamma/2\ll 1,$ (30)
and diffusive motion
$\tau\gg 2/\gamma.$ (31)
Due to (29), we may neglect the last term in the potential energy (5) and
adopt the Hamiltonian solutions for (7) between any two acts of SE. The
evolution of $u$ is described by the approximate solutions (8) for the regular
Raman-Nath motion and (9) for the chaotic motion. It follows from the
condition of moderate atomic velocity (30) and solutions (8), (9) that $|u|\ll
1$. In other words, $u$ cannot go far from zero between acts of SE after each
of which $u=0$. Under the conditions (29) and $|u|\ll 1$, the mean kinetic
atomic energy is much greater than the potential one and in this weak Raman-
Nath approximation
$|u\sin x+\frac{\Delta z}{2}|\ll\frac{\omega_{r}p^{2}}{2},$ (32)
the solution (7) for $z$ is simplified
$\displaystyle z(\tau)\simeq\mp\sin\left(2\int\limits_{0}^{\tau}\cos
xd\tau^{\prime}+\chi_{0}\right).$ (33)
In the diffusion regime (31) and in the weak Raman-Nath approximation (32) the
momentum fluctuations between two SE are small in comparison with its
fluctuations over the time scale of atomic transport. So, the atomic momentum
just before SE is equal to its value at the node
$p\simeq p(\tau_{j}^{-})\simeq p_{\rm node}.$ (34)
Now we can get simplified solutions for $u$. In the exact Raman-Nath
condition, $x=\omega_{r}p\tau$, we have from (8) the approximate deterministic
map written in a two-step form
$\displaystyle u_{m}^{RN}\approx 2\Delta\sqrt{\frac{\pi}{\omega_{r}p}}\
v_{0}\cos\left(\frac{2}{\omega_{r}p}-\frac{\pi}{4}\right)+u_{m-2}^{RN}.$ (35)
In the chaotic regime we have from $(\ref{u_m})$ the stochastic map
$\displaystyle
u_{m}^{ch}=|\Delta|\sqrt{\frac{\pi}{\omega_{r}p}}\sin\phi_{m}+u_{m-1}^{ch}.$
(36)
The maps derived enable us to estimate the values of $u(\tau_{j}^{-})$ just
before $j$th act of SE. After the $(j-1)$th SE $u_{0}=u(\tau_{j-1}^{+})=0$,
$u(\tau_{j}^{-})=u_{M}$ is an accumulated value of $u$ after passing $M$ nodes
in the interval $\tau_{j-1}<\tau<\tau_{j}$. The average number of node
crossings can be estimated to be ${\lang M\rang}\simeq
2|\omega_{r}p|/\gamma\pi$. In the exact Raman-Nath limit,
$u_{M}=M(u_{m}-u_{m-2})/2$, and the mean squared value in the regular regime
${\lang u^{2}(\tau_{j}^{-})\rang}_{RN}={\lang
u^{2}_{M}\rang}_{RN}\simeq\frac{4\Delta^{2}\omega_{r}p}{\pi\gamma^{2}}\ {\lang
v^{2}_{0}\rang}\cos^{2}\left(\frac{2}{\omega_{r}p}-\frac{\pi}{4}\right).$ (37)
Beyond the strong Raman-Nath limit, $u(\tau_{j}^{-})=u_{M}$ is a sum of $M$
random numbers which are proportional to $\sin\phi_{m}$. From the probability
theory we get
${\lang u^{2}(\tau_{j}^{-})\rang}_{ch}={\lang u^{2}_{M}\rang}_{ch}\simeq{\lang
M\rang}\Delta^{2}\frac{\pi}{2\omega_{r}p}\simeq\frac{\Delta^{2}}{\gamma}.$
(38)
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|
arxiv-papers
| 2012-01-01T03:05:00 |
2024-09-04T02:49:25.842639
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. Yu. Argonov and S. V. Prants",
"submitter": "Michael Uleysky",
"url": "https://arxiv.org/abs/1201.0319"
}
|
1201.0323
|
GEOMETRIC OPTICS WITH ATOMIC BEAMS
---
SCATTERED BY A DETUNED STANDING LASER WAVE
S.V. Prants, V.O. Vitkovsky, L.E. Konkov
Laboratory of Nonlinear Dynamical Systems,
Pacific Oceanological Institute of the Russian Academy of Sciences,
690041 Vladivostok, Russia, URL: dynalab.poi.dvo.ru
∗Corresponding author e-mail: prants@poi.dvo.ru
###### Abstract
We report on theoretical and numerical study of propagation of atomic beams
crossing a detuned standing-wave laser beam in the geometric optics limit. The
interplay between external and internal atomic degrees of freedom is used to
manipulate the atomic motion along the optical axis by light. By adjusting the
atom-laser detuning, we demonstrate how to focus, split and scatter atomic
beams in a real experiment. The novel effect of chaotic scattering of atoms at
a regular near-resonant standing wave is found numerically and explained
qualitatively. Some applications of the effects found are discussed.
Keywords: atomic scattering, standing wave, optical nanolithography
## 1 Introduction
Manipulation of atoms by light becomes possible due to the dipole forces which
are well described by the semiclassical model with quantum description of
internal atomic transitions induced by a near resonant laser field and
classical description of their center-of-mass motion [1]. For the first time,
the ideas to trap and channel cold atoms with the help of standing laser waves
(SLW) have been proposed by V. Letokhov and his co-workers [2, 3, 4]. The
ability of a SLW to deflect, channel and split atomic beams [5, 6] has been
used for a variety of applications including atom microscopy, interferometry,
isotope separation and optical lithography [6, 7, 8, 9]. Lasers can be used to
manipulate atomic trajectories to create atomic analogues of such familiar
optical phenomena as focusing of light, beam splitting and light scattering.
It is remarkable that now we are able to reverse the roles of light and matter
from their familiar roles. The semiclassical description, used in this paper,
is similar to the geometric optics limit in conventional optics. Atomic
trajectories play the role of light rays with the SLW being a light mask.
In the present paper we intend to demonstrate theoretically and numerically
that adjusting in an experiment only one parameter, the detuning between the
frequencies of a working atomic transition and the SLW, one can explore a
variety of the regimes of the atom-laser interaction to focus, split and
scatter atomic beams. Near the atom-field resonance, where the interaction
between the internal and external atomic degrees of freedom is intense, there
is a possibility to create conditions for chaotic scattering of atoms [10, 11,
12] without any additional efforts like a SLW modulation. It becomes possible
due to the peculiarities of the dipole force in a near-resonant optical
lattice [13, 14, 15, 16].
## 2 Focusing, splitting, bunching and scattering of atomic beams
### 2.1 Equations of motion
A beam of two-level atoms in the $z$ direction crosses a SLW laser field with
optical axis in the $x$ direction. The laser beam has the Gaussian profile
$\exp[-(z-z_{0})^{2}/r^{2}]$ with $r$ being the $e^{-2}$ radius at the laser
beam waist. The characteristic length of the atom-field interaction is
supposed to be $\pm 1.5r$ because the light intensity at $z=\pm 1.5r$ is two
orders of magnitude smaller than the peak value. The longitudinal velocity of
atoms, $v_{z}$, is much larger than their transversal velocity $v_{x}$ and is
supposed to be constant. Thus, the spatial laser profile may be replaced by
the temporal one. The Hamiltonian of a two-level atom in the one-dimensional
SLW can be written in the frame rotating with the angular laser frequency
$\omega_{f}$ as follows:
$\hat{H}=\frac{P^{2}}{2m_{a}}+\frac{\hbar}{2}(\omega_{a}-\omega_{f})\hat{\sigma}_{z}-\\\
\hbar\Omega_{0}\exp[-(t-\frac{3}{2}\sigma_{t})^{2}/\sigma^{2}_{t}]\left(\hat{\sigma}_{-}+\hat{\sigma}_{+}\right)\cos{k_{f}X_{a}}-\frac{i\hbar\Gamma}{2}\hat{\sigma}_{+}\hat{\sigma}_{-},$
(1)
where $\hat{\sigma}_{\pm,z}$ are the Pauli operators for the internal atomic
degrees of freedom, $X_{a}$ and $P$ are the classical atomic position and
momentum, $\Gamma$, $\omega_{a}$, and $\Omega_{0}$ are the decay rate, the
atomic transition and maximal Rabi frequencies, respectively. The simple wave
function for the electronic degree of freedom is
${\left|\Psi(t)\right>}=a(t){\left|2\right>}+b(t){\left|1\right>}$, where
$a\equiv A+i\alpha$ and $b\equiv B+i\beta$ are the complex-valued probability
amplitudes to find the atom in the excited, ${\left|2\right>}$, and ground,
${\left|1\right>}$, states, respectively.
In the semiclassical approximation, atom with quantized internal dynamics is
treated as a point-like particle to be described by the Hamilton–Schrödinger
equations of motion written for the real and imaginary parts of the
probability amplitudes
$\begin{gathered}\dot{x}=\omega_{r}p,\,\dot{p}=-2\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}](AB+\alpha\beta)\sin
x,\\\ \dot{A}=\frac{1}{2}(\omega_{r}p^{2}-\Delta)\alpha-\frac{1}{2}\gamma
A-\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]\beta\cos x,\\\
\dot{\alpha}=-\frac{1}{2}(\omega_{r}p^{2}-\Delta)A-\frac{1}{2}\gamma\alpha+\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]B\cos
x,\\\
\dot{B}=\frac{1}{2}(\omega_{r}p^{2}+\Delta)\beta-\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]\alpha\cos
x,\\\
\dot{\beta}=-\frac{1}{2}(\omega_{r}p^{2}+\Delta)B+\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]A\cos
x,\end{gathered}$ (2)
where $x\equiv k_{f}X_{a}$ and $p\equiv P/\hbar k_{f}$ are scaled atomic
center-of-mass position and transversal momentum, respectively and dot denotes
differentiation with respect to the dimensionless time
$\tau\equiv\Omega_{0}t$. The recoil frequency, $\omega_{r}\equiv\hbar
k_{f}^{2}/m_{a}\Omega_{0}\ll 1$, the atom-laser detuning,
$\Delta\equiv(\omega_{f}-\omega_{a})/\Omega_{0}$, the decay rate
$\gamma=\Gamma/\Omega_{0}$, and the characteristic interaction time,
$\sigma_{\tau}\equiv r\Omega_{0}/v_{z}$, are the control parameters.
Let us introduce instead of the complex-valued probability amplitudes $a$ and
$b$ the following real-valued variables:
$u\equiv 2\operatorname{Re}\left(ab^{*}\right),\quad
v\equiv-2\operatorname{Im}\left(ab^{*}\right),\quad
z\equiv\left|a\right|^{2}-\left|b\right|^{2},$ (3)
where $u$ and $v$ are synchronized and quadrature components of the atomic
electric dipole moment, respectively, and $z$ is the atomic population
inversion. In the absence of any losses ($\gamma=0$), Eqs. (2) can be cast in
the form
$\begin{gathered}\dot{x}=\omega_{r}p,\quad\dot{p}=-u\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]\sin
x,\quad\dot{u}=\Delta v,\\\ \dot{v}=-\Delta
u+2\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]z\cos
x,\quad\dot{z}=-2\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]v\cos
x.\end{gathered}$ (4)
The system (4) has two integrals of motion, namely, the total energy
$H\equiv\frac{\omega_{r}}{2}p^{2}-u\cos x-\frac{\Delta}{2}z,$ (5)
and the length of the Bloch vector, $u^{2}+v^{2}+z^{2}=1$, whose conservation
follows immediately from Eqs. (3).
Equations (4) constitute a nonlinear Hamiltonian autonomous system with two
and half degrees of freedom which, owing to the two integrals of motion, move
on a three-dimensional hypersurface with a given energy value $H$. In general,
motion in a three-dimensional phase space in characterized by a positive
Lyapunov exponent $\lambda$, a negative exponent equal in magnitude to the
positive one, and zero exponent [17]. The maximal Lyapunov exponent
characterizes the mean rate of the exponential divergence of initially close
trajectories and serves as a quantitative measure of dynamical chaos in the
system. The values of the maximal Lyapunov exponent in dependence on the
detuning, the recoil frequency and the initial atomic momentum have been
computed in Refs. [14, 15].
There are different regimes of the center-of-mass motion along the SLW optical
axis [12, 15]. In dependence on the initial conditions and the values of the
control parameters, atoms may oscillate in a regular or a chaotic way in wells
of the optical potential or move ballistically over its hills with regular or
chaotic variations of their velocity. Chaotic motion with a positive value of
the maximal Lyapunov exponent becomes possible in a narrow range of the
detuning values, $0<|\Delta|<1$ [15]. At $\Delta=0$, the synchronized
electric-dipole component, $u$, becomes a constant. That implies the
additional integral of motion in the Hamiltonian version (4) of Eqs. (2) and
the regular motion with zero maximal Lyapunov exponent. Far off the resonance,
at $|\Delta|>1$, the motion is regular both in the trapping and ballistic
modes.
It is remarkable that there is a specific type of motion, chaotic walking in a
deterministic optical potential, when atoms can change the direction of motion
alternating between flying through the SLW and being trapped in its potential
wells. We would like to stress that the local instability produces chaotic
center-of-mass motion in a rigid SLW without any modulation of its parameters.
Chaotic walking occurs due to the specific behavior of the Bloch-vector
component of a moving atom $u$ whose shallow oscillations between the SLW
nodes are interrupted by sudden jumps with different amplitudes while atom
crosses each node of the SLW [15]. It looks like a random like shots happened
in a fully deterministic environment. It follows from the second equation in
the set (4) that those jumps result in jumps of the atomic momentum while
crossing a node of the SLW. If the value of the atomic energy is close to the
separatrix one, the atom after the corresponding jump-like change in $p$ can
either overcome the potential barrier and leave a potential well or it will be
trapped by the well, or it will move as before. The jump-like behavior of $u$
is the ultimate reason of chaotic atomic walking along a rigid SW.
The total atomic energy (5) consists of the kinetic one,
$K=\omega_{r}p^{2}/2$, and the potential one, $U=-u\cos x-z\Delta/2$. The
optical potential changes its depth in course of time. Averaging over fast
oscillations of the internal atomic variables, we get the averaged potential
$\bar{U}=-\bar{u}\cos x-\bar{z}\Delta/2$ that can be used to explain why atoms
move in such or another way.
At small detunings $|\Delta|\ll 1$, the potential is approximately $U\simeq
u\cos x$. If $K(\tau=0)>|U_{\rm max}|=1$, then the atom will move
ballistically. This occurs if the initial atomic momentum, $p_{0}$, satisfies
to the condition $p_{0}>\sqrt{2/\omega_{r}}$. If the initial conditions are
chosen to give $0\leq K(\tau=0)+U(\tau=0)\leq 1$, the corresponding atoms with
$0\lesssim p_{0}\lesssim\sqrt{2/\omega_{r}}$ are expected to move chaotically
at the appropriate values of $\Delta$.
### 2.2 Focusing and splitting
In this section we demonstrate how to focus and split atomic beams crossing a
Gaussian laser beam by varying only one of the control parameters, the atom-
field detuning $\Delta$. Firstly, we perform simulation with a negligible
probability of spontaneous emission and solve the Hamiltonian equations of
motion (4) at comparatively large value of the detuning, $\Delta=1$. To be
concrete we take as an example calcium atoms with the working intercombination
transition $4^{1}S_{0}-4^{3}P_{1}$ at $\lambda_{a}=657.5$ nm, the recoil
frequency $\nu_{\rm rec}\simeq 10$ KHz, and the lifetime of the excited state
$T_{\rm sp}=0.4$ ms. Taking the maximal Rabi frequency to be
$\Omega_{0}/2\pi=2\cdot 10^{7}$ Hz, the radius of the laser beam $r=0.3$ cm,
and the mean longitudinal velocity $v_{z}=10^{3}$ m/s, the interaction time is
estimated to be $0.9$ ms, longer than the atomic lifetime. The normalized
recoil frequency is $\omega_{r}=4\pi\nu_{\rm rec}/\Omega_{0}=10^{-3}$ and the
normalized characteristic time is $\sigma_{\tau}=400$.
Trajectories for 50 calcium atoms to be prepared in the ground states
($u_{0}=v_{0}=0$, $z_{0}=-1$) with the same initial momentum, $p_{0}=10$, and
initial positions in the range $-\pi/10\leq x\leq\pi/10$ are shown in Fig. 1.
In units of the optical wavelength, $X=x/2\pi$, this range is $-0.05\leq X\leq
0.05$. The focusing occurs at those moments of time when the average
transverse momentum in the atomic beam is approximately equal to zero. If one
turns off the laser at one of these moments, it becomes possible to reduce the
beam width practically in ten times. The reason of focusing is simple. It is
well known [4] that at positive blue detunings atoms are attracted to the
nodes of the SLW where the minima of the optical potential are situated at
$\Delta>0$. The first node the atoms reach at $p_{0}>0$ is situated at
$X=1/4$. The initial kinetic atomic energy, $K_{0}=0.05$, is not enough to
overcome the potential barrier whose depth can be estimated to be $\simeq
0.35$ because the simulation gives $\bar{u}\simeq 0$ and $\bar{z}\simeq-0.7$.
So, all the atoms in the beam oscillate in the first potential well in the
$x$-direction around the first node. The initial width of the beam, $\delta
X_{0}=0.1$, is gradually reduced because in course of time the atoms with
initial negative positions catch up with the ones with positive $X_{0}$ near
the first turning point where the average beam momentum is close to zero. The
time interval of the atomic interaction with the SLW field is estimated to be
$3\sigma_{\tau}=1200$. So, the atoms leave the potential well after that time
and move freely (see Fig. 1).
In order to take into account spontaneous emission we use the standard
stochastic wave-function technique [18, 19, 20] for solving Eqs. (2). The
integration time is divided into a large number of small time intervals
$\delta\tau$. At the end of the first interval, $\tau=\tau_{1}$, the
probability of spontaneous emission,
$s_{1}=\gamma\delta\tau|a_{\tau_{1}}|^{2}/(|a_{\tau_{1}}|^{2}+|b_{\tau_{1}}|^{2})$,
is computed and compared with a random number, $\varepsilon$, from the
interval $[0,1]$. If $s_{1}<\varepsilon_{1}$, then one prolongs the
integration but renormalizes the state vector in the end of the first interval
at $\tau=\tau_{1}^{+}$:
$a_{\tau_{1}^{+}}=a_{\tau_{1}}/\sqrt{|a_{\tau_{1}}|^{2}+|b_{\tau_{1}}|^{2}}$
and
$b_{\tau_{1}^{+}}=b_{\tau_{1}}/\sqrt{|a_{\tau_{1}}|^{2}+|b_{\tau_{1}}|^{2}}$.
If $s_{1}\geq\varepsilon_{1}$, then the atom emits a spontaneous photon and
jumps to the ground state at $\tau=\tau_{1}$ with
$A_{\tau_{1}}=\alpha_{\tau_{1}}=\beta_{\tau_{1}}=0$, $B_{\tau_{1}}=1$. Its
momentum in the $x$ direction changes for a random number from the interval
$[0,1]$ due to the photon recoil effect, and the next time step commences.
We simulate lithium atoms with the relevant transition $2S_{1/2}-2P_{3/2}$,
the corresponding wavelength $\lambda_{a}=670.7$ nm, recoil frequency
$\nu_{\rm rec}=63$ KHz, and the decay time $T_{\rm sp}=2.73\cdot 10^{-8}$ s.
With the maximal Rabi frequency $\Omega_{0}/2\pi\simeq 126$ MHz and the radius
of the laser beam $r=0.05$ cm one gets $\omega_{r}=10^{-3}$,
$\sigma_{\tau}=400$, and $\gamma=0.05$. Trajectories for 50 spontaneously
emitting atoms under the same conditions as in Fig. 1a are shown in Fig. 1b.
As expected, spontaneous emission destroys in part the effect of focusing.
However, the atoms move more or less coherently because spontaneous emission
events are comparatively rare at $\Delta=1$.
Figure 1: (a) Focusing the atomic beam with a long lifetime of the excited
state. (b) The effect of spontaneous emission on the focusing. The detuning is
$\Delta=1$ in both the cases. The atomic position $X$ is in units of the
optical wavelength.
Figure 2: Splitting the atomic beam (a) without and (b) with spontaneous
emission. The detuning is $\Delta=-1$.
The other effect, we would like to demonstrate with atomic beams crossing the
SLW, is a splitting of the beam. To do this one needs to choose such the value
of the detuning in order that some atoms in the beam would be trapped in the
first well of the optical potential but another ones could overcome the
barrier and leave that well. It is possible to split atomic beams as at
positive and negative values of the detuning. As an example, we demonstrate in
Fig. 2 the effect of splitting at $\Delta=-1$ for atoms without and with
spontaneous emission. It is seen that spontaneous emission changes slightly
the effect because a few atoms may leave the potential wells due to random
recoils.
## 3 Bunching and chaotic scattering of atoms
The ability of blue and red detuned lasers to attract atoms to the nodes and
antinodes of the SLW, respectively, can be used to create periodic structures
composed of atoms deposited on substrates in the process of optical
nanolithography [7, 8, 9]. To simulate a real experiment we consider a beam
with $N_{0}=10^{5}$ calcium atoms with the initial Gaussian distribution (with
the rms $\sigma_{x}=\sigma_{p}=2$ and the average values $x_{0}=0$ and
$p_{0}=10$) and compute their distribution against the SLW at a fixed moment
of time $\tau=1000$. The bunching of atoms at the SLW nodes at $\Delta=1$
(blue detuning) is shown in Fig. 3a where the atomic density, $n=N(X)/N_{0}$,
is plotted along the optical axis $X$ at $\tau=1000$. The same effect, but
with the atoms bunching around the SLW antinodes (red detuning,
$\Delta=-0.2$), is shown in Fig. 3b. In both the cases we get a periodic
atomic relief with the period $\lambda_{f}/2$ the width of which is restricted
by the time the atoms interact with the Gaussian laser beam.
Figure 3: The effect of bunching of $10^{5}$ calcium atoms around (a) the SLW
nodes (blue detuning, $\Delta=1$) and (b) the antinodes (red detuning,
$\Delta=-0.2$). The plot of atomic density $n=N(X)/N_{0}$ at the fixed moment
of time $\tau=1000$.
The problem we consider resembles the scattering process with particles
entering an interaction region along completely regular trajectories and
leaving it along asymptotically regular trajectories. It is known from many
studies in celestial mechanics, fluid dynamics and other disciplines that
under certain conditions the motion inside the interaction region may have
features that are typical for dynamical chaos, (homoclinic and heteroclinic
tangles, fractals, strange invariant sets, positive finite-time Lyapunov
exponents, etc.) although the particle’s trajectories are not chaotic in a
rigorous sense because chaos is strictly defined as an irregular motion over
infinite time. It has been found [21, 22, 23, 24] that transient Hamiltonian
chaos in the interaction region occurs due to existence of, at least, one
nonattractive chaotic invariant set consisting of an infinite number of
localized unstable periodic orbits and aperiodic orbits. This set possesses
stable and unstable manifolds extending in the phase space into the regions of
regular motion. The particles with the initial positions close to the stable
manifold follow the chaotic-set trajectories for a comparatively long time,
then deviate from them, and leave the interaction region along the unstable
manifold. It is the common mechanism of chaotic scattering that in our problem
causes the chaotic walking of atoms along the SLW.
Figure 4: The distributions of $10^{5}$ calcium atoms at $\tau=1000$ in (a)
the real and (b) momentum space under the conditions of chaotic scattering at
$\Delta=0.2$.
Figure 5: Comparison of the distributions of $10^{5}$ calcium atoms at
$\tau=1000$ over the phase plane in the regimes of (a) chaotic ($\Delta=0.2$)
and (b) regular scattering ($\Delta=-0.2$).
In Fig. 4a we show the atomic position distribution at $\tau=1000$ in the
regime of the chaotic scattering at $\Delta=0.2$ with $10^{5}$ calcium atoms.
This plot should be compared with Fig. 3a where the atomic position
distribution is shown for regularly scattered atoms at $\Delta=1$. First of
all, the distribution of chaotically scattered atoms has a prominent pedestal
and is much broader. Moreover, it has no such a periodic structure as shown in
Fig. 3a. Only the peaks around the first two SLW nodes are prominent in Fig.
4a. The atomic position distribution in the momentum space in Fig. 4b is much
broader than the one for regularly scattered atoms at $\Delta=1$ (not shown).
Thus, we predict that under the conditions of chaotic scattering there should
appear less contrast and more broadened atomic reliefs as compared to the case
of regular scattering because a large number of atoms are expected to be
deposited between the nodes as a result of chaotic walking along the SLW axis.
The effect is expected to be more prominent under the coherent evolution but
it seems to be observable with spontaneously emitting atoms as well. The
difference between chaotic and regular scattering of atoms at a rigid SLW is
especially prominent on the corresponding phase space portraits shown in Fig.
5 where positions and momenta of $10^{5}$ calcium atoms are plotted at the
fixed time moment.
## 4 Conclusion
We have simulated some geometric optics effects with atomic beams crossing a
SLW in the limit of long relaxation time and with spontaneous emission taken
into account. Trajectories of spontaneously emitting atoms have been simulated
with the help of the standard stochastic wave-function technique [18, 19, 20].
It has been shown that by adjusting the detuning it is possible to focus,
split and scatter atoms. The effects have been explained by a coupling between
external and internal atomic degrees of freedom. The depth of the optical
potential depends on the sign and value of the detuning. Varying $\Delta$, one
can create conditions for focusing, splitting and bunching the atoms. It is
remarkable that near the atom-field resonance we have found the new type of
atomic diffraction at a SLW without any modulation of its parameters that can
be observed in real experiments. That would be the prove of existence of the
novel type of atomic motion, chaotic walking in a deterministic environment.
The effects found could be used in optical nanolithography to fabricate
complex atomic structures on substrates.
We predict that experiments on the scattering of atomic beams at a SLW can
directly image chaotic walking of atoms along the SLW. In a real experiment
the final spatial distribution can be recorded via fluorescence or absorption
imaging on a CCD, commonly used methods in atom optics experiments yielding
information on the number of atoms and the cloud’s spatial size. The other
possibility is a nanofabrication where the atoms after the interaction with
the SW are deposited on a silicon substrate in a high vacuum chamber. In this
case the spatial distribution can be analyzed with an atomic force microscope.
As to the momentum distribution, it can be measured, for example, by a time-
of-flight technique [25]. The modern tools of atom optics enable to create
narrow initial atomic distributions in position and momentum, reduce coupling
to the environment and technical noise, create one-dimensional optical
potentials, and to measure spatial and momentum distributions with high
sensitivity and accuracy.
## Acknowledgments
This work was supported by the Russian Foundation for Basic Research (projects
nos. 09-02-00358 and 09-02-01258), by the Integration grant from the Far-
Eastern and Siberian branches of the Russian Academy of Sciences, and by the
Program “Fundamental Problems of Nonlinear Dynamics”.
## References
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* [3] V.I. Balykin, V.S. Letokhov, Opt. Comm., 64, 151 (1987).
* [4] V. Letokhov, Laser control of atoms and molecules (Oxford University Press, New York, 2007).
* [5] E. Arimondo, A. Bambini, S. Stenholm, Phys. Rev., 24, 898 (1981).
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* [12] S.V. Prants, V.Yu. Sirotkin, Phys. Rev. A, 64, 033412 (2001).
* [13] V. Yu. Argonov, S. V. Prants, JETP, 96, 832 (2003) [ZhETF, 123, 946 (2003)].
* [14] V. Yu. Argonov, S. V. Prants, J. Russ. Laser Res., 27, 360 (2006)
* [15] V. Yu. Argonov, S. V. Prants, Phys. Rev. A, 75, art. 063428 (2007).
* [16] V. Yu. Argonov, S. V. Prants, Phys. Rev. A, 78, art. 043413 (2008).
* [17] L.E. Kon’kov, S. V. Prants, JETP Letters, 65, 833 (1997) [Pis’ma ZhETF, 65, 801 (1997)].
* [18] H. J. Carmichael, An open systems approach to quantum optics (Berlin, Springer, 1993).
* [19] J. Dalibard, Y. Castin, K. Mölmer, Phys. Rev. Lett., 68, 580 (1992).
* [20] R. Dum, P. Zoller, H. Ritsch, Phys. Rev. A, 45, 4879 (1992).
* [21] P. Gaspard, Chaos, Scattering and Statistical Mechanics, Cambridge University Press, Cambridge (1998).
* [22] K.A. Mitchell, J.P. Handley, B. Tighe, J.B. Delos, S.K. Knudson, Chaos, 13, 880 (2003).
* [23] M. Budyansky, M. Uleysky, S. Prants, Physica D, 195, 369 (2004).
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|
arxiv-papers
| 2012-01-01T03:30:11 |
2024-09-04T02:49:25.853118
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. V. Prants, V. O. Vitkovsky, L. E. Konkov",
"submitter": "Michael Uleysky",
"url": "https://arxiv.org/abs/1201.0323"
}
|
1201.0324
|
# Group-theoretical approach to study atomic motion in a laser field
S V Prants Laboratory of Nonlinear Dynamical Systems,
Pacific Oceanological Institute of the Russian Academy of Sciences,
43 Baltiiskaya St., 690041 Vladivostok, Russia,
www.dynalab.poi.dvo.ru prants@poi.dvo.ru
###### Abstract
Group-theoretical approach is applied to study behavior of lossless two-level
atoms in a standing-wave laser field. Due to the recoil effect, the internal
and external atomic degrees of freedom become coupled. The internal dynamics
is described quantum mechanically in terms of the $SU(2)$ group parameters.
The evolution operator is found in an explicit way after solving a single ODE
for one of the group parameters. The translational motion in a standing wave
is governed by the classical Hamilton equations which are coupled to the
$SU(2)$ group equations. It is shown that the full set of equations may be
chaotic in some ranges of the control parameters and initial conditions. It
means physically that there are regimes of motion with chaotic center-of-mass
motion and irregular internal dynamics. It is established that the chaotic
regime is specified by the character of oscillations of the group parameter
characterizing the mean interaction energy between the atom and the laser
field. It is shown that the effect of chaotic walking can be observed in a
real experiment with cold atoms crossing a standing-wave laser field.
###### pacs:
2.20.Sv, 03.65.Fd, 05.45.Mt, 37.10.Jk
## 1 Introduction
In quantum physics the unitary time evolution of a driven quantum system is
described by the evolution operator equation
$i\hbar\frac{d}{dt}\hat{U}(t,t_{0})=\hat{H}\left[{\bf
h}(t)\right]\hat{U}(t,t_{0}),\quad\hat{U}(t_{0},t_{0})=\hat{I},$ (1)
where $\hat{U}_{{\bf h}}(t,t_{0})$ is a time evolution operator, $\hat{H}$ is
a Hamiltonian and ${\bf h}(t)$ is a vector-function of the system’s control
parameters. From the abstract point of view, the evolution equation (1) can be
regarded as a differential equation on the group of dynamical symmetry. By
dynamical symmetry we mean simply that the Hamiltonian can be expressed as a
linear combination of operators belonging to a finite-dimensional Lie algebra
with $n$ basic elements. The parameters, $g_{k}\,,k=1,2,\ldots,n$, of the
respective Lie group satisfy to a set of $n$ first-order ordinary differential
nonlinear equations which depend only on the structure of the algebra and on
$c$-number coefficients of the system’s Hamiltonian or the other governing
operator [1, 2, 3, 4]. Thus, the dynamical group itself may be considered as a
dynamical system.
The dynamical-symmetry and Lie-algebraic approach has been successfully
applied to describe the time evolution of numerous physical systems in
different disciplines extending from classical mechanics [5], classical optics
[6, 7, 8, 9] and quantum mechanics [10, 11, 12, 13, 14] to physics of neutrino
oscillations [15]. As to study of dynamics of laser driven atoms, this
approach has been applied to get Lie algebraic solution of the Bloch equations
[16, 17].
The evolution of an isolated quantum system is regular, and the overlap of any
two different quantum state vectors is a constant in course of time. All the
expectation values of the quantum variables evolve in a quasiperiodic way at
most. It does no matter how complicated is a dynamical symmetry of the quantum
system under consideration and the corresponding Lie algebra. On the other
hand, it is well known that even simple classical systems may be unstable and
demonstrate chaotic behavior [19, 20]. Classical instability is usually
defined as an exponential separation of two initially close trajectories in
time with an asymptotic rate given by the maximal Lyapunov exponent $\lambda$.
Such a behavior is possible because of the continuity of the classical phase
space where the system’s states can be arbitrary close to each other. The
trajectory concept is absent in quantum mechanics, and the quantum phase space
is not continuous due to the Heisenberg uncertainty principle. Perfectly
isolated quantum systems are unitary, and there can be no chaos in the sense
of exponential instability even if their classical limits are chaotic. What is
usually understood under quantum chaos is, in essence, the special features of
the quantum unitary evolution of the system under consideration (no matter how
complicated the evolution is) in the region of its control parameter values
and/or initial conditions at which its classical analogue is chaotic [21, 22,
23, 24]. In fact, it is not a special quantum problem. Any type of propagating
waves (electromagnetic, sound or others), satisfying to a linear wave equation
(which is an analogue of the Scrodinger equation), has the same property. Wave
chaos is the special features of the wave field in the region of control
parameters and/or initial conditions at which its ray analogue is chaotic [25,
26]. Thus, the quantum (wave) chaos problem is partly the problem of quantum
(wave)-classical (ray) correspondence.
Let us describe briefly the interconnection between the dynamical symmetries
and dynamical chaos in physics of the atom-field interaction.
1. 1.
The simplest problem is dynamics of a two-level atom at rest in an external
laser field. From the dynamical symmetry point of view, the $SU(2)$ group,
generated by the corresponding Hamiltonian, is driven by an external force
that is not considered to be a dynamical system. It is the case of an external
driving. The problem has been studied in Ref. [4] where it has been shown that
the evolution of atomic internal variables in a linearly polarized bichromatic
laser field with incommensurate frequencies may be very complicated on the
Bloch sphere albeit regular. It is simply because the dynamics takes place on
the two-dimensional surface of the Bloch sphere.
2. 2.
If we deal with a two-level atom at rest in an ideal cavity and take into
account the response of the atom to the cavity radiation field, the
semiclassical evolution of the coupled atom-field system may be chaotic in the
sense of exponential sensitivity to small variations in initial conditions
and/or parameters [27, 28, 29, 30, 31, 32, 33]. This is the case of so-called
dynamical driving [4] when the $SU(2)$ group, generated by the atomic
Hamiltonian, is driven by another dynamical system, the field one. We deal now
with a quantum system, the atom, which is coupled with a classical system, the
radiation field governed by the Maxwell equations. The resulting Maxwell-
Scrodinger (Bloch) equations constitute the five-dimensional set of nonlinear
ordinary differential equations with two integrals of motion, the total atom-
field energy (which is a constant in the absence of any losses) and the length
of the Bloch vector. The motion in the phase space takes place on a three-
dimensional manifold and may be chaotic due to transverse intersection of
stable and unstable manifolds of hyperbolic points in some ranges of the
control parameters, the values of the maximal Rabi frequency and the coupling
strength [33].
3. 3.
If a two-level atom moves within a standing-wave laser field in an open space,
not in a cavity, the field may be considered as an external driving but one
needs to take into account the atomic recoil effect, i.e. changes of the
atomic momentum after absorption or emission photons. If the atom is not
especially cold, we may treat its translation degree of freedom classically.
It is again the case of the dynamical driving with the $SU(2)$ group driven by
the dynamical system which is now the classical atomic degree of freedom. The
governing Hamilton-Scrodinger equations constitute the five-dimensional set of
nonlinear ordinary differential equations with two integrals of motion, the
atomic total energy, including the kinetic one, and the length of the Bloch
vector. The motion in the phase space takes place on a three-dimensional
manifold and may be chaotic in some ranges of the control parameters, the
values of the maximal Rabi and atomic recoil frequencies. A number of
nonlinear dynamical effects have been analytically and numerically
demonstrated with this system: chaotic Rabi oscillations [34, 35], Hamiltonian
chaotic atomic transport and dynamical fractals [36, 37, 38], Lévy flights and
anomalous diffusion [39, 35]. These effects are caused by local instability of
the center-of-mass motion in a laser field. A set of atomic trajectories under
certain conditions becomes exponentially sensitive to small variations in
initial quantum internal and classical external states or/and in the control
parameters, mainly, the atom-laser detuning.
In this paper we consider the physical situation mentioned in the third part
of our nomenclature to focus at the ultimate reasons of chaotic atomic
external and internal motion and its connection with the $SU(2)$ dynamical
symmetry.
## 2 Lie algebraic solution for the evolution operator with the $SU(2)$
dynamical symmetry
In a variety of physical problems $SU(2)$ appears to be a group of dynamical
symmetry. It is known [1, 4] that the set of three equations for the $SU(2)$
group parameters can be reduced to a single second-order differential
equation. The form of this governing equation depends on the choice of the
basis and its exponential ordering. The appropriate choice of parameterization
of the dynamical group is especially important if we need to solve explicitly
the governing equation for a given physical Hamiltonian.
The Hermitian Hamiltonian of a quantum system with the $SU(2)$ dynamical
symmetry can be cast in the general form
$\hat{H}(t)=h_{0}(t)\hat{R}_{0}+h^{*}(t)\hat{R}_{-}+h(t)\hat{R}_{+},$ (2)
where $\hat{R}_{0}$ and $\hat{R}_{\pm}$ are the generators that satisfy the
commutation relations
$\left[\hat{R}_{-},\hat{R}_{+}\right]=-2\hat{R}_{0},\quad\left[\hat{R}_{0},\hat{R}_{\pm}\right]=\pm\hat{R}_{\pm}.$
(3)
It is convenient to choose the following noncanonical parameterization of the
$SU(2)$ group
$\hat{U}=\exp\Bigl{[}\Bigl{(}g_{0}-i\int\limits_{0}^{t}h_{0}\,d\tau\Bigr{)}\hat{R}_{0}\Bigr{]}\;\exp
g_{-}\hat{R}_{-}\;\exp g_{+}\hat{R}_{+}\;.$ (4)
Substituting Eq.(4) into Eq.(1), one finds the set of differential equations
for the group parameters that can be reduced to the single equation for the
group parameter $g\equiv\exp(g_{0}/2)$
$\frac{d^{2}g}{dt^{2}}-\Bigl{(}\frac{{dh}/{dt}}{h}+ih_{0}\Bigr{)}\frac{dg}{dt}+\mid
h\mid^{2}g=0\;,\;g(0)=1\;,\;\frac{dg}{dt}(0)=0\;.$ (5)
Once Eq.(5) is solved analytically, all the other parameters in the product
(4) may be expressed in terms of the parameter $g$ as follows:
$g_{-}=\frac{ig({dg}/{dt})}{h}\exp\Bigl{(}-i\int\limits_{0}^{t}h_{0}\,d\tau\Bigr{)}\;,\quad\frac{{dg}_{+}}{dt}=-\frac{ih}{g^{2}}\exp\Bigl{(}i\int\limits_{0}^{t}h_{0}\,d\tau\Bigr{)}\;.$
(6)
It is convenient to introduce the new variable
$\tilde{g}\equiv g_{-}/g.$ (7)
Then any group element in the parameterization (4) can be described by a pair
of complex numbers $g$ and $\tilde{g}$ obeying the condition
$\mid{g\mid}^{2}+\mid{\tilde{g}\mid}^{2}=1\;.$ (8)
It should be noted that all these formulas are valid within any representation
and within any realization of the $SU(2)$ group. It is well known that the
unitary irreducible representations of $SU(2)$ are characterized by half-
integer and integer numbers $j$. The dimensionality of the $j$th
representation is equal to $2j+1$. In the $(2j+1)$-dimensional space of
representation there is a canonical basis
$\left|j,m\right\rangle,\quad m=-j,-j+1,...,j\;.$ (9)
The representation matrix elements in the noncanonical parameterization (4)
are given by [4]
$\begin{array}[]{c}U_{m^{\prime}m}^{(j)}=\exp\left[-im^{\prime}\int\limits_{0}^{t}h_{0}(\tau)d\tau\right]\sum\limits_{l=-j}^{j}\left[\frac{(j-m^{\prime})!(j-m)!}{(j+m^{\prime})!(j+m)!}\right]^{1/2}\times\\\
\frac{(j+l)!}{(j-l)!(l-m)!(l-m^{\prime})!}g^{m+m^{\prime}}({\tilde{g}})^{l-m^{\prime}}(-{\tilde{g}^{*}})^{l-m}\;.\end{array}$
(10)
To analyze the dynamics of a two-level quantum system we need the two-
dimensional representation of the $SU(2)$ group. In this case the generators
$R$’s are connected with the familiar Pauli matrices
$\hat{R}_{0}=\frac{1}{2}\hat{\sigma}_{z}=\frac{1}{2}\left|\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right|,\quad\hat{R}_{-}=\hat{\sigma}_{-}=\left|\begin{array}[]{cc}0&0\\\
1&0\end{array}\right|,\quad\hat{R}_{+}=\hat{\sigma}_{+}=\left|\begin{array}[]{cc}0&1\\\
0&0\end{array}\right|\;,$ (11)
where
$[\hat{\sigma}_{+},\,\hat{\sigma}_{-}]=\hat{\sigma}_{z}\;,\quad[\hat{\sigma}_{z},\,\hat{\sigma}_{\pm}]=\pm
2\hat{\sigma}_{\pm}\;.$ (12)
In this representation the Hamiltonian of a driven two-level system has the
form
$\hat{H}(t)=\frac{1}{2}\hbar\omega_{a}\hat{\sigma}_{z}+\hbar\Omega^{*}(t)\hat{\sigma}_{-}+\hbar\Omega(t)\hat{\sigma}_{+}\;,$
(13)
where $\Omega(t)$ is a time-dependent function which is, in general, a
complex-valued one. The temporal evolution of the two-level system is now
governed by the equation
$\frac{d^{2}g}{dt^{2}}-\Bigl{(}\frac{{d\Omega}/{dt}}{\Omega}+i\omega_{a}\Bigr{)}\frac{dg}{dt}+\mid\Omega\mid^{2}g=0\;,\;g(0)=1\;,\;\frac{dg}{dt}(0)=0\;.$
(14)
The evolution matrix in the basis
${\left|1\right>}={\left|\frac{1}{2}\;,-\frac{1}{2}\right>},{\left|2\right>}={\left|\frac{1}{2}\;,\frac{1}{2}\right>}$
(15)
is given by
$\hat{U}^{(1/2)}=\left(\begin{array}[]{cc}e^{-i\omega_{a}t/2}&0\\\
0&e^{i\omega_{a}t/2}\end{array}\right)\left(\begin{array}[]{cc}g&-{\tilde{g}}^{*}\\\
{\tilde{g}}&g^{*}\end{array}\right)\;.$ (16)
## 3 The $SU(2)$ group–Hamilton equations for a two-level atom moving in a
standing-wave laser field
We consider a two-level atom with mass $m_{a}$ and transition frequency
$\omega_{a}$, moving with the momentum $P$ along the axis $X$ in a one-
dimensional classical laser standing wave with the frequency $\omega_{f}$ and
the wave vector $k_{f}$. In the frame, rotating with the frequency
$\omega_{f}$, the model Hamiltonian is the following:
$\hat{H}=\frac{P^{2}}{2m_{a}}+\frac{1}{2}\hbar(\omega_{a}-\omega_{f})\hat{\sigma}_{z}-\hbar\Omega_{0}\left(\hat{\sigma}_{-}+\hat{\sigma}_{+}\right)\cos{k_{f}X},$
(17)
where $\Omega_{0}$ is the maximal Rabi frequency which is proportional to the
square root of the number of photons in the wave. The laser field is assumed
to be strong enough, so we can treat the field classically.
In the process of emitting and absorbing photons, atoms not only change their
internal electronic states but their external translational states change as
well due to the photon recoil. If the atomic mean momentum is large as
compared to the photon momentum $\hbar k_{f}$, one can describe the
translational degree of freedom classically. The position and momentum of a
point-like atom satisfy classical Hamilton equations of motion which we
represent in the normalized form
$\dot{x}=\omega_{r}p,\quad\dot{p}=-<\hat{\sigma}_{-}(t)+\hat{\sigma}_{+}(t)>\sin
x,$ (18)
where $x\equiv k_{f}X$ and $p\equiv P/\hbar k_{f}$ are normalized classical
atomic center-of-mass position and momentum, respectively. The dot denotes
differentiation with respect to the dimensionless time $\tau\equiv\Omega_{0}t$
and $\omega_{r}\equiv\hbar k_{f}^{2}/m_{a}\Omega_{0}\ll 1$ is the normalized
recoil frequency. To compute the quantum expectation value
$<\hat{\sigma}_{-}(t)+\hat{\sigma}_{+}(t)>$ we need to use the solution for
the evolution operator (16). Supposing that the atom is initially in the
ground state ${\left|1\right>}$, we get
$<\hat{\sigma}_{-}(t)+\hat{\sigma}_{+}(t)>={\left<1\right|}\hat{U}^{\dagger}(t)\hat{U}(t){\left|1\right>}=-(gG^{*}+g^{*}G),$
(19)
where we introduce for convenience the new complex-valued variable
$G\equiv-\frac{i\dot{g}^{*}}{\cos x}.$ (20)
The internal atomic dynamics is governed by Eq. (14) that can be rewritten in
the form of two first-order equations for the complex-valued group parameters
$g$ and $G$. The self-consistent set of equations for the coupled external and
internal atomic degrees of freedom now reads as
$\dot{x}=\omega_{r}p,\quad\dot{p}=(gG^{*}+g^{*}G)\sin x,\\\ \dot{g}=iG\cos
x,\quad\dot{G}=-i\Delta G+ig\cos x,$ (21)
where the normalized recoil frequency $\omega_{r}$ and the atom-field
detuning, $\Delta\equiv(\omega_{f}-\omega_{a})/\Omega_{0}$, are the control
parameters. The six-dimensional dynamical system (21) has two independent
integrals of motion, the total energy,
$H\equiv\frac{\omega_{r}}{2}p^{2}+(gG^{*}+g^{*}G)\cos
x-\frac{\Delta}{2}(GG^{*}-gg^{*}),$ (22)
and the integral,
$\mid{g\mid}^{2}+\mid{G\mid}^{2}=1$ (23)
reflecting conservation of the norm of the atomic wave function. It is evident
from the second integral (23) that the squared absolute values of the $SU(2)$
group parameters, $\mid{g\mid}^{2}$ and $\mid{G\mid}^{2}$, have the sense of
the probability amplitudes to find the atom in the ground and excited states,
respectively.
The equations of motion (21) describe the mixed quantum-classical dynamics of
a two-level atom in a one-dimensional standing-wave laser field. The dynamical
$SU(2)$ group is responsible for internal atomic dynamics caused by the
interaction of the atomic electric dipole moment with the strength of the
electric component of the field. The quantum expectation value of the
corresponding interaction energy is given by the combination of the $SU(2)$
group parameters (19). The classical translational degree of freedom is
described by the Hamilton equations (see the first two equations in the set
(21)) governed by the interaction energy. In Introduction we called such a
situation as a dynamical driving when the $SU(2)$ group, generated by the
atomic quantum Hamiltonian, is driven by another dynamical system, the
classical atomic degree of freedom. In fact, we deal not with a fully quantum
system but with a quantum-classical hybrid which is described by the c-number
nonlinear dynamical system (21) that may be chaotic in the strict sense of
this term in some ranges of the control parameters and/or initial conditions.
## 4 Dynamical chaos in the group-theoretical picture
Equations (21) constitute a nonlinear autonomous dynamical system with three
degrees of freedom and, in general, with the two integrals of motion, (22) and
(23). Thus, the dynamical system (21) may be chaotic in the sense of
exponential sensitivity to small variation in initial conditions and/or the
control parameters, $\omega_{r}$ and $\Delta$. The common test to confirm that
is to compute the maximum Lyapunov exponent characterizing the mean rate of
exponential divergence of initially close trajectories which serves as a
quantitative measure of dynamical chaos [40, 41]:
$\lambda({\bf Q}_{0},\Delta{\bf q}_{0})=\lim_{t\to\infty,\Delta{\bf q}_{0}\to
0}\frac{1}{t}\ln\frac{\left\|\Delta{\bf q}({\bf
Q}_{0},t)\right\|}{\left\|\Delta{\bf q}_{0}\right\|},$ (24)
where $\Delta{\bf q}$ is the vector in the phase space with the components
$\left\\{\Delta q_{j},j=1,...,N\right\\}$ and the norm $\left\|\Delta{\bf
q}\right\|$. In Eq.(24), $\Delta{\bf q}_{0}$ and $\Delta{\bf q}({\bf
Q}_{0},t)$ denote the separation between two initially adjacent trajectories
at the initial moment $t=0$ and at time $t$, respectively, ${\bf Q}_{0}$ is
the initial position. If, at least, one of the Lyapunov exponents of the
dynamical system under question is positive, then trajectories, starting close
together in the phase space, separate exponentially as time grows. This very
sensitive dependence on initial conditions is one of the main indicator of
dynamical chaos.
The result of computation of the maximum Lyapunov exponent with the equations
of motion (21) at $\omega_{r}=10^{-3}$ in dependence on the detuning $\Delta$
and the initial atomic momentum $p_{0}$ is shown in Fig. 1. Color in the plot
codes the value of the maximum Lyapunov exponent $\lambda$. In white regions
in Fig. 1 the values of $\lambda$ are almost zero, and the atomic motion is
regular in the corresponding ranges of $\Delta$ and $p_{0}$. In shadowed
regions positive values of $\lambda$ imply unstable motion. The atoms with
zero $\lambda$’s either oscillate in a regular way in a well of the optical
potential or move ballistically over the hills of the potential with a regular
variation of their velocity.
Figure 1: Maximum Lyapunov exponent, $\lambda$, vs the atom-field detuning
$\Delta$ (in units of the maximal Rabi frequency $\Omega$) and the initial
atomic momentum $p_{0}$ (in units of the photon momentum $\hbar k_{f}$) at
$\omega_{r}=10^{-3}$. Color codes the values of $\lambda$. Figure 2: Regimes
of motion of two-level atoms in a one-dimensional deterministic standing-wave
laser field. Trajectories in the real space at $\omega_{r}=10^{-3}$: regular
flight (RF, $\Delta=0.8$, $p_{0}=45$), chaotic flight (CF, $\Delta=0.2$,
$p_{0}=45$), chaotic walking (CW, $\Delta=0.2$, $p_{0}=10$) and trapping in a
potential well (T, $\Delta=-0.2$, $p_{0}=5$). $x$ is in units of the
wavelength $\lambda_{f}$.
Figure 3: Plots with 50 atomic trajectories with different values of the
initial atomic momentum $0\leq p_{0}\leq 50$ but with the same initial
position $x_{0}=0$ and the same other initial conditions. (a) Real space. (b)
Momentum space.
At exact resonance, the equations of motion (21) become integrable due to an
additional integral of motion, $gG^{*}+g^{*}G={\rm const}$, and we get
$\lambda=0$. Thus at $\Delta=0$, the center-of-mass motion and the motion in
the space of the $SU(2)$ group parameters are regular.
Figure 4: Behavior of the mean atom-field interaction energy
$g_{1}G_{1}+g_{2}G_{2}$ in the regimes of (a) regular oscillations in a well
of the optical potential and (b) regular flight.
Figure 5: The same as in Fig. 4 but in the regimes of (a) chaotic flight and
(b) chaotic walking.
There are three possible chaotic types of motion of a two-level atom in a one-
dimensional standing-wave laser field. In dependence on the initial conditions
and the parameter values atoms may oscillate chaotically in a well of the
optical potential, move ballistically over the hills of the potential with
chaotic variations of their velocity or perform a chaotic walking. In the
regime of the chaotic walking an atom in a deterministic standing-wave field
alternates between flying through the standing-wave and being trapped in the
wells of the optical potential. Moreover, it may change the direction of
motion in a random-like way [38]. We would like to stress that local
instability produces chaotic center-of-mass motion in a rigid standing wave
without any modulation of its parameters in difference from the situation with
atoms in a periodically kicked optical lattice [42, 43, 44]. To illustrate
different types of motion we plot in Fig. 2 four trajectories of the atoms in
the real space at $\omega_{r}=10^{-3}$ corresponding to a regular flight (RF),
chaotic flight (CF), chaotic walking (CW) and trapping in a potential well
(T).
Figure 6: Projections of the single trajectory of a trapped atom in the six-
dimensional phase space on the plane of the complex-valued $SU(2)$ group
parameter $g=g_{1}+ig_{2}$ at (a) $\tau=100$, (b) $\tau=500$ and (c)
$\tau=1000$.
Figure 7: The same as in Fig. 6 but for a regular flight.
Let us estimate the values of the control parameters and the initial
conditions under which atoms oscillate in the first well of the optical
potential, move ballistically or walk chaotically. At small detunings,
$\Delta\ll 1$, the total energy (22) consists of the kinetic one,
$K=\omega_{r}p^{2}/2$, and the potential one, $U=(gG^{*}+g^{*}G)\cos
x=(g_{1}G_{1}+g_{2}G_{2})\cos x$, the sum of which is conserved approximately
in course of time. The maximal absolute value of the optical potential energy
is 1. Let the atom is prepared in the ground state ${\left|1\right>}$, i.e.,
$g_{1}(\tau=0)=1$, $g_{2}(\tau=0)=G_{1}(\tau=0)=G_{2}(\tau=0)=0$ and
$U_{0}=0$. If $K_{0}>|U_{\rm max}|=1$, then the atom will move ballistically.
This occurs if the initial atomic momentum, $p_{0}$, satisfies to the
condition $p_{0}>\sqrt{2/\omega_{r}}\simeq 44$ at $\omega_{r}=10^{-3}$. If the
initial conditions are chosen to give $0\leq H_{0}=K_{0}+U_{0}\leq 1$, then
the atom performs a chaotic walking. This occurs at $0\leq p_{0}\leq 44$. The
atom will be trapped in the first well of the optical potential if $H_{0}<0$.
It is posiible with the initial conditions chosen only if $\Delta<0$.
To demonstrate strong dependence of the atomic motion on initial conditions we
compute 50 trajectories with different values of the initial atomic momentum,
$0\leq p_{0}\leq 50$, but with the same initial position, $x_{0}=0$, and the
same other initial conditions. Figure 3 gives an impressive image of dynamical
chaos with atoms in a laser field both in the real and momentum spaces. Most
of the atoms in this bunch (with $0\leq p_{0}\leq 44$) walks chaotically,
changing the direction of motion in course of time. Atomic trajectories with
close initial conditions diverge in the real one-dimensional space in such a
way that it is practically impossible to predict their final position after
the predictability time
$\tau_{p}\approx\frac{1}{\lambda}\ln\frac{\Delta x}{\Delta x(0)},$ (25)
where $\Delta x$ is the confidence interval and $\Delta x_{(}0)$ is the
practically inevitable error in measuring the initial atomic position.
It follows from (21) that the translational motion is described by the
equation for a nonlinear physical pendulum with the frequency modulation
$\ddot{x}-2\omega_{r}(g_{1}G_{1}+g_{2}G_{2})\sin x=0.$ (26)
It is clear that the regime of the center-of-mass motion is specified by the
character of oscillations of the group parameter, $g_{1}G_{1}+g_{2}G_{2}$,
which has the sense of the mean interaction energy between the atom and the
laser field (see the integral of motion (22)). If the atom is trapped in the
first well of the optical potential, its center of mass oscillates between the
first negative and positive nodes, $-\pi/2<x<\pi/2$. If, in addition, the
control parameters are chosen in appropriate way, it will oscillate
periodically. This case is shown in Fig. 4 a ($\Delta=-0.2$, $p_{0}=5$) with
periodic albeit modulated oscillations of the quantity
$g_{1}G_{1}+g_{2}G_{2}$. Figure 4 b is plotted for another regime of the
center-of-mass motion, a regular ballistic flight with $\Delta=0.8$ and
$p_{0}=45$. The quantity $g_{1}G_{1}+g_{2}G_{2}$ again oscillates periodically
but with the modulation period that is equal to the flight time between two
adjacent nodes of the laser standing wave,
$T_{f}\simeq\pi/\omega_{r}p_{0}\simeq 70$.
Behavior of the group parameter $g_{1}G_{1}+g_{2}G_{2}$ is absolutely
different in the chaotic regimes of motion, CF and CW. In the regime of
chaotic ballistic flight (see Fig. 5 a with $\Delta=0.2$ and $p_{0}=45$),
shallow oscillations of that quantity are interrupted by jumps of different
amplitudes that occur when the atom crosses each node of the standing wave. In
the regime of chaotic center-of-mass walking (see Fig. 5 b with $\Delta=0.2$
and $p_{0}=10$), oscillations of the quantity $g_{1}G_{1}+g_{2}G_{2}$ look
even more complicated. We may conclude that namely the chaotic oscillations of
the mean interaction energy between the atom and the laser field,
$g_{1}G_{1}+g_{2}G_{2}$, in some ranges of the control parameters,
$\omega_{r}$ and $\Delta$, and initial atomic momentum $p_{0}$ are responsible
for the chaotic center-of-mass motion.
Figure 8: The same as in Fig. 6 but for a chaotic flight.
Figure 9: The same as in Fig. 6 but for a chaotic walking.
The equations of motion (21) can be recast in the form of the two second-order
differential equations, the classical one (26), describing the center-of-mass
motion, and the quantum one
$\ddot{g}+(i\Delta+\dot{x}\tan x)\dot{g}+g\cos^{2}x=0,$ (27)
describing the internal atomic dynamic in terms of the complex-valued $SU(2)$
group parameter $g=g_{1}+ig_{2}$. In order to illustrate how different may be
behavior of the quantum degree of freedom of the quantum-classical hybrid, we
compute the evolution of the real and imaginary parts of $g$ in course of
time. The results are shown in Figs. 6– 9 with different regimes of motion.
The plots give projections of the single atomic trajectory in the six-
dimensional phase space on the plane of the complex-valued $SU(2)$ group
parameter $g$ at the time moments $\tau=100$, $\tau=500$ and $\tau=1000$.
The plots with a periodically oscillating atom in a trap (Fig. 6) and with a
regular flight (Fig. 7) demonstrate the strictly periodic patterns on the
$g_{1}$–$g_{2}$ plane with forbidden regions in the center. Internal dynamics
of the atoms in the chaotic center-of-mass regimes of motion, chaotic flight
in Fig. 8 and chaotic walking in Fig. 9, is much more complicated. In both the
cases, the trajectories on the $g_{1}$–$g_{2}$ visit in course of time all the
accessible part of the plane with $\mid g\mid<1$.
## 5 How to observe chaotic walking of atoms in a real experiment
In this section we propose the scheme of a real experiment to observe the
effect of chaotic walking of atoms in a deterministic standing wave described
in the previous section. A beam of two-level atoms in the $z$ direction
crosses a standing-wave laser field with optical axis in the $x$ direction
(Fig. 10a). One measures either the atomic density on a substrate as in the
atom-lithography experiments [46, 47] or the spatial atomic distribution as in
the atom optics experiments [42, 43, 44]. In each type of the experiments the
results are expected to be different in the regimes of chaotic atomic walking
and regular motion. To switch between the regimes it is enough to vary the
value of the detuning in the appropriate way. The laser beam has the Gaussian
profile $\exp[-(z-z_{0})^{2}/r^{2}]$ with $r$ being the $e^{-2}$ radius at the
laser beam waist. The longitudinal velocity of atoms, $v_{z}$, is much larger
than their transversal velocity $v_{x}$ and is supposed to be constant.
Therefore, the spatial laser profile may be replaced by the temporal one.
The Hamiltonian (17) now takes the time-dependent form
$\hat{H}=\frac{\hat{P}^{2}}{2m_{a}}+\frac{\hbar}{2}(\omega_{a}-\omega_{f})\hat{\sigma}_{z}-\\\
\hbar\Omega_{0}\exp[-(v_{z}t-\frac{3}{2}r)^{2}/r^{2}]\left(\hat{\sigma}_{-}+\hat{\sigma}_{+}\right)\cos{k_{f}\hat{X}}$
(28)
with the same dynamical symmetry. Using the same normalization as before, we
get the equations of motion
$\displaystyle\ddot{x}-\omega_{r}\Omega(\tau)(gG^{*}+g^{*}G)\sin x=0$ (29)
$\displaystyle\ddot{g}+\left[i\Delta+\dot{x}\tan
x-\frac{\dot{\Omega}(\tau)}{\Omega(\tau)}\right]\dot{g}+g[\Omega(\tau)]^{2}\cos^{2}x=0$
(30)
with the time-dependent coefficient
$\Omega(\tau)=\exp[-(\tau-\frac{3}{2}\sigma_{\tau})^{2}/\sigma^{2}_{\tau}]$,
where $\sigma_{\tau}\equiv r\Omega_{0}/v_{z}$ is the normalized characteristic
interaction time.
Figure 10: (a) Scheme of the proposed experiment on observation of chaotic
walking (cw) of atoms scattered at a Gaussian standing laser wave. (b) The
distributions of $10^{4}$ lithium atoms at $\tau=1000$ ($z=200$ microns) under
the conditions of chaotic walking at $\Delta=0.2$ (bold curve) and regular
motion (rm) at $\Delta=1$ (dashed curve).
To be concrete let us take lithium atoms with the relevant transition
$2S_{1/2}-2P_{3/2}$, the corresponding wavelength $\lambda_{a}=670.7$ nm and
the recoil frequency $\nu_{\rm rec}=63$ KHz. With the maximal Rabi frequency
$\Omega_{0}/2\pi\simeq 126$ MHz and the radius of the laser beam $r=0.05$ cm
one gets $\omega_{r}=10^{-3}$ and $\sigma_{\tau}=400$. To simulate a real
experiment let us consider a beam of $10^{4}$ lithium atoms with the initial
Gaussian position and momentum distributions (the rms
$\sigma_{x}=\sigma_{p}=2$, the average values, $x_{0}=0$, and $p_{0}=10$) and
compute their position distribution at a fixed moment of time. In Fig. 10 b we
compare the atomic position distributions at $\tau=1000$ ($z=200$ microns) for
the chaotic walking at $\Delta=0.2$ (bold curve) and the regular motion at
$\Delta=1$ (dashed curve). The difference is evident. In the chaotic regime
atoms are distributed more or less homogeneously over a large distance of 8
wavelengths along the $x$-axis whereas in the regime of the regular motion
they form a few peaks in a much more narrow interval. Thus, we predict that
under the conditions of chaotic walking there should appear a less contrast
and more broadened atomic relief as compared to the case of regular motion
because a large number of atoms are expected to be deposited between the nodes
as a result of chaotic walking along the standing-wave axis.
## 6 Conclusion
We have studied behavior of lossless two-level atoms in a one-dimensional
standing-wave laser field in the group-theoretical picture. In this picture we
have represented the internal quantum atomic dynamics in terms of the
dynamical $SU(2)$ group parameters and the center-of-mass motion by the
classical Hamilton equations. Thus, we have modeled the system by a quantum-
classical hybrid with coupled quantum and classical degrees of freedom. We
have derived the corresponding set of the $SU(2)$ group-Hamilton equations of
motion with, in general, two integrals of motion. This set has been
numerically shown to be chaotic in some ranges of the control parameters and
initial conditions. We have found five different regimes of the center-of-mass
motion including chaotic walking when an atom in an absolutely deterministic
standing-wave field may change the direction of motion in a random-like way
alternating between flying in the optical potential and being trapped in its
wells. All the regimes have been illustrated by the trajectory plots in the
real and momentum spaces. It has been established that the instability of
motion and dynamical chaos are caused by the character of oscillations of the
group parameter characterizing the mean interaction energy between the atom
and the laser field. Projections of atomic trajectories in the six-dimensional
phase space on the plane of the complex-valued $SU(2)$ group parameter $g$
have been shown to form regular and irregular patterns in the regimes of
regular and chaotic center-of-mass motion, respectively.
We proposed the scheme of an experiment on the scattering of atomic beams at a
standing-wave laser field that could directly image chaotic walking of atoms
along the optical axis. In a real experiment the final spatial distribution
can be recorded via fluorescence or absorption imaging on a CCD, commonly used
methods in atom optics experiments yielding information on the number of atoms
and the cloud’s spatial size. The other possibility is a nanofabrication where
the atoms after the interaction with the standing wave are deposited on a
silicon substrate in a high vacuum chamber. In this case the spatial
distribution can be analyzed with an atomic force microscope. The modern tools
of atom optics enable to create narrow initial atomic distributions in
position and momentum, reduce coupling to the environment and technical noise,
create one-dimensional optical potentials, and to measure spatial and momentum
distributions with high sensitivity and accuracy [42, 43, 44].
The results obtained can be applied to other models of the atom-field
interaction as well. In particular, relaxation processes in two-level atoms
can be described within the framework of the $SO(3)$ dynamical-symmetry
approach to solving the Bloch equations [17]. Moreover, one may consider by
the method developed in this paper the dynamics not only of two-level atoms
but of three-, four- and multilevel atoms excited by a few laser fields at
different atomic transitions. If the corresponding model Hamiltonian has the
$SU(2)$ dynamical symmetry, then one may use the solution obtained in Sec. 2
that is valid for any representation of the $SU(2)$ group.
The model considered can be generalized to the case with two-level atoms
inside a high-quality cavity with a quantized field. In the rotating wave
approximation the state space of the corresponding Jaynes-Cummings model
splits up into an infinite class of two-dimensional non-communicating
subspaces each of which being labeled by eigenvalues of the Casimir operator.
The system evolves in such a way that transitions between the subspaces with
different eigenvalues are forbidden. The solution for the time-evolution
operator in each of these subspaces is given by the matrix (15) with the group
parameter satisfying to the equation similar to (5). The resulting equations
of motion for the coupled atom-field system are expected to constitute an
infinite-dimensional set of the type (21) with the group equation (15) acting
in each of the subspaces labeled by its own eigenvalue. This set is expected
to admit very different regimes of motion including chaotic ones.
## Acknowledgments
This work was supported by the Russian Foundation for Basic Research (project
no. 09-02-00358), by the Integration grant from the Far-Eastern and Siberian
branches of the Russian Academy of Sciences and by the Program “Fundamental
Problems of Nonlinear Dynamics” of the Russian Academy of Sciences.
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* [42] Moore F, Robinson J, Bharucha C, Sundaram B and Raizen M 1995 Phys. Rev. Lett. 75 4598
* [43] Steck D, Oskay W and Raizen M 2001 Science 293 274
* [44] Hensinger W, Heckenberg N, Milburn G and Rubinsztein-Dunlop H 2003 J. Opt. B: Quantum Semiclass. Opt. 5 83
* [45] Raizen M G 1999 Adv. At. Mol. Opt. Phys. 41 43
* [46] Timp G et al Phys. Rev. Lett. 69 1636
* [47] Jürgens D et al Phys. Rev. Lett. 93 237402\.
|
arxiv-papers
| 2012-01-01T03:55:50 |
2024-09-04T02:49:25.860008
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. V. Prants",
"submitter": "Michael Uleysky",
"url": "https://arxiv.org/abs/1201.0324"
}
|
1201.0326
|
# Matter-wave chaos with a cold atom in a standing-wave laser field
S.V. Prants prants@poi.dvo.ru, tel.007-4232-312602, fax 007-4232-312573
Laboratory of Nonlinear Dynamical Systems, Pacific Oceanological Institute of
the Russian Academy of Sciences, Baltiiskaya St., 43, 690041 Vladivostok,
Russia
###### Abstract
Coherent motion of cold atoms in a standing-wave field is interpreted as a
propagation in two optical potentials. It is shown that the wave-packet
dynamics can be either regular or chaotic with transitions between these
potentials after passing field nodes. Manifestations of de Broglie-wave chaos
are found in the behavior of the momentum and position probabilities and the
Wigner function. The probability of those transitions depends on the ratio of
the squared detuning to the Doppler shift and is large in that range of the
parameters where the classical motion is shown to be chaotic.
###### keywords:
cold atom; matter wave; quantum chaos
###### PACS:
03.75.-b, 42.50.Md
## 1 Introduction
Cold atoms in a standing-wave laser field [1, 2, 3, 4, 5] are ideal objects
for studying fundamental principles of quantum physics, quantum-classical
correspondence, and quantum chaos. The proposal [6] to study atomic dynamics
in a far-detuned modulated standing wave made atomic optics a testing ground
for quantum chaos. A number of impressive experiments have been carried out in
accordance with this proposal [7, 8, 9]. Furthermore, cold atoms have been
recently used to study different phenomena in statistical physics including
ratchet effect of a directed transport of atoms in the absence of a net bias
[10, 11, 12] and Brillouin-like propagation modes in optical lattices [13].
New possibilities are opened if one works near the atom-field resonance where
the interaction between the internal and external atomic degrees of freedom is
intense. A number of nonlinear dynamical effects have been found [14, 15, 16]
with point-like atoms in a near-resonant rigid optical lattice: chaotic Rabi
oscillations, chaotic walking, dynamical fractals, Lévy flights, and anomalous
diffusion.
Dynamical chaos in classical mechanics is a special kind of random-like motion
without any noise and/or random parameters. It is characterized by exponential
sensitivity of trajectories in the phase space to small variations in initial
conditions and/or control parameters. Such sensitivity does not exist in
isolated quantum systems because their evolution is unitary, and there is no
well-defined notion of a quantum trajectory. Thus, there is a fundamental
problem of emergence of classical dynamical chaos from more profound quantum
mechanics which is known as quantum chaos problem and the related problem of
quantum-classical correspondence. In a more general context it is a problem of
wave chaos. It is clear now that quantum chaos, microwave, optical, and
acoustic chaos have much in common (see [17] for a review). The common
practice is to construct an analogue for a given wave object in a
semiclassical (ray) approximation and study its chaotic properties (if any) by
well-known methods of dynamical system theory. Then, it is necessary to solve
the corresponding linear wave equation in order to find manifestations of
classical chaos in the wave-field evolution in the same range of the control
parameters. If one succeeds in that a quantum-classical or wave-ray
correspondence is announced to be established.
In this paper we perform this program with cold two-level atoms in a one-
dimensional standing-wave laser field and show that coherent dynamics of the
atomic matter waves is really complicated in that range of the control
parameters where the corresponding classical point-like atomic motion can be
strictly characterized as a chaotic one. The effect is explained by a
proliferation of atomic wave packets at the nodes of the standing wave.
## 2 Order and chaos in dynamics of atomic wave packets in a laser standing
wave
The Hamiltonian of a two-level atom, moving in a one-dimensional classical
standing-wave laser field, can be written in the frame rotating with the laser
frequency $\omega_{f}$ as follows:
$\hat{H}=\frac{\hat{P}^{2}}{2m_{a}}+\frac{\hbar}{2}(\omega_{a}-\omega_{f})\hat{\sigma}_{z}-\hbar\Omega\left(\hat{\sigma}_{-}+\hat{\sigma}_{+}\right)\cos{k_{f}\hat{X}},$
(1)
where $\hat{\sigma}_{\pm,z}$ are the Pauli operators for the internal atomic
degrees of freedom, $\hat{X}$ and $\hat{P}$ are the atomic position and
momentum operators, $\omega_{a}$ and $\Omega$ are the atomic transition and
Rabi frequencies, respectively. We will work in the momentum representation
and expand the state vector as follows:
${|\Psi(t)\closeket}=\int[a(P,t){|2\closeket}+b(P,t){|1\closeket}]{|P\closeket}dP,$
(2)
where $a(P,t)$ and $b(P,t)$ are the probability amplitudes to find atom at
time $t$ with the momentum $P$ in the excited and ground states, respectively.
The Schrödinger equation for these amplitudes is
$\displaystyle i\dot{a}(p)$
$\displaystyle=\frac{1}{2}(\omega_{r}p^{2}-\Delta)a(p)-\frac{1}{2}[b(p+1)+b(p-1)],$
(3) $\displaystyle i\dot{b}(p)$
$\displaystyle=\frac{1}{2}(\omega_{r}p^{2}+\Delta)b(p)-\frac{1}{2}[a(p+1)+a(p-1)],$
where dot denotes differentiation with respect to dimensionless time
$\tau\equiv\Omega t$, and the atomic momentum $p$ is measured in units of the
photon momentum $\hbar k_{f}$. The normalized recoil frequency,
$\omega_{r}\equiv\hbar k_{f}^{2}/m_{a}\Omega$, and the atom-field detuning,
$\Delta\equiv(\omega_{f}-\omega_{a})/\Omega$, are the control parameters.
We will treat the wave-packet motion in the dressed-state basis [18]
${|+\closeket}_{\Delta}=\sin{\Theta}{|2\closeket}+\cos{\Theta}{|1\closeket},\
{|-\closeket}_{\Delta}=\cos{\Theta}{|2\closeket}-\sin{\Theta}{|1\closeket}.$
(4)
The dressed states are eigenstates of an atom at rest in a laser field with
the eigenvalues of the quasienergy $E_{\Delta}^{(\pm)}$ and the mixing angle
$\Theta$:
$E_{\Delta}^{(\pm)}=\pm\sqrt{\frac{\Delta^{2}}{4}+\cos^{2}{x}},\
\tan{\Theta}\equiv\frac{\Delta}{2\cos{x}}-\sqrt{\left(\frac{\Delta}{2\cos{x}}\right)^{2}+1}.$
(5)
The depth of the nonresonant optical potential is
$U_{\Delta}=\left|\sqrt{\frac{\Delta^{2}}{4}+1}-\frac{|\Delta|}{2}\right|.$
(6)
At $\Delta=0$, the depth changes abruptly from $1$ to $2$. The resonant
$E_{0}^{(\pm)}$ ($\Delta=0$) and non-resonant $E_{\Delta}^{(\pm)}$
($\Delta\not=0$) potentials of an atom in a standing-wave field are drawn in
Fig. 1. The ground atomic state can be written as a superposition of the
dressed states
${|1\closeket}={|+\closeket}_{\Delta}\frac{1}{\sqrt{1+\tan^{2}{\Theta}}}-{|-\closeket}_{\Delta}\frac{\tan{\Theta}}{\sqrt{1+\tan^{2}{\Theta}}}.$
(7)
In general, an atom in the ground state, placed initially at $x_{0}=0$, will
move along two trajectories simultaneously because it is situated
simultaneously at the top of $E_{\Delta}^{(+)}$ and the bottom of
$E_{\Delta}^{(-)}$ (see Fig.1).
In the dressed-state basis, the probability amplitudes to find the atom at the
point $x$ in the potentials $E_{\Delta}^{(+)}$ and $E_{\Delta}^{(-)}$ are,
respectively,
$C_{+}(x)=a(x)\sin{\Theta}+b(x)\cos{\Theta},\
C_{-}(x)=a(x)\cos{\Theta}-b(x)\sin{\Theta},$ (8)
where the amplitudes in the bare-state basis, $a(x)$ and $b(x)$, are computed
in the position representation with the help of the Fourier transform
$a(x)={\rm
const}\int_{-\infty}^{\infty}dp^{\prime}e^{ip^{\prime}x}a(p^{\prime}),\
b(x)={\rm
const}\int_{-\infty}^{\infty}dp^{\prime}e^{ip^{\prime}x}b(p^{\prime}).$ (9)
To clarify the character of the bipotential motion we write down the
Hamiltonian of the internal degrees of freedom in the basis
${|\pm\closeket}=({|1\closeket}\pm{|2\closeket})/\sqrt{2}$:
$\hat{H}_{\rm int}=\hat{\sigma}_{z}\cos x+\frac{\Delta}{2}\hat{\sigma}_{x}.$
(10)
Let us linearize the potential in the vicinity of a node of the standing wave
and estimate a small distance the atom makes when crossing the node as
follows: $\delta x\simeq\omega_{r}|p_{\rm node}|\tau$, where $|p_{\rm node}|$
is the mean atomic momentum near the node. The quantity $\omega_{r}|p_{\rm
node}|$ is a normalized Doppler shift for an atom moving with the momentum
$|p_{\rm node}|$, i.e., $\omega_{D}\equiv\omega_{r}|p_{\rm node}|\equiv
k_{f}|v_{\rm node}|/\Omega$. We arrive now at the famous Landau–Zener problem
[19] to find a probability of transition between the states (4) when the
energy difference varies linearly in time. In other words it is a a
probability of transition from one potential to another or from one trajectory
of motion to another. The asymptotic solution is
$P_{\rm LZ}=\exp\left(-\frac{\pi\Delta^{2}}{\omega_{D}}\right).$ (11)
There are three possibilities.
1\. $\Delta^{2}\gg\omega_{D}$. The transition probability is exponentially
small and an atom moves adiabatically along, in general, two trajectories
without any transitions between them.
2\. $\Delta^{2}\ll\omega_{D}$. The probability of a Landau-Zener transition is
close to unity, and an atom changes the potential each time upon crossing any
node, i.e., it moves mainly in the resonant potentials $E_{0}^{(\pm)}$.
3\. $\Delta^{2}\simeq\omega_{D}$. The probabilities to change the potential or
to remain in the same one upon crossing a node are of the same order. In this
regime one may expect strong complexification of the wave function.
At large and small detunings, the translational motion splits into two
independent motions in the potentials $E_{\Delta}^{(\pm)}$, and the wave-
packet motion is regular in the first two cases. In the third case, the motion
is complex because of a proliferation of wave packets at the nodes of the
standing wave. We call such a motion as ”a chaotic” one by the reasons which
will be clear in section 3. The switch between the regular and chaotic regimes
of atomic motion can be easily performed by changing the detuning.
Our normalization enables to change dimensionless values of the recoil
frequency $\omega_{r}$ and the detuning $\Delta$ varying the Rabi frequency
$\Omega$. Working with a cesium atom at the transition $6S_{1/2}$ – $6P_{3/2}$
($m_{a}=133$ a. u., $\lambda_{a}=852.1$ nm and $\nu_{\rm rec}\simeq 2$ KHz
[20]), we have $\omega_{r}=10^{-3}$ at $\Omega=1$ MHz. Let the atom is
initially prepared in the ground state as a Gaussian wave packet in the
momentum space with $p_{0}=55$. The probability to find the atom with the
momentum $p$ at time $\tau$ is ${\it
P}(p,\tau)=|a(p,\tau)|^{2}+|b(p,\tau)|^{2}$.
To illustrate the difference between the regular and chaotic regimes of the
wave-packet motion we take two values of the detuning, $\Delta=1$ and
$\Delta=0.2$. At $\Delta=1$, the motion is expected to be adiabatic and
regular because $\Delta^{2}\gg\omega_{D}=0.055$, and the Landau–Zener
probability, $P_{\rm LZ}$, is exponentially small. At $\Delta=0.2$, one
expects a much more complicated wave-packet motion with nonadiabatic
transitions between the two potentials at the field nodes because
$\Delta^{2}=0.04\simeq\omega_{D}$. Figure 2 shows the dependence of the mean
atomic momentum $<p>$ over a large time scale in those two cases. In both the
cases $<p>$ oscillates in a rather irregular way. The difference is that for
an adiabatically moving wave packet, which we refer as a regular motion, the
mean atomic momentum oscillates in a narrow range around $p_{0}=55$ (the upper
curve in the figure). Whereas the range of its oscillations for a wave packet
moving with nonadiabatic transitions at the field nodes, which we refer as a
chaotic motion, is much more broad (the lower curve in the figure). It is a
simple illustration of the two different regimes of the wave-packet
propagation in terms of the classical variable.
In Fig. 3a we plot the dependence of the momentum probability-density on time
at $\Delta=1$. The initial wave packet splits from the beginning to a few
components because the initial ground state is a superposition of the dressed
states (4). The initial kinetic energy is enough to perform a ballistic
motion. The momentum changes in a comparatively small range, from 40 to 70 of
the photon-momentum units. The packet does not split at the nodes of the
standing wave but it, on the contrary, recollects in the momentum space at the
nodes and spreads in between. However, this recollection smears out in course
of time.
At $\Delta=0.2$, the atomic ground state is the following superposition of the
dressed states: ${|1\closeket}\simeq
0.74{|+\closeket}_{\Delta}+0.66{|-\closeket}_{\Delta}$. The
${|+\closeket}_{\Delta}$-component of the initial wave packet, i.e., that one,
starting from the top of the potential $E_{\Delta}^{(+)}$, overcomes the
barriers of that potential and moves in the positive direction of the axis $x$
proliferating at the nodes. As to the ${|-\closeket}$-component with decreased
values of $p$, it will be trapped in the potential well performing
oscillations in the momentum and position spaces. The period of those
oscillations is about $T\simeq 280$ which is equal approximately to the period
of revivals of the Rabi oscillations for the population inversion. Figure 3b
illustrates the effect of simultaneous trapping and ballistic motion of the
atomic wave packet in the chaotic regime resulting in a broad momentum
distribution, from $p=-60$ to $p\simeq 80$.
To illustrate the nonadiabatic transitions from one potential to another and
their absence at the nodes more explicitly, we go to the position space and
compute the probabilities $|C_{\pm}(x,\tau)|^{2}$ (8) to be at the point $x$
at time $\tau$ in the potentials $E_{\Delta}^{(+)}$ and $E_{\Delta}^{(-)}$,
respectively. In Fig. 4 we plot the evolution of the probability density
$|C_{-}(x,\tau)|^{2}$ in the frame moving with the initial atomic velocity
$\omega_{r}p_{0}=0.055$. The slope straight lines in the figure mark positions
of the nodes in the moving frame. In the case of the regular motion at
$\Delta=1$ (Fig. 4a), no transitions happen when the atom crosses the nodes.
In the chaotic regime at $\Delta=0.2$, one observes visible changes in the
probability-density $|C_{-}(x)|^{2}$ exactly at the node lines (see Fig. 4b).
It means transitions from one trajectory to another at the field nodes that
should occur in a specific range of the control parameters if
$\Delta^{2}\simeq\omega_{D}$. This results in a proliferation of the
components of the wave packet at the nodes and, therefore, a complexification
of the wave function (see Fig. 3b).
The Wigner function can be used to visualize complexity of the wave function
in the chaotic regime of the atomic motion. We compute the evolution of the
Wigner function of the ground state in the momentum space
$W_{b}(p,x,\tau)={\rm
const}\int_{-\infty}^{\infty}dp^{\prime}e^{-ip^{\prime}x}a(p-p^{\prime}/2)a^{*}(p+p^{\prime}/2).$
(12)
This quantity gives a quasi-probability distribution corresponding to a
general quantum state (2). Figure 5 shows a contour plot of the Wigner
function (12) at two moments of time, $\tau=50$ and $\tau=200$, when the atom
moves in a regular way ($\Delta=1$). Figure 6 is a contour plot of this
function at the same times, but with an atom making nonadiabatic transitions
at the field nodes ($\Delta=0.2$). In the chaotic regime (Fig. 6b) we see a
dust-like distribution of nonzero values of the Wigner function at $\tau=200$
which occupy much more larger area in the phase space than the function for
the regular motion (Fig. 5b).
## 3 Quantum-classical correspondence
In this section we compare the results of the quantum treatment with those
obtained for the same problem in the semiclassical approximation when the
translational motion has been treated as a classical one [14, 15]. We must
compare quantum results for a single atomic wave packet not with a single
point-like atom but with an ensemble of point-like atoms. Dynamical chaos has
been found and analyzed in detail in Refs. [14, 15] in the semiclassical
approximation. Both the internal and external degrees of freedom of a two-
level atom in a standing wave field have been shown to be chaotic in a
specific range of values of the detuning $\Delta$, the recoil frequency
$\omega_{r}$, and the initial momentum $p_{0}$.
Coherent semiclassical evolution of a point-like two-level atom is governed by
the Hamilton-Schrödinger equations with the same normalization as in the
quantum case [15]
$\begin{gathered}\dot{x}=\omega_{r}p,\quad\dot{p}=-u\sin x,\quad\dot{u}=\Delta
v,\\\ \dot{v}=-\Delta u+2z\cos x,\quad\dot{z}=-2v\cos x,\end{gathered}$ (13)
where
$u\equiv 2\operatorname{Re}(a_{0}b_{0}^{*}),\
v\equiv-2\operatorname{Im}(a_{0}b_{0}^{*}),\ z\equiv|a_{0}|^{2}-|b_{0}|^{2}$
(14)
are the atomic-dipole components ($u$ and $v$) and population-inversion ($z$),
and $a_{0}$ and $b_{0}$ are the complex-valued probability amplitudes to find
the atom in the excited, ${|2\closeket}$, and ground, ${|1\closeket}$, states,
respectively. The system (13) has two integrals of motion, the total energy
$W\equiv\frac{\omega_{r}}{2}p^{2}-u\cos x-\frac{\Delta}{2}z,$ (15)
and the length of the Bloch vector, $u^{2}+v^{2}+z^{2}=1$. Equations (13)
constitute a nonlinear Hamiltonian autonomous system with two and half degrees
of freedom and two integrals of motion.
In classical mechanics there is a qualitative criterion of dynamical chaos,
the maximal Lyapunov exponent $\lambda$, which measures the mean rate of the
exponential divergence of initially closed trajectories in the phase space. In
Fig. 7 we plot this quantity, computed with semiclassical equations of motion
(13) vs the detuning $\Delta$ at $\omega_{r}=10^{-3}$ and $p_{0}=55$. At zero
detuning, the set of semiclassical equations acquires an additional integral
of motion and becomes integrable. The center-of-mass motion is regular at
small values of the detuning, $\Delta\ll 1$, and at large ones, $\Delta>0.8$.
Positive values of $\lambda$ at $0<\Delta<0.8$ characterize unstable motion.
Local instability of the center-of-mass motion produce chaotic motion of an
atom in a rigid standing wave without any modulation of its parameters in
difference from the situation with atoms in a periodically kicked optical
lattice [7, 8]. There is a range of initial conditions and the control
parameters where the center-of-mass motion in an absolutely deterministic
standing wave resembles a random walking. It means that a point-like atom
alternates between flying through the lattice, and being trapped in its wells
changing the direction of motion in a random-like way [15, 16].
In Fig. 8 we illustrate semiclassical chaos with the Poincaré mapping for a
number of atomic ballistic trajectories in the western ($u<0$) and eastern
($u>0$) hemispheres of the Bloch sphere $(u,v,z)$ on the plane $v-z$. One can
see a typical structure with regions of regular motion in the form of islands
and chains of islands filled by regular trajectories. The islands are imbedded
into a stochastic sea, and they are produced by nonlinear resonances of
different orders. Increasing the resolution of the mapping, one can see that
large islands are surrounded by islands of a smaller size each of which, in
turn, is surrounded by a chain of even more smaller islands, and so on.
In quantum mechanics there is no well-defined notion of a trajectory in the
phase space, the very phase space is not continuous due to the Heisenberg
uncertainty relation, and, hence, the Lyapunov exponents can not be computed
(however, see Ref. [21] where a notion of a Lyapunov exponent for quantum
dynamics has been discussed). The main result of this paper is the
establishment of the fact, that chaotically-like complexification of the wave
function, caused by nonadiabatic transitions at the field nodes, occurs
exactly at the same range of the control parameters where the semiclassical
dynamics has been shown to be chaotic in Refs. [14, 15]. It should be stressed
that quantum motion of a wave packet with nonadiabatic transitions between the
two optical potentials is compared with the center-of-mass motion of an
ensemble of atoms each of which moves in a single optical potentials. So, when
we say about a quantum-classical correspondence we mean a correspondence
between the wave function of a single quantum atom and the trajectories of the
ensemble of classical atoms with different values of the initial momentum
$p_{0}$ and at the other equal conditions.
## 4 Conclusion
We have studied coherent dynamics of cold atomic wave packets in a one-
dimensional standing-wave laser field. The problem has been considered in the
momentum representation and in the dressed-state basis where the motion of a
two-level atom was interpreted as a propagation in two optical potentials. The
character of that motion has been shown to depend strongly on the ratio of the
squared detuning, $\Delta^{2}$, to the normalized Doppler shift $\omega_{D}$.
In the regular regime, when $\Delta$ is comparatively large or small, wave
packets move in a simple way. The chaotic regime occurs if
$\Delta^{2}\simeq\omega_{D}$ when the probability for an atom to make
nonadiabatic transitions while crossing the nodes of the standing wave is
large. Atom in this regime of motion simultaneously moves ballistically and is
trapped in a well of the optical potential. This type of motion and
proliferation of wave packets at the nodes result in a complexification of the
wave function both in the momentum and position spaces manifesting itself in
the irregular behavior of the Wigner function.
Comparing the results of the quantum treatment with those obtained in the
semiclassical approximation, when the translational motion has been treated as
a classical one [14, 15], we have found that the wave-packet dynamics is
complicated exactly in that range of the atom-field detuning and recoil
frequency where the classical center-of-mass motion has been shown to be
chaotic in the sense of exponential sensitivity to small variations in initial
conditions or parameters.
As to possible practical applications of the results obtained we mention
atomic lithography to produce small-scale complex prints of cold atoms (see,
for example, beautiful experiments on coherent matter-wave manipulation [22,
23, 24]), new ways to manipulate atomic motion in optical lattices by varying
the atom-filed detuning and atomic ratchets with cold atoms.
## Acknowledgments
This work was supported by the Russian Foundation for Basic Research (project
no. 09-02-00358), the Integration grant from the Far-Eastern and Siberian
branches of the RAS, and the Program “Fundamental Problems of Nonlinear
Dynamics” of the RAS. I would like to thank L. Konkov for preparing Figs.6 and
7.
## References
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Figure 1: The resonant $E_{0}^{(\pm)}$ ($\Delta=0$, dotted curves) and non-
resonant $E_{\Delta}^{(\pm)}$ ($\Delta\not=0$, solid curves) potentials of a
two-level atom in a standing-wave laser field. An atomic wave packet, centered
at $x_{0}=0$, and its initial evolution in the upper and lower potentials are
shown schematically. Figure 2: Mean atomic momentum $\left<p\right>$ vs time
in the regular ($\Delta=1$, the upper bold curve) and chaotic ($\Delta=0.2$)
regimes of motion.
Figure 3: Momentum probability-density distribution ${\it P}(p,\tau)$ vs time
in (a) the regular and (b) chaotic regimes of motion. The color codes the
corresponding values of ${\it P}(p)$.
Figure 4: The probability $|C_{-}(x)|^{2}$ to find the atom in the potential
$E_{\Delta}^{(-)}$ in the moving frame of reference in (a) the regular and (b)
chaotic regimes of motion. The slopes mark positions of the nodes in the
moving frame and the color codes the values of $|C_{-}(x)|^{2}$.
Figure 5: Contour plots of the Wigner function at (a) $\tau=50$ and (b)
$\tau=200$ for the regular regime of the atomic motion. Color online: red and
blue areas show positive and negative values of the Wigner function,
respectively.
Figure 6: The same as in Fig. 5 for the chaotic regime of motion. Figure 7:
Maximal Lyapunov exponent $\lambda$, computed with semiclassical equations of
motion (13), vs the detuning $\Delta$ at $\omega_{r}=10^{-3}$ and $p_{0}=55$.
Figure 8: Poincaré sections of the Bloch sphere illustrating the effect of
semiclassical chaos with point-like atoms at $\omega_{r}=10^{-5}$,
$\Delta=-0.05$ and the total atomic energy $W=36.45$. (a) $u<0$ (western Bloch
hemisphere), (b) $u>0$ (eastern Bloch hemisphere).
|
arxiv-papers
| 2012-01-01T04:14:10 |
2024-09-04T02:49:25.868226
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. V. Prants",
"submitter": "Michael Uleysky",
"url": "https://arxiv.org/abs/1201.0326"
}
|
1201.0334
|
# Introduction to the MSSM
Sudhir K. Vempati Centre for High Energy Physics, Indian Institute of Science,
Bangalore 560 012, India vempati@cts.iisc.ernet.in
###### Abstract
These lecture notes are based on a first course on the Minimal Supersymmetric
Standard Model. The level of the notes is introductory and pedagogical.
Standard Model, basic supersymmetry algebra and its representations are
considered as prerequisites. The topics covered include particle content,
structure of the lagrangian, supersymmetry breaking soft terms, electroweak
symmetry breaking and the sparticle mass spectrum. Popular supersymmetry
breaking models like minimal supergravity and gauge mediation are also
introduced.
## I Prerequisites
These lectures111 Based on lectures presented at SERC school, held at IIT-
Bombay, Mumbai. are devised as an introduction to the Minimal Supersymmetric
Standard Model. In the course of these lectures, we will introduce the basic
features of the Supersymmetric Standard Model, the particle content, the
structure of the lagrangian, feynman rules, supersymmetric breaking soft
terms, Electroweak symmetry breaking and the mass spectrum of the MSSM.
Supersymmetry is a vast subject and these lectures are definitely not a
comprehensive review and in fact, they are also not what one could term as an
introduction to supersymmetric algebra and supersymmetric gauge theories. The
prerequisites for this course are a good knowledge of the Standard Model and
also supersymmetry say, at the level of first eight chapters of Wess and
Baggerwessbagger : supersymmetry transformations, representations,
superfields, supersymmetric gauge theories and elements of supersymmetry
breaking. It is strongly recommended that readers keep a text book on basic
course of supersymmetry westetal ; sohnius ; onethousand with them all the
time for consulting, while going through this lecture notes.
The lectures are organised as follows : in the next section, we will give a
lightening introduction to the Standard Model and the structure of its
lagrangian. The reason for this being that we will like to introduce the MSSM
(Minimal Supersymmetric Standard Model ) in a similar organisational fashion,
which makes it easier to remember the MSSM lagrangian - as well as arranging
the differences and similarities , one expects in the supersymmetric theories
in a simpler way, if possible. The next section would introduce the basic form
of the MSSM lagrangian - the three functions of the chiral/vector superfields
- the superpotential, the Kahler potential and the field strength superfield
and the particle spectrum. The fourth section will be devoted to R-parity and
some sample feynman rules. Supersymmetry breaking and electroweak symmetry
breaking will be introduced in section 5 and the physical supersymmetric
particle mass spectrum will be done in section 6. Higgs sector will be
reviewed in section 7, while we close with some ‘standard’ models of
supersymmetry breaking in section 8.
Finally for the students not completely familiar with Standard Model, we point
out at some references with increasing order of difficulty in reading and
requirements of pre-requisites. These are : (a) Aitchison and Heyathey , Gauge
Field Theories, Vol I and Vol II (b)A good functional introduction to field
theory required for understanding Standard Model can be found in : M.
Srednicki, Quantum Field Theory srednicki (c) M. E. Peskin and D. Schroeder,
Quantum Field Theory peskin (d) E. Abers and B. W. Lee, Physics Reports on
Gauge Field Theories aberslee (e) T. Cheng and L. Li, Gauge theory of
elementary particle physicschengli (f) S. Weinberg, Quantum Theory of Fields,
Vol I -IIweinberg , (g) S. Pokorski, Gauge Field Theories pokorski and (h)
Donoghue, Golowich and Holstein, Dynamics of the Standard Model dynamics .
## II Step 0 : A lightening recap of the Standard Model
The Standard Model (SM) is a spontaneously broken Yang-Mills quantum field
theory describing the strong and electroweak interactions. The theoretical
assumption on which the Standard Model rests on is the principle of local
gauge invariance with the gauge group given by
$G_{SM}\equiv SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y},$ (1)
where the subscript $c$ stands for color, $L$ stands for the ‘left-handed’
chiral group whereas $Y$ is the hypercharge. The particle spectrum and their
transformation properties under these gauge groups are given as,
$\displaystyle Q_{i}\equiv\left(\begin{array}[]{c}{u_{L}}_{i}\\\
{d_{L}}_{i}\end{array}\right)\sim\left(3,~{}2,~{}{1\over 6}\right)$
$\displaystyle U_{i}\equiv{u_{R}}_{i}\sim\left(\bar{3},~{}1,~{}{2\over
3}\right)$ $\displaystyle
D_{i}\equiv{d_{R}}_{i}\sim\left(\bar{3},~{}1,~{}-{1\over 3}\right)$
$\displaystyle L_{i}\equiv\left(\begin{array}[]{c}{\nu_{L}}_{i}\\\
{e_{L}}_{i}\end{array}\right)\sim\left(1,~{}2,~{}-{1\over 2}\right)$
$\displaystyle E_{i}\equiv{e_{R}}_{i}\sim\left(1,~{}1,~{}-1\right)$ (7)
In the above $i$ stands for the generation index, which runs over the there
generations $i=1,2,3$. $Q_{i}$ represents the left handed quark doublets
containing both the up and down quarks of each generation. Similarly, $L_{i}$
represents left handed lepton doublet, $U_{i},~{}D_{i},~{}E_{i}$ represent
right handed up-quark, down-quark and charged lepton singlets respectively.
The numbers in the parenthesis represent the transformation properties of the
particles under $G_{SM}$ in the order given in eq.(1). For example, the quark
doublet $Q$ transforms a triplet (3) under $SU(3)$ of strong interactions, a
doublet (2) under weak interactions gauge group and carry a hypercharge
$(Y/2)$ of 1/6 222Note that the hypercharges are fixed by the Gellman-
Nishijima relation $Y/2~{}=~{}Q-T_{3}$, where $Q$ stands for the charge of the
particle and $T_{3}$ is the eigenvalue of the third generation of the particle
under $SU(2)$.. In addition to the fermion spectra represented above, there is
also a fundamental scalar called Higgs whose transformation properties are
given as
$H\equiv\left(\begin{array}[]{c}H^{+}\\\
H^{0}\end{array}\right)\sim\left(1,~{}2,~{}1/2\right).$ (8)
However, the requirement of local gauge invariance will not be fulfilled
unless one includes the gauge boson fields also. Including them, the total
lagrangian with the above particle spectrum and gauge group can be represented
as,
${\cal L}_{SM}={\cal L}_{F}+{\cal L}_{YM}+{\cal L}_{yuk}+{\cal L}_{S}.$ (9)
The fermion part ${\cal L}_{F}$ gives the kinetic terms for the fermions as
well as their interactions with the gauge bosons. It is given as,
${\cal L}_{F}=i\bar{\Psi}\gamma^{\mu}{\cal{D}}_{\mu}\Psi,$ (10)
where $\Psi$ represents all the fermions in the model,
$\Psi=\left(Q_{i}~{}U_{i},~{}D_{i},~{}L_{i},~{}E_{i}\right)$ (11)
where ${\cal{D}}_{\mu}$ represents the covariant derivative of the field given
as,
${\cal{D}}_{\mu}=\partial\mu-ig_{s}G_{\mu}^{A}\lambda^{A}-i{g\over
2}W_{\mu}^{I}\tau^{I}-ig^{\prime}B_{\mu}Y$ (12)
Here $A=1,..,8$ with $G_{\mu}^{A}$ representing the $SU(3)_{c}$ gauge bosons,
$I=1,2,3$ with $W_{\mu}^{I}$ representing the $SU(2)_{L}$ gauge bosons. The
$U(1)_{Y}$ gauge field is represented by $B_{\mu}$. The kinetic terms for the
gauge fields and their self interactions are given by,
${\cal L}_{YM}=-{1\over 4}G^{\mu\nu A}G_{\mu\nu}^{A}-{1\over 4}W^{\mu\nu
I}W_{\mu\nu}^{I}-{1\over 4}B^{\mu\nu}B_{\mu\nu}$ (13)
with
$\displaystyle G_{\mu\nu}^{A}$ $\displaystyle=$
$\displaystyle\partial_{\mu}G_{\nu}^{A}-\partial_{\nu}G_{\mu}^{A}+g_{s}~{}f_{ABC}G_{\mu}^{B}G_{\nu}^{C}$
$\displaystyle F_{\mu\nu}^{I}$ $\displaystyle=$
$\displaystyle\partial_{\mu}W_{\nu}^{I}-\partial_{\nu}W_{\mu}^{I}+g~{}f_{IJK}W_{\mu}^{J}W_{\nu}^{K}$
$\displaystyle B_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu},$ (14)
where $f_{ABC(IJK)}$ represent the structure constants of the $SU(3)(SU(2))$
group.
In addition to the gauge bosons, the fermions also interact with the Higgs
boson, through the dimensionless Yukawa couplings given by
${\cal
L}_{yuk}=h^{u}_{ij}\bar{Q}_{i}U_{j}\tilde{H}+h^{d}_{ij}\bar{Q}_{i}D_{j}H+h^{e}_{ij}\bar{L}_{i}E_{j}H+H.c$
(15)
where $\tilde{H}=i\sigma^{2}H^{\star}$. These couplings are responsible for
the fermions to attain masses once the gauge symmetry is broken from
$G_{SM}~{}\rightarrow~{}SU(3)_{c}\times U(1)_{em}$. This itselves is achieved
by the scalar part of the lagrangian which undergoes spontaneous symmetry
breakdown. The scalar part of the lagrangian is given by,
${\cal L}_{S}=\left({\cal{D}}_{\mu}H\right)^{\dagger}{\cal{D}}_{\mu}H-V(H),$
(16)
where
$V(H)=\mu^{2}H^{\dagger}H+\lambda\left(H^{\dagger}H\right)^{2}$ (17)
For $\mu^{2}~{}<~{}0$, the Higgs field attains a vacuum expectation value
(vev) at the minimum of the potential. The resulting goldstone bosons are
‘eaten away’ by the gauge bosons making them massive through the so-called
Higgs mechanism. Only one degree of the Higgs field remains physical, the only
scalar particle of the SM - the Higgs boson. The fermions also attain their
masses through their Yukawa couplings, once the Higgs field attains a vev. The
only exception is the neutrinos which do not attain any mass due to the
absence of right handed neutrinos in the particle spectrum and thus the
corresponding Yukawa couplings. Finally, the Standard Model is renormalisable
and anomaly free. We would also insist that the Supersymmetric version of the
Standard Model keeps these features of the Standard Model intact.
#### II.0.1 Think it Over
Here are some important aspects of the Standard Model which have not found a
mention in the above. These are formulated in some sort of a problem mode,
which would require further study.
* •
What is the experiment that showed that there are only three generations of
particles in the Standard Model ? Can one envisage a fourth generation ? If
so, what are the constraints this generation of particles expected to satisfy
?
* •
The gauge bosons ‘mix’ at the tree level by an angle
$\tan~{}\theta_{W}=({g^{\prime}\over g})^{2}$. What happens at the 1-loop
level ? However are the relevant observables classified ? Some relevant
information can be found athollik .
* •
What are the theoretical limits on the Higgs boson mass ? How sensitive is the
upper bound on the Higgs mass from precision measurements to the top quark
mass ? What is the lower limit on the Higgs mass from the LEP experiment ?
Some relevant information can be found atlephwg .
* •
The LHC experiment has been rapidly constraining the allowed parameter space
of the Higgs boson. For latest information have a look at cerntwiki .
* •
What is the CKM mixing ? How well are these angles measured ? What is the
present status after the results from various B-factories about the CP phase
in the SM ? What is the analogous mixing in the leptonic sector called ? In
comparison to the CKM matrix, how well are these angles measured ?
## III Step 1 : Particle Spectrum of MSSM
What we aim to build over the course of next few lectures is a supersymmetric
version of the Standard Model, which means the lagrangian we construct should
not only be gauge invariant under the Standard Model gauge group $G_{SM}$ but
also now be supersymmetric invariant. Such a model is called Minimal
Supersymmetric Standard Model with the the word ’Minimal’ referring to minimal
choice of the particle spectrum required to make it work. Furthermore, we
would also like the MSSM to be renormalisable and anomaly free, just like the
Standard Model is.
Before we proceed to discuss about the particle spectrum, let us remind
ourselves that ordinary quantum fields are upgraded in supersymmetric333All
through this set of lectures, whenever we mention supersymmetry we mean N=1
SUSY ; only one set of SUSY generators. theories to so-called supermultiplets
or superfields444Superfields are functions (fields) written over a
‘superspace’ made of ordinary space ($x_{\mu}$) and two fermionic ‘directions’
($\theta$,$\bar{\theta}$); they are made up of quantum fields whose spins
differ by 1/2. To build interaction lagrangians one normally resorts to this
formalism, originally given by Salam and Strathdeesalamstrahdee , as
superfields simplify addition and multiplication of the representations. It
should be noted however that the component fields may always be recovered from
superfields by a power series expansion in grassman variable, $\theta$.
A chiral superfield has an expansion :
$\Phi=\phi+\sqrt{2}\theta\psi+\theta\theta F,$ (18) where $\phi$ is the scalar
component, $\psi$, the two component spin 1/2 fermion and $F$ the auxiliary
field. A vector superfield in (Wess-Zumino gauge) has an expansion :
$V=-\theta\sigma^{\mu}\bar{\theta}A_{\mu}+i\theta\theta\bar{\theta}\bar{\lambda}-i\bar{\theta}\bar{\theta}\theta\lambda+{1\over
2}\theta\theta\bar{\theta}\bar{\theta}D$ (19) . Given that supersymmetry
transforms a fermion into a boson and vice-versa, supermultiplets or
superfields are multiplets which collect fermion-boson pairs which transform
in to each other. We will deal with two kinds of superfields - vector
superfields and chiral superfields. A chiral superfield555Here we are
presenting the particle content in the off-shell formalism. contains a weyl
fermion, a scalar and and an auxiliary scalar field generally denoted by F. A
vector superfield contains a spin 1 boson, a spin 1/2 fermion and an auxiliary
scalar field called D.
The minimal supersymmetric extension of the Standard Model is built by
replacing every standard model matter field by a chiral superfield and every
vector field by a vector superfield. Thus the existing particle spectrum of
the Standard Model is doubled. The particle spectrum of the MSSM and their
transformation properties under $G_{SM}$ is given by,
$\displaystyle
Q_{i}\equiv\left(\begin{array}[]{cc}u_{L_{i}}&{\tilde{u}}_{L_{i}}\\\
d_{L_{i}}&\tilde{d}_{L_{i}}\end{array}\right)\sim\left(3,~{}2,~{}{1\over
6}\right)$ $\displaystyle
U_{i}^{c}\equiv\left(\begin{array}[]{cc}u_{i}^{c}&\tilde{u}^{c}_{i}\end{array}\right)\sim\left(\bar{3},~{}1,~{}-{2\over
3}\right)$ (23) $\displaystyle
D_{i}\equiv\left(\begin{array}[]{cc}d_{i}^{c}&\tilde{d}^{c}_{i}\end{array}\right)\sim\left(\bar{3},~{}1,~{}{1\over
3}\right)$ (25) $\displaystyle
L_{i}\equiv\left(\begin{array}[]{cc}\nu_{L_{i}}&\tilde{\nu}_{L_{i}}\\\
e_{L_{i}}&\tilde{e}_{L_{i}}\end{array}\right)\sim\left(1,~{}2,~{}-{1\over
2}\right)$ $\displaystyle
E_{i}\equiv\left(\begin{array}[]{cc}e_{i}^{c}&\tilde{e}^{c}_{i}\end{array}\right)\sim\left(1,~{}1,~{}1\right)$
(29)
The scalar partners of the quarks and the leptons are typically named as
‘s’quarks and ‘s’leptons. Together they are called sfermions. For example, the
scalar partner of the top quark is known as the ‘stop’. In the above, these
are represented by a ‘tilde’ on their SM counterparts. As in the earlier case,
the index $i$ stands for the generation index.
There are two distinct features in the spectrum of MSSM : (a) Note that we
have used the conjugates of the right handed particles, instead of the right
handed particles themselves. There is no additional conjugation on the
superfield itselves, the $c$ in the superscript just to remind ourselves that
this chiral superfield is made up of conjugates of SM quantum fields. In
eq.(23), $u^{c}~{}=~{}u_{R}^{\dagger}$ and
$\tilde{u}^{c}~{}=~{}\tilde{u}_{R}^{\star}$. This way of writing down the
particle spectrum is highly useful for reasons to be mentioned later in this
section. Secondly (b) At least two Higgs superfields are required to complete
the spectrum - one giving masses to the up-type quarks and the other giving
masses to the down type quarks and charged leptons. As mentioned earlier, this
is the minimal number of Higgs particles required for the model to be
consistent from a quantum field theory point of view666The Higgs field has a
fermionic partner, higgsino which contributes to the anomalies of the SM. At
least two such fields with opposite hyper-charges ($U(1)_{Y}$) should exist to
cancel the anomalies of the Standard Model.. These two Higgs superfields have
the following transformation properties under $G_{SM}$:
$\displaystyle H_{1}$ $\displaystyle\equiv$
$\displaystyle\left(\begin{array}[]{cc}H^{0}_{1}&\tilde{H}_{1}^{0}\\\
H^{-}_{1}&\tilde{H}_{1}^{-}\end{array}\right)\sim\left(1,~{}2,~{}-{1\over
2}\right)$ (33) $\displaystyle H_{2}$ $\displaystyle\equiv$
$\displaystyle\left(\begin{array}[]{cc}H^{+}_{2}&\tilde{H}_{2}^{+}\\\
H^{0}_{2}&\tilde{H}_{2}^{0}\end{array}\right)\sim\left(1,~{}2,~{}{1\over
2}\right)$ (36)
The Higgsinos are represented by a $\tilde{}$ on them. This completes the
matter spectrum of the MSSM. Then there are the gauge bosons and their super
particles. Remember that in supersymmetric theories, the gauge symmetry is
imposed by the transformations on matter superfields as :
$\Phi^{\prime}=e^{i\Lambda_{l}t_{l}}\Phi$ (37)
where $\Lambda_{l}$ is an arbitrary chiral superfield and $t_{l}$ represent
the generators of the gauge group which are $l$ in number and the index $l$ is
summed over777To be more specific, $t_{l}$ is just a number for the abelian
groups. For non-abelian groups, $t_{l}$ is a matrix and so is $\Lambda_{l}$,
with $\Lambda_{ij}=t^{l}_{ij}\Lambda_{l}$ Note that $V$ is also becomes a
matrix in this case.. The gauge invariance is restored in the kinetic part by
introducing a (real) vector superfield, $V$ such that the combination
$\Phi^{\dagger}e^{gV}\Phi$ (38)
remains gauge invariant. For this to happen, the vector superfield $V$
itselves transforms under the gauge symmetry as
$\delta V=i(\Lambda-\Lambda^{\dagger})$ (39)
The supersymmetric invariant kinetic part of the lagrangian is given by:
$\mathcal{L}_{kin}=\int
d\theta^{2}d\bar{\theta}^{2}\Phi^{\dagger}e^{gV}\Phi=\Phi^{\dagger}e^{gV}\Phi|_{\theta\theta\bar{\theta}\bar{\theta}}$
(40)
In the MSSM, corresponding to three gauge groups of the SM and for each of
their corresponding gauge bosons, we need to add a vector superfield which
transforms as the adjoint under the gauge group action. Each vector superfield
contains the gauge boson and its corresponding super partner called gaugino.
Thus in MSSM we have the following vector superfields and their corresponding
transformation properties under the gauge group, completing the particle
spectrum of the MSSM:
$\displaystyle V_{s}^{A}$ $\displaystyle:$
$\displaystyle\left(\begin{array}[]{cc}G^{\mu
A}&\tilde{G}^{A}\end{array}\right)~{}~{}\sim~{}~{}(8,1,0)$ (42) $\displaystyle
V_{W}^{I}$ $\displaystyle:$ $\displaystyle\left(\begin{array}[]{cc}W^{\mu
I}&\tilde{W}^{I}\end{array}\right)~{}~{}~{}\sim~{}~{}(1,3,0)$ (44)
$\displaystyle V_{Y}$ $\displaystyle:$
$\displaystyle\left(\begin{array}[]{cc}B^{\mu}&\tilde{B}\end{array}\right)~{}~{}\sim~{}~{}(1,1,0)$
(46)
The $G$’s ($G$ and $\tilde{G}$) represent the gluonic fields and their
superpartners called gluinos, the index $A$ runs from $1$ to $8$. The $W$’s
are the $SU(2)$ gauge bosons and their superpartners ‘Winos’, the index $I$
taking values from $1$ to $3$ and finally $B$s represents the $U(1)$ gauge
boson and its superpartner ‘Bino’. Together all the superpartners of the gauge
bosons are called ‘gauginos’. This completes the particle spectrum of the
MSSM.
## IV Step 2: The superpotential and R-parity
The supersymmetric invariant lagrangian is constructed from functions of
superfields. In general there are three functions which are: (a) The Kähler
potential, $K$, which is a real function of the superfields (b) The
superpotential $W$, which is a holomorphic (analytic) function of the
superfields, and (c) the gauge kinetic function $f_{\alpha\beta}$ which
appears in supersymmetric gauge theories. This is the coefficient of the
product of field strength superfields,
$\mathcal{W}_{\alpha}\mathcal{W}^{\beta}$. The field strength superfield is
derived from the vector superfields contained in the model. $f_{\alpha\beta}$
determines the normalisation for the gauge kinetic terms. In MSSM,
$f_{\alpha\beta}=\delta_{\alpha\beta}$. The lagrangian of the MSSM is thus
given in terms of $G_{SM}$ gauge invariant functions $K$, $W$ and add the
field strength superfield $\mathcal{W}$, for each of the vector superfields in
the spectrum.
The gauge invariant Kähler potential has already been discussed in the
eqs.(40). For the MSSM case, the Kähler potential will contain all the three
vector superfields corresponding to the $G_{SM}$ given in the eq.(46). Thus we
have :
$\mathcal{L}_{kin}=\int
d\theta^{2}d\bar{\theta}^{2}\sum_{\scriptstyle{\text{SU(3)},\text{SU(2)},\text{U(1)}}}\Phi_{\beta}^{\dagger}~{}e^{gV}\Phi_{\beta}$
(47)
where the index $\beta$ runs over all the matter fields
$\Phi_{\beta}~{}=~{}\\{Q_{i},U^{c}_{i},D^{c}_{i},L_{i},e^{c}_{i},H_{1},H_{2}\\}$888The
indices $i,j,k$ always stand for the three generations through out this notes,
taking values between 1 and 3. in appropriate representations. Corresponding
to each of the gauge groups in $G_{SM}$, all the matter fields which transform
non-trivially under this gauge group999As given in the list of representations
in eqs. (23,33) are individually taken and the grassman
($d\theta^{2}d{\bar{\theta}}^{2}$) integral is evaluated with the
corresponding vector superfields in the exponential101010Remember that the
function $e^{gV}$ truncates at ${1\over 2}g^{2}V^{2}$ in the Wess-Zumino
gauge. In fact, in this gauge, this function can be determined by noting:
$\exp
V_{WZ}=1-\theta\sigma^{\mu}\bar{\theta}A_{\mu}+i\theta\theta\bar{\theta}\bar{\lambda}-i\bar{\theta}\bar{\theta}\theta\lambda+{1\over
2}~{}\theta\theta\bar{\theta}\bar{\theta}~{}(D-{1\over 2}A^{\mu}A_{\mu}),$
(48) for an abelian Vector superfield. Here as usual $\lambda$ denotes the
gaugino field while $A_{\mu}$ represents the gauge field. D represents the
auxiliary field of the Vector multiplet. The extension to the non-abelian case
is straight forward.. After expanding and evaluating the integral, we get the
lagrangian which is supersymmetric invariant in terms of the ordinary quantum
fields - the SM particles and the superparticles. This part of the lagrangian
would give us the kinetic terms for the SM fermions, kinetic terms for the
sfermions and their interactions with the gauge bosons and in addition also
the interactions of the type: fermion-sfermion-gaugino which are structurally
like the Yukawa interactions ($ff\phi$), but carry gauge couplings. Similarly,
for the Higgs fields, this part of the lagrangian gives the kinetic terms for
the Higgs fields and their fermionic superpartners Higgsinos and the
interaction of the gauge bosons with the Higgs fields and Higgs-Higgsino-
gaugino vertices.
The second possible function of the superfields is the analytic or holomorphic
function111111This would mean that $W$ is purely a function of complex fields
($z_{1}z_{2}z_{3}$) or its conjugates
($z_{1}^{\star}z_{2}^{\star}z_{3}^{\star}$). of the superfields called the
superpotential, $W$. This function essentially gives the interaction part of
the lagrangian which is independent of the gauge couplings, like the Yukawa
couplings. If renormalisability is demanded, the dimension of the
superpotential is restricted to be less than or equal to three,
$[W]~{}\leq~{}3$ i.e, only products of three or less number of chiral
superfields are allowed. Imposing this restriction of renormalisability the
most general $G_{SM}$ gauge invariant form of the $W$ for the matter spectrum
of MSSM (23,33) is given as
$W=W_{1}+W_{2},\\\ $ (49)
where
$\displaystyle W_{1}$ $\displaystyle=$ $\displaystyle
h^{u}_{ij}Q_{i}U_{j}^{c}H_{2}+h^{d}_{ij}Q_{i}D_{j}^{c}H_{1}+h_{ij}^{e}L_{i}E_{j}^{c}H_{1}+\mu
H_{1}H_{2}$ (50) $\displaystyle W_{2}$ $\displaystyle=$
$\displaystyle\epsilon_{i}L_{i}H_{2}+\lambda_{ijk}L_{i}L_{j}E_{k}^{c}+\lambda^{\prime}_{ijk}L_{i}Q_{j}D_{k}^{c}+\lambda^{\prime\prime}_{ijk}U_{i}^{c}D_{j}^{c}D_{k}^{c}.$
(51)
Here we have arranged the entire superpotential in to two parts, $W_{1}$ and
$W_{2}$ with a purpose. Though both these parts are gauge invariant, $W_{2}$
also violates the global lepton number and baryon quantum numbers. The
simultaneous presence of both these set of operators can lead to rapid proton
decay and thus can make the MSSM phenomenologically invalid. For these
reasons, one typically imposes an additional symmetry called R-parity in MSSM
which removes all the dangerous operators in $W_{2}$. We will deal with
R-parity in greater detail in the next section. For the present, let us just
set $W_{2}$ to be zero due to a symmetry called R-parity and just call $W_{1}$
as $W$. The lagrangian can be derived from the superpotential containing
(mostly) gauge invariant product of the three superfields by taking the
$\theta\theta$ component, which can be represented in the integral form as
$\mathcal{L}_{yuk}=\int d\theta^{2}~{}~{}W(\Phi)+\int
d\bar{\theta}^{2}~{}~{}\bar{W}(\bar{\Phi})$ (52)
This part gives121212The $\theta\theta$ components of the product of three
chiral superfields is given aswessbagger
$\Phi_{i}\Phi_{j}\Phi_{k}|_{\theta\theta}=-\psi_{i}\psi_{j}\phi_{k}-\psi_{j}\psi_{k}\phi_{i}-\psi_{k}\psi_{i}\phi_{j}+F_{i}\phi_{j}\phi_{k}+F_{j}\phi_{k}\phi_{i}+F_{k}\phi_{i}\phi_{j},$
(53) where as earlier, $\psi_{i}$ represents the fermionic, $\phi_{i}$ the
scalar and $F_{i}$ the auxiliary component of the chiral superfield
$\Phi_{i}$. Similarly for the product of two superfields on has :
$\Phi_{i}\Phi_{j}|_{\theta\theta}=-\psi_{i}\psi_{j}+F_{i}\phi_{j}+F_{j}\phi_{i}$
(54) the standard Yukawa couplings for the fermions with the Higgs, in
addition also give the fermion-sfermion-higgsino couplings and scalar terms.
For example, the coupling $h^{u}_{ij}~{}Q_{i}u_{j}^{c}H_{2}$ in the
superpotential has the following expansion in terms of the component fields :
$\displaystyle\mathcal{L}_{yuk}$ $\displaystyle\supset$ $\displaystyle
h^{u}_{ij}~{}Q_{i}u_{j}^{c}H_{2}~{}|_{\theta\theta}$ (55)
$\displaystyle\supset$ $\displaystyle
h^{u}_{ij}~{}(~{}u_{i}u_{j}^{c}H_{2}^{0}-d_{i}u_{j}^{c}H_{2}^{+}~{})~{}|_{\theta\theta}$
$\displaystyle\supset$ $\displaystyle
h^{u}_{ij}(\psi_{u_{i}}\psi_{u_{j}^{c}}\phi_{H_{2}^{0}}+\phi_{\tilde{u}_{i}}\psi_{u_{j}^{c}}\psi_{\tilde{H}_{2}^{0}}+\psi_{u_{i}}\phi_{\tilde{u}_{j}^{c}}\psi_{\tilde{H}_{2}^{0}}-\psi_{d_{i}}\psi_{u_{j}^{c}}\phi_{H_{2}^{+}}-\phi_{\tilde{d}_{i}}\psi_{u_{j}^{c}}\psi_{\tilde{H}_{2}^{+}}-\psi_{d_{i}}\phi_{\tilde{u}_{j}^{c}}\psi_{\tilde{H}_{2}^{+}})$
$\displaystyle\equiv$ $\displaystyle
h^{u}_{ij}~{}(~{}u_{i}u_{j}^{c}H_{2}^{0}+~{}\tilde{u}_{i}u_{j}^{c}\tilde{H}_{2}^{0}+u_{i}\tilde{u}_{j}^{c}\tilde{H}_{2}^{0}-d_{i}u_{j}^{c}H_{2}^{+}~{}-\tilde{d}_{i}u_{j}^{c}\tilde{H}_{2}^{+}~{}-d_{i}\tilde{u}_{j}^{c}\tilde{H}_{2}^{+}~{}),$
(56)
where in the last equation, we have used the same notation for the chiral
superfield as well as for its lowest component namely the scalar component.
Note that we have not written the F-terms which give rise to the scalar terms
in the potential. Similarly, there is the $\mu$ term which gives ‘Majorana’
type mass term for the Higgsino fields.
Finally, for every vector superfield (or a set of superfields) we have an
associated field strength superfield $\mathcal{W}^{\alpha}$, which gives the
kinetic terms for the gauginos and the field strength tensors for the gauge
fields. Given that it is a chiral superfield, the component expansion is given
by taking the $\theta\theta$ component of ‘square’ of that superfield131313In
the Wess-Zumino gauge, $\mathcal{W}_{\alpha}=-{1\over
4}\bar{\mathcal{D}}\bar{\mathcal{D}}\mathcal{D}_{\alpha}V_{WZ}$ wessbagger
($\mathcal{D}$ is the differential operator on superfields) and the lagrangian
has the form : $\mathcal{L}\supset{1\over
4}\left(\mathcal{W}^{\alpha}\mathcal{W}_{\alpha}|_{\theta\theta}+\mathcal{W}^{\dot{\alpha}}\mathcal{W}_{\dot{\alpha}}|_{\bar{\theta}\bar{\theta}}\right)={1\over
2}D^{2}-{1\over
4}F_{\mu\nu}F^{\mu\nu}-i\lambda\sigma^{\mu}\partial_{\mu}\bar{\lambda}$ (57)
$D$ represents the auxiliary component of the vector superfields. The
extension to non-abelian vector superfields in straight forward.. In the MSSM,
we have to add the corresponding field strength $\mathcal{W}$ superfields for
electroweak vector superfields, $W$ and $B$ as well as for the gluonic $G$
vector superfields of eqs.(46).
So far we have kept the auxiliary fields ($D$ and $F$) of various chiral and
vector superfields in the component form of our lagrangian. However, given
that these fields are unphysical, they have to be removed from the lagrangian
to go “on-shell”. To eliminate the $D$ and $F$ fields, we have to use the
equations of motions of these fields which have simple solutions for the $F$
and $D$ as :
$F_{i}={\partial
W\over\partial\phi_{i}}~{}~{}~{}~{}\;;\;~{}~{}~{}~{}D_{A}=-g_{A}~{}~{}\phi^{\star}_{i}~{}T^{A}_{ij}~{}\phi_{j},$
(58)
where $\phi_{i}$ represents all the scalar fields present in MSSM. The index
$A$ runs over all the gauge groups in the model. For example, for $U(1)_{Y}$,
$T^{A}_{ij}=~{}(Y^{2}/2)\delta_{ij}$. The $F$ and $D$ terms together form the
scalar potential of the MSSM141414Later we will see that there are also
additional terms which contribute to the scalar potential which come from the
supersymmetry breaking sector. which is given as
$V=\sum_{i}~{}|F_{i}~{}|^{2}~{}+~{}{1\over 2}~{}D^{A}D_{A}$ (59)
Putting together, we see that the lagrangian of the MSSM with SUSY unbroken is
of the form :
${\cal L}_{MSSM}^{(0)}=\int\left(d\theta^{2}~{}W(\Phi)+H.c\right)+\int
d\theta^{2}~{}d{\bar{\theta}}^{2}~{}\Phi_{i}^{\dagger}~{}e^{gV}~{}\Phi_{i}+\int\left(d\theta^{2}~{}{\cal
W}^{\alpha}{\cal W}_{\alpha}+H.c\right).$ (60)
where all the functions appearing in (60) have been discussed in eqs.(47,50)
and (57).
#### IV.0.1 Think it over
* •
The full supersymmetric lagrangian of the Standard Model can be constructed
from the prescription given in the above section. Identify the dominant one-
loop contributions for the Higgs particle. Note that SUSY is still unbroken.
What are the dominant 1-loop contributions for other scalar particles, say the
stop ? Compute the processes $\mu~{}\to~{}e+~{}\gamma$ and $K^{0}-\bar{K}^{0}$
mixing in this limit.
* •
As we have seen, W is a holomorphic function and that there are two Higgs
doublets giving masses to up type and down type quarks separately. (a) Give
examples of operators which are gauge invariant but non-holomorphic ? (b) Show
that such operators involving the Higgs fields will lead to Yukawa like
couplings with the “wrong” Higgs. Study the implications of such couplings.
Historical Note
Supersymmetries were first introduced in the context of string theories by
Ramond. In quantum field theories, this symmetry is realised through fermionic
generators, thus escaping the no-go theorems of Coleman and Mandula weinbergv3
. The simplest Lagrangian realising this symmetry in four dimensions was built
by Wess and Zumino which contains a spin ${1\over 2}$ fermion and a scalar.
In particle physics, supersymmetry plays an important role in protecting the
Higgs mass. To understand how it protects the Higgs mass, let us consider the
hierarchy problem once again. The Higgs mass enters as a bare mass parameter
in the Standard Model lagrangian, eq.(16). Contributions from the self energy
diagrams of the Higgs are quadratically divergent pushing the Higgs mass up to
cut-off scale. In the absence of any new physics at the intermediate energies,
the cut-off scale is typically $M_{GUT}$ or $M_{planck}$. Cancellation of
these divergences with the bare mass parameter would require fine-tuning of
order one part in $10^{-36}$ rendering the theory ‘unnatural’natural . In a
complete GUT model like SU(5) this might reflect as a severe problem of
doublet-triplet splitting buras ; gildener . On the other hand, if one has
additional contributions, say, for example, for the diagram with the Higgs
self coupling, there is an additional contribution from a fermionic loop, with
the fermion carrying the same mass as the scalar, the contribution from this
additional diagram would now cancel the quadratically divergent part of the SM
diagram, with the total contribution now being only logarithmically divergent.
If this mechanism needs to work for all the diagrams, not just for the Higgs
self-coupling and for all orders in perturbation theory, it would require a
symmetry which would relate a fermion and a boson with same mass.
Supersymmetry is such a symmetry.
### IV.1 R-parity
In the previous section, we have seen that there are terms in the
superpotential, eq.(51) which are invariant under the Standard Model gauge
group $G_{SM}$ but however violate baryon ($\mathrm{B}$) and individual lepton
numbers ($\mathrm{L}_{e,\mu,\tau}$). At the first sight, it is bit surprising
: the matter superfields carry the same quantum numbers under the $G_{SM}$
just like the ordinary matter fields do in the Standard Model and $\mathrm{B}$
and $\mathrm{L}_{e,\mu,\tau}$ violating terms are not present in the Standard
Model. The reason can be traced to the fact that in the MSSM, where matter
sector is represented in terms of superfields, there is no distinction between
the fermions and the bosons of the model. In the Standard Model, the Higgs
field is a boson and the leptons and quarks are fermions and they are
different representations of the Lorentz group. This distinction is lost in
the MSSM, the Higgs superfield, $H_{1}$ and the lepton superfields $L_{i}$
have the same quantum numbers under $G_{SM}$ and given that they are both
(chiral) superfields, there is no way of distinguishing them. For this reason,
the second part of the superpotential $W_{2}$ makes an appearance in
supersymmetric version of the Standard Model. In fact, the first three terms
of eq.(51) can be achieved by replacing $H_{1}~{}\to~{}L_{i}$ in the terms
containing $H_{1}$ of $W_{1}$.
The first three terms of the second part of the superpotential $W_{2}$
(eq.(51)), are lepton number violating whereas the last term is baryon number
violating. The simultaneous presence of both these interactions can lead to
proton decay, for example, through a squark exchange. An example of such an
process in given in Figure 1. Experimentally the proton is quite stable. In
fact its life time is pretty large
$\stackrel{{\scriptstyle\scriptstyle>}}{{\scriptstyle\sim}}~{}\mathcal{O}(10^{33})$
years skproton . Thus products of these couplings ($\lambda^{\prime\prime}$
and one of ($\lambda^{\prime}~{},\epsilon,~{}\lambda$) which can lead to
proton decay are severely constrained to be of the order of
$\mathcal{(}O)(10^{-20})$151515The magnitude of these constraints depends also
on the scale of supersymmetry breaking, which we will come to discuss only in
the next section. For a list of constraints on R-violating couplings, please
see G. Bhattacharyya gautamB .. Thus to make the MSSM phenomenologically
viable one should expect these couplings in $W_{2}$ to take such extremely
small values.
$u$$u$$u$$d$$\lambda^{{}^{\prime\prime}}$$\tilde{s}$$\lambda^{\prime}$$\bar{l}$$\bar{u}$
Figure 1: A sample diagram showing the decay of the proton in the presence of
R-parity violating couplings.
A more natural way of dealing with such small numbers for these couplings
would be to set them to be zero. This can be arrived at by imposing a discrete
symmetry on the lagrangian called R-parity. R-parity has been originally
introduced as a discrete R-symmetry 161616R-symmetries are symmetries under
which the $\theta$ parameter transform non-trivially. by Ferrar and Fayet
fayetfarrar and then later realised to be of the following form by Ferrar and
Weinberg weinfarrar acting on the component fields:
$R_{p}=(-1)^{3(B-L)+2s},$ (61)
where B and L represent the Baryon and Lepton number respectively and s
represents the spin of the particle. Under R-parity the transformation
properties of various superfields can be summarised as:
$\displaystyle\\{V_{s}^{A},V_{w}^{I},V_{y}\\}$ $\displaystyle\rightarrow$
$\displaystyle\\{V_{s}^{A},V_{w}^{I},V_{y}\\}$ $\displaystyle\theta$
$\displaystyle\rightarrow$ $\displaystyle-\theta^{\star}$
$\displaystyle\\{Q_{i},U^{c}_{i},D^{c}_{i},L_{i},E_{i}^{c}\\}$
$\displaystyle\rightarrow$
$\displaystyle-\\{Q_{i},U^{c}_{i},D_{i}^{c},L_{i},E_{i}^{c}\\}$
$\displaystyle\\{H_{1},H_{2}\\}$ $\displaystyle\rightarrow$
$\displaystyle\\{H_{1},H_{2}\\}$ (62)
Imposing these constraints on the superfields will now set all the couplings
in $W_{2}$ to zero.
Imposing R-parity has an advantage that it provides a natural candidate for
dark matter. This can be seen by observing that R-parity distinguishes a
particle from its superpartner. This ensures that every interaction vertex has
at least two supersymmetric partners when R-parity is conserved. The lightest
supersymmetric particle (LSP) cannot decay in to a pair of SM particles and
remains stable. R-parity can also be thought of as a remnant symmetry theories
with an additional $U(1)$ symmetry, which is natural in a large class of
supersymmetric Grand Unified theories. Finally, one curious fact about
R-parity : it should be noted that R-parity constraints baryon and lepton
number violating couplings of dimension four or rather only at the
renormalisable level. If one allows for non-renormalisable operators in the
MSSM, i.e that is terms of dimension more than three in the superpotential,
they can induce dim 6 operators which violate baryon and lepton numbers at the
lagrangian level and are still allowed by R-parity. Such operators are
typically suppressed by high mass scale $\sim M_{Pl}$ or $M_{GUT}$ and thus
are less dangerous. In the present set of lectures, we will always impose
R-parity in the MSSM so that the proton does not decay, though there are
alternatives to R-parity which can also make proton stable.
#### IV.1.1 Think it over
* •
Is imposing R-parity the only way to get rid of the terms which lead to proton
decay ? (Hint: For proton decay to occur both $\mathrm{L}$ and $\mathrm{B}$
violating operators are required. R-parity removes both these sets of
operators which is unnecessary. We can think of discrete symmetries which can
remove only either $\mathrm{B}$ or $\mathrm{L}$ type of operators.) See for
examplerosshall .
## V STEP 3: Supersymmetry breaking
So far, we have seen that the Supersymmetric Standard Model lagrangian can
also be organised in a similar way like the Standard Model lagrangian though
one uses functions of superfields now to get the lagrangian rather than the
ordinary fields. In the present section we will cover the last part (term) of
the total MSSM lagrangian
$\mathcal{L}_{\mbox{MSSM}}=\mathcal{L}_{\mbox{gauge/kinetic}}\left(K(\Phi,V)\right)+\mathcal{L}_{\mbox{yukawa}}\left(W(\Phi)\right)+\mathcal{L}_{\mbox{scalar}}\left(F^{2},D^{2}\right)+\mathcal{L}_{\mbox{SSB}}$
(63)
which we have left out so far and that concerns supersymmetry breaking (SSB).
Note that the first three terms are essentially from
$\mathcal{L}_{\mbox{MSSM}}^{(0)}$ of eq.(60). In Nature, we do not observe
supersymmetry. Supersymmetry breaking has to be incorporated in the MSSM to
make it realistic. In a general lagrangian, supersymmetry can be broken
spontaneously if the auxiliary fields F or D appearing in the definitions of
the chiral and vector superfields respectively attain a vacuum expectation
value (vev). If the $F$ fields get a vev, it is called $F$-breaking whereas if
the $D$ fields get a vev, it is called $D$-breaking.
Incorporation of spontaneous SUSY breaking in MSSM would mean that at least
one (or more) of the F-components corresponding to one ( or more) of the MSSM
chiral (matter) superfields would attain a vacuum expectation value. However,
this approach fails as this leads to phenomenologically unacceptable
prediction that at least one of the super-partner should be lighter (in mass)
than the ordinary particle. This is not valid phenomenologically as such a
light super partner (of SM particle) has been ruled out experimentally. One
has to think of a different approach for incorporating supersymmetry breaking
in to the MSSM Luty .
Figure 2: A schematic diagram showing SUSY breaking using Hidden sector models
One of the most popular and successful approaches has been to assume another
sector of the theory consisting of superfields which are not charged under the
Standard Model gauge group. Such a sector of the theory is called ‘Hidden
Sector’ as they cannot been ”seen” like the Standard Model particles and
remain hidden. Supersymmetry can be broken spontaneously in this sector. This
information is communicated to the visible sector or MSSM through a messenger
sector. The messenger sector can be made up of gravitational interactions or
ordinary gauge interactions. The communication of supersymmetry breaking leads
to supersymmetry breaking terms in MSSM. Thus, supersymmetry is not broken
spontaneously within the MSSM, but explicitly by adding supersymmetry breaking
terms in the lagrangian.
However, not all supersymmetric terms can be added. We need to add only those
terms which do not re-introduce quadratic divergences back into the
theory171717Interaction terms and other couplings which do not lead to
quadratically divergent (in cut-off $\Lambda$) terms in the theory once loop
corrections are taken in to consideration. It essentially means we only add
dimensional full couplings which are supersymmetry breaking.. It should be
noted that in most models of spontaneous supersymmetry breaking, only such
terms are generated. These terms which are called “soft” supersymmetry
breaking terms can be classified as follows:
* •
a) Mass terms for the gauginos which are a part of the various vector
superfields of the MSSM.
* •
b) Mass terms for the scalar particles,
$m^{2}_{\phi_{ij}}~{}\phi_{i}^{\star}\phi_{j}$ with $\phi_{i,j}$ representing
the scalar partners of chiral superfields of the MSSM.
* •
c) Trilinear scalar interactions, $A_{ijk}\phi_{i}\phi_{j}\phi_{k}$
corresponding to the cubic terms in the superpotential.
* •
d) Bilinear scalar interactions, $B_{ij}\phi_{i}\phi_{j}$ corresponding to the
bilinear terms in the superpotential.
Note that all the above terms are dimensionful. Adding these terms would make
the MSSM non-supersymmetric and thus realistic. The total MSSM lagrangian is
thus equal to
${\cal L}_{total}={\cal L}_{MSSM}^{(0)}+{\cal L}_{SSB}$ (64)
with ${\cal L}_{MSSM}^{(0)}$ given in eq.(60). Sometimes in literature we have
${\cal L}_{SSB}={\cal L}_{soft}$. Let us now see the complete list of all the
soft SUSY breaking terms one can incorporate in the MSSM:
1. 1.
Gaugino Mass terms: Corresponding to the three vector superfields (for gauge
groups $U(1)$, $SU(2)$ and $SU(3)$) we have $\tilde{B},\tilde{W}$ and
$\tilde{G}$) we have three gaugino mass terms which are given as
$M_{1}\tilde{B}\tilde{B}$, $M_{2}\tilde{W}_{I}\tilde{W}_{I}$ and
$M_{3}\tilde{G}_{A}\tilde{G}_{A}$, where $I(A)$ runs over all the
$SU(2)(SU(3))$ group generators.
2. 2.
Scalar Mass terms: For every scalar in each chiral (matter) superfield , we
can add a mass term of the form $m^{2}~{}\phi_{i}^{\star}\phi_{j}$. Note that
the generation indices $i,j$ need not be the same. Thus the mass terms can
violate flavour. Further, given that SUSY is broken prior to $SU(2)\times
U(1)$ breaking , all these mass terms for the scalar fields should be written
in terms of their ‘unbroken’ $SU(2)\times U(1)$ representations. Thus the
scalar mass terms are : $m_{Q_{ij}}^{2}\tilde{Q}_{i}^{\dagger}\tilde{Q}_{j}$ ,
$m_{u_{ij}}^{2}\tilde{u^{c}}_{i}^{\star}\tilde{u^{c}}_{j}$ ,
$m_{d_{ij}}^{2}\tilde{d^{c}}_{i}^{\star}\tilde{d^{c}}_{j}$ ,
$m_{L_{ij}}^{2}\tilde{L}_{i}^{\dagger}\tilde{L}_{j}$ ,
$m_{e_{ij}}^{2}\tilde{e^{c}}_{i}^{\star}\tilde{e^{c}}_{j}$ ,
$m_{H_{1}}^{2}H_{1}^{\dagger}H_{1}$ and $m_{H_{2}}^{2}H_{2}^{\dagger}H_{2}$.
3. 3.
Trilinear Scalar Couplings: As mentioned again, there are only three types of
trilinear scalar couplings one can write which are $G_{SM}$ gauge invariant.
In fact, their form exactly follows from the Yukawa couplings. These are :
$A^{u}_{ij}\tilde{Q}_{i}\tilde{u}^{c}_{j}H_{2}$,
$A^{d}_{ij}\tilde{Q}_{i}\tilde{d}^{c}_{j}H_{1}$ and
$A^{e}_{ij}\tilde{L}_{i}\tilde{e}^{c}_{j}H_{1}$.
4. 4.
Bilinear Scalar Couplings: Finally, there is only one bilinear scalar coupling
(other than the mass terms) which is gauge invariant. The form of this term
also follows from the superpotential. It is given as : $BH_{1}H_{2}$.
Adding all these terms completes the lagrangian for the MSSM. However, the
particles are still not in their ‘physical’ basis as $SU(2)\times U(1)$
breaking is not yet incorporated. Once incorporated the physical states of the
MSSM and their couplings could be derived.
## VI STEP 4: $SU(2)\times U(1)$ breaking
As a starting point, it is important to realize that the MSSM is a two Higgs
doublet model i.e, SM with two Higgs doublets instead of one, with a different
set of couplings higgshunter . Just as in Standard Model, spontaneous breaking
of $SU(2)_{L}\times U(1)_{Y}~{}\to~{}U(1)_{EM}$ can be incorporated here too.
Doing this leads to constraints relating various parameters of the model. To
see this, consider the neutral Higgs part of the total scalar potential
including the soft terms. It is given as
$\displaystyle V_{scalar}$ $\displaystyle=$
$\displaystyle(m_{H_{1}}^{2}+\mu^{2})|H_{1}^{0}|^{2}+(m_{H_{2}}^{2}+\mu^{2})|H_{2}^{0}|^{2}-(B_{\mu}\mu
H_{1}^{0}H_{2}^{0}+H.c)$ (65) $\displaystyle+$ $\displaystyle{1\over
8}(g^{2}+g^{\prime 2})({H_{2}^{0}}^{2}-{H_{1}^{0}}^{2})^{2}+\ldots,$
where $H_{1}^{0},H_{2}^{0}$ stand for the neutral Higgs scalars and we have
parameterised the bilinear soft term $B\equiv B_{\mu}\mu$. Firstly, we should
require that the potential should be bounded from below. This gives the
condition (in field configurations where the D-term goes to zero, i.e, the
second line in eq.(65)):
$2B_{\mu}<2|\mu|^{2}~{}+~{}m_{H_{2}}^{2}~{}+m_{H_{1}}^{2}$ (66)
Secondly, the existence of a minima for the above potential would require at
least one of the Higgs mass squared to be negative giving the condition,
(determinant of the $2\times 2$ neutral Higgs mass squared matrix should be
negative)
$B_{\mu}^{2}>(|\mu|^{2}~{}+~{}m_{H_{2}}^{2}~{})~{}(|\mu|^{2}+m_{H_{1}}^{2})$
(67)
In addition to ensuring the existence of a minima, one would also require that
the minima should be able to reproduce the standard model relations i.e,
correct gauge boson masses. We insist that both the neutral Higgs attain
vacuum expectation values :
$<H^{0}_{1}>={v_{1}\over\sqrt{2}}\;\;\;\;;\;\;\;<H_{2}^{0}>={v_{2}\over\sqrt{2}}$
(68)
and furthermore we define
$~{}v_{1}^{2}+v_{2}^{2}~{}=~{}v^{2}~{}=~{}246^{2}~{}\mbox{GeV}^{2},$
where $v$ represents the vev of the Standard Model (SM) Higgs field. However,
these vevs should correspond to the minima of the MSSM potential. The minima
are derived by requiring $\partial V/\partial H_{1}^{0}~{}=~{}0$ and $\partial
V/H_{2}^{0}~{}=~{}0$ at the minimum, where the form of $V$ is given in
eq.(65). These derivative conditions lead to relations between the various
parameters of the model at the minimum of the potential. We have, using the
Higgs vev (68) and the formulae for181818In this lecture note, we will be
using $g_{2}~{}=g~{}=~{}g_{W}$ for the SU(2) coupling, whereas
$g^{\prime}=g_{1}$ for the $U(1)_{Y}$ coupling and $g_{s}=g_{3}$ for the SU(3)
strong coupling. $M_{Z}^{2}~{}=~{}{1\over 4}(g^{2}+g^{{}^{\prime}~{}2})v^{2}$,
the minimisation conditions can rewritten as
$\displaystyle{1\over 2}M_{Z}^{2}$ $\displaystyle=$
$\displaystyle{m_{H_{1}}^{2}-\tan^{2}\beta~{}m_{H_{2}}^{2}\over\tan^{2}\beta-1}-\mu^{2}$
$\displaystyle\mbox{Sin}2\beta$ $\displaystyle=$
$\displaystyle{2B_{\mu}~{}\mu~{}\over m_{H_{2}}^{2}+m_{H_{1}}^{2}+2\mu^{2}},$
(69)
where we have used the definition $\tan\beta=v_{2}/v_{1}$ as the ratio of the
vacuum expectation values of $H_{2}^{0}$ and $H_{1}^{0}$ respectively. Note
that the parameters $m_{H_{1}}^{2},~{}m_{H_{2}}^{2},B_{\mu}$ are all
supersymmetry breaking ‘soft’ terms. $\mu$ is the coupling which comes in the
superpotential giving the supersymmetry conserving masses to the Higgs
scalars. These are related to the Standard Model parameters $M_{Z}$ and a
ratio of vevs, parameterised by an angle tan$\beta$. Thus these conditions
relate SUSY breaking soft parameters with the SUSY conserving ones and the
Standard Model parameters. For any model of supersymmetry to make contact with
reality, the above two conditions (VI )need to be satisfied.
The above minimisation conditions are given for the ‘tree level’ potential
only. 1-loop corrections a ’la Coleman-Weinberg can significantly modify these
minima. We will discuss a part of them in later sections when we discuss the
Higgs spectrum. Finally we should mention that, in a more concrete approach,
one should consider the entire scalar potential including all the scalars in
the theory, not just confining ourselves to the neutral Higgs scalars. For
such a potential one should further demand that there are no deeper minima
which are color and charge breaking (which effectively means none of the
colored and charged scalar fields get vacuum expectation values). These
conditions lead to additional constraints on parameters of the MSSMcasasdimo .
#### VI.0.1 Think it over
* •
In the MSSM, we have considered here contains two Higgs doublets. In addition
to $H_{1}$ and $H_{2}$, consider an additional Higgs field field $S$, which
transforms as a singlet under all the gauge groups of $G_{SM}$. Write down the
superpotential including the singlet field $S$ invariant under $G_{SM}$.
Derive the corresponding scalar potential including the soft SUSY breaking
terms. Minimise the neutral Higgs potential and derive the electro-weak
minimisation conditions. How many are there and what are they? (Hint: Assume
the $S$ field also develops a vev and that its vev is much larger than $v_{1}$
and $v_{2}$. )
## VII Step 5: Mass spectrum
We have seen in the earlier section, supersymmetry breaking terms introduce
mass-splittings between ordinary particles and their super-partners. Given
that particles have zero masses in the limit of exact $G_{SM}$, only
superpartners are given soft mass terms. After the $SU(2)~{}\times U(1)$
breaking, ordinary particles as well as superparticles attain mass terms. For
the supersymmetric partners, these mass terms are either additional
contributions or mixing terms between the various super-particles. Thus, just
like in the case of ordinary SM fermions, where one has to diagonalise the
fermion mass matrices to write the lagrangian in the ‘on-shell’ format or the
physical basis, a similar diagonalisation has to be done for the super-
symmetric particles and their mass matrices.
#### VII.0.1 The Neutralino Sector
To begin with lets start with the gauge sector. The superpartners of the
neutral gauge bosons (neutral gauginos) and the fermionic partners of the
neutral higgs bosons (neutral higgsinos) mix to form Neutralinos. The
neutralino mass matrix in the basis
$\mathcal{L}~{}\supset~{}{1\over
2}~{}\Psi_{N}\mathcal{M}_{N}\Psi_{N}^{T}~{}+H.c$
where
$\Psi_{N}=\\{\tilde{B},~{}\tilde{W}^{0},\tilde{H}_{1}^{0},\tilde{H}_{2}^{0}\\}$
is given as :
$\mathcal{M}_{N}~{}=~{}\left(\matrix{M_{1}&0&-M_{Z}c\beta~{}s\theta_{W}&M_{Z}s\beta~{}s\theta_{W}\cr
0&M_{2}&M_{Z}c\beta~{}c\theta_{W}&M_{Z}s\beta~{}c\theta_{W}\cr-
M_{Z}c\beta~{}s\theta_{W}&M_{Z}c\beta~{}c\theta_{W}&0&-\mu\cr
M_{Z}s\beta~{}s\theta_{W}&-M_{Z}s\beta~{}c\theta_{W}&-\mu&0}\right),$ (70)
with $c\beta(s\beta)$ and $c\theta_{W}(s\theta_{W})$ standing for
$\cos\beta(\sin\beta)$ and $\cos\theta_{W}(\sin\theta_{W})$ respectively. As
mentioned earlier, $M_{1}$ and $M_{2}$ are the soft parameters, whereas $\mu$
is the superpotential parameter, thus SUSY conserving. The angle $\beta$ is
typically taken as a input parameter, $tan\beta={v_{2}/v_{1}}$ whereas
$\theta_{W}$ is the Weinberg angle given by the inverse tangent of the ratio
of the gauge couplings as in the SM. Note that the neutralino mass matrix
being a Majorana mass matrix is complex symmetric in nature. Hence it is
diagonalised by a unitary matrix $N$,
$\displaystyle N^{*}\cdot M_{\tilde{N}}\cdot
N^{\dagger}=\mbox{Diag.}(m_{\chi_{1}^{0}},m_{\chi_{2}^{0}},m_{\chi_{3}^{0}},m_{\chi_{4}^{0}})$
(71)
The states are rotated by $\chi_{i}^{0}=N^{\star}\Psi$ to go the physical
basis.
#### VII.0.2 The Chargino Sector
In a similar manner to the neutralino sector, all the fermionic partners of
the charged gauge bosons and of the charged Higgs bosons mix after electroweak
symmetry breaking. However, they combine in a such a way that a Wino-Higgsino
Weyl fermion pair forms a Dirac fermion called the chargino. This mass matrix
is given as
$\mathcal{L}\supset-\frac{1}{2}\left(\matrix{\tilde{W}^{-}&\tilde{H}_{1}^{-}}\right)\;\left(\matrix{M_{2}&\sqrt{2}M_{W}\sin\beta\cr\sqrt{2}M_{W}\cos\beta&\mu}\right)\left(\matrix{\tilde{W}^{+}\cr\tilde{H}_{2}^{+}}\right),$
(72)
Given the non-symmetric (non-hermitian) matrix nature of this matrix, it is
diagonalised by a bi-unitary transformation, $U^{*}\cdot M_{C}\cdot
V^{\dagger}=\mbox{Diag.}(m_{\chi_{1}^{+}},m_{\chi_{2}^{+}})$. The chargino
eigenstates are typically represented by $\chi^{\pm}$ with mass eigenvalues
$m_{\chi^{\pm}}$. The explicit forms for $U$ and $V$ can be found by the
eigenvectors of $M_{C}M_{C}^{\dagger}$ and $M_{C}^{\dagger}M_{C}$ respectively
haberkane .
#### VII.0.3 The Sfermion Sector
Next let us come to the sfermion sector. Remember that we have added different
scalar fields for the right and left handed fermions in the Standard Model.
After electroweak symmetry breaking, the sfermions corresponding to the left
fermion and the right fermion mix with each other. Furthermore while writing
down the mass matrix for the sfermions, we should remember that these terms
could break the flavour i.e, we can have mass terms which mix different
generation. Thus, in general the sfermion mass matrix is a $6\times 6$ mass
matrix given as :
$\xi^{\dagger}~{}M_{\tilde{f}}^{2}\xi~{}~{};~{}~{}~{}\xi=\\{{\tilde{f}_{L_{i}}},{\tilde{f}_{R_{i}}}\\}$
From the total scalar potential, the mass matrix for these sfermions can be
derived using standard definition given as
$m_{ij}^{2}=\left(\begin{array}[]{cc}{\partial^{2}V\over\partial\phi_{i}\partial\phi_{j}^{\star}}&{\partial^{2}V\over\partial\phi_{i}\partial\phi_{j}}\\\
{\partial^{2}V\over\partial\phi_{i}^{\star}\partial\phi_{j}^{\star}}&{\partial^{2}V\over\partial\phi_{i}^{\star}\partial\phi_{j}}\end{array}\right)$
(73)
Using this for sfermions, we have :
$M_{\tilde{f}}^{2}\;=\;\left(\matrix{m_{\tilde{f}_{\mbox{\scriptsize
LL}}}^{2}&m_{\tilde{f}_{\mbox{\scriptsize
LR}}}^{2}\cr{m_{\tilde{f}_{\mbox{\scriptsize
LR}}}^{2\,\dagger}}&m_{\tilde{f}_{\mbox{\scriptsize RR}}}^{2}}\right),$ (74)
where each of the above entries represents $3\times 3$ matrices in the
generation space. More specifically, they have the form (as usual, $i,j$ are
generation indices):
$\displaystyle m^{2}_{\tilde{f}_{L_{i}L_{j}}}$ $\displaystyle=$ $\displaystyle
M^{2}_{\tilde{f}_{L_{i}L_{j}}}+m^{2}_{f}\delta_{ij}+M_{Z}^{2}\cos
2\beta(T_{3}+\sin^{2}\theta_{W}Q_{\mbox{\scriptsize em}})\delta_{ij}$
$\displaystyle m_{{\tilde{f}}_{L_{i}R_{j}}}^{2}$ $\displaystyle=$
$\displaystyle\left(\big{(}Y^{A}_{f}\cdot^{v_{2}}_{v_{1}}-m_{f}\mu^{\tan\beta}_{\cot\beta}\big{)}\;\;\mbox{for}\;f=^{e,d}_{u}\right)\delta_{ij}$
$\displaystyle m_{\tilde{f}_{\mbox{\scriptsize RR}}}^{2}$ $\displaystyle=$
$\displaystyle M^{2}_{\tilde{f}_{R_{ij}}}+\left(m^{2}_{f}+M_{Z}^{2}\cos
2\beta\sin^{2}\theta_{W}Q_{\mbox{\scriptsize em}}\right)\delta_{ij}$ (75)
In the above, $M^{2}_{\tilde{f}_{L}}$ represents the soft mass term for the
corresponding fermion ($L$ for left, $R$ for right), $T_{3}$ is the eigenvalue
of the diagonal generator of $SU(2)$, $m_{f}$ is the mass of the fermion with
$Y$ and $Q_{em}$ representing the hypercharge and electromagnetic charge (in
units of the charge of the electron ) respectively. The sfermion mass matrices
are hermitian and are thus diagonalised by a unitary rotation,
$R_{\tilde{f}}R_{\tilde{f}}^{\dagger}=1$:
$R_{\tilde{f}}\cdot M_{\tilde{f}}\cdot
R_{\tilde{f}}^{\dagger}=\mbox{Diag.}(m_{\tilde{f}_{1}},m_{\tilde{f}_{2}},\dots,m_{\tilde{f}_{6}})$
(76)
#### VII.0.4 The Higgs sector
Now let us turn our attention to the Higgs fields. We will use again use the
standard formula of eq.(73), to derive the Higgs mass matrices. The eight
Higgs degrees of freedom form a $8\times 8$ Higgs mass matrix which breaks
down diagonally in to three $2\times 2$ mass matrices191919The discussion in
this section closely follows from the discussion presented in Ref.rohinibook .
The mass matrices are divided in to charged sector, CP odd neutral and CP even
neutral. This helps us in identifying the goldstone modes and the physical
spectrum in an simple manner. Before writing down the mass matrices, let us
first define the following parameters :
$m_{1}^{2}=m_{H_{1}}^{2}+\mu^{2},\;\;\;m_{2}^{2}=m_{H_{2}}^{2}+\mu^{2},\;\;\;m_{3}^{2}=B_{\mu}\mu.$
In terms of these parameters, the various mass matrices and the corresponding
physical states obtained after diagonalising the mass matrices are given
below:
Charged Higgs and Goldstone Modes:
$\left(\begin{array}[]{cc}H_{1}^{+}&H_{2}^{+}\end{array}\right)\left(\begin{array}[]{cc}m_{1}^{2}+{1\over
8}(g_{1}^{2}+g_{2}^{2})(v_{1}^{2}-v_{2}^{2})+{1\over
4}g_{2}^{2}v_{2}^{2}&m_{3}^{2}+{1\over 4}g_{2}^{2}v_{1}v_{2}\\\
m_{3}^{2}+{1\over 4}g_{2}^{2}v_{1}v_{2}&m_{2}^{2}-{1\over
8}(g_{1}^{2}+g_{2}^{2})(v_{1}^{2}-v_{2}^{2})+{1\over
4}g_{2}^{2}v_{2}^{2}\end{array}\right)\left(\begin{array}[]{c}H_{1}^{-}\\\
H_{2}^{-}\end{array}\right)$ (77)
Using the minimisation conditions (VI), this matrix becomes,
$\left(\begin{array}[]{cc}H_{1}^{+}&H_{2}^{+}\end{array}\right)({m_{3}^{2}\over
v_{1}v_{2}}+{1\over
4}g_{2}^{2})\left(\begin{array}[]{cc}v_{2}^{2}&v_{1}v_{2}\\\
v_{1}v_{2}&v_{1}^{2}\\\ \end{array}\right)\left(\begin{array}[]{c}H_{1}^{-}\\\
H_{2}^{-}\end{array}\right)$ (78)
which has determinant zero leading to the two eigenvalues as :
$\displaystyle m_{G^{\pm}}^{2}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle m_{H^{\pm}}^{2}$ $\displaystyle=$
$\displaystyle\left({m_{3}^{2}\over v_{1}v_{2}}+{1\over
4}g_{2}^{2}\right)(v_{1}^{2}+v_{2}^{2}),$ (79) $\displaystyle=$
$\displaystyle{2m_{3}^{2}\over sin2\beta}+M_{W}^{2}$ (80)
where $G^{\pm}$ represents the Goldstone mode. The physical states are
obtained just by rotating the original states in terms of the $H_{1},~{}H_{2}$
fields by an mixing angle. The mixing angle in the present case (in the
unitary gauge) is just tan$\beta$:
$\left(\begin{array}[]{c}H^{\pm}\\\
G^{\pm}\end{array}\right)=\left(\begin{array}[]{cc}sin\beta&cos\beta\\\
-cos\beta&sin\beta\end{array}\right)\left(\begin{array}[]{c}H^{\pm}\\\
G^{\pm}\end{array}\right)$ (81)
CP odd Higgs and Goldstone Modes:
Let us now turn our attention to the CP-odd Higgs sector. The mass matrices
can be written in a similar manner but this time for imaginary components of
the neutral Higgs.
$\left(\begin{array}[]{cc}ImH_{1}^{0}&ImH_{2}^{0}\end{array}\right)\left(\begin{array}[]{cc}m_{1}^{2}+{1\over
8}(g_{1}^{2}+g_{2}^{2})(v_{1}^{2}-v_{2}^{2})&m_{3}^{2}\\\
m_{3}^{2}&m_{2}^{2}-{1\over
8}(g_{1}^{2}+g_{2}^{2})(v_{1}^{2}-v_{2}^{2})\end{array}\right)\left(\begin{array}[]{c}ImH_{1}^{0}\\\
ImH_{2}^{0}\end{array}\right)$ (82)
As before, again using the minimisation conditions, this matrix becomes,
$\left(\begin{array}[]{cc}ImH_{1}^{0}&ImH_{2}^{0}\end{array}\right)m_{3}^{2}\left(\begin{array}[]{cc}v_{2}/v_{1}&1\\\
1&v_{1}/v_{2}\\\ \end{array}\right)\left(\begin{array}[]{c}ImH_{1}^{0}\\\
ImH_{2}^{0}\end{array}\right)$ (83)
which has determinant zero leading to the two eigenvalues as :
$\displaystyle m_{G^{0}}^{2}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle m_{A^{0}}^{2}$ $\displaystyle=$
$\displaystyle\left({m_{3}^{2}\over
v_{1}v_{2}}\right)(v_{1}^{2}+v_{2}^{2})~{}~{}=~{}~{}{2m_{3}^{2}\over
sin2\beta}$ (84)
Similar to the charged sector, the mixing angle between these two states in
the unitary gauge is again just tan$\beta$.
${1\over\sqrt{2}}\left(\begin{array}[]{c}A^{0}\\\
G^{0}\end{array}\right)=\left(\begin{array}[]{cc}sin\beta&cos\beta\\\
-cos\beta&sin\beta\end{array}\right)\left(\begin{array}[]{c}ImH^{0}_{1}\\\
ImH_{2}^{0}\end{array}\right)$ (85)
CP even Higgs:
Finally, let us come to the real part of the neutral Higgs sector. The mass
matrix in this case is given by the following.
$\left(\begin{array}[]{cc}ReH_{1}^{0}&ReH_{2}^{0}\end{array}\right)~{}{1\over
2}~{}\left(\begin{array}[]{cc}2m_{1}^{2}+{1\over
4}(g_{1}^{2}+g_{2}^{2})(3v_{1}^{2}-v_{2}^{2})&-2m_{3}^{2}-{1\over
4}v_{1}v_{2}(g_{1}^{2}+g_{2}^{2})\\\ -2m_{3}^{2}-{1\over
4}v_{1}v_{2}(g_{1}^{2}+g_{2}^{2})&2m_{2}^{2}+{1\over
4}(g_{1}^{2}+g_{2}^{2})(3v_{2}^{2}-v_{1}^{2})\end{array}\right)\left(\begin{array}[]{c}ReH_{1}^{0}\\\
ReH_{2}^{0}\end{array}\right)$ (86)
Note that in the present case, there is no Goldstone mode. As before, we will
use the minimisation conditions and further using the definition of
$m_{A}^{2}$ from eq.(84), we have :
$\left(\begin{array}[]{cc}ReH_{1}^{0}&ReH_{2}^{0}\end{array}\right)\left(\begin{array}[]{cc}m_{A}^{2}sin^{2}\beta+M_{z}^{2}cos\beta&-(m_{A}^{2}+m_{Z}^{2})sin\beta
cos\beta\\\ -(m_{A}^{2}+m_{Z}^{2})sin\beta
cos\beta&m_{A}^{2}cos^{2}\beta+M_{z}^{2}sin\beta\end{array}\right)\left(\begin{array}[]{c}ReH_{1}^{0}\\\
ReH_{2}^{0}\end{array}\right)$ (87)
The matrix has two eigenvalues which are given by the two signs of the
following equation:
$m^{2}_{H,h}={1\over
2}\left[m_{A}^{2}+m_{Z}^{2}\pm\\{(m_{A}^{2}+m_{Z}^{2})^{2}-4m_{Z}^{2}m_{A}^{2}cos^{2}2\beta\\}^{1/2}\right]$
(88)
The heavier eigenvalue $m_{H}^{2}$, is obtained by taken the positive sign,
whereas the lighter eigenvalue $m_{h}^{2}$ is obtained by taking the negative
sign respectively. The mixing angle between these two states can be read out
from the mass matrix of the above202020The mixing angle for a $2\times 2$
symmetric matrix, $C_{ij}$ is given by $tan2\theta=2C_{12}/(C_{22}-C_{11}).$
as :
$tan~{}2\alpha={m_{A}^{2}+m_{Z}^{2}\over m_{A}^{2}-m_{Z}^{2}}~{}tan~{}2\beta$
(89)
Tree Level Catastrophe:
So far we have seen that out of the eight Higgs degrees of freedom, three of
them form the Goldstone modes after incorporating $SU(2)\times U(1)$ breaking
and there are five physical Higgs bosons fields in the MSSM spectrum. These
are the charged Higgs ($H^{\pm}$) a CP-odd Higgs ($A$) and two CP-even Higgs
bosons ($h,H$). From the mass spectrum analysis above, we have seen that the
mass eigenvalues of these Higgs bosons are related to each other. In fact,
putting together all the eigenvalue equations, we summarise the relations
between them as follows :
$\displaystyle m^{2}_{H^{\pm}}$ $\displaystyle=$ $\displaystyle
m_{A}^{2}+m_{W}^{2}>\max(M_{W}^{2},m_{A}^{2})$ $\displaystyle
m_{h}^{2}+m_{H}^{2}$ $\displaystyle=$ $\displaystyle m_{A}^{2}+m_{Z}^{2}$
$\displaystyle m_{H}$ $\displaystyle>$ $\displaystyle max(m_{A},m_{Z})$
$\displaystyle m_{h}$ $\displaystyle<$ $\displaystyle
min(m_{A},M_{Z})|cos2\beta|<min(m_{A},m_{Z})$ (90)
Let us concentrate on the last relation of the above eq.(VII.0.4). The
condition on the lightest CP even Higgs mass, $m_{h}$, tell us that it should
be equal to $m_{Z}$ in the limit tan$\beta$ is saturated to be maximum, such
that cos$2\beta~{}\rightarrow~{}1$ and $m_{A}~{}\rightarrow~{}\infty$. If
these limits are not saturated, it is evident that the light higgs mass is
less that $m_{Z}$. This is one of main predictions of MSSM which could make it
easily falsifiable from the current generation of experiments like LEP,
Tevatron and the upcoming LHC. Given that present day experiments have not
found a Higgs less that Z-boson mass, it is tempting to conclude that the MSSM
is not realised in Nature. However caution should be exercised before taking
such a route as our results are valid only at the tree level. In fact, in a
series of papers in the early nineties oneloophiggs , it has been shown that
large one-loop corrections to the Higgs mass can easily circumvent this limit.
The light Higgs Spectrum at 1-loop
As mentioned previously, radiative corrections can significantly modify the
mass relations which we have presented in the previous section. As is evident,
these corrections can be very important for the light Higgs boson mass. Along
with the 1-loop corrections previously, in the recent years dominant parts of
two-loop corrections have also been available slavich2loop with a more
complete version recently givenmartin2loop . In the following we will present
the one-loop corrections to the light Higgs mass and try to understand the
implications for the condition eq.(VII.0.4). Writing down the 1-loop
corrections to the CP-even part of the Higgs mass matrix as :
$M^{2}_{Re}=M^{2}_{Re}(0)+\delta M^{2}_{Re},$ (91)
where $M^{2}_{Re}(0)$ represents the tree level mass matrix given by eq.(87)
and $\delta M^{2}_{Re}$ represents its one-loop correction. The dominant one-
loop correction comes from the top quark and stop squark loops which can be
written in the following form:
$\delta M^{2}_{Re}=\left(\begin{array}[]{cc}\Delta_{11}&\Delta_{12}\\\
\Delta_{12}&\Delta_{22}\end{array}\right),$ (92)
where
$\displaystyle\Delta_{11}$ $\displaystyle=$
$\displaystyle{3G_{F}m_{t}^{4}\over
2\sqrt{2}\pi^{2}sin^{2}\beta}\left[{\mu(A_{t}+\mu cot\beta)\over
m_{\tilde{t}_{1}}^{2}-m_{\tilde{t}_{2}}^{2}}\right]^{2}\left(2-{m_{\tilde{t}_{1}}^{2}+m_{\tilde{t}_{2}}^{2}\over
m_{\tilde{t}_{1}}^{2}-m_{\tilde{t}_{2}}^{2}}ln{m_{\tilde{t}_{1}}^{2}\over
m_{\tilde{t}_{2}}^{2}}\right)$ $\displaystyle\Delta_{12}$ $\displaystyle=$
$\displaystyle{3G_{F}m_{t}^{4}\over
2\sqrt{2}\pi^{2}sin^{2}\beta}\left[{\mu(A_{t}+\mu cot\beta)\over
m_{\tilde{t}_{1}}^{2}-m_{\tilde{t}_{2}}^{2}}\right]ln{m_{\tilde{t}_{1}}^{2}\over
m_{\tilde{t}_{2}}^{2}}+{A_{t}\over\mu}\Delta_{11}$ $\displaystyle\Delta_{22}$
$\displaystyle=$
$\displaystyle{3G_{F}m_{t}^{4}\over\sqrt{2}\pi^{2}sin^{2}\beta}\left[ln{m_{\tilde{t}_{1}}^{2}m_{\tilde{t}_{2}}^{2}\over
m_{t}^{2}}+{A_{t}(A_{t}+\mu cot\beta)\over
m_{\tilde{t}_{1}}^{2}-m_{\tilde{t}_{2}}}ln{m_{\tilde{t}_{1}}^{2}\over
m_{\tilde{t}_{2}}^{2}}\right]+{A_{t}\over\mu}\Delta_{11}$ (93)
In the above $G_{F}$ represents Fermi Decay constant, $m_{t}$, the top mass,
$m_{\tilde{t}_{1}}^{2},~{}m_{\tilde{t}_{1}}^{2}$ are the eigenvalues of the
stop mass matrix and $A_{t}$ is the trilinear scalar coupling (corresponding
to the top Yukawa coupling) in the stop mass matrix. $\mu$ and the angle
$\beta$ have their usual meanings. Taking in to account these corrections, the
condition (VII.0.4) takes the form:
$m_{h}^{2}~{}<~{}m_{Z}^{2}\text{cos}^{2}2\beta+\Delta_{11}\text{cos}^{2}\beta+\Delta_{12}\text{sin}2\beta+\Delta_{22}\text{sin}^{2}\beta$
(94)
Given that $m_{t}$ is quite large, almost twice the $m_{Z}$ mass, for suitable
values of the stop masses, it is clear that the tree level upper limit on the
light Higgs mass is now evaded. However, a reasonable upper limit can still be
got by assuming reasonable values for the stop mass. For example assuming stop
masses to be around 1 TeV and maximal mixing the stop sector, one attains an
upper bound on the light Higgs mass as:
$m_{h}~{}\stackrel{{\scriptstyle\scriptstyle<}}{{\scriptstyle\sim}}~{}135~{}\text{GeV}.$
(95)
#### VII.0.5 Feynman Rules
In this section, we have written down all the mass matrices of the
superpartners, their eigenvalues and finally the eigenvectors which are
required to transform the superpartners in to their physical basis. The
feynman rules corresponding to the various vertices have to be written down in
this basis. Thus various soft supersymmetry breaking and supersymmetry
conserving parameters entering these mass matrices would now determine these
couplings as well as the masses, which in turn determine the strength of
various physical processes like crosssections and decay rates. A complete list
of the Feynman rules in the mass basis can be found in various references like
Physics Reports like Haber & Kane haberkane and D Chung et. alKaneking and
also in textbooks like Sparticles rohinibook and Baer & Tata baertata . A
complete set of Feynman rules is out of reach of this set of lectures. Here I
will just present two examples to illustrate the points I have been making
here.
$l_{i}$$\tilde{\chi}^{+}_{A}$$\tilde{v}_{X}$$\tilde{C}_{iAX}$ | $l_{i}$$\tilde{\chi}^{0}_{A}$$\tilde{l}_{X}$$\tilde{D}_{iAX}$
---|---
Figure 3: lepton-slepton-chargino and lepton-slepton-neutralino vertices.
Due to the mixing between the fermionic partners of the gauge bosons and the
fermionic partners of the Higgs bosons, the gauge and the yukawa vertices get
mixed in MSSM. We will present here the vertices of fermion-sfermion-chargino
and fermion-sfermion-neutralino where this is evident. These are presented in
Figure 3.
(i) Fermion-Sfermion-Chargino :
This is the first vertex on the left of the figure. The explicit structure of
this vertex is given by:
$\tilde{C}_{iAX}=C^{R}_{iAX}P_{R}+C^{L}_{iAX}P_{L}$ (96)
where $P_{L}(P_{R})$ are the project operators212121$P_{L}=(1-\gamma_{5})/2$
and $P_{R}=(1+\gamma_{5})/2$. and $C^{R}$ and $C^{L}$ are given by
$\displaystyle c^{R}_{iAX}$ $\displaystyle=$ $\displaystyle-
g_{2}(U)_{A1}R^{\nu}_{Xi}$ (97) $\displaystyle C^{L}_{iAX}$ $\displaystyle=$
$\displaystyle g_{2}{m_{l_{i}}\over\sqrt{2}m_{W}cos\beta}(V)_{A2}R^{\nu}_{Xi}$
(98)
In the above $U$ and $V$ are the diagonalising matrices of chargino mass
matrix $M_{C}$, $R^{\nu}$ is the diagonalising matrix of the sneutrino mass
matrix, $M_{\tilde{\nu}}^{2}$. And the indices $A$ and $X$ runs over the
dimensions of the respective matrices ($A=1,2$ for Charginos, $X=1,2,3$ for
sneutrinos), whereas $i$ as usual runs over the generations, $m_{l_{i}}$ is
the mass of the $i$ th lepton and rest of the parameters carry the standard
definitions.
(ii) Fermion-Sfermion-Neutralino :
In a similar manner, the fermion-sfermion-neutralino vertex is given by:
$\tilde{D}_{iAX}=D^{R}_{iAX}P_{R}+D^{L}_{iAX}P_{L}$ (99)
where $D^{L}$ and $D^{R}$ have the following forms:
$\displaystyle D^{R}_{iAX}$ $\displaystyle=$
$\displaystyle-{g_{2}\over\sqrt{2}}\left\\{\left[-N_{A2}-N_{A1}tan\theta_{W}\right]R^{l}_{Xi}+{m_{l_{i}}\over
m_{W}cos\beta}N_{A3}R^{l}_{X,i+3}\right\\}$ (100) $\displaystyle D^{L}_{iAX}$
$\displaystyle=$ $\displaystyle-{g_{2}\over\sqrt{2}}\left\\{{m_{l_{i}}\over
m_{W}cos\beta}N_{A3}R^{l}_{Xi}+2N_{A1}tan\theta_{W}R^{l}_{X,i+3}\right\\}$
(101)
In the above $N$ is diagonalising matrices of neutralino mass matrix $M_{N}$,
$R^{l}$ is the diagonalising matrix of the slepton mass matrix,
$M_{\tilde{l}}^{2}$. And the indices $A$ and $X$ runs over the dimensions of
the respective matrices ($A=1,..,4$ for neutralinos, $X=1,..,6$ for sleptons),
whereas $i$ as usual runs over the generations.
#### VII.0.6 Think it Over:
The LEP experiment at CERN searched for a light Higgs boson which has SM like
couplings through the process $e^{+}e^{-}~{}\to~{}ZH$ and has a put a limit on
the lightest Higgs boson mass as
$m_{h}~{}\stackrel{{\scriptstyle\scriptstyle>}}{{\scriptstyle\sim}}~{}114.2$GeV.
This limit applies to the light Higgs boson of the MSSM (except in some range
and in the presence of CP violation in the Higgs sector). Take the formula of
the 1-loop Higgs mass given by eq.(94) and simplify it by assuming the stop
masses are of the similar order $~{}M_{S}$ and the mixing between the stops is
maximal. Find out what is the least value of the $M_{S}$ which is consistent
with the Higgs mass. Now compute the 1-loop corrections to the minimisation
conditions and check what is the amount of fine-tuning required to obtain the
correct $M_{Z}$ mass. Show that a few percent fine tuning is already required
to satisfy the LEP limit on the light Higgs mass. The fine tuning rapidly
increases with increasing Higgs mass. This goes under the name Little
Hierarchy Problem.
## VIII ‘Standard’ Models of Supersymmetry breaking
So far we have included supersymmetry breaking within the MSSM through a set
of explicit supersymmetry breaking soft terms however, at a more fundamental
we would like to understand the origins of these soft terms as coming from a
theory where supersymmetry is spontaneously broken. In a previous section, we
have mentioned that supersymmetry needs to be broken spontaneously in a hidden
sector and then communicated to the visible sector through a messenger sector.
In the below we will consider two main models for the messenger sector (a) the
gravitational interactions and (b) the gauge interactions. But before we
proceed to list problems with the general form soft supersymmetry breaking
terms as discussed in the previous section. This is essential to understand
what kind of constructions of supersymmetric breaking models are likely to be
realised in Nature and thus are consistent with phenomenology.
The way we have parameterised supersymmetry breaking in the MSSM, using a set
of gauge invariant soft terms, at the first sight, seems to be the most
natural thing to do in the absence of a complete theory of supersymmetry
breaking. However, this approach is itselves laden with problems as we realise
once we start confronting this model with phenomenology. The two main problems
can be listed as below:
(i). Large number of parameters
Compared to the SM, in MSSM, we have a set of more than 50 new particles;
writing down all possible gauge invariant and supersymmetry breaking soft
terms, limits the number of possible terms to about 105. All these terms are
completely arbitrary, there is no theoretical input on their magnitudes,
relative strengths, in short there is no theoretical guiding principle about
these terms. Given that these are large in number, they can significantly
effect the phenomenology. In fact, the MSSM in its softly broken form seems to
have lost predictive power except to say that there are some new particles
within a broad range in mass(energy) scale. The main culprit being the large
dimensional parameter space $\sim$ 105 dimensional space which determines the
couplings of the supersymmetric particles and their the masses. If there is a
model of supersymmetry breaking which can act as a guiding principle and
reduce the number of free parameters of the MSSM, it would only make MSSM more
predictive.
(ii). Large Flavour and CP violations. As mentioned previously, the soft mass
terms $m_{ij}^{2}$ and the trilinear scalar couplings $A_{ijk}$ can violate
flavour. This gives us new flavour violating structures beyond the standard
CKM structure of the quark sector which can also be incorporated in the MSSM.
Furthermore, all these couplings can also be complex and thus could serve as
new sources of CP violation in addition to the CKM phase present in the
Standard Model. Given that all these terms arbitrary and could be of any
magnitude close to weak scale, these terms can contribute dominantly compared
to the SM amplitudes to various flavour violating processes at the weak scale,
like flavour violating decays like $b\to s+\gamma$ or flavour oscillations
like $K^{0}\leftrightarrow\bar{K}^{0}$ etc and even flavour violating decays
which do not have any Standard Model counterparts like $\mu\to e+\gamma$ etc.
The CP violating phases can also contribute to electric dipole moments (EDM)s
which are precisely measured at experiments.
To analyse the phenomenological impact of these processes on these terms, an
useful and powerful tool is the so called Mass Insertion (MI) approximation.
In this approximation, we use flavour diagonal gaugino vertices and the
flavour changing is encoded in non-diagonal sfermion propagators. These
propagators are then expanded assuming that the flavour changing parts are
much smaller than the flavour diagonal ones. In this way we can isolate the
relevant elements of the sfermion mass matrix for a given flavour changing
process and it is not necessary to analyse the full $6\times 6$ sfermion mass
matrix. Using this method, the experimental limits lead to upper bounds on the
parameters (or combinations of)
$\delta_{ij}^{f}\equiv\Delta^{f}_{ij}/m_{\tilde{f}}^{2}$, known as mass
insertions; where $\Delta^{f}_{ij}$ is the flavour-violating off-diagonal
entry appearing in the $f=(u,d,l)$ sfermion mass matrices and
$m_{\tilde{f}}^{2}$ is the average sfermion mass. In addition, the mass-
insertions are further sub-divided into LL/LR/RL/RR types, labeled by the
chirality of the corresponding SM fermions. The limits on various $\delta$’s
coming from various flavour violating processes have been computed and
tabulate in the literature and can be found for instance in Ref.ourlectures .
These limits show that the flavour violating terms should be typically at
least a couple of orders of magnitude suppressed compared to the flavour
conserving soft terms222222The flavour problem could also be alleviated by
considering decoupling soft masses or alignment mechanisms.. While this is
true for the first two generations of soft terms, the recent results from
B-factories have started constraining flavour violating terms involving the
third generation too. In light of this stringent constraint, it is more
plausible to think that the fundamental supersymmetry breaking mechanism some
how suppresses these flavour violating entries. Similarly, this mechanism
should also reduce the number of parameters such that the MSSM could be easily
be confronted with phenomenology and make it more predictive. We will consider
two such models of supersymmetry breaking below which will use two different
kinds of messenger sectors.
### VIII.1 Minimal Supergravity
In the minimal supergravity framework, gravitational interactions play the
role of messenger sector. Supersymmetry is broken spontaneously in the hidden
sector. This information is communicated to the MSSM sector through
gravitational sector leading to the soft terms. Since gravitational
interactions play an important role only at very high energies, $M_{p}\sim
O(10^{19})$ GeV, the breaking information is passed on to the visible sector
only at those scales. The strength of the soft terms is characterised roughly
by, $m_{\tilde{f}}^{2}~{}\approx~{}M_{S}^{2}/M_{planck}$, where $M_{S}$ is the
scale of supersymmetry breaking. These masses can be comparable to weak scale
for $M_{S}\sim 10^{10}$ GeV. This $M_{S}^{2}$ can correspond to the F-term vev
of the Hidden sector. The above mechanism of supersymmetry breaking is called
supergravity (SUGRA) mediated supersymmetry breaking.
A particular class of supergravity mediated supersymmetry breaking models are
those which go under the name of ”minimal” supergravity. This model has
special features that it reduces to total number of free parameters
determining the entire soft spectrum to five. Furthermore, it also removes the
dangerous flavour violating soft terms in the MSSM. The classic features of
this model are the following boundary conditions to the soft terms at the high
scale $\sim~{}M_{Planck}$ :
* •
All the gaugino mass terms are equal at the high scale.
$M_{1}=M_{2}=M_{3}=M_{1/2}$
* •
All the scalar mass terms at the high scale are equal.
$m_{\phi_{ij}}^{2}=m_{0}^{2}\delta_{ij}$
* •
All the trilinear scalar interactions are equal at the high scale.
$A_{ijk}=Ah_{ijk}$
* •
All bilinear scalar interactions are equal at the high scale.
$B_{ij}=B$
Using these boundary conditions, one evolves the soft terms to the weak scale
using renormalisation group equations. It is possible to construct
supergravity models which can give rise to such kind of strong universality in
soft terms close to Planck scale. This would require the Kahler potential of
the theory to be of the canonical form. As mentioned earlier, the advantage of
this model is that it drastically reduces the number of parameters of the
theory to about five, $m_{0},M$ (or equivalently $M_{2}$), ratio of the vevs
of the two Higgs, tan$\beta$, $A$, $B$. Thus, these models are also known as
‘Constrained’ MSSM in literature. The supersymmetric mass spectrum of these
models has been extensively studied in literature. The Lightest Supersymmetric
Particle (LSP) is mostly a neutralino in this case.
### VIII.2 Gauge Mediated Supersymmetry breaking
In a more generic case, the Kahler potential need not have the required
canonical form. In particular, most low energy effective supergravities from
string theories do not posses such a Kahler potential. In such a case, large
FCNC’s and again large number of parameters are expected from supergravity
theories. An alternative mechanism has been proposed which tries to avoid
these problems in a natural way. The key idea is to use gauge interactions
instead of gravity to mediate the supersymmetry breaking from the hidden (also
called secluded sector sometimes) to the visible MSSM sector. In this case
supersymmetry breaking can be communicated at much lower energies $\sim 100$
TeV.
A typical model would contain a susy breaking sector called ‘messenger sector’
which contains a set of superfields transforming under a gauge group which
‘contains’ $G_{SM}$. Supersymmetry is broken spontaneously in this sector and
this breaking information is passed on to the ordinary sector through gauge
bosons and their fermionic partners in loops. The end-effect of this mechanism
also is to add the soft terms in to the lagrangian. But now these soft terms
are flavour diagonal as they are generated by gauge interactions. The soft
terms at the messenger scale also have simple expressions in terms of the susy
breaking parameters. In addition, in minimal models of gauge mediated
supersymmetry breaking, only one parameter can essentially determine the
entire soft spectrum.
In a similar manner as in the above, the low energy susy spectrum is
determined by the RG scaling of the soft parameters. But now the high scale is
around 100 TeV instead of $M_{GUT}$ as in the previous case. The mass spectrum
of these models has been studied in many papers. The lightest supersymmetric
particle in this case is mostly the gravitino in contrast to the mSUGRA case.
#### VIII.2.1 Think it Over
* •
In both gravity mediated as well as gauge mediated supersymmetry breaking
models, we have seen that RG running effects have to included to study the
soft terms at the weak scale. Typically, the soft masses which appear at those
scales are positive at the high scale. But radiative corrections can
significantly modify the low scale values of these parameters; in particular,
making one of the Higgs mass to be negative at the weak scale leading to
spontaneous breaking of electroweak symmetry. This mechanism is called
radiative electroweak symmetry breaking. Consider two hypothetical situations
when (a) the top mass is twice its present value $m_{t}~{}=2~{}m_{t}$ (b) the
top mass is 1/10 th its present value $m_{t}~{}=~{}m_{t}/10$. In which case
there would be more efficient Electroweak symmetry breaking ?
* •
The recent limits from LHC already put severe constraints on the lightest
squarks and gluino masses. They push their masses to be greater than 800 GeV -
1 TeV. In fact, this has severe constraints on mSUGRA model. For latest limits
have a look at cerntwiki .
## IX Remarks
The present set of lectures are only a set of elementary introduction to the
MSSM. More detailed accounts can be found in various references which we have
listed at various places in the text. In preparing for these set of lectures,
I have greatly benefitted from various review articles and text books. I have
already listed some of them at various places in the text. Martin’s review
martin is perhaps the most comprehensive and popular references. It is also
constantly updated. Some other excellent reviews are peskin1 and bagger . A
concise introduction can also be found in csaki . For more formal aspects of
supersymmetry including a good introduction to supergravity please have a look
at vanproeyen and west . For Grand Unified theories and supersymmetry, please
have a look at mohapatra , yanagida , ramond and rosstextbook . For a
comprehensive introduction to supersymmetric dark matter, please see
kamionkowski . Finally, I would also recommend the original papers of anomaly
mediated supersymmetry breaking amsb . Happy Susying.
Acknowledgements We would like to thank Urjit Yajnik for detailed discussions
and consultations through out the teaching period. We would also like to thank
Ranjan Laha and Manimala Mitra for going through these lecture notes several
times and pointing out various typographical errors. The author is supported
by DST Ramanujan fellowship and DST project ”Complementarity between direct
and indirect searches of supersymmetry”.
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|
arxiv-papers
| 2012-01-01T07:17:19 |
2024-09-04T02:49:25.876426
|
{
"license": "Public Domain",
"authors": "Sudhir K. Vempati",
"submitter": "Sudhir Vempati",
"url": "https://arxiv.org/abs/1201.0334"
}
|
1201.0757
|
# Concordance models of reionization: implications for faint galaxies and
escape fraction evolution
Michael Kuhlen, Claude-André Faucher-Giguère
Theoretical Astrophysics Center, University of California, Berkeley, CA 94720
mqk@astro.berkeley.eduMiller Fellow; cgiguere@berkeley.edu
###### Abstract
Recent observations have constrained the galaxy ultra-violet (UV) luminosity
function up to $z\sim 10$. However, these observations alone allow for a wide
range of reionization scenarios due to uncertainties in the abundance of faint
galaxies and the escape fraction of ionizing photons. We show that requiring
continuity with post-reionization ($z<6$) measurements, where the Ly$\alpha$
forest provides a complete probe of the cosmological emissivity of ionizing
photons, significantly reduces the permitted parameter space. Models that are
simultaneously consistent with the measured UV luminosity function, the
Thomson optical depth to the microwave background, and the Ly$\alpha$ forest
data require either: 1) extrapolation of the galaxy luminosity function down
to very faint UV magnitudes $M_{\rm lim}\sim-10$, corresponding roughly to the
UV background suppression scale; 2) an increase of the escape fraction by a
factor $\gtrsim 10$ from $z=4$ (where the best fit is 4%) to $z=9$; or 3) more
likely, a hybrid solution in which undetected galaxies contribute
significantly and the escape fraction increases more modestly. Models in which
star formation is strongly suppressed in low-mass, reionization-epoch haloes
of mass up to $M_{\rm h}\sim 10^{10}$ M⊙ (e.g., owing to a metallicity
dependence) are only allowed for extreme assumptions for the redshift
evolution of the escape fraction. However, variants of such models in which
the suppression mass is reduced (e.g., assuming an earlier or higher
metallicity floor) are in better agreement with the data. Interestingly,
concordance scenarios satisfying the available data predict a consistent
redshift of 50% ionized fraction $z_{\rm reion}(50\%)\sim 10$. On the other
hand, the duration of reionization is sensitive to the relative contribution
of bright versus faint galaxies, with scenarios dominated by faint galaxies
predicting a more extended reionization event. Scenarios relying too heavily
on high-redshift dwarfs are disfavored by kinetic Sunyaev-Zeldovich
measurements, which prefer a short reionization history.
###### keywords:
cosmology: theory – intergalactic medium – reionization – galaxies: high-
redshift – galaxies: formation – galaxies: dwarfs
## 1 Introduction
The installation of the Wide Field Camera 3 (WFC-3) on the Hubble Space
Telescope (HST) has recently improved the efficiency of searches for faint
$z\gtrsim 7$ galaxies by more than an order of magnitude (e.g., Bouwens et al.
2010a; McLure et al. 2010; Bunker et al. 2010). As a result, deep WFC-3
observations have provided new measurements of the rest frame ultra-violet
(UV, $\sim$1,500 Å) galaxy luminosity function at these redshifts. These
measurements are particularly important since galaxies are the most likely
sources of hydrogen reionization (e.g., Madau et al. 1999; Faucher-Gigère et
al. 2008a,b). Nevertheless, it is difficult to robustly translate these
measurements into predictions of the reionization history, because of
significant uncertainties in the spectral energy distribution (SED) of the
galaxies, the fraction of ionizing photons that escape into the intergalactic
medium (IGM), and in the contribution of fainter, as of yet undetected
galaxies.
Because of these uncertainties, it has been unclear whether star-forming
galaxies can actually reionize the Universe by $z\sim 6$ (as required by the
transmission of the Ly$\alpha$ forest at lower redshifts; Fan et al. 2002;
Becker et al. 2007, although see McGreer et al. 2011) and account for the
Thomson scattering optical depth to the microwave background implied by the
latest, 7-year Wilkinson Microwave Anisotropy Probe analysis (WMAP-7; Komatsu
et al. 2011), corresponding to a redshift of instantaneous reionization
$z_{\rm reion}=10.4\pm 1.2$.111In reality, the epoch of reionization is
expected to be extended in time and the Thomson scattering optical depth only
provides an integral constraint on reionization. Even if galaxies are in fact
the dominant re-ionizing sources, it is not clear to what extent faint sources
below the detection limit of existing observations are needed.
The amount of star formation taking place in low-mass dark matter haloes is
not only relevant for reionization, but also for our understanding of galaxy
formation and evolution in general. Indeed, several lines of evidence suggest
that star formation in such haloes is suppressed, at least in certain regimes.
For instance, it is well known at lower redshifts that the baryonic mass
fraction in low-mass haloes is strongly suppressed relative to $\Omega_{\rm
b}/\Omega_{\rm m}$ (e.g., Conroy & Wechsler 2009; Guo et al. 2010). This
baryon deficiency is commonly attributed to a combination of feedback
processes, such as galactic winds, and suppression by the photo-ionizing
background (e.g., Dekel & Silk 1986; Efstathiou 1992; Murray et al. 2005;
Faucher-Giguère et al. 2011). Observationally, there also appear to be far
fewer dwarf galaxies in the haloes of the Milky Way and M31 than the number of
dark matter sub-haloes capable of hosting them predicted in $N-$body
simulations (e.g., Bullock et al. 2000; Madau et al. 2008), suggesting that
some process inhibited star formation in the dark sub-haloes. Recently,
theoretical models have also suggested that star formation may be specifically
suppressed in low-mass haloes at early times due to a metallicity dependence
of the star formation efficiency (Robertson & Kravtsov 2008; Gnedin & Kravtsov
2010; Krumholz & Dekel 2011; Kuhlen et al. 2012). If star formation is indeed
strongly suppressed in early dwarf galaxies, then it may not be possible to
rely on them to reionize the Universe. It is thus necessary to clarify the
importance of those galaxies for reionization.
The primary goal of this paper is to examine the existing observational
constraints on hydrogen reionization and its sources, and to systematically
determine which scenarios are (and are not) allowed by the data. A main
distinction of our study relative to recent analyses (e.g., Bouwens et al.
2011c; Shull et al. 2011; Jaacks et al. 2012) is the inclusion of lower-
redshift Ly$\alpha$ forest data (see also Miralda-Escudé 2003; Bolton &
Haehnelt 2007; Faucher-Giguère et al. 2008a; Pritchard et al. 2010; Haardt &
Madau 2011). The mean transmission of the Ly$\alpha$ forest, which is set by a
balance between the ionizing background and recombinations, has the advantage
of being a complete probe of the ionizing sources. The total instantaneous
rate of injection of ionizing photons into the IGM can be measured from the
Ly$\alpha$ forest without recourse to assumptions on the escape fraction or
extrapolating the contribution of faint sources, two of the principal
uncertainties affecting traditional analyses based on the galaxy UV luminosity
function. We also include recent constraints on the duration of reionization
from measurements of the kinetic Sunyaev-Zeldovich (kSZ) effect by the _South
Pole Telescope_ 222http://pole.uchicago.edu (SPT) high-resolution microwave
background experiment (Zahn et al. 2011; for a recent parameter space study of
the kSZ signal from patchy reionization, see Mesinger et al. 2011).
While the Ly$\alpha$ forest data are mostly restricted to $z\leq 6$, when
reionization is probably complete,333Because reionization is predicted to be
highly inhomogeneous, existing constraints have not ruled out that some
regions of the Universe may have been reionized as late as $z\sim 5$ (e.g.,
McGreer et al. 2011), but as we show in this paper various data taken
collectively suggest that the bulk of reionization occured significantly
earlier. they provide valuable constraints in two ways. First, realistic
reionization scenarios should continuously connect to the post-reionization
IGM probed by the forest. Second, measurements of the galaxy UV luminosity
function (analogous to those directly probing the epoch of reionization) are
available over the full redshift interval covered by the Ly$\alpha$ forest
data (e.g., Bouwens et al. 2007; Reddy & Steidel 2009). Where the data
overlap, comparison of the Ly$\alpha$ forest and the UV luminosity function
allows us to constrain the escape fraction and limiting magnitude (minimum
luminosity) down to which the luminosity function must be integrated in order
to account for all the ionizing photons measured using the forest (Faucher-
Giguère et al. 2008a). Since these parameters are constrained where the data
overlap, we can test whether they must evolve with redshift in order to
accommodate the reionization constraints from WMAP and galaxy surveys. Such
evolution, in particular in the escape fraction, is sometimes invoked to
support the hypothesis that galaxies can indeed reionize the Universe (e.g.,
Haardt & Madau 2011), but there is little direct evidence for the required
change because direct measurements of escaping Lyman continuum photons are
prohobitive during the epoch of reionization.
The plan of this paper is as follows. In §2, we review how UV luminosity
function measurements can be converted into predictions for the reionization
history. We show how uncertainties in the SED of galaxies, their escape
fraction, and the limiting magnitude introduce large degeneracies and allow a
wide range of scenarios to be consistent with the standard WMAP constraint. In
§3, we introduce the Ly$\alpha$ forest constraints on the ionizing background
at $2\leq z\leq 6$ and explain how these constraints relate to the ionizing
sources. In §4, we compare with the galaxy UV luminosity function and
Ly$\alpha$ forest data to constrain the escape fraction and limiting magnitude
at $z=4$. We then combine these constraints with the higher-redshift galaxy
survey data and the measured WMAP optical depth to quantify the allowed
scenarios, parameterized by the required limiting magnitude and evolution of
the escape fraction. We conclude with a discussion of the implications for
galaxy formation and experiments aimed at probing the epoch of reionization in
§5.
Throughout, we assume cosmological parameters consistent with the WMAP 7-year
data in combination with supernovae and baryonic acoustic oscillations:
$(\Omega_{\rm m},~{}\Omega_{\rm
b},~{}\Omega_{\Lambda},~{}h)=(0.28,~{}0.046,~{}0.72,~{}0.7)$ (abbreviated
WMAP-7; Komatsu et al. 2011). We adopt hydrogen and helium mass fractions
$X=0.75$ and $Y=0.25$, respectively. All magnitudes are in the AB system (Oke
& Gunn 1983). Unless otherwise noted, all errors are $1\sigma$.
Figure 1: Fits to Schechter galaxy UV luminosity function parameters versus
redshift. Data points with error bars are from Bouwens et al. (2011c). The
solid line is our best fit linear model (FIT). The dashed and dotted lines
show the MIN and MAX models, in which the parameters were adjusted within the
linear fit formal 1$\sigma$ errors to minimize (MIN) or maximize (MAX) the
contribution from faint galaxies (see text for details and Table 1 for
numerical values).
## 2 Galaxy survey and WMAP reionization constraints
The two most basic observational constraints on hydrogen reionization are the
high-redshift galaxy UV luminosity function (LF) and the Thomson (electron)
scattering optical depth to the microwave background measured by WMAP-7,
$\tau_{\rm e}=0.088\pm 0.015$ (Komatsu et al. 2011). In the following, we
review how these measurements can be combined to constrain parameters of the
ionizing source population, in particular the limiting UV magnitude $M_{\rm
lim}$ and the escape fraction of ionizing photons $f_{\rm esc}$ from star-
forming galaxies. The procedure is based on calculating, for a given set of
assumptions on the galaxy population, the predicted evolution of the IGM
ionized fraction versus redshift and evaluating the corresponding $\tau_{\rm
e}$. As we show in §2.3, the escape fraction is degenerate with the ratio of
$1,500$ Å UV continuum to ionizing flux, a quantity sensitive to the SED of
the galaxies and whose effect we encapsulate in a dimensionless parameter
$\zeta_{\rm ion}$ defined below. Thus, our analysis formally constrains the
combination $\zeta_{\rm ion}f_{\rm esc}$. For simplicity, though, we will
occasionally summarize our results in terms of $f_{\rm esc}$ (for values of
$\zeta_{\rm ion}$ motivated by stellar population synthesis models), since it
is the most uncertain of the two factors.
In this paper, we assume that the majority of the ionizing photons are
produced by star-forming galaxies dominated by ordinary Pop II stars. In
principle, other sources such as massive Pop III stars (e.g., Bromm et al.
1999; Yoshida et al. 2004), accreting black holes (e.g., Haiman & Loeb 1998;
Madau et al. 2004; Kuhlen & Madau 2005), or annihilating dark matter (e.g.,
Belikov & Hooper 2009) could also contribute ionizing photons. However, there
is essentially no observational support for these more exotic scenarios. In
particular, the luminosity function of luminous quasars drops sharply beyond
$z\sim 2$ (Hopkins et al. 2007) and theoretical models suggest that only one
or two supernovae from Pop III stars suffice to trigger the transition to Pop
II in an early halo (Wise et al. 2010). This is supported by IGM metallicity
measurements at $z-5-6$, which show that the relative abundances are
consistent with measurements down to $z\sim 2$, and thus that there is no
evidence for significant metal production from Pop III stars in the first
billion years (Becker et al. 2011b). In contrast, star-forming galaxies are
now routinely observed at $z\gtrsim 7$ and we show explicitly in this work
that scenarios in which they are solely responsible for hydrogen reionization
are consistent with the available data (for a recent review, see also
Robertson et al. 2010).
If sources other than star-forming galaxies dominated hydrogen reionization,
then the constraints on $M_{\rm lim}$ and $f_{\rm esc}$ that follow would be
arbitrarily weakened. However, following Occam’s razor, we do not consider
such scenarios further here.
### 2.1 Calculation of the HII volume filling fraction and of the Thomson
optical depth
The evolution of the volume filling fraction of ionized hydrogen, $Q_{\rm
HII}(z)$, is given by the differential equation
$\frac{dQ_{\rm HII}}{dt}=\frac{\dot{n}_{\rm ion}}{\bar{n}_{\rm
H}}-\frac{Q_{\rm HII}}{\bar{t}_{\rm rec}},$ (1)
consisting of a source term proportional to the ionizing emissivity and a sink
term due to recombinations (Madau et al. 1998).
Under the assumption that galaxies provide the bulk of the ionizing photons,
the comoving ionizing emissivity (in units of photons per unit time, per unit
volume) can be expressed as an integral over the galaxy UV LF, $\phi(M_{\rm
UV})$:
$\dot{n}_{\rm ion}^{\rm com}=\int_{M_{\rm lim}}^{\infty}\\!\\!\\!dM_{\rm
UV}\,\phi(M_{\rm UV})\gamma_{\rm ion}(M_{\rm UV})\,f_{\rm esc}\,.$ (2)
We denote by $\gamma_{\rm ion}(M_{\rm UV})$ the ionizing luminosity (in units
of photons per unit time) of a galaxy with absolute rest-frame UV (1500 Å)
magnitude $M_{\rm UV}$. $f_{\rm esc}$ denotes the effective escape fraction,
which by definition we treat as a function of $z$ only (see §2.4). The volume
averaged recombination time is given by
$\displaystyle\bar{t}_{\rm rec}$ $\displaystyle=$
$\displaystyle\frac{1}{C_{\rm HII}\alpha_{\rm B}(T_{0})\,\bar{n}_{\rm
H}(1+Y/4X)\,(1+z)^{3}\,}$ $\displaystyle\approx$ $\displaystyle 0.93\;{\rm
Gyr}\,\left(\frac{C_{\rm HII}}{3}\right)^{-1}\left(\frac{T_{0}}{2\times
10^{4}\,{\rm K}}\right)^{0.7}\\!\\!\left(\frac{1+z}{7}\right)^{-3},$
where $\alpha_{B}$ is the case B hydrogen recombination coefficient, $T_{0}$
is the IGM temperature at mean density, $C_{\rm HII}$ is the effective
clumping factor in ionized gas, and $\bar{n}_{\rm H}$ is the mean comoving
hydrogen number density. We assume that helium is singly ionized at the same
time as hydrogen, but only fully ionized later through the action of quasars
(e.g., Faucher-Giguère et al. 2008a). We use the effective clumping factor to
account for both the actual clumpiness of the gas and for the fact that the
IGM temperature (and hence the proper recombination coefficient) in general
depends on density, so that formally an average over the temperature
distribution should be performed.
The clumping factor must be selected with care, since formal averages $C_{\rm
HII}\sim\langle n_{\rm HII}^{2}\rangle/\langle n_{\rm HII}\rangle^{2}$ over
simulation volumes yield large values $\sim 30$ (e.g., Gnedin & Ostriker 1997;
Springel & Hernquist 2003) that imply very demanding requirements on the
ionizing sources. These large clumping factors arise because the average
includes very dense galaxy halo gas. However, absorption of ionizing photons
by gas inside (or in the immediate vicinity of) galaxies is already accounted
for by the escape fraction. Thus, the correct clumping factor to use is one
that accounts only for recombinations occurring in the more diffuse IGM.
Although some ambiguity is inherent in this definition, recent studies suggest
that values $C_{\rm HII}=1-3$ are appropriate during the epoch of reionization
(e.g., Pawlik et al. 2009; Shull et al. 2011; McQuinn et al. 2011). The IGM
temperature $T_{0}$ is also uncertain, but the fiducial value $T_{0}=2\times
10^{4}$ K is reasonable for freshly reionized gas (Hui & Haiman 2003).
The Thomson optical depth to microwave background is then obtained by
integrating $Q_{\rm HII}$,
$\tau_{\rm e}=\int_{0}^{\infty}dz\frac{c(1+z)^{2}}{H(z)}Q_{\rm
HII}(z)\,\sigma_{\rm T}\,\bar{n}_{\rm H}\,(1+\eta Y/4X),$ (4)
where $H(z)$ is the Hubble parameter, $\sigma_{\rm T}$ is the Thomson cross
section, and we consider helium to be only singly ionized ($\eta=1$) at $z>4$
and doubly ionized ($\eta=2$) at lower redshift.
The main uncertainties in these calculations, which we discuss next, are
1. 1.
the extrapolation of the LF to magnitudes and redshifts for which no direct
measurement exists (§ 2.2),
2. 2.
the conversion from $M_{\rm UV}$ to ionizing photon luminosity ($\gamma_{\rm
ion}$; § 2.3),
3. 3.
and the escape fraction of ionizing photons ($f_{\rm esc}$; § 2.4).
### 2.2 High redshift galaxy luminosity functions
We base our analysis on recent observational determinations of the rest-frame
UV LF at $z\geq 4$ in the HUDF09 (Beckwith et al. 2006; Oesch et al. 2007),
ERS (Windhorst et al. 2011), and CANDELS fields (Grogin et al. 2011; Koekemoer
et al. 2011) by Bouwens et al. (2007, 2011b, 2011c). The best-fit Schechter
function parameters ($\phi^{*}$, $M^{*}$, and $\alpha$) are summarized in
Bouwens et al. (2011c, hereafter B11). $M^{*}$ quantifies the characteristic
magnitude, $\phi^{*}$ measures the comoving number density, and $\alpha$ is
the faint-end slope.
Figure 2: Comparison of our FIT, MIN, and MAX luminosity function models to
the data of Bouwens et al. (2011a) at $z=4$ and 7, and to the updated limits
from Oesch et al. (2011) at $z=10$. The $z=10$ data points, obtained from a
single galaxy candidate, were not included in the fits. Table 1: UV
luminosity function evolution models
Model | $M^{*}$ | $\log_{10}\phi^{*}$ | $\alpha$
---|---|---|---
| $A$ | $B$ | $A$ | $B$ | $A$ | $B$
FIT | $-20.42\pm 0.05$ | $0.27\pm 0.03$ | $-3.01\pm 0.04$ | $-0.07\pm 0.02$ | $-1.84\pm 0.04$ | $-0.06\pm 0.02$
MIN | $-20.37$ | $0.30$ | $-3.05$ | $-0.09$ | $-1.80$ | $-0.04$
MAX | $-20.47$ | $0.24$ | $-2.97$ | $-0.05$ | $-1.88$ | $-0.08$
FIT denotes the best linear fit of the form
$\\{M^{*},~{}\log_{10}{\phi^{*}},~{}\alpha\\}=A+B(z-6)$ to the Schechter
parameters reported in Bouwens et al. (2011c) at $z=4,~{}5,~{}6,~{}7$ and 8.
The parameters of the MAX and MIN models are adjusted within 1$\sigma$ of the
best fit (independently) so as to maximize and minimize the contribution of
faint galaxies.
To interpolate between redshift bins and extrapolate to redshifts not directly
probed by the data, we fit the redshift evolution of the three Schechter
parameters to a simple linear model of the form
$\\{M^{*},~{}\log_{10}{\phi^{*}},~{}\alpha\\}=A+B(z-6)$. The best-fit
parameters, denoted FIT, are given in Table 1. In order to explore the
uncertainties in the extrapolation to very faint galaxies, we adopt two
additional models, in which we vary the redshift evolution within the formal
1$\sigma$ errors of our linear fits to either maximize (MAX) or minimize (MIN)
the contribution from faint galaxies. Compared to our FIT model, the MAX model
has a slightly brighter and less rapidly dimming $M^{*}$, a slightly larger
and more slowly decreasing $\phi^{*}$, and a steeper and more quickly
steepening faint end slope $\alpha$; and vice-versa for the MIN model. Figure
1 shows our three fits for the redshift evolution of the Schechter parameters.
Figure 2 shows how these fits compare to the actual LF data from Bouwens et
al. (2011a) at $z=4$ and $z=7$, and to the 1$\sigma$ upper limits at $z\sim
10$ obtained from the detection of a single galaxy candidate by Oesch et al.
(2011). While the FIT and MIN models are in good agreement with the $z\sim 10$
limits, the MAX model predicts more galaxies than observed at $M_{\rm
UV}=-19.6$ by a $\sim 2.5\sigma$. Given the substantial uncertainties in
estimating limits from a single candidate in a relatively small field, we
however consider the MAX model to represent a valid limiting case.
Figure 3: Abundance matching between the dark matter halo mass function and
the UV luminosity functions from Bouwens et al. (2010a) at $z=4,~{}7$ and
$10$.
For comparison with theoretical predictions, it is useful to relate the UV
magnitudes to the total mass of the haloes likely to host these galaxies.
Since direct mass determinations from gravitational lensing or clustering are
not available at very high redshift, we attempt to establish such a relation
via an abundance matching technique, by equating the cumulative dark matter
halo mass function to the cumulative UV luminosity function over the redshifts
of interest. At lower redshift the validity of the abundance matching
technique has been demonstrated by its ability to reproduce the spatial
clustering of galaxies in the SDSS/LRG catalog (Conroy et al. 2006; Moster et
al. 2010; Guo et al. 2010). Here, we use the UV luminosity, which traces star
formation rather than stellar mass, and we should expect a larger scatter in
its relation to total halo mass. Nevertheless, the relation is likely to still
be monotonic on average and we therefore expect the abundance matching results
to be valid at the order-of-magnitude level. The results are shown in Figure.
3, which reveals that the faint values of $M_{\rm lim}$ advocated by B11
correspond to total halo masses below $10^{9}\,\rm M_{\sun}$. These results
are in good agreement with a similar determination by Trenti et al. (2010).
### 2.3 Conversion from UV magnitude to ionizing luminosity
To evaluate equation (2), it is necessary to convert from the measured UV
magnitudes to ionizing luminosity (the $\gamma_{\rm ion}$ term). To do so, we
adopt a simple double power-law model for the galaxy SED in the relevant
range,444Some authors first convert the UV magnitude to a star formation rate,
then convert the star formation rate to a rate of production of ionizing
photons. This however introduces extraneous steps.
$L_{\nu}=L_{\nu_{1500}}\begin{cases}\left(\frac{\nu}{\nu_{1500}}\right)^{\beta_{\nu}}&h\nu<1~{}{\rm
Ry}\\\ f_{\rm
LyC}\left(\frac{\nu_{912}}{\nu_{1500}}\right)^{\\!\beta_{\nu}}\left(\frac{\nu}{\nu_{912}}\right)^{-\gamma}&1~{}{\rm
Ry}\leq h\nu<4\,{\rm Ry}\\\ 0&h\nu>4~{}{\rm Ry}.\end{cases}$ (5)
This simple form is adequate to capture the main features of more detailed
stellar population synthesis models over the limited energy range of interest
(cf. Leitherer et al. 1999; Schaerer 2003). We use the notation
$\nu_{\lambda}$ to denote the frequency corresponding to wavelength
$\lambda$/Å, e.g. $\nu_{912}$ is the frequency at the 912 Å Lyman edge. Note
that the “$\beta$” slopes often discussed in the literature (e.g., Bouwens et
al. 2010b) are usually defined in terms of wavelength,
$L_{\lambda}\propto\nu^{-\beta_{\lambda}}$, so that we have the relation
$\beta_{\nu}=-(\beta_{\lambda}+2)$.
The hydrogen ionizing photon luminosity ($\gamma_{\rm ion}$) is then given by
$\gamma_{\rm
ion}=\int_{\nu_{912}}^{\infty}\\!\frac{d\nu}{h\nu}L_{\nu}\,\equiv\,2\times
10^{25}\,{\rm s}^{-1}\,\left(\frac{L_{\nu_{1500}}}{{\rm
erg\,s^{-1}\,Hz^{-1}}}\right)\zeta_{\rm ion}.$ (6)
To express this as a function of UV magnitude, we use the standard AB relation
$\log_{10}(L_{\nu_{1500}}/({\rm erg\,s^{-1}\,Hz^{-1}}))=0.4\,(51.63-M_{\rm
UV})$. Using equation 5, we can solve for the dimensionless parameter
$\zeta_{\rm ion}$:
$\zeta_{\rm ion}=1.5\left(\frac{f_{\rm
LyC}}{0.2}\right)\,(1.65)^{\beta_{\nu}}\left(\frac{1-4^{-\gamma}}{\gamma}\right).$
(7)
This parameter, a function of the stellar spectrum characteristics,
encapsulates all the information necessary to convert from UV magnitude to
ionizing photon luminosity.
In order to bracket the uncertainties in the spectral parameters $(f_{\rm
LyC},\beta_{\nu},\gamma)$, we consider three different models: a fiducial
model (FID) with $\zeta_{\rm ion}=1$, a harder spectrum model (HARD) with
$\zeta_{\rm ion}=2$, and a softer spectrum model (SOFT) with $\zeta_{\rm
ion}=0.5$. This range is representative of Pop II star-forming galaxies with
continuous star formation histories and age $\sim 10-100$ Myr (Leitherer et
al. 1999). Note that converting from $M_{\rm UV}$ to ionizing luminosity via
the star formation rate as done in B11 corresponds to $\zeta_{\rm ion}=1$ (our
FID model). Our three $L_{\nu}$ models thus span a factor of two variation (up
and down) around the hydrogen-ionizing luminosity used by B11.
### 2.4 The escape fraction of ionizing photons
Some fraction of the ionizing radiation produced by stellar populations is
absorbed by dust and neutral hydrogen within their host galaxies, and thus
does not contribute to ionizing the IGM. We capture this suppression by a
simple multiplicative prefactor, $f_{\rm esc}$, applied in equation (2). Since
our calculations are tied to the observed rest-frame UV LF, our $f_{\rm esc}$
is strictly speaking a relative escape fraction, capturing the additional
suppression of photons blueward of the Lyman edge compared to 1500 Å photons.
While neutral hydrogen only absorbs the ionizing photons, dust extinguishes
1500 Å and ionizing photons similarly. Because of this broad band extinction
by dust, $f_{\rm esc}$ is not equal to the fraction of all ionizing photons
produced by stars which are absorbed in the galaxy. Evaluating the latter
would require knowledge of dust extinction, but is not actually required for
our purposes. Similar relative definitions of the escape fraction are often
adopted observationally as well (e.g., Steidel et al. 2001; Shapley et al.
2006; Inoue et al. 2006).
The true escape fraction may well vary with galaxy mass, age, star formation
history, or other properties. Such dependences are however essentially unknown
at this time. We therefore assume in this work that $f_{\rm esc}$ is a
function of $z$ only, i.e. we use $f_{\rm esc}(z)$ to represent an effective
escape fraction averaged over the galaxy population at redshift $z$, suitably
weighted by the (unabsorbed) ionizing luminosity. A time dependence of $f_{\rm
esc}$ could thus arise from either a genuine time evolution in the escape
fraction of galaxies (e.g., owing to an evolution in the star formation rate
and its associated feedback), or from a redshift evolution in the make up of
the galaxy population, with the escape fraction of galaxies with certain
properties remaining constant. In §4, we quantify the redshift evolution
required of $f_{\rm esc}$ required by the data for different scenarios.
Figure 4: Volume fraction filling of HII regions as a function of redshift for
a set of representative models that satisfy the measured galaxy UV LF and the
WMAP-7 Thomson scattering optical depth. LF evolution fits FIT, MIN, and MAX
are shown in blue, cyan, and magenta, respectively. The FID, SOFT, and HARD
spectral hardness models are indicated with solid, dotted, and dashed lines.
The line thickness corresponds to $f_{\rm esc}=5$%, 20%, and 50% (from thin to
thick).
### 2.5 Range of models allowed by the UV LF and WMAP-7 constraints alone
Figure 5: Thomson scattering optical depth to the microwave background versus
limiting UV magnitude. The colors represent our three different galaxy UV LF
parameterizations: FIT (blue), MIN (cyan), and MAX (magenta). The solid line
corresponds to the FID ($\zeta_{\rm ion}=1$) $L_{\nu}$-model, and the shaded
regions are bounded by the SOFT ($\zeta_{\rm ion}=0.5$) and HARD ($\zeta_{\rm
ion}=2$) models. The WMAP-7 $\tau_{\rm e}=0.088\pm 0.015$ (Komatsu et al.
2011) is indicated with a gray band. The top panel is for $f_{\rm esc}=20\%$,
the bottom left for $f_{\rm esc}=5\%$ and the bottom right for $f_{\rm
esc}=50\%$. $M_{\rm lim}$, $\zeta_{\rm ion}$, and $f_{\rm esc}$ are assumed
constant in these calculations, and we used a clumping factor of $C_{\rm
HII}=3$.
In Figure 5 we show the Thomson optical depth, $\tau_{\rm e}$, for different
reionization scenarios consistent with the measured UV luminosity function.
The models explored correspond to varying assumptions for $M_{\rm lim}$,
$\zeta_{\rm ion}$, and $f_{\rm esc}$, which are further assumed here to be
constant with redshift.
For the best-fit UV LF evolution parameterization (FIT), fiducial
$L_{\nu}$-model ($\zeta_{\rm ion}=1$, FID), and $f_{\rm esc}=0.2$ (solid blue
line in the top panel), we recover the result of B11 that a very faint
limiting magnitude, $M_{\rm lim}\ga-11$, is required in order to produce an
optical depth in agreement with WMAP-7. However, many other solutions are
possible. For example, the same LF model with a harder spectrum (upper edge of
blue shaded region) is consistent with the WMAP-7 data for $M_{\rm lim}=-14$,
and with the MAX LF model (magenta band) the WMAP-7 $\tau_{\rm e}$ constraint
can accommodate values of $M_{\rm lim}$ ranging from $-11$ to as bright as
$-16$, depending on the spectral hardness. The escape fraction provides yet
another degree of freedom. With a constant $f_{\rm esc}$ of 5% (bottom left
panel), most models cannot satisfy the WMAP-7 constraint. On the other hand,
if $f_{\rm esc}=50\%$ even the MIN LF fit or models with very soft spectra can
result in a sufficiently high $\tau_{\rm e}$.
In Figure 4 we show the volume filling factor of HII regions, $Q_{\rm
HII}(z)$, for a few representative models that all satisfy the WMAP-7
$\tau_{\rm e}$ constraint. Interestingly, these scenarios have limiting
magnitudes ranging from $-10$ to $-16$ and include models that extrapolate the
contribution of faint galaxies quite differently. We conclude that the
existing measurements of the high-redshift galaxy luminosity function (still
limited to relatively luminous sources) and of the Thomson optical depth to
the microwave background do not uniquely determine how reionization proceeded.
In particular, these constraints do not suffice to determine the role played
by low-luminosity galaxies.
## 3 Ly$\alpha$ forest constraints on the ionizing sources
The Ly$\alpha$ forest provides complementary constraints on the cosmological
emissivity of ionizing photons (Miralda-Escudé 2003; Bolton & Haehnelt 2007;
Faucher-Giguère et al. 2008a). Although saturation prevents accurate
measurements of the Ly$\alpha$ forest at $z\ga 6$ (e.g., Fan et al. 2002), it
has the advantage of being a complete probe, in the sense that it includes the
contribution of all ionizing sources, even if they are individually too faint
to be detected in galaxy surveys. Furthermore, the ionizing emissivity implied
by the Ly$\alpha$ forest does not depend on an assumed escape fraction. Thus,
the Ly$\alpha$ forest is not subject to the two main uncertainties affecting
the inference of the ionizing emissivity from the galaxy UV luminosity
function, namely $M_{\rm lim}$ and $f_{\rm esc}$. By assuming continuity
between the post-reionization epochs probed by the Ly$\alpha$ forest and the
reionization epoch probed by high-redshift galaxy surveys, it is therefore
possible to significantly reduce the permitted parameter space. In particular,
comparison of the UV luminosity function and Ly$\alpha$ data where they
overlap allows us to constrain a combination of escape fraction, the limiting
magnitude, and the conversion factor from 1,500 Å UV to ionizing luminosity
(§4).
### 3.1 Total ionization rate from the Ly$\alpha$ forest
The basic quantity constrained by the Ly$\alpha$ forest is the hydrogen
photoionization rate $\Gamma_{\rm HI}$,
$\Gamma_{\rm
HI}(z)=4\pi\int_{\nu_{912}}^{\infty}\frac{d\nu}{h\nu}J_{\nu}(z)\sigma_{\rm
HI}(\nu),$ (8)
where $J_{\nu}$ is the average specific intensity of the ultra-violet
background, $\sigma_{\rm HI}(\nu)$ is the photoionization cross section of
hydrogen, and the integral is from the Lyman limit to infinity. Indeed, the
mean level of transmission of the Ly$\alpha$ forest is set by the equilibrium
between the ionizing background and recombinations in the IGM. Thus, given a
model of the density fluctuations in the IGM and knowledge of the
intergalactic gas temperature-density relation, the mean transmission of the
Ly$\alpha$ forest can be inverted to give $\Gamma_{\rm HI}$ (e.g., Rauch et
al. 1997).
In this work, we use principally the $\Gamma_{\rm HI}$ data points from
Faucher-Giguère et al. (2008a,b) based on the mean transmission measurement of
Faucher-Giguère et al. (2008d). This mean transmission measurement, based on
86 high-resolution and high-signal-to-noise quasar spectra covering Ly$\alpha$
redshifts $2\leq z\leq 4.2$, was corrected for absorption by metal ions and
for biases in the continuum fits, an important effect at $z\gtrsim 4$. At
$z=5$ and $z=6$, we use the constraints on $\Gamma_{\rm HI}$ from Bolton &
Haehnelt (2007), also from mean transmission data. We do not use proximity
effect measurements, as they are typically of lower statistical precision and
affected by more severe systematics (e.g., Faucher-Giguère et al. 2008c).
Nevertheless, at $z=5-6$, where some of these effects are mitigated, the
proximity effect measurements of Calverley et al. (2011) are consistent with
Bolton & Haehnelt (2007).
In Table 2, we summarize the $\Gamma_{\rm HI}$ measurements and other inputs
used in our Ly$\alpha$ forest analysis.
Table 2: Ly$\alpha$ forest constraints on the ionizing emissivity
$z$ | $\Gamma_{\rm HI}$ | $\lambda_{\rm mfp}^{912}$ | $\dot{n}_{\rm ion}^{\rm com}$ | References
---|---|---|---|---
| $10^{-12}$ s-1 | pMpc | $10^{50}$ s-1 cMpc-3 |
2.0 | 0.64$\pm$0.18 | 303$\pm$84 | 2.0$\pm$0.8 (${}^{+2.1}_{-1.4}$) | FG08, SC10
2.2 | 0.51$\pm$0.10 | 227$\pm$61 | 1.7$\pm$0.6 (${}^{+1.7}_{-1.2}$) | FG08, SC10
2.4 | 0.50$\pm$0.08 | 174$\pm$45 | 1.8$\pm$0.6 (${}^{+1.8}_{-1.2}$) | FG08, SC10
2.6 | 0.51$\pm$0.07 | 135$\pm$34 | 2.0$\pm$0.6 (${}^{+1.9}_{-1.3}$) | FG08, SC10
2.8 | 0.51$\pm$0.06 | 106$\pm$26 | 2.2$\pm$0.6 (${}^{+2.0}_{-1.4}$) | FG08, SC10
3.0 | 0.59$\pm$0.07 | 84.4$\pm$21 | 2.7$\pm$0.7 (${}^{+2.5}_{-1.8}$) | FG08, SC10
3.2 | 0.66$\pm$0.08 | 67.9$\pm$16 | 3.3$\pm$0.9 (${}^{+3.0}_{-2.2}$) | FG08, SC10
3.4 | 0.53$\pm$0.05 | 55.2$\pm$13 | 2.8$\pm$0.7 (${}^{+2.5}_{-1.8}$) | FG08, SC10
3.6 | 0.49$\pm$0.05 | 49.5$\pm$2.1 | 2.6$\pm$0.3 (${}^{+1.7}_{-1.5}$) | FG08, P09
3.8 | 0.51$\pm$0.04 | 41.7$\pm$2.4 | 2.8$\pm$0.3 (${}^{+1.8}_{-1.6}$) | FG08, P09
4.0 | 0.55$\pm$0.05 | 34.0$\pm$3.1 | 3.2$\pm$0.4 (${}^{+2.2}_{-1.9}$) | FG08, P09
4.2 | 0.52$\pm$0.08 | 26.2$\pm$3.9 | 3.5$\pm$0.8 (${}^{+2.9}_{-2.2}$) | FG08, P09
5.0 | 0.52${}^{+0.35}_{-0.21}$ | 13.9$\pm$3.6 | 4.3$\pm$2.6 ($\pm$2.6) | B07, SC10
6.0 | $<$0.19 | 7.0$\pm$2.0 | $<$2.6 ($<2.6$) | B07, SC10
The HI photoionization rates measurements are taken from Faucher-Gigère et al.
(2008a) (FG08) and Bolton & Haehnelt (2007) (B07); the mean free paths are
taken from the fits of Prochaska et al. (2009) (P09) and Songaila & Cowie
(2010) (SC10). Errors on $\Gamma_{\rm HI}$ and $\lambda_{\rm mfp}^{912}$ are
$1\sigma$ and predominantly statistical (except for the B07 $\Gamma_{\rm HI}$
points, which include a systematic error budget). Total uncertainties on
$\dot{n}_{\rm ion}^{\rm com}$, including systematic effects arising from the
spectral shape of the UV background and the thermal history of the IGM, are
given in parentheses and shown by the light gray band in Figure 6 (see the
text). The prefixes ‘p’ and ‘c’ indicate proper and comoving units,
respectively.
### 3.2 From ionization rate to ionizing emissivity
The quantity most directly related to the sources of ionizing photons is their
spatially-averaged emissivity, $\epsilon_{\nu}$ (here in proper, specific
units). Assuming that the ionizing background has a power-law spectrum
$J_{\nu}=J_{\nu_{912}}(\nu/\nu_{912})^{-\gamma_{\rm bg}}$ between the HI and
HeII ionizing edges (and zero beyond),
$J_{\nu_{912}}=\frac{\Gamma_{\rm HI}h(\gamma_{\rm bg}+3)}{4\pi\sigma_{\rm
HI}(\nu_{912})}\left[1-\frac{1}{4^{\gamma_{\rm bg}+3}}\right]^{-1}.$ (9)
Since
$\epsilon_{\nu}(z)\approx 4\pi\frac{J_{\nu}(z)}{\lambda_{\rm mfp}(\nu,~{}z)},$
(10)
where $\lambda_{\rm mfp}$ is the mean free path of ionizing photons in proper
units (denoted $\lambda_{\rm mfp}^{912}$ at the Lyman limit),555This
approximation is valid at $z\geq 2$, where the mean free path is much smaller
than the Hubble scale.
$\epsilon_{\nu_{912}}(z)\approx\frac{\Gamma_{\rm HI}(z)h(\gamma_{\rm
bg}+3)}{\sigma_{\rm HI}\lambda_{\rm
mfp}^{912}(z)}\left[1-\frac{1}{4^{\gamma_{\rm bg}+3}}\right]^{-1}.$ (11)
Assuming similarly that
$\epsilon_{\nu}=\epsilon_{\nu_{912}}(\nu/\nu_{912})^{-\gamma}$ between the HI
and HeII ionizing edges (and zero beyond),666Since the ionizing background
spectrum is affected by filtering by the IGM, in general
$\gamma\neq\gamma_{\rm bg}$ (e.g., Haardt & Madau 1996; Faucher-Giguère et al.
2009).
$\displaystyle\dot{n}_{\rm ion}^{\rm com}(z)$
$\displaystyle=\frac{1}{(1+z)^{3}}\int_{\nu_{912}}^{\infty}\frac{d\nu}{h\nu}\epsilon_{\nu}(z)$
(12) $\displaystyle=\frac{1}{(1+z)^{3}}\frac{\Gamma_{\rm HI}(z)h}{\sigma_{\rm
HI}\lambda_{\rm mfp}^{912}(z)}\frac{(\gamma_{\rm
bg}+3)}{\gamma}\left[1-\frac{1}{4^{\gamma}}\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left[1-\frac{1}{4^{\gamma_{\rm
bg}+3}}\right]^{-1}$
(compare with eq. (2)). Equation (12) shows how the total comoving emissivity
of ionizing photons can be derived from the photoionization rate measured from
the Ly$\alpha$ forest and knowledge of the mean free path of the ionizing
photons, given a model for the spectral shape of the ionizing sources and
their integrated background.
At $3.6\leq z\leq 4.2$, we use the mean free path measured by Prochaska et al.
(2009) using a stacking analysis. This approach avoids the usual uncertainties
in calculating the mean free path from the column density distribution
stemming from the difficulty of measuring the column density of systems near
the Lyman limit (on the flat part of the curve of growth). At the other
redshifts $2\leq z\leq 6$, we use the mean free path derived by Songaila &
Cowie (2010) based on a new analysis of the column density distribution. These
expressions agree well with the mean free path inferred previously by Faucher-
Giguère et al. (2008a), but have significantly reduced uncertainties. On the
other hand, this mean free path is larger than that assumed by Madau et al.
(1999) by a factor $\sim 2.5$. Furthermore, these mean free path measurements
are significantly more accurate than the simple model based on the mean
spacing between Lyman limit systems assumed by Bolton & Haehnelt (2007).
Figure 6: Ly$\alpha$ constraints on the rate at which ionizing photons are
injected into the IGM (see §3 for details and Table 2 for numerical values).
The light gray band indicates instantaneous constraints from the measured mean
transmission of the Ly$\alpha$ forest, including systematic effects. The dark
gray band indicates the minimum value necessary to keep the Universe ionized,
assuming that reionization is complete, for a fiducial IGM temperature
$T_{0}=2\times 10^{4}$ K and effective clumping factor $C_{\rm HII}=1-3$.
Models of the ionizing background indicate that the ionizing emissivity is
dominated by star-forming galaxies at $z\gtrsim 3$, but that quasars may
dominate at lower redshifts (Faucher-Giguère et al. 2008a, 2009).
Figure 6 summarizes the IGM observational constraints on $\dot{n}_{\rm
ion}^{\rm com}$. The error bars on the data points account for the statistical
uncertainty on the photonization rate and on the mean free path. The $z=5$ and
$z=6$ error bars also include a systematic error budget on $\Gamma_{\rm HI}$,
as quantified by Bolton & Haehnelt (2007). Total uncertainties, including
systematics, are indicated by the light gray band and estimated as follows.
First, we allow for a 50% systematic error on the $\Gamma_{\rm HI}$ data
points from Faucher-Giguère et al. (2008a) to account for uncertainties in the
thermal state of the IGM and the probability distribution function of density
fluctuations, which enter in the mean transmission method (Bolton et al. 2005;
Faucher-Giguère et al. 2008a). Second, we vary the source spectral index
$\gamma$ from 1 to 3. The harder value $\gamma=1$ is preferred by optical line
ratio diagnostics in local starbursts (Kewley et al. 2001), while many stellar
population synthesis models predict $\gamma\approx 3$ (e.g., Leitherer et al.
1999). This range of slopes is also consistent with the possibility that
quasars, with mean spectral index $\sim 1.6$ (Telfer et al. 2002), contribute
significantly at the lower redshift end. The fiducial value assumed in our
calculations is $\gamma=1$; since stellar population synthesis models
generally predict softer spectra, we do not explore harder values. The
spectral index of the background in the ionizing regime is not independent but
instead satisfies $\gamma_{\rm bg}=\gamma-3(\beta-1)$,777This approximation
(arising from the frequency dependence of the mean free path; eq. (10)) is
valid at least up to $z\sim 4$, where the column density distribution has been
measured to be well approximated by a series of power laws (Prochaska et al.
2010). However, it may break down at earlier times, where optically thick
absorbers could be relatively more numerous and dominate the opacity. This
provides additional motivation for focusing our post-reionization analysis at
$z=4$ (§4). where $\beta$ is the slope of the HI column density distribution
(Faucher-Giguère et al. 2008a). We adopt $\beta=1.3$ (Songaila & Cowie 2010).
To indicate the total uncertainty, we first calculate the range of
$\dot{n}_{\rm ion}^{\rm com}$ values allowed by simultaneously varying the
systematically uncertain parameters to their extremes. We then add the
statistical uncertainty to the minimum and maximum values in each redshift
bin. We believe that this procedure conservatively captures the constraints on
$\dot{n}_{\rm ion}^{\rm com}$.
The mean transmission data points are instantaneous constraints that must be
satisfied by the galaxy population. Models of the ionizing background at
intermediate redshifts indicate that star-forming galaxies dominate at
$z\gtrsim 3$ (Haehnelt et al. 2001; Bolton et al. 2005; Faucher-Giguère et al.
2008a, 2009). However, quasars may dominate the hydrogen photoionization rate
at later times, so that the total background should be regarded as an upper
limit to the contribution of star-forming galaxies alone. It should also be
noted that any redshift evolution in $\dot{n}_{\rm ion}^{\rm com}$ contained
in the light gray band in Figure 6 is allowed. In particular, it is possible
that the true redshift evolution is titled in slope relative to that suggested
by the fiducial data points. This is because the uncertain parameters could
evolve significantly with redshift. For instance, measurements indicate that
the IGM temperature peaks at $T_{0}\gtrsim 2\times 10^{4}$ K around $z\sim
3.4$ (Lidz et al. 2010), possibly owing to re-heating from HeII reionization,
but could be less than $\sim 10^{4}$ K between HI and HeII reionization (Hui &
Haiman 2003; Becker et al. 2011a; Bolton et al. 2012). Recent observations
also indicate that the UV slopes of $z\sim 7$ galaxies are significantly bluer
than their $z\sim 3$ counterparts (Bouwens et al. 2010b), so that the relevant
spectral indexes could also evolve.
It is apparent from Figure 6 that the best-fit comoving ionizing photon
emissivity, $\dot{n}^{\rm com}_{\rm ion}$, increases from $z=2$ to $z=4.2$,
and perhaps even to higher redshift. This basic behavior stems from the fact
that while the photoionization rate, $\Gamma_{\rm HI}$, is approximately
constant over this redshift interval, the mean free path decreases rapidly
with increasing redshift (Faucher-Giguère et al. 2008a; McQuinn et al. 2011).
Thus, an increasing ionizing emissivity is required to maintain the observed
photoionization rate (eq. 2).
### 3.3 Keeping the Universe ionized
Figure 6 also shows (as the dark gray band) $\dot{n}_{\rm ion}^{\rm
com,crit}$, the minimum $\dot{n}_{\rm ion}^{\rm com}$ required to keep the IGM
ionized once it has already been reionized. This number is obtained by
balancing the global recombination rate in the fully ionized IGM with the rate
at which ionizing photons escape galaxies:
$\displaystyle\dot{n}_{\rm ion}^{\rm com,crit}$ $\displaystyle=C_{\rm
HII}\alpha_{\rm A}(T_{0})\bar{n}_{\rm H}(1+Y/4X)(1+z)^{3}$ (13)
$\displaystyle\approx 3\times 10^{50}~{}{\rm
s^{-1}~{}cMpc^{-3}}\left(\frac{C_{\rm HII}}{3}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left(\frac{T_{0}}{2\times
10^{4}\,{\rm K}}\right)^{-0.7}\\!\\!\left(\frac{1+z}{7}\right)^{3}.$
Here, $\alpha_{\rm A}$ is the case A recombination coefficient of hydrogen.
Although we used the case B coefficient for the calculation of the HII volume
filling factor during reionization (eq. (2.1)), a large fraction of the
recombinations directly to the ground state at later times do not actually
contribute to the ionizing background (Faucher-Giguère et al. 2009).
The $\dot{n}_{\rm ion}^{\rm com,crit}$ band in Figure 6 covers the range
$C_{\rm HI}=1-3$ and assumes a fiducial IGM temperature $T_{0}=2\times 10^{4}$
K. This temperature is consistent with measurements at $z\sim 2-4$ (e.g., Lidz
et al. 2010), but could be higher or lower by $\sim 10^{4}$ K depending on the
spectrum of the re-ionizing sources and the time elapsed since reionization
(Hui & Haiman 2003). Estimates of the IGM temperature at $z>4$ in fact suggest
$T_{0}<10,000$ K at $z\sim 5-6$ (Becker et al. 2011a; Bolton et al. 2012).
Fortunately, this uncertainty does not substantially affect our arguments,
since the recombination coefficient is a relatively weak function of
temperature.
Note that the $\dot{n}_{\rm ion}^{\rm com,crit}$ lower limit only applies
after reionization is complete. Since we do not a priori know the redshift of
reionization, the plotted constraint (extending to $z=7.5$) need not
necessarily be satisifed everywhere. However, any viable reionization scenario
must satisfy this constraint at all redshifts following the time when an
ionized fraction $\sim 1$ is reached.
## 4 Combining Ly$\alpha$ forest, galaxy survey, and WMAP constraints
Figure 7: Top: Value of $\zeta_{\rm ion}f_{\rm esc}$ at $z=4$ needed to
simultaneously match the total comoving emissivity of ionizing photons
measured from the Ly$\alpha$ forest, $\dot{n}_{\rm ion}^{\rm com}$, and the
observed UV luminosity function at the same redshift, as a function of the
limiting UV magnitude. Since our LF fits are almost identical at $z=4$, we
only show the FIT case. Because $\zeta_{\rm ion}=1$ for the fiducial spectral
model, the values directly quantify the implied escape fraction. Bottom:
Power-law index $\kappa$ of the redshift evolution of $\zeta_{\rm ion}f_{\rm
esc}$ (see eq. 15) needed to simultaneously match the $z=4$ Ly$\alpha$ forest
and WMAP-7 Thomson optical depth constraints, as a function of the limiting UV
magnitude (assumed constant here), for our three LF evolution fits. The solid
(dotted) lines correspond to the median value ($\pm 1\sigma$) of the WMAP-7
Thomson optical depth. The shaded regions encompass the total (including
systematic) uncertainty in $\dot{n}_{\rm ion}^{\rm com}(z\\!=\\!4)$. Note that
some models with bright $M_{\rm lim}$ do not admit solutions for the entire
$\tau_{\rm e}$ range. Figure 8: Evolution of $\zeta_{\rm ion}f_{\rm esc}$
versus redshift required to simultaneously satisfy the $z=4$ Ly$\alpha$ forest
and WMAP-7 Thomson optical depth constraints, for $M_{\rm
lim}=-10,\,-13,\,{\rm and}\,-16$ (models corresponding to the solid lines in
Fig. 7). A ceiling of $\zeta_{\rm ion}f_{\rm esc}<2$ is imposed in our
calculations, corresponding to $f_{\rm esc}=1$ for $\zeta_{\rm ion}=2$ (our
HARD spectral model).
We showed in §2 that many different reionization scenarios are consistent with
the existing galaxy survey and WMAP constraints, principally due to
uncertainties in $M_{\rm lim}$ and $f_{\rm esc}$, and also $\zeta_{\rm ion}$.
We now combine these constraints with the Ly$\alpha$ forest data at lower
redshifts (§3), which allow us to break certain degeneracies and quantify
possible redshift evolution in the relevant parameters. The key idea is that
for any $M_{\rm lim}$, comparison of the Ly$\alpha$ forest and galaxy UV
luminosity function data where they overlap imply a unique $\zeta_{\rm
ion}f_{\rm esc}$ value (assuming that galaxies dominate the ionizing
background). We choose to make this comparison at $z=4$, because at this
redshift we expect the hydrogen ionizing background to in fact be dominated by
star-forming galaxies (Faucher-Giguère et al. 2008a, 2009). Furthermore, at
this redshift the observational constraints on the Ly$\alpha$ forest
transmission and the mean free path of ionizing photons are quite good.
Additionally, we do not expect this redshift to be strongly affected by large
inhomogeneities in the ionizing background. At higher redshifts (especially at
$z\sim 6$), interpretation of the Ly$\alpha$ forest data becomes more
uncertain because of the small number of sight lines available and because
reionization may not be 100% complete (McGreer et al. 2011). Because only
certain combination $(M_{\rm lim},~{}\zeta_{\rm ion}f_{\rm esc})$ are allowed
at $z=4$, we can quantify whether redshift evolution in these parameters is
required in order to simultaneously satisfy the higher-redshift constraints
from galaxy surveys and WMAP.
Figure 9: Redshift at which reionization is 20% (top left), 50% (top right),
90% (bottom left), and 100% complete (bottom right), as a function of the
limiting UV magnitude, for models in which $\zeta_{\rm ion}f_{\rm esc}(z)$ is
tuned to reproduce both the WMAP-7 Thomson optical depth and the $z=4$
Ly$\alpha$ forest constraints. The colors represent our three LF evolution
fits, and the shaded region encompasses both the WMAP-7 $\tau_{e}$ 1$\sigma$
region and the total (including systematic) uncertainty in $\dot{n}_{\rm
ion}^{\rm com}(z=4)$. Figure 10: Same as Figure 6, but with the models from
Figure 4 overplotted, for the constant $\zeta_{\rm ion}f_{\rm esc}$ case (top)
and modified to allow for redshift evolution in $\zeta_{\rm ion}f_{\rm esc}$
(bottom). For the models with redshift evolution in $\zeta_{\rm ion}f_{\rm
esc}$, the luminosity function parameterization and $M_{\rm lim}$ are fixed to
the values from Figure 4, $\zeta_{\rm ion}f_{\rm esc}(z=4)$ is set by the
Ly$\alpha$ forest data at $z=4$, and we solve for $\kappa$ (eq. (14)) such
that the Thomson optical depth matches the central WMAP-7 measurement. In the
same order as the legend in Figure 4, the best-fit values are
$\kappa=2.2,\,3.3,\,3.0,\,1.3,\,6.7,\,2.1,\,3.0$. The model lines are greyed-
out at $z<4$, since we do not utilize galaxy luminosity function data at those
redshifts.
In principle, $M_{\rm lim}$, $\zeta_{\rm ion}$, and $f_{\rm esc}$ can all be
arbitrary functions of redshift. However, the limited data do not allow us to
explore the entire range of possibilities. Instead, we adopt simple, single-
parameter power-law models to quantify the required redshift evolution:
$\displaystyle(\zeta_{\rm ion}f_{\rm esc})(z)\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!(\zeta_{\rm ion}f_{\rm
esc})(z\\!=\\!4)\,\left(\frac{1+z}{5}\right)^{\kappa}$ (14) $\displaystyle
10^{-0.4M_{\rm lim}(z)}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!10^{-0.4M_{\rm
lim}(z\\!=\\!4)}\,\left(\frac{1+z}{5}\right)^{-\lambda}.$ (15)
The last expression encodes a power-law parameterization in the UV luminosity,
which is logarithmically related to the UV magnitude. In order to prevent
unphysical values of $f_{\rm esc}$, we impose $\zeta_{\rm ion}f_{\rm esc}\leq
2$, corresponding to a ceiling of $f_{\rm esc}=1$ for the case $\zeta_{\rm
ion}=2$ (our HARD spectral model; §2.3). There is also a physical lower limit
on the luminosity of the faintest galaxies, corresponding to the minimum halo
mass in which baryons can collapse and form stars. However, this limit is not
as well understood and could depend significantly on redshift. We impose an
extremely faint ceiling of $M_{\rm lim}=0$ ($\sim 4$ orders of magnitude below
the UV suppression sacle of $M=-10$), and discuss the viability of different
scenarios later. Note that we have defined the signs of the $\kappa$ and
$\lambda$ power-law indexes so that positive values correspond to increasing
efficiency of ionizing photon production going to higher redshifts.
In the following, we consider possible evolution first in $\zeta_{\rm
ion}f_{\rm esc}$ (§4.1), and then in $M_{\rm lim}$ (§4.2). In principle, there
could be simultaneous evolution in both $\zeta_{\rm ion}f_{\rm esc}$ and
$M_{\rm lim}$, but the data do not allow us to discriminate between such mixed
scenarios. Furthermore, we will show that the strong redshift evolution
required in models relying only on relatively bright galaxies is most
plausibly accounted for by evolution in the escape fraction. In §4.3, we
consider the redshift and duration of reionization in different allowed
scenarios, showing that the redshift of 50% ionized fraction is consistently
at $z_{\rm reion}(50\%)\sim 10$ among the different models allowed by the
data, but that the duration of reionization, $\Delta z_{\rm reion}\equiv
z_{\rm reion}(100\%)-z_{\rm reion}(20\%)$ (where $z_{\rm reion}(x)$ is the
redshift such that $Q_{\rm HII}(z_{\rm reion}(x))=x$), is on the other hand
sensitive to the contribution of faint galaxies.
### 4.1 Redshift evolution of $\zeta_{\rm ion}f_{\rm esc}$
We first focus on the redshift evolution of $\zeta_{\rm ion}f_{\rm esc}$ from
$z=4$ toward higher redshifts.
Our goal is to determine as a function of $M_{\rm lim}$ (here assumed to be
independent of $z$) what values of $\kappa$ are consistent with both the
$z\approx 4$ Ly$\alpha$ forest and WMAP-7 Thomson optical depth constraints,
while also satisfying the measurements of the galaxy UV LF at the bright end.
Operationally, we first determine for a given $M_{\rm lim}$ what range of
$\zeta_{\rm ion}f_{\rm esc}(z=4)$ is required to give $\dot{n}_{\rm ion}^{\rm
com}(z=4)=3.2^{+2.2}_{-1.4}\times 10^{50}\,{\rm s}^{-1}\,{\rm cMpc}^{-3}$. The
results of this calculation are shown in the top panel of Figure 7. For
$M_{\rm lim}=-10$ to $-16$, the Ly$\alpha$ constraints require $\zeta_{\rm
ion}f_{\rm esc}(z=4)$ to lie between 2% and 8% (since the luminosity function
has already been measured down to $M_{\rm UV}=-16$ at $z=4$, cases with
brighter $M_{\rm lim}$ at this redshift are not allowed; Bouwens et al. 2007).
As $\zeta_{\rm ion}=1$ for the fiducial spectral model, this directly
quantifies the implied escape fraction. This result is independent of which LF
fit we employ, since at $z=4$ the parameters of our three fits are nearly
identical.
In the second step, we determine for each $M_{\rm lim}$ and $\zeta_{\rm
ion}f_{\rm esc}(z=4)$ what values of the power-law index $\kappa$ yield a
Thomson optical depth in the range allowed by WMAP-7. The resulting range of
allowed $\kappa$ values is shown in the bottom panel of Figure 7, with the
different color bands corresponding to our three LF fits. The width of the
bands encompasses both the 1$\sigma$ uncertainty of the WMAP-7 Thomson optical
depth measurement and the total (including systematic) uncertainty in the
$z=4$ Ly$\alpha$ forest data. Figure 8 shows the curves of $\zeta_{\rm
ion}f_{\rm esc}$ versus $z$ corresponding to allowed values of $\kappa$, for
representative choices of $M_{\rm lim}$. For each $\kappa$ solution, Figure 9
shows $z_{\rm reion}(20\%)$, $z_{\rm reion}(50\%)$, $z_{\rm reion}(90\%)$, and
$z_{\rm reion}(100\%)$.
Lastly, Figure 10 shows explicit examples of how redshift evolution in
$\zeta_{\rm ion}f_{\rm esc}$ allows models to simultaneously satisfy
reionization-epoch constraints and the $z<6$ Ly$\alpha$ forest data.
Specifically, we consider the same models as in Figure 4 (top) and modify them
to allow for redshift evolution in $\zeta_{\rm ion}f_{\rm esc}$ (bottom). For
the models with redshift evolution in $\zeta_{\rm ion}f_{\rm esc}$, the
luminosity function parameterization and $M_{\rm lim}$ are fixed to the values
from Figure 4, $\zeta_{\rm ion}f_{\rm esc}(z=4)$ is set by the Ly$\alpha$
forest data at $z=4$, and we solve for $\kappa$ such that the Thomson optical
depth matches the central WMAP-7 measurement. While the original models
without evolution in $\zeta_{\rm ion}f_{\rm esc}$ overproduce the ionizing
emissivity probed by the Ly$\alpha$ forest at $z=4$, the modified models
simultaneously satisfy all the constraints. Furthermore, models exist in which
the extrapolation to $z=2$ is also in good agreement with the lower-redshift
Ly$\alpha$ forest data.
At the $1\sigma$ level, only the MAX model with very faint $M_{\rm
lim}\gtrsim-11$ can accommodate no redshift evolution in $\zeta_{\rm
ion}f_{\rm esc}$. However, these scenarios are disfavored by external
constraints on the duration of reionization from the kinetic Sunyaev-Zeldovich
effect (Zahn et al. 2011), which in combination with the WMAP-7 optical depth
constrains the timing of its beginning and end. For the most conservative case
of arbitrary correlations between the thermal Sunyaev-Zeldovich (tSZ) effect
and the cosmic infrared background (CIB), Zahn et al. (2011) find that $z_{\rm
reion}(20\%)<13.1$ at 95% confidence level (CL), and $z_{\rm
reion}(99\%)>5.8$, also at 95% CL. In this work, we take these constraints at
face value. It is important to bear in mind, however, that the templates on
which they are based assume that reionization occurs primarily via star-
forming galaxies. Furthermore, the limits on the kSZ signal rely critically on
accurate subtraction of contaminating point sources. It will thus be important
to confirm these findings with refined analyses.
For the best-fit parameterization of the UV LF (the FIT model), models with no
redshift evolution in $\zeta_{\rm ion}f_{\rm esc}$ are disfavored even for
$M_{\rm lim}=-10$. Models that rely only on brighter galaxies formally satisfy
all the present constraints but only for strong redshift evolution in
$\zeta_{\rm ion}f_{\rm esc}$. For example, the case of $M_{\rm lim}=-16$ for
the FIT parameterization requires an evolution in $\zeta_{\rm ion}f_{\rm esc}$
by a factor $\approx 20$ from $z=4$ to $z=9$. As we will discuss at greater
length in §5, models that rely too heavily on fainter galaxies may be in
tension with theoretical models that suppress star formation in early, low-
mass systems (e.g., Krumholz & Dekel 2011; Kuhlen et al. 2012), which are
helpful in explaining some properties of the cosmic star formation history. If
star formation is indeed suppressed in those early dwarfs, then the existing
data would imply strong evolution in the escape fraction.
We also explored constraints on the redshift evolution $\zeta_{\rm ion}f_{\rm
esc}$ from $z=2$ to $z=4$ by comparing the Ly$\alpha$ forest data to the
galaxy UV luminosity function from Reddy & Steidel (2009) at $z=2$. Over that
redshift interval, a wide range $\kappa\sim 0-4.5$ is allowed, almost
independent of the assumed $M_{\rm lim}$ owing to the relatively shallow
faint-end slope of the luminosity function. In particular, the combination of
the UV luminosity function and Ly$\alpha$ forest data alone do not require any
significant evolution. Note that such evolution is nonetheless allowed by the
data, and in fact suggested by direct Lyman continuum observations (e.g.,
Inoue et al. 2006; Siana et al. 2010). Although this is not necessary on
physical grounds, it is interesting that most of the $\kappa$ values implied
from $z=4$ and up (Fig. 7) are also allowed from $z=2$ to $z=4$.
### 4.2 Redshift evolution of $M_{\rm lim}$
Figure 11: Top: Power-law index $\lambda$ of the redshift evolution of the
limiting UV magnitude $M_{\rm lim}$ (see eq. 14) needed to simultaneously
match the $z=4$ Ly$\alpha$ forest and the WMAP-7 Thomson optical depth
constraints, as a function of $M_{\rm lim}(z=4)$ ($\zeta_{\rm ion}\,f_{\rm
esc}$ is assumed constant). As before, the solid (dotted) lines correspond to
the median value ($\pm 1\sigma$) of the WMAP-7 Thomson optical depth, and the
shaded regions encompass the total (including systematic) uncertainty in
$\dot{n}_{\rm ion}^{\rm com}(z\\!=\\!4)$. Models with the MIN LF fit do not
admit any solutions with evolution only in $M_{\rm lim}$. Middle: Redshift at
which reionization is completed ($Q_{\rm HII}=1$). Most of the $M_{\rm
lim}$-only evolution models do not complete reionization by $z=5.8$, as
required by the combination of kSZ and WMAP-7 data (Zahn et al. 2011). Bottom:
Median $M_{\rm lim}$ redshift evolution (corresponding to the solid lines in
the top panel), for $M_{\rm lim}=-10,\,-13,\,{\rm and}\,-16$. Very faint
limiting magnitudes $M_{\rm lim}\gg-10$ are also likely excluded on physical
grounds.
We now turn to the possibility of redshift evolution in $M_{\rm lim}$.
Starting from the same values of $\zeta_{\rm ion}f_{\rm esc}(z=4)$ for a given
$M_{\rm lim}(z=4)$ as in the previous section, we determine what values of
$\lambda$ can produce agreement between the Ly$\alpha$ forest constraints at
$z=4$, and higher-redshift LF and WMAP-7 constraints. The top panel of Figure
11 demonstrates that substantial evolution is necessary to match the WMAP-7
Thomson optical depth measurement. Many models actually do not allow for a
solution: for the MIN LF fit, the contribution of dwarfs is suppressed to such
a degree that no amount of $M_{\rm lim}$ evolution is able to raise $\tau_{\rm
e}$ into the range allowed by WMAP-7. The FIT model only has solutions with
very steep $M_{\rm lim}$ evolution, requiring $M_{\rm lim}$ significantly
below $-10$ at high redshift. Only the MAX models are able to provide
solutions with a moderate amount of $M_{\rm lim}$ evolution. Most importantly,
as the middle panel of Figure 11 demonstrates, the majority of the models with
evolving $M_{\rm lim}$ (but constant $\zeta_{\rm ion}f_{\rm esc}$) do not
reach complete reionization by $z=5.8$, as required by the combination of the
kSZ and WMAP constraints (Zahn et al. 2011).
Except for small, extreme corners of parameter space, it is therefore not
possible to simultaneously match $z=4$ Ly$\alpha$ forest and reionization
constraints, and complete reionization in time, by allowing only evolution in
$M_{\rm lim}$. This provides further evidence for the need for a significant
redshift evolution in $\zeta_{\rm ion}f_{\rm esc}$ from $z=4$ toward higher
redshifts, as discussed in the previous section. In contrast, the data do not
provide conclusive evidence for a significant redshift evolution in $M_{\rm
lim}$ (although some evolution may be expected on physical grounds). We will
therefore concentrate the following discussion on scenarios with constant
$M_{\rm lim}$ but evolving $\zeta_{\rm ion}f_{\rm esc}$. It is important to
keep in mind, however, that evolution in $M_{\rm lim}$ could reduce the
required amount of evolution in $\zeta_{\rm ion}f_{\rm esc}$ somewhat.
### 4.3 Redshift and duration of reionization
Interestingly, Figure 9 shows that the redshift of 50% ionized fraction is
consistently at $z\sim 10$ between the different allowed scenarios. This is
consistent with the redshift of instantaneous reionization implied by WMAP-7,
$z_{\rm reion}=10.4\pm 1.2$. However, the predicted duration of the
reionization process is more extended for scenarios that include a larger
contribution from faint galaxies. This is a consequence of the shape of the
LF: the brighter and closer to $M^{*}$ (the knee of the LF) $M_{\rm lim}$ is,
the fewer galaxies contribute to reionization at high $z$. The reionization
process then quickly sets in once the exponential cutoff of the LF has shifted
to bright enough galaxies that $M_{\rm lim}$ galaxies become common. In
contrast, with a faint $M_{\rm lim}$ abundant faint galaxies contribute to
reionization even at very high redshifts, and therefore the overall evolution
of the process is slowed down.
As shown in Figure 9, the kSZ data favor relatively short reionization
histories and thus disfavor models that rely too heavily on high-redshift
dwarfs, as in the MAX models. A range of MIN and FIT models are however
allowed. In particular, our conclusions are consistent with theoretical
analyses anchored in the predicted dark matter halo mass function, rather than
to the observed luminosity function (Trenti et al. 2010; Mesinger et al. 2011;
Ciardi et al. 2011), indicating that as of yet undetected galaxies must
contribute significantly to reionization. However, our analysis suggest
alternative possibilities when strong redshift evolution in $\zeta_{\rm
ion}f_{\rm esc}$ is allowed.
Recently, two significant observational advances in using astrophysical
sources to probe the epoch of reionization have been reported. First, Mortlock
et al. (2011) discovered a luminous quasar at $z=7.085$ in which the
Ly$\alpha$ transmission profile is consistent with an IGM neutral fraction
$\sim 10$% at that time. Although this is compatible with most of the
scenarios allowed by our analysis (Fig. 9), excluding those relying on a
maximal contribution from dwarf galaxies, alternative interpretations exist in
the context of inhomogeneous reionization (Bolton et al. 2011).
Second, recent surveys for Ly$\alpha$ emitting galaxies at $z\geq 6$ have
found a decreasing fraction of Lyman break-selected galaxies with detected
Ly$\alpha$ emission of rest-frame equivalent width $\geq 20$ Å from $z\sim 6$
to $z\sim 7$ (Schenker et al. 2011; Pentericci et al. 2011; Ono et al. 2012).
The decline is such that existing models of Ly$\alpha$ propagation through
galactic winds and the intervening IGM indicate $Q_{\rm HII}\gtrsim 50$% at
$z\sim 7$ (Dijkstra et al. 2011). Such a high neutral fraction at $z=7$ is in
tension with the concordance scenarios summarized in Figure 9. Furthermore, if
reionization is essentially complete by $z\sim 6$ as the combination of kSZ
and WMAP-7 measurements indicate, and as suggested by Gunn-Peterson troughs in
the Ly$\alpha$ forest (e.g., Fan et al. 2002), then a very rapid evolution in
the neutral fraction would be implied. These findings are not easy to
reconcile, and highlight the need for more detailed modeling of Ly$\alpha$
radiative transfer in order to fully exploit the new and upcoming high-quality
data.
## 5 Summary and Discussion
Measurements of the galaxy UV ($\sim$1,500 Å) luminosity function at redshifts
$z\gtrsim 6$, recently improved by more than an order of magnitude thanks to
the WFC-3 camera on HST, provide constraints on the likely sources of hydrogen
reionization. However, these observations only directly reveal the sources
luminous enough to be individually detected. Furthermore, converting the
luminosity function measurements to IGM ionization rates involves large
uncertainties, in addition to the extrapolation necessary to model sources too
faint to be detected, including the SED of the star-forming galaxies and their
escape fraction of ionizing photons. These latter uncertainties are
encapsulated in the dimensionless factor $\zeta_{\rm ion}f_{\rm esc}$ used to
convert from a luminosity at 1,500 Å to a rate of production of ionizing
photons escaping into the IGM.
Given these uncertain parameters, we showed in §2 that many scenarios exist in
which star-forming galaxies are the dominant ionizing sources and which
satisfy both the galaxy survey constraints and the Thomson optical depth
implied by the WMAP-7 data. Such scenarios include ones with escape fraction
ranging from $f_{\rm esc}=5$% to $f_{\rm esc}=50$%, and limiting UV magnitude
ranging from $M_{\rm lim}=-16$ to $M_{\rm lim}=-10$, even when these are
assumed to be constant. Thus, these constraints alone (which have been the
focus of many analyses; e.g., Bouwens et al. 2011c; Bunker et al. 2010) are
not sufficient to determine the role of faint galaxies in reionizing the
Universe, and whether such galaxies are even present in significant number.
In §3 we used the Ly$\alpha$ forest at redshifts $2\leq z\leq 6$ to measure
the total instantaneous rate at which ionizing photons are injected into the
IGM. Although these measurements cover redshifts past the epoch of
reionization, they provide significant leverage over galaxy surveys. In
particular, the total ionizing emissivity implied by the mean transmission of
the Ly$\alpha$ forest does not rely on assuming an escape fraction or a
limiting magnitude, two of the main uncertainties limiting the predictive
power of UV luminosity function measurements alone. At $z=4$, where the
Ly$\alpha$ forest data is both abundant and free of large systematic effects
due to inhomogeneities in the ionizing background, comparison with the UV
luminosity function allowed us to determine $\zeta_{\rm ion}f_{\rm esc}$
almost independently of $M_{\rm lim}$, owing to the comparatively shallow
faint-end slope of the LF at $z\la 4$. Since $M_{\rm lim}\geq-16$ at $z=4$
(the UV luminosity function having already been measured down to this
magnitude), $f_{\rm esc}(z=4)=2-8\%$ (median $f_{\rm esc}(z=4)=4\%$), for the
fiducial spectral model $\zeta_{\rm ion}=1$.
Combining the Ly$\alpha$ forest, WMAP-7, and galaxy survey data and assuming
that galaxies are the main ionizing sources requires either: 1) extrapolation
of the galaxy luminosity function down to very faint UV magnitudes $M_{\rm
lim}\sim-10$, corresponding roughly to the UV background suppression scale
(e.g., Faucher-Giguère et al. 2011; but see Dijkstra et al. 2004); 2) an
increase of the escape fraction by a factor $\gtrsim 10$ from $z=4$ to $z=9$;
or 3) more likely, a hybrid solution in which undetected galaxies contribute
significantly and the escape fraction increases more modestly.
The present data do not allow us to select a unique viable reionization
scenario. Quantitatively, a range of combinations of limiting magnitudes and
redshift evolution of the parameters affecting the conversion of $\sim 1,500$
Å UV luminosity functions to rates of production of ionizing photons are
allowed and summarized in Figures 7, 8, and 11. Redshift evolution in the
limiting magnitude alone requires extreme assumptions in order to satisfy both
the Ly$\alpha$ forest and WMAP-7 constraints without appealing to very faint
galaxies. Even so, such scenarios predict that reionization ends at $z\lesssim
6$, in tension with recent measurements of the kinetic Sunyaev-Zeldovich
effect by SPT, which indicate that reionization ends earlier than $z=5.8$ at
95% CL (Zahn et al. 2011). On the other hand, significant redshift evolution
in $\zeta_{\rm ion}f_{\rm esc}$ is more plausible. In fact, $\zeta_{\rm ion}$
can vary by a factor $\sim 4$ owing to changes in the age, metallicity, and
IMF of the stellar populations (corresponding to the range $\zeta_{\rm
ion}=0.5-2$ assumed in this paper; §2.3).888Consistency between star formation
rate and stellar mass density measurements at $z\sim 7-8$ suggests that the
IMF of reionization-epoch galaxies is not very different from the local
Universe (Bouwens et al. 2011b). More importantly, $f_{\rm esc}$ can in
principle increase from $f_{\rm esc}\sim 4$% at $z=4$ to $f_{\rm esc}\sim 1$
at earlier times. A similar strong redshift evolution of the ionizing
luminosity-weighted escape fraction was also recently found to be required in
the “minimal cosmic reionization model” of Haardt & Madau (2011).
Although there are at present no direct constraints on the escape fraction
from faint galaxies during the epoch of reionization, deep searches for
escaping Lyman continuum radiation at lower redshifts do show some evidence
for redshift evolution (Steidel et al. 2001; Inoue et al. 2006; Shapley et al.
2006; Cowie et al. 2009; Siana et al. 2010; Nestor et al. 2011). Such
evolution, in which the escape fraction increases with redshift, could owe to
increased feedback at earlier times when star formation was more vigorous
(e.g., Wise & Cen 2009). This picture, in which ionizing photons escape
galaxies along lines of sight cleared of obscuring gas, would be consistent
with Lyman continuum observations suggesting “on/off” escape, possibly
connected to the viewing geometry (e.g., Shapley et al. 2006; Nestor et al.
2011; Vanzella et al. 2012). Another possibility is that faint galaxies may
typically have higher escape fraction than more massive galaxies (e.g. Yajima
et al. 2011), in which case the larger relative abundance of faint galaxies at
high redshift would result in an increase in the population-averaged escape
fraction. The extremely blue UV continuum slopes recently reported for $z\sim
7$ galaxies (Bouwens et al. 2010b) are also suggestive of weak nebular
recombination emission, which would be consistent with very high escape
fractions of ionizing photons (but see Dunlop et al. 2011 for a critical
analysis of the UV continuum slopes).
Recent models predict that star formation is suppressed in low-mass, high-
redshift galaxies owing to the metallicity dependence of the transition from
warm HI to dense molecular gas. In the fiducial implementations of Krumholz &
Dekel (2011) and Kuhlen et al. (2012), metallicity effects can strongly
suppress star formation in reionization-epoch galaxies in haloes of mass of
$M_{\rm h}\sim 10^{9}-10^{10}$ M⊙, corresponding to $M_{\rm UV}\sim-13$ to
$-16$ at $z\sim 7$. Formally, even the brighter $M_{\rm lim}$ of these models
can satisfy the existing reionization constraints, but only for strong
redshift evolution in $\zeta_{\rm ion}f_{\rm esc}$ (Fig. 7). Since such a
H2-regulated star formation suppression threshold is not far from the current
limits of HST observations at $z\sim 7$, deeper integrations have the
potential to significantly constrain those models. As Kuhlen et al. (2012)
showed, the exact halo mass below which metal-poor dwarfs are suppressed is
however sensitive to the details of the model implementation. In particular,
the relevant mass scale depends significantly on the metallicity floor assumed
to model the unresolved effects of metal enrichment by early Pop III stars. If
the correct halo mass threshold is lower by an order of magnitude relative to
the fiducial models of Kuhlen et al. (2012) and closer to the predictions of
Krumholz & Dekel (2011), i.e. $M_{\rm h}\sim 10^{9}$ M⊙ at $z\sim 7$, then
abundance matching suggests that the turn over in the luminosity function
would occur instead at $M_{\rm UV}\sim-13$ (see Fig. 3). For limiting
magnitudes in this neighborhood, more modest redshift evolution in $\zeta_{\rm
ion}f_{\rm esc}$ can satisfy the galaxy survey, WMAP-7, and Ly$\alpha$ forest
data. Thus, given the present implementation uncertainties, H2-regulated star
formation in high-redshift dwarfs is consistent with galaxies reionizing the
Universe.
For each scenario satisfying the observational constraints considered in this
work, we evaluated the timing and duration of the corresponding reionization
history (Fig. 9). Interestingly, the redshift at which the ionized fraction
reaches 50% is consistent among the different allowed scenarios, $z_{\rm
reion}(50\%)\sim 10$. This redshift is consistent with the redshift of
instantaneous reionization $z_{\rm reion}=10.4\pm 1.2$ implied by the WMAP-7
analysis (Komatsu et al. 2011). On the other hand, the predicted duration of
reionization is quite sensitive to the fractional contribution of faint
galaxies. For instance, our MAX parameterization with $M_{\rm lim}=-10$ (i.e.
with a heavy dwarf contribution) allows scenarios with $z_{\rm
reion}(20\%)=17$ and $z_{\rm reion}(100\%)<4$. On the other hand, models which
rely on bright galaxies are predicted to reionize the Universe much more
sharply, with $z_{\rm reion}(20\%)-z_{\rm reion}(100\%)\approx 3$ for our FIT
parameterization with ($M_{\rm lim}=-16$). Thus, experiments capable of
measuring the duration of reionization will have direct implications for faint
galaxies. In fact, the more extended reionization histories which rely on a
heavy contribution from faint galaxies are already ruled out by recent
constraints from the kinetic Sunyaev-Zeldovich effect by SPT. In the next few
years, expanded data sets from high-resolution microwave background
experiments in combination with more precise measurements of the integrated
Thomson optical depth with _Planck_ 999http://www.rssd.esa.int will improve
these constraints further. Refined analyses should also improve the accuracy
with which contaminating point sources are subtracted, and thus solidify the
results.
On the theoretical front, there is much room for improving our understanding
of how to reliably use astrophysical sources such as high-redshift Ly$\alpha$
emitting galaxies, luminous quasars, and $\gamma-$ray bursts to measure the
neutral fraction in the IGM. Finally, efforts aimed at detecting 21 cm
emission from high-redshift intergalactic neutral gas (e.g., Bowman et al.
2007; Parsons et al. 2010; Bowman & Rogers 2010) are poised to eventually
directly map the reionization epoch.
## Acknowledgments
CAFG is supported by a fellowship from the Miller Institute for Basic Research
in Science and NASA grant 10-ATP10-0187.
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|
arxiv-papers
| 2012-01-03T21:00:03 |
2024-09-04T02:49:25.906540
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Kuhlen and C.-A. Faucher-Giguere (UC Berkeley)",
"submitter": "Michael Kuhlen",
"url": "https://arxiv.org/abs/1201.0757"
}
|
1201.0846
|
# Using complex surveys to estimate the $L_{1}$-median of a functional
variable: application to electricity load curves
Mohamed Chaouch 1 and Camelia Goga2
1 EDF Recherche & Développement, Département ICAME,
Clamart - France
2Institut de Mathématiques de Bourgogne, Université de Bourgogne,
DIJON - France
email : $mohamed.chaouch$@edf.fr, $camelia.goga$@u-bourgogne.fr
###### Abstract
Mean profiles are widely used as indicators of the electricity consumption
habits of customers. Currently, in Électricité De France (EDF), class load
profiles are estimated using point-wise mean function. Unfortunately, it is
well known that the mean is highly sensitive to the presence of outliers, such
as one or more consumers with unusually high-levels of consumption. In this
paper, we propose an alternative to the mean profile: the $L_{1}$-median
profile which is more robust. When dealing with large datasets of functional
data (load curves for example), survey sampling approaches are useful for
estimating the median profile avoiding storing the whole data. We propose here
estimators of the median trajectory using several sampling strategies and
estimators. A comparison between them is illustrated by means of a test
population. We develop a stratification based on the linearized variable which
substantially improves the accuracy of the estimator compared to simple random
sampling without replacement. We suggest also an improved estimator that takes
into account auxiliary information. Some potential areas for future research
are also highlighted.
Key Words: Horvitz-Thompson estimator, k-means algorithm, poststratification,
stratified sampling, substitution estimator, variance estimation.
## 1 Introduction
The French electricity company, Électricité De France (EDF), uses extensively
the customer class load profiles in distribution network calculation. Load
profiles are also used to predict future loads in distribution network
planning or to estimate the daily load curve of a new customer. The customer
data usually includes information about the type of the electricity
connection, the customer class, the consumption type and some other additional
information. The combination of the individual customer informations and the
class load profiles allows us to estimate its load curve.
At EDF, the mean profile is used as an indicator of the electricity
consumption of the customers. Nevertheless, it is well known that the mean is
highly sensitive to the presence of outliers, for instance consumers with high
level of consumption. As Small (1990) states, ”it suffices to have only one
customer contaminating a data set and going off to infinity to send the mean
curve to infinity as well. By contrast, at least 50% of the data must be moved
to infinity to force the median curve to do the same“. More precisely, the
median is robust to punctually extreme electricity consumptions of some
customers and from a practical point of view, this can help to manage the
electricity supply. Moreover, in the context of the electricity open market,
new customers join the EDF company while other ones leave it and it is
important to know the amount of electricity demand. Since the load profiles
are not known for new customers, it is more difficult to predict their impact
on the global electricity demand. Based on individual customer information,
new customers will belong to a specific class and will be allowed the
synthetic profile that describes the consumption behavior of its class. In
these situations, robust profiles should be used and this is why, we suggest
using the median profile besides the mean curve as a robust indicator for
analyzing the population of electricity load curves.
The median of a sample of univariate observations is a natural and useful
characteristic of central position. Multivariate data, on the other hand, have
no natural ordering. There are several ways to generalize the univariate
median to multivariate data and they all have their advantages and
disadvantages (see Small, 1990 for a survey of multidimentional medians).
First uses of the multivariate median were limited to two-dimentional vectors
and were motivated mainly by problems of quantitative geography (namely, of
centro-graphical analysis) which were dealt with by the U.S.A. Bureau of
census in the late 19th and early 20th century.
We focus here on the $L_{1}$-median, also called the geometric or spatial
median. Early work on the spatial median is due to Hayford (1902) and Gini &
Galvani (1929) among others. Its definition is a direct generalization of the
real median proposed by Haldane (1948) and properties of the spatial median
have been studied in details by Kemperman (1987). Iterative estimation
algorithms have been developed by Gower (1974) and Vardi and Zhang (2000).
In the next few years, the French company EDF intends to install over 30
millions electricity smart meters, in each firm and household. A meter is an
electronic device constructed for measuring the electricity consumption. These
meters will be able to send individual electricity consumption measures on
very fine time scales. The new smart electricity meters will provide accurate
and up-to-date electricity consumption data that can be used to model
distribution network loads. In view of this new setting, the interest
variables such as the consumption curve for example, may be considered as
realizations of functional variables depending on a continuous time index $t$
that belongs to $[0,T]$ rather than multivariate vectors. Kemperman (1987),
Cadre (2001) and Gervini (2008) studied the properties of the median with
functional data. We cite also the very recent work of Cardot et al. (2011).
Another important issue is data storage. The amount of load data will be
enormous when all or almost all of the customers have smart meters.
Collecting, saving and analyzing all this information, would be very
expensive. For example, if measures are taken every 10 minutes during one year
and if we are interested in estimating the total electricity consumption for
residential customers, the data storage is of about 100 terabytes.
We suggest using survey sampling techniques in order to get estimates of the
median profile that are as accurate as possible at reasonable cost. The reader
is referred to Fuller (2009) for a very recent monograph on survey sampling
theory. Nevertheless, the idea of selecting randomly a sample from a
population of curves is rather new. Chiky & Hébrail (2008) compare two
approaches for estimating the mean curve. The first one consists in using
signal compression methods for the whole population of curves and the second
approach suggests taking a simple random sampling of the actual curves. Their
conclusion is that the results are better in the latter situation even with
rather simple sampling designs. Very recently, Cardot et al. (2010) developed
the estimation of functional principal component analysis with survey data and
Cardot and Josserand (2011) studied the properties of the mean curve estimator
with stratified sampling. We cite also Chaouch and Goga (2010) who treated the
estimation of geometric quantiles, the generalization of the spatial median,
with survey data. As far as we know nothing has been done in the estimation of
the $L_{1}$-median in a functional framework with survey data whereas it might
have great practical interest. This is why, we investigate in this paper the
median curve estimator when several sampling designs and estimators are used.
It is worth mentioning that the results presented in this paper may be applied
for other functional data which are not necessarily related to time as it was
the case of the electricity data. Nowadays, functional data may arise in
various other domains such as chemometrics or remote sensing and then the
functional response variables depend on index $t$ that may be a frequency and
not necessary a time index.
The paper is structured as follows: section 2 gives the main results
concerning the median curve estimation with survey data. A weighted estimator
for the median curve is suggested and its asymptotic variance function is
exhibited by means of the linearization technique developed by Deville (1999).
A variance estimator is also proposed. Section 3 gives the estimation of the
median curve and of its variance function for a firm population of $N=18902$
load electricity curves. We consider several sampling designs: the simple
random sampling without replacement, the systematic sampling, the stratified
sampling with optimal and proportional allocation, and the with replacement
proportional-to-size design. In the case of the stratified sampling, we
suggest using the k-means algorithm to construct homogeneous strata with
respect to the linearized variables. We illustrate through simulations that a
substantial reduction compared to simple random sampling is obtained. We adapt
to the functional framework the selection of a sample when auxiliary
information is used at the sampling stage as for the with replacement
proportional-to-size design. Finally, we improve the Horvitz-Thompson
estimator of the functional median by considering the poststratified
estimator.
## 2 Functional Median in a Survey Sampling Setting
Let us consider the finite population $U=\\{1,\dots,N\\}$ of size $N$ and a
functional variable $\mathcal{Y}$ defined for each element $k$ of the
population $U:$ $Y_{k}(t),$ for $t\in[0,T],$ with $T<\infty.$ Let
$<\cdot,\cdot>,$ respectively $||\cdot||,$ be the inner product, respectively
the norm, defined on $L^{2}[0,T].$ The empirical median trajectory calculated
from $Y_{1},\ldots,Y_{N}$ is defined as (Chaudhuri, 1996 and Gervini, 2008)
$\displaystyle m_{N}=\mathop{\mathrm{arg\,min}}_{y\in
L^{2}[0,T]}\sum_{k=1}^{N}||Y_{k}-y||.$ (1)
Supposing that $Y_{k},$ for all $k=1,\ldots,N,$ are not concentrated on a
straight line, the median exists and is unique (Kemperman, 1987) and is the
solution of the following estimating equation,
$\displaystyle\sum_{k=1}^{N}\frac{Y_{k}-y}{||Y_{k}-y||}=0$ (2)
provided that $m_{N}\neq Y_{k}$ for all $k=1,\ldots,N.$
For $Y_{1},\ldots,Y_{N}\in\mathbb{R}^{d},$ the median $m_{N}$ defined by the
formula (1) arises as a natural generalization of the well-known
characterization of the univariate median (Koenker and Basset, 1978),
$q=\arg\min_{\theta}\sum_{k=1}^{N}|Y_{k}-\theta|$
and it was called the spatial median by Brown (1983) or the $L_{1}$-median by
Small (1990). Weber (1909) considered $m_{N}$ as a solution to a problem in a
location theory in which the $Y_{1},\ldots,Y_{N}$ are the planar coordinates
of $N$ customers, who are served by a company that wants to find an optimal
location for its warehouse. It is also known as the Fermat-Weber point. A
geometrical interpretation of the median defined by (2) is that the centroid
of the vectors $\displaystyle\frac{Y_{k}-m_{N}}{||Y_{k}-m_{N}||}$ is the
origin in $\mathbb{R}^{d}.$ With only three points and bidimensional data, the
median $m_{N}$ is known to be the Steiner point of the triangle
$Y_{1}Y_{2}Y_{3}.$ The spatial median has also origins in the early work
during the twelve census in the United Sates in 1900 concerned by finding the
geographical center of the population over time. Hayford (1902) proposed the
point-wise median as the geographical center but explicitly noted the drawback
of the fact that the point-wise median depends on the choice of the orthogonal
coordinates and it is not equivariant under orthogonal transformations. Brown
(1983) goes further with this idea and states that when dealing with spatial
data where variables possess isometry and require statistical techniques that
have rotational invariance, it is more appropriate to use a median that shares
these properties. We recall that with observations $Y_{1},\ldots,Y_{N}$ that
lie in $\mathbb{R}^{d},$ the point-wise median is the $d$-dimensional vector
of medians computed from the univariate components and for functional
variables, the empirical point-wise median is obtained if the $L^{1}$ norm is
used in (1) instead of the $L^{2}$ norm,
$\mbox{med}(t)=\mathop{\mathrm{arg\,min}}_{y(t)\in\mathbb{R}}\sum_{k=1}^{N}|Y_{k}(t)-y(t)|,\quad\mbox{for
all }t\in[0,T].$
To illustrate the mean curve and the point-wise median versus the spatial
median, we plot in Figure 1 the three curves for the test population of
$N=18902$ companies considered in section 3. The electricity consumption is
measured every 30 minutes.
Figure 1: The spatial median profile is plotted in dashed line, the point-wise
median profile in dotted line and the mean profile in solid line.
Chaudhuri (1996) shows that the geometric quantiles defined in formula (3)
from below and which are a generalization of the median defined by (1) are
equivariant under orthogonal transformations unlike the point-wise median.
Moreover, Chaudhuri showed also that the spatial or the $L_{1}$ median is
equivariant under any homogeneous scale transformation of the coordinates of
the multivariate observations which is appropriate when one needs to
standardize the coordinate variables appropriately before computing the
median.
The main arguments that play in favor of the spatial median are the uniqueness
(see e.g. Chaudhuri, 1996 for the $d$-dimensional case with $d\geq 2$ and
Kemperman, 1987 for the functional case) and the fact that it is a global and
central indicator of the distribution of the data. More exactly, the spatial
median takes into account all instants making the spatial median a central
indicator of the distribution of the data while the point-wise median is a
central indicator but only for each instant. Consider for example that we have
consumption electricity data recorded during two weeks: one working week and
one holiday week such as the Christmas week. We compute first the spatial,
respectively the point-wise median, by considering only the working week time
measurements. Next, we consider the two week consumption electricity and we
compute again the spatial, respectively the point-wise median. It can be
noticed that the coordinates of the point-wise median that correspond to the
working week are the same in both situations while they are changed for the
spatial median. Since is due to the fact that the spatial median is computed
by taking into account all the time measurements while the point-wise median
is computed instant by instant.
Moreover, Brown (1983) shows that there is an asymptotic efficiency from using
the spatial median instead of the point-wise median. In fact, one can see that
the objective function is differentiable in the case of the spatial median
while this property is not fulfilled for the point-wise median.
As noted by Serfling (2002), the median defined by (2) and
$Y_{1},\ldots,Y_{N}\in\mathbb{R}^{d},$ has a nice robustness property in the
sense that $m_{N}$ depends only on its direction towards $Y_{i}.$ More
exactly, $m_{N}$ remains unchanged if the $Y_{i}$ are moved outward along
these rays while it is obvious that the point-wise median will change.
Remark: Chaudhuri (1996) extends the definition (1) to geometric quantiles by
using the geometry of data clouds. In a functional setting, its definition
indexes the quantiles by the elements $v\in L^{2}[0,T]$ with $||v||<1,$
$\displaystyle\mathcal{Q}(v)=\mathop{\mathrm{arg\,min}}_{y\in
L^{2}[0,T]}\sum_{k=1}^{N}\left(||Y_{k}-y||+<Y_{k}-y,v>\right).$ (3)
In this way, functional quantiles are characterized by a direction and
magnitude specified by $v\in L^{2}[0,T]$ with $||v||<1.$ Nevertheless, except
the case $v=0,$ it is difficult to interpret the functional quantile defined
in this way. This is why, our discussion is limited to the case $v=0.$
### 2.1 The design-based estimator for the median $m_{N}$
Algorithms have been proposed to solve the equation (2) (Vardi and Zhang,
2000, Gervini, 2008) but they need important computational efforts especially
when the number of time measurements is large. In this work, we suggest
estimating the median curve $m_{N}$ by taking only a sample $s$ from $U$
according to a sampling design. A probability measure $p(\cdot)$ on the set of
subsets of $U,$ henceforth denoted $\mathcal{P}(U)$, is called a sampling
design. Any random variable $S$ with values in $\mathcal{P}(U)$ and
distribution $p$, is called a random sample associated to the sampling design
$p.$ Let $s$ be a realization of $S.$ For any $k\in U$, the inclusion
probability of $k$ is given by
$\pi_{k}=\mathbb{P}(k\in S)=\sum_{k\in s}p(s),$
where the sum is considered over all samples $s$ containing the individual
$k.$ If $k\neq l$ are two elements of $U$, the second-order inclusion
probability of $k$ and $l$ is given by
$\pi_{kl}=\mathbb{P}(k,l\in S)=\sum_{k,l\in s}p(s),$
where the sum is considered over all samples $s$ containing both $k$ and $l.$
In practice, a wide variety of selection schemes are used. We distinguish
direct element sampling designs such as the simple random sampling without
replacement (SRSWOR), stratified sampling (STRAT) or proportional-to-size
sampling designs (with or without replacement). Most of these designs are used
extensively in practice. However, such designs require having a sampling frame
list identifying every population element which may be difficult, expensive or
even impossible to realize. In order to avoid it, more complex designs such as
cluster or multi-stage designs can then be used. This is for example
appropriate when the population is widely distributed geographically or may
occur in natural clusters. Using such designs saves money and human efforts
but entails a loss of efficiency. A detailed presentation of the survey
sampling theory and many practical illustrations can be found in Korn and
Graubard (1999), Lehtonen and Pahkinen (2004) and the reference book of
Särndal, Swensson & Wretman (1992).
The median $m_{N}$ given by (2) is a nonlinear parameter of finite population
totals defined by an implicit equation. In order to estimate $m_{N},$ we use
the functional substitution approach proposed by Deville (1999) for
multivariate variables $\mathcal{Y}$ and extended to functional variables
$\mathcal{Y}$ by Cardot et al. (2010). Let $M$ be the discrete measure defined
on $L^{2}[0,T]$ assigning the unity mass to each curve $Y_{k}$ with $k\in U$
and zero elsewhere, namely
$M=\sum_{k\in U}\delta_{Y_{k}},$
where $\delta_{Y_{k}}$ is the Dirac function in $Y_{k}.$ The total mass of $M$
is $N,$ the population size. Let $T$ be the functional with respect to $M$ and
depending of $y$ as follows
$\displaystyle
T(M;y)=-\int\frac{\mathcal{Y}-y}{||\mathcal{Y}-y||}dM=-\sum_{k\in
U}\frac{Y_{k}-y}{||Y_{k}-y||}.$ (4)
Remark that $T$ defined in this way is the derivative with respect to $y$ of
the objective function given in (1). The median $m_{N}$ is then defined as an
implicit functional with respect to $M,$
$\displaystyle T(M;m_{N})=0$ (5)
or equivalently,
$\displaystyle\int\frac{\mathcal{Y}-m_{N}}{||\mathcal{Y}-m_{N}||}dM=0.$ (6)
Let $\widehat{M}$ be a weighted estimator of $M$ based on the sample $s,$
$\displaystyle\widehat{M}=\sum_{k\in s}w_{k}\delta_{Y_{k}}=\sum_{k\in
U}I_{k}w_{k}\delta_{Y_{k}},$ (7)
where $I_{k}=\mathbf{1}_{\\{k\in s\\}}$ is the sample membership indicator of
element $k\in U$ (Särndal et al., 1992). In fact, $\widehat{M}$ is also a
discrete and finite measure assigning the weight $w_{k}$ for each $Y_{k}$ with
$k\in s$ and zero elsewhere. Usually, one take $w_{k}=1/\pi_{k},$ the Horvitz-
Thompson weights. In this case, we obtain the Horvitz-Thompson (1952) of $M$
which estimates unbiasedly the measure $M$ since $E_{p}(I_{k})=\pi_{k},$ for
all $k\in U$ for $E_{p}(\cdot)$ the expectation with respect to the sampling
design $p(\cdot).$ The reader is referred to Cardot et al. (2010) and Cardot
and Josserand (2011) for more details about the Horvitz-Thompson estimation
with functional data. However, for $Y_{1},\ldots,Y_{N}$ lying in
$\mathbb{R}^{d},$ weights that take into account auxiliary information have
been suggested. We mention Deville (1999) for calibration weights or the very
recent work of Goga and Ruiz-Gazen (2011) for nonparametric weights.
Nevertheless, the extension to the functional case is not straightforward and
it will be treated elsewhere. For the rest of the paper, we consider
$w_{k}=1/\pi_{k}$ and in section 3.1, we suggest the poststratified estimator
of $M.$
Plugging $\widehat{M}$ into the functional expression of $m_{N}$ given by (5),
yields the design-based estimator $\widehat{m}_{n}$ of $m_{N}.$ Hence,
$\widehat{m}_{n}$ verifies
$T(\widehat{M};\widehat{m}_{n})=0,$
namely, $\widehat{m}_{n}$ is the solution of the design-based estimating
equation,
$\displaystyle\sum_{k\in
s}\frac{1}{\pi_{k}}\frac{Y_{k}-\widehat{m}_{n}}{||Y_{k}-\widehat{m}_{n}||}=0.$
(8)
Supposing now that for all $k\in s,$ $Y_{k}\neq\widehat{m}_{n}$ and that
$Y_{k}$ are not concentrated on a straight line, we obtain that the solution
$\widehat{m}_{n}$ exists and is unique following the same arguments as in
Kemperman (1987) or Chaudhuri (1996). The median estimator $\widehat{m}_{n}$
is also called the substitution estimator of $m_{N}$ and it is defined by a
non-linear implicit function of Horvitz-Thompson estimators. As a consequence,
the variance as well as the variance estimator of $\widehat{m}_{n}$ can not be
obtained directly using Horvitz-Thomson formulas. We will give in the next a
first-order expansion of $\widehat{m}_{n}$ in order to approximate
$\widehat{m}_{n}$ by the Horvitz-Thompson estimator for the finite population
total of appropriate artificial variables.
### 2.2 Asymptotic properties
The functional $T$ given by (4) is Fréchet differentiable (Serfling, 1980)
with respect to the measure $M$ and $y.$ Let $\Gamma$ be the Jacobian operator
of $T$ with respect to $y$ and given by (Gervini, 2008)
$\displaystyle\Gamma=\sum_{k\in
U}\frac{1}{||Y_{k}-m_{N}||}\left[\mathbf{I}-\frac{(Y_{k}-m_{N})\otimes(Y_{k}-m_{N})}{||Y_{k}-m_{N}||^{2}}\right],$
(9)
where $\mathbf{I}$ is the identity operator defined by $\mathbf{I}y=y$ and the
tensor product of two elements $a$ and $b$ of $L^{2}[0,T]$ is the rank one
operator such that $a\otimes b(y)=<a,y>b$ for all $y\in L^{2}[0,T].$ One can
easily obtain that $\Gamma$ is a strictly positive operator, namely $<\Gamma
y,y>>0$ and supposing that $N^{-1}\sum_{k\in U}||Y_{k}-m_{N}||^{-1}<\infty,$
we can get following the same arguments as in Cardot et al. (2011), that
$\Gamma/N$ is a bounded operator, namely $||\Gamma/N||_{\infty}<\infty$ with
$||\Gamma||_{\infty}=\mbox{sup}_{||y||\leq 1}||\Gamma y||.$ We recall that for
the operator $\Gamma:L^{2}[0,T]\longrightarrow L^{2}[0,T],$ we have
$\Gamma y=\sum_{k\in
U}\frac{1}{||Y_{k}-m_{N}||}\left(y-\frac{<Y_{k}-m_{N},y>}{||Y_{k}-m_{N}||^{2}}(Y_{k}-m_{N})\right)\quad\mbox{for
all}\quad y\in L^{2}[0,T]$
which gives
$\displaystyle\Gamma y(t)=\sum_{k\in
U}\frac{y(t)}{||Y_{k}-m_{N}||}-\int_{0}^{T}\gamma(r,t)y(r)dr$ (10)
where
$\gamma(r,t)=\sum_{k\in
U}\frac{(Y_{k}(r)-m_{N}(r))(Y_{k}(t)-m_{N}(t))}{||Y_{k}-m_{N}||^{3}}.$
The median is defined by the implicit equation (6) and using then the implicit
function theorem, we obtain that it exists a unique functional $\widetilde{T}$
such that
$\widetilde{T}(M)=m_{N}$
and
$\widetilde{T}(\widehat{M})=\widehat{m}_{n}.$
Moreover, the functional $\widetilde{T}$ is also Fréchet differentiable with
respect to $M$ and the derivative of $\widetilde{T}$ with respect to $M$ is
called the influence function and defined, when it exists, as follows
$\displaystyle I\widetilde{T}(M,\xi)=\mbox{lim}_{\lambda\rightarrow
0}\frac{\widetilde{T}(M+\lambda\delta_{\xi})-\widetilde{T}(M)}{\lambda}$
where $\delta_{\xi}$ is the Dirac function at $\xi\in L^{2}[0,T].$ Note that
this definition suggested by Deville (1999) and extended to the functional
case by Cardot et al. (2010) is slightly different from the usual definition
of the influence function used in robust statistics (see e.g. Hampel, 1974 or
Serfling, 1980), which is based on a probability distribution instead of a
finite measure $M.$ A nonstandardized measure $M$ is used in survey sampling
because the total mass $N$ may be unknown.
Under the asymptotic framework from Deville (1999), we may give a first-order
von-Mises (1947) expansion of $\tilde{T}$ in $\widehat{M}/N$ around $M/N,$
$\displaystyle\widetilde{T}\left(\frac{\widehat{M}}{N}\right)=\widetilde{T}\left(\frac{M}{N}\right)+\int
I\widetilde{T}\left(\frac{M}{N},\xi\right)d\left(\frac{\widehat{M}}{N}-\frac{M}{N}\right)(\xi)+o_{p}(n^{-1/2})$
(11)
which may be written in the equivalent form,
$\displaystyle\widetilde{T}(\widehat{M})=\widetilde{T}(M)+\int
I\widetilde{T}\left(M,\xi\right)d(\widehat{M}-M)(\xi)+o_{p}(n^{-1/2})$ (12)
since $\widetilde{T}$ is a functional of degree zero, namely
$\widetilde{T}(M/N)=\widetilde{T}(M)$ and in this case,
$I\widetilde{T}\left(\frac{M}{N},\xi\right)=N\cdot
I\widetilde{T}\left(M,\xi\right)$ (Deville, 1999).
Let $u_{k},$ for all $k\in U,$ be the linearized variables of
$\widetilde{T}(M)=m_{N}$ and defined as the value of the influence function
$I\widetilde{T}$ at $\xi=Y_{k},$ namely
$\displaystyle u_{k}$ $\displaystyle=$ $\displaystyle
I\widetilde{T}(M,Y_{k})=\Gamma^{-1}\left(\frac{Y_{k}-m_{N}}{||Y_{k}-m_{N}||}\right).$
(13)
We have used here the fact that for fixed $y,$ the functional
$T(M;y)=-\displaystyle\sum_{U}\frac{Y_{k}-y}{||Y_{k}-y||}$ is a finite
population total with influence function at $Y_{k}$ given by
$IT(M,Y_{k};y)=-(Y_{k}-y)/||Y_{k}-y||$ (Deville, 1999). From the Riesz’s
theorem, we have that for all bounded $h\in L^{2}[0,T]$ there is a unique
$f\in L^{2}[0,T]$ such that $\Gamma f=h$ and $\Gamma f(g)=<h,g>$ for all $g\in
L^{2}[0,T].$ This unique $f$ will denote $\Gamma^{-1}h$ for a given $h\in
L^{2}[0,T].$
Hence, the expansion (12) becomes
$\displaystyle\widehat{m}_{n}$ $\displaystyle=$ $\displaystyle
m_{N}+\sum_{k\in s}\frac{u_{k}}{\pi_{k}}-\sum_{k\in U}u_{k}+o_{p}(n^{-1/2}).$
(14)
The above formula shows that the nonlinear estimator $\widehat{m}_{n}$ may be
approximated by the Horvitz-Thompson estimator for the total of the linearized
variables $u_{k}.$ In this way, $u_{k}$ is an artificial variable used to
compute the approximative variance of $\widehat{m}_{n}.$ Now, the linearized
variable $u_{k}$ is also a functional defined on $L^{2}[0,T]$ and it is
unknown since $m_{N}$ and $\Gamma$ are unknown. We suggest estimating $u_{k}$
by
$\displaystyle\hat{u}_{k}=\widehat{\Gamma}^{-1}\left(\frac{Y_{k}-\widehat{m}_{n}}{||Y_{k}-\widehat{m}_{n}||}\right),$
(15)
where $\widehat{\Gamma}$ is given by
$\displaystyle\widehat{\Gamma}=\sum_{k\in
s}\frac{1}{\pi_{k}||Y_{k}-\widehat{m}_{n}||}\left[\mathbf{I}-\frac{(Y_{k}-\widehat{m}_{n})\otimes(Y_{k}-\widehat{m}_{n})}{||Y_{k}-\widehat{m}_{n}||^{2}}\right].$
(16)
Using relation (14), one can obtain the asymptotic variance function of
$\widehat{m}_{n}$ calculated under the sampling design,
$\displaystyle var(t)=\sum_{k\in U}\sum_{k\in
U}(\pi_{kl}-\pi_{k}\pi_{l})\cdot\frac{u_{k}(t)}{\pi_{k}}\cdot\frac{u_{l}(t)}{\pi_{l}}=\mathbf{u}^{\prime}(t)\mathbf{\Delta}\mathbf{u}(t)\quad\mbox{for
all}\quad t\in[0,T]$ (17)
where $\mathbf{u}(t)=(u_{k}(t))_{k\in U}$ with $u_{k}(t)$ is given by (13) and
$\displaystyle\mathbf{\Delta}=\left(\frac{\pi_{kl}-\pi_{k}\pi_{l}}{\pi_{k}\pi_{l}}\right)_{k,l\in
U}$. The variance is estimated by
$\displaystyle\widehat{var}(t)=\sum_{k\in s}\sum_{k\in
s}\frac{\pi_{kl}-\pi_{k}\pi_{l}}{\pi_{kl}}\cdot\frac{\hat{u}_{k}(t)}{\pi_{k}}\cdot\frac{\hat{u}_{l}(t)}{\pi_{l}}=\mathbf{\widehat{u}}^{\prime}_{s}(t)\widehat{\mathbf{\Delta}}\mathbf{\widehat{u}}_{s}(t),$
(18)
where $\mathbf{\widehat{u}}_{s}(t)=(\hat{u}_{k}(t))_{k\in s}$ with
$\hat{u}_{k}$ given by (15) and
$\displaystyle\widehat{\mathbf{\Delta}}=\left(\frac{\pi_{kl}-\pi_{k}\pi_{l}}{\pi_{k}\pi_{l}\pi_{kl}}\right)_{k,l\in
s}$.
Remark 1: It is worth mentioning that the linearized variable $u_{k}$ plays a
central role for the estimation of the median. More exactly, the efficiency of
any sampling design used for estimating the median curve depends on how well
it estimates the total of the linearized variable $u_{k}.$ For example, a
stratified strategy will be efficient if the strata are homogeneous with
respect to $u_{k}$ as it will be showed below. Nevertheless, to put in
practice such a design requires knowing all $u_{k}$ which may not be readily
available.
Remark 2: In practice, we observe the curves $Y_{k}$ at $D$ discretized
points, say $0\leq t_{1}<t_{2}<\ldots<t_{D}\leq T$ that we suppose to be the
same for all the curves. When the discretization points vary to one curve to
another, methods described in Ramsay and Silverman (2005) may be employed. In
order to compute numerical approximations to integrals and inner products,
quadrature rules are used.
With discretized points, curves may be viewed as multidimensional vectors, in
our case, ${\bf Y}^{\prime}_{k}=\left(Y_{k}(t_{1}),\dots,Y_{k}(t_{D})\right)$
and
$\mathbf{\widehat{u}}^{\prime}_{k}=(\hat{u}_{k}(t_{1}),\ldots,\hat{u}_{k}(t_{D})).$
For each $k\in s,$ we need to compute the estimated linearized variable in
points $t_{1},\ldots,t_{D}.$ Let
$\mathbf{\widehat{u}}_{s}=(\mathbf{\widehat{u}}^{\prime}_{k})_{k\in s}$ be the
sample vector of estimated linearized variables which can be derived by
solving the $D\times n$ dimensional system
$\widehat{\Gamma}\mathbf{\widehat{u}}^{\prime}_{s}=\left(\frac{{\bf
Y}_{1}-\widehat{m}_{n}}{||{\bf Y}_{1}-\widehat{m}_{n}||},\ldots,\frac{{\bf
Y}_{N}-\widehat{m}_{n}}{||{\bf Y}_{N}-\widehat{m}_{n}||}\right),$
where $\widehat{\Gamma}$ given by (16) is replaced by a $D\times D$ symmetric
matrix. The variance estimator is then derived directly using (18).
## 3 Application to the EDF load curves
### 3.1 General settings
The volume of data treated and analyzed by Électricité De France is increasing
greatly. In fact, in the next few years Electricité De France plans to install
millions of smart electricity meters that will be able to send, on request,
electricity consumption measurements every second. Obviously, it will be
difficult to store and analyse online all these information. The
statistician’s challenge is to find a strategy, meaning indicators and
estimation methods, capable to give a good description of data and to used it
for forecasting. While working with huge data, methods not being time-
consuming are highly desirable.
Our proposal consist in considering the median curve as a robust indicator of
the data and estimating it with probability sampling designs. As Lohr stated
in ”Sampling: Design and Analysis” (1999): If a probability sampling design is
implemented well, an investigator can use a relatively small sample to make
inferences about an arbitrarily large population.
Let $U$ be a population of $N=18902$ electricity meters installed in small and
large companies sending every 30 minutes the electricity consumption during a
period of two weeks. We aim at estimating the median curve of the electricity
consumption during the second week whereas the consumption recorded during the
first week will be used as auxiliary information. This means that we have
$336$ time measures. So, our study population of curves is a set of $N=18902$
vectors ${\bf Y}^{\prime}_{k}=\left(Y_{k}(t_{1}),\dots,Y_{k}(t_{D})\right)$
with $D=336.$ Let $X_{k}$ be the consumption curve for the $k$th firm and
recorded during the first week. The consumption curves present low peaks
corresponding to night time measurements and high peaks corresponding to
middle day measurements. The electricity consumption decreases roughly around
the 250th time measurement which corresponds to the beginning of weekend time.
The mean and median curves present the same effect as we can see in Figure 1.
We consider several strategies of fixed size $n=2000$ and we compare them
through simulations. We distinguish two kinds of sampling designs whether they
use or do not use auxiliary information. If auxiliary information is used at
the sampling stage, some changing are needed because the variables involved
now are curves. On the opposite situation, the selection of the sample is
realized from the sampling frame list as for classical multivariate surveys.
Finally, the frame list of French firms is well-constructed being very often
updated and most of the designs considered below are usually used in practice.
1. 1.
Simple random sampling without replacement (SRSWOR).
The SRSWOR sampling is a very simple design easy to put into practice. Every
possible subset of $n$ units in the population has the same chance to be the
sample. In a functional framework, the selection of a sample of $n=2000$
curves is performed as for the multivariate surveys, namely $n$ labels are
drawn from the list of $N$ companies. The estimation of the median curve with
SRSWOR is obtained from equation (8) for $\pi_{k}=n/N,$ namely
$\widehat{m}_{n}$ is the unique solution of the following equation
$\displaystyle\sum_{k\in
s}\frac{Y_{k}-\widehat{m}_{n}}{||Y_{k}-\widehat{m}_{n}||}=0.$ (19)
The asymptotic variance function is equal to
$var_{SRSWOR}(t)=\displaystyle
N^{2}\left(\frac{1}{n}-\frac{1}{N}\right)S^{2}_{u(t),U},\quad\mbox{for all t}$
where $S^{2}_{u(t),U}=\sum_{k\in U}(u_{k}(t)-\overline{u}(t))^{2}/(N-1)$ with
$\overline{u}(t)=\sum_{k\in U}u_{k}(t)/N.$ This variance is estimated by
$\displaystyle\widehat{var}_{SRSWOR}(t)=\displaystyle
N^{2}\left(\frac{1}{n}-\frac{1}{N}\right)S^{2}_{\hat{u}(t),s},\quad\mbox{for
all t}$ (20)
where $S^{2}_{\hat{u}(t),s}=\sum_{k\in
s}(\hat{u}_{k}(t)-\overline{\hat{u}}_{s}(t))^{2}/(n-1)$ with
$\overline{\hat{u}}_{s}(t)=\sum_{k\in s}\hat{u}_{k}(t)/n$ and $\hat{u}_{k}(t)$
given by (15).
2. 2.
Systematic sampling (SYS).
We consider the systematic design in its basic form (Särndal et al., 1992).
The inclusion probabilities are $\pi_{k}=n/N,$ so the median estimator is
obtained according to the same equation (19). It is well-known that the
systematic sampling may be very inefficient compared to the SRSWOR sampling if
the systematic samples are homogeneous. One way to improve the efficiency of
SYS sampling is to order the sampling frame list according to an auxiliary
variable highly correlated with the variable of interest. In this way,
adjacent elements tend to be more similar than elements that are farther
apart. In our study, we ordered the frame according to the mean electricity
consumption during the first week, namely the variable
$\tilde{X}_{k}=\sum_{d=1}^{D}X_{k}(t_{d})/D,$ for all $k\in U$ and $D$ is the
number of discretization points in $[0,T].$ Another trade-off for the
simplicity of SYS sampling is that there is no unbiased estimator of the
design variance function since $\pi_{kl}=0$ for all $k$ and $l$ not belonging
to the same systematic sample. However, using the ordering according to the
variable $\tilde{X}_{k}$, the SYS is at least as good as the SRSWOR sampling.
So, we might use the variance estimator appropriate for the SRSWOR design
given in (20).
Systematic sample is really a special case of cluster sampling, so it is often
used when it is difficult to construct a sampling frame in advance.
3. 3.
Stratified sampling with simple random sampling without replacement within
strata (STRAT).
In this case, the population is divided into $H$ nonoverlapping strata denoted
$U_{h}$ and a simple random sample without replacement is selected
independently in each stratum. Let $n_{h}$ be the sample size within stratum
$h$ and $N_{h}$ be the stratum size. To obtain the median estimator, we solve
the estimation equation
$\sum_{h=1}^{H}\sum_{s_{h}}\frac{Y_{k}-\widehat{m}_{n}}{\pi_{k}^{h}||Y_{k}-\widehat{m}_{n}||}=0,$
where $s_{h}=s\cap U_{h}$ and $\pi_{k}^{h}=n_{h}/N_{h}.$ It is well-known that
stratification may substantially improve the quality of estimates compared to
simple random sampling without replacement and systematic sampling if the
strata are well-constructed. More exactly, the more homogeneous the strata are
the more efficient the stratification is. It is worth mentioning that
improving the estimation of the median curve means constructing strata
homogeneous with respect to the linearized variables, $u_{k}.$ Indeed,
relation (17) gives us that the asymptotic variance function of
$\widehat{m}_{n}$ with STRAT is
$var_{STRAT}(t)=\sum_{h=1}^{H}N_{h}^{2}\left(\frac{1}{n_{h}}-\frac{1}{N_{h}}\right)S^{2}_{u(t),U_{h}},$
where $S^{2}_{u(t),U_{h}}$ is the population variance within stratum $h$ of
$u(t)=(u_{k}(t))_{k\in U_{h}}.$ That is, the lower the variation of the
linearized variable within stratum, the lower the asymptotic variance of
$\widehat{m}_{n}.$ The variance estimator is the sum of variance estimators
(20) computed within each stratum.
Usually, one builds the stratification using a variable known on the whole
population and strongly correlated with the variable of interest. In our case,
we suggest two stratification variables computed using the first week: the
first one is the linearized variable $u_{k}$ and the second one is the
consumption $Y_{k}$ (Cardot and Josserand, 2011). The following two sample
allocations are used:
* $\bullet$
the proportional allocation (PROP): $n_{h}=nN_{h}/N$ for all $h=1,\ldots H.$
* $\bullet$
the $u^{(1)}$-optimal allocation ($u^{1}$-OPTIM) as suggested by Cardot and
Josserand (2011) and computed here with respect to the variance
$S^{2}_{u^{(1)}(t),U_{h}}$ of the linearized variable computed during the
first week and denoted by $u_{k}^{(1)},$
$n_{h}=n\frac{N_{h}\sqrt{\int_{0}^{T}S^{2}_{u^{(1)}(t),U_{h}}dt}}{\sum_{h=1}^{H}N_{h}\sqrt{\int_{0}^{T}S^{2}_{u^{(1)}(t),U_{h}}dt}}\quad
h=1,\dots,H.$
The $u^{(1)}$-optimal allocation is similar to the Neyman optimal allocation
but computed using $u^{(1)}$ instead of $u.$ The $x$-optimal allocation is
obtained when the consumption during the first week $X_{k}$ is used.
Stratification based on the linearized variable during the first week
The proposed strategy can be split into two steps:
Step 1: we calculate the linearized variables $u^{(1)}_{k}$ for all $k\in U$
during the first week.
Step 2: we stratify the population $U$ using the k-means clustering algorithm
with the euclidean distance and applied to the linearized variables
$u_{k}^{(1)}$ for $k\in U.$ According to within cluster variance
considerations, we decide to keep $H=4$ different clusters. The strata sizes
as well as both the proportional and $u^{(1)}$-optimal allocation are given in
Table1.
Stratum number | 1 | 2 | 3 | 4
---|---|---|---|---
Stratum size $N_{h}$ | 6767 | 2420 | 2503 | 7212
PROP allocation | 716 | 256 | 265 | 763
$u^{(1)}$-OPTIM allocation | 525 | 395 | 428 | 652
Table 1: Strata sizes, proportional and $u^{(1)}$-optimal allocations when
$n=2000.$
We plot in Figure 2 (a), the mean of $u_{k}$ computed during the second week
and within the $H=4$ strata. Differences among the strata means are noticeable
accounting for a significant gain in efficiency if the proportional allocation
is used. Now, to better see what kind of consumers the four strata are built
of, we plot in Figure 2 (b) the mean of the consumption $Y_{k}.$ We remark
that the stratification based on $u_{k}^{(1)}$ induces a stratification for
the consumption curves also.
|
---|---
(a) | (b)
Figure 2: Stratification based on the linearized variable: (a) Mean of
linearized variables $u_{k}$ within each stratum. (b) Mean of the consumption
curve $Y_{k}$ within each stratum
Stratification based on the consumption curve during the first week
Cardot and Josserand (2011) suggested taking $H=4$ strata corresponding to the
maximum level of consumption during the first week $X_{k}$ and based on
quartiles so that all strata have the same size. We denote the allocation
obtained in this way by $x$-OPTIM. The strata sizes as well as both the
proportional and the $x$-optimal allocation are given in Table 2.
Stratum number | 1 | 2 | 3 | 4
---|---|---|---|---
Stratum size $N_{h}$ | 4725 | 4726 | 4725 | 4726
PROP allocation | 500 | 500 | 500 | 500
$x$-OPTIM allocation | 126 | 212 | 333 | 1329
Table 2: Strata sizes, proportional and $x$-optimal allocations for $n=2000.$
We plot in Figure 3 (b), the consumption mean within strata and during the
second week. We notice that the stratum 4 corresponds to consumers with high
global levels of consumption, whereas stratum 1, corresponds to consumers with
low global of consumption. Figure 3 (a) gives the mean curves of the
linearized variable within strata and computed for the second week. As for the
first stratification, the population of the linearized variable curves is also
stratified.
|
---|---
(a) | (b)
Figure 3: Stratification based on the consumption curve: (a) Mean of
linearized variables $u_{k}$ within each stratum. (b) Mean of the consumption
curve $Y_{k}$ within each stratum
4. 4.
Proportional-to-size sampling (PPS)
Unequal probability designs are widely used in practice because they are
usually more efficient than the equal probability designs. In PPS sampling,
the sampling is with-replacement and the probability $p_{k}$ with which the
individual $k$ is selected is proportional to a positive measure $X_{k},$
where $X_{k}$ is an auxiliary variable roughly proportional to the study
variable $Y_{k}.$ The probability of selection has the expression
$p_{k}=\frac{X_{k}}{\sum_{k\in U}X_{k}}.$
In our situation, the study variable is a curve and so is the auxiliary
information. To cope with this problem, we suggest using $p_{k}$ proportional
to the mean of $X_{k}(t)$ over all $t=1,\ldots,D$ where $D$ is the number of
discretization points in the interval $[0,T].$ This means that
$p_{k}=\frac{\tilde{X}_{k}}{\sum_{k\in U}\tilde{X}_{k}},$
where $\tilde{X}_{k}=\sum_{t=1}^{D}X_{k}(t)/D.$ For our study, we consider
again $X_{k}$ as being the electricity consumption for the $k$th firm recorded
during the first week. The inclusion probabilities are given by
$\pi_{k}=1-(1-p_{k})^{n}.$ The Horvitz-Thompson estimator of the median is
obtained by solving the equation
$\displaystyle\sum_{k\in\tilde{s}}\frac{Y_{k}-\widehat{m}_{n}}{\pi_{k}||Y_{k}-\widehat{m}_{n}||}=0,$
(21)
where $\tilde{s}$ is the set of distinct elements of $s.$ In with-replacement
designs, one may use the Hansen and Hurwitz (1943) estimator which presents
the advantage that the variance formula is easier (no double sums are needed).
5. 5.
Poststratification (POST)
Let consider now the poststratification which is one of the simplest way to
take into account auxiliary information in order to improve the Horvitz-
Thompson estimator of the median. We suppose that the population is
partitioned into subpopulations $U_{1},\ldots,U_{G}$ according to a given
classification principle. These subpopulations are called poststrata since
they do not serve for performing stratified sampling as described before.
Practical considerations may favor some other (perhaps simpler or less costly)
designs, such as SRSWOR from the whole population $U.$ After the sample
selection, $Y_{k}$ is observed for the elements $k\in s$ and the sampling
frame is used to establish the group each individual belongs to. Remark that
group memberships may be unknown before the sample selection which makes
impossible to perform the stratified sampling. Nevertheless, the group
membership totals $N_{g}$ are known for all $g=1,\ldots,G$ and this auxiliary
information may be used to construct an improved estimator of $m_{N}.$ The
weights used in this case are given by
$w_{ks}=N_{g}/(\hat{N}_{g}\pi_{k})\quad\mbox{for all}\quad k\in s_{g}=s\cap
U_{g}$
where $\hat{N}_{g}=\sum_{k\in s_{g}}1/\pi_{k}.$ Hence, the poststratified
estimator of $m_{N}$ is obtained by solving the following equation
$\displaystyle\sum_{g=1}^{G}\sum_{k\in
s_{g}}\frac{N_{g}}{\hat{N}_{g}\pi_{k}}\frac{Y_{k}-\widehat{m}_{n}}{||Y_{k}-\widehat{m}_{n}||}=0.$
(22)
It is important to notice that the weights used here depend on the sample and
are more general than the Horvitz-Thompson weights used before in the sense
that they include the auxiliary information given by the group size $N_{g}.$
In this case, the auxiliary information is used at the estimation stage and
not at the design stage.
With SRSWOR sampling from $U,$ the poststratified weights become
$w_{k}=N_{g}/n_{g}$ for all $k\in s_{g}$ and $n_{g}$ the size of $s_{g}.$ The
median estimator verifies
$\displaystyle\sum_{g=1}^{G}\sum_{k\in
s_{g}}\frac{N_{g}}{n_{g}}\frac{Y_{k}-\widehat{m}_{n}}{||Y_{k}-\widehat{m}_{n}||}=0.$
(23)
One should remark that in the case of the poststratification, the samples
sizes $n_{g}$ are random. The poststratified estimator of the median given by
(23) cannot be computed if some sample sizes $n_{g}$ are equal to zero.
However, if the total sample size $n$ is large enough, and if no group
accounts for a very small proportion of the whole population, then the
probability of having $n_{g}=0$ is very small. Särndal et al. (1992) suggest
aggregating small groups in order to guarantee that $n_{g}$ are at least 20.
The poststratified estimator is a calibrated estimator (Deville and Särndal,
1992) when the auxiliary information is the group memberships with known
totals. Using the same arguments as in Deville (1999), the expansion given in
(14) remains valid and as a consequence, the asymptotic variance of
$\widehat{m}_{n}$ is equal to the variance of residuals
$E_{k}=u_{k}-\overline{u}_{g},$ $\overline{u}_{g}=\sum_{k\in
U_{g}}u_{k}/N_{g}$ and with SRSWOR, we obtain
$var(t)=N^{2}\left(\frac{1}{n}-\frac{1}{N}\right)\sum_{g=1}^{G}\frac{N_{g}-1}{N-1}S_{u(t),U_{g}}^{2},$
where $S_{u(t),U_{g}}^{2}$ is the group variance. We can remark that the
variance agrees very nearly with the asymptotic variance for the stratified
sampling with proportional allocation.
We can conclude that the SRSWOR with poststratification is essentially as
efficient as STRAT sampling with proportional allocation unless the sample is
very small. This result is well-known for multivariate variables $Y_{k}$ (see
e.g., Särndal et al. 1992). We have developed above a strategy based on the
linearized variables to obtain well-constructed strata. One drawback with that
method is that the so-constructed strata reduce the variance for the median
estimator but could be inefficient for many other variables. Therefore, using
SRSWOR with poststratification will often improve overall efficiency.
### Consistency of $\widehat{m}_{n}(t)$ from survey data
Focus is now on the estimation of the median curve of the consumption recorded
during the second week. We consider for that the following sampling designs of
size $n=2000$ with the Horvitz-Thompson estimator: SRSWOR, STRAT based on the
two stratification variables and with proportional and optimal allocation,
SYS, PPS. We consider for our study the POST estimator also. Figure (4) shows
the estimation of the median curve computed from one sample selected according
to four sampling designs.
Figure 4: One sample estimation of the median trajectory for $n=2000$ with 4
sampling strategies (SRSWOR, SYS, STRAT.OPT, STRAT.PROP).
In order to compare these designs, we made 500 replications and considered the
following loss criteria,
$\displaystyle
R(\widehat{m}_{n})=\int_{0}^{T}|\widehat{m}_{n}(t)-m_{N}(t)|dt.$
Since in our study we have equally spaced discretized time measurements, the
above loss criterion is approximated due to quadrature rules by
$\frac{1}{D}\sum_{d=1}^{D}|\widehat{m}_{n}(t_{d})-m_{N}(t_{d})|.$ Basic
statistics, respectively boxplots, for the estimation errors of the median
function estimator are given in Table 3, respectively Figure 5. The
stratification variable used here is the one based on the linearized variables
$u_{k}^{(1)}.$ First, we can observe that clustering the space of functions by
performing stratified sampling leads to an important gain in terms of accuracy
of the estimators, dividing by at least two times the mean error compared to
simple random sampling without replacement. We note that the
poststratification gives results similar to those given by stratified sampling
with proportional allocation and that the SYS design with ordering on the
first week mean consumption is almost as good as the SRSWOR design. A rather
surprising result is obtained with PPS sampling. Simulations results not
reported here show that this design performs very well for estimating the mean
consumption curve, being as good as the stratified sampling but fails for the
median curve. We believe that this fact is due to numerical problems
encountered in the resolution of the implicit equation (21) when a large
number of small probabilities of selection $p_{k}$ are used to estimate
$m_{N}.$ More research is needed to better clarify this issue and to find a
way to improve it.
| Mean | $1^{st}$ quartile | median | $3^{rd}$ quartile
---|---|---|---|---
SRSWOR | 2.531 | 1.322 | 1.982 | 3.351
SYS | 2.625 | 1.355 | 2.412 | 3.087
PROP | 1.060 | 0.8498 | 1.017 | 1.234
$u^{(1)}$-OPTIM | 1.000 | 0.7946 | 0.9552 | 1.142
POST | 1.041 | 0.8275 | 0.9785 | 1.203
PPS | 7.1410 | 2.7880 | 6.1370 | 9.5600
Table 3: Estimation errors for $m_{N}$. Figure 5: Comparison of the
distribution of estimation errors of the median curve for SRSWOR, PROP, OPTIM,
SYS.
We also performed simulations for the second stratification based on the first
week consumption $X_{k}$ and the results are given in Table 4.
| Mean | $1^{st}$ quartile | median | $3^{rd}$ quartile
---|---|---|---|---
PROP | 1.7370 | 1.0470 | 1.4860 | 2.2480
$x$-OPTIM | 2.2940 | 1.4660 | 1.9790 | 2.7830
Table 4: Estimation errors for $m_{N}$ with stratification based on $Y_{k}$.
We can remark that STRAT with proportional allocation and stratification based
on $u^{(1)}$ gives better results than STRAT with $x$-optimal allocation
stratification based on $X.$ This result is not surprising since in the latter
case, the strata have been constructed taken into account the consumption
variable and the optimal allocation has been computed by minimizing the
variance for the mean estimator while we are interested here in estimating the
median curve. This is why, the proportional allocation is usually advisable
with multipurpose surveys. Moreover, if we compare the two stratifications, we
remark that the stratification based on the consumption variable is less
efficient than the stratification based on the linearized variable but it
remains still better than the SRSWOR or SYS designs.
Both stratifications used in this paper need the consumption curve $Y_{k}$
computed during the first week for all the individuals from the population.
Sometimes, this can be too costly to obtain or even impossible because of
storage or confidentiality constraints. In such situations, some other
stratification variables may be considered such as for example, the
electricity power given by the subscribed contract between one firm and EDF.
### Consistency of the variance function estimation from survey data
We analyze in the following the estimator for the variance function $var(t)$
when the SRSWOR and STRAT designs are used. To judge the quality of the
estimators, we use the following criterion
$\displaystyle R(\widehat{var})=\int_{0}^{T}|\widehat{var}(t)-var(t)|\;dt.$
We give in Table 5 statistics for the estimation errors of the variance
function estimation with SRSWOR and stratified sampling with proportional and
$u^{(1)}$-optimal allocations. Figure 6 gives the theoretical standard
deviation function curves of $\widehat{m}_{n}$, $\sqrt{var(t)}$, with the
considered designs.
| Mean | $1^{st}$ quartile | median | $3^{rd}$ quartile
---|---|---|---|---
SRSWOR | 0.599 | 0.339 | 0.506 | 0.750
PROP | 0.068 | 0.055 | 0.064 | 0.076
$u^{(1)}$-OPTIM | 0.056 | 0.047 | 0.053 | 0.062
Table 5: Statistics about the estimation errors for $var(t)$. Figure 6:
Theoretical standard deviation function of $\widehat{m}_{n}(t)$ for simple
random sampling without replacement (solid line), stratified sampling with
proportional allocation (dotted line) and stratified sampling with
$u^{(1)}$-optimal allocation (dashed line).
One can remark that the theoretical variance is much smaller, at all instants
$t$, for the stratified sampling with optimal allocation rule. The stratified
sampling with optimal allocation gives more accurate estimation of $var(t)$
than the other strategies. We can observe that clustering the space of
functions by performing stratified sampling may leads to a considerable gain
in terms of accuracy of the estimators of the variance function, dividing by
ten the mean error compared to simple random sampling without replacement.
Moreover, there is also a difference between proportional and optimal
allocations rules, for example the third quartile in optimal case is lower
than the median loss in the proportional case.
## 4 Conclusion and perspectives
In this paper, we have developed a survey sampling approach for estimating the
median of a functional variable. From a practical point of view, an appealing
consequence of the new methodology is that the proposed estimators are faster
to calculate. The experimental results on a test population of electricity
consumption curves confirm that even with high dimensional data,
stratification associated with the optimal allocation rule leads to important
reduction of the variance estimators. Having appropriate strata is the key for
getting more accurate estimators and the k-means algorithm is well adapted in
this situation. Nevertheless, choosing the stratification variables is a
rather complex issue and more work is needed in this direction.
A challenging future research avenue concerns the use of auxiliary information
at the estimation stage. While, in this paper, we have concentrated on the
estimation of the median using the Horvitz-Thompson estimator or the
poststratified estimator, more complex estimators using functional regression
models can be developed. For example, it is possible to set a linear
functional model which explains the functional variable $Y_{k}$ using a scalar
$X_{k}$ and to develop a regression estimator for the median curve. Developing
a general framework for regression estimators for the median curve is left for
future studies.
Acknowledgments
The authors thank the two anonymous referees, and the associate editor for
their constructive remarks that helped to improve the manuscript.
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|
arxiv-papers
| 2012-01-04T08:53:42 |
2024-09-04T02:49:25.923423
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mohamed Chaouch and Camelia Goga",
"submitter": "Camelia Goga",
"url": "https://arxiv.org/abs/1201.0846"
}
|
1201.0865
|
# Parker Winds Revisited: An Extension to Disk Winds
Timothy R. Waters,1 Daniel Proga,1,2
###### Abstract
A simple, one-dimensional dynamical model of thermally driven disk winds, one
in the spirit of the original Parker (1958) model, is presented. We consider
two different axi-symmetric streamline geometries: geometry (i) is commonly
used in kinematic models to compute synthetic spectra, while geometry (ii),
which exhibits self-similarity and more closely resembles the geometry found
by many numerical simulations of disk winds, is likely unused for this purpose
— although it easily can be with existing kinematic models. We make the case
that it should be, i.e. that geometry (ii) leads to transonic wind solutions
with substantially different properties.
1Department of Physics, University of Nevada, Las Vegas, NV 89154
2Princeton University Observatory, Peyton Hall, Princeton, NJ 08544
## 1 Introduction
Developing baseline disk wind models analogous to the spherically symmetric
Parker model (Parker, 1958) has proven to be a difficult task. A major
roadblock has been the uncertainty in the streamline geometry. Another obvious
and related difficulty is posed by the fact that accretion disks span many
more orders of magnitude in physical size than do stars, and they can host
radically different, spatially and temporally variable, thermodynamic
environments. It should come as no surprise then, that despite clear
observational evidence of outflows from many systems, identifying the actual
driving mechanisms, as well as determining the wind geometry, remains a
challenge.
Studies of disk winds therefore rely heavily on kinematic models in order to
quickly explore the parameter space without assuming a particular driving
mechanism. A popular choice of geometry, one that has been used in conjunction
with sophisticated radiative transfer simulations to model accretion disk
spectra from many systems, including AGN (Sim et al. 2008), is the Converging
model — geometry (i) in Figure 1. Recent multi-dimensional, time-dependent
simulations of a thermally driven wind carried out by Luketic et al. (2010)
suggest that the Converging model may not be well-suited for sampling the
entire wind, but rather only the inner portions of it. The outer portion is
better approximated by a model in which streamlines emerge at a constant
inclination angle $i$ to the midplane (hence the name, the CIA model —
geometry (ii).)
We have generalized the isothermal and polytropic Parker wind solutions so
that they apply to geometries (i) and (ii). Our solutions amount to a simple
dynamical disk wind model (see Waters & Proga 2012). Rather than positing a
velocity law as is done for kinematic models, the purpose of a dynamical model
is to impose the physical conditions and solve for the wind velocity as a
function of distance along a streamline. Here we summarize our findings for
how the streamline geometry alone can result in winds with substantially
different flow properties, limiting our attention to the isothermal case.
## 2 Results & Conclusions
The long-dashed and solid curves in the plot in Figure 1 depict the steady-
state flow properties of a Parker-like disk wind traversing geometries (i) and
(ii), respectively. Specifically, we plot the equivalent nozzle function
(denoted $\mathcal{N}$) along a streamline, in units of the gravitational
radius. Also shown are $\mathcal{N}$ for the spherically symmetric (bottom
dotted curve) and Keplerian (a radial Parker wind with a Keplerian azimuthal
velocity component; topmost dashed-dotted curve) Parker winds. Revolving
$\mathcal{N}$ about the horizontal axis sweeps out the shape of a de Laval
Nozzle that yields steady-state flow properties identical to that of the wind;
this shape is exponentially dependent on the effective potential and the
squared ratio of the local escape velocity to the sound speed (the HEP).
Comparing the throat locations and corresponding magnitudes of $\mathcal{N}$
for geometries (i) and (ii), it is clear that the CIA model has a sonic point
distance about twice that of the Converging model (implying a smaller
acceleration) and an initial Mach number
$\mathcal{M}_{o}=\mathcal{V}_{o}/c_{s}$ that is smaller by nearly an order of
magnitude. Since $\mathcal{M}_{o}$ is a direct gauge of the mass flux density,
the total mass loss rate for a CIA wind will be smaller in general. These
differences all result from the confined expansion area of the CIA model, due
to its lack of adjacent streamline divergence.
Both winds experience a reduced centrifugal force at $i=60^{\circ}$ compared
to a Keplerian Parker wind, explaining why the latter has a significantly
higher initial Mach number. We can therefore arrive at the result that the
mass flux densities of our disk wind models are always bounded from below by
that of the spherically symmetric Parker wind and above by that of the
Keplerian Parker wind.
In summary, the different properties of the CIA and Converging models are
solely due to geometric effects. If, for a given HEP and $i$, the resulting
velocity profiles were approximated by a beta-law, the parameters
$\mathcal{V}_{o}$ and $\beta$ (the slope) might differ by an order of
magnitude. Kinematic models that make use of a beta-law are therefore
sensitive to the type of wind geometry. The implication is that employing the
Converging model may lead to significant overestimates of the flow
acceleration if the true streamline geometry more closely resembles the CIA
model. The synthetic line profiles will be affected, especially if the
ionization balance of the wind is assumed to depend upon the density or
temperature profiles, which significantly differ for these geometries.
Figure 1.: Adjacent streamlines diverge from each other in the Converging
model but not in the CIA model. The plot of equivalent nozzle functions was
calculated by taking HEP $=11$ and $i=60^{\circ}$. We have normalized
$\mathcal{N}$ such that $\mathcal{N}\approx\mathcal{M}_{o}$ at the nozzle
throat; the horizontal lines mark the exact values of $\mathcal{M}_{o}$.
## References
* Luketic et al. (2010) Luketic, S., Proga, D., Kallman, T. R., Raymond, J. C., & Miller, J. M. 2010, ApJ, 719, 515. 1003.3264
* Parker (1958) Parker, E. N. 1958, ApJ, 128, 664
* Sim et al. (2008) Sim, S. A., Long, K. S., Miller, L., & Turner, T. J. 2008, MNRAS, 388, 611. 0805.2251
* Waters & Proga (2012) Waters, T. R. & Proga, D. 2012, submitted
|
arxiv-papers
| 2012-01-04T10:27:44 |
2024-09-04T02:49:25.933106
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tim Waters, Daniel Proga",
"submitter": "Timothy Waters",
"url": "https://arxiv.org/abs/1201.0865"
}
|
1201.0907
|
# A Geometrical Method of Decoupling
C. Baumgarten Paul Scherrer Institute, Switzerland
christian.baumgarten@psi.ch
###### Abstract
The computation of tunes and matched beam distributions are essential steps in
the analysis of circular accelerators. If certain symmetries – like midplane
symmetrie – are present, then it is possible to treat the betatron motion in
the horizontal, the vertical plane and (under certain circumstances) the
longitudinal motion separately using the well-known Courant-Snyder theory, or
to apply transformations that have been described previously as for instance
the method of Teng and Edwards Teng ; EdwardsTeng . In a preceeding paper it
has been shown that this method requires a modification for the treatment of
isochronous cyclotrons with non-negligible space charge forces cyc_paper .
Unfortunately the modification was numerically not as stable as desired and it
was still unclear, if the extension would work for all conceivable cases.
Hence a systematic derivation of a more general treatment seemed advisable.
In a second paper the author suggested the use of real Dirac matrices as basic
tools for coupled linear optics and gave a straightforward recipe to decouple
positive definite Hamiltonians with imaginary eigenvalues rdm_paper . In this
article this method is generalized and simplified in order to formulate a
straightforward method to decouple Hamiltonian matrices with eigenvalues on
the real and the imaginary axis. The decoupling of symplectic matrices which
are exponentials of such Hamiltonian matrices can be deduced from this in a
few steps. It is shown that this algebraic decoupling is closely related to a
geometric “decoupling” by the orthogonalization of the vectors $\vec{E}$,
$\vec{B}$ and $\vec{P}$, that were introduced with the so-called
“electromechanical equivalence” rdm_paper .
A mathematical analysis of the problem can be traced down to the task of
finding a structure-preserving block-diagonalization of symplectic or
Hamiltonian matrices. Structure preservation means in this context that the
(sequence of) transformations must be symplectic and hence canonical.
When used iteratively, the decoupling algorithm can also be applied to
n-dimensional systems and requires ${\cal O}(n^{2})$ iterations to converge to
a given precision.
Hamiltonian mechanics, coupled oscillators, beam optics, Lorentz
transformation
###### pacs:
45.20.Jj, 05.45.Xt, 41.85.-p, 03.30.+p
## I Introduction
The significance of the symplectic groups in Hamiltonian dynamics has been
emphasized for instance by A. Dragt Dragt , and it has long been known DGL
that the Dirac matrices are generators of the symplectic group $Sp(4,R)$. In
Ref. rdm_paper the author presented a toolbox for the treatment of two
coupled harmonic oscillators that is based on the use of the real Dirac
matrices (RDMs) as generators of the symplectic group $Sp(4,R)$ and a
systematic survey of symplectic transformations in two dimensions. This
toolbox enabled the developement of a straightforward recipe for the
decoupling of positive definite two-dimensional harmonic oscillators. Here we
present an improvement of the method that is based on geometric arguments,
i.e. on the orthogonalization of 3-dimensional vectors associated via the
electromechanical equivalence (EMEQ) to certain linear combinations of matrix
elements.
There is a long history of publications covering the diagonalization (and
related) problems in linear algebra as well as in linear coupled optics,
linear Hamiltonian dynamics and control theory. A (non-exhaustive) list is
given in the bibliography (see Refs. LMC ; PvL ; vanLoan ; BMX ; BMX2 ;
Coleman ; Dieci ; SR ; YQXX ; BKM ; Luo ; CM ; MK ; MSW ; ABK ; BFS ; CS , but
also Ref. cyc_paper ; rdm_paper and references therein). However, none of the
previous works (known to the author) takes full advantage of the group
structure of the generators of $Sp(4)$. The conceptually closest approach uses
“quaternions”, the representations of which seems to be identical to the RDMs
FMM , but seems to be limited to orthogonal symplectic transformations. The
decoupling method of Teng and Edwards has been the starting point for this
work, as it turned out to fail in some special cases (see Ref. cyc_paper and
App. D).
The method that we present here, is based on a survey of all symplectic
similarity transformations. We do not make specific assumptions about the
Hamiltonian other than that it is a symmetric quadratic form and we present a
geometric interpretation via the EMEQ, which provides a physical notation of
otherwise complicated and non-descriptive algebraic expressions 111Compare for
instance Ref. CMC .. Furthermore we believe that the use of the EMEQ is an
interesting example of how elements of classical physics, quantum mechanics,
special relativity, electrodynamics, group theory, geometric algebra,
statistics stat_paper and last but not least symplectic theory fit together
and allow to use a common formalism.
The simplest classical linear dynamical system with interaction (coupling) has
two degrees of freedom and hence a 4-dimensional phase space. It can be
considered as “fundamental” and a detailed analysis of its properties will
likely be instructive also for $n>2$. Indeed it turns out, that the decoupling
technique of a two-dimensional system can be iteratively applied to systems
with more than two degrees of freedom. A Jacobi-like iteration with pivot-
search and is sketched in Sec. V.3.
From the viewpoint of coupled linear optics the problem is solved if a
symplectic transformation is derived that transforms (constant) Hamiltonian
matrices to $2\times 2$-block-diagonal form (see below). It has been shown in
Ref. rdm_paper , that the same transformation method can be applied to
symplectic matrices as well. The arguments will be briefly reported below.
When applied to symplectic matrices, the method is equivalent to the
computation of the matrix logarithm. A solution for the couterpart, i.e. the
computation of the matrix exponential with emphasis on the use of Dirac
matrices, has been presented by Barut, Zeni and Laufer in 1994 BZL .
If $2\times 2$-block-diagonal form has been achieved, the remaining task is
completely analogous to the application of the Courant-Snyder theory for one
degree of freedom. Nevertheless some arguments require awareness of the
eigenvalues and their relation to the properties of the Dirac matrices so that
a reference to a complete diagonalization seemed appropriate.
## II Coupled Linear Optics
The Hamiltonian of a $n$-dimensional harmonic oscillator with arbitrary
coupling terms can be written in the form
$H={1\over 2}\,\psi^{T}\,{\bf A}\,\psi\,,$ (1)
where ${\bf A}$ is a symmetric matrix and $\psi$ is a state-vector or “spinor”
of the form $\psi=(q_{1},p_{1},q_{2},p_{2},\dots,q_{n},p_{n})^{T}$. Even
though the matrix ${\bf A}$ is time-dependent in the general case, it is well-
known practice to use the Floquet-transformation to reduce it to a constant
matrix for the treatment of periodic systems (see app. B and for instance Ref.
Talman ; MHO ). The symplectic unit matrix (usually labeled ${\bf J}$ or ${\bf
S}$) is a skew-symmetric matrix that squares to the negative unit matrix. For
$n=2$ it is identified with the real Dirac matrix $\gamma_{0}$. As described
in Ref. rdm_paper , it is possible to freely choose the order of the variables
in the state vector. However, the order of the variables fixes the form of the
symplectic unit matrix $\gamma_{0}$ 222 For $n=2$, this also follows from the
fundamental theorem of the Dirac matrices (see for instance Ref. AJM ; DGL )..
We prefer the use of a ordering system in which the phase space coordinates
$(q_{i},p_{i})$ are grouped as pairs of canonical conjugate variables, so that
$\gamma_{0}$ has the form
$\gamma_{0}=\left(\begin{array}[]{ccccc}0&1&&0&0\\\ -1&0&\dots&0&0\\\
&\vdots&&\vdots&\\\ 0&0&\dots&0&1\\\ 0&0&&-1&0\\\ \end{array}\right)\,,$ (2)
Using the over-dot to indicate the derivative with respect to time (or path
length), the equations of motion (EQOM) have the familiar form
$\begin{array}[]{rclp{5mm}rcl}\dot{q}_{i}&=&{\partial H\over\partial
p_{i}}&&\dot{p}_{i}&=&-{\partial H\over\partial q_{i}}\,,\end{array}$ (3)
or in vector notation:
$\dot{\psi}=\gamma_{0}\,\nabla_{\psi}\,H={\bf F}\,\psi\\\ $ (4)
where the force matrix ${\bf F}$ is given as
${\bf F}=\gamma_{0}\,{\bf A}\,.$ (5)
From the definition of ${\bf F}$ one quickly finds that Talman ; MHO
${\bf F}^{T}=\gamma_{0}\,\,{\bf F}\,\gamma_{0}\,,$ (6)
where the superscript “T” denotes the transposed matrix. Matrices that obey
Eqn. (6) are usually called “infinitesimally symplectic” or “Hamiltonian”
Talman . Both terms are - in the opinion of the author - misleading: The
former because ${\bf F}$ is neither symplectic nor is it infinitesimal, the
latter since ${\bf F}$ does not appear in the Hamiltonian while the symmetric
matrix ${\bf A}$ does. In addition ${\bf A}$ and not ${\bf F}$ is in the view
of the author the classical counterpart of the Hamiltonian operator (see App.
A in Ref. rdm_paper ). Furthermore, Eqn. (6) is a purely formal property and
not necessarily connected to a Hamiltonian. Therefore the author decided to
use the term “symplex” (plural “symplices”) when referring to its formal
definition (i.e. Eqn. (6)) and its relation to the symplectic transfer matrix
and to call it “force matrix” when referring to its physical content -
especially with respect to the EMEQ (see Ref. rdm_paper and below).
Accordingly we speak of an anti-symplex or cosymplex (i.e. “skew-Hamiltonian”
matrix), if a matrix ${\bf C}$ fulfills the equation
${\bf C}^{T}=-\gamma_{0}\,\,{\bf C}\,\gamma_{0}\,.$ (7)
If we write ${\bf S}$ (${\bf C}$) for (co-)symplices, respectively, optionally
with a subscript, then it is easy to prove that
$\left.\begin{array}[]{c}{\bf S}_{1}\,{\bf S}_{2}-{\bf S}_{2}\,{\bf S}_{1}\\\
{\bf C}_{1}\,{\bf C}_{2}-{\bf C}_{2}\,{\bf C}_{1}\\\ {\bf C}\,{\bf S}+{\bf
S}\,{\bf C}\,,\end{array}\right\\}\Rightarrow\mathrm{symplex}\,,$ (8)
and
$\left.\begin{array}[]{c}{\bf S}_{1}\,{\bf S}_{2}+{\bf S}_{2}\,{\bf S}_{1}\\\
{\bf C}_{1}\,{\bf C}_{2}+{\bf C}_{2}\,{\bf C}_{1}\\\ {\bf C}\,{\bf S}-{\bf
S}\,{\bf C}\\\ \end{array}\right\\}\Rightarrow\mathrm{cosymplex}\,.$ (9)
### II.1 Dirac Matrices
In the following we focus on two degrees of freedom ($n=2$), i.e. to a four-
dimensional phase space and the use of the real Dirac matrices to describe its
dynamics and transformation properties. Often the term “Dirac matrices” is
used more restrictively and designates only four matrices, namely
$\gamma_{k}\,,\,k\in\,[0\dots 3]$. Here we consider the four basic Dirac
matrices as the four basic elements of a Clifford algebra $Cl(3,1)$ with $16$
elements derived from the basic matrices (see app. A). For further details see
for instance Ref. Okubo ; Scharnhorst ; Hestenes .
Any real $4\times 4$-matrix ${\bf M}$ can be written as a linear combination
of the RDMs
${\bf M}=\sum\limits_{k=0}^{15}\,m_{k}\,\gamma_{k}\,.$ (10)
The RDM-coefficients $m_{k}$ are given by 333Eqn. (11) is based on the fact
that all RDMs except the unit matrix have zero trace.
$m_{k}=\mathrm{Tr}(\gamma_{k}^{2})\,\mathrm{Tr}\left({{\bf
M}\,\gamma_{k}+\gamma_{k}\,{\bf M}\over 32}\right)\,,$ (11)
where $\mathrm{Tr}({\bf X})$ is the trace of the matrix ${\bf X}$. Only the
first ten RDMs are symplices and since symplices obey the superposition
principle rdm_paper ; MHO ; Dragt , any force matrix (symplex) can be written
as
${\bf F}=\sum\limits_{k=0}^{9}\,f_{k}\,\gamma_{k}\,.$ (12)
The solution of Eqn. (4) is known to be
$\psi(s)=\exp{({\bf F}\,s)}\,\psi(0)\,,$ (13)
where the matrix
${\bf M}=\exp{({\bf F}\,s)}$ (14)
is called transfer matrix, which can be shown to fulfill the symplectic
condition, if ${\bf F}$ is a symplex rdm_paper ; Dragt ; MHO :
${\bf M}\,\gamma_{0}\,{\bf M}^{T}=\gamma_{0}\,.$ (15)
Vice versa it is known that symplectic matrices can be written in the form of
Eqn. (14) Talman ; MHO .
Transfer matrices can be split into two parts, one ($M_{s}$) being a symplex,
the other ($M_{c}$) being a cosymplex rdm_paper ; Parzen :
$\begin{array}[]{rcl}{\bf M}_{c}&=&({\bf M}-\gamma_{0}\,{\bf
M}^{T}\,\gamma_{0})/2\\\ {\bf M}_{s}&=&({\bf M}+\gamma_{0}\,{\bf
M}^{T}\,\gamma_{0})/2\,,\end{array}$ (16)
which is in case of a symplectic matrix ${\bf M}$ identical to
$\begin{array}[]{rcl}{\bf M}_{c}&=&({\bf M}+{\bf M}^{-1})/2\\\ {\bf
M}_{s}&=&({\bf M}-{\bf M}^{-1})/2\,.\end{array}$ (17)
It has been shown in Ref. rdm_paper , that the decoupling of the symplex-part
${\bf M}_{s}$ of a symplectic matrix ${\bf M}$ automatically decouples the
corresponding cosymplex ${\bf M}_{c}$. Hence it is sufficient to derive a
method to decouple symplices of the above mentioned type. In cases where only
the one-turn-transfer matrix is available, Eqn. (16) is used beforehand to
extract the symplex-part of the transfer matrix. The decoupling algorithm can
then be applied to this matrix (see also the detailed discussion in Ref.
rdm_paper ).
## III Block-Diagonalization and Eigenvalues
The force matrix ${\bf F}$ is by definition a product of a symmetric matrix
${\bf A}$ and of a skew-symmetric matrix $\gamma_{0}$. Hence it has zero trace
and the sum of all eigenvalues is zero. We restrict ourselves to systems with
real-valued force matrices and therefore real-valued transfer matrices. The
eigenvalues of real-valued $2\times 2$-symplices are either both real or both
purely imaginary (since they are the square root of a real expression). Block-
diagonalization (in the case of the variable ordering as described above)
means to find a symplectic similarity transformation ${\bf R}$ such that the
matrix ${\bf\tilde{F}}={\bf R}\,{\bf F}\,{\bf R}^{-1}$ has the form
${\bf\tilde{F}}=\left(\begin{array}[]{cc}{\bf\tilde{F}}_{1}&0\\\
0&{\bf\tilde{F}}_{2}\end{array}\right)\,,$ (18)
where ${\bf\tilde{F}}_{k}$ are real $2\times 2$-matrices. Since similarity
transformations preserve the eigenvalues, a symplex is block-diagonalizable in
the form that we are going to describe, if the (pairs of) eigenvalues are
either real or imaginary. In case of imaginary eigenvalues, the corresponding
degree of freedom (i.e. pair $(q_{i},p_{i})$) is stable (or focused), while a
pair of real eigenvalues belongs to an unstable (non-focused) degree of
freedom. The corresponding betatron motion is unstable in the sense, that no
sufficient focusing is present.
However – in the general coupled case without further assumptions – ${\bf F}$
is a general $4\times 4$-symplex (or larger). Using the RDMs it is relatively
easy to construct matrices with complex eigenvalues. An example is
${\bf F}=E_{x}\,\gamma_{4}+B_{x}\,\gamma_{7}\,,$ (19)
which has the complex eigenvalues $\pm i\,(B_{x}\pm i\,E_{x})$. Since the
eigenvalues are complex, also the $2\times 2$-blocks are complex. They can be
block-diagonalized, but the generalization to the $2\,n\times 2\,n$-case
requires a general treatment of the complex case, which goes beyond the scope
of this paper.
As in Ref. rdm_paper the author speaks of regular or massive systems, if the
Hamiltonian is positive definite and of irregular or magnetic systems in case
of indefinite Hamiltonian, respectively. Both types may be stable or unstable
and this distinction should not be confused with the question of stability. A
detailed discussion of stability would go beyond the scope of this paper and
we refer the reader for instance to Ref. MHO or Ref. FMM and references
therein.
### III.1 The $\bf S$-matrix
The matrix of second moments $\sigma$ of a charged particle distribution
$\sigma=\langle\psi\,\psi^{T}\rangle\,,$ (20)
has the time derivative
$\dot{\sigma}={\bf F}\,\sigma+\sigma\,{\bf F}^{T}\,.$ (21)
Multiplication from the left with $\gamma_{0}$ and the use of Eqn. (6) leads
to
${\bf\dot{S}}={\bf F}\,{\bf S}-{\bf S}\,{\bf F}\,,$ (22)
where the matrix ${\bf S}$ is defined by
${\bf S}=\sigma\,\gamma_{0}\,.$ (23)
If Eqn. (23) is compared to Eqn. (5), then it is obvious that ${\bf S}$ is
also a symplex as it is also the product of a symmetric and a skew-symmetric
matrix and obeys Eq. 6. From Eqns. (13), (14) and (20) it follows that
$\sigma(s)={\bf M}(s)\,\sigma(0)\,{\bf M}^{T}(s)\,.$ (24)
The second moments of a matched distribution are unchanged after one turn (or
sector) of period $L$ so that $\sigma(L)=\sigma(0)$ so that one obtains in a
few steps 444See common textbooks on linear Hamiltonian dynamics or Ref.
rdm_paper .:
${\bf M}\,{\bf S}-{\bf S}\,{\bf M}=0\,.$ (25)
### III.2 The Eigensystems and Matching
Hence one finds that the matrices ${\bf M}$, ${\bf F}$ and ${\bf S}$ have the
same eigenvectors - but in general different eigenvalues HMMG ; Wolski :
$\begin{array}[]{rclp{5mm}rcl}{\bf F}&=&{\bf E}\,\lambda\,{\bf E}^{-1}&&{\bf
M}&=&{\bf E}\,\Lambda\,{\bf E}^{-1}\\\ {\bf S}&=&{\bf E}\,{\bf D}\,{\bf
E}^{-1}&&\\\ \end{array}$ (26)
where Wolski
$\begin{array}[]{rcl}\lambda&=&\mathrm{Diag}(i\,\omega_{1},-i\,\omega_{1},i\,\omega_{2},-i\,\omega_{2})\\\
\Lambda&=&\mathrm{Diag}(e^{i\,\omega_{1}},e^{-i\,\omega_{1}},e^{i\,\omega_{2}},e^{-i\,\omega_{2}})\\\
{\bf
D}&=&\mathrm{Diag}(-i\,\varepsilon_{1},i\,\varepsilon_{1},-i\,\varepsilon_{2},i\,\varepsilon_{2})\,.\end{array}$
(27)
$\omega_{i}$ are the oscillation frequencies and $\varepsilon_{i}$ the
emittances. If $\bf E$ is known, the second moments of the matched
distribution can be computed by replacing the eigenfrequencies by the
emittances. If a sympletic transformation ${\bf R}$ is known, that brings
${\bf F}$ (and hence ${\bf S}$ and ${\bf M}$) to block-diagonal form, then one
can simply use the usual Courant Snyder theory for one-dimensional systems
Hinterberger . In this case an explicit computation of the eigenvectors is not
required.
## IV The Electromechanical Equivalence
It was shown in Ref. rdm_paper , that the ten coefficients of the force matrix
${\bf F}$ or the ${\bf S}$-matrix can be identified with energy ${\cal E}$ and
momentum $\vec{P}$ of a particle and with electric and magnetic field
($\vec{E}$ and $\vec{B}$, respectively) seen by a charged particle in external
fields. The meaning of this identification is, that the corresponding
coefficients of ${\bf F}$ or ${\bf S}$ transform under symplectic
transformations in the exact same way as the fields and the momentum transform
under the corresponding boosts and rotations.
It was also shown that the envelope equations of coupled linear optics are
isomorphic to the Lorentz force equation. The Lorentz group was found to be a
subset of the two-dimensional symplectic group. The so defined “fields”
($\vec{E}$ and $\vec{B}$) of the EMEQ should not be confused with the real
fields of the beamline elements or accelerator components.
This isomorphism has been named electromechanical equivalence (EMEQ). The ten
possible symplectic transformations are identified with spatial and phase-
rotations, Lorentz boosts and so-called “phase boosts”. The transformation
properties are analogous to those in Minkowski space-time.
This structural analogy is the basic idea behind the electromechanical
equivalence (EMEQ). Naturally, $\gamma_{0}$ is associated with the time-like
components of 4-vectors (i.e. energy), the spatial matrices
$\vec{\gamma}=(\gamma_{1},\gamma_{2},\gamma_{3})^{T}$ are associated with the
momentum, the matrices $\gamma_{0}\,\vec{\gamma}$ with the electric field and
$\gamma_{14}\,\gamma_{0}\,\vec{\gamma}$ with the magnetic field. The
pseudoscalar has been named
$\gamma_{14}=\gamma_{0}\,\gamma_{1}\,\gamma_{2}\,\gamma_{3}$ (instead of
$\gamma_{5}$, as convention in QED). The remaining six matrices are
$\gamma_{10}$, which is the time-component of the pseudo-vector,
$(\gamma_{11},\gamma_{12},\gamma_{13})^{T}=\gamma_{14}\,\vec{\gamma}$ are the
spatial components of the pseudo-vector and $\gamma_{15}={\bf 1}$ is the unit
matrix. A complete list is given in App. A, further details in Ref. rdm_paper
and in textbooks on quantum electrodynamics.
The EMEQ is given by the following nomenclature:
$\begin{array}[]{rcl}{\cal E}&\equiv&f_{0}\\\
\vec{P}&\equiv&(f_{1},f_{2},f_{3})^{T}\\\
\vec{E}&\equiv&(f_{4},f_{5},f_{6})^{T}\\\
\vec{B}&\equiv&(f_{7},f_{8},f_{9})^{T}\,,\end{array}$ (28)
with the $f_{k}$ given by Eqn. (12). Using the EMEQ, the eigenvalues of ${\bf
F}$ (Eqn. LABEL:eq_eigen and Eqn. 27) can be expressed by:
$\begin{array}[]{rcl}K_{1}&=&{\cal
E}^{2}+\vec{B}^{2}-\vec{E}^{2}-\vec{P}^{2}\\\ K_{2}&=&-2\,{\cal
E}\,\vec{P}\cdot(\vec{E}\times\vec{B})+{\cal
E}^{2}\,\vec{B}^{2}+\vec{E}^{2}\,\vec{P}^{2}\\\
&-&(\vec{E}\cdot\vec{P})^{2}-(\vec{E}\cdot\vec{B})^{2}-(\vec{P}\cdot\vec{B})^{2}\\\
&=&({\cal
E}\,\vec{B}+\vec{E}\times\vec{P})^{2}-(\vec{E}\cdot\vec{B})^{2}-(\vec{P}\cdot\vec{B})^{2}\\\
\omega_{1}&=&\sqrt{K_{1}+2\,\sqrt{K_{2}}}\\\
\omega_{2}&=&\sqrt{K_{1}-2\,\sqrt{K_{2}}}\\\ \mathrm{Det}({\bf
F})&=&K_{1}^{2}-4\,K_{2}\\\ \end{array}$ (29)
Force matrices of stable systems have purely imaginary eigenvalues Arnold , so
that for stable systems one has $K_{2}>0$ and $K_{1}>2\,\sqrt{K_{2}}$.
Using the notation of the EMEQ a general symplex ${\bf F}$ is given
explicitely by
$\begin{array}[]{rcl}{\bf
F}&=&\left(\begin{array}[]{cccc}-E_{x}&E_{z}+B_{y}&E_{y}-B_{z}&B_{x}\\\
E_{z}-B_{y}&E_{x}&-B_{x}&-E_{y}-B_{z}\\\
E_{y}+B_{z}&B_{x}&E_{x}&E_{z}-B_{y}\\\
-B_{x}&-E_{y}+B_{z}&E_{z}+B_{y}&-E_{x}\\\ \end{array}\right)\\\
&+&\left(\begin{array}[]{cccc}-P_{z}&{\cal E}-P_{x}&0&P_{y}\\\ -{\cal
E}-P_{x}&P_{z}&P_{y}&0\\\ 0&P_{y}&-P_{z}&{\cal E}+P_{x}\\\ P_{y}&0&-{\cal
E}+P_{x}&P_{z}\\\ \end{array}\right)\,,\end{array}$ (30)
Note that ${\bf F}$ is block-diagonal, if $B_{x}=B_{z}=E_{y}=P_{y}=0$.
## V Decoupling of 2-dimensional systems
### V.1 The geometrical approach
In the following we describe a geometrical approach of decoupling that is
inspired by the observation, that in the decoupled force matrix, the scalar
products $\vec{E}\cdot\vec{B}$ and $\vec{P}\cdot\vec{B}$ vanish rdm_paper . In
Hamiltonian form (see Eqn. 38 below), also the product $\vec{P}\cdot\vec{E}$
is zero and only the components ${\cal E}$, $P_{x}$, $E_{z}$ and $B_{y}$
remain. It is therefore instructive to analyze the symplectic transformation
properties of these scalar products. The product $\vec{E}\cdot\vec{B}$ is
known to be invariant under rotations and Lorentz boosts. Formally it is a
pseudo-scalar in contrast to the scalar component representing the mass. Hence
one might loosely speak of “mass components” and use the abbreviations:
$\begin{array}[]{rcl}M_{r}&=&\vec{E}\cdot\vec{B}\\\
M_{g}&=&\vec{B}\cdot\vec{P}\\\ M_{b}&=&\vec{E}\cdot\vec{P}\\\ \end{array}$
(31)
The “mass components” are invariant under spatial rotations. We may therefore
proceed with phase rotations and boosts. We introduce the following auxiliary
vectors:
$\begin{array}[]{rcl}\vec{r}&\equiv&{\cal E}\,\vec{P}+\vec{B}\times\vec{E}\\\
\vec{g}&\equiv&{\cal E}\,\vec{E}+\vec{P}\times\vec{B}\\\ \vec{b}&\equiv&{\cal
E}\,\vec{B}+\vec{E}\times\vec{P}\,,\end{array}$ (32)
so that $K_{2}$ from Eqn. LABEL:eq_eigenfreq can be written as
$K_{2}=\vec{b}^{2}-M_{r}^{2}-M_{g}^{2}\,.$ (33)
It is easy to see that ${\vec{g}}$, ${\vec{r}}$ and ${\vec{b}}$ transform
under spatial rotations just like usual vectors. It is also quite obvious that
the vector $\vec{g}$ equals the usual Lorentz force and the vector $\vec{b}$
equals the “Lorentz force” of a particle with magnetic charge, as the role of
$\vec{E}$ and $\vec{B}$ is exchanged compared to $\vec{g}$ in the algebraic
way that corresponds to a duality rotation through an angle of ${\pi\over 2}$
rdm_paper .
One finds the following products:
$\begin{array}[]{rcl}\vec{g}^{2}&=&-2\,{\cal
E}\,\vec{P}\cdot(\vec{E}\times\vec{B})+{\cal
E}^{2}\,\vec{E}^{2}+\vec{B}^{2}\,\vec{P}^{2}-M_{g}^{2}\\\
\vec{r}^{2}&=&-2\,{\cal E}\,\vec{P}\cdot(\vec{E}\times\vec{B})+{\cal
E}^{2}\,\vec{P}^{2}+\vec{B}^{2}\,\vec{E}^{2}-M_{r}^{2}\\\
\vec{b}^{2}&=&-2\,{\cal E}\,\vec{P}\cdot(\vec{E}\times\vec{B})+{\cal
E}^{2}\,\vec{B}^{2}+\vec{E}^{2}\,\vec{P}^{2}-M_{b}^{2}\\\
\vec{g}\cdot\vec{r}&=&({\cal E}^{2}-\vec{B}^{2})\,M_{b}+M_{r}\,M_{g}\\\
\vec{g}\cdot\vec{b}&=&({\cal E}^{2}-\vec{P}^{2})\,M_{r}+M_{g}\,M_{b}\\\
\vec{r}\cdot\vec{b}&=&({\cal E}^{2}-\vec{B}^{2})\,M_{g}+M_{r}\,M_{b}\\\
\end{array}$ (34)
We introduce the following abbreviations for a better readability
$\begin{array}[]{rclp{5mm}rcl}c&=&\cos{(\varepsilon)}&&s&=&\sin{(\varepsilon)}\\\
c_{2}&=&\cos{(2\,\varepsilon)}&&s_{2}&=&\sin{(2\,\varepsilon)}\\\
C&=&\cosh{(\varepsilon)}&&S&=&\sinh{(\varepsilon)}\\\
C_{2}&=&\cosh{(2\,\varepsilon)}&&S_{2}&=&\sinh{(2\,\varepsilon)}\\\
\end{array}$ (35)
The phase rotation generated by $\gamma_{0}$ yields:
$\begin{array}[]{rcl}{\vec{g}}^{\prime}&=&\vec{g}\,c+\vec{r}\,s\\\
{\vec{r}}^{\prime}&=&\vec{r}\,c-\vec{g}\,s\\\ {\vec{b}}^{\prime}&=&\vec{b}\\\
\end{array}$ (36)
The transformation of the mass components is listed in Tab. 1.
| $M_{r}^{\prime}$ | $M_{g}^{\prime}$ | $M_{b}^{\prime}$
---|---|---|---
$\gamma_{0}$ | $M_{r}\,c+M_{g}\,s$ | $M_{g}\,c-M_{r}\,s$ | $M_{b}\,c_{2}+{\vec{P}^{2}-\vec{E}^{2}\over 2}\,s_{2}$
$\gamma_{1}$ | $M_{r}\,C-(\vec{b})_{x}\,S$ | $M_{g}$ | $M_{b}\,C-(\vec{r})_{x}\,S$
$\gamma_{2}$ | $M_{r}\,C-(\vec{b})_{y}\,S$ | $M_{g}$ | $M_{b}\,C-(\vec{r})_{y}\,S$
$\gamma_{3}$ | $M_{r}\,C-(\vec{b})_{z}\,S$ | $M_{g}$ | $M_{b}\,C-(\vec{r})_{z}\,S$
$\gamma_{4}$ | $M_{r}$ | $M_{g}\,C+(\vec{b})_{x}\,S$ | $M_{b}\,C+(\vec{g})_{x}\,S$
$\gamma_{5}$ | $M_{r}$ | $M_{g}\,C+(\vec{b})_{y}\,S$ | $M_{b}\,C+(\vec{g})_{y}\,S$
$\gamma_{6}$ | $M_{r}$ | $M_{g}\,C+(\vec{b})_{z}\,S$ | $M_{b}\,C+(\vec{g})_{z}\,S$
Table 1: Table of transformed “mass components” for symplectic transformations
in 2 dimensions. Compare Eqns. LABEL:eq_aux_masses, 32 and
LABEL:eq_aux_angles.
From the discussion of the normal form of the force matrix in Ref. rdm_paper
it follows, that decoupling to block-diagonal form is done by a transformation
that makes $P_{y}=E_{y}=B_{x}=B_{z}=0$. Geometrically this means, that
$\vec{B}$ has to be aligned along the y-axis and the vectors $\vec{P}$ and
$\vec{E}$ should be in the plane perpendicular to $\vec{B}$. In a first step,
the decoupling of a two-dimensional harmonic oscillator requires the (partial)
orthogonalization of the (3-dimensional) “vectors” $\vec{E}$, $\vec{B}$ and
$\vec{P}$:
$\begin{array}[]{rcl}M_{r}&=&\vec{E}\cdot\vec{B}\to 0\\\
M_{g}&=&\vec{P}\cdot\vec{B}\to 0\,,\end{array}$ (37)
which can be interpreted as a geometrical “decoupling”. The alignment of
$\vec{B}$ along the y-axis in a second step is simple. A transformation to
what we call “Hamiltonian” form
${\bf F}_{d}=\left(\begin{array}[]{cccc}0&\alpha&0&0\\\ -\beta&0&0&0\\\
0&0&0&\gamma\\\ 0&0&-\delta&0\end{array}\right)\,,$ (38)
requires additionally to make $E_{x}=P_{z}=0$, which can again by done in two
steps, orthogonalization
$M_{b}=\vec{E}\cdot\vec{P}\to 0\,,$ (39)
and subsequent alignment of $\vec{E}$ and $\vec{P}$. The general form of
symplectic transformations has been described in some detail in Ref. rdm_paper
, here we give only a brief summary. A symplectic transformation matrix ${\bf
R}_{b}$ is generated by a basic symplex $\gamma_{b}$ with $b\in[0\dots 9]$ and
controlled by a parameter $\varepsilon$:
$\begin{array}[]{rcl}{\bf R}_{b}&=&\exp{(\gamma_{b}\,{\varepsilon\over 2})}\\\
{\bf R}_{b}^{-1}&=&\exp{(-\gamma_{b}\,{\varepsilon\over 2})}\\\ {\bf
F}&\to&{\bf R}_{b}\,{\bf F}\,{\bf R}_{b}^{-1}\\\ \end{array}$ (40)
The effect of a basic symplex $\gamma_{b}$ depends on its “signature”, which
is positive for symmetric and negative for skew-symmetric $\gamma_{b}$:
$\begin{array}[]{rcl}{\bf R}_{b}&=&\left\\{\begin{array}[]{lcl}{\bf
1}\,\cos{(\varepsilon/2)}+\gamma_{b}\,\sin{(\varepsilon/2)}&\mathrm{for}&\gamma_{b}^{2}=-{\bf
1}\\\ {\bf
1}\,\cosh{(\varepsilon/2)}+\gamma_{b}\,\sinh{(\varepsilon/2)}&\mathrm{for}&\gamma_{b}^{2}={\bf
1}\\\ \end{array}\right.\,,\end{array}$ (41)
where the bold printed ${\bf 1}$ is the unity matrix. Note that
transformations with $\gamma_{b}^{2}=-{\bf 1}$ ($+{\bf 1}$) are called
rotations (boosts), respectively. Explicitely, $\gamma_{0}$ is the generator
of a “phase rotation”, $\gamma_{b}\,\,\,b\in[7,8,9]$ are “spatial rotations“
with respect to the $x$, $y$ and $z$-axis and $\gamma_{b}\,\,\,b\in[4,5,6]$
generate “Lorentz boosts” with respect to the $x$, $y$ and $z$-axis. The
“phase boosts” generated by $\gamma_{b}\,\,\,b\in[1,2,3]$ are combinations of
phase rotations and Lorentz boosts. The parameter $\varepsilon$ is called
“angle” in case of rotations and “rapidity” in case of boosts. As the
decoupling requires a sequence of transformations, we emphasize that the RDM-
coefficients have to be updated according to Eq. 11 after each transformation.
Inspection of Tab. 1 shows that a straightforward strategy is the following:
* •
$M_{g}\to 0$: Make a phase rotation generated by $\gamma_{0}$ with angle
$\varepsilon=\arctan{({M_{g}\over M_{r}})}$. This will always work independent
on the size of $M_{i}$.
* •
$\vec{b}\to|\vec{b}|\,\vec{e}_{y}$: Align the vector $\vec{b}$ along the
$y$-axis by the spatial rotations with ${\bf R}_{7}$ and an angle of
$\varepsilon=\arctan{({b_{z}\over b_{y}})}$ and with ${\bf R}_{9}$ through an
angle of $\varepsilon=-\arctan{({b_{x}\over b_{y}})}$. Such rotations can
always be done.
* •
$M_{r}\to 0$: Boost using $\gamma_{2}$ and angle
$\varepsilon=\mathrm{arctanh}{({M_{r}\over b_{y}})}$.
The last transformation is only possible, if $|M_{r}|<|b_{y}|=|\vec{b}|$:
$\begin{array}[]{rcl}(\vec{E}\cdot\vec{B})^{2}&\leq&-2\,{\cal
E}\,\vec{P}\cdot(\vec{E}\times\vec{B})+{\cal
E}^{2}\,\vec{B}^{2}+\vec{E}^{2}\,\vec{P}^{2}-(\vec{E}\cdot\vec{P})^{2}\\\
\end{array}$ (42)
The first transformations result in $\vec{P}\cdot\vec{B}=0$, so that Eq.
LABEL:eq_transcond is identical to the requirement that $K_{2}\geq 0$ (see Eq.
LABEL:eq_eigenfreq). This means that the eigenvalues are either located on the
real or imaginary axis, but not off-axis in the complex plane. If this
condition is fulfilled, then the vector-components $(\vec{g})_{y}$ and
$(\vec{r})_{y}$, $(\vec{b})_{x}$ and $(\vec{b})_{z}$ are zero after the
decoupling transformations have been applied. It follows from
$M_{r}=\vec{E}\cdot\vec{B}=0$ and $M_{g}=\vec{P}\cdot\vec{B}=0$ and Eq. 32
that $\vec{E}\cdot\vec{b}=0$ and $\vec{P}\cdot\vec{b}=0$, and since we aligned
$\vec{b}$ along the $y$-axis, we have $E_{y}=0$ and $P_{y}=0$, so that with
$\vec{b}$ also $\vec{B}$ is aligned along the $y$-axis and $B_{x}=B_{z}=0$. If
we compare this with Eq. 30, then we note that the matrix ${\bf F}$ is now
block-diagonal.
That is: we found a symplectic decoupling algorithm for both - systems with
purely imaginary eigenvalues, which are called “strongly” stable Arnold , and
unfocused systems with purely real eigenvalues. That the algorithm works in
both cases equally well, is important for instance in the case of transverse-
longitudinal coupling with space charge in cyclotrons cyc_paper .
We continue the discussion of force matrices with eigenvalues off axis in the
complex plane in Sec. V.2 and assume for now, that $K_{2}>0$. Using the
abbreviations
$\begin{array}[]{rcl}M_{x}&=&\sqrt{M_{r}^{2}+M_{g}^{2}}\\\
b_{yz}&=&\sqrt{b_{y}^{2}+b_{z}^{2}}\,,\end{array}$ (43)
the RDM-coefficients of the block-diagonal (decoupled) force matrix are given
by:
$\begin{array}[]{rcl}{\cal E}^{\prime}&=&{\cal
E}\,\sqrt{1-{M_{x}^{2}\over\vec{b}^{2}}}\\\
P_{x}^{\prime}&=&{P_{x}\,M_{r}-E_{x}\,M_{g}\over
M_{x}}\,{\sqrt{\vec{b}^{2}-M_{x}^{2}}\over b_{yz}}\\\
P_{z}^{\prime}&=&{\sqrt{\vec{b}^{2}-M_{x}^{2}}\over\vec{b}^{2}\,M_{x}\,b_{yz}}\,\left[M_{g}\,(b_{z}\,E_{y}-b_{y}\,E_{z})+M_{r}\,(b_{y}\,P_{z}-b_{z}\,P_{y})\right]\\\
E_{x}^{\prime}&=&{\vec{b}^{2}\,(M_{r}\,E_{x}+M_{g}\,P_{x})-{\cal
E}\,b_{x}\,M_{x}^{2}\over M_{x}\,b_{yz}\,|b|}\\\
E_{z}^{\prime}&=&{M_{r}\,(b_{y}\,E_{z}-b_{z}\,E_{y})+M_{g}\,(b_{y}\,P_{z}-b_{z}\,P_{y})\over
M_{x}\,b_{yz}}\\\ B_{y}^{\prime}&=&{{\cal
E}\,\vec{B}^{2}-\vec{P}\cdot(\vec{E}\times\vec{B})\over|\vec{b}|}\\\
B_{x}^{\prime}&=&B_{z}^{\prime}=E_{y}^{\prime}=P_{y}^{\prime}=0\\\
\end{array}$ (44)
In order to bring the block-diagonal force matrix to Hamiltonian form, one may
apply the following transformations:
* •
$M_{b}\to 0$: Use another phase rotation with $\gamma_{0}$ with
$\varepsilon={1\over 2}\,\arctan{({2\,M_{b}\over\vec{E}^{2}-\vec{P}^{2}})}$
* •
$P_{z}\to 0$: Use rotation about $y$-axis with $\gamma_{8}$ with
$\varepsilon=-\arctan{({P_{z}\over P_{x}})}$.
After these two rotations, the matrix has Hamiltonian form, if $K_{2}>0$
holds. In charged particle optics this is usually the case and therefore we
consider this method as a generally applicable decoupling algorithm.
### V.2 Complex Eigenvalues
Even though the problem of complex eigenvalues has not yet been solved for the
general $2\,n\times 2\,n$ case, it is possible to give a solution for the
$4\times 4$ case as we are going to describe here. The more general case of
arbitrary $2\,n\times 2\,n$-symplices with arbitrary (complex) eigenvalues can
presumably be solved by a block-diagonalization with $4\times 4$-blocks for
each set of complex conjugate eigenvalues and $2\times 2$-blocks for each pair
of real or imaginary eigenvalues.
If $K_{2}<0$ the eigenvalues are complex and a block-diagonalization with
$2\times 2$-blocks is not possible (within the reals). However a
simplification of the matrix is possible with the aim, that the RDM-
coefficients of the transformed matrix have the following structure:
$\begin{array}[]{rclp{5mm}rclp{5mm}rcl}P_{x}&=&0&&P_{y}&=&0&&P_{z}&=&0\\\
E_{x}&=&0&&B_{x}&=&0&&B_{z}&=&0\\\ E_{z}&\neq&0&&E_{y}&\neq&0&&B_{y}&\neq&0\\\
{\cal E}&=&0&&M_{g}&=&0&&M_{b}&=&0\,,\end{array}$ (45)
so that one finds $\vec{g}=0$ and $\vec{b}=0$ and the auxiliary vector
$\vec{r}$ has only a single non-vanishing component $r_{x}$. We distinguish
two cases, the first with ${\cal E}^{2}<\mathrm{Max}(\vec{P}^{2},\vec{E}^{2})$
and the second with ${\cal E}^{2}>\mathrm{Min}(\vec{P}^{2},\vec{E}^{2})$. In
both cases the goal is to let “energy” and “momentum” vanish by appropriate
Lorentz or phase boosts. Then one may align $\vec{B}$ along the $y$-axis and
rotate about the $y$-axis to make $E_{x}=0$. Then the conditions of Eqn. 45
are fulfilled.
#### V.2.1 The Low Energy Case
The decoupling strategy for the first case, i.e. for ${\cal
E}^{2}<\mathrm{Max}(\vec{P}^{2},\vec{E}^{2})$:
* •
$M_{g}\to 0$: Apply a phase rotation ${\bf R}_{0}$ with angle
$\varepsilon_{1}=\arctan{({M_{g}\over M_{r}})}$. Note that this maximizes
$M_{r}=\vec{E}\cdot\vec{B}$.
* •
$\vec{E}\to|\vec{E}|\,\vec{e}_{y}$: Align the vector $\vec{E}$ along the
$y$-axis by the spatial rotations with ${\bf R}_{7}$ and an angle of
$\varepsilon_{2}=\arctan{({E_{z}\over E_{y}})}$ and (after an update of the
RDM-coefficients and a recomputation of the auxiliary vector and mass
components) with ${\bf R}_{9}$ about an angle of
$\varepsilon_{3}=-\arctan{({E_{x}\over E_{y}})}$.
* •
${\cal E}\to 0$: Boost using ${\bf R}_{2}$ and rapidity
$\varepsilon_{4}=\mathrm{arctanh}{({{\cal E}\over E_{y}})}$. According to the
assumptions, this is possible and does not change $E_{x}=0$ or $E_{z}=0$.
* •
$P_{x}\to 0$: Boost using ${\bf R}_{3}$ and rapidity
$\varepsilon_{5}=-\mathrm{arctanh}{({P_{x}\over B_{y}})}$.
* •
$P_{z}\to 0$: Boost using ${\bf R}_{1}$ and rapidity
$\varepsilon_{6}=\mathrm{arctanh}{({P_{z}\over B_{y}})}$. Since ${\cal
E}=E_{z}=E_{x}=0$, the energy ${\cal E}$ as well as $\vec{E}$ are unchanged by
the boost.
* •
$\vec{B}\to|\vec{B}|\,\vec{e}_{y}$: Align the vector $\vec{B}$ along the
$y$-axis by the spatial rotations with ${\bf R}_{7}$ and an angle of
$\varepsilon_{7}=\arctan{({B_{z}\over B_{y}})}$ and (after an update of the
RDM-coefficients and a recomputation of the auxiliary vector and mass
components) with ${\bf R}_{9}$ about an angle of
$\varepsilon_{8}=-\arctan{({B_{x}\over B_{y}})}$.
* •
$E_{x}\to 0$: Rotate about the $y$-axis with ${\bf R}_{8}$ with an angle of
$\varepsilon_{9}=\arctan{({E_{x}\over E_{z}})}$.
#### V.2.2 The Intermediate Energy Case
The case where ${\cal E}^{2}>\mathrm{Min}(\vec{P}^{2},\vec{E}^{2})$ but
$K_{2}<0$ might be called “intermediate”, since the energy is large compared
to the “low energy” case, but not large enough to make $K_{2}>0$. The
following procedure leads to the state described by Eqn. 45:
* •
$M_{b}\to 0$: Apply a phase rotation ${\bf R}_{0}$ with angle
$\varepsilon_{1}={1\over
2}\,\arctan{({2\,M_{b}\over\vec{E}^{2}-\vec{P}^{2}})}$. Note that this
transformation minimizes $\vec{P}^{2}$.
* •
$\vec{P}\to|\vec{P}|\,\vec{e}_{y}$: Align the vector $\vec{P}$ along the
$y$-axis by the spatial rotations with ${\bf R}_{7}$ and an angle of
$\varepsilon_{2}=\arctan{({P_{z}\over P_{y}})}$ and (after an update of the
RDM-coefficients and a recomputation of the auxiliary vector and mass
components) with ${\bf R}_{9}$ about an angle of
$\varepsilon_{3}=-\arctan{({P_{x}\over P_{y}})}$. Since
$M_{b}=\vec{E}\cdot\vec{P}=0$ one also has now $E_{y}=0$.
* •
$P_{y}\to 0$: Lorentz boost using ${\bf R}_{5}$ and rapidity
$\varepsilon_{4}=\mathrm{arctanh}{({P_{y}\over{\cal E}})}$.
* •
$\vec{B}\to|\vec{B}|\,\vec{e}_{y}$: Align the vector $\vec{B}$ along the
$y$-axis by the spatial rotations with ${\bf R}_{7}$ and an angle of
$\varepsilon_{6}=\arctan{({B_{z}\over B_{y}})}$ and (after an update of the
RDM-coefficients and a recomputation of the auxiliary vector and mass
components) with ${\bf R}_{9}$ about an angle of
$\varepsilon_{7}=-\arctan{({B_{x}\over B_{y}})}$.
* •
${\cal E}\to 0$: Boost using ${\bf R}_{2}$ and rapidity
$\varepsilon_{8}=\mathrm{arctanh}{({{\cal E}\over E_{y}})}$.
* •
$E_{x}\to 0$: Rotate about the $y$-axis with ${\bf R}_{8}$ with an angle of
$\varepsilon_{9}=\arctan{({E_{x}\over E_{z}})}$.
In both cases the transformed matrix ${\bf F}$ then has the form
${\bf F}=E_{y}\,\gamma_{5}+E_{z}\,\gamma_{6}+B_{y}\,\gamma_{8}\,.$ (46)
In order to bring it to Hamiltonian form, one applies the transformation ${\bf
R}_{2}$ with an “angle” of $i\,\pi/2$:
${\bf R}=\exp{(i\,{\pi/4}\,\gamma_{2})}={1\over\sqrt{2}}\,({\bf
1}+i\,\gamma_{2})\,.$ (47)
so that ${\bf F}\to{\bf R}\,{\bf F}\,{\bf R}^{-1}$ is:
${\bf F}=-i\,E_{y}\,\gamma_{0}+E_{z}\,\gamma_{6}+B_{y}\,\gamma_{8}\,.$ (48)
Note that the complex eigenvalues of a force matrix with $K_{2}<0$ all lie on
a circle of radius $\rho=(K_{1}^{2}+4\,|K_{2}|)^{1/4}$ in the complex plane.
### V.3 Decoupling n-dimensional Symplices
The general eigenvalue problem of symplices (Hamiltonian matrices) is an area
of intense research. The algorithm presented above is based on a physical and
geometrical analysis of 2-dimensional linear symplectic systems. As described
before, the algorithm is limited to symplices that have real or imaginary
eigenvalues, but a generalization to include complex eigenvalues might be
possible - even though not urgently required in charged particle optics 555The
${\bf S}$-matrix for instance will never have complex eigenvalues as it is
derived from the matrix of second moments. Complex eigenvalues would only be
possible for correlations with a modulus greater than one..
In order to decouple symplectic systems with more than two degrees of freedom,
the described algorithm can be used in an iterative scheme analogous to the
Jacobi method for symmetric matrices666 Jacobi introduced a method to
iteratively diagonalize real symmetric matrices by a sequence of orthogonal
transformations each of which diagonalizes a $2\times 2$ submatrix Jacobi ..
If all eigenvalues are real or imaginary, it is possible to avoid computations
using complex numbers. The $2\,n\,\times 2\,n$-symplex is then regarded as a
$n\times n$ matrix of $2\times 2$-blocks. We tested a pivot search that picks
the maximum average square amplitude of all non-diagonal blocks ${\bf
B}_{ij}$. The blocks ${\bf B}_{ii}$, ${\bf B}_{ij}$, ${\bf B}_{ji}$ and ${\bf
B}_{jj}$ are then analyzed as $4\times 4$-symplices and the symplectic
similarity transformation that block-diagonalizes this submatrix is applied,
so that ${\bf\tilde{B}}_{ij}={\bf\tilde{B}}_{ji}=0$ holds. This iterative
scheme allows to compute simultaneously the symplectic transformation matrix
and the resulting block-diagonal or Hamiltonian form symplex with high
precision.
Given $\\{x\\}$ is a sequence of random numbers between zero and one, then one
may construct random symmetric $2\,n\times 2\,n$-matrices ${\bf A}$ according
to the rule:
${\bf A}_{ij}={\bf A}_{ji}=\left\\{\begin{array}[]{rcl}x-{1\over
2}&\mathrm{for}&i\neq j\\\ n+x&\mathrm{for}&i=j\\\ \end{array}\right.$ (49)
The increase of the diagonal terms helps to avoid complex eigenvalues. The
symplex to decouple is then given by ${\bf F}=\gamma_{0}\,{\bf A}$. We tested
the algorithm with these random matrices up to $n=12$ and logged the number of
$4\times 4$-diagonalization steps.
Figure 1: Solid line: Number of iterations required to bring a $2\,n\times
2\,n$ symplex (Hamiltonian matrix) to Hamiltonian form. Dashed line:
Approximation by $5\,{n\,(n-2)\over 2}$. The number $n_{b}$ of non-diagonal
$2\times 2$-blocks is $n_{b}={n\,(n-1)\over 2}$.
Fig. 1 shows the average number of iterations that is required to compute the
transformation that brings a $2\,n\times 2\,n$ symplex to Hamiltonian form,
i.e. into the form:
$\left(\begin{array}[]{ccccc}0&\beta_{1}&\dots&0&0\\\
-\gamma_{1}&0&\dots&0&0\\\ \vdots&\vdots&&\vdots&\vdots\\\
0&0&\dots&0&\beta_{n}\\\ 0&0&\dots&-\gamma_{n}&0\\\ \end{array}\right)$ (50)
### V.4 Diagonalization
In order to proceed from Eqn. 38 towards diagonalization, the matrix is scaled
using the generators $\gamma_{3}$ and $\gamma_{4}$:
$\begin{array}[]{rcl}{\bf R}&=&\exp{[(\gamma_{3}+\gamma_{4}){s\over
2}+(\gamma_{3}-\gamma_{4}){t\over 2}]}\\\
&=&\mathrm{Diag}(\exp{(-s)},\exp{(s)},\exp{(-t)},\exp{(t)})\\\
s&=&\exp{(\log{(|{\alpha\over\beta}|)}/4)}\\\
t&=&\exp{(\log{(|{\gamma\over\delta}|)}/4)}\\\ \end{array}$ (51)
so that one obtains (for stable systems), what we call “normal” form:
$\begin{array}[]{rcl}{\bf F}&\to&{\bf R}\,{\bf F}\,{\bf R}^{-1}\\\
&=&\left(\begin{array}[]{cccc}0&\omega_{1}&0&0\\\ -\omega_{1}&0&0&0\\\
0&0&0&\omega_{2}\\\ 0&0&-\omega_{2}&0\\\ \end{array}\right)\,,\end{array}$
(52)
where the signs of the frequencies $\omega_{1}$ and $\omega_{2}$ can both be
positive and negative, depending on the signs of $\alpha$, $\beta$, $\gamma$
and $\delta$. At this stage, all components of $\vec{g}$ and $\vec{r}$ as well
as $(\vec{b})_{x}$ and $(\vec{b})_{z}$ are zero. Only $(\vec{b})_{y}$ is non-
zero.
$\begin{array}[]{rcl}{\bf F}&=&{\bf
E}_{0}\,\mathrm{Diag}(i\,\omega_{1},-i\,\omega_{1},i\,\omega_{2},-i\,\omega_{2})\,{\bf
E}_{0}^{-1}\\\ {\bf E}_{0}&=&{1\over
2}\,\left(\begin{array}[]{cccc}1-i&-1+i&0&0\\\ 1+i&1+i&0&0\\\ 0&0&1-i&-1+i\\\
0&0&1+i&1+i\\\ \end{array}\right)\\\ &=&{1\over 2}\,\left({\bf
1}-\gamma_{0}+i\,\gamma_{3}+i\,\gamma_{6}\right)\\\ {\bf
E}_{0}\,\gamma_{0}\,{\bf E}_{0}^{T}&=&\gamma_{0}\\\ {\bf E}_{0}^{-1}&=&{\bf
E}_{0}^{\dagger}\\\ \end{array}$ (53)
That is - the last transformation matrix that is required for diagonalization
is not only symplectic - it is also unitary.
### V.5 Example
A simplified and idealized cyclotron model with space charge was described,
which served as an example for an irregular system cyc_paper ; rdm_paper .
Without repeating all details, the constant force matrix has the following
form:
${\bf F}=\left(\begin{array}[]{cccc}0&1&0&0\\\ -k_{x}+K_{x}&0&0&h\\\
-h&0&0&{1\over\gamma^{2}}\\\ 0&0&K_{z}\,\gamma^{2}&0\\\ \end{array}\right)\,,$
(54)
The RDM-coefficients are then given by:
$\begin{array}[]{rcl}{\cal E}&=&{1\over
4}\,\left(1+k_{x}-K_{x}+{1\over\gamma^{2}}-\gamma^{2}\,K_{z}\right)\\\
P_{x}&=&{1\over
4}\,\left(-1+k_{x}-K_{x}+{1\over\gamma^{2}}+\gamma^{2}\,K_{z}\right)\\\
P_{y}&=&P_{z}=0\\\ E_{x}&=&B_{x}=0\\\ E_{y}&=&B_{z}=-{h\over 2}\\\
E_{z}&=&{1\over
4}\,\left(1-k_{x}+K_{x}+{1\over\gamma^{2}}+\gamma^{2}\,K_{z}\right)\\\
B_{y}&=&{1\over
4}\left(1+k_{x}-K_{x}-{1\over\gamma^{2}}+\gamma^{2}\,K_{z}\right)\\\
\end{array}$ (55)
From this one finds for the “mass” terms and the vectors $\vec{g}$, $\vec{r}$
and $\vec{b}$:
$\begin{array}[]{rcl}M_{r}&=&\vec{E}\cdot\vec{B}=-{h\over
4}\,(1+K_{z}\,\gamma^{2})\\\ M_{g}&=&\vec{P}\cdot\vec{B}=0\\\
M_{b}&=&\vec{E}\cdot\vec{P}=0\\\ \vec{g}&=&(0,{h\over
4}\,(K_{z}\,\gamma^{2}-1),{1+\gamma^{2}\,(k_{x}-K_{x})\,K_{z}\over
4\,\gamma^{2}})^{T}\\\ \vec{r}&=&({k_{x}-K_{x}+\gamma^{2}\,(K_{z}-h^{2})\over
4\,\gamma^{2}},0,0)^{T}\\\ \vec{b}&=&(0,{K_{z}+k_{x}-K_{x}\over
4},{h\,(\gamma^{2}\,K_{z}-1)\over 4})^{T}\\\ \end{array}$ (56)
According to the geometrical approach, the first transformation can be
omitted, since the “mass” $M_{g}$ is zero. The second transformation using
$\gamma_{7}$ aligns $\vec{b}$ along the $y$-axis. The second rotation may
again be omitted, since the vector $\vec{r}$ is already aligned along the
$x$-axis. The last transformation is a phase boost using $\gamma_{2}$ and is
sufficient to bring ${\bf F}$ into block-diagonal form. This transformation
would usually change the value of $M_{b}$, but here it does not, since
$M_{b}=(\vec{r})_{y}=0$ as can be seen from Tab. 1. Hence $M_{b}$ remains zero
- $M_{g}$ is invariant under both transformations. Hence, all “mass terms” are
then zero after the described two transformations so that the system is
decoupled.
### V.6 Operators, Expectation Values and Lax Pairs
Coupled linear optics is in its essence (as quantum mechanics) a statistical
theory. Since the reference trajectory is fixed, the coordinates are always
taken relative to the local reference frame and the geometry is (only) locally
euclidean. Even though the starting point is the description of single
particle motion, the orbits of single particles are usually both, hard to
access experimentally and of low practical value. The description of the beam
by average values in contrast is both - measureable and of high value. The use
of symplectic transformations leaves the expectation values unchanged. We can
therefore evaluate the expectation values of any operator ${\bf O}$ in an
arbitrary reference frame:
$\begin{array}[]{rcl}\langle{\bf O}\rangle&\equiv&\langle\bar{\psi}\,{\bf
O}\,\psi\,\rangle\\\ &=&\langle\psi^{T}\,\gamma_{0}\,{\bf R}^{-1}\,{\bf
R}\,{\bf O}\,{\bf R}^{-1}\,{\bf R}\,\psi\,\rangle\\\
&=&\langle\psi^{T}\,\gamma_{0}\,{\bf
R}^{-1}\,{\bf\tilde{O}}\,\tilde{\psi}\,\rangle\\\ &=&\langle\psi^{T}\,{\bf
R}^{T}\,\gamma_{0}\,{\bf\tilde{O}}\,\tilde{\psi}\,\rangle\\\
&=&\langle\tilde{\bar{\psi}}\,{\bf\tilde{O}}\,\tilde{\psi}\,\rangle\,,\end{array}$
(57)
since for symplectic ${\bf R}$ we have
$\begin{array}[]{rcl}{\bf R}^{T}\,\gamma_{0}&=&\gamma_{0}\,{\bf R}^{-1}\\\
{\bf R}^{T}\,\gamma_{0}\,{\bf R}&=&\gamma_{0}\\\ \end{array}$ (58)
The time derivative of the expectation value of an arbitrary operator ${\bf
O}$, that does not explicitely depend on time, is:
$\begin{array}[]{rcl}{d\over d\tau}\left(\bar{\psi}\,{\bf
O}\,\psi\right)&=&\dot{\bar{\psi}}\,{\bf O}\,\psi+\bar{\psi}\,{\bf
O}\,\dot{\psi}\\\ &=&\psi^{T}\,{\bf F}^{T}\,\gamma_{0}\,{\bf
O}\,\psi+\bar{\psi}\,{\bf O}\,{\bf F}\,\psi\\\ &=&\bar{\psi}\,({\bf O}\,{\bf
F}-{\bf F}\,{\bf O})\,\psi\\\ \end{array}$ (59)
Equations of the form (here ${\bf S}=\sigma\,\gamma_{0}$)
${\bf\dot{S}}={\bf F}\,{\bf S}-{\bf S}\,{\bf F}\,,$ (60)
appear frequently in the theory of coupled linear optics and it is worth
mentioning that Eq. 60 is a so-called Lax representation and the operators
${\bf S}$ and ${\bf F}$ are a so-called Lax pair Lax ; Lax2 . As a
consequence, the expressions
$I_{k}=Tr({\bf S}^{k})$ (61)
are first integrals of motion, where $Tr()$ is the trace. Using again the EMEQ
to express the elements of ${\bf S}$, one finds:
$\begin{array}[]{rcl}I_{1}&=&Tr({\bf S})=0\\\ I_{2}&=&Tr({\bf
S}^{2})=-4\,\left({\cal
E}^{2}-\vec{P}^{2}+\vec{B}^{2}-\vec{E}^{2}\right)=-4\,K_{1}\\\ I_{3}&=&Tr({\bf
S}^{3})=0\\\ I_{4}&=&Tr({\bf S}^{4})=4\,(K_{1}^{2}+4\,K_{2})\\\ \end{array}$
(62)
The values of $K_{1}$ and $K_{2}$ are (as expected) first integrals and
constants of motion. The complete expression for ${\bf S}^{4}$ is
${\bf S}^{4}=(K_{1}^{2}+4\,K_{2})\,{\bf
1}-4\,K_{1}\,\left(M_{g}\,\gamma_{10}+M_{r}\,\gamma_{14}+\vec{b}\,\gamma_{14}\,\vec{\gamma}\right)\,.$
(63)
Another derivation of Eqn. 61 has been given in DNR .
## VI Summary and Outlook
A powerful method for symplectic decoupling of the n-dimensional non-
dissipative harmonic oscillator has been developed. The method apparently is
stable, of the order ${\cal O}(n^{2})$ and works with purely real or purely
imaginary eigenvalues, for which a Hamiltonian Schur form does not always
exists CM . The resulting block-diagonal symplex can be used to compute the
$\sigma$-matrix of matched beam ellipsoids of linear coupled systems in
charged particle optics Wolski ; rdm_paper ; cyc_paper . Another application
is the production of multivariate gaussian distributions for a given
covariance matrix stat_paper .
The presented parametrization gives deep insight into the general nature of
coupling and might be instructive also in other areas of physics. The
algebraic problem of finding the eigenvalues and eigenvectors of a two-
dimensional symplectic system was solved using geometrical arguments based on
the use of the real Dirac matrices and the electromechanical equivalence.
###### Acknowledgements.
We would like to mention the work of D. Hestenes, who emphasised the
geometrical significance of the Dirac algebra that he called space-time
algebra Hestenes . The idea to introduce the EMEQ is inspired by his work.
Mathematica® has been used for some of the symbolic calculations. Additional
software has been written in “C” and been compiled with the GNU©-C++ compiler
3.4.6 on Scientific Linux. The CERN library (PAW) was used to generate the
figure.
## Appendix A The $\gamma$-Matrices
To complete the list of the real $\gamma$-matrices used throughout this paper:
$\begin{array}[]{rclp{4mm}rcl}\gamma_{0}&=&\left(\begin{array}[]{cccc}0&1&0&0\\\
-1&0&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\\\
\end{array}\right)&&\gamma_{1}&=&\left(\begin{array}[]{cccc}0&-1&0&0\\\
-1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\ \end{array}\right)\\\
\gamma_{2}&=&\left(\begin{array}[]{cccc}0&0&0&1\\\ 0&0&1&0\\\ 0&1&0&0\\\
1&0&0&0\\\
\end{array}\right)&&\gamma_{3}&=&\left(\begin{array}[]{cccc}-1&0&0&0\\\
0&1&0&0\\\ 0&0&-1&0\\\ 0&0&0&1\\\ \end{array}\right)\\\
\gamma_{14}&=&\gamma_{0}\,\gamma_{1}\,\gamma_{2}\,\gamma_{3};&&\gamma_{15}&=&{\bf
1}\\\
\gamma_{4}&=&\gamma_{0}\,\gamma_{1};&&\gamma_{7}&=&\gamma_{14}\,\gamma_{0}\,\gamma_{1}=\gamma_{2}\,\gamma_{3}\\\
\gamma_{5}&=&\gamma_{0}\,\gamma_{2};&&\gamma_{8}&=&\gamma_{14}\,\gamma_{0}\,\gamma_{2}=\gamma_{3}\,\gamma_{1}\\\
\gamma_{6}&=&\gamma_{0}\,\gamma_{3};&&\gamma_{9}&=&\gamma_{14}\,\gamma_{0}\,\gamma_{3}=\gamma_{1}\,\gamma_{2}\\\
\gamma_{10}&=&\gamma_{14}\,\gamma_{0}&=&\gamma_{1}\,\gamma_{2}\,\gamma_{3}&&\\\
\gamma_{11}&=&\gamma_{14}\,\gamma_{1}&=&\gamma_{0}\,\gamma_{2}\,\gamma_{3}&&\\\
\gamma_{12}&=&\gamma_{14}\,\gamma_{2}&=&\gamma_{0}\,\gamma_{3}\,\gamma_{1}&&\\\
\gamma_{13}&=&\gamma_{14}\,\gamma_{3}&=&\gamma_{0}\,\gamma_{1}\,\gamma_{2}&&\\\
\end{array}$ (64)
## Appendix B Floquet Theorem
If the matrix ${\bf A}$ in Eqn. (1) and hence the forces are not constant, but
periodic (${\bf F}(t+T)={\bf F}(t)$), then Floquet’s theorem can be applied
and the solution has the general form Talman ; MHO :
${\bf M}(t)={\bf K}(t)\,\exp{({\bf\bar{F}}\,t)}\,,$ (65)
where ${\bf K}(t)$ is symplectic and periodic with period $T$.
$\begin{array}[]{rcl}{\bf M}(0)&=&{\bf 1}\,\,\Rightarrow\,\,{\bf K}(0)={\bf
1}\\\ {\bf K}(t+T)&=&{\bf K}(t)\,\,\Rightarrow\,\,{\bf K}(T)={\bf 1}\\\
\end{array}$ (66)
The transfer matrix of one period of length $T$ (“one-turn-transfer-matrix”)
${\bf M}(T)={\bf M}_{T}=\exp{({\bf\bar{F}}\,T)}$ (67)
is identical to the transfer matrix for a system with the constant force
matrix ${\bf\bar{F}}$ and length $T$. In this sense ${\bf\bar{F}}$ is the
“average” or “effective” force matrix with respect to one turn and can
formally be written as Talman :
${\bf\bar{F}}={1\over T}\,\ln{({\bf M}_{T})}\,.$ (68)
From Eqs. 65 one derives in a few steps Talman ; Leach :
${\bf\dot{K}}={\bf F}\,{\bf K}-{\bf K}\,{\bf\bar{F}}\,.$ (69)
If the canonical transformation represented by ${\bf K}$ has been applied to
the state vector, then with ${\bf K}(0)={\bf 1}$ it follows:
$\tilde{\psi}(t)={\bf
K}^{-1}\,\psi(t)=\exp{({\bf\bar{F}}\,t)}\,\tilde{\psi}(0)={\bf\tilde{M}}\,\tilde{\psi}(0)\,.$
(70)
Note that the knowledge of ${\bf K}$ is not required to solve the decoupling
problem, as long as the one-turn-transfer matrix ${\bf M}_{T}$ is known. ${\bf
M}_{T}$ can either be obtained as a product of the transfer matrices of all
beamline elements or simply by numerical integration. If the matched beam
distribution has been found at an arbitrary (known) position $s=0$ along the
closed reference orbit, then the matched distribution can be computed for any
position $s$ using:
$\sigma(s)={\bf M}(s)\,\sigma(0)\,{\bf M}^{T}(s)\,.$ (71)
## Appendix C Quick Guide to Decoupling
To start with it is required to have either the average or constant force
matrix ${\bf F}$ or the symplectic transfer matrix ${\bf M}$ that represents a
complete turn or (cyclotron) sector. In the latter case one computes an
auxiliary force matrix by
${\bf M}_{s}={1\over 2}\,({\bf M}+\gamma_{0}\,{\bf M}^{T}\,\gamma_{0})\,,$
(72)
while the usual (effective) force matrix has the form
${\bf F}={\bf
E}\,\mathrm{Diag}(i\,\omega_{1},-i\,\omega_{1},i\,\omega_{2},-i\,\omega_{2})\,{\bf
E}^{-1}\,,$ (73)
$\omega_{i}$ being the betatron frequencies. The auxiliary matrix has the same
structure
${\bf M}_{s}={\bf
E}\,\mathrm{Diag}(i\,s_{1},-i\,s_{1},i\,s_{2},-i\,s_{2})\,{\bf E}^{-1}\,,$
(74)
but different eigenvalues $s_{i}=\sin{(\omega_{i}\,\tau)}$, where
$\omega_{i}\,\tau=2\pi\,Q_{i}$ with the betatron tunes $Q_{i}$.
Now compute the RDM-coefficients according to:
$\begin{array}[]{rcl}{\cal E}&=&-Tr({\bf F}\,\gamma_{0}+\gamma_{0}\,{\bf
F})/8\\\ P_{x}&=&Tr({\bf F}\,\gamma_{1}+\gamma_{1}\,{\bf F})/8\\\
P_{y}&=&Tr({\bf F}\,\gamma_{2}+\gamma_{2}\,{\bf F})/8\\\ P_{z}&=&Tr({\bf
F}\,\gamma_{3}+\gamma_{3}\,{\bf F})/8\\\ E_{x}&=&Tr({\bf
F}\,\gamma_{4}+\gamma_{4}\,{\bf F})/8\\\ E_{y}&=&Tr({\bf
F}\,\gamma_{5}+\gamma_{5}\,{\bf F})/8\\\ E_{z}&=&Tr({\bf
F}\,\gamma_{6}+\gamma_{6}\,{\bf F})/8\\\ B_{x}&=&-Tr({\bf
F}\,\gamma_{7}+\gamma_{7}\,{\bf F})/8\\\ B_{y}&=&-Tr({\bf
F}\,\gamma_{8}+\gamma_{8}\,{\bf F})/8\\\ B_{z}&=&-Tr({\bf
F}\,\gamma_{9}+\gamma_{9}\,{\bf F})/8\\\ \end{array}$ (75)
Note that the coefficients for $\gamma_{k}$ with $k\in[10,\dots,15]$ must be
zero \- otherwise the system is not symplectic. Then compute the eigenvalues
and auxiliary vectors $\vec{r},\vec{g},\vec{b}$ according to Eq.
LABEL:eq_eigenfreq, LABEL:eq_aux_masses and 32. Construct the transformation
matrices ${\bf R}_{b}$ according to:
$\begin{array}[]{rcl}{\bf R}_{b}&=&\left\\{\begin{array}[]{lcr}{\bf
1}\,\cos{(\varepsilon/2)}+\gamma_{b}\,\sin{(\varepsilon/2)}&\mathrm{for}&b\in[0,7,8,9]\\\
{\bf
1}\,\cosh{(\varepsilon/2)}+\gamma_{b}\,\sinh{(\varepsilon/2)}&\mathrm{for}&b\in[1,\dots,6]\\\
\end{array}\right.\\\ {\bf R}_{b}^{-1}&=&\left\\{\begin{array}[]{lcr}{\bf
1}\,\cos{(\varepsilon/2)}-\gamma_{b}\,\sin{(\varepsilon/2)}&\mathrm{for}&b\in[0,7,8,9]\\\
{\bf
1}\,\cosh{(\varepsilon/2)}-\gamma_{b}\,\sinh{(\varepsilon/2)}&\mathrm{for}&b\in[1,\dots,6]\\\
\end{array}\right.\end{array}$ (76)
Transform with $\gamma_{0}$ and $\varepsilon=\arctan{\left({M_{g}\over
M_{r}}\right)}$:
$\begin{array}[]{rcl}{\bf F}\to{\bf R}_{0}\,{\bf F}\,{\bf
R}_{0}^{-1}\,.\end{array}$ (77)
Recompute RDM-coefficients, then transform using $\gamma_{7}$ with
$\varepsilon=\arctan{\left({b_{z}\over b_{y}}\right)}$. Recompute RDM-
coefficients, then transform using $\gamma_{9}$ with
$\varepsilon=\arctan{\left({b_{x}\over b_{y}}\right)}$. Recompute RDM-
coefficients, then transform using $\gamma_{2}$ with
$\varepsilon=\mathrm{arctanh}{\left({M_{r}\over b_{y}}\right)}$. The
(auxiliary) force matrix should now be block-diagonal. Recompute RDM-
coefficients, then transform with $\gamma_{0}$ and
$\varepsilon=\arctan{\left({2\,M_{b}\over\vec{E}^{2}-\vec{P}^{2}}\right)}$.
Recompute RDM-coefficients, then transform with $\gamma_{8}$ and
$\varepsilon=-\arctan{\left({P_{z}\over P_{x}}\right)}$. Now the (auxiliary)
force matrix should have normal form, so that the frequencies (or their sines)
are given by:
$\begin{array}[]{rcl}\omega_{1}&=&\sqrt{-F_{1,2}\,F_{2,1}}\\\
\omega_{2}&=&\sqrt{-F_{3,4}\,F_{4,3}}\\\ \end{array}$ (78)
The complete transformation is given by:
$\begin{array}[]{rcl}{\bf R}^{-1}&=&{\bf R}_{0}^{-1}\cdot{\bf
R}_{1}^{-1}\dots{\bf R}_{n}^{-1}\\\ {\bf R}&=&{\bf R}_{n}\cdot{\bf
R}_{n-1}\dots{\bf R}_{0}\\\ {\bf F}_{d}&=&{\bf R}\,{\bf F}\,{\bf R}^{-1}\\\
\end{array}$ (79)
If the auxiliary matrix has been used, then compute the matrix
${\bf\tilde{M}}_{c}$ according to
${\bf\tilde{M}}_{c}={1\over 2}\,{\bf R}\,({\bf M}-\gamma_{0}\,{\bf
M}^{T}\,\gamma_{0})\,{\bf R}^{-1}\,.$ (80)
The cosines of the tunes are then given by:
$\begin{array}[]{rcl}\cos{(\omega_{1}\,\tau)}+\cos{(\omega_{2}\,\tau)}&=&Tr({\bf\tilde{M}})/2\\\
\cos{(\omega_{1}\,\tau)}-\cos{(\omega_{2}\,\tau)}&=&Tr({\bf\tilde{M}}\,\gamma_{12}+\gamma_{12}\,{\bf\tilde{M}})/4\\\
\end{array}$ (81)
## Appendix D The Teng and Edwards Ansatz
Assume that we have an even number of DOF, so that a $4\,n\times 4\,n$
symplectic matrix ${\bf R}$ can be written in block-form according to Teng ;
EdwardsTeng :
${\bf R}=\left(\begin{array}[]{cc}{\bf A}&{\bf a}\\\ {\bf b}&{\bf B}\\\
\end{array}\right)$ (82)
where all quadratic submatrices are of size $2\,n\times 2\,n$, then the matrix
${\bf R}$ is symplectic, if
$\begin{array}[]{rcl}\gamma_{0}&=&\left(\begin{array}[]{cc}{\bf A}&{\bf a}\\\
{\bf b}&{\bf B}\\\
\end{array}\right)\,\gamma_{0}\,\left(\begin{array}[]{cc}{\bf A}^{T}&{\bf
b}^{T}\\\ {\bf a}^{T}&{\bf B}^{T}\\\ \end{array}\right)\\\
&=&\left(\begin{array}[]{cc}{\bf A}\,\gamma_{0}\,{\bf A}^{T}+{\bf
a}\,\gamma_{0}\,{\bf a}^{T}&{\bf A}\,\gamma_{0}\,{\bf b}^{T}+{\bf
a}\,\gamma_{0}\,{\bf B}^{T}\\\ {\bf b}\,\gamma_{0}\,{\bf A}^{T}+{\bf
B}\,\gamma_{0}\,{\bf a}^{T}&{\bf b}\,\gamma_{0}\,{\bf b}^{T}+{\bf
B}\,\gamma_{0}\,{\bf B}^{T}\end{array}\right)\\\ \end{array}$ (83)
which yields:
$\begin{array}[]{rcl}\gamma_{0}&=&{\bf A}\,\gamma_{0}\,{\bf A}^{T}+{\bf
a}\,\gamma_{0}\,{\bf a}^{T}\\\ \gamma_{0}&=&{\bf b}\,\gamma_{0}\,{\bf
b}^{T}+{\bf B}\,\gamma_{0}\,{\bf B}^{T}\\\ 0&=&{\bf A}\,\gamma_{0}\,{\bf
b}^{T}+{\bf a}\,\gamma_{0}\,{\bf B}^{T}\,,\end{array}$ (84)
where $\gamma_{0}$ has - in dependence of the context - to be taken as
$2\,n\times 2\,n$ or $4\,n\times 4\,n$.
If one now assumes that ${\bf A}={\bf B}=C\,{\bf 1}$, then it follows that
$\begin{array}[]{rcl}\gamma_{0}\,(1-C^{2})&=&{\bf a}\,\gamma_{0}\,{\bf
a}^{T}\\\ \gamma_{0}\,(1-C^{2})&=&{\bf b}\,\gamma_{0}\,{\bf b}^{T}\\\ {\bf
b}&=&\gamma_{0}\,{\bf a}^{T}\,\gamma_{0}\,.\end{array}$ (85)
If one assumes furthermore with Teng and Edwards, that $C=\cos{(\phi)}$, then
may define ${\bf a}=\sin{(\phi)}\,{\bf a}_{s}$ and ${\bf b}=\sin{(\phi)}\,{\bf
b}_{s}$ with symplectic matrizes ${\bf a}_{s}$ and ${\bf b}_{s}$,
respectively:
$\begin{array}[]{rcl}\gamma_{0}&=&{\bf a}_{s}\,\gamma_{0}\,{\bf a}_{s}^{T}\\\
\gamma_{0}&=&{\bf b}_{s}\,\gamma_{0}\,{\bf b}_{s}^{T}\\\ \end{array}$ (86)
It has been shown in Ref. cyc_paper , that $C=\cos{(\phi)}$ is not the general
case, since one might also choose $C=\cosh{(\phi)}$, ${\bf
a}=\sinh{(\phi)}\,{\bf a}_{s}$ and ${\bf b}=\sinh{(\phi)}\,{\bf b}_{s}$. In
this case one finds
$\begin{array}[]{rcl}-\gamma_{0}&=&{\bf a}_{s}\,\gamma_{0}\,{\bf a}_{s}^{T}\\\
-\gamma_{0}&=&{\bf b}_{s}\,\gamma_{0}\,{\bf b}_{s}^{T}\,,\end{array}$ (87)
i.e. the matrizes ${\bf a}_{s}$ and ${\bf b}_{s}$ can also be antisymplectic
(symplectic with multiplier $-1$). Still the matrix ${\bf R}$ remains
symplectic. Hence Teng and Edwards limited their treatment in two ways: First,
they assumed that $C=\cos{(\psi)}$ such that ${\bf R}$ must be a rotation
matrix and secondly, they considered only the case that ${\bf a}$ and ${\bf
b}$ are symplectic.
## Appendix E Cosymplices
The geometric approach is based on the second order terms, i.e. products of
the RDM coefficients. It is therefore instructive to see where else these
terms appear. For instance one quickly finds the “mass” terms and vectors
$\vec{g}$, $\vec{r}$, $\vec{b}$ in the following products:
$\begin{array}[]{rcl}{\bf F}\,{\bf F}&=&-({\cal
E}^{2}-\vec{P}^{2}+\vec{B}^{2}-\vec{E}^{2})\,{\bf 1}+2\,M_{r}\,\gamma_{14}\\\
&+&2\,M_{g}\,\gamma_{10}+2\,\vec{b}\,\gamma_{14}\,\vec{\gamma}\\\ {\bf
F}\,\gamma_{0}\,{\bf F}&=&(3\,{\cal
E}^{2}-\vec{P}^{2}-\vec{E}^{2}-\vec{B}^{2})\,\gamma_{0}-4\,{\cal E}\,{\bf
F}\\\
&+&2\,\vec{r}\,\vec{\gamma}+2\,\vec{g}\,\gamma_{0}\,\vec{\gamma}+2\,\vec{b}\,\gamma_{14}\,\gamma_{0}\,\vec{\gamma}\\\
{\bf F}\,\gamma_{14}\,{\bf F}&=&2\,M_{b}\,\gamma_{10}-2\,M_{r}\,{\bf 1}\\\
&+&({\cal
E}^{2}-\vec{P}^{2}+\vec{E}^{2}-\vec{B}^{2})\,\gamma_{14}+2\,\vec{g}\,\gamma_{14}\vec{\gamma}\\\
{\bf F}\,\gamma_{10}\,{\bf F}&=&2\,M_{b}\,\gamma_{14}-2\,M_{g}\,{\bf 1}\\\
&+&({\cal
E}^{2}+\vec{P}^{2}-\vec{E}^{2}-\vec{B}^{2})\,\gamma_{10}+2\,\vec{r}\,\gamma_{14}\vec{\gamma}\\\
\end{array}$ (88)
So that in the decoupled and normalized case (see Eq. 52), these products are:
$\begin{array}[]{rcl}{\bf F}\,{\bf F}&=&-({\cal
E}^{2}-\vec{P}^{2}+\vec{B}^{2}-\vec{E}^{2})\,{\bf
1}+2\,(\vec{b})_{y}\,\gamma_{12}\\\ {\bf F}\,\gamma_{0}\,{\bf F}&=&(3\,{\cal
E}^{2}-\vec{P}^{2}-\vec{E}^{2}-\vec{B}^{2})\,\gamma_{0}-4\,{\cal E}\,{\bf
F}\\\ &+&2\,(\vec{b})_{y}\,\gamma_{8}\\\ {\bf F}\,\gamma_{14}\,{\bf
F}&=&({\cal E}^{2}-\vec{P}^{2}+\vec{E}^{2}-\vec{B}^{2})\,\gamma_{14}\\\ {\bf
F}\,\gamma_{10}\,{\bf F}&=&({\cal
E}^{2}+\vec{P}^{2}-\vec{E}^{2}-\vec{B}^{2})\,\gamma_{10}\\\ \end{array}$ (89)
## Appendix F Expectation Values (Complement)
In Ref. rdm_paper it has been shown that the expectation values of the RDMs,
$f_{k}$, defined by
$f_{k}={1\over 2}\,\bar{\psi}\,\gamma_{k}\,\psi\,,$ (90)
vanish for all cosymplices, i.e. for $\gamma_{k}$ with $k\in[10,\dots,15]$. It
was also shown that for all symplices (i.e. $\gamma_{k}$ with
$k\in[0,\dots,9]$ or linear combinations thereof) the expectation values
$g_{k}\equiv\bar{\psi}(\gamma_{k}\,{\bf F}+{\bf F}\,\gamma_{k})\psi$ vanish.
Nevertheless nothing was mentioned about the $g_{k}$ for $k\in[10,\dots,15]$.
The complement is given in the following:
$\begin{array}[]{rcl}g_{10}&=&2\,\left(P_{x}\,f_{7}+P_{y}\,f_{8}+P_{z}\,f_{9}-B_{x}\,f_{1}-B_{y}\,f_{2}+B_{z}\,f_{3}\right)\\\
g_{11}&=&2\,\left(-{\cal
E}\,f_{7}+B_{x}\,f_{0}+P_{z}\,f_{5}+E_{y}\,f_{3}-P_{y}\,f_{6}-E_{z}\,f_{2}\right)\\\
g_{12}&=&2\,\left(-{\cal
E}\,f_{8}+B_{y}\,f_{0}+P_{x}\,f_{6}+E_{z}\,f_{1}-P_{z}\,f_{4}-E_{x}\,f_{3}\right)\\\
g_{13}&=&2\,\left(-{\cal
E}\,f_{9}+B_{z}\,f_{0}+P_{y}\,f_{4}+E_{x}\,f_{2}-P_{x}\,f_{5}-E_{y}\,f_{1}\right)\\\
g_{14}&=&2\,\left(E_{x}\,f_{7}+E_{y}\,f_{8}+E_{z}\,f_{9}-B_{x}\,f_{4}-B_{y}\,f_{5}+B_{z}\,f_{6}\right)\\\
g_{15}&=&2\,\langle{\bf F}\,\rangle\\\ \end{array}$ (91)
According to Eq. LABEL:eq_OpEx the expectation values of the operators $g_{k}$
are:
$\dot{g}_{k}=\bar{\psi}\,(\gamma_{k}\,{\bf F}^{2}-{\bf
F}^{2}\,\gamma_{k})\,\psi\,.$ (92)
The square of the force matrix is given in Eq. LABEL:eq_squares. Now we insert
this into Eq. 92. The scalar part commutes with all $\gamma_{k}$ and hence
contributes nothing. Since all commutators of symplices with cosymplices
result in cosymplices, we obtain $\dot{g}_{k}=0$ for all symplices. This had
to be expected as for all symplices we had $g_{k}=0$. Hence the remaining
terms are:
$\begin{array}[]{rcl}\dot{g}_{10}&=&4\,\bar{\psi}\,(M_{r}\,\gamma_{0}+\vec{b}\,\gamma_{0}\,\vec{\gamma})\,\psi\\\
&=&4\,(M_{r}\,f_{0}+b_{x}\,f_{4}+b_{y}\,f_{5}+b_{z}\,f_{6})\\\
\dot{g}_{11}&=&2\,(4\,M_{r}\,f_{1}-4\,M_{g}\,f_{4}+b_{y}\,f_{9}-b_{z}\,f_{8})\\\
\dot{g}_{12}&=&2\,(4\,M_{r}\,f_{2}-4\,M_{g}\,f_{5}+b_{z}\,f_{7}-b_{x}\,f_{9})\\\
\dot{g}_{13}&=&2\,(4\,M_{r}\,f_{3}-4\,M_{g}\,f_{6}+b_{x}\,f_{8}-b_{y}\,f_{7})\\\
\dot{g}_{14}&=&-4\,\bar{\psi}\,(M_{g}\,\gamma_{0}+\vec{b}\,\vec{\gamma})\,\psi\\\
&=&-4\,(M_{g}\,f_{0}+b_{x}\,f_{1}+b_{y}\,f_{2}+b_{z}\,f_{3})\\\ \end{array}$
(93)
## References
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|
arxiv-papers
| 2012-01-04T14:58:34 |
2024-09-04T02:49:25.939514
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Christian Baumgarten",
"submitter": "Christian Baumgarten",
"url": "https://arxiv.org/abs/1201.0907"
}
|
1201.1098
|
# Can Self–Organizing Maps accurately predict photometric redshifts?
M.J. Way11affiliation: NASA Goddard Institute for Space Studies, 2880
Broadway, New York, New York 10025, USA 22affiliation: NASA Ames Research
Center, Space Sciences Division, MS 245–6, Moffett Field, California 94035,
USA 33affiliation: Department of Astronomy and Space Physics, Uppsala, Sweden
, C.D. Klose44affiliation: Think Geohazards, 205 Vernon Street, Suite A
Roseville, CA 95678, USA
###### Abstract
We present an unsupervised machine learning approach that can be employed for
estimating photometric redshifts. The proposed method is based on a vector
quantization approach called Self–Organizing Mapping (SOM). A variety of
photometrically derived input values were utilized from the Sloan Digital Sky
Survey’s Main Galaxy Sample, Luminous Red Galaxy, and Quasar samples along
with the PHAT0 data set from the PHoto-z Accuracy Testing project. Regression
results obtained with this new approach were evaluated in terms of root mean
square error (RMSE) to estimate the accuracy of the photometric redshift
estimates. The results demonstrate competitive RMSE and outlier percentages
when compared with several other popular approaches such as Artificial Neural
Networks and Gaussian Process Regression. SOM RMSE–results (using
$\Delta$z=zphot–zspec) for the Main Galaxy Sample are 0.023, for the Luminous
Red Galaxy sample 0.027, Quasars are 0.418, and PHAT0 synthetic data are
0.022. The results demonstrate that there are non–unique solutions for
estimating SOM RMSEs. Further research is needed in order to find more robust
estimation techniques using SOMs, but the results herein are a positive
indication of their capabilities when compared with other well-known methods.
###### Subject headings:
methods: data analysis, methods: statistical, galaxies: distances and
redshifts
## 1\. Introduction
There is a pressing need for accurate estimates of galaxy photometric
redshifts (photo–z’s) as demonstrated by the increasing number of papers on
this topic and especially by recent attempts to objectively compare methods
(e.g. Hildebrandt et al., 2010; Abdalla et al., 2011). The need for photo-z’s
will only increase as larger and deeper surveys such as Pan-STARRS111Panoramic
Survey Telescope & Rapid Response System(Kaiser, 2004), LSST222Large Synoptic
Survey Telescope(Ivezic et al., 2008) and Euclid (Sorba & Sawicki, 2011) come
on–line in the coming decade. The photometric–only surveys (Pan-STARRS, LSST)
will have relatively small numbers of follow-up spectroscopic redshifts and
will rely upon either template-fitting methods such as Bayesian Photo-z’s
(Benítez, 2000) Le Phare (Ilbert et al., 2006), or training-set methods such
as those discussed herein. The Euclid mission may include a slitless
spectrograph offering far more training–set galaxies.
A diverse set of regression techniques using training–set methods have been
applied to the problem of estimating photometric redshifts in the past 10
years. These include Artificial Neural Networks (Firth et al., 2003;
Tagliaferri et al., 2003; Ball et al., 2004; Collister & Lahav, 2004; Vanzella
et al., 2004), Decision Trees (Suchkov et al., 2005), Gaussian Process
Regression (Way & Srivastava, 2006; Foster et al., 2009; Way et al., 2009;
Bonfield et al., 2010; Way, 2011), Support Vector Machines (Wadadekar, 2005),
Ensemble Modeling (Way et al., 2009), Random Forests Carliles et al. (2008),
and Kd–Trees (Csabai et al., 2003) to name but a few.
On the other hand, even though Self–Organizing Maps (SOMs) have been used
extensively in a number of other scientific fields (the paper that opened the
field, Kohonen (1982), currently has over 2000 citations) they have been used
sparingly thus far in Astronomy (e.g. Mahdi, 2011; Naim et al., 1997; Way,
Gazis & Scargle, 2011), and only this year in estimating photometric redshifts
(Geach, 2011).
In this work we attempt to use SOMs to estimate photometric redshifts for
several Sloan Digital Sky Survey (SDSS, York et al., 2000) derived catalogs of
different galaxy types, including Quasars along with the PHAT0 data set of
Hildebrandt et al. (2010). In Section 2 we describe the input data sets used,
in Section 3 we give an overview of SOMs, and some conclusions in Section 4.
## 2\. Data
Three different data sets derived from the SDSS Data Release Seven (DR7,
Abazajian et al., 2009) were used. They include the Main Galaxy Sample (MGS,
Strauss et al., 2002) the Luminous Red Galaxy Sample (LRG, Eisenstein et al.,
2001), and the Quasar sample (QSO, Schneider et al., 2007). Data from the
Galaxy Zoo333http://www.galaxyzoo.org (Lintott et al., 2008) Data Release 1
(Lintott et al., 2011) survey results were used to segregate galaxies as
Spiral or Elliptical in the case of the MGS and LRG samples. Details of how
this was done are given in Way (2011). Dereddened magnitudes (u,g,r,i,z) were
used as inputs in all scenarios. The same SDSS photometric and redshift
quality flags on the input variables were used as in Way (2011). In addition
we used the simulation–based PHAT0 data set (see Hildebrandt et al., 2010)
which was constructed to to test a variety of different photo–z estimation
methods. The PHAT0 data set consists of 5 SDSS like filters (u,g,r,i,z) used
on MEGACAM at CFHT (Boulade et al., 2003) with an additional 6 input filters
(Y,J,H,K,Spitzer IRAC [3.6], Spitzer IRAC [4.5]) giving a total of 11 filters
spanning a range of 4000Å to 50,000Å. This large range should help to avoid
color–redshift degeneracies that can occur if ultraviolet or infrared
bandpasses are not used (Benítez, 2000). The PHAT0 synthetic photometry was
created from the Le Phare photo-z code (Arnouts et al., 2002; Ilbert et al.,
2006). Initially Le Phare creates noise free data, but given the desire to
test more real–world conditions we utilized the PHAT0 data with added noise. A
parametric form was used for the signal–to–noise as a function of magnitude
where it acts as an exponential at fainter magnitudes and a power–law a
brighter ones. The magnitude cut between these two regimes is filter dependent
and is given in Table 2 of Hildebrandt et al. (2010). The larger of two
catalogs was used herein (as suggested for training–set methods) that contains
$\sim$ 170,000 objects.
Since we use a training–set method our original data sets are split into
training=89%, testing=10% and validation=1%. Validation was only used in the
Artificial Neural Network algorithm discussed in the next section. The full
size of each input data set are listed in parentheses in column 1 of Table 1.
## 3\. Methods
Several methods in use for calculating photometric redshifts were compared
with the SOM results: the Artificial Neural Network code of Collister & Lahav
(2004) (ANNz), the Gaussian Process Regression code of Foster et al. (2009)
(GPR), as well as simple Linear and Quadratic regression. The latter is
comparable to that of the Polynomial fits used by Li & Yee (2008). Both the
ANNz and GPR codes are freely downloadable444GPR:
http://dashlink.arc.nasa.gov/algorithm/stableGP and ANNz:
http://www.star.ucl.ac.uk/ lahav/annz.html. Details on the ANNz and GPR
algorithms can be found in their respective citations above.
Table 1Results DataaaMGS=Main Galaxy Sample (Strauss et al., 2002), LRG=Luminous Red Galaxies (Eisenstein et al., 2001), SP=Classified as spiral by Galaxy Zoo, ELL=Classified as elliptical by Galaxy Zoo, QSO=Quasar sample (Schneider et al., 2007) | MethodbbGPR=Gaussian Process Regression (Foster et al., 2009), ANNz=Artificial Neural Network (Collister & Lahav, 2004), SOM=Self–Organizing Maps (SOM-4100 and SOM-5100 see Figure 2 for details), phat0=PHAT synthetic sample | | $\sigma_{RMSE}$ccWe quote the bootstrapped 50%, 10% and 90% confidence levels as in Way et al. (2009) for the root mean square error (RMSE) when available. | | OutlierddPercentage of points defined as outliers. Following a prescription similar to that of Hildebrandt et al. (2010) we quote the percentage of points outside of $\Delta$z=zphot–zspec $\pm$ 0.1
---|---|---|---|---|---
| | 50% | 10% | 90% |
MGS | GPR | 0.02087 | 0.02072 | 0.02096 | 0.11629
(455803) | ANNz | 0.02044 | – | – | 0.14482
– | SOM | 0.02339 | – | – | 0.1689
– | Linear | 0.02742 | 0.02729 | 0.02758 | 0.35986
– | Quadratic | 0.02494 | 0.02412 | 0.02762 | 0.29184
LRG | GPR | 0.02278 | 0.02256 | 0.02309 | 0.41898
(143221) | ANNz | 0.02138 | – | – | 0.41176
– | SOM | 0.02689 | – | – | 0.64292
– | Linear | 0.02896 | 0.02896 | 0.02897 | 0.71516
– | Quadratic | 0.02382 | 0.02376 | 0.02402 | 0.45510
MGS–ELL | GPR | 0.01455 | 0.01434 | 0.01473 | 0.06591
(45521) | ANNz | 0.01442 | – | – | 0.06591
– | SOM | 0.02044 | – | – | 0.10984
– | Linear | 0.01745 | 0.01731 | 0.01756 | 0.19772
– | Quadratic | 0.01612 | 0.01609 | 0.01629 | 0.10984
MGS–SP | GPR | 0.02078 | 0.02061 | 0.02093 | 0.13305
(120266) | ANNz | 0.01991 | – | – | 0.05821
– | SOM | 0.02426 | – | – | 0.04158
– | Linear | 0.02539 | 0.02529 | 0.02555 | 0.28272
– | Quadratic | 0.02326 | 0.02296 | 0.02607 | 0.20788
LRG–SP | GPR | 0.01416 | 0.01397 | 0.01436 | 0.00000
(13708) | ANNz | 0.01516 | – | – | 0.00000
– | SOM | 0.01848 | – | – | 0.07299
– | Linear | 0.01635 | 0.01627 | 0.01649 | 0.07299
– | Quadratic | 0.01469 | 0.01462 | 0.01477 | 0.00000
LRG–ELL | GPR | 0.01186 | 0.01162 | 0.01224 | 0.00000
(27378) | ANNz | 0.01298 | – | – | 0.10961
– | SOM | 0.01568 | – | – | 0.00000
– | Linear | 0.01362 | 0.01361 | 0.01364 | 0.10961
– | Quadratic | 0.01263 | 0.01254 | 0.01274 | 0.07307
QSO | GPR | 0.37342 | 0.03967 | 0.37626 | 50.96627
(56923) | ANNz | 0.65802 | – | – | 88.54533
– | SOM | 0.41821 | – | – | 54.23401
– | Linear | 0.57061 | 0.57010 | 0.57102 | 84.64512
– | Quadratic | 0.53972 | 0.53679 | 0.54539 | 81.27196
phat0 | GPR | 0.01487 | 0.01436 | 0.01532 | 0.03539
(169520) | ANN | 0.01805 | – | – | 0.05309
– | SOM | 0.02236 | – | – | 0.37754
– | Linear | 0.08703 | 0.08702 | 0.08704 | 19.34875
– | Quadratic | 0.02436 | 0.02433 | 0.02438 | 0.19467
The main purpose of Self–Organized mapping is the ability of SOMs to transform
a feature vector of arbitrary dimension drawn from the given feature space of
photometric inputs (e.g., the SDSS u,g,r,i,z magnitudes) into simplified 1– or
2–dimensional discrete maps. The method was originally developed by Kohonen
(1982, 2001) to organize information in a logical manner. This type of machine
learning utilizes an unsupervised learning scheme of vector quantization,
known as competitive learning in the field of neural information processing.
It is useful for analyzing complex data with a–priori unknown relationships
that are visualized by the self-organization process (Kohonen, 2001).
A SOM is structured in two layers: an input layer and a Kohonen layer (Figure
1). For example, the Kohonen layer could represent a structure with a single
2–dimensional map (lattice) consisting of neurons arranged in rows and
columns. Each neuron of this discrete lattice is fixed and is fully connected
with all source neurons in the input layer. For the given task of estimating
photometric redshifts, a 5–dimensional feature vector of the u,g,r,i,z
magnitudes is defined. One feature vector (u,g,r,i,z) is presented to 5 input
layer neurons. This typically activates (stimulates) one neuron in the Kohonen
layer. Learning occurs during the self–organizing procedure as feature vectors
drawn from a training data set are presented to the input layer of the SOM
network (Figure 1a). These feature vectors are also referred to as input
vectors. Neurons of the Kohonen layer compete to see which neuron will be
activated by the weight vectors that connect the input neurons and Kohonen
neurons. In other words, the weight vectors identify which input vector can
represented by a single Kohonen neuron. Hence, they are used to determine only
one activated neuron in the Kohonen layer after the winner–takes–all principle
(Figure 1b).
The SOM is considered as trained after learning, at which time the weights of
the neurons have stored the inter–relations of all 5–dimensional u,g,r,i,z
feature vectors. Then, known spectroscopic redshift values for all input
vectors of a test data set that are represented by a single Kohonen neuron are
averaged (Fig.1b). The redshift mean value represents all 5–D u,g,r,i,z
vectors that are similar to the weight vector of the activated Kohonen neuron.
The more Kohonen neurons there are the more precisely each input vector can be
represented by a weight vector. However, the total number of Kohenen neurons
are optimized for each data set (see Figure 2). A practical overview about the
learning/training process is described by Klose (2006); Klose et al. (2008,
2010) and in much greater detail by Kohonen (2001).
After training, the u,g,r,i,z input vectors of a test data set are presented
to a trained SOM. At the end of a classification step, every Kohonen neuron
approximates an input vector whereby similar/dissimilar input data were
represented by neighboring/distant neurons. One neuron could even classify
several input vectors, if these input vectors were very similar compared to
the other given input vectors. Results from the photometric redshift
approximations are then compared to known spectroscopic redshift data.
Regression performance is estimated based on the root mean square error (RMSE)
of the predicted photometric redshifts and the known spectroscopic redshifts
(using $\Delta$z=zphot–zspec). To reiterate, during the training phase, each
Kohonen neuron identifies a certain number of galaxies that are characterized
by similar u,g,r,i,z intensities. Photometric redshift data were then averaged
for these intensity values.
The SOM approximates the input feature space and maps it into an output space.
The output space shows the SOM approximation as a 2-D map (Haykin, 2009). Best
results can be obtained with an optimization scheme such that the RMSE of the
test data set is minimal as illustrated in Figure 2. Accuracy (e.g. RMSE)
depends on the size of the Kohonen map. The number of neurons in the Kohonen
map can be considered a regularization parameter ($\xi$) as shown in Figure 2.
Figure 2 shows that RMSE is high when the number of Kohonen neurons is too
small ($\xi<$2000) or too large ($\xi>$10000) and hence that the 5–dimensional
u,g,r,i,z–input space is underfit or overfit. Theoretically, a global minimum
of the RMSE–curve might exist. However, the input feature space for the given
photometric redshift problem shows a very rough RMSE–curve (Figure 2) with at
least 2 local minima. In this case it is clear that SDSS redshift estimation
tends to have several local minima, which makes is important to chose the
right optimization method to determine the SOM network size. On the other
hand, the smoother the RMSE–curve is the better gradient methods can be
utilized. Evolution strategies or genetic programming could be applied to
rougher curves with many local minima. This in turn can make it cumbersome to
find fast back–propagation Artificial Neural Network (ANN) structures,
especially when data sets are small.
Another advantage of SOMs in comparison to ANNs is that there is no need to
optimize the structure of SOMs (e.g., number of hidden layers), since it is
based on unsupervised learning.
Only the size of the Kohonen map needs to be optimized for each data set. SOMs
also allow non–experts to visualize the redshift estimates in relation to the
multi–dimensional input space. This eliminates the often criticized “black
box” problem of ANNs. As mentioned previously, SOMs approximate the input
feature space while ANNs typically separate them into sub–regions. Finally,
SOMs are known to be powerful when very small data sets are available for
training (see, Haykin, 2009).
Figure 1.— Schematic illustration of the structure (a) and functionality (b) of a Self–Organizing Map with $I$ input neurons and $M\times N$ Kohonen neurons. The SOM visualizes the structure of the $I$–dimensional input space. In this case, the SOM illuminates a certain redshift$\pm$error within the Kohonen map and as a function of the input space. Figure 2.— Accuracy (RMSE) versus regularization parameter value $\xi$ for the LRG–ELL data set (see Table 1). Different classifications will result from different choices of the $\xi$ value. The regularization value is defined by the number of Kohonen neurons, which is maximum with respect to the training data set. The convex curve has a two local minima at $\xi$=4100 and $\xi$=5100. The roughness of this RMSE cost function shows that traditional gradient based optimization strategies, e.g. deterministic annealing, might result in sub–optimal solutions. Other methods, such as, genetic programming might find the global minimum much faster. |
---|---
|
Figure 3.— Results from the three methods using SDSS u–g–r–i–z dereddened
magnitudes as inputs for the SDSS DR7 Luminous Red Galaxies classified as
ellipticals by the GalaxyZoo team. The bottom two plots show the SOM results
for the two local minima described in Section 3 and shown in Figure 2
## 4\. Conclusion
SOMs offer a competitive choice in terms of low RMSE, algorithm comprehension
(also see Göppert & Rosenstiel (1993)) and percentage of outliers. The final
results are presented in Table 1 and plots for the LRG–ELL data set for the
SOM, ANNz and GPR methods are shown in Figure 3
As mentioned previously, obtaining the global minimum is important and, not
surprisingly, can affect the results. Figure 2 shows the two local minima
($\xi$=4100 and 5100) listed for the LRG–ELL (Luminous Red Galaxies classified
as Ellipticals by GalaxyZoo) data set in Table 1. Clearly there are a number
of other $\xi$–values and the RMSE will be greatly affected by the choice as
seen on the y–axis of Figure 2 for a given $\xi$–value. Given these facts, the
roughness of the RMSE cost function in Figure 2 shows that traditional
gradient based optimization strategies, e.g, deterministic annealing, might
yield sub–optimal solutions. Other methods, such as, genetic programming might
find the “global” minimum much faster, if a global minima exists with respect
to the uncertainties of the RMSE.
During completion of this manuscript another paper using SOMs for
classification and photometric estimation was released (Geach, 2011). Our work
differs in that we mostly focus on a wider variety of low–redshift samples
drawn from the SDSS, while (Geach, 2011) focuses more on the higher redshift
samples akin to those used in Hildebrandt et al. (2010).
We have shown that SOMs are a powerful tool for estimating photometric
redshifts and that with additional work they are sure to be useful in future
surveys with limited numbers of follow–up spectroscopic redshifts.
M.J.W would like to thank the Astrophysics Department at Uppsala University
for their generous hospitality while part of this work was completed. C.D.K.
thanks Think Geohazards for providing the computational resources needed for
estimating photometric redshifts via Self–Organizing Mapping. Thanks goes to
Joe Bredekamp and the NASA Applied Information Systems Research Program for
support and encouragement. Funding for the SDSS has been provided by the
Alfred P. Sloan Foundation, the Participating Institutions, the National
Aeronautics and Space Administration, the National Science Foundation, the
U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck
Society. 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 University of Chicago, Fermilab, the
Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins
University, Los Alamos National Laboratory, the Max–Planck–Institute for
Astronomy, the Max–Planck–Institute for Astrophysics, New Mexico State
University, University of Pittsburgh, Princeton University, the United States
Naval Observatory, and the University of Washington. This research has made
use of NASA’s Astrophysics Data System Bibliographic Services. This research
has also utilized the viewpoints (Gazis, Levit, & Way, 2010) software package.
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|
arxiv-papers
| 2012-01-05T10:02:30 |
2024-09-04T02:49:25.956983
|
{
"license": "Public Domain",
"authors": "M. J. Way (NASA/GISS) and C. D. Klose (Think GeoHazards)",
"submitter": "Michael Way",
"url": "https://arxiv.org/abs/1201.1098"
}
|
1201.1154
|
11institutetext: GSI Helmholtz Centre for Heavy Ion Research, Darmstadt,
Germany
# Ferrite cavities
H. Klingbeil
###### Abstract
Ferrite cavities are used in synchrotrons and storage rings if the maximum RF
frequency is in the order of a few MHz. We present a simple model for
describing ferrite cavities. The most important parameters are defined, and
the material properties are discussed. Several practical aspects are
summarized, and the GSI SIS18 ferrite cavity is presented as an example.
## 0.1 Introduction
The revolution frequency of charged particles in synchrotrons or storage rings
is usually lower than $10\UMHz$. Even if we consider comparatively small
synchrotrons (e.g., HIT/HICAT in Heidelberg, Germany, or CNAO in Pavia, Italy,
of about $20\text{--}25\Um$ diameter, both used for tumour therapy), the
revolution time will be greater than $200\;\rm ns$ since the particles cannot
reach the speed of light. Since, according to
$f_{\text}{RF}=h\cdot f_{\text}{rev},$
the RF frequency is an integer multiple of the revolution frequency, the RF
frequency will typically be lower than $10\UMHz$ if only small harmonic
numbers $h$ are desired. For such an operating frequency, the spatial
dimensions of a conventional RF resonator would be far too large to be used in
a synchrotron. One way to solve this problem is to reduce the wavelength by
filling the cavity with magnetic material. This is the basic idea of ferrite-
loaded cavities [1]. Furthermore, this type of cavity offers a simple means to
modify the resonant frequency in a wide range (typically up to a factor of
$10$) and in a comparatively short time (typically at least $10\Ums$ cycle
time). Therefore, ferrite cavities are suitable for ramped operation in a
synchrotron.
Owing to the low operating frequencies, the transit-time factor of traditional
ferrite-loaded cavities is almost $1$ and therefore not of interest.
## 0.2 Permeability of magnetic materials
In this article, all calculations are based on permeability quantities $\mu$
for which
$\mu=\mu_{r}\mu_{0}$
holds. In material specifications, the relative permeability $\mu_{r}$ is
given which means that we have to multiply with $\mu_{0}$ to obtain $\mu$.
This comment is also valid for the incremental/differential permeability
introduced in the following.
In RF cavities, only so-called soft magnetic materials which have a narrow
hysteresis loop are of interest since their losses are comparatively low (in
contrast to hard magnetic materials which are used for permanent magnets111No
strict separation exists between hard and soft magnetic materials.).
Figure 1: Hysteresis loop
Figure 1 shows the hysteresis loop of a ferromagnetic material. It is well
known that the hysteresis loop leads to a residual induction $B_{r}$ if no
magnetizing field $H$ is present and that some coercive magnetizing field
$H_{c}$ is needed to set the induction $B$ to zero.
Let us now assume that some cycles of the large hysteresis loop have already
passed and that $H$ is currently increasing. We now stop increasing the
magnetizing field $H$ in the upper right part of the diagram. Then, $H$ is
decreased by a much smaller amount $2\cdot\Delta H$, afterwards increased
again by that amount $2\cdot\Delta H$, and so forth222The factor of $2$ was
assumed in order to have the same total change of $2\cdot\Delta H$ as in the
equation $H_{\text}{AC}(t)=\Delta H\;\cos\;\omega t$ which is usually used for
harmonic oscillations.. As the diagram shows, this procedure will lead to a
much smaller hysteresis loop where $B$ changes by $2\cdot\Delta B$. We may
therefore define a differential or incremental permeability333In a strict
sense, the differential permeability is the limit $\mu_{\Delta}=\frac{dB}{dH}$
for $\Delta H,\Delta B\rightarrow 0$.
$\framebox{$\displaystyle\mu_{\Delta}=\frac{\Delta B}{\Delta H}$}$
which describes the slope of the local hysteresis loop. It is this quantity
$\mu_{\Delta}$ which is relevant for RF applications. One can see that
$\mu_{\Delta}$ can be decreased by increasing the DC component of $H$. Since
$H$ is generated by currents, one speaks of a bias current that is applied in
order to shift the operating point to higher inductions $B$ leading to a lower
differential permeability $\mu_{\Delta}$.
If no biasing is applied, the maximum $\mu_{\Delta}$ is obtained which is
typically in the order of a few hundred or a few thousand times $\mu_{0}$.
The hysteresis loop and the AC permeability of ferromagnetic materials can be
described in a phenomenological way by the so-called Preisach model which is
explained in the literature (cf. [2]). Unfortunately, the material properties
are even more complicated since they are also frequency-dependent. One usually
uses the complex permeability
$\framebox{$\displaystyle\underline{\mu}=\mu^{\prime}_{s}-j\mu^{\prime\prime}_{s}$}$
(1)
in order to describe losses (hysteresis loss, eddy current loss and residual
loss). The parameters $\mu^{\prime}_{s}$ and $\mu^{\prime\prime}_{s}$ are
frequency-dependent. In the following, we will assume that the complex
permeability $\underline{\mu}$ describes the material behaviour in rapidly
alternating fields as does the above-mentioned real quantity $\mu_{\Delta}$
when a biasing field $H_{\text}{bias}$ is present. However, we will omit the
index $\Delta$ for the sake of simplicity.
## 0.3 Magnetostatic analysis of a ferrite cavity
Figure 2 shows the main elements of a ferrite-loaded cavity. The beam pipe is
interrupted by a ceramic gap. This gap ensures that the beam pipe may still be
evacuated but it allows a voltage $V_{\text}{gap}$ to be induced in
longitudinal direction. Several magnetic ring cores are mounted in a
concentric way around the beam and beam pipe (five ring cores are drawn here
as an example). The whole cavity is surrounded by a metallic housing which is
connected to the beam pipe.
Figure 2: Simplified 3D sketch of a ferrite-loaded cavity Figure 3: Simplified
model of a ferrite cavity
Figure 3 shows a cross-section through the cavity. The dotted line represents
the beam which is located in the middle of a metallic beam pipe (for analysing
the influence of the beam current, this dotted line is regarded as a part of a
circuit that closes outside the cavity, but this is not relevant for
understanding the basic operation principle). The ceramic gap has a parasitic
capacitance, but additional lumped-element capacitors are usually connected in
parallel — leading to the overall capacitance $C$. Starting at the generator
port located at the bottom of the figure, an inductive coupling loop surrounds
the ring core stack. This loop was not shown in Fig. 2.
Note that due to the cross-section approach, we get a wire model of the cavity
with two wires representing the cavity housing. This is sufficient for the
practical analysis, but one should remember that the currents are distributed
in reality.
All voltages, currents, field, and flux quantities used in the following are
phasors, i.e., complex amplitudes/peak values for a given frequency
$f=\omega/2\pi$.
Let us consider a contour which surrounds the lower left ring core stack.
Based on Maxwell’s second equation in the time domain (Faraday’s law)
$\oint_{\partial S}\vec{E}\cdot d\vec{l}=-\int_{S}\dot{\vec{B}}\cdot d\vec{S}$
we find
$V_{\text}{gen}=+j\omega\Phi_{\text}{tot}$ (2)
in the frequency domain. If we now use the complete lower cavity half as
integration path, one obtains
$V_{\text}{gap}=+j\omega\Phi_{\text}{tot}.$
Hence we find
$V_{\text}{gap}=V_{\text}{gen}.$ (3)
Here we assumed that the stray field $B$ in the air region is negligible in
comparison with the field inside the ring cores (due to their high
permeability). Finally, we consider the beam current contour:
$V_{\text}{beam}=+j\omega\Phi_{\text}{tot}=V_{\text}{gap}.$
For negligible displacement current we have Maxwell’s first equation (Ampère’s
law)
$\oint_{\partial S}\vec{H}\cdot d\vec{l}=\int_{S}\vec{J}\cdot d\vec{S}.$
We use a concentric circle with radius $r$ around the beam as integration
path:
$H\;2\pi r=I_{\text}{tot}.$ (4)
This leads to
$B=\underline{\mu}\frac{I_{\text}{tot}}{2\pi r}$ (5)
with
$I_{\text}{tot}=I_{\text}{gen}-I_{C}-I_{\text}{beam}.$ (6)
For the flux through one single ring core we get
$\Phi_{1}=\int\vec{B}\cdot
d\vec{S}=t\;\int_{r_{i}}^{r_{o}}B\;dr=\frac{t\underline{\mu}I_{\text}{tot}}{2\pi}\;ln\frac{r_{o}}{r_{i}}.$
With the complex permeability
$\underline{\mu}=\mu^{\prime}_{s}-j\mu^{\prime\prime}_{s}$
and assuming that $N$ ring cores are present, one finds
$V_{\text}{gap}=j\omega\Phi_{\text}{tot}=j\omega
N\;\Phi_{1}=j\omega\frac{Nt(\mu^{\prime}_{s}-j\mu^{\prime\prime}_{s})I_{\text}{tot}}{2\pi}\;ln\frac{r_{o}}{r_{i}}.$
Therefore we obtain
$V_{\text}{gap}=I_{\text}{tot}(j\omega L_{s}+R_{s})=I_{\text}{tot}Z_{s},$ (7)
if
$Z_{s}=\frac{1}{Y_{s}}=j\omega L_{s}+R_{s},$ (8) $\framebox{$\displaystyle
L_{s}=\frac{Nt\mu^{\prime}_{s}}{2\pi}\;ln\frac{r_{o}}{r_{i}}$},$
$\framebox{$\displaystyle
R_{s}=\omega\frac{Nt\mu^{\prime\prime}_{s}}{2\pi}\;ln\frac{r_{o}}{r_{i}}=\omega\frac{\mu^{\prime\prime}_{s}}{\mu^{\prime}_{s}}L_{s}=\frac{\omega
L_{s}}{Q}$}$ (9)
are defined. Here,
$Q=\frac{\mu^{\prime}_{s}}{\mu^{\prime\prime}_{s}}=\frac{1}{\tan\;\delta_{\mu}}$
(10)
is the quality factor (or Q factor) of the ring core material. Using Eq. (6)
we find
$V_{\text}{gap}Y_{s}=I_{\text}{tot}=I_{\text}{gen}-I_{\text}{beam}-V_{\text}{gap}\;j\omega
C$ $\Rightarrow\framebox{$\displaystyle
V_{\text}{gap}=\frac{I_{\text}{gen}-I_{\text}{beam}}{Y_{s}+j\omega
C}=Z_{\text}{tot}(I_{\text}{gen}-I_{\text}{beam})$}\,.$ (11)
This equation corresponds to the equivalent circuit shown in Fig. 5. In the
last step we defined
$Y_{\text}{tot}=\frac{1}{Z_{\text}{tot}}=Y_{s}+j\omega C.$
In the literature one often finds a different version of Eq. (11) where
$I_{\text}{beam}$ has the same sign as $I_{\text}{gen}$. This corresponds to
both currents having the same direction (flowing into the circuits in Figs. 5
and 5). In any case, one has to make sure that the correct phase between beam
current and gap voltage is established.
Figure 4: Series equivalent circuit
Figure 5: Parallel equivalent circuit
## 0.4 Parallel and series lumped element circuit
In the vicinity of the resonant frequency, it is possible to convert the
lumped element circuit shown in Fig. 5 into a parallel circuit as shown in
Fig. 5. The admittance of both circuits shall be equal:
$Y_{\text}{tot}=j\omega C+\frac{1}{R_{s}+j\omega L_{s}}=j\omega
C+\frac{1}{R_{p}}+\frac{1}{j\omega L_{p}}$ $\Rightarrow\frac{R_{s}-j\omega
L_{s}}{R_{s}^{2}+(\omega L_{s})^{2}}=\frac{1}{R_{p}}+\frac{1}{j\omega L_{p}}.$
A comparison of real and imaginary part yields:
$\displaystyle R_{p}$ $\displaystyle=$ $\displaystyle\frac{R_{s}^{2}+(\omega
L_{s})^{2}}{R_{s}}$ (12) $\displaystyle\omega L_{p}$ $\displaystyle=$
$\displaystyle\frac{R_{s}^{2}+(\omega L_{s})^{2}}{\omega L_{s}}.$ (13)
For the inverse relation, we modify the first equation according to
$(\omega L_{s})^{2}=R_{s}(R_{p}-R_{s})$
and use this result in the second equation:
$\omega L_{p}\sqrt{R_{s}(R_{p}-R_{s})}=R_{s}R_{p}$ $\Rightarrow(\omega
L_{p})^{2}(R_{p}-R_{s})=R_{s}R_{p}^{2}$ $\Rightarrow\framebox{$\displaystyle
R_{s}=\frac{(\omega L_{p})^{2}}{R_{p}^{2}+(\omega L_{p})^{2}}R_{p}$}\,.$
Equations (12) and (13) directly provide
$R_{p}R_{s}=(\omega L_{p})(\omega L_{s})$ (14)
which leads to
$\framebox{$\displaystyle\omega L_{s}=\frac{R_{p}}{\omega
L_{p}}R_{s}=\frac{R_{p}^{2}}{R_{p}^{2}+(\omega L_{p})^{2}}\omega L_{p}$}\,.$
Since it is suitable to use both types of lumped element circuit, it is also
convenient to define the complex $\underline{\mu}$ in a parallel form:
$\framebox{$\displaystyle\frac{1}{\underline{\mu}}=\frac{1}{\mu^{\prime}_{p}}+j\frac{1}{\mu^{\prime\prime}_{p}}$}\,.$
(15)
This is an alternative representation for the series form shown in Eq. (1)
which leads to
$\frac{1}{\underline{\mu}}=\frac{\mu^{\prime}_{s}+j\mu^{\prime\prime}_{s}}{{\mu^{\prime}_{s}}^{2}+{\mu^{\prime\prime}_{s}}^{2}}.$
Comparing the real and imaginary parts of the last two equations, we find:
$\mu^{\prime}_{p}=\frac{{\mu^{\prime}_{s}}^{2}+{\mu^{\prime\prime}_{s}}^{2}}{\mu^{\prime}_{s}},$
(16)
$\mu^{\prime\prime}_{p}=\frac{{\mu^{\prime}_{s}}^{2}+{\mu^{\prime\prime}_{s}}^{2}}{\mu^{\prime\prime}_{s}}.$
(17)
These two equations lead to
$\mu^{\prime}_{p}\mu^{\prime}_{s}=\mu^{\prime\prime}_{p}\mu^{\prime\prime}_{s}.$
Together with Eqs. (9), (10), and (14) we conclude:
$\framebox{$\displaystyle
Q=\frac{\mu^{\prime}_{s}}{\mu^{\prime\prime}_{s}}=\frac{\omega
L_{s}}{R_{s}}=\frac{R_{p}}{\omega
L_{p}}=\frac{\mu^{\prime\prime}_{p}}{\mu^{\prime}_{p}}$}\,.$ (18)
With these expressions, we may write Eqs. (16) and (17) in the form
$\framebox{$\displaystyle\mu^{\prime}_{p}=\mu^{\prime}_{s}\left(1+\frac{1}{Q^{2}}\right)$}$
(19)
$\framebox{$\displaystyle\mu^{\prime\prime}_{p}=\mu^{\prime\prime}_{s}\left(1+Q^{2}\right)$}\,.$
(20)
If we use Eq. (18)
$Q=\frac{\omega L_{s}}{R_{s}},$
we may rewrite Eqs. (12) and (13) in the form
$\displaystyle R_{p}$ $\displaystyle=$ $\displaystyle R_{s}(1+Q^{2})$ (21)
$\displaystyle L_{p}$ $\displaystyle=$ $\displaystyle
L_{s}\left(1+\frac{1}{Q^{2}}\right).$ (22)
By combining Eqs. (21) and (9) we find
$R_{p}=(1+Q^{2})\omega\frac{Nt\mu^{\prime\prime}_{s}}{2\pi}\;ln\frac{r_{o}}{r_{i}}.$
With the help of Eqs. (18) and (19) one gets
$\mu^{\prime\prime}_{s}=\frac{\mu^{\prime}_{s}}{Q}=\frac{\mu^{\prime}_{p}}{Q+\frac{1}{Q}}=\frac{\mu^{\prime}_{p}Q}{1+Q^{2}}.$
The last two equations lead to
$R_{p}=\omega\frac{Nt\mu^{\prime}_{p}Q}{2\pi}\;ln\frac{r_{o}}{r_{i}}=Nt\mu^{\prime}_{p}Qf\;ln\frac{r_{o}}{r_{i}}.$
This shows that $R_{p}$ is proportional to the product $\mu^{\prime}_{p}Qf$
which is a material property. The other parameters refer to the geometry.
Therefore, the manufacturers of ferrite cores sometimes specify the
$\bf\mu_{r}Qf$ product (for the sake of simplicity, we define
$\mu_{r}:=\mu^{\prime}_{p,r}$).
For $Q\geq 5$, we may use the approximations
$R_{p}\approx R_{s}\;Q^{2},\mbox{\qquad}L_{p}\approx
L_{s},\mbox{\qquad}\mu^{\prime}_{p}\approx\mu^{\prime}_{s},\mbox{\qquad}\mu^{\prime\prime}_{p}\approx\mu^{\prime\prime}_{s}\;Q^{2}$
which then have an error of less than $4\%$.
## 0.5 Frequency dependence of material properties
As an example, the frequency dependence of the permeability is shown in Figs.
7 and 7 for the special ferrite material Ferroxcube 4 assuming small magnetic
RF fields without biasing. All the data presented for this material are taken
from Ref. [3]. It is obvious that the behaviour depends significantly on the
choice of the material. Without biasing, a constant
$\mu^{\prime}_{s}\approx\mu^{\prime}_{p}$ may only be assumed up to a certain
frequency (see Fig. 7). Increasing the frequency from $0$ upwards, the Q
factor will decrease (compare Figs. 7 and 7). Figure 8 shows the resulting
frequency dependence of the $\mu_{r}Qf$ product.
If the magnetic RF field is increased, both $Q$ and $\mu_{r}Qf$ will decrease
in comparison with the diagrams in Figs. 7 to 8. The effective incremental
permeability $\mu_{r}$ will increase for rising magnetic RF fields as one can
see by interpreting Fig. 1. Therefore, it is important to consider the
material properties under realistic operating conditions (the maximum RF
B-field is usually in the order of $\Unit{10\text{--}20}{mT}$).
Figure 6: $\mu^{\prime}_{s,r}$ versus frequency for three different types of
ferrite material (1: Ferroxcube 4A, 2: Ferroxcube 4C, 3: Ferroxcube 4E). Data
adopted from Ref. [3].
Figure 7: $\mu^{\prime\prime}_{s,r}$ versus frequency for three different
types of ferrite material (1: Ferroxcube 4A, 2: Ferroxcube 4C, 3: Ferroxcube
4E). Data adopted from Ref. [3].
Figure 8: $\mu^{\prime}_{s,r}Qf$ product versus frequency for three different
types of ferrite material (1: Ferroxcube 4A, 2: Ferroxcube 4C, 3: Ferroxcube
4E). No bias field is present, and small magnetic RF field amplitudes are
assumed. Data adopted from Ref. [3].
If biasing is applied, the $\mu_{r}Qf$ curve shown in Fig. 8 will be shifted
to the lower right side; this effect may approximately compensate the increase
of $\mu_{r}Qf$ with frequency [3]. Therefore, the $\mu_{r}Qf$ product may
sometimes approximately be regarded as a constant if biasing is used to keep
the cavity at resonance for all frequencies under consideration.
## 0.6 Quality factor of the cavity
The quality factor of the equivalent circuit shown in Fig. 5 is obtained if
the resonant (angular) frequency
$\framebox{$\displaystyle\omega_{0}=2\pi f_{0}=\frac{1}{\sqrt{L_{p}C}}$}$
is inserted into Eq. (18):
$\framebox{$\displaystyle Q_{0}=R_{p}\sqrt{\frac{C}{L_{p}}}$}\,.$
In general, all parameters $\mu_{s}^{\prime}$, $\mu_{s}^{\prime\prime}$,
$\mu_{p}^{\prime}$, $\mu_{p}^{\prime\prime}$, $R_{s}$, $L_{s}$, $R_{p}$,
$L_{p}$, $Q$ and $Q_{0}$ are frequency-dependent. It depends on the material
whether the parallel or the series lumped element circuit is the better
representation in the sense that its parameters may be regarded as
approximately constant in the relevant operating range. In the following, we
will use the parallel representation.
We briefly show that $Q_{0}$ is in fact the quality factor defined by
$Q_{0}=\omega\frac{W_{\text}{tot}}{P_{\text}{loss}}$
where $W_{\text}{tot}$ is the stored energy and $P_{\text}{loss}$ is the power
loss (both time-averaged):
$\framebox{$\displaystyle
P_{\text}{loss}=\frac{|V_{\text}{gap}|^{2}}{2R_{p}}$}$ (23)
$W_{\text}{el}=\frac{1}{4}\;C\;|V_{\text}{gap}|^{2}$
$W_{magn}=\frac{1}{4}\;L_{p}\;|I_{L}|^{2}=\frac{1}{4}\;L_{p}\;\frac{|V_{\text}{gap}|^{2}}{\omega^{2}L_{p}^{2}}=\frac{|V_{\text}{gap}|^{2}}{4\omega^{2}L_{p}}$
At resonance, we have $W_{\text}{el}=W_{magn}$ which leads to
$Q_{0}=2\omega\frac{W_{\text}{el}}{P_{\text}{loss}}=2\omega\frac{R_{p}C}{2}=R_{p}\sqrt{\frac{C}{L_{p}}}$
as expected. The parallel resistor $R_{p}$ defined by Eq. (23) is often called
shunt impedance.
## 0.7 Impedance of the cavity
The impedance of the cavity
$Z_{\text}{tot}=\frac{1}{\frac{1}{R_{p}}+j\left(\omega C-\frac{1}{\omega
L_{p}}\right)}=\frac{\sqrt{\frac{L_{p}}{C}}}{\frac{1}{R_{p}}\sqrt{\frac{L_{p}}{C}}+j\left(\omega\sqrt{L_{p}C}-\frac{1}{\omega\sqrt{L_{p}C}}\right)}$
may be written as
$Z_{\text}{tot}=\frac{\frac{R_{p}}{Q_{0}}}{\frac{1}{Q_{0}}+j\left(\frac{\omega}{\omega_{0}}-\frac{\omega_{0}}{\omega}\right)}$
$\Rightarrow\framebox{$\displaystyle
Z_{\text}{tot}=\frac{R_{p}}{1+j\;Q_{0}\left(\frac{\omega}{\omega_{0}}-\frac{\omega_{0}}{\omega}\right)}$}\,.$
The Laplace transformation yields
$Z_{\text}{tot}(s)=\frac{R_{p}}{1+s\frac{Q_{0}}{\omega_{0}}+\frac{Q_{0}\omega_{0}}{s}}=\frac{R_{p}\frac{\omega_{0}}{Q_{0}}s}{s\frac{\omega_{0}}{Q_{0}}+s^{2}+\omega_{0}^{2}},$
which may be found in the literature in the form
$Z_{\text}{tot}(s)=\frac{2R_{p}\sigma\;s}{s^{2}+2\sigma s+\omega_{0}^{2}}$
if
$\sigma=\frac{\omega_{0}}{2Q_{0}}$
is defined.
## 0.8 Length of the cavity
In the previous sections, we assumed that the ferrite ring cores can be
regarded as lumped-element inductors and resistors. This is of course only
true if the cavity is short in comparison with the wavelength.
As an alternative to the transformer model introduced above, one may therefore
use a coaxial transmission line model. For example, the section of the cavity
that is located on the left side of the ceramic gap in Fig. 3 may be
interpreted as a coaxial line that is homogeneous in longitudinal direction
and that has a short-circuit at the left end. The cross section consists of
the magnetic material of the ring cores, air between the ring cores and the
beam pipe, and air between the ring cores and the cavity housing. This is of
course an idealization since cooling disks, conductors and other air regions
are neglected. Taking the SIS18 cavity at GSI as an example, the ring cores
have $\mu_{r}=28$ at an operating frequency of $2.5\UMHz$. The ring cores have
a relative dielectric constant of $10\text{--}15$, but this is reduced to an
effective value of $\epsilon_{r,eff}=1.8$ since the ring cores do not fill the
full cavity cross section. These values lead to a wavelength of
$\lambda=16.9\Um$. Since 64 ring cores with a thickness of $25\Umm$ are used,
the effective length of the magnetic material is $1.6\Um=0.095\;\lambda$
(which corresponds to a phase of $34^{\circ}$). In this case, the transmission
line model leads to deviations of less than 10% with respect to the lumped-
element model. The transmission line model also shows that the above-mentioned
estimation for the wavelength is too pessimistic; it leads to $\lambda=24\Um$
which corresponds to a cavity length of only $24^{\circ}$.
This type of model makes it understandable why the ferrite cavity is sometimes
referred to as a shortened quarter-wavelength resonator.
Of course, one may also use more detailed models where subsections of the
cavity are modeled as lumped elements. In this case, computer simulations can
be performed to calculate the overall impedance. In case one is interested in
resonances which may occur at higher operating frequencies, one should perform
full electromagnetic simulations.
In any case, one should always remember that some parameters are difficult to
determine, especially the permeability of the ring core material under
different operating conditions. This uncertainty may lead to larger errors
than simplifications of the model. Measurements of full-size ring cores in the
requested operating range are inevitable when a new cavity is developed. Also
parameter tolerances due to the manufacturing process have to be taken into
account.
In general, one should note that the total length and the dimensions of the
cross-section of the ferrite cavity are not determined by the wavelength as
for a conventional RF cavity. For example, the SIS18 ferrite cavity has a
length of $3\Um$ flange-to-flange although only $1.6\Um$ are filled with
magnetic material. This provides space for the ceramic gap, the cooling disks,
and further devices like the bias current bars. In order to avoid resonances
at higher frequencies, one should not waste too much space, but there is no
exact size of the cavity housing that results from the electromagnetic
analysis.
## 0.9 Cavity filling time
The equivalent circuit shown in Fig. 5 was derived in the frequency domain. As
long as no parasitic resonances occur, this equivalent circuit may be
generalized. Therefore, we may also analyse it in the time domain (allowing
slow changes of $L_{p}$ with time):
$I_{C}=C\cdot\frac{dV_{\text}{gap}}{dt}\mbox{,\qquad}V_{\text}{gap}=L_{p}\cdot\frac{dI_{L}}{dt}\mbox{,\qquad}V_{\text}{gap}=(I_{\text}{gen}-I_{L}-I_{C}-I_{\text}{beam})\;R_{p}$
$\Rightarrow
I_{L}=-\frac{V_{\text}{gap}}{R_{p}}+I_{\text}{gen}-I_{C}-I_{\text}{beam}$ (24)
$\Rightarrow
V_{\text}{gap}=L_{p}\left(-\frac{1}{R_{p}}\frac{dV_{\text}{gap}}{dt}+\frac{d}{dt}(I_{\text}{gen}-I_{\text}{beam})-C\frac{d^{2}V_{\text}{gap}}{dt^{2}}\right)$
$\Rightarrow
L_{p}C\;\ddot{V}_{\text}{gap}+\frac{L_{p}}{R_{p}}\;\dot{V}_{\text}{gap}+V_{\text}{gap}=L_{p}\frac{d}{dt}(I_{\text}{gen}-I_{\text}{beam})$
$\Rightarrow\framebox{$\displaystyle\ddot{V}_{\text}{gap}+\frac{2}{\tau}\;\dot{V}_{\text}{gap}+\omega_{0}^{2}V_{\text}{gap}=\frac{1}{C}\frac{d}{dt}(I_{\text}{gen}-I_{\text}{beam}).$}$
(25)
Here we used the definition
$\framebox{$\displaystyle\tau=2R_{p}C$}\,.$
The product $R_{p}C$ is also present in the expression for the quality factor:
$Q_{0}=R_{p}\sqrt{\frac{C}{L_{p}}}=\frac{R_{p}C}{\sqrt{L_{p}C}}=\frac{1}{2}\tau\omega_{0}$
$\Rightarrow\framebox{$\displaystyle\tau=\frac{2Q_{0}}{\omega_{0}}=\frac{Q_{0}}{\pi
f_{0}}$}\,.$
Under the assumption $\omega_{0}>\frac{1}{\tau}$ ($Q_{0}>\frac{1}{2}$), the
approach $V_{\text}{gap}=V_{0}e^{\alpha t}$ (with a complex constant $\alpha$)
for the homogeneous solution of Eq. (25) actually leads to
$\alpha=-\frac{1}{\tau}\pm j\omega_{x}$
with the exponential decay time $\tau$ and the oscillation frequency
$\omega_{x}=\omega_{0}\sqrt{1-\frac{1}{(\tau\omega_{0})^{2}}}=\omega_{0}\sqrt{1-\frac{1}{4Q_{0}^{2}}}.$
This leads to $\omega_{x}\approx\omega_{0}$ even for moderately high $Q>2$
(error less than 4%).
The time $\tau$ is the time constant for the cavity which also determines the
cavity filling time. Furthermore, the time constant $\tau$ is relevant for
phase jumps of the cavity (see, e.g., Ref. [4]).
## 0.10 Power amplifier
Up to now, we only dealt with the so-called ‘unloaded Q factor’ $Q_{0}$ of the
cavity. An RF power amplifier that feeds the cavity may often be represented
by a voltage-controlled current source (e.g., in the case of a tetrode
amplifier). The impedance of this current source will be connected in parallel
to the equivalent circuit thereby reducing the ohmic part $R_{p}$. Therefore,
the loaded Q factor will be reduced in comparison with the unloaded Q factor.
Also the cavity filling time will be reduced due to the impedance of the power
amplifier.
It has to be emphasized that for ferrite cavities $50\leavevmode\nobreak\
\Omega$ impedance matching is not necessarily used in general. The cavity
impedance is usually in the order of a few hundred ohms or a few kilo-ohms.
Therefore, it is often more suitable to directly connect the tetrode amplifier
to the cavity. Impedance matching is not mandatory if the amplifier is located
close to the cavity. Short cables have to be used since they contribute to the
overall impedance/capacitance. Cavity and RF power amplifier must be
considered as one unit; they cannot be developed individually in the sense
that the impedance curves of the cavity and the power amplifier influence each
other.
## 0.11 Cooling
Both the power amplifier and the ferrite ring cores need active cooling. Of
course, the Curie temperature of the ferrite material (typically
$>100^{\circ}\rm C$) must never be reached. Depending on the operating
conditions (e.g., CW or pulsed operation), forced air cooling may be
sufficient or water cooling may be required. Cooling disks in-between the
ferrite cores may be used. In this case, one has to make sure that the thermal
contact between cooling disks and ferrites is good.
## 0.12 Cavity tuning
We already mentioned in Section 0.2 that a DC bias current may be used to
decrease $\mu_{\Delta}$ which results in a higher resonant frequency. This is
one possible way to realize cavity tuning. Strictly speaking, one deals with a
quasi-DC bias current since the resonant frequency must be modified during a
synchrotron machine cycle if it is equal to the variable RF frequency. Such a
tuning of the resonant frequency $f_{0}$ to the RF frequency $f_{RF}$ is
usually desirable since the large $Z_{\text}{tot}$ allows one to generate
large voltages with moderate RF power consumption.
Sometimes, the operating frequency range is small enough in comparison with
the bandwidth of the cavity that no tuning is required.
If tuning is required, one has at least two possibilities to realize it:
1. 1.
Bias current tuning
2. 2.
Capacitive tuning
The latter may be realized by a variable capacitor (see, e.g., Refs. [5, 6])
whose capacitance may be varied by a stepping motor. This mechanical
adjustment, however, is only possible if the resonant frequency is not changed
from machine cycle to machine cycle or even within one machine cycle.
In the case of bias current tuning, one has two different choices, namely
perpendicular biasing (also denoted as transverse biasing) and parallel
biasing (also denoted as longitudinal biasing). The terms parallel and
perpendicular refer to the orientation of the DC field $H_{\text}{bias}$ in
comparison with the RF field $H$.
Parallel biasing is simple to realize. One adds bias current loops which may
in principle be located in the same way as the inductive coupling loop shown
in Fig. 3. If only a few loops are present, current bars with large cross
sections are needed to withstand the bias current of several hundred amperes.
The required DC current may of course be reduced if the number
$N_{\text}{bias}$ of loops is increased accordingly (keeping the ampere-turns
constant). This increase of the number of bias current windings may be limited
by resonances. On the other hand, a minimum number of current loops is usually
applied to guarantee a certain amount of symmetry which leads to a more
homogeneous flux in the ring cores.
Perpendicular biasing is more complicated to realize; it requires more space
between the ring-cores, and the permeability range is smaller than for
parallel biasing. The main reason for using perpendicular biasing is that
lower losses can be reached (see, e.g., Ref. [7]). One can also avoid the so-
called Q-loss effect or high loss effect. The Q-loss effect often occurs when
parallel biasing is applied and if the bias current is constant or varying
only slowly. After a few milliseconds, one observes that the induced voltage
breaks down by a certain amount even though the same amount of RF power is
still applied (see, e.g., Refs. [8, 9]). For perpendicular biasing, the Q-loss
effect was not observed. The Q-loss effect is not fully understood yet.
However, there are strong indications that it may be caused by mechanical
resonances of the ring cores induced by magnetostriction effects [10]. It was
possible to suppress the Q-loss effect by mechanical damping. For example, in
some types of ferrite cavities, the ring cores are embedded in a sealing
compound [11] which should damp mechanical oscillations. Not only the Q-loss
effect but also further anomalous loss effects have been observed [8].
When the influence of biasing is described, one usually defines an average
bias field $H_{\text}{bias}$ for the ring cores. For this purpose, one may use
the magnetic field
$H_{\text}{bias}=\frac{N_{\text}{bias}I_{\text}{bias}}{2\pi\bar{r}}$
located at the mean radius
$\bar{r}=\sqrt{r_{i}r_{o}}.$
Of course, this choice is somewhat arbitrary from a theoretical point of view,
but it is based on practical experience.
A combination of bias current tuning and capacitive tuning has also been
applied to extend the frequency range [12].
## 0.13 Further complications
We already mentioned that the effective differential permeability depends on
the hysteresis behaviour of the material, i.e., on the history of bias and RF
currents. It was also mentioned that, owing to the spatial dimensions of the
cavity, we have to deal with ranges between lumped-element circuits and
distributed elements. The anomalous loss effects are a third complication.
There are further points which make the situation even more complicated in
practice:
* •
If no biasing is applied, the maximum of the magnetic field is present at the
inner radius $r_{i}$. One has to make sure that the maximum ratings of the
material are not exceeded.
* •
Bias currents lead to an $r^{-1}$ dependency of the induced magnetic field
$H_{\text}{bias}$. Therefore, biasing is more effective in the inner parts of
the ring cores than in the outer parts resulting in a $\mu_{\Delta}$ which
increases with $r$. According to Eq. (5), this will modify the $r^{-1}$
dependency of the magnetic RF field. As a result, the dependence on $r$ may be
much weaker than without bias field.
* •
The permeability depends not only on the frequency, on the magnetic RF field,
and on the biasing. It is also temperature-dependent.
* •
Depending on the thickness of the ferrite cores, on the conductivity of the
ferrite, on the material losses and on the operating frequency, the magnetic
field may decay from the surface to the inner regions reducing the effective
volume.
* •
At higher operating frequencies with strong bias currents, the differential
permeability will be rather low. This means that the magnetic flux will not be
perfectly guided by the ring cores anymore. The fringe fields in the air
regions will be more important, and resonances may occur.
## 0.14 Cavity configurations
A comparison of different types of ferrite cavities can be found in Refs. [13,
14, 15]. We just summarize a few aspects here that lead to different
solutions.
* •
Instead of using only one stack of ferrite ring cores and only one ceramic gap
as shown in Fig. 3, one may also use more sections with ferrites (e.g., one
gap with half the ring cores on the left side and the other half on the right
side of the gap — for reasons of symmetry) or more gaps. Sometimes, the
ceramic gaps belong to different independent cavity cells which may be coupled
by copper bars (e.g., by connecting them in parallel). Connections of this
type must be short to allow operation at high frequencies.
* •
One configuration that is often used is a cavity consisting of only one
ceramic gap and two ferrite stacks on both sides. Figure-of-eight windings
surround these two ferrite stacks (see, e.g., Ref. [16]). With respect to the
magnetic RF field, this leads to the same magnetic flux in both stacks. In
this way, an RF power amplifier that feeds only one of the two cavity halves
will indirectly supply the other cavity half as well. This corresponds to a
1:2 transformation ratio. Hence, the beam will see four times the impedance
compared with the amplifier load. Therefore, the same RF input power will lead
to higher gap voltages (but also to a higher beam impedance). The
transformation law may be derived by an analysis that is similar to the one in
Section 0.3.
* •
Instead of the inductive coupling shown in Fig. 3, one may also use capacitive
coupling if the power amplifier is connected to the gap via capacitors. If a
tetrode power amplifier is used, one still has to provide it with a high anode
voltage. Therefore, an external inductor (choke coil) is necessary which
allows the DC anode current but which blocks the RF current from the DC power
supply. Often a combination of capacitive and inductive coupling is used
(e.g., to influence parasitic resonances). The coupling elements will
contribute to the equivalent circuit.
* •
Another possibility is inductive coupling of individual ring cores. This leads
to lower impedances which ideally allow a $50\leavevmode\nobreak\ \Omega$
impedance matching to a standard solid-state RF power amplifier (see, e.g.,
Ref. [17]).
* •
In case a small relative tuning range is required, it is not necessary to use
biasing for the ferrite ring cores inside the cavity. One may use external
tuners (see, e.g., Refs. [18, 19]) which can be connected to the gap. For
external tuners, both parallel and perpendicular biasing may be applied [20].
No general strategy can be defined as to how a new cavity is designed. Many
compromises have to be found. A certain minimum capacitance is given by the
gap capacitance and the parasitic capacitances. In order to reach the upper
limit of the frequency range, a certain minimum inductance has to be realized.
If biasing is used, this minimum inductance must be reached using the maximum
bias current but the effective permeability should still be high enough to
reduce stray fields. Also the lower frequency limit should be reachable with a
minimum but non-zero bias current. There is a maximum RF field $B_{RF,max}$
(about $\Unit{15}{mT}$) which should not be exceeded for the ring cores. This
imposes a lower limit for the number of ring cores. The required tuning range
in combination with the overall capacitance will also restrict the number of
ring cores. As mentioned above, the amplifier design should be taken into
account from the very beginning, especially with respect to the impedance. The
maximum beam impedance that is tolerable is defined by beam dynamics
considerations. This impedance budget also defines the power that is required.
If more ring cores can be used, the impedance of the cavity will increase, and
the power loss will decrease for a given gap voltage.
## 0.15 The GSI ferrite cavities in SIS18
As an example for a ferrite cavity, we summarize the main facts about GSI’s
SIS18 ferrite cavities (see Figs. 9 and 10). Two identical ferrite cavities
are located in the synchrotron SIS18.
Figure 9: SIS18 ferrite cavity Figure 10: Gap area of the SIS18 ferrite cavity
The material Ferroxcube FXC 8C12m is used for the ferrite ring cores. In
total, $N=64$ ring cores are used per cavity. Each core has the following
dimensions:
$d_{o}=2\;r_{o}=498\;{\rm mm}\mbox{,\quad}d_{i}=2\;r_{i}=270\;{\rm
mm}\mbox{,\quad}t=25\;{\rm mm}$ $\bar{r}=\sqrt{r_{i}r_{o}}=183\Umm.$
For biasing,
$N_{\text}{bias}=6$
figure-of-eight copper windings are present. The total capacitance amounts to
$C=\Unit{740}{pF},$
including the gap, the gap capacitors, the cooling disks, and other parasitic
capacitances. The maximum voltage that is reached under normal operating
conditions is $V_{\text}{gap}=16\UkV$.
Table 1: Equivalent circuit parameters for SIS18 ferrite cavities (without influence of tetrode amplifiers) Resonant frequency $f_{0}$ | $620\UkHz$ | $2.5\UMHz$ | $5\UMHz$
---|---|---|---
Relative permeability $\mu^{\prime}_{p,r}$ | $450$ | $28$ | $7$
Magnetic bias field at mean radius $H_{\text}{bias}$ | $25\UA/\UmZ$ | $700\UA/\UmZ$ | $2750\UA/\UmZ$
Bias current $I_{\text}{bias}$ | $4.8\UA$ | $135\UA$ | $528\UA$
$\mu^{\prime}_{p,r}Qf$ product | $4.2\cdot 10^{9}\Us^{-1}$ | $3.7\cdot 10^{9}\Us^{-1}$ | $3.3\cdot 10^{9}\Us^{-1}$
Q-factor $Q$ | $15$ | $53$ | $94$
$L_{s}$ | $88.2\leavevmode\nobreak\ \rm\mu H$ | $5.49\leavevmode\nobreak\ \rm\mu H$ | $1.37\leavevmode\nobreak\ \rm\mu H$
$L_{p}$ | $88.5\leavevmode\nobreak\ \rm\mu H$ | $5.49\leavevmode\nobreak\ \rm\mu H$ | $1.37\leavevmode\nobreak\ \rm\mu H$
$R_{s}$ | $22.8\leavevmode\nobreak\ \rm\Omega$ | $1.63\leavevmode\nobreak\ \rm\Omega$ | $0.46\leavevmode\nobreak\ \rm\Omega$
$R_{p}$ | $5200\leavevmode\nobreak\ \rm\Omega$ | $4600\leavevmode\nobreak\ \rm\Omega$ | $4100\leavevmode\nobreak\ \rm\Omega$
Cavity time constant $\tau$ | $7.7\leavevmode\nobreak\ \rm\mu s$ | $6.7\leavevmode\nobreak\ \rm\mu s$ | $6.0\leavevmode\nobreak\ \rm\mu s$
Table 1 shows the main parameters for three different frequencies. All these
values are consistent with the formulas presented in the paper at hand. It is
obvious that both $\mu^{\prime}_{p,r}Qf$ and $R_{p}$ do not vary strongly with
frequency justifying the parallel equivalent circuit. This compensation effect
was mentioned at the end of Section 0.5.
All the parameters mentioned here refer to the beam side of the cavity. The
cavity is driven by an RF amplifier which is coupled to only one of two
ferrite core stacks (consisting of 32 ring cores each). The two ring core
stacks are coupled by the bias windings. Therefore, a transformation ratio of
1:2 is present from amplifier to beam. This means that the amplifier has to
drive a load of about $R_{p}/4=1.1\leavevmode\nobreak\ \rm k\Omega$. For a
full amplitude of $U_{\text}{gap}=16\UkV$ at $f=5\UMHz$ the power loss in the
cavity amounts to $\Unit{31}{kW}$.
The SIS18 cavity is supplied by a single-ended tetrode power amplifier using a
combination of inductive and capacitive coupling.
It has to be emphasized that the values in Table 1 do not contain the
amplifier influence. Depending on the working point of the tetrode, $R_{p}$
will be reduced significantly, and all related parameters vary accordingly.
## 0.16 Further practical considerations
For measuring the gap voltage, one needs a gap voltage divider in order to
decrease the high-voltage RF to a safer level. This can be done by capacitive
voltage dividers. Gap relays are used to short-circuit the gap if the cavity
is temporarily unused. This reduces the impedance seen by the beam which may
be harmful for beam stability. If cycle-by-cycle switching is needed,
semiconductor switches may be used instead of vacuum relays. Another
possibility to temporarily reduce the beam impedance is to detune the cavity.
The capacitance/impedance of the gap periphery devices must be considered when
the overall capacitance $C$ and the other elements in the equivalent circuit
are calculated. Also further parasitic elements may be present.
On the one hand, the cavity dimensions should be as small as possible since
space in synchrotrons and storage rings is valuable and since undesired
resonances may be avoided. On the other hand certain minimum distances have to
be kept in order to prevent high-voltage spark-overs. For EMC reasons, RF
seals are often used between conducting metal parts of the cavity housing to
reduce electromagnetic emission.
In order to fulfil high vacuum requirements, it may be necessary to allow a
bakeout of the vacuum chamber. This can be realized by integrating a heating
jacket that surrounds the beam pipe. One has to guarantee that the ring cores
are not damaged by heating and that safety distances (for RF purposes and
high-voltage requirements) are kept.
In case the cavity is used in a radiation environment, the radiation hardness
of all materials is an important topic.
## 0.17 Magnetic materials
A large variety of magnetic materials is available. Nickel-Zinc (NiZn)
ferrites may be regarded as the traditional standard material for ferrite-
loaded cavities. The following material properties are of interest for the
material selection and may differ significantly for different types of
material:
* •
permeability
* •
magnetic losses
* •
saturation induction (typically $\Unit{200\text{--}300}{mT}$ for NiZn
ferrites)
* •
maximum RF inductions (typically $10\text{--}20$ mT for NiZn ferrites)
* •
relative dielectric constant (in the order of $10\text{--}15$ for NiZn
ferrites but very high for MnZn ferrites, for example) and dielectric losses
(usually negligible for typical NiZn applications)
* •
maximum operating temperature, thermal conductivity, and temperature
dependence in general
* •
magnetostriction
* •
specific resistance (very high for NiZn ferrites, very low for MnZn ferrites).
In order to determine the RF properties under realistic operating conditions
(large magnetic flux, biasing), thorough reproducible measurements in a fixed
test setup are inevitable.
Amorphous and nanocrystalline magnetic alloy (MA) materials have been used to
build very compact cavities that are based on similar principles to those of
classical ferrite cavities (see, e.g., Refs. [21, 15, 22, 6, 23]). These
materials allow a higher induction and have a very high permeability. This
means that a smaller number of ring cores is needed for the same inductance.
MA materials typically have lower Q factors in comparison with ferrite
materials. Low Q factors have the advantage that frequency tuning is often not
necessary and that it is possible to generate signal forms including higher
harmonics instead of pure sine signals. MA cavities are especially of interest
for pulsed operation at high field gradients. In case a low Q-factor is not
desired, it is also possible to increase it by cutting the MA ring cores.
Microwave garnet ferrites have been used at frequencies in the range
$40\text{--}60$ MHz in connection with perpendicular biasing since they
provide comparatively low losses (see, e.g., Refs. [24, 25, 26]).
## Acknowledgements
The author would like to thank all the GSI colleagues with whom he discussed
several RF cavity issues during the past years, especially Priv.-Doz. Dr.
Peter Hülsmann, Dr. Hans Günter König, Dr. Ulrich Laier, and Dr. Gerald
Schreiber. He is also grateful to the former staff members of the ring RF
group, especially Dr. Klaus Blasche, Dipl.-Phys. Martin Emmerling, and Dr.
Klaus Kaspar for transferring their RF cavity know-how to their successors.
Last but not least, the author thanks Dr. Rolf Stassen (FZ Jülich) for
reviewing the manuscript.
It is impossible to provide a complete list of references. The following list
cites only a few references regarding the most important aspects. Many other
important publications exist.
## References
* [1] J. M. Brennan, _in_ Handbook of Accelerator Physics and Engineering, Eds. A. W. Chao and M. Tigner (World Scientific, Singapore, 1999), pp. 570–2.
* [2] I. D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, 2nd ed., e-book (Elsevier, San Diego, 2003).
* [3] F. G. Brockman, H. van der Heide, and M. W. Louwerse, Ferroxcube für Protonensynchrotrons, _Philips Technische Rundschau_ , 30 (1969/70) 323–342.
* [4] S. Papureanu, Ch. Hamm, A. Schnase, and H. Meuth, Performance test of a ferrite-loaded cavity under operation conditions, _in_ 15th IEEE Particle Accelerator Conference, Washington, DC, USA, 1993 (IEEE, Piscataway, NJ, 1994), pp. 962–4.
* [5] M. Morvillo, R. Garoby, D. Grier, M. Haase, A. Krusche, P. Maesen, M. Paoluzzi, and C. Rossi, The PS 13.3–20 MHz RF system for LHC, _in_ 20th IEEE Particle Accelerator Conference, Portland, OR, USA, 2003, Ed. J. Chew (JACoW, Geneva, 2003), pp. 1724–6.
* [6] P. Hülsmann, G. Hutter, and W. Vinzenz, The bunch compressor system for SIS18 at GSI, _in_ EPAC, Luzern, 2004, pp. 1165–7.
* [7] W. R. Smythe and T. G. Brophy, RF cavities with transversely biased ferrite tuning, _IEEE Trans. Nucl. Sci._ NS-32 (1985) 2951–3.
* [8] J. E. Griffin and G. Nicholls, A review of some dynamic loss properties of Ni-Zn accelerator RF system ferrite, _IEEE Trans, Nucl. Sci._ 26 (1979) 3965–7.
* [9] K. Kaspar, H. G. König, and T. Winnefeld, Studies on maximum RF voltages in ferrite-tuned accelerating cavities, _in_ EPAC, Luzern, 2004, pp. 985–7.
* [10] H. G. König and S. Schäfer, Reduction of Q-loss effects in ferrite-loaded cavities, _in_ EPAC, Genoa, Italy, 2008, pp. 985–7.
* [11] V. S. Arbuzov et al., Accelerating RF station for HIRFL-CSR, Lanzhou, China, _in_ RuPAC XIX, Dubna, 2004, pp. 332–4.
* [12] X. Pei, S. Anderson, D. Jenner, D. McCammon, and T. Sloan, A wide tuning range RF cavity with external ferrite biasing, _in_ 15th IEEE Particle Accelerator Conference, Washington, DC, USA, 1993 (IEEE, Piscataway, NJ, 1994), pp. 1421–3.
* [13] A. Susini, Low frequency ferrite cavities, _in_ EPAC, Rome, 1988, Ed. S. Tazzari (World Scientific, Singapore, 1989), pp. 1416–7.
* [14] I. S. K. Gardner, Ferrite dominated cavities, _in_ CAS - CERN Accelerator School: RF Engineering for Particle Accelerators, Oxford, 1991, S. Turner (Ed.) CERN 92-3, pp. 349–374.
* [15] A. Schnase, Cavities with a swing, _in_ CAS - CERN Accelerator School: Radio Frequency Engineering, Seeheim, 2000, J. Miles (Ed.) CERN 2005-3, pp. 236–272.
* [16] A. Krusche and M. Paoluzzi, The new low frequency accelerating systems for the CERN PS Booster, _in_ EPAC, Stockholm, 1998 (IOP, Bristol, 1998), pp. 1782–3.
* [17] J. Dey, I. Kourbanis, and D. Wildman, A new RF system for bunch coalescing in the Fermilab Main Ring, _in_ 16th IEEE Particle Accelerator Conference, Dallas, TX, USA, 1995 (IEEE, Piscataway, NJ, 1995), pp. 1672–4.
* [18] R. M. Hutcheon, A perpendicular-biased ferrite tuner for the 52 MHz PETRA II cavities, _in_ 12th IEEE Particle Accelerator Conference, Washington, DC, USA, 1987 (IEEE, New York, 1987), pp. 1543–5.
* [19] C. C. Friedrichs, R. D. Carlini, G. Spalek, and W. R. Smythe, Test results of the Los Alamos ferrite-tuned cavity, _in_ 12th IEEE Particle Accelerator Conference, Washington, DC, USA, 1987 (IEEE, New York, 1987), pp. 1896–7.
* [20] R. L. Poirier, Perpendicular biased ferrite-tuned cavities, _in_ 15th IEEE Particle Accelerator Conference, Washington, DC, USA, 1993 (IEEE, Piscataway, NJ, 1994), pp. 753–7.
* [21] C. Fougeron, P. Ausset, D. de Menezes, J. Peyromaure, and G. Charruau, Very wide range and short accelerating cavity for MIMAS, _in_ 15th IEEE Particle Accelerator Conference, Washington, DC, USA, 1993 (IEEE, Piscataway, NJ, 1994), pp. 858–61.
* [22] K. Saito, K. Matsuda, H. Nishiuchi, M. Umezawa, K. Hiramoto, and R. Shinagawa, RF accelerating system for a compact ion synchrotron, _in_ 19th IEEE Particle Accelerator Conference, Chicago, IL, USA, 2001 (IEEE, New York, 2002), pp. 966–8.
* [23] R. Stassen, K. Bongardt, F. J. Etzkorn, H. Stockhorst, S. Papureanu, and A. Schnase, The HESR RF system and tests in COSY, _in_ EPAC, Genoa, 2008 (JACoW, Geneva, 2008), pp. 361–3.
* [24] L. M. Earley, H. A. Thiessen, R. Carlini, and J. Potter, A high-Q ferrite-tuned cavity, IEEE Trans. Nucl. Sci. NS-30 (1983) 3460–2.
* [25] K. Kaspar, Design of Ferrite-Tuned Accelerator Cavities Using Perpendicular-Biased High-Q Ferrites, Technical report, Los Alamos National Laboratory, New Mexico, USA, November 1984, LA-10277-MS.
* [26] G. Schaffer, Improved ferrite biasing scheme for Booster RF cavities, _in_ EPAC, Berlin, 1992 (Éditions Frontières, Gif-sur-Yvette, 1992), pp. 1234–6.
|
arxiv-papers
| 2012-01-05T12:50:29 |
2024-09-04T02:49:25.964859
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. Klingbeil (GSI)",
"submitter": null,
"url": "https://arxiv.org/abs/1201.1154"
}
|
1201.1245
|
# The influence of transition metal solutes on dislocation core structure and
values of Peierls stress and barrier in tungsten.
G. D. Samolyuk Materials Science and Technology Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831, USA Y. N. Osetsky Materials Science and
Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
R. E. Stoller Materials Science and Technology Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831, USA
(4 February 2009)
###### Abstract
Several transition metals were examined to evaluate their potential for
improving the ductility of tungsten. The dislocation core structure and
Peierls stress and barrier of $1/2\langle 111\rangle$ screw dislocations in
binary tungsten-transition metal alloys (W1-xTMx) were investigated using
first principles electronic structure calculations. The periodic quadrupole
approach was applied to model the structure of $1/2\langle 111\rangle$
dislocation. Alloying with transition metals was modeled using the virtual
crystal approximation and the applicability of this approach was assessed by
calculating the equilibrium lattice parameter and elastic constants of the
tungsten alloys. Reasonable agreement was obtained with experimental data and
with results obtained from the conventional supercell approach. Increasing the
concentration of a transition metal from the VIIIA group, i.e. the elements in
columns headed by Fe, Co and Ni, leads to reduction of the $C^{\prime}$
elastic constant and increase of elastic anisotropy A=$C_{44}/C^{\prime}$.
Alloying W with a group VIIIA transition metal changes the structure of the
dislocation core from symmetric to asymmetric, similar to results obtained for
W1-xRex alloys in the earlier work of Romaner et al (Phys. Rev. Lett. 104,
195503 (2010)). In addition to a change in the core symmetry, the values of
the Peierls stress and barrier are reduced. The latter effect could lead to
increased ductility in a tungsten-based alloy. Our results demonstrate that
alloying with any of the transition metals from the VIIIA group should have
similar effect as alloying with Re.
###### pacs:
74.70.Dd,72.15.-v,74.25.-q
## I Introduction
Tungsten is the prime candidate for use in the divertor of future fusion
reactors because of its high melting temperature and resistance to sputtering
El-Guebaly et al. (2011); Nygren et al. (2011). However, its lack of ductility
is an impediment to its use. The low-temperature brittleness is a common
problem for all metals from the VIA group, such as chromium, molybdenum and
tungsten Klopp (1969). The ductility of these metals can be improved by
alloying with rhenium, leading to the so-called ”Re effect” Geach and Hughes
(1956); Klopp (1969); Savitskii et al. (1965). However, Re, is a very rare and
expensive element. It is therefore desirable to find alternate elements which
provide a similar increase in ductility at lower cost. An experimental
investigation of the range of candidates in the periodic table would be rather
expensive, but computational materials science methods based on accurate
first-principles calculations provide a very promising way to narrow the range
of possible candidates.
Several possible mechanisms for the Re-effect have been discussed Edington et
al. (1966); Klopp (1969); Raffo (1969); Luo et al. (1991); Gornostyrev et al.
(1991); Trefilov et al. (1975); Kurdyumova et al. (1980), and the following
two mechanisms are selected as most promising in application to monocrystals,
i) solid solution softening in which an impurity improves mobility of 1/2(111)
screw dislocation and ii) enhancement of cross-slip in which an impurity
modifies the dislocation core structure making it easier to cross-slip and
increasing the number of possible slip planes. Both of these are related to
the effect of impurities on the dislocation core and may be amenable to
investigation by first-principles computational methods even though such
calculations cannot be used to directly estimate the mechanical properties.
Therefore, it should be possible to define one or more calculable figures of
merit that are related to a material’s elastic properties and potentially to
ductility. Among the potential figures of merit are the material’s individual
elastic constants, Poisson’s ratio Gao et al. (2008), and Peierls stress; and
there is strong evidence that these parameters are also directly related to
the electronic structure of particular impurities.
The experimentally observed strong correlation between the number of valence
electrons on the solute atom and the degree of softening Hiraoka et al.
(2004); Klopp (1975); Stephens and Witzke (1975) points to the importance of
electronic factors in solution softening Medvedeva et al. (2005, 2007). First
principles calculation Romaner et al. (2010); Li et al. (2012) indicate that
the ductilizing effect of Re could originate from ”direct” improvement of the
mobility of $1/2\langle 111\rangle$ screw dislocations Raffo (1969); Trinkle
and Woodward (2005); Medvedeva et al. (2005, 2007), either by decreasing the
Peierls stress, or enhancing cross-slip by changing the slip plane from
$\\{110\\}$ to $\\{112\\}$ which would increase the number of available slip
planes from 6 to 12 Edington et al. (1966); Garfinkle (1966). First principles
density functional (DFT) calculations demonstrated that alloying with Re in a
W1-xRex alloy leads to a transition of the $1/2\langle 111\rangle$ screw
dislocation structure from the symmetric core to asymmetric core, and to a
reduction in Peierls stress Romaner et al. (2010). Closely related results
were obtained from first-principles calculation for Mo alloys with 5$d$
transition metals Trinkle and Woodward (2005). The authors Trinkle and
Woodward (2005) placed solutes in a row along the $1/2\langle 111\rangle$
dislocation core and calculated the change of stiffness associated with moving
the row along the $\langle 111\rangle$ direction. According to the results,
solutes having fewer $d$ electrons (Hf and Ta) increase the stiffness, which
authors infer strengthens the Mo alloy, whereas those having more $d$
electrons (Re, Os and Ir) decrease stiffness, leading to softening.
The goal of the present investigation is to study possible substitutes for Re
in alloys that will result in a similar ductilizing effect as Re. The
properties of W-transition metal (TM) alloys, were modeled using the virtual
crystal approximation Faulkner (1982) (VCA). The applicability of this
approach to modeling elastic properties, structural stability and phonon
properties of a W-Re alloy has been demonstrated Gornostyrev et al. (1991);
Persson et al. (1999); Ekman et al. (2000). The approach we use can be
described as follows. First, a dislocation is placed in the effective media
representing the W-TM alloy. The properties of the dislocation in this medium
are calculated exactly but for a W-TM alloy described by a pseudopotentail
which is the weights average of the actual pseudopotentials of the impurity
and host atoms and the averaged number of electrons. A periodic lattice is
used with the lattice parameter chosen by obtain zero pressure condition. This
approach corresponds to a zero order contribution to the electronic system
energy expansion with respect to the difference between real atomic potentials
and the virtual atom. Such an approach allows separating the so-called ”band
structure effects”, in this particular case filling of $d$-states, from the
effect of local modification of the lattice due to W substitution. The latter
is not possible in a direct super cell calculation. It was demonstrated
Romaner et al. (2010) that dislocation properties are sensitive to
modifications of this effective media and alloying with Re leads to a sizable
reduction of Peierls stress and barrier. Within this approach, a comparison of
elastic constants calculated using the VCA and super-cell methods provides a
verification of the VCA accuracy as we demonstrate below.
We show that alloying with transition metals with a higher number of $d$
electrons (VIIIA group) reduces the Peierls stress and barrier for a
$1/2\langle 111\rangle$ screw dislocation. Similar to W-Re alloys, the
dislocation core symmetry is reduced by alloying with TM from group VIIIA. It
is demonstrated that the scale of Peierls barrier reductions are similar for
all W1-xTMx alloys with the same ratio of electrons per atom ($e/a$).
Although, the VCA approach describes an influence of band-structure on the
properties of screw dislocations, it does not describe the discrete nature of
alloy structure. However, this approach permits an assessment that can
significantly reduce the range of possible solute candidates suitable for Re
substitution. This paper is organized as follows. In section II we briefly
review computational details used in the calculations. Section III.1 describes
the electronic structure of tungsten and its alloys with transition metals
from IVA, VA and VIIA groups. This is an almost complete list of possible
binary solid solutions Lassner and Schubert (1999). The results of
modifications to the elastic constants with alloying are discussed in section
III.2. Section III.3 presents our prediction of core structure, Peierls
barrier and Peierls stress for W1-xTMx alloys. Finally, the conclusions are
given in section IV.
## II Computational approaches
The electronic structure within the generalized gradient approximation (GGA)
of density functional theory (DFT) was calculated using the QUANTUM ESPRESSO
(QE) package Giannozzi et al. (2009). The calculation was done using a plane-
wave basis set and ultrasoft pseudo-potentials optimized in the RRKJ scheme
Rappe et al. (1990). We used the Perdew-Wang Perdew and Wang (1992) exchange-
correlation functional. The Brillouin zone (BZ) summations were carried out
over a $24\times 24\times 24$ BZ grid for the system with one unit cell and
$16\times 16\times 16$ grid for the supercell containing $2\times 2\times 2$
unit cells, with electronic smearing with a width of 0.02 Ry applied according
to the Methfessel-Paxton method. The plane wave energy cut off of 42 Ry allows
reaching an accuracy of 0.2 mRy/atom. As a realization of VCA for the pseudo
potential method, we used the scheme proposed by Ramer and Rappe Ramer and
Rappe (2000). The elastic constants were calculated from the total energies
obtained for the set of unit cell deformations Mehl et al. (1990).
We use a periodic quadrupolar arrangement for a $1/2\langle 111\rangle$ screw
dislocation Bigger et al. (1992) in the cell with basis vectors
$\vec{b}_{1}=9\vec{u}_{1}$, $\vec{b}_{2}=5\vec{u}_{2}$ and
$\vec{b}_{3}=\vec{u}_{3}$, where $\vec{u}_{1}=[\bar{1}10]$,
$\vec{u}_{2}=[\bar{1}\bar{1}2]$ and $\vec{u}_{3}=1/2[111]$. An appropriate
choice of lattice vectors Bigger et al. (1992) reduces the quadrupole cell to
half the size, and we therefore use a cell with basis vectors equal
$\vec{h}_{1}=(\vec{u}_{1}+\vec{u}_{2}+\vec{u}_{3})/2$,
$\vec{h}_{2}=\vec{b}_{2}$ and $\vec{h}_{3}=\vec{b}_{3}$ and 135 atoms. This
unit cell contains only two dislocations with opposite Burgers vectors (Fig.
1). It was demonstrated earlier that this cell size is large enough to
reproduce such characteristics of the dislocations as Peierls stress and
barrier Segall et al. (2001a); Ismail-Beigi and Arias (2000); Segall et al.
(2003); Frederiksena and Jacobsena (2003); Li et al. (2004); Ventelon and
Willaime (2007); Odbadrakh et al. (2011) reasonably well if the elastic
interaction correction is included. We are interested in how the barriers
change with solute concentration and therefore the above correction is not
included since it’s the same for all concentrations. For the dislocation
calculation, the BZ summation was carried out over a $1\times 2\times 8$ BZ
grid Romaner et al. (2010) and the initial structure was relaxed until the
forces were smaller than 0.0005 Ry/Å.
## III Results
### III.1 Electronic structure
The electronic density of states (DOS) calculated using the VCA are presented
in Figure 2 (colored online) for pure tungsten (blue solid line) and two
tungsten alloys, one with 6.25 % Re (red dashed line) and the second one with
6.25 % Zr (green dash-dot line). Zero energy on this plot corresponds to
occupation of electronic bands by six electrons. Thus, for pure tungsten, zero
energy corresponds to the Fermi level. The Fermi energy is placed in the
pseudo gap between bonding and antibonding $d$-states. As can be seen from
Fig. 2, alloying with Re or Zr doesn’t produce sizable changes in the DOS, at
least in VCA approach. The substitution of tungsten with 6.25 % Re leads to an
upwards shift of the Fermi energy, shown by a red vertical line. The area
under DOS between zero energy and the W0.9375Re0.0625 Fermi energy corresponds
to 0.0625 additional electrons, i.e., occupation of each virtual atom by an
additional 0.0625 electrons. Correspondingly, alloying with atoms that have a
lower number of electrons, such as Zr, leads to a downward shift of the Fermi
energy, shown by vertical green line, and each virtual atom is occupied by
0.0625 fewer electrons.
Such a modification of electronic structure in tungsten alloys can be
described using the rigid band approximation Faulkner (1982). This approach
assumes that alloying does not change the DOS but shifts the Fermi level, so
the occupation of each virtual atom corresponds to the average number of
electrons per atom $(e/a)$ in the alloy. Thus, within this approximation, all
alloys with the same $(e/a)$ value have the same properties. The concentration
of different alloying elements which have the same $(e/a)$ value can be
calculated through the simple expression
$x=\frac{(e/a)}{Z_{TM}-Z_{W}}$ (1)
, where $Z_{TM}$ is the number of valence electrons of the alloying atom and
$Z_{W}$ is the number for tungsten. Although this approach has limited
accuracy, it can be very useful for qualitatively estimating the change in
such properties as elastic constants or Peierls barrier. It should be
mentioned that alloying of tungsten with TM from VIIA and VIIIA groups, within
their solubility limits, fills bonding $d$ states with additional electrons
and increases strength of the bonds. It reflects in the increase of cohesive
energy and reduction of Wigner-Seitz radius for 4$d$ TM with filling of
bonding $d$ states and reduction of cohesive energy and increase of Wigner-
Seitz radius with filling of antibonding states Moruzzi et al. (1977).
The VCA approach could be quite inaccurate for describing the properties of
disordered substitutional metallic alloys, especially for transition metals
Faulkner (1982) and therefore the accuracy of this approach should be analyzed
in each particular case. In order to verify the accuracy of VCA we compared
its results with those obtained using a supercell approach. The supercell
contained $2\times 2\times 2$ cubic unit cells with one tungsten atom
substituted by a solute atom. Thus, the solute concentration corresponds to
6.25 %. In Figure 3, the DOS calculated using the supercell (blue line) and
VCA models (red line) are presented for W15Fe. The DOS calculated using the
VCA approach is very close to the supercell result. Comparisons of elastic
properties calculated using the two approaches will be discussed below.
### III.2 Elastic constants
The modification of elastic properties in W-Re alloys has been widely
investigated both experimentally Ayres et al. (1974) and theoretically Persson
et al. (1999); Ekman et al. (2000). Here we expand these investigations to a
wide set of tungsten-based alloys. The results of equilibrium lattice
constant, $a$ , bulk modulus, $B$, and elastic constants, $C_{11}$, $C_{12}$,
$C_{44}$ and $C^{\prime}=(C_{11}-C_{12})/2$, calculated by both VCA and
supercell methods, are presented in Table LABEL:elst_cnst_vca_vs_sc. For the
elastic constants, the largest difference between calculated and experimental
values was obtained for $C_{44}$ and is about 13 % for the VCA calculations
and slightly larger for $2\times 2\times 2$ super cell calculations (SC). The
agreement between VCA and SC results W alloys with Zr, Ta, and Re is very good
and differences for elastic constants do not exceed 2 %. Hoever this
difference increases for alloys with solute atoms with a larger number of
valence electron. For example, for the Fe-family (Fe, Ru and Os) the
difference is about 5 %, while for the Co- (Co, Rh and Ir) and for Ni-families
(Ni, Pd and Pt) the difference increases to 12-15 %. Since we investigate the
reduction of Peierls stress and barrier for alloys in which the number of
valence electrons per atom $(e/a)$ corresponds to 10 % of Re or smaller the
elastic properties of the alloys should be described with an accuracy similar
to Re of 5 % or less. As discussed above, alloying of tungsten with transition
metals with a higher number of $d$ electrons, metals from the VIIA and VIIIA
groups, fills bonding $d$ states. According to the general results obtained
for transition metals Moruzzi et al. (1977); Faulkner (1982), this filling
leads to a decrease of the lattice parameter. The calculated results for
W1-xRex alloys reproduce this general tendency (see Table LABEL:elst_cnst) and
agrees with experiment Ayres et al. (1974). The same tendency was obtained in
tungsten alloyed with other TM except Pd and Pt. For these two elements, the
lattice parameter calculated in SC slightly increases. Another general trend
for all tungsten alloys with solute atoms having a higher number of $d$
electrons is reduction of the $C^{\prime}$ elastic constant. The calculated
results in Table LABEL:elst_cnst are also in good agreement with experimental
values for the W-Re alloy Ayres et al. (1974) and reflect a reduction in the
stability of bcc structure. At concentrations higher than x=0.25% the bcc
W1-xRex transforms to the $\sigma$-phase Massalski et al. (1986). Similar to
previous results Persson et al. (1999), the $C^{\prime}$ modulus decreases
with Re concentration until it becomes negative at 85 % Re. This change in the
sign of $C^{\prime}$ corresponds to the dynamic loss of stability of the bcc
structure. The $C^{\prime}$ elastic constant corresponds to the long-wave
transversal phonon branch in $[\xi\xi 0]$ directions (T $[1\bar{1}0][\xi\xi]$)
and softening of this phonon mode provides a transition path from the bcc to
dhcp structure (see discussion in Persson et al. (1999)).
According to the rigid band model discussed in the previous section, the
parameter which determines the elastic properties of W1-xTMx alloys is the
number of electrons per atom $e/a=(1-x)Z_{W}+xZ_{TM}$ (see explanation of
Eq.1). In Fig. 4 the elastic constant $C^{\prime}$ and elastic anisotropy
$A=C_{44}/C^{\prime}$ are presented as a function of $e/a$. The $C^{\prime}$
and $A$ values are very similar for all TM from group VIIA and VIIIA except
for Fe and Ir. However, even for these two elements the deviation from the
average values is around 7 %. Additional evidence for the importance of the
$e/a$ parameter follows from the correlation between the width of the stable
bcc solid solution region of W1-xTMx and the number of valence electrons for
TM elements from groups VIIA and VIIIA Massalski et al. (1986). Alloys with
solute atoms having a higher number of valence electrons have a narrower solid
solution region.
### III.3 Dislocation core structure, Peierls stress and Peierls barrier.
In bcc structures $1/2\langle 111\rangle$ screw dislocations have three
nonequivalent core configurations, that are referred to as ”easy”, ”hard” and
”split” Vitek (1974); Xu and Moriarty (1996); Ismail-Beigi and Arias (2000);
Takeuchi (1979) \- stable and two metastable, respectively. These
configurations can be obtained for a given Burgers vector $\vec{b}$ by placing
the core origin in specific positions. Recently, it was demonstrated that in
bcc Fe the lowest energy is the symmetric ”easy” core configuration, then
symmetric ”hard” core and the highest energy is the ”split” core configuration
Itakura et al. (2012). In a contrast to Fe, for the case of W we determined
that the symmetric ”hard” core configuration decays into the ”split” core.
Thus, in the present publication we assume that the dislocation migrates
between the ”easy” and the ”split” core configurations.
Following, Xu and Moriarty Xu and Moriarty (1998) we marked these sites as 1,
4 or 5 for ”easy” cores and site 2 or 3 for ”split” cores in Figure 5. The
calculated structure of the ”easy” core configuration for a $1/2\langle
111\rangle$ dislocation is shown in Fig. 5 for pure W, W0.75Re0.25 and
W0.88Fe0.12, where the concentrations of transition metals were chosen to give
the same $(e/a)$ value. The circles in Fig. 5 represent W atoms looking in the
$\langle 111\rangle$ direction and the dislocation structure is illustrated by
differential displacement (DD) maps Vitek (1974). On a DD map the displacement
of atoms in the [111] direction relative to a neighbor is plotted as an arrow
connecting the atom with its neighbors. The length of an arrow is normalized
so that arrows connecting two atoms corresponds to displacement by $b/3$.
Thus, summing arrows in the circuit around a dislocation gives the Burgers
vector. For tungsten alloys with VIIIA group TM having the same $(e/a)$
value, the core configurations looks exactly the same as for the Re and Fe
alloys. On the contrary, alloying with Ta or Zr produces the same core
symmetry as pure tungsten. The same result was obtained for W-Ta by Li et al
Li et al. (2012). The core structure of pure W is symmetric, as shown in
Figure 5a, i.e., the dislocation expands equally along the six $\langle
112\rangle$ directions, similar to results obtained earlier Romaner et al.
(2010).
Alloying with Re or any other TM with a higher number of valence electrons
leads to a change in dislocation core structure from symmetric to asymmetric.
The core is spread out in three $\langle 112\rangle$ directions on $\\{110\\}$
planes Xu and Moriarty (1998). There are a six possible $\langle 112\rangle$
orientations and the core is thus double degenerate. The transition from
symmetric to asymmetric core can change the dislocation slip plane Romaner et
al. (2010). The symmetric core dislocations glide uniformly on $\\{110\\}$
planes; asymmetric ones glide in a zigzag manner Hirth and Lothe (1982) and
the slip plane changes to $\\{112\\}$. Using the core site notation in Figure
5, the symmetric core glide path $1\rightarrow 2\rightarrow 4$, which is
shown by the gray band in Figure 5a, will be changed to glide path
$1\rightarrow 2\rightarrow 4\rightarrow 3\rightarrow 5$, shown by the gray
band in Figure 5c Gröger (2007) . In the present work, the transition from
symmetric to asymmetric core was obtained in the VCA calculations and this
result was confirmed for the W-Re alloy in a super cell model calculation
reported earlier Romaner et al. (2010); Li et al. (2012).
Addition of Re or any TM from group VIIIA leads to the change of dislocation
slip plane and decrease in the value of Peierls stress $\sigma_{p}$ Romaner et
al. (2010). Following a widely used technique Segall et al. (2001b), we apply
pure shear strain in the $\langle 111\rangle$ direction which results in a
stress that has $\sigma_{zx}$ as its main component, the influence of
$\sigma_{yx}$ can be neglected since $\sigma_{yx}/\sigma_{zx}=0.04$ for pure
W. The corresponding strain is produced by modification of the basis vector
$\vec{h}_{1}$ along $\vec{u}_{3}$ direction
$\vec{h}_{1}=1/2\vec{u}_{1}+1/2\vec{u}_{2}+(1/2+\varepsilon)\vec{u}_{3}$.
Figure 6 shows the dependence of total energy (see inset) per dislocation per
Burgers vector and shear stress as a function of strain, $\varepsilon$.
For small strains, i. e. in the elastic regime, the energy increases as
$\varepsilon^{2}$. At larger $\varepsilon$ the energy dependence deviates from
square dependence and abruptly drops. This drop in energy is caused by a jump
of the dislocation core to the next stable ”easy” core neighboring position
($1\rightarrow 4$). The corresponding stress and strain are considered to be
the Peierls stress $\sigma_{p}$ and strain $\varepsilon_{p}$. Earlier it was
demonstrated Romaner et al. (2010) that the cell size of 135 atoms as used in
the present work is enough to reach convergence in $\sigma_{p}$ values.
Alloying with both Re and Fe reduces values of $\varepsilon_{p}$ and
$\sigma_{p}$. For Fe this reduction is even larger than for Re although the
solute concentrations corresponds to the same $e/a$ value in both cases. The
absolute value of $\sigma_{p}$ equal to 1.71 GPa for pure tungsten (Table
LABEL:PS)) is somewhat lower than those obtained earlier Romaner et al.
(2010), which were 2.09 GPa using QE and 2.37 GPa using VASP. The difference
is attributed to the use of different types of pseudo-potentials, ultra-soft
in our case and norm-conserving (QE) and PAW (VASP) in Romaner et al. Romaner
et al. (2010), and the higher accuracy obtained using our larger plane wave
energy cut-off of 42 Ry compared to 30 Ry (QE calculations) and 16.4 Ry (VASP
calculations) Romaner et al. (2010). Moreover, the norm-conserving
pseudopotential would require an even larger cut-off energy than the ultra-
soft one. The relative reduction of $\sigma_{p}$ is 20 % for Re and 36 % for
Fe. A larger reduction of $\sigma_{p}$ in tungsten-iron alloys correlates with
the larger reduction of $C^{\prime}$ modulus caused by alloying with Fe
(Figure 4 and Table LABEL:elst_cnst). These results demonstrate that even if
the reduction of Peierls stress and $C^{\prime}$ values can be qualitatively
understood as arising from a filling of $d$-states in tungsten by additional
electrons from group VII and VIII solutes, some other element-specific
mechanism exists. It should be mentioned that the solubility limit of Fe in W
is around 2% and in the concentration of Fe in limited to this value the
reduction of Peierls barrier is about one half of that of the alloy with 5%
Fe. For the W-Ru alloy with a solubility limit of 3 %, $\sigma_{p}$ is
reduced by 6 %. This result is consistent with the experimentally observed
reduction of ductile-brittle transition temperature in W-Ru alloy O’Dell et
al. (2012).
The second method used in present work to estimate Peierls stress was
originally proposed by Nabarro Nabarro (1947), and has been discussed in
publications by Gröger et al. Gröger (2007); Gröger and Vitek (2012).
Following Nabarro, we measure the system energy as a function of the
dislocation location as it moves from position 1 to position 2 in Figure 5.
It is easy to see that both $1\rightarrow 2\rightarrow 4$ ($\\{110\\}$ slip
plane) and $1\rightarrow 2\rightarrow 4\rightarrow 3\rightarrow 5$
($\\{112\\}$ slip plane) paths contain a set of dislocation core jumps between
”easy” and ”split” core configurations with the same $1\rightarrow 2$ barrier.
The energy dependence obtained is associated with the Peierls barrier and the
Peierls stress is given by the maximum gradient of this function
$\sigma_{p}=\frac{1}{b}\left(\frac{dV}{dR_{c}}\right)_{max},$ (2)
where $b$ is the Burgers vector, $V$ is energy per unit length of a straight
dislocation and $R_{c}$ is the distance along the dislocation core path on the
move from 1 to 2. $R_{c}$ equals 0 for position 1 and $\sqrt{2}a/3$, where $a$
is the lattice parameter, for the position 2. We use the drag method, also
called the reaction coordinate method Xu and Moriarty (1998), to move the
dislocation from 1 (”easy”) to 2 (”split”) core configuration. According to
this method, the $z$-coordinates of atoms in two columns around both cores in
the modeling cell, shown by purple circles in Fig. 5a, are fixed so that
$z=r_{c}z_{0}+(1-r_{c})z_{1}$, where $z_{0}$ and $z_{1}$ correspond to the
$z$-coordinate of the specified atoms in the ”easy” and ”split” core
configurations respectively, and $r_{c}=R_{c}/(\sqrt{2}a/3$) is the so called
reaction coordinate. In the initial implementation Xu and Moriarty (1998), the
authors fix the $z$-coordinate of atom which has the largest displacement
resulting from the movement of the dislocation core. However from our
experience the atomic relaxation convergence increases if we fix the
$z$-coordinate of any two of the three atoms surrounding site 2 in Fig. 5a. In
order to avoid any changes in the elastic interaction between dislocations,
both cores are moved simultaneously. The results obtained for the Peierls
barrier changes are presented in Figure 7 for pure tungsten and alloys of
tungsten with Re, Fe, Ru and Zr. In order to demonstrate that the ”split” core
configuration corresponds to a metastable state, we calculated the potential
barrier using a small reaction coordinate step for pure tungsten around
$r_{c}=1$. As can be seen from the results presented in the upper insert in
Figure 7, the ”split” core configuration corresponds to a local minimum in the
potential energy. When normalized using the values for pure tungsten, the
shape of the Peierls barrier for the alloys is shown to be similar to tungsten
and quite insensitive to solute type and concentration, see lower insert of
Figure 7. In all these alloys, the results yield a single barrier. For alloys
with Os, Ir, Pt, Ru, Rh, Pd, Co and Ni the total energy was calculated for the
”easy” core (reaction coordinate, $r_{c}=0$) and ”split” core ($r_{c}=1$).
Since the shape of Peierls barrier is insensitive to type of solute, the
difference between these energies provides an estimate of the value of the
Peierls barrier as shown in Figure 8. The reduction of the Peierls barrier
observed in these calculations correlates well with the reduction in Peierls
stress estimated in the deformation simulation described above. Alloying with
Re or any group VIII transition metal at a concentration which corresponds to
$(e/a)=6.10$ reduces the barrier by 25 %. Fe reduces the Peierls barrier even
more significantly, whereas Zr increases the barrier. This is consistent with
the effect obtained for Mo alloys with 5$d$ transition metals in which placing
a row of solute atoms with higher number of valence $d$ electrons along the
dislocation core reduce stiffness and a row of atoms with lower number
increase stiffness Trinkle and Woodward (2005).
The Peierls stress calculated using the direct deformation method,
${{\sigma}}_{p}$, and from the Peierls barrier using Eq.2,
${{\tilde{\sigma}}}_{p}$ are presented in Table LABEL:PS. For all alloys the
Peierls stress values obtained from the Peierls barrier are lower than those
from the direct method. The difference is largest, 14 %, for W alloys with 10
% Re, or 2 % Fe. Thus, the ${{\tilde{\sigma}}}_{p}$ results, introduction of 2
% Fe to tungsten reduces the Peierls stress by 12 %. While for the pure W and
W alloy with 5 % of Fe it is below 5 %. Gröger and Vitek Gröger and Vitek
(2012) connected the discrepancy in the results of two methods with the fact
that both drag and Nudged Elastic Band Jonsson et al. (1998) (NEB) methods
garantee that images of the system are distributed uniformly along the minimum
energy path and do not implay that dislocation position is distributed along
this path. In order to overcome this problem, the authors introduced a
modified NEB Gröger and Vitek (2012) method which gives a Peierls stress that
agrees within 8 % of the directly calculated stress. However we believe that
the 14 % accuracy obtained in our calculation is reasonably good considering
that maintains the same trend in ${{\sigma}}_{p}$. It should be metioned that
problems with direct application of drag or NEB methods can be more
substentional. According to modeling result obtained in the deformed cell the
dislocation core structure is slightly modified with increase of deformation
and then jumps directly to next ”easy” core position. The dislocation core is
not observed in metastable, ”split” core position. Thus the agreemnt between
data presented in Table LABEL:PS gives a support to application of drag method
to calculation of Peierls stress.
The reduction of Peierls stress/barrier due to alloying with transition metals
with a higher number of valence electrons and the increase for lower number of
valence electrons is supported by existing experimental data. Hardening was
observed in W-Zr alloys Pod’yachev and Gavrilyuk (1975) and softening in
W-Re/Ir alloys Luo et al. (1991). Very interesting experimental results for Mo
alloys with group VIIIA $3d$ transition metals were presented by Hiraoka et al
Hiraoka et al. (2004). The authors report a decrease of yield strength with
addition of any element from group VIIIA, Fe, Co, Ni or $4d$ Pd. The author
concluded that this effect could be well understood in the terms number of
valence electrons of alloying elements, and that the effect of atomic size
mismatch is secondary and minor. Although the value of yield strength is not
directly related to ductility, the observed correlation of properties with
$e/a$ value supports the validity of our calculation using the VCA approach.
## IV Conclusions
The influence of alloying tungsten with transition metal solutes on the
elastic properties and properties of $1/2\langle 111\rangle$ screw dislocation
based on electronic structure calculations. In comparison with a standard
supercell method, we demonstrated that the virtual crystal approximation gives
a fairly good description of such alloying, especially in the case of 4$d$ and
5$d$ transition metals. For the case of alloying with transition metals from
group VIIIA, the modification of elastic constants, Peierls stress and barrier
can be understood within the rigid band approximation. This means that alloys
with TM concentrations leading to the same number of electrons per atoms
exhibit a similar reduction of $C^{\prime}$ modulus and elastic anisotropy
$A$. Together with this modification of elastic moduli the value of the
Peierls stress and barrier are reduced by alloying with transition metals from
group VIIIA. In addition to the Peierls barrier reduction, alloying with any
metal from this group changes the ”easy” core dislocation structure from the
symmetric to asymmetric configuration. This similarity allows us to conclude
that the search for an alternative to Re could be reduced to transition metals
from the group VIIIA.
## V Acknowledgments
Work was supported by the U.S. Department of Energy Office of Fusion Energy
Sciences. This research used resources of the National Energy Research
Scientific Computing Center, which is supported by the Office of Science of
the U.S. Department of Energy.
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* Jonsson et al. (1998) H. Jonsson, G. Mills, and K. W. Jacobsen, in _Classical and Quantum Dynamics in Condensed Phase Simulations_ , edited by E. B. J. Berne, G. Ciccotti, and D. F. Coker (World Scientific, 1998), p. 385\.
* Pod’yachev and Gavrilyuk (1975) V. N. Pod’yachev and M. I. Gavrilyuk, Metal Science and Heat Treatment 17, 299 (1975).
###### List of Figures
1. 1 The [111] projection of dislocation dipole unit cell, where atoms are shown by circles and arrows corresponds to differential displacements.
2. 2 DOS for pure W, blue solid line, W0.9375Re0.0625, red dashed line, and W0.9375Zr0.0625, green dash-dotted line, calculated using VCA.
3. 3 W0.9375Fe0.0625 DOS per atom calculated using a $2\times 2\times 2$ supercell model, blue dashed line, and VCA, red solid line. The Fermi energy corresponds to zero.
4. 4 $C^{\prime}$ and $A$ as a function of number of electrons per atom $e/a$ calculated for W1-xTMx using VCA.
5. 5 The DD map of ”easy” core dislocation in W1-xTMx calculated using the VCA approach. Some of arrows are highlighted in green to illustrate a change of core symmetry. The sites 1, 4 and 5 correspond to the stable ”easy” core dislocation center, while sites 2 and 3 indicate the metastable ”split” core center. The $z$-coordinate of atoms in columns shown by filled circles are fixed during calculation of Peierls barrier within the reaction coordinate method. The symmetric and asymmetric core glide paths are shown by gray bands.
6. 6 Shear stress and total energy, $\Delta E$, per dislocation per Burgers vector $\vec{b}$ (shown in the inset) as a function of strain, $\varepsilon$, for pure $W$ (shown by squares), W0.90Re0.10 (shown by circles), W0.95Fe0.05 (shown by up pointing triangles) and W0.98Fe0.02 (shown by down pointing triangles).
7. 7 Calculated Peierls barrier, where reaction coordinate, $r_{c}$, equal 0 for ”easy” core configuration and 1 for ”split” core. The curves are normalized in the center insert. The detailed structure of the calculated Peierls barrier in pure W around maximum is shown in the insert at upper right.
8. 8 The height of Peierls barrier for the set of W alloys.
###### List of Tables
1. 1 Comparison of lattice parameter ($a$) in Å, bulk modulus ($B$) and elastic constants ($C_{ij}$) in GPa calculated using the VCA and supercell (SC) approaches for W1-xTMx. The concentration $x=0.0625$ corresponds to W15TM.
2. 2 Experimental and calculated(based on VCA) values of lattice parameter ($a$) in Å, bulk modulus ($B$) and elastic constants ($C_{ij}$) in GPa for W1-xTMx. The concentration for all TM except Re is chosen to keep number of electrons per atom $(e/a)=6.10$.
3. 3 In order to compare the reduction of Peierls stress calculated using the direct deformation method, ${\sigma}_{p}$ in GPa, and from the Peierls barrier using Eq.2, ${\tilde{\sigma}}_{p}$ in W, W0.90Re0.10, W0.95Fe0.05, W0.98Fe0.02 and W0.97Ru0.03.
Figure 1: The [111] projection of dislocation dipole unit cell, where atoms
are shown by circles and arrows corresponds to differential displacements.
Figure 2: DOS for pure W, blue solid line, W0.9375Re0.0625, red dashed line,
and W0.9375Zr0.0625, green dash-dotted line, calculated using VCA.
Figure 3: W0.9375Fe0.0625 DOS per atom calculated using a $2\times 2\times 2$
supercell model, blue dashed line, and VCA, red solid line. The Fermi energy
corresponds to zero.
Figure 4: $C^{\prime}$ and $A$ as a function of number of electrons per atom
$e/a$ calculated for W1-xTMx using VCA.
Figure 5: The DD map of ”easy” core dislocation in W1-xTMx calculated using
the VCA approach. Some of arrows are highlighted in green to illustrate a
change of core symmetry. The sites 1, 4 and 5 correspond to the stable ”easy”
core dislocation center, while sites 2 and 3 indicate the metastable ”split”
core center. The $z$-coordinate of atoms in columns shown by filled circles
are fixed during calculation of Peierls barrier within the reaction coordinate
method. The symmetric and asymmetric core glide paths are shown by gray bands.
Figure 6: Shear stress and total energy, $\Delta E$, per dislocation per
Burgers vector $\vec{b}$ (shown in the inset) as a function of strain,
$\varepsilon$, for pure $W$ (shown by squares), W0.90Re0.10 (shown by
circles), W0.95Fe0.05 (shown by up pointing triangles) and W0.98Fe0.02 (shown
by down pointing triangles).
Figure 7: Calculated Peierls barrier, where reaction coordinate, $r_{c}$,
equal 0 for ”easy” core configuration and 1 for ”split” core. The curves are
normalized in the center insert. The detailed structure of the calculated
Peierls barrier in pure W around maximum is shown in the insert at upper
right.
Figure 8: The height of Peierls barrier for the set of W alloys.
Table 1: Comparison of lattice parameter ($a$) in Å, bulk modulus ($B$) and elastic constants ($C_{ij}$) in GPa calculated using the VCA and supercell (SC) approaches for W1-xTMx. The concentration $x=0.0625$ corresponds to W15TM. TM | Approach | $a$ | $B$ | $C^{\prime}$ | $C_{44}$ | $C_{11}$ | $C_{12}$
---|---|---|---|---|---|---|---
pure W | VCA | 3.1903 | 304 | 160 | 141 | 518 | 197
| SC | 3.1896 | 308 | 157 | 139 | 517 | 203
Zr | VCA | 3.2095 | 287 | 153 | 126 | 491 | 185
| SC | 3.2079 | 286 | 150 | 127 | 489 | 186
Ta | VCA | 3.1946 | 298 | 161 | 136 | 513 | 191
| SC | 3.1963 | 308 | 157 | 135 | 517 | 203
Fe | VCA | 3.1802 | 284 | 132 | 128 | 459 | 196
| SC | 3.1716 | 300 | 140 | 132 | 487 | 207
Co | VCA | 3.1788 | 284 | 125 | 115 | 450 | 200
| SC | 3.1725 | 297 | 129 | 131 | 469 | 211
Ni | VCA | 3.1830 | 279 | 116 | 126 | 434 | 202
| SC | 3.1743 | 295 | 126 | 131 | 463 | 211
Ru | VCA | 3.1772 | 308 | 142 | 142 | 498 | 213
| SC | 3.1842 | 301 | 135 | 137 | 481 | 211
Rh | VCA | 3.1710 | 309 | 135 | 141 | 490 | 219
| SC | 3.1862 | 298 | 124 | 133 | 462 | 215
Pd | VCA | 3.1639 | 310 | 129 | 139 | 482 | 224
| SC | 3.1901 | 293 | 116 | 131 | 448 | 216
Re | VCA | 3.1857 | 308 | 151 | 144 | 510 | 207
| SC | 3.1859 | 308 | 150 | 143 | 508 | 208
Os | VCA | 3.1809 | 312 | 143 | 146 | 503 | 216
| SC | 3.1848 | 307 | 136 | 144 | 489 | 216
Ir | VCA | 3.1736 | 316 | 147 | 148 | 513 | 218
| SC | 3.1865 | 304 | 125 | 140 | 470 | 221
Pt | VCA | 3.1687 | 317 | 132 | 148 | 494 | 229
| SC | 3.1917 | 299 | 113 | 135 | 449 | 223
Table 2: Experimental and calculated(based on VCA) values of lattice parameter ($a$) in Å, bulk modulus ($B$) and elastic constants ($C_{ij}$) in GPa for W1-xTMx. The concentration for all TM except Re is chosen to keep number of electrons per atom $(e/a)=6.10$. TM | $x$ | $a$ | $B$ | $C^{\prime}$ | $C_{44}$ | $C_{11}$ | $C_{12}$
---|---|---|---|---|---|---|---
experiment Ayres et al. (1974)
W | 0.0000 | 3.1659 | 314 | 164 | 163 | 533 | 205
calculated, virtual crystal approximation
Re | 0.0000 | 3.1903 | 304 | 160 | 141 | 518 | 197
| 0.0300 | 3.1880 | 306 | 156 | 142 | 514 | 202
| 0.0500 | 3.1866 | 307 | 153 | 143 | 512 | 205
| 0.1000 | 3.1831 | 310 | 146 | 145 | 504 | 212
| 0.3000 | 3.1689 | 322 | 130 | 156 | 495 | 235
| 1.0000 | 3.1317 | 350 | -30 | 163 | 309 | 370
Zr | 0.0250 | 3.1972 | 297 | 161 | 136 | 511 | 190
Ta | 0.0500 | 3.1936 | 299 | 162 | 137 | 515 | 191
Fe | 0.0250 | 3.1849 | 296 | 148 | 136 | 494 | 198
Ru | 0.0250 | 3.1846 | 306 | 153 | 141 | 510 | 204
Rh | 0.0167 | 3.1846 | 306 | 153 | 141 | 509 | 204
Os | 0.0250 | 3.1862 | 307 | 153 | 143 | 511 | 205
Ir | 0.0167 | 3.1849 | 308 | 154 | 143 | 512 | 205
Table 3: In order to compare the reduction of Peierls stress calculated using the direct deformation method, ${\sigma}_{p}$ in GPa, and from the Peierls barrier using Eq.2, ${\tilde{\sigma}}_{p}$ in W, W0.90Re0.10, W0.95Fe0.05, W0.98Fe0.02 and W0.97Ru0.03. | $\sigma_{p}$ | ${\tilde{\sigma}}_{p}$
---|---|---
W | 1.71 | 1.64
W0.95Zr0.05 | 2.18 | 1.82
W0.90Re0.10 | 1.37 | 1.15
W0.95Fe0.05 | 1.09 | 1.05
W0.98Fe0.02 | 1.65 | 1.40
W0.97Ru0.03 | 1.60 | 1.35
|
arxiv-papers
| 2012-01-05T18:13:04 |
2024-09-04T02:49:25.975922
|
{
"license": "Public Domain",
"authors": "G. D. Samolyuk and Y. N. Osetsky and R. E. Stoller",
"submitter": "German Samolyuk Dr.",
"url": "https://arxiv.org/abs/1201.1245"
}
|
1201.1285
|
# WorldWide Telescope in Research and Education
Alyssa Goodman1, Jonathan Fay2, August Muench1, Alberto Pepe1, Patricia
Udompraseret1, and Curtis Wong2 1Harvard-Smithsonian Center for Astrophysics,
Cambridge, MA 2Microsoft Research, Redmond, WA
###### Abstract
The WorldWide Telescope computer program, released to researchers and the
public as a free resource in 2008 by Microsoft Research, has changed the way
the ever-growing Universe of online astronomical data is viewed and
understood. The WWT program can be thought of as a scriptable, interactive,
richly visual browser of the multi-wavelength Sky as we see it from Earth, and
of the Universe as we would travel within it. In its web API format, WWT is
being used as a service to display professional research data. In its desktop
format, WWT works in concert (thanks to SAMP and other IVOA standards) with
more traditional research applications such as ds9, Aladin and TOPCAT. The WWT
Ambassadors Program (founded in 2009) recruits and trains astrophysically-
literate volunteers (including retirees) who use WWT as a teaching tool in
online, classroom, and informal educational settings. Early quantitative
studies of WWTA indicate that student experiences with WWT enhance science
learning dramatically. Thanks to the wealth of data it can access, and the
growing number of services to which it connects, WWT is now a key linking
technology in the Seamless Astronomy environment we seek to offer researchers,
teachers, and students alike.
## 1 Introduction
The “WorldWide Telescope” was originally envisioned, by Jim Gray and Alex
Szalay in 2001, as “an Archetype for Online Science” (Szalay & Gray 2001).
When the program called “WordWide Telescope” was released by Microsoft
Research in 2008, it was received by the press primarily as an amazing new
tool for outreach–offering access to the world’s best astronomical imagery and
expertise to all. Since its release, though, the free WWT software has become
both an essential new piece of the research ecosystem and the amazing
educational tool the press perceived (Goodman & Wong 2009). All versions of
WWT discussed in this article are available at no cost for non-commercial use
at worldwidetelesope.org.
## 2 WWT in Astronomical Research
The full version of WWT (WWT-desktop) runs as a standalone desktop application
in Windows (either on a Windows-only system, or on a Mac running a Windows
Virtual Machine). In addition, WWT runs within a web browser on any machine
capable of running Silverlight (e.g. almost any Mac or PC in use today). On
the web, users and developers have a choice of: 1) a menu-driven version of
WWT that looks identical to WWT-desktop for Sky-based work; or 2) an API
offering a fully-functional data-viewing window plus added functionality (e.g.
“Finder Scope”) as-desired (see Figure 1).
Figure 1.: Annotated screen shot of the WorldWide Telescope application, as it
appears in either its Windows desktop version or as a Silverlight-based Web
application. Notice in particular the Finder Scope functionality associated
with the (movable) cross-hair.
At present, WWT-desktop is SAMP-compliant, and near-future compliance with
WebSAMP (Taylor 2012, this volume) is planned. In concert with SAMP, and other
VO tools, WWT is a fantastic viewer for manipulating, overlaying, and cross-
matching image and catalog-based information. In its API form, WWT makes a
powerful visualization tool, and a clear example of how it can be used to show
survey coverage and user-selectable data layers is given at the COMPLETE
Survey’s data-coverage page, at
www.worldwidetelescope.org/COMPLETE/WWTCoverageTool.htm.
The fast, smooth, panning and zooming in and out possible within all forms of
WWT, in combination with its “context globe” (see Figure 1), scale indicators,
and built-in all-sky multi-wavelength views offers a contextual perspective on
astronomical data that has not been possible before. This kind of context is
typically missing in professionals’ views of their data, and its presence
allows for better understanding of the potential interactions amongst various
physical processes.
The context WWT offers is now also accessible from within ADS Labs
(adslabs.org), under the “SIMBAD Objects” facet there, making the linkages
between WWT and ADS now truly two-way. (WWT has, since its initial release,
offered a direct link to ADS articles about any point in the sky via the
“Research, Information, Look up publications on ADS” option in the Finder
Scope, and direct links to CDS/SIMBAD are accessible similarly via “Research,
Information, Look up on SIMBAD.”) Soon, the just-funded NASA ADS All Sky
Survey will employ WWT as one of a few all-sky viewers capable of showing
image holdings extracted from ADS articles in-context on the sky, filterable
by subject, object, author, time, and more (Pepe et al. 2012, this volume).
## 3 WWT in Teaching and Learning
In education, WWT is being used at all age and expertise levels. The WorldWide
Telescope Ambassadors Program (WWTA), founded as a Harvard-Microsoft
collaboration in 2009, trains astronomy experts to use WWT in both informal
and formal (classroom) environments (Udomprasert et al. 2012). Ambassadors are
trained to create “Tours” within WWT, and to facilitate the use of the program
and the Tours. Tours are interactive paths through WWT’s Sky or 3D content
designed to make a point. Sample tours include Galileo’s New Order, which
explains how Galileo’s discovery of Jupiter’s moons led to the adoption of our
current heliospheric view of our Solar System, and John Huchra’s Universe, a
tribute to John Huchra that explains the significance of redshift surveys to
our understanding of our Universe. Tours created and/or vetted by experts are
served at the WWT Ambassadors website (wwtambassadors.org, and many are also
accessible from the “Tours” menu tab within the program). The WWTA website
provides a faceted education-friendly view of all Tour content, as well as an
area where students and teachers working on their own Tours can share them.
Figure 2.: Results from the Clarke Middle School Pilot of the WorldWide
Telescope Ambassadors Program (2010).
In the 2010 pilot of WWTA at Clarke Middle School in Lexington, MA, two groups
of $\sim 75$ sixth-grade students were studied: the “treatment” group
participated in WWTA and the other group was a control, with access only to
the standard curriculum. Treatment students all created their own Tours, in
groups of three or four. Ambassadors in the classroom facilitated the Tour
creation, mostly by pointing students to additional online astrophysical
resources, as these 11-year-olds typically did not need much help learning to
use the program. Each group of students was surveyed before and after six-
weeks of astronomy study, and the comparative results are shown in Figure 2.
WWTA had dramatic effects across the board, increasing knowledge,
understanding, and interest in science, and even increasing student interest
in using real, physical, telescopes! (Goodman et al. 2011). In 2011, the WWTA
Program was successfully expanded to several more Boston-area schools, and we
(in collaboration with Stephen Strom of NOAO) are actively seeking funding to
facilitate much-asked-for US and international expansion of the Program.
In higher-education, WWT is now part of several University courses and labs,
and we (in collaboration with Edwin Ladd of Bucknell University) have
additional proposals pending to develop curricular materials appropriate to
these higher levels. Since the best of today’s university students are the
near-future’s most important researchers, the “educational” environment of
universities actually offers the greatest potential for expanding WWT’s use in
research in the near-term future.
Astronomy has been aptly called a “gateway drug” for STEM (Science,
Technology, Engineering and Math) learning, and as such we feel a
responsibility to expand the WWTA site and programs to include all ages of
learners, from pre-K to retirees.
## 4 WWT in the Future
WWT is a key part of a larger program based at the Harvard-Smithsonian Center
for Astrophysics called “Seamless Astronomy” (for a list of collaborators, see
projects.iq.harvard.edu/seamlessastronomy/). The vision of Seamless Astronomy
is shared by many ADASS participants: astronomical research tools should
interoperate so well that boundaries between data archives, countries, and
program(s) functionality all but disappear. WWT has demonstrated how several
forms of the same tool, accessing many different data archives, and connecting
“seamlessly” (thanks to SAMP and other IVOA standards) to many other tools and
services, can make astronomy research and STEM education easier, and so much
more fun.
## References
* Goodman et al. (2011) Goodman, A. A., Udomprasert, P. S., Kent, B., Sathiapal, H., & Smareglia, R. 2011, in Astronomical Data Analysis Software and Systems XX, edited by I. N. Evans, A. Accomazzi, D. J. Mink, & A. H. Rots, vol. 442 of Astronomical Society of the Pacific Conference Series, 659
* Goodman & Wong (2009) Goodman, A. A., & Wong, C. G. 2009, Bringing the night sky closer: discoveries in the data deluge (Redmond, WA: Microsoft Research)
* Szalay & Gray (2001) Szalay, A., & Gray, J. 2001, Science, 293, 2037
* Udomprasert et al. (2012) Udomprasert, P. S., Goodman, A., & Wong, C. 2012, WWT Ambassadors: Worldwide Telescope For Interactive Learning (ASP Conference series, in press)
|
arxiv-papers
| 2012-01-05T16:39:23 |
2024-09-04T02:49:25.985880
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alyssa Goodman, Jonathan Fay, August Muench, Alberto Pepe, Patricia\n Udomprasert, Curtis Wong",
"submitter": "Alyssa A. Goodman",
"url": "https://arxiv.org/abs/1201.1285"
}
|
1201.1309
|
# $q$-Analogue of $p$-Adic $\log$ $\Gamma$ type functions associated with
Modified $q$-extension of Genocchi numbers with weight $\alpha$ and $\beta$
Serkan Aracı University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com and Mehmet
Acikgoz University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY acikgoz@gantep.edu.tr
(Date: January 20, 2012)
###### Abstract.
The fundamental aim of this paper is to describe $q$-Analogue of $p$-adic
$\log$ gamma functions with weight alpha and beta. Moreover, we give
relationship between $p$-adic $q$-$\log$ gamma funtions with weight
($\alpha,\beta$) and $q$-extension of Genocchi numbers with weight alpha and
beta and modified $q$-Euler numbers with weight $\alpha$
###### Key words and phrases:
Modified $q$-Genocchi numbers with weight alpha and beta, Modified $q$-Euler
numbers with weight alpha and beta, $p$-adic log gamma functions.
###### 2000 Mathematics Subject Classification:
Primary 46A15, Secondary 41A65
## 1\. Introduction
Assume that $p$ be a fixed odd prime number. Throughout this paper
$\mathbb{Z},$ $\mathbb{Z}_{p},$ $\mathbb{Q}_{p}$ and $\mathbb{C}_{p}$ will
denote by the ring of integers, the field of $p$-adic rational numbers and the
completion of the algebraic closure of $\mathbb{Q}_{p},$ respectively. Also we
denote $\mathbb{N}^{\ast}=\mathbb{N}\cup\left\\{0\right\\}$ and
$\exp\left(x\right)=e^{x}.$ Let
$v_{p}:\mathbb{C}_{p}\rightarrow\mathbb{Q}\cup\left\\{\infty\right\\}\left(\mathbb{Q}\text{
is the field of rational numbers}\right)$ denote the $p$-adic valuation of
$\mathbb{C}_{p}$ normalized so that $v_{p}\left(p\right)=1$. The absolute
value on $\mathbb{C}_{p}$ will be denoted as $\left|.\right|_{p}$, and
$\left|x\right|_{p}=p^{-v_{p}\left(x\right)}$ for $x\in\mathbb{C}_{p}.$ When
one talks of $q$-extensions, $q$ is considered in many ways, e.g. as an
indeterminate, a complex number $q\in\mathbb{C},$ or a $p$-adic number
$q\in\mathbb{C}_{p},$ If $q\in\mathbb{C}$ we assume that $\left|q\right|<1.$
If $q\in\mathbb{C}_{p},$ we assume $\left|1-q\right|_{p}<p^{-\frac{1}{p-1}},$
so that $q^{x}=\exp\left(x\log q\right)$ for $\left|x\right|_{p}\leq 1.$ We
use the following notation
(1.1) $\left[x\right]_{q}=\frac{1-q^{x}}{1-q},\text{ \
}\left[x\right]_{-q}=\frac{1-\left(-q\right)^{x}}{1+q}$
where $\lim_{q\rightarrow 1}\left[x\right]_{q}=x;$ cf. [1-24].
For a fixed positive integer $d$ with $\left(d,f\right)=1,$ we set
$\displaystyle X$ $\displaystyle=$ $\displaystyle
X_{d}=\lim_{\overleftarrow{N}}\mathbb{Z}/dp^{N}\mathbb{Z},$ $\displaystyle
X^{\ast}$ $\displaystyle=$
$\displaystyle\underset{\underset{\left(a,p\right)=1}{0<a<dp}}{\cup}a+dp\mathbb{Z}_{p}$
and
$a+dp^{N}\mathbb{Z}_{p}=\left\\{x\in X\mid x\equiv
a\left(\mathop{\mathrm{m}od}dp^{N}\right)\right\\},$
where $a\in\mathbb{Z}$ satisfies the condition $0\leq a<dp^{N}.$
It is known that
$\mu_{q}\left(x+p^{N}\mathbb{Z}_{p}\right)=\frac{q^{x}}{\left[p^{N}\right]_{q}}$
is a distribution on $X$ for $q\in\mathbb{C}_{p}$ with
$\left|1-q\right|_{p}\leq 1.$
Let $UD\left(\mathbb{Z}_{p}\right)$ be the set of uniformly differentiable
function on $\mathbb{Z}_{p}.$ We say that $f$ is a uniformly differentiable
function at a point $a\in\mathbb{Z}_{p},$ if the difference quotient
$F_{f}\left(x,y\right)=\frac{f\left(x\right)-f\left(y\right)}{x-y}$
has a limit $f^{{\acute{}}}\left(a\right)$ as
$\left(x,y\right)\rightarrow\left(a,a\right)$ and denote this by $f\in
UD\left(\mathbb{Z}_{p}\right).$ The $p$-adic $q$-integral of the function
$f\in UD\left(\mathbb{Z}_{p}\right)$ is defined by
(1.2)
$I_{q}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{q}\left(x\right)=\lim_{N\rightarrow\infty}\frac{1}{\left[p^{N}\right]_{q}}\sum_{x=0}^{p^{N}-1}f\left(x\right)q^{x}$
The bosonic integral is considered by Kim as the bosonic limit $q\rightarrow
1,$ $I_{1}\left(f\right)=\lim_{q\rightarrow 1}I_{q}\left(f\right).$ Similarly,
the $p$-adic fermionic integration on $\mathbb{Z}_{p}$ defined by Kim as
follows:
$I_{-q}\left(f\right)=\lim_{q\rightarrow-q}I_{q}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{-q}\left(x\right)$
Let $q\rightarrow 1,$ then we have $p$-adic fermionic integral on
$\mathbb{Z}_{p}$ as follows:
$I_{-1}\left(f\right)=\lim_{q\rightarrow-1}I_{q}\left(f\right)=\lim_{N\rightarrow\infty}\sum_{x=0}^{p^{N}-1}f\left(x\right)\left(-1\right)^{x}.$
Stirling asymptotic series are defined by
(1.3)
$\log\left(\frac{\Gamma\left(x+1\right)}{\sqrt{2\pi}}\right)=\left(x-\frac{1}{2}\right)\log
x+\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n+1}}{n\left(n+1\right)}\frac{B_{n+1}}{x^{n}}-x$
where $B_{n}$ are familiar $n$-th Bernoulli numbers cf. [6, 8, 9, 25].
Recently, Araci et al. defined modified $q$-Genocchi numbers and polynomials
with weight $\alpha$ and $\beta$ in [4, 5] by the means of generating
function:
(1.4)
$\sum_{n=0}^{\infty}g_{n,q}^{\left(\alpha,\beta\right)}\left(x\right)\frac{t^{n}}{n!}=t\int_{\mathbb{Z}_{p}}q^{-\beta\xi}e^{\left[x+\xi\right]_{q^{\alpha}}t}d\mu_{-q^{\beta}}\left(\xi\right)$
So from above, we easily get Witt’s formula of modified$\ q$-Genocchi numbers
and polynomials with weight $\alpha$ and $\beta$ as follows:
(1.5)
$\frac{g_{n+1,q}^{\left(\alpha,\beta\right)}\left(x\right)}{n+1}=\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left[x+\xi\right]_{q^{\alpha}}^{n}d\mu_{-q^{\beta}}\left(\xi\right)$
where
$g_{n,q}^{\left(\alpha,\beta\right)}\left(0\right):=g_{n,q}^{\left(\alpha,\beta\right)}$
are modified $q$ extension of Genocchi numbers with weight $\alpha$ and
$\beta$ cf. [4,5].
In [21], Rim and Jeong are defined modified $q$-Euler numbers with weight
$\alpha$ as follows:
(1.6)
$\widetilde{\xi}_{n,q}^{\left(\alpha\right)}=\int_{\mathbb{Z}_{p}}q^{-t}\left[t\right]_{q^{\alpha}}d\mu_{-q}\left(t\right)$
From expressions of (1.5) and (1.6), we get the following Proposition 1:
###### Proposition 1.
The following
(1.7)
$\widetilde{\xi}_{n,q}^{\left(\alpha\right)}=\frac{g_{n+1,q}^{\left(\alpha,1\right)}}{n+1}$
is true.
In previous paper [6], Araci, Acikgoz and Park introduced weighted
$q$-Analogue of $p$-Adic $\log$ gamma type functions and they derived some
interesting identities in Analytic Numbers Theory and in $p$-Adic Analysis.
They were motivated from paper of T. Kim by ”On a $q$-analogue of the $p$-adic
log gamma functions and related integrals, J. Number Theory, 76 (1999), no. 2,
320-329.” We also introduce $q$-Analogue of $p$-Adic $\log$ gamma type
function with weight $\alpha$ and $\beta.$ We derive in this paper some
interesting identities this type of functions.
On p-adic $\log$ $\Gamma$ function with weight $\alpha$ and $\beta$
In this part, from (1.2), we begin with the following nice identity:
(1.8) $I_{-q}^{\left(\beta\right)}\left(q^{-\beta
x}f_{n}\right)+\left(-1\right)^{n-1}I_{-q}^{\left(\beta\right)}\left(q^{-\beta
x}f\right)=\left[2\right]_{q^{\beta}}\sum_{l=0}^{n-1}\left(-1\right)^{n-1-l}f\left(l\right)$
where $f_{n}\left(x\right)=f\left(x+n\right)$ and $n\in\mathbb{N}$ (see [4]).
In particular for $n=1$ into (1.8), we easily see that
(1.9) $I_{-q}^{\left(\beta\right)}\left(q^{-\beta
x}f_{1}\right)+I_{-q}^{\left(\beta\right)}\left(q^{-\beta
x}f\right)=\left[2\right]_{q^{\beta}}f\left(0\right).$
With the simple application, it is easy to indicate as follows:
(1.10)
$\left(\left(1+x\right)\log\left(1+x\right)\right)^{{\acute{}}}=1+\log\left(1+x\right)=1+\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n+1}}{n\left(n+1\right)}x^{n}$
where
$\left(\left(1+x\right)\log\left(1+x\right)\right)^{{\acute{}}}=\frac{d}{dx}\left(\left(1+x\right)\log\left(1+x\right)\right)$
By expression of (1.10), we can derive
(1.11)
$\left(1+x\right)\log\left(1+x\right)=\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n+1}}{n\left(n+1\right)}x^{n+1}+x+c,\text{
where }c\text{ is constant.}$
If we take $x=0,$ so we get $c=0.$ By expression of (1.10) and (1.11), we
easily see that,
(1.12)
$\left(1+x\right)\log\left(1+x\right)=\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n+1}}{n\left(n+1\right)}x^{n+1}+x.$
It is considered by T. Kim for $q$-analogue of $p$ adic locally analytic
function on $\mathbb{C}_{p}\backslash\mathbb{Z}_{p}$ as follows:
(1.13)
$G_{p,q}\left(x\right)=\int_{\mathbb{Z}_{p}}\left[x+\xi\right]_{q}\left(\log\left[x+\xi\right]_{q}-1\right)d\mu_{-q}\left(\xi\right)\text{
(for detail, see[5,6]).}$
By the same motivation of (1.13), in previous paper [6], $q$-analogue of
$p$-adic locally analytic function on $\mathbb{C}_{p}\backslash\mathbb{Z}_{p}$
with weight $\alpha$ is considered
(1.14)
$G_{p,q}^{\left(\alpha\right)}\left(x\right)=\int_{\mathbb{Z}_{p}}\left[x+\xi\right]_{q^{\alpha}}\left(\log\left[x+\xi\right]_{q^{\alpha}}-1\right)d\mu_{-q}\left(\xi\right)\text{
}$
In particular $\alpha=1$ into (1.14), we easily see that,
$G_{p,q}^{\left(1\right)}\left(x\right)=G_{p,q}\left(x\right).$
With the same manner, we introduce $q$-Analoge of $p$-adic locally analytic
function on $\mathbb{C}_{p}\backslash\mathbb{Z}_{p}$ with weight $\alpha$ and
$\beta$ as follows:
(1.15)
$G_{p,q}^{\left(\alpha,\beta\right)}\left(x\right)=\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left[x+\xi\right]_{q^{\alpha}}\left(\log\left[x+\xi\right]_{q^{\alpha}}-1\right)d\mu_{-q^{\beta}}\left(\xi\right)\text{
}$
From expressions of (1.9) and (1), we state the following Theorem:
###### Theorem 1.
The following identity holds:
$G_{p,q}^{\left(\alpha,\beta\right)}\left(x+1\right)+G_{p,q}^{\left(\alpha,\beta\right)}\left(x\right)=\left[2\right]_{q^{\beta}}\left[x\right]_{q^{\alpha}}\left(\log\left[x\right]_{q^{\alpha}}-1\right).$
It is easy to show that,
$\displaystyle\left[x+\xi\right]_{q^{\alpha}}$ $\displaystyle=$
$\displaystyle\frac{1-q^{\alpha\left(x+\xi\right)}}{1-q^{\alpha}}$
$\displaystyle=$ $\displaystyle\frac{1-q^{\alpha x}+q^{\alpha
x}-q^{\alpha\left(x+\xi\right)}}{1-q^{\alpha}}$ $\displaystyle=$
$\displaystyle\left(\frac{1-q^{\alpha x}}{1-q^{\alpha}}\right)+q^{\alpha
x}\left(\frac{1-q^{\alpha\xi}}{1-q^{\alpha}}\right)$ $\displaystyle=$
$\displaystyle\left[x\right]_{q^{\alpha}}+q^{\alpha
x}\left[\xi\right]_{q^{\alpha}}$
Substituting $x\rightarrow\frac{q^{\alpha
x}\left[\xi\right]_{q^{\alpha}}}{\left[x\right]_{q^{\alpha}}}$ into (1.12) and
by using (1), we get interesting formula:
(1.17)
$\left[x+\xi\right]_{q^{\alpha}}\left(\log\left[x+\xi\right]_{q^{\alpha}}-1\right)=\left(\left[x\right]_{q^{\alpha}}+q^{\alpha
x}\left[\xi\right]_{q^{\alpha}}\right)\log\left[x\right]_{q^{\alpha}}+\sum_{n=1}^{\infty}\frac{\left(-q^{\alpha
x}\right)^{n+1}}{n(n+1)}\frac{\left[\xi\right]_{q^{\alpha}}^{n+1}}{\left[x\right]_{q^{\alpha}}^{n}}-\left[x\right]_{q^{\alpha}}$
If we substitute $\alpha=1$ into (1.17), we get Kim’s $q$-Analogue of $p$-adic
$\log$ gamma fuction (for detail, see[8]).
From expression of (1.2) and (1.17), we obtain worthwhile and interesting
theorems as follows:
###### Theorem 2.
For $x\in\mathbb{C}_{p}\backslash\mathbb{Z}_{p}$ the following
(1.18)
$G_{p,q}^{\left(\alpha,\beta\right)}\left(x\right)=\left(\frac{\left[2\right]_{q^{\beta}}}{2}\left[x\right]_{q^{\alpha}}+q^{\alpha
x}\frac{g_{2,q}^{\left(\alpha,\beta\right)}}{2}\right)\log\left[x\right]_{q^{\alpha}}+\sum_{n=1}^{\infty}\frac{\left(-q^{\alpha
x}\right)^{n+1}}{n\left(n+1\right)\left(n+2\right)}\frac{g_{n+1,q}^{\left(\alpha,\beta\right)}}{\left[x\right]_{q^{\alpha}}^{n}}-\left[x\right]_{q^{\alpha}}\frac{\left[2\right]_{q^{\beta}}}{2}$
is true.
###### Corollary 1.
Taking $q\rightarrow 1$ into (1.18), we get nice identity:
$G_{p,1}^{\left(\alpha,\beta\right)}\left(x\right)=\left(x+\frac{G_{2}}{2}\right)\log
x+\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n+1}}{n\left(n+1\right)\left(n+2\right)}\frac{G_{n+1}}{x}-x$
where $G_{n}$ are called famous Genocchi numbers.
###### Theorem 3.
The following nice identity
(1.19)
$G_{p,q}^{\left(\alpha,1\right)}\left(x\right)=\left(\frac{\left[2\right]_{q}}{2}\left[x\right]_{q^{\alpha}}+q^{\alpha
x}\widetilde{\xi}_{1,q}^{\left(\alpha\right)}\right)\log\left[x\right]_{q^{\alpha}}+\sum_{n=1}^{\infty}\frac{\left(-q^{\alpha
x}\right)^{n+1}}{n\left(n+1\right)}\frac{\widetilde{\xi}_{n,q}^{\left(\alpha\right)}}{\left[x\right]_{q^{\alpha}}^{n}}-\frac{\left[2\right]_{q}}{2}\left[x\right]_{q^{\alpha}}$
is true.
###### Corollary 2.
Putting $q\rightarrow 1$ into (1.19), we have the following identity:
$G_{p,1}^{\left(\alpha,\beta\right)}\left(x\right)=\left(x+E_{1}\right)\log
x+\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n+1}}{n\left(n+1\right)}\frac{E_{n}}{x^{n}}-x$
where $E_{n}$ are familiar Euler numbers.
## References
* [1] Araci, S., Acikgoz, M., and Seo, J-J., A note on the weighted $q$-Genocchi numbers and polynomials with Their Interpolation Function, Accepted in Honam Mathematical Journal.
* [2] Araci, S., Erdal, D., and Seo., J-J., 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.
* [3] Araci, S., Seo, J-J., and Erdal, D., New construction weighted ($h$,$q$)-Genocchi numbers and polynomials related to Zeta type function, Discrete Dynamics in nature and Society, Volume 2011, Article ID 487490, 7 pages.
* [4] Araci, S., Acikgoz, M., Qi, Feng., and Jolany, H., A note on the modified $q$-Genocchi numbers and polynomials with weight ($\alpha,\beta$) and Their Interpolation function at negative integer, submitted.
* [5] Araci, S., Acikgoz, M., and Ryoo, C-S., A note on the values of the weighted $q$-Bernstein Polynomials and Modified $q$-Genocchi Numbers with weight $\alpha$ and $\beta$ via the $p$-adic $q$-integral on $\mathbb{Z}_{p},$ submitted
* [6] Araci, S., Acikgoz, M., Park, K-H., A note on the $q$-Analogue of Kim’s $p$-adic $\log$ gamma functions associated with q-extension of Genocchi and Euler polynomials with weight $\alpha,$ submitted.
* [7] Acikgoz, M., and Simsek, Y., On multiple interpolation functions of the Nörlund type $q$-Euler polynomials, Abstract and Applied Analysis, Volume 2009, Article ID 382574, 14 pages.
* [8] Kim, T., A note on the $q$-analogue of $p$-adic $\log$ gamma function, arXiv:0710.4981v1 [math.NT].
* [9] Kim, T., On a $q$-analogue of the $p$-adic log gamma functions and related integrals, J. Number Theory, 76 (1999), no. 2, 320-329.
* [10] Kim, T., On the $q$-extension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007) 1458-1465.
* [11] Kim, T., On the multiple $q$-Genocchi and Euler numbers, Russian J. Math. Phys. 15 (4) (2008) 481-486. arXiv:0801.0978v1 [math.NT]
* [12] Kim, T., On the weighted $q$-Bernoulli numbers and polynomials, Advanced Studies in Contemporary Mathematics 21(2011), no.2, p. 207-215, http://arxiv.org/abs/1011.5305.
* [13] Kim, T., $q$-Volkenborn integration, Russ. J. Math. phys. 9$\left(2002\right),$ 288-299.
* [14] Kim, T., An invariant $p$-adic $q$-integrals on $\mathbb{Z}_{p}$, Applied Mathematics Letters, vol. 21, pp. 105-108,2008.
* [15] Kim, T., $q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals, J. Nonlinear Math. Phys., 14 (2007), no. 1, 15–27.
* [16] Kim, T., New approach to $q$-Euler polynomials of higher order, Russ. J. Math. Phys., 17 (2010), no. 2, 218–225.
* [17] Kim, T., Some identities on the $q$-Euler polynomials of higher order and $q$-Stirling numbers by the fermionic $p$-adic integral on $\mathbb{Z}_{p}$, Russ. J. Math. Phys., 16 (2009), no.4,484–491.
* [18] Kim, T. and Rim, S.-H., On the twisted $q$-Euler numbers and polynomials associated with basic $q$-$l$-functions, Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 738–744, 2007.
* [19] Kim, T., On $p$-adic $q$-$l$-functions and sums of powers, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa.2006.07.071
* [20] Ryoo. C. S., A note on the weighted $q$-Euler numbers and polynomials, Advan. Stud. Contemp. Math. 21(2011), 47-54.
* [21] Rim, S-H., and Jeong, J., A note on the modified $q$-Euler numbers and Polynomials with weight $\alpha,$ International Mathematical Forum, Vol. 6, 2011, no. 65, 3245-3250.
* [22] Simsek, Y., Theorems on twisted $L$-function and twisted Bernoulli numbers, Advan. Stud. Contemp. Math., 11(2005), 205-218.
* [23] Simsek, Y., Twisted $(h,q)$-Bernoulli numbers and polynomials related to twisted $(h,q)-$zeta function and $L$-function, J. Math. Anal. Appl., 324(2006), 790-804.
* [24] Simsek, Y., On $p$-Adic Twisted $q$-$L$-Functions Related to Generalized Twisted Bernoulli Numbers, Russian J. Math. Phys., 13(3)(2006), 340-348.
* [25] Zill, D., and Cullen, M. R., Advanced Engineering Mathematics, Jones and Bartlett, 2005.
|
arxiv-papers
| 2012-01-05T21:35:44 |
2024-09-04T02:49:25.992094
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Mehmet Acikgoz",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1201.1309"
}
|
1201.1315
|
# A UNIFICATION OF THE MULTIPLE TWISTED EULER AND GENOCCHI NUMBERS AND
POLYNOMIALS ASSOCIATED WITH $p$-ADIC $q$-INTEGRAL ON $\mathbb{Z}_{p}$ AT
$q=-1$
Serkan Aracı 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 , Kyoung-Ho Park
Division of General Education-Mathematics, Kwangwoon University, Seoul
139-171, Republic of Korea sagamath@yahoo.co.kr and Hassan Jolany School of
Mathematics, Statistics and Computer Science, University of Tehran, Iran
hassan.jolany@khayam.ut.ac.ir
(Date: October 26, 2011)
###### Abstract.
The present paper deals with unification of the multiple twisted Euler and
Genocchi numbers and polynomials associated with $p$-adic $q$-integral on
$\mathbb{Z}_{p}$ at $q=-1$. Some earlier results of Ozden’s papers in terms of
unification of the multiple twisted Euler and Genocchi numbers and polynomials
associated with $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1$ can be
deduced. We apply the method of generating function and $p$-adic $q$-integral
representation on $\mathbb{Z}_{p}$, which are exploited to derive further
classes of Euler polynomials and Genocchi polynomials. To be more precise we
summarize our results as follows, we obtain some relations between H.Ozden’s
generating function and fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at
$q=-1$. Furthermore we derive Witt’s type formula for the unification of
twisted Euler and Genocchi polynomials. Also we derive distribution formula
(Multiplication Theorem) for multiple twisted Euler and Genocchi numbers and
polynomials associated with $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at
$q=-1$ which yields a deeper insight into the effectiveness of this type of
generalizations. Furthermore we define unification of multiple twisted zeta
function and we obtain an interpolation formula between unification of
multiple twisted zeta function and unification of the multiple twisted Euler
and Genocchi numbers at negative integer. Our new generating function possess
a number of interesting properties which we state in this paper.
###### Key words and phrases:
###### 1991 Mathematics Subject Classification:
05A10, 11B65, 28B99, 11B68, 11B73.
## 1\. Introduction, Definitions and Notations
Bernoulli numbers introduced by Jacques Bernoulli (1654-1705), in the second
part of his treatise published in 1713, Ars conjectandi, at the time,
Bernoulli numbers were used for writing the infinite series expansions of
hyperbolic and trigonometric functions. Van den berg was the first to discuss
finding recurrence formulae for the Bernoulli numbers with arbitrary sized
gaps (1881). Ramanujan showed how gaps of size 7 could be found, and
explicitly wrote out the recursion for gaps, of size 6. Lehmer in 1934
extended these methods to Euler numbers, Genocchi numbers, and Lucas numbers
(1934), and calculated the 196-th Bernoulli numbers. The study of generalized
Bernoulli, Euler and Genocchi numbers and polynomials and their combinatorial
relations has received much attention [20], [21], [22]-[25], [26], [27], [28],
[29], [30]. Generalized Bernoulli polynomials, generalized Euler polynomials
and generalized Genocchi numbers and polynomials are the signs of very strong
bond between elementary number theory, complex analytic number theory,
Homotopy theory (stable Homotopy groups of spheres), differential topology
(differential structures on spheres), theory of modular forms (Eisenstein
series), $p$-adic analytic number theory ($p$-adic $L$-functions), quantum
physics(quantum Groups). $p$-adic numbers were invented by Kurt Hensel around
the end of the nineteenth century. In spite of their being already one hundred
years old, these numbers are still today enveloped in an aura of mystery
within the scientific community. The $p$-adic integral was used in
mathematical physics, for instance, the functional equation of the $q$-zeta
function, $q$-stirling numbers and $q$-Mahler theory of integration with
respect to the ring $\mathbb{Z}_{p}$ together with Iwasawa’s $p$-adic $q$-$L$
functions. Also the $p$-adic interpolation functions of the Bernoulli and
Euler polynomials have been treated by Tsumura [31] and Young [32]. Professor
T.Kim [3]-[17] also studied on $p$-adic interpolation functions of these
numbers and polynomials. In [33], Carlitz originally constructed $q$-Bernoulli
numbers and polynomials. These numbers and polynomials are studied by many
authors (see cf. [3]-[19], [34], [35], [38]). In the last decade, a surprising
number of papers appeared proposing new generalizations of the Bernoulli,
Euler and Genocchi polynomials to real and complex variables. In [3]-[18], Kim
studied some families of multiple Bernoulli, Euler and Genocchi numbers and
polynomials. By using the fermionic $p$-adic invariant integral on
$\mathbb{Z}_{p}$, he constructed $p$-adic Bernoulli, Euler and Genocchi
numbers and polynomials of higher order. A unification (and generalization) of
Bernoulli polynomials and Euler polynomials with a,b and c parameters first
was introduced and investigated by Q.-M.Luo [22], [23], [24], [25]. After he
with H.M.Srivastava defined unification (and generalization) of Apostol type
Bernoulli polynomials with a, b and c parameters of higher order [25]. After
Hacer Ozden et al [35]. unified and extended the generating functions of the
generalized Bernoulli polynomials, the generalized Euler polynomials and the
generalized Genocchi polynomials associated with the positive real parameters
a and b and the complex parameter. Also they by applying the Mellin
transformation to the generating function of the unification of Bernoulli,
Euler and Genocchi polynomials, constructed a unification of the zeta
functions. Actually their definition provides a generalization and unification
of the Bernoulli, Euler and Genocchi polynomials and also of the
Apostol–Bernoulli, Apostol–Euler and Apostol–Genocchi polynomials, which were
considered in many earlier investigations by (among others) Srivastava et al.
[39], [40], [41], Karande [42]. Also they by using a Dirichlet character
defined unification of the generating functions of the generalized Bernoulli,
Euler and Genocchi polynomials and numbers . T. Kim in [13], constructed
Apostol-Euler numbers and polynomials by using fermionic expression of
$p$-adic $q$-integral at $q=-1$. In this paper by his method we derive several
properties for unification of the multiple twisted Euler and Genocchi numbers
and polynomials.
Let $p$ be a fixed odd prime number. Throughout this paper we use the
following notations, by $\mathbb{Z}_{p}$ denotes the ring of $p$-adic rational
integers, $\mathbb{Q}$ denotes the field of rational numbers, $\mathbb{Q}_{p}$
denotes the field of $p$-adic rational numbers, and $\mathbb{C}_{p}$ denotes
the completion of algebraic closure of $\mathbb{Q}_{p}$. Let $\mathbb{N}$ be
the set of natural numbers and
$\mathbb{N}^{\ast}=\mathbb{N}\cup\left\\{0\right\\}.$ The $p$-adic absolute
value is defined by $\left|p\right|_{p}=\frac{1}{p}.$ In this paper we assume
$\left|q-1\right|_{p}<1$ as an indeterminate. $\left[x\right]_{q}$ is a
$q$-extension of $x$ which is defined by
$\left[x\right]_{q}=\frac{1-q^{x}}{1-q}$, we note that $\lim_{q\rightarrow
1}\left[x\right]_{q}=x$.
We say that $f$ is a uniformly differntiable function at a point
$a\in\mathbb{Z}_{p},$ if the difference quotient
$F_{f}\left(x,y\right)=\frac{f\left(x\right)-f\left(y\right)}{x-y}$
has a limit $f{\acute{}}\left(a\right)$ as
$\left(x,y\right)\rightarrow\left(a,a\right)$ and denote this by $f\in
UD\left(\mathbb{Z}_{p}\right).$
Let $UD\left(\mathbb{Z}_{p}\right)$ be the set of uniformly differentiable
function on $\mathbb{Z}_{p}.$ For $f\in UD\left(\mathbb{Z}_{p}\right),$ let us
begin with the expressions
$\frac{1}{\left[p^{N}\right]}\sum_{0\leq
x<p^{N}}f\left(x\right)q^{x}=\sum_{0\leq
x<p^{N}}f\left(x\right)\mu_{q}\left(x+p^{N}\mathbb{Z}_{p}\right),$
represents $p$-adic $q$-analogue of Riemann sums for $f.$ The integral of $f$
on $\mathbb{Z}_{p}$ will be defined as the limit
$\left(N\rightarrow\infty\right)$ of these sums, when it exists. The $p$-adic
$q$-integral of function $f\in UD\left(\mathbb{Z}_{p}\right)$ is defined by
$Kim$
(1.1)
$I_{q}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{q}\left(x\right)=\lim_{N\rightarrow\infty}\frac{1}{\left[p^{N}\right]_{q}}\sum_{x=0}^{p^{N}-1}f\left(x\right)q^{x}\text{
}$
The bosonic integral is considered by $Kim$ as the bosonic limit $q\rightarrow
1,$ $I_{1}\left(f\right)=\lim_{q\rightarrow 1}I_{q}\left(f\right).$ Similarly,
the fermionic $p$-adic integral on $\mathbb{Z}_{p}$ is considered by $Kim$ as
follows:
$I_{-q}\left(f\right)=\lim_{q\rightarrow-q}I_{q}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{-q}\left(x\right)$
Assume that $q\rightarrow 1,$ then we have fermionic $p$-adic fermionic
integral on $\mathbb{Z}_{p}$ as follows
(1.2)
$I_{-1}\left(f\right)=\lim_{q\rightarrow-1}I_{q}\left(f\right)=\lim_{N\rightarrow\infty}\sum_{x=0}^{p^{N}-1}f\left(x\right)\left(-1\right)^{x}.$
If we take $f_{1}\left(x\right)=f\left(x+1\right)$ in (1.2), then we have
(1.3) $I_{-1}\left(f_{1}\right)+I_{-1}\left(f\right)=2f\left(0\right).$
Let $p$ be a fixed prime. For a fixed positive integer $d$ with
$\left(p,d\right)=1,$ we set
$\displaystyle X$ $\displaystyle=$ $\displaystyle
X_{d}=\lim_{\overset{\leftarrow}{N}}\mathbb{Z}/dp^{N}\mathbb{Z},$
$\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle\mathbb{Z}_{p},$
$\displaystyle X^{\ast}$ $\displaystyle=$
$\displaystyle\underset{\underset{\left(a,p\right)=1}{0<a<dp}}{\cup}a+dp\mathbb{Z}_{p}$
and
$a+dp^{N}\mathbb{Z}_{p}=\left\\{x\in X\mid x\equiv
a\left(\mathop{\mathrm{m}od}dp^{N}\right)\right\\},$
where $a\in\mathbb{Z}$ satisfies the condition $0\leq a<dp^{N}.$
###### Definition 1.
(see, cf. [36]). A unification $y_{n,\beta}\left(x:k,a,b\right)$ of the
Bernoulli, Euler and Genochhi polynomials is given by the following generating
function:
(1.4) $\displaystyle F_{a,b}\left(x;t;k,\beta\right)$ $\displaystyle=$
$\displaystyle\frac{2\left(\frac{t}{2}\right)^{k}}{\beta^{b}e^{t}-a^{b}}e^{xt}=\sum_{n=0}^{\infty}y_{n,\beta}\left(x:k,a,b\right)\frac{t^{n}}{n!}\text{
\ }\left(\left|t+\log\left(\frac{\beta}{a}\right)\right|<2\pi;\text{
}x\in\mathbb{R}\right)$
$\displaystyle\left(k\in\mathbb{N}^{\ast};a,b\in\mathbb{R}^{+};\beta\in\mathbb{C}\right),$
where as usual $\mathbb{R}^{+},$ and $\mathbb{C}$ denote the sets of positive
real numbers and complex numbers, respectively, $\mathbb{R}$ being the set of
real numbers.
Observe that, if we put $x=0$ in the generating function (1.4), then we obtain
the corresponding unification of the generating functions of Bernoulli, Euler
and Genocchi numbers. Then we have
$y_{n,\beta}\left(0:k,a,b\right)=y_{n,\beta}\left(k,a,b\right).$
We are now ready to give relationship between the Ozden’s generating function
and the fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1$ with the
following theorem:
###### Theorem 1.
The following relationship holds:
(1.5)
$a^{-b}\left(\frac{t}{2}\right)^{k}\int_{\mathbb{Z}_{p}}\left(-1\right)^{x+1}\left(\frac{\beta}{a}\right)^{bx}e^{tx}d\mu_{-1}\left(x\right)=\sum_{n=0}^{\infty}y_{n,\beta}\left(k,a,b\right)\frac{t^{n}}{n!}.$
###### Proof.
We set
$f\left(x\right)=a^{-b}\left(\frac{t}{2}\right)^{k}\left(-1\right)^{x+1}\left(\frac{\beta}{a}\right)^{bx}e^{tx}$
in (1.3), it is easy to show
$\displaystyle
a^{-b}\left(\frac{t}{2}\right)^{k}\left(-\left(\frac{\beta}{a}\right)^{b}e^{t}+1\right)\int_{\mathbb{Z}_{p}}\left(-1\right)^{x+1}\left(\frac{\beta}{a}\right)^{bx}e^{tx}d\mu_{-1}\left(x\right)$
$\displaystyle=$ $\displaystyle-\frac{2\left(\frac{t}{2}\right)^{k}}{a^{b}}$
$\displaystyle
a^{-b}\left(\frac{t}{2}\right)^{k}\int_{\mathbb{Z}_{p}}\left(-1\right)^{x+1}\left(\frac{\beta}{a}\right)^{bx}e^{tx}d\mu_{-1}\left(x\right)$
$\displaystyle=$
$\displaystyle\frac{2\left(\frac{t}{2}\right)^{k}}{\beta^{b}e^{t}-a^{b}}$
So, we complete the proof.
###### Theorem 2.
Then the following identity holds:
$\int_{\mathbb{Z}_{p}}\left(-1\right)^{x+1}\left(\frac{\beta}{a}\right)^{bx}x^{n-k}d\mu_{-1}\left(x\right)=2^{k}a^{b}\frac{\left(n-k\right)!}{n!}y_{n,\beta}\left(k,a,b\right).$
###### Proof.
From (1.5) and by using the taylor expansion of $e^{tx},$ we readily see that,
$\sum_{n=0}^{\infty}\left(2^{-k}a^{-b}\int_{\mathbb{Z}_{p}}\left(-1\right)^{x+1}\left(\frac{\beta}{a}\right)^{bx}x^{n}d\mu_{-1}\left(x\right)\right)\frac{t^{n+k}}{n!}=\sum_{n=0}^{\infty}y_{n,\beta}\left(k,a,b\right)\frac{t^{n}}{n!}$
By comparing coefficients of $t^{n}$ in the both sides of the above equation,
we arrive at the desired result.
Similarly, we obtain te following theorem for a unification of the Euler and
Genocchi polynomials as follows:
###### Theorem 3.
Then the following identity holds:
(1.6)
$\int_{\mathbb{Z}_{p}}\left(-1\right)^{y+1}\left(\frac{\beta}{a}\right)^{by}\left(x+y\right)^{n}d\mu_{-1}\left(y\right)=2^{k}a^{b}\frac{n!}{\left(n+k\right)!}y_{n+k,\beta}\left(x:k,a,b\right).$
From the binomial theorem in (1.6), we possess the following theorem:
###### Theorem 4.
The following relation holds:
$\frac{y_{n+k,\beta}\left(x:k,a,b\right)}{\binom{n+k}{k}}=\sum_{m=0}^{n}\frac{\binom{n}{m}}{\binom{m+k}{k}}y_{m+k,\beta}\left(k,a,b\right)x^{n-m}$
###### Proof.
By using (1.6) and binomial theorem, we express the following relation
$\sum_{m=0}^{n}\binom{n}{m}\left(\int_{\mathbb{Z}_{p}}\left(-1\right)^{y+1}\left(\frac{\beta}{a}\right)^{by}y^{m}d\mu_{-1}\left(y\right)\right)x^{n-m}=2^{k}a^{b}\frac{n!}{\left(n+k\right)!}y_{n+k,\beta}\left(x:k,a,b\right)$
By using $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1,$ we arrive at the
desired proof of the theorem.
Now, we consider symmetric properties of this type of polynomials as follows:
###### Theorem 5.
The following relation holds:
$y_{n,\beta^{-1}}\left(1-x:k,a^{-1},b\right)=\left(-1\right)^{k+n+1}\beta^{b}a^{b}y_{n,\beta}\left(x:k,a,b\right).$
###### Proof.
We set $x\rightarrow 1-x,$ $\beta\rightarrow\beta^{-1}$ and $a\rightarrow
a^{-1}$ into (1.6), that is
$\displaystyle\int_{\mathbb{Z}_{p}}\left(-1\right)^{y+1}\left(\frac{\beta^{-1}}{a^{-1}}\right)^{by}\left(1-x+y\right)^{n}d\mu_{-1}\left(y\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n}\int_{\mathbb{Z}_{p}}\left(-1\right)^{y+1}\left(\frac{\beta}{a}\right)^{-by}\left(x-1+y\right)^{n}d\mu_{-1}\left(y\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{k+n+1}\beta^{b}a^{b}y_{n,\beta}\left(x:k,a,b\right)$
Thus, we complete proof of the theorem.
Ozden has obtained distribution formula for $y_{n,\beta}\left(x:k,a,b\right).$
We will also obtain distribution formula by using $p$-adic $q$-integral on
$\mathbb{Z}_{p}$ at $q=-1.$
###### Theorem 6.
The following identity holds:
$y_{n,\beta}\left(x:k,a,b\right)=a^{b\left(d-1\right)}d^{n-k}\sum_{j=0}^{d-1}\left(\frac{\beta}{a}\right)^{bj}y_{n,\beta^{d}}\left(\frac{x+j}{d}:k,a^{d},b\right).$
###### Proof.
By using definition of the $p$-adic integral on $\mathbb{Z}_{p},$ we compute
$\displaystyle
2^{k}a^{b}\frac{n!}{\left(n+k\right)!}y_{n+k,\beta}\left(x:k,a,b\right)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\left(-1\right)^{y+1}\left(\frac{\beta}{a}\right)^{by}\left(x+y\right)^{n}d\mu_{-1}\left(y\right)$
$\displaystyle=$
$\displaystyle\lim_{N\rightarrow\infty}\sum_{y=0}^{dp^{N}-1}\left(-1\right)^{y+1}\left(\frac{\beta}{a}\right)^{by}\left(x+y\right)^{n}\left(-1\right)^{y}$
$\displaystyle=$ $\displaystyle
d^{n}\sum_{j=0}^{d-1}\left(\frac{\beta}{a}\right)^{bj}\lim_{N\rightarrow\infty}\sum_{y=0}^{p^{N}-1}\left(-1\right)^{y+1}\left(\frac{\beta}{a}\right)^{bdy}\left(\frac{x+j}{d}+y\right)^{n}\left(-1\right)^{y}$
$\displaystyle=$ $\displaystyle
d^{n}\sum_{j=0}^{d-1}\left(\frac{\beta}{a}\right)^{bj}\int_{\mathbb{Z}_{p}}\left(-1\right)^{y+1}\left(\frac{\beta^{d}}{a^{d}}\right)^{by}\left(\frac{x+j}{d}+y\right)^{n}d\mu_{-1}\left(y\right)$
$\displaystyle=$ $\displaystyle
d^{n}\sum_{j=0}^{d-1}\left(\frac{\beta}{a}\right)^{bj}2^{k}a^{db}\frac{n!}{\left(n+k\right)!}y_{n+k,\beta^{d}}\left(\frac{x+j}{d}:k,a^{d},b\right)$
Substituting $n$ by $n-k$, we will be completed the proof of theorem.
###### Remark 1.
This distribution for $y_{n,\beta}\left(x:k,a,b\right)$ is also introduced by
Ozden cf.[36].
###### Definition 2.
(see, for detail [35])Let $\chi$ be a Dirichlet character with conductor
$d\in\mathbb{N}.$ The generating functions of the generalized Bernoulli, Euler
and Genocchi polynomials with parameters $a,$ $b,$ $\beta$ and $k$ have been
defined by Ozden, Simsek and Srivastava as follows:
$\displaystyle\tciFourier_{\chi,\beta}\left(t,k,a,b\right)$ $\displaystyle=$
$\displaystyle
2\left(\frac{t}{2}\right)^{k}\sum_{j=1}^{d}\frac{\chi\left(j\right)\left(\frac{\beta}{a}\right)^{j}e^{jt}}{\beta^{bd}e^{dt}-a^{bd}}$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}y_{n,\chi,\beta}\left(x:k,a,b\right)\frac{t^{n}}{n!},\text{
}\left(\left|t+b\log\left(\frac{\beta}{a}\right)\right|<2\pi;\text{
}d,k\in\mathbb{N};\text{ }a,b\in\mathbb{R}^{+};\text{
}\beta\in\mathbb{C}\right)$
By using $p$-adic integral on $\mathbb{Z}_{p},$ we can obtain (2) with the
following theorem:
###### Theorem 7.
Let $\chi$ be a Dirichlet’s character with conductor $d\in\mathbb{N}.$ Then
the following relation holds
(1.8)
$a^{b\left(1-d\right)}\left(\frac{t}{2}\right)^{k}\int_{\mathbb{Z}_{p}}\chi\left(x\right)\left(-1\right)^{x+1}\left(\frac{\beta}{a}\right)^{bx}e^{tx}d\mu_{-1}\left(x\right)=2^{1-k}t^{k}\sum_{j=1}^{d}\frac{\chi\left(j\right)\left(\frac{\beta}{a}\right)^{bj}e^{tj}}{\beta^{db}e^{dt}-a^{db}}$
###### Proof.
From the definition of $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1,$ we
compute
$\displaystyle
a^{b\left(1-d\right)}\left(\frac{t}{2}\right)^{k}\int_{\mathbb{Z}_{p}}\chi\left(x\right)\left(-1\right)^{x+1}\left(\frac{\beta}{a}\right)^{bx}e^{tx}d\mu_{-1}\left(x\right)$
$\displaystyle=$ $\displaystyle
a^{b\left(1-d\right)}\left(\frac{t}{2}\right)^{k}\lim_{N\rightarrow\infty}\sum_{x=0}^{dp^{N}-1}\chi\left(x\right)\left(-1\right)^{x+1}\left(\frac{\beta}{a}\right)^{bx}e^{tx}\left(-1\right)^{x}$
$\displaystyle=$
$\displaystyle\frac{1}{d^{k}}\sum_{j=1}^{d}\chi\left(j\right)\left(\frac{\beta}{a}\right)^{bj}e^{tj}\left(\frac{1}{a^{db}}\left(\frac{td}{2}\right)^{k}\lim_{N\rightarrow\infty}\sum_{x=0}^{p^{N}-1}\left(-1\right)^{x+1}\left(\frac{\beta^{d}}{a^{d}}\right)^{bx}e^{tdx}\left(-1\right)^{x}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{d^{k}}\sum_{j=1}^{d}\chi\left(j\right)\left(\frac{\beta}{a}\right)^{bj}e^{tj}\left(\frac{2\left(\frac{td}{2}\right)^{k}}{\beta^{db}e^{dt}-a^{db}}\right)$
$\displaystyle=$ $\displaystyle
2^{1-k}t^{k}\sum_{j=1}^{d}\frac{\chi\left(j\right)\left(\frac{\beta}{a}\right)^{bj}e^{tj}}{\beta^{db}e^{dt}-a^{db}}$
Thus, we arrive at the desired result.
By expression of (1.8), we get the following equation
(1.9)
$a^{b\left(1-d\right)}\left(\frac{t}{2}\right)^{k}\int_{\mathbb{Z}_{p}}\chi\left(x\right)\left(-1\right)^{x+1}\left(\frac{\beta}{a}\right)^{bx}e^{tx}d\mu_{-1}\left(x\right)=\sum_{n=0}^{\infty}y_{n,\chi,\beta}\left(x:k,a,b\right)\frac{t^{n}}{n!}.$
We are now ready to give distribution formula for generalized Euler and
Genocchi polynomials by using $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at
$q=-1$ by means of theorem.
###### Theorem 8.
For any $n,k,d\in\mathbb{N}$ $a,b\in\mathbb{R}^{+};$ $\beta\in\mathbb{C},$ we
have
$y_{n,\chi,\beta}\left(x:k,a,b\right)=d^{n-k}\sum_{j=0}^{d-1}\chi\left(j\right)\left(\frac{\beta}{a}\right)^{bj}y_{n,\beta^{d}}\left(\frac{x+j}{d}:k,a^{d},b\right).$
###### Proof.
By expression of (1.9), we compute as follows:
$\displaystyle\sum_{n=0}^{\infty}y_{n,\chi,\beta}\left(x:k,a,b\right)\frac{t^{n}}{n!}$
$\displaystyle=$ $\displaystyle
a^{b\left(1-d\right)}\left(\frac{t}{2}\right)^{k}\int_{\mathbb{Z}_{p}}\chi\left(y\right)\left(-1\right)^{y+1}\left(\frac{\beta}{a}\right)^{by}e^{t\left(x+y\right)}d\mu_{-1}\left(y\right)$
$\displaystyle=$ $\displaystyle
a^{b\left(1-d\right)}\left(\frac{t}{2}\right)^{k}\lim_{N\rightarrow\infty}\sum_{y=0}^{dp^{N}-1}\chi\left(y\right)\left(-1\right)^{y+1}\left(\frac{\beta}{a}\right)^{by}e^{t\left(x+y\right)}\left(-1\right)^{y}$
$\displaystyle=$
$\displaystyle\frac{1}{d^{k}}\sum_{j=0}^{d-1}\chi\left(j\right)\left(\frac{\beta}{a}\right)^{bj}\left(\frac{1}{a^{db}}\left(\frac{dt}{2}\right)^{k}\lim_{N\rightarrow\infty}\sum_{y=0}^{p^{N}-1}\left(-1\right)^{y+1}\left(\frac{\beta^{d}}{a^{d}}\right)^{by}e^{dt\left(\frac{x+j}{d}+y\right)}\left(-1\right)^{y}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{d^{k}}\sum_{j=0}^{d-1}\chi\left(j\right)\left(\frac{\beta}{a}\right)^{bj}\left(\frac{1}{a^{db}}\left(\frac{dt}{2}\right)^{k}\int_{\mathbb{Z}_{p}}\left(-1\right)^{y+1}\left(\frac{\beta^{d}}{a^{d}}\right)^{by}e^{dt\left(\frac{x+j}{d}+y\right)}d\mu_{-1}\left(y\right)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{d^{k}}\sum_{j=0}^{d-1}\chi\left(j\right)\left(\frac{\beta}{a}\right)^{bj}\left(\sum_{n=0}^{\infty}d^{n}y_{n,\beta^{d}}\left(\frac{x+j}{d}:k,a^{d},b\right)\frac{t^{n}}{n!}\right)$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(d^{n-k}\sum_{j=0}^{d-1}\chi\left(j\right)\left(\frac{\beta}{a}\right)^{bj}y_{n,\beta^{d}}\left(\frac{x+j}{d}:k,a^{d},b\right)\right)\frac{t^{n}}{n!}.$
So, we complete the proof of theorem.
## 2\.
New properties on the unification of multiple twisted Euler and Genocchi
polynomials
In this section, we introduce a unification of the twisted Euler and Genocchi
polynomials. We assume that $q\in\mathbb{C}_{p}$ with
$\left|1-q\right|_{p}<1.$ For $n\in\mathbb{N},$ by the definition of the
$p$-adic integral on $\mathbb{Z}_{p},$ we have
(2.1)
$I_{-1}\left(f_{n}\right)+\left(-1\right)^{n-1}I_{-1}\left(f\right)=2\sum_{x=0}^{n-1}f\left(x\right)\left(-1\right)^{n-1-x}$
where $f_{n}\left(x\right)=f\left(x+n\right).$
Let $T_{p}=\underset{n\geq
1}{\cup}C_{p^{n}}=\lim_{n\rightarrow\infty}C_{p^{n}}=C_{p^{\infty}}$ be the
locally constant space, where $C_{p^{n}}=\left\\{w\mid w^{p^{n}}=1\right\\}$
is the cylic group of order $p^{n}.$ For $w\in T_{p},$ we denote the locally
constant function by
(2.2) $\phi_{w}:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{p},\text{ }x\rightarrow
w^{x},\text{ }$
If we set
$f\left(x\right)=\phi_{w}\left(x\right)a^{-b}\left(\frac{t}{2}\right)^{k}\left(-1\right)^{x+1}\left(\frac{\beta}{a}\right)^{bx}e^{tx},$
then we have
(2.3)
$a^{-b}\left(\frac{t}{2}\right)^{k}\int_{\mathbb{Z}_{p}}\phi_{w}\left(x\right)\left(-1\right)^{x+1}\left(\frac{\beta}{a}\right)^{bx}e^{tx}d\mu_{-1}\left(x\right)=\frac{2\left(\frac{t}{2}\right)^{k}}{w\beta^{b}e^{t}-a^{b}}$
We now define unification of twisted Euler and Genocchi polynomials as
follows:
$\frac{2\left(\frac{t}{2}\right)^{k}}{w\beta^{b}e^{t}-a^{b}}=\sum_{n=0}^{\infty}y_{n,w,\beta}\left(k,a,b\right)\frac{t^{n}}{n!},$
We note that by substituting $w=1,$ we obtain Ozden’s generating function
(1.4). From (2.2) and (2.3), we obtain witt’s type formula for a unificaton of
twisted Euler and Genocchi polynomials as follows:
(2.4)
$a^{-b}2^{-k}\int_{\mathbb{Z}_{p}}\phi_{w}\left(x\right)\left(-1\right)^{x+1}\left(\frac{\beta}{a}\right)^{bx}x^{n}d\mu_{-1}\left(x\right)=\frac{y_{n+k,w,\beta}\left(k,a,b\right)}{k!\binom{n+k}{k}}$
for each $w\in T_{p}$ and $n\in\mathbb{N}.$
We now establish Witt’s type formula for the unification of multiple twisted
Euler and Genocchi polynomials by the following theorem.
###### Definition 3.
Let be $w\in T_{p},$ $n,h,k\in\mathbb{N}$ $a,b\in\mathbb{R}^{+};$
$\beta\in\mathbb{C},$ we define
$\displaystyle
a^{-hb}2^{-hk}\underset{h-times}{\underbrace{\int_{\mathbb{Z}_{p}}...\int_{\mathbb{Z}_{p}}}}\phi_{w}\left(x_{1}+...+x_{h}\right)\left(-1\right)^{x_{1}+...+x_{h}+h}$
$\displaystyle\times\left(\frac{\beta}{a}\right)^{b\left(x_{1}+...+x_{h}\right)}\left(x_{1}+...+x_{h}\right)^{n}d\mu_{-1}\left(x_{1}\right)...d\mu_{-1}\left(x_{h}\right)$
$\displaystyle=$
$\displaystyle\frac{y_{n+kh,w,\beta}^{\left(h\right)}\left(k,a,b\right)}{\left(kh\right)!\binom{n+kh}{kh}}.$
###### Remark 2.
Taking $h=1$ into (3), we get the unification of the twisted Euler and
Genocchi polynomials $y_{n,w,\beta}\left(k,a,b\right).$
###### Remark 3.
By substituting $h=1$ and $w=1,$ we obtain a special case of the unification
of Euler and Genocchi polynomials $y_{n,\beta}\left(k,a,b\right).$
###### Theorem 9.
For any $w\in T_{p},$ $n,h,k\in\mathbb{N}$ $a,b\in\mathbb{R}^{+};$
$\beta\in\mathbb{C},$
$\frac{y_{n+kh,w,\beta}^{\left(h\right)}\left(k,a,b\right)}{\left(kh\right)!\binom{n+kh}{kh}}=\sum_{\underset{l_{1},...,l_{h}\geq
0}{l_{1}+...+l_{h}=n}}\frac{n!}{l_{1}!...l_{h}!}\mathop{\displaystyle\prod}\limits_{i=1}^{h}\frac{y_{l_{i}+kh,w,\beta}^{\left(h\right)}\left(k,a,b\right)}{\left(kh\right)!\binom{l_{i}+kh}{kh}}$
###### Proof.
By using definition of the multiple twisted a unification of Euler and
Genocchi numbers and polynomials, and, definition of
$\left(x_{1}+x_{2}+...+x_{h}\right)^{n}=\sum_{\underset{l_{1},...,l_{h}\geq
0}{l_{1}+...+l_{h}=n}}\frac{n!}{l_{1}!...l_{h}!}x_{1}^{l_{1}}x_{2}^{l_{2}}...x_{h}^{l_{h}},$
we see that,
$\displaystyle
a^{-hb}2^{-hk}\underset{h-times}{\underbrace{\int_{\mathbb{Z}_{p}}...\int_{\mathbb{Z}_{p}}}}\phi_{w}\left(x_{1}+...+x_{h}\right)\left(-1\right)^{x_{1}+...+x_{h}+h}\left(\frac{\beta}{a}\right)^{b\left(x_{1}+...+x_{h}\right)}$
$\displaystyle\times\left(x+x_{1}+...+x_{h}\right)^{n}d\mu_{-1}\left(x_{1}\right)...d\mu_{-1}\left(x_{h}\right)$
$\displaystyle=$ $\displaystyle\sum_{\underset{l_{1},...,l_{h}\geq
0}{l_{1}+...+l_{h}=n}}\frac{n!}{l_{1}!...l_{h}!}\left(a^{-b}2^{-k}\int_{\mathbb{Z}_{p}}w^{x_{1}}\left(\frac{\beta}{a}\right)^{bx_{1}}x_{1}^{l_{1}}d\mu_{-1}\left(x_{1}\right)\right)\times$
$\displaystyle...\times\left(a^{-b}2^{-k}\int_{\mathbb{Z}_{p}}w^{x_{h}}\left(\frac{\beta}{a}\right)^{bx_{h}}x_{h}^{l_{h}}d\mu_{-1}\left(x_{h}\right)\right)$
$\displaystyle=$ $\displaystyle\sum_{\underset{l_{1},...,l_{h}\geq
0}{l_{1}+...+l_{h}=n}}\frac{n!}{l_{1}!...l_{h}!}\mathop{\displaystyle\prod}\limits_{j=1}^{h}\frac{y_{l_{i}+kh,w,\beta}^{\left(h\right)}\left(k,a,b\right)}{\left(kh\right)!\binom{l_{i}+kh}{kh}}$
Thus, we arrive at the desired result.
From these formulas, we can define the unification of the twisted Euler and
Genocchi polynomials as follows:
(2.6)
$\left(\frac{2\left(\frac{t}{2}\right)^{k}}{w\beta^{b}e^{t}-a^{b}}\right)^{h}e^{xt}=\sum_{n=0}^{\infty}y_{n,w,\beta}^{\left(h\right)}\left(x:k,a,b\right)\frac{t^{n}}{n!},$
So from above, we get the Witt’s type formula for
$y_{n,w,\beta}^{\left(h\right)}\left(x:k,a,b\right)$ as follows.
###### Theorem 10.
For any $w\in T_{p},$ $n,h,k\in\mathbb{N}$ $a,b\in\mathbb{R}^{+};$
$\beta\in\mathbb{C},$ we get
(2.7) $\displaystyle
a^{-hb}2^{-hk}\underset{h-times}{\underbrace{\int_{\mathbb{Z}_{p}}...\int_{\mathbb{Z}_{p}}}}\phi_{w}\left(x_{1}+...+x_{h}\right)\left(-1\right)^{x_{1}+...+x_{h}+h}\left(\frac{\beta}{a}\right)^{b\left(x_{1}+...+x_{h}\right)}$
$\displaystyle\times\left(x+x_{1}+...+x_{h}\right)^{n}d\mu_{-1}\left(x_{1}\right)...d\mu_{-1}\left(x_{h}\right)$
$\displaystyle=$
$\displaystyle\frac{y_{n+kh,w,\beta}^{\left(h\right)}\left(x:k,a,b\right)}{\left(kh\right)!\binom{n+kh}{kh}}$
Note that
(2.8)
$\left(x+x_{1}+x_{2}+...+x_{h}\right)^{n}=\sum_{\underset{l_{1},...,l_{h}\geq
0}{l_{1}+...+l_{h}=n}}\frac{n!}{l_{1}!...l_{h}!}x_{1}^{l_{1}}x_{2}^{l_{2}}...\left(x+x_{h}\right)^{l_{h}}$
We obtain the sum of powers of consecutive a unification of multiple twisted
Euler and Genocchi polynomials as follows:
###### Theorem 11.
For any $w\in T_{p},$ $n,h,k\in\mathbb{N}$ $a,b\in\mathbb{R}^{+};$
$\beta\in\mathbb{C},$ we get
$\frac{y_{n+kh,w,\beta}^{\left(h\right)}\left(x:k,a,b\right)}{\left(kh\right)!\binom{n+kh}{kh}}=\sum_{\underset{l_{1},...,l_{h}\geq
0}{l_{1}+...+l_{h}=n}}\frac{n!}{l_{1}!...l_{h}!}\frac{y_{l_{h}+kh,w,\beta}^{\left(h\right)}\left(x:k,a,b\right)}{\left(kh\right)!\binom{l_{h}+kh}{kh}}\mathop{\displaystyle\prod}\limits_{j=1}^{h-1}\frac{y_{l_{i}+kh,w,\beta}^{\left(h\right)}\left(k,a,b\right)}{\left(kh\right)!\binom{l_{i}+kh}{kh}}.$
###### Proof.
By (2.7) and (2.8), we see that,
$\displaystyle
a^{-hb}2^{-hk}\underset{h-times}{\underbrace{\int_{\mathbb{Z}_{p}}...\int_{\mathbb{Z}_{p}}}}\phi_{w}\left(x_{1}+...+x_{h}\right)\left(-1\right)^{x_{1}+...+x_{h}+h}\left(\frac{\beta}{a}\right)^{b\left(x_{1}+...+x_{h}\right)}$
$\displaystyle\times\left(x+x_{1}+...+x_{h}\right)^{n}d\mu_{-1}\left(x_{1}\right)...d\mu_{-1}\left(x_{h}\right)$
$\displaystyle=$ $\displaystyle\sum_{\underset{l_{1},...,l_{h}\geq
0}{l_{1}+...+l_{h}=n}}\frac{n!}{l_{1}!...l_{h}!}\left(a^{-b}2^{-k}\int_{\mathbb{Z}_{p}}w^{x_{1}}\left(\frac{\beta}{a}\right)^{bx_{1}}x_{1}^{l_{1}}d\mu_{-1}\left(x_{1}\right)\right)\times$
$\displaystyle...\times\left(a^{-b}2^{-k}\int_{\mathbb{Z}_{p}}w^{x_{h}}\left(\frac{\beta}{a}\right)^{bx_{h}}\left(x+x_{h}\right)^{l_{h}}d\mu_{-1}\left(x_{h}\right)\right)$
$\displaystyle=$ $\displaystyle\sum_{\underset{l_{1},...,l_{h}\geq
0}{l_{1}+...+l_{h}=n}}\frac{n!}{l_{1}!...l_{h}!}\frac{y_{l_{h}+kh,w,\beta}^{\left(h\right)}\left(x:k,a,b\right)}{\left(kh\right)!\binom{l_{h}+kh}{kh}}\mathop{\displaystyle\prod}\limits_{j=1}^{h-1}\frac{y_{l_{i}+kh,w,\beta}^{\left(h\right)}\left(k,a,b\right)}{\left(kh\right)!\binom{l_{i}+kh}{kh}}$
So, we complete the proof of the theorem.
## 3\. A unification of multiple twisted Zeta functions
Our goal in this section is to establish a unification of multiple twisted
zeta functions which interpolates of a unification of multiple twisted Euler
and Genocchi polynomials at negative integers. For $q\in\mathbb{C},$
$\left|q\right|<1$ and $w\in T_{p},$ a unification of multiple twisted Euler
and Genocchi polynomials are considered as follows:
(3.1)
$\left(\frac{2\left(\frac{t}{2}\right)^{k}}{w\beta^{b}e^{t}-a^{b}}\right)^{h}=\sum_{n=0}^{\infty}y_{n,w,\beta}^{\left(h\right)}\left(k,a,b\right)\frac{t^{n}}{n!},\text{
}\left|t+\log\left(w\left(\frac{\beta}{a}\right)^{b}\right)\right|<2\pi.$
By (3.1), we easily see that,
$\displaystyle\sum_{n=0}^{\infty}y_{n,w,\beta}^{\left(h\right)}\left(k,a,b\right)\frac{t^{n}}{n!}$
$\displaystyle=$ $\displaystyle
2^{h}\left(\frac{t}{2}\right)^{kh}\left(\frac{1}{w\beta^{b}e^{t}-a^{b}}\right)...\left(\frac{1}{w\beta^{b}e^{t}-a^{b}}\right)$
$\displaystyle=$ $\displaystyle
2^{h}\left(\frac{t}{2}\right)^{kh}\left(-1\right)^{h}\sum_{n_{1}=0}^{\infty}w^{n_{1}}\left(\frac{\beta}{a}\right)^{bn_{1}}e^{n_{1}t}...\sum_{n_{h}=0}^{\infty}w^{n_{h}}\left(\frac{\beta}{a}\right)^{bn_{h}}e^{n_{h}t}$
$\displaystyle=$ $\displaystyle
2^{h}\left(\frac{t}{2}\right)^{kh}\left(-1\right)^{h}\sum_{n_{1},...,n_{h}=0}^{\infty}\phi_{w}\left(n_{1}+...+n_{h}\right)\left(\frac{\beta}{a}\right)^{b\left(n_{1}+...+n_{h}\right)}e^{(n_{1}+...+n_{h})t}$
By using the taylor expansion of $e^{(n_{1}+...+n_{h})t}$ and by comparing the
coefficients of $t^{n}$ in the both side of the above equation, we obtain that
(3.2)
$\frac{y_{n+kh,w,\beta}^{\left(h\right)}\left(k,a,b\right)}{(kh)!\binom{n+kh}{kh}}=2^{h\left(1-k\right)}\left(-1\right)^{h}\sum_{\underset{n_{1}+...+n_{h}\neq
0}{n_{1},...,n_{h}\geq
0}}^{\infty}\phi_{w}\left(n_{1}+...+n_{h}\right)\left(\frac{\beta}{a}\right)^{b\left(n_{1}+...+n_{h}\right)}(n_{1}+...+n_{h})^{n}$
From (3.2), we can define a unification of multiple twisted zeta functions as
follows:
$\zeta_{\beta,w}^{\left(h\right)}\left(s:k,a,b\right)=2^{h\left(1-k\right)}\left(-1\right)^{h}\sum_{\underset{n_{1}+...+n_{h}\neq
0}{n_{1},...,n_{h}=0}}^{\infty}\frac{\phi_{w}\left(n_{1}+...+n_{h}\right)\left(\frac{\beta}{a}\right)^{b\left(n_{1}+...+n_{h}\right)}}{(n_{1}+...+n_{h})^{s}}$
for all $s\in\mathbb{C}.$ We also obtain the following theorem in which a
unification of multiple twisted zeta functions interpolate a unification of
multiple twisted Euler and Genocchi polynomials at negative integer.
###### Theorem 12.
For any $w\in T_{p},$ $n,h,k\in\mathbb{N}$ $a,b\in\mathbb{R}^{+};$
$\beta\in\mathbb{C},$ we obtain
$\zeta_{\beta,w}^{\left(h\right)}\left(-n:k,a,b\right)=\frac{y_{n+kh,w,\beta}^{\left(h\right)}\left(k,a,b\right)}{(kh)!\binom{n+kh}{kh}}.$
## References
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* [2] Araci, S., Seo, J-J., and Erdal, D., New construction weighted ($h,q$)-Genocchi numbers and polynomials related to Zeta type functions, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 487490, 7 pages, doi:10.1155/2011/487490.
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|
arxiv-papers
| 2012-01-05T22:10:10 |
2024-09-04T02:49:25.998174
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Mehmet Acikgoz, Kyoung Ho Park, Hassan Jolany",
"submitter": "Serkan Araci mtsrkn",
"url": "https://arxiv.org/abs/1201.1315"
}
|
1201.1319
|
# A study involving the Completion of Quasi 2-normed space
Mehmet Kır Atatürk University, Faculty of Science, Department of Mathematics,
25000 Erzurum, TURKEY mehmet_040465@yahoo.com and Mehmet Acikgoz University
of Gaziantep, Faculty of Science and Arts, Department of Mathematics, 27310
Gaziantep, TURKEY acikgoz@gantep.edu.tr
(Date: January 5, 2012)
###### Abstract.
The fundamental aim of this paper is to introduce and investigate a new
property of quasi $2$-normed space based on a question given by C. Park (2006)
[2] for the completion quasi $2$-normed space. Finally, we also find an answer
for a question Park’s.
###### Key words and phrases:
2-normed spaces, quasi normed space, completion
###### 2000 Mathematics Subject Classification:
Primary 46A16, 46Bxx, 54D35
## 1\. Introduction, Definitions and Notations
In 1928, K. Menger introduced the notion called n-metrics (or generalized
metric). But many mathematicians had not paid attentions to Menger’s theory
about generalized metrics. But several mathematicians, A. Wald, L. M.
Blumenthal, W. A. Wilson etc. have developed Menger’s idea.
In 1963, S. Gähler limits Menger’s considerations to $n=2$. Gähler’s study is
more complete in view of the fact that he developes the topological properties
of the spaces in question. Gähler also proves that if the space is a linear
normed space, then it is possible to define 2-norm.
Since 1963, S. Gähler, Y. J. Cho, R. W. Frees, C. R. Diminnie, R. E. Ehret, K.
Iséki, A. White and many others have studied on 2-normed spaces and 2-metric
spaces. It is well-known that $\mathbb{R}$ is complete but $\mathbb{Q}$ is not
complete. Since $\mathbb{Q}$ is dense in $\mathbb{R}$ it is said that R is
completion of $\mathbb{Q}$. It is very important that an incomplete space can
be completed in similar sense. Complete spaces, in other words Banach spaces,
play quite important role in many branches of mathematics and its
applications. Many mathematicians showed the existince of completion of normed
spaces (for more information, see [1], [2], [6]). We shall also show
completion of quasi 2-normed spaces via similar sense.
###### Definition 1.
Let $X$ be a real linear space with $dim\geq 2$ and
$\left\|.,.\right\|:X^{2}\rightarrow[0,\infty)$ a function. Then
$\left(X,\left\|.,.\right\|\right)$ is called linear 2-normed spaces if
$2N_{1})\left\|x,y\right\|=0\Longleftrightarrow$ $x$ and $y$ linearly
dependent,
$2N_{2})\left\|x,y\right\|=\left\|y,x\right\|,$
$2N_{3})\left\|\alpha x,y\right\|=\left|\alpha\right|\left\|x,y\right\|,$
$2N_{4})\left\|x+y,z\right\|=\left\|x,z\right\|+\left\|y,z\right\|,$
for all $\alpha\in\mathbb{R}$ and all $x,$ $y,$ $z\in X.$
###### Example 1.
Let $E_{3}$ denotes Euclidean vector three spaces. Let $x=ai+bj+ck$ and
$y=di+ej+fk$ define
$\displaystyle\left\|x,y\right\|$ $\displaystyle=$ $\displaystyle\left|x\times
y\right|=abs\left|\begin{array}[]{ccc}i&j&k\\\ a&b&c\\\
d&e&f\end{array}\right|$ $\displaystyle=$ $\displaystyle\left|\left(bf-
ce\right)^{2}i+\left(cd-af\right)^{2}j+\left(ae-
db\right)^{2}k\right|^{\frac{1}{2}}$
Then $\left(E_{3},\left\|.,.\right\|\right)$ is a 2-normed space and this
space is complete (for more information, see [6]).
Also
1) In addition to $2N_{1}),$ $2N_{2}),$ $2N_{3}),$ if there is a constant
$K\geq 1$ such that
$2N_{4}^{\ast})\text{ }\left\|x+y,z\right\|\leq
K\left(\left\|x,z\right\|+\left\|y,z\right\|\right)\text{ for all }x,\text{
}y,\text{ }z\in X$
is called quasi 2-normed space.
2) A $2$-norm $\left\|.,.\right\|$ defined on a linear space $X$ is said to be
uniformly continuous in both variables if for any $\varepsilon>0$ there exist
a neighbourhood $U_{\varepsilon}$ of $0$ such that
$\left|\left\|a,b\right\|-\left\|a{\acute{}},b{\acute{}}\right\|\right|<\varepsilon$
whenever $a$-$a{\acute{}}$ and $b$-$b{\acute{}}$ are in $U_{\varepsilon},$
which is independent of the choice of $a$, $a{\acute{}}$, $b$, $b{\acute{}}.$
3) A pseudo $2$-norm is defined to be real-valued function having all the
properties of $2$-norm $\left\|.,.\right\|$ except the condition that
$\left\|a,b\right\|=0$ implies the linear dependence of $a$ and $b$ (for
details, see [1]).
###### Example 2.
Let $X$ be a linear space with $\dim X\geq 2$ and $\left\|.,.\right\|$ be
$2$-norm on $X.$
$\left\|x,y\right\|_{q}=2\left\|x,y\right\|$
is quasi $2$-norm on $X$ and $\left(X,\left\|.,.\right\|_{q}\right)$ is quasi
$2$-normed space.
###### Solution 1.
By using conditions $2N_{1}),$ $2N_{2}),$ $2N_{3})$ in $2$-normed spaces,
however, using $2N_{4}^{\ast})$, we show that as follows:
$2N_{1})\left\|x,y\right\|_{q}=0\text{ if and only if
}2\left\|x,y\right\|=0,\text{ namely, }x\text{ and }y\text{ linearly
dependent,}$
It is easy to see for $2N_{2}),$ that is,
$\left\|x,y\right\|_{q}=\left\|y,x\right\|_{q},$ Using property of $2N_{3}),$
we readily see the following applications: For all $\alpha\in\mathbb{R},$
$\displaystyle\left\|\alpha x,y\right\|_{q}$ $\displaystyle=$ $\displaystyle
2\left\|\alpha
x,y\right\|=\left|\alpha\right|\left(2\left\|x,y\right\|\right)$
$\displaystyle=$ $\displaystyle\left|\alpha\right|\left\|x,y\right\|_{q}$
We are now ready to prove property of $2N_{4}^{\ast}),$ That is, For all $x$,
$y$, $z\in X$
$\displaystyle\left\|x+y,z\right\|_{q}$ $\displaystyle=$ $\displaystyle
2\left\|x+y,z\right\|$ $\displaystyle\leq$ $\displaystyle
2\left(\left\|x,z\right\|+\left\|y,z\right\|\right)$ $\displaystyle=$
$\displaystyle\left\|x,z\right\|_{q}+\left\|y,z\right\|_{q}$
So, we complete solution of Example $2$.
###### Theorem 1.
Let $X$ be a linear space with $\dim X\geq 2$ and $\left\|.,.\right\|$ be
$2$-norm on $X,$ for constant $a$, $b\in\mathbb{R}$ which are
$a\geq\frac{1}{2}$ and $b\geq\frac{1}{2}.$ There exists
$\left\|x,y\right\|_{q}$ quasi $2$-norm on $X$ defined as
$\left\|x,y\right\|_{q}=a\left\|x,y\right\|+b\left\|x,y\right\|$
###### Proof.
It is evident to show conditions $2N_{1}),$ $2N_{2})$ and $2N_{3}),$
Therefore, It is sufficient to prove condition of $2N_{4}^{\ast})$ as follows:
For all $x$, $y$, $z\in X$
$\displaystyle\left\|x+y,z\right\|_{q}$ $\displaystyle=$ $\displaystyle
a\left\|x+y,z\right\|+b\left\|x+y,z\right\|$ $\displaystyle\leq$
$\displaystyle
a\left(\left\|x,z\right\|+\left\|y,z\right\|\right)+b\left(\left\|x,z\right\|+\left\|y,z\right\|\right)$
$\displaystyle=$
$\displaystyle\left(a+b\right)\left\|x,z\right\|+\left(a+b\right)\left\|y,z\right\|$
$\displaystyle=$ $\displaystyle
K\left(\left\|x,z\right\|+\left\|y,z\right\|\right)$
since there exists a constant $K\geq 1,$ namely, by substituting $K:=a+b,$ we
show that $\left\|.,.\right\|_{q}$ is a quasi $2$-norm on $X.$
###### Definition 2.
Let $\left(X,\left\|.,.\right\|\right)$ be a quasi $2$-normed space.
a) A sequence $\left\\{x_{n}\right\\}$ is a Cauchy sequence in a linear quasi
$2$-normed space $\left(X,\left\|.,.\right\|\right)$ if and only if
$\lim_{n,m\rightarrow\infty}\left\|x_{n}-x_{m},z\right\|=0$ for every $z$ in
$X.$
b) A sequence $\left\\{x_{n}\right\\}$ in $X$ is called a convergent sequence
if there is an $x\in X$ such that
$\lim_{n,m\rightarrow\infty}\left\|x_{n}-x,z\right\|=0$ for every $z$ in $X.$
c) A quasi $2$-normed space in which every Cauchy sequence converges is called
complete.
###### Definition 3.
Let $\left(X,\left\|.,.\right\|_{X}\right)$ and
$\left(Y,\left\|.,.\right\|_{Y}\right)$ be quasi $2$-normed spaces.
a´) A mapping $T:X\rightarrow Y$ is said to be isometric or isometry if for
all $x,$ $y\in X$
$\left\|Tx,Ty\right\|_{Y}=\left\|x,y\right\|_{X}$
b´) The space $X$ is said to be isometry with the space $Y$ if there exists a
bijective isometry of $X$ onto $Y.$ The spaces $X$ and $Y$ are called
isometric spaces.
###### Theorem 2.
If a sequence $\left\\{x_{n}\right\\}$ is a Cauchy sequence in a linear
$2$-normed space $\left(X,\left\|.,.\right\|\right),$ then
$\lim_{n\rightarrow\infty}\left\|x_{n},z\right\|$ exists for every $z$ in $X$
(for proof, see [1]).
###### Theorem 3.
If $X$ is a linear space having a uniformly continuous $2$-norm
$\left\|.,.\right\|$ defined on it, then for any two Cauchy sequences
$\left\\{x_{n}\right\\}$ and $\left\\{y_{n}\right\\}$ in $X,$
$\lim_{n\rightarrow\infty}\left\|x_{n},y_{n}\right\|$ (for proof, see [1]).
###### Definition 4.
Two Cauchy sequences $\left\\{x_{n}\right\\}$ and $\left\\{y_{n}\right\\}$ in
a linear $2$-normed space $\left(X,\left\|.,.\right\|\right)$ are said to be
equivalent, denoted by
$\left\\{x_{n}\right\\}\thicksim\left\\{y_{n}\right\\},$ if for every
neighbourhood $U$ of $0$ there is an integer $N\left(U\right)$ such that
$n\geq N\left(U\right)$ implies that $x_{n}-y_{n}\in U$ cf. [1].
## 2\. COMPLETION OF QUASI 2-NORMED SPACE
In [2], C. Park introduced quasi $2$-normed spaces and gave some results on
$p$-normed spaces. Also he introduced a question which was ”Construct a
completion of a quasi -2-norm”. In this section, we give an answer to this
question.
###### Theorem 4.
The relation $\thicksim$ on the set of Cauchy sequences in $X$ is an
equivalence relation on $X.$
###### Proof.
It is clear that $\left\\{x_{n}\right\\}\thicksim\left\\{y_{n}\right\\}$
$\left(\text{reflexivity}\right)$ and
$\left\\{y_{n}\right\\}\thicksim\left\\{x_{n}\right\\}$ when
$\left\\{x_{n}\right\\}\thicksim\left\\{y_{n}\right\\}$
$\left(\text{symmetry}\right).$
Let $\left\\{x_{n}\right\\}\thicksim\left\\{y_{n}\right\\}$ and
$\left\\{y_{n}\right\\}\thicksim\left\\{z_{n}\right\\},$ $z\in X$
$\displaystyle\lim_{n\rightarrow\infty}\left\|x_{n}-z_{n},z\right\|$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\left\|x_{n}-y_{n}+y_{n}-z_{n},z\right\|$
$\displaystyle\leq$
$\displaystyle\lim_{n\rightarrow\infty}K\left(\left\|x_{n}-y_{n},z\right\|+\left\|y_{n}-z_{n},z\right\|\right),\text{
}K\geq 1$ $\displaystyle\leq$ $\displaystyle
K\lim_{n\rightarrow\infty}\left\|x_{n}-y_{n},z\right\|+\left\|y_{n}-z_{n},z\right\|$
$\displaystyle=$ $\displaystyle K\left(0+0\right)=0.$
Then $\left\\{x_{n}\right\\}\thicksim\left\\{z_{n}\right\\}$
$\left(\text{transitivity}\right).$ So $\thicksim$ is a equivalence relation
on $X.$
###### Theorem 5.
$\left\\{x_{n}\right\\}$ is equivalent to $\left\\{a_{n}\right\\}$ in a linear
$2$-normed space $\left(X,\left\|.,.\right\|\right)$ if and only if
$\lim_{n\rightarrow\infty}\left\|x_{n}-a_{n},z\right\|=0$
for every $z$ in $X$ (for proof, see [1]).
###### Theorem 6.
If $\left\\{a_{n}\right\\}$ and $\left\\{b_{n}\right\\}$ are equivalent to
$\left\\{x_{n}\right\\}$ and $\left\\{y_{n}\right\\}$ in a linear 2-normed
space $\left(X,\left\|.,.\right\|\right),$ respectively, then
$\left\\{a_{n}+b_{n}\right\\}$ is equivalent to $\left\\{x_{n}+y_{n}\right\\}$
and $\left\\{\alpha a_{n}\right\\}$ is equivalent to $\left\\{\alpha
x_{n}\right\\}$ (for proof, see [1]).
Denote by $\widehat{X}$ the set of all equivalence classes of Cauchy sequences
in $X.$ Let $\widehat{x},$ $\widehat{y}$, $\widehat{z}$, etc., denote the
elements of $\widehat{X}.$ Define an addition and scalar multiplication on
$\widehat{X}$ as follows:
† $\widehat{x}+\widehat{y}=$ the set of sequences of equivalent to
$\left\\{x_{n}+y_{n}\right\\},$ where $\left\\{x_{n}\right\\}$ is in
$\widehat{x}$ and $\left\\{y_{n}\right\\}$ in $\widehat{y},$ and
‡$\alpha\widehat{x}=$ the set of sequences equivalent to $\left\\{\alpha
x_{n}\right\\},$ where $\left\\{x_{n}\right\\}$ is in $\widehat{x}.$ It is
clear that these two operations are well defined since they are independent of
the choice of elements from $\widehat{x}$ and $\widehat{y}.$ So $\widehat{X}$
is a linear space with operations.
###### Theorem 7.
If $X$ is linear space having a uniformly continuous $2$-norm
$\left\|.,.\right\|$ defined on it, then for pairs of equivalent Cauchy
sequences and $\left\\{x_{n}\right\\}\thicksim\left\\{a_{n}\right\\}$ and
$\left\\{y_{n}\right\\}\thicksim\left\\{b_{n}\right\\}$, Then
$\lim_{n\rightarrow\infty}\left\|x_{n},y_{n}\right\|=\left\|a_{n},b_{n}\right\|$
###### Theorem 8.
If $\left\\{x_{n}\right\\}$ and $\left\\{y_{n}\right\\}$ are Cauchy sequences
in a linear 2-normed space $\left(X,\left\|.,.\right\|\right)$, then
$\left\\{x_{n}-y_{n}\right\\}$ is a Cauchy sequence in $X.$
###### Proof.
We see that as follows:
$\displaystyle\left\|\left(x_{n}-y_{n}\right)-\left(x_{m}-y_{m}\right),z\right\|$
$\displaystyle=$
$\displaystyle\left\|\left(x_{n}-x_{m}\right)-\left(y_{n}-y_{m}\right),z\right\|$
$\displaystyle\leq$ $\displaystyle
K\left(\left\|x_{n}-x_{m},z\right\|+\left\|y_{n}-y_{m},z\right\|\right)$
we can readily see that, when $n\rightarrow\infty,$
$\left\\{x_{n}-y_{n}\right\\}$ is a Cauchy sequence in $X.$
Whenever $X$ is a space having a uniformly continuous $2$-norm defined it
which is possible to define real-valued function on the space $\widehat{X}.$
The function is defined as follows:
For any two elements $\widehat{x}$ and $\widehat{y}$ in $\widehat{X},$
$\left\|\widehat{x},\widehat{y}\right\|=\lim_{n\rightarrow\infty}\left\|x_{n},y_{n}\right\|,$
Where $\left\\{x_{n}\right\\}\in$ $\widehat{x}$ and
$\left\\{y_{n}\right\\}\in\widehat{y}.$
Since the limit is exist and independent of the choice of the elements in
$\widehat{x}$ and $\widehat{y}$. The function is well defined.
###### Theorem 9.
If $X$ is a linear space having a uniformly continuous $2$-norm
$\left\|.,.\right\|$ defined on it and $\left\\{x_{n}\right\\}$ and
$\left\\{y_{n}\right\\}$ are Cauchy sequences in $\widehat{x}$ and
$\widehat{y}$, respectively, then the function defined by
$\left\|\widehat{x},\widehat{y}\right\|=\lim_{n\rightarrow\infty}\left\|x_{n},y_{n}\right\|$
is a pseudo quasi $2$-norm on $\widehat{X}.$
###### Proof.
By using definition of $2$-normed spaces, we see that,
$2N_{1})$ Let $\widehat{x}=\alpha\widehat{y}$
$\displaystyle\left\|\widehat{x},\widehat{y}\right\|$ $\displaystyle=$
$\displaystyle\left\|\alpha\widehat{y},\widehat{y}\right\|$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\left(\left|\alpha\right|\left\|x_{n},y_{n}\right\|\right)$
$\displaystyle=$ $\displaystyle 0.$
$2N_{2})$ It is easy to see as follows:
$\displaystyle\left\|\widehat{x},\widehat{y}\right\|$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\left\|x_{n},y_{n}\right\|$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\left\|y_{n},x_{n}\right\|$
$\displaystyle=$ $\displaystyle\left\|\widehat{y},\widehat{x}\right\|$
$2N_{3})$ On account of definition of $2$-normed space, that is,
$\displaystyle\left\|\alpha\widehat{x},\widehat{y}\right\|$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\left\|\alpha
x_{n},y_{n}\right\|=\lim_{n\rightarrow\infty}\left|\alpha\right|\left\|x_{n},y_{n}\right\|$
$\displaystyle=$
$\displaystyle\left|\alpha\right|\lim_{n\rightarrow\infty}\left\|x_{n},y_{n}\right\|$
$\displaystyle=$
$\displaystyle\left|\alpha\right|\left\|\widehat{x},\widehat{y}\right\|.$
$2N_{4}^{\ast})$ $\widehat{x},$ $\widehat{y},$ $\widehat{z}\in\widehat{X},$
$\left\\{x_{n}\right\\}\in\widehat{x},$
$\left\\{y_{n}\right\\}\in\widehat{y},$ $\left\\{z_{n}\right\\}\in\widehat{z}$
are Cauchy sequences
$\displaystyle\left\|\widehat{x},\widehat{y}+\widehat{z}\right\|$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\left\|x_{n},y_{n}+z_{n}\right\|$
$\displaystyle\leq$
$\displaystyle\lim_{n\rightarrow\infty}\left(K\left(\left\|x_{n},y_{n}\right\|+\left\|x_{n},z_{n}\right\|\right)\right)$
$\displaystyle=$ $\displaystyle
K\lim_{n\rightarrow\infty}\left\|x_{n},y_{n}\right\|+K\lim_{n\rightarrow\infty}\left\|x_{n},z_{n}\right\|$
$\displaystyle=$ $\displaystyle
K\left(\left\|\widehat{x},\widehat{y}\right\|+\left\|\widehat{x},\widehat{z}\right\|\right)$
This shows that
$\left\|\widehat{x},\widehat{y}\right\|=\lim_{n\rightarrow\infty}\left\|x_{n},y_{n}\right\|$
is a pseudo quasi $2$-norm on $\widehat{X}.$
Let $\widehat{X}$ be the subset of $X$ consisting of those equivalence classes
which contain a Cauchy sequence $\left\\{x_{n}\right\\}$ for which
$x_{1}=x_{2}=...=x_{n}=\cdots$. At most one sequence of this kind can be in
each equivalence class. If $\widehat{x}$ and $\widehat{y}$ are in
$\widehat{X}_{0}$ and if corresponding Cauchy sequence are
$\left\\{x_{n}\right\\}$ and $\left\\{y_{n}\right\\}$ with $x_{n}=x$ and
$y_{n}=y$ for every $n$, then we have
$\left\|\widehat{x},\widehat{y}\right\|=\lim_{n\rightarrow\infty}\left\|x_{n},y_{n}\right\|=\left\|x,y\right\|.$
Thus $\widehat{X}_{0}$ and $\widehat{X}$ are isometrics. This isometry will be
used to show that $\widehat{X}_{0}$ is dense in $\widehat{X}$ (for details,
see[1]).
###### Theorem 10.
If $X$ is a linear space having a uniformly continuous quasi $2$-norm
$\left\|.,.\right\|$ defined on it,
$\overline{\left(\widehat{X}_{0}\right)}=\widehat{X}$ (for proof, see [1]).
###### Theorem 11.
If $X$ is a linear space having a uniformly continuous quasi $2$-norm which is
defined as
$\left\|\widehat{x},\widehat{y}\right\|=\lim_{n\rightarrow\infty}\left\|x_{n},y_{n}\right\|$
then $\widehat{X}$ is complete and the pair
$\left(\widehat{X},\left\|.,.\right\|\right)$ is called completion of quasi
$2$-normed space.
###### Proof.
In order to see that $\left(\widehat{X},\left\|.,.\right\|\right)$ is
complete, we have to show every Cauchy sequence in $\widehat{X}$ is convergent
in $\widehat{X}.$ Let $\left\\{a_{n}\right\\}$ be a Cauchy sequence in
$\widehat{X}$ and $\widehat{b}_{n}\in\widehat{X}_{0},$
$\widehat{c}_{n}\in\widehat{X}_{0}.$
Because of $\overline{\left(\widehat{X}_{0}\right)}=\widehat{X}$ then we can
write $\left\|\widehat{a}_{n}-\widehat{c}_{n},\widehat{b}\right\|<\frac{1}{n}$
for each $n$, Also we have,
$\displaystyle\left\|\widehat{c}_{n}-\widehat{c}_{m},\widehat{b}\right\|$
$\displaystyle=$
$\displaystyle\left\|\widehat{c}_{n}-\widehat{a}_{n}+\widehat{a}_{n}-\widehat{c}_{m},\widehat{b}\right\|$
$\displaystyle\leq$ $\displaystyle
K\left(\left\|\widehat{c}_{n}-\widehat{a}_{n},\widehat{b}\right\|+\left\|\widehat{a}_{n}-\widehat{c}_{m},\widehat{b}\right\|\right)$
$\displaystyle=$ $\displaystyle
K\left(\left\|\widehat{c}_{n}-\widehat{a}_{n},\widehat{b}\right\|+\left\|\widehat{a}_{n}-\widehat{a}_{m}+\widehat{a}_{m}-\widehat{c}_{m},\widehat{b}\right\|\right)$
$\displaystyle\leq$ $\displaystyle
K\left\|\widehat{a}_{n}-\widehat{c}_{n},\widehat{b}\right\|+K\left(K\left\|\widehat{a}_{n}-\widehat{a}_{m},\widehat{b}\right\|+\left\|\widehat{a}_{m}-\widehat{c}_{m},\widehat{b}\right\|\right),\text{
for }K\geq 1$ $\displaystyle<$
$\displaystyle\frac{K}{n}+K^{2}\left\|\widehat{a}_{n}-\widehat{a}_{m},\widehat{b}\right\|+\frac{K^{2}}{m}$
Last from inequality when $n,m\rightarrow\infty$ right hand side will be equal
to $0.$ Thus
$\lim_{n,m\rightarrow\infty}\left\|\widehat{c}_{n}-\widehat{c}_{m},\widehat{b}\right\|=0$
this shows us that $\left\\{\widehat{c}_{n}\right\\}$ is a Cauchy sequence in
$\widehat{X}.$
Use of $\widehat{X}$ and $\widehat{X}_{0}$ are isometric there is a Cauchy
sequence $\left\\{c_{n}\right\\}$ in $X$ that corresponding
$\left\\{\widehat{c}_{n}\right\\}.$
On the other hand there is $\widehat{a}\in\widehat{X}$ such that
$\widehat{a}\in\left\\{\widehat{c}_{n}\right\\}$
$\displaystyle\left\|\widehat{a}_{n}-\widehat{a},\widehat{b}\right\|$
$\displaystyle=$
$\displaystyle\left\|\widehat{a}_{n}-\widehat{c}_{n}+\widehat{c}_{n}-\widehat{a},\widehat{b}\right\|$
$\displaystyle\leq$ $\displaystyle
K\left(\left\|\widehat{a}_{n}-\widehat{c}_{n},\widehat{b}\right\|+\left\|\widehat{c}_{n}-\widehat{a},\widehat{b}\right\|\right)$
$\displaystyle<$
$\displaystyle\frac{K}{n}+K\left\|\widehat{c}_{n}-\widehat{a},\widehat{b}\right\|$
Last from inequality as $n\rightarrow\infty$ and $\widehat{X}_{0}$ is dense in
$\widehat{X},$
$\lim_{n\rightarrow\infty}\left\|\widehat{a}_{n}-\widehat{a},\widehat{b}\right\|=0$
so arbitrary a Cauchy sequence $\left\\{\widehat{a}_{n}\right\\}$ convergent
to $\widehat{a}\in\widehat{X}.$ Then
$\left(\widehat{X},\left\|.,.\right\|\right)$ is complete.
## References
* [1] R. W. Freese, Y. J. Cho, Geometry of linear 2-normed spaces, Huntington N. Y. Nova Puplishes, (2001).
* [2] Park, C., Generalized quasi-Banach spaces and quasi -(2; p) normed spaces, Journal of the Chungcheong Matematical Society, vol. 19, no. 2, June 2006.
* [3] S. Gähler, Lineare 2-normierte Räume, Diese Nachr. 28, 1-43 (1965).
* [4] K. Menger, Untersuchungen Veber allgeine Metrik, Math. Ann. 100 (1928).
* [5] K. Isekı, Mathematics on two normed spaces, Bull. Korean Math. Soc. vol. 13, no. 2, 1976.
* [6] Albert G. White, 2-Banach spaces, Jr. of St. Bonaventure, (1967), New York.
|
arxiv-papers
| 2012-01-05T22:33:57 |
2024-09-04T02:49:26.005244
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mehmet Kir, Mehmet Acikgoz",
"submitter": "Serkan Araci mtsrkn",
"url": "https://arxiv.org/abs/1201.1319"
}
|
1201.1344
|
# An Invariant of Algebraic Curves from the Pascal Theorem††thanks: The
project is supported by NNSFC(Nos. 10771028, 60533060), Program of New Century
Excellent Fellowship of NECC, and is partially funded by a DoD fund
(DAAD19-03-1-0375).
Zhongxuan Luo Corresponding author: zxluo@dlut.edu.cn School of Software,
School of Mathematical Sciences, Dalian University of Technology, Dalian,
116024, China
(Sept 5, 2007)
###### Abstract
In 1640’s, Blaise Pascal discovered a remarkable property of a hexagon
inscribed in a conic - Pascal Theorem, which gave birth of the projective
geometry. In this paper, a new geometric invariant of algebraic curves is
discovered by a different comprehension to Pascal s mystic hexagram or to the
Pascal theorem. Using this invariant, the Pascal theorem can be generalized to
the case of cubic (even to algebraic curves of higher degree), that is, For
any given 9 intersections between a cubic $\Gamma_{3}$ and any three lines
$a,b,c$ with no common zero, none of them is a component of $\Gamma_{3}$, then
the six points consisting of the three points determined by the Pascal mapping
applied to any six points (no three points of which are collinear) among those
9 intersections as well as the remaining three points of those 9 intersections
must lie on a conic. This generalization differs quite a bit and is much
simpler than Chasles’s theorem and Cayley-Bacharach theorems.
Keywords: Algebraic curve; Pascal theorem; Characteristic ratio;
Characteristic mapping; Characteristic number; spline.
## 1 Introduction
Algebraic curve is a classical and an important subject in algebraic geometry.
An algebraic plane curve is the solution set of a polynomial equation
$P(x,y)=0$, where $x$ and $y$ are real or complex variables, and the degree of
the curve is the degree of the polynomial $P(x,y)$. Let $\mathbb{P}^{2}$ be
the projective plane and $\mathbb{P}_{n}$ be the space of all homogeneous
polynomials in homogeneous coordinates $(x,y,z)$ of total degree $\leq n$. An
algebraic curve $\Gamma_{n}$ in the projective plane is defined by the
solution set of a homogeneous polynomial equation $P(x,y,z)=0$ of degree $n$.
In 1640, Blaise Pascal discovered a remarkable property of a hexagon inscribed
in a circle, shortly thereafter Pascal realized that a similar property holds
for a hexagon inscribed in an ellipse even a conic. As the birth of the
projective geometry, Pascal theorem assert: If six points on a conic section
is given and a hexagon is made out of them in an arbitrary order, then the
points of intersection of opposite sides of this hexagon will all lie on a
single line. The generalizations of Pascal’s theorem have a glorious history.
It has been a subject of active and exciting research. As generalizations of
the Pascal theorem, Chasles’s theorem and Cayley-Bacharach theorems in various
versions received a great attention both in algebraic geometry and in
multivariate interpolation. A detailed introduction to Cayley-Bacharach
theorems as well as conjectures can be found in [16, 19, 30].
The Pascal theorem can be comprehended in the following several aspect: first,
it is easy to verify that Pascal’s theorem can be proved by Chasles’s
theorem[16] and therefore, probably, Chasles’s theorem has been regarded as a
generalization of Pascal’s theorem in the literature. However, the Chasles’s
theorem and Cayley-Bacharach theorems have not formally inherited the
appearance of the Pascal theorem, that is the three points joined a line are
obtained from intersections of three pair of lines in which each line was
determined by two points lying on a conic; Secondly, the Pascal theorem can be
used to (geometrically) judge whether or not any six points simultaneously lie
on a conic. Another interesting observation to the Pascal theorem is that it
plays a key role in revealing the instability of a linear space
$S_{2}^{1}(\Delta_{MS})$ (the set of all piecewise polynomials of degree 2
with global smoothness 1 over Morgan-Scott triangulation, see figure 5.1 and
refer to Appendix 5.1). That is, the Pascal theorem gives an equivalent
relationship between the algebraic and geometric conditions to the instability
of $S_{2}^{1}(\Delta_{MS})$(refer to Appendix 5.1).
Actually, readers will see that the Pascal theorem contains a geometric
invariant of algebraic curves, which is exactly reason that we rake up the
Pascal theorem in this paper. In order to get this new invariant of algebraic
curves, one must observe the Pascal theorem from a different viewpoint in
which “arbitrary six points are given by intersections of a conic and any
three lines without no common zero” instead of “six points on a conic section
is given” (a historical viewpoint) in the Pascal theorem. This slight
different comprehension to the Pascal theorem makes us easily generalize the
Pascal theorem to algebraic curves of higher degrees and discover an invariant
of algebraic curves. Similar to the source of this paper in which all involved
points are the set of intersections between lines and a curve, [20] has given
some interesting results to the special case of the following classical
problem: Let $X$ be the intersection set of two plane algebraic curves
$\mathcal{D}$ and $\mathcal{E}$ that do not share a common component. If $d$
and $e$ denote the degrees of $\mathcal{D}$ and $\mathcal{E}$, respectively,
then $X$ consists of at most $d\cdot e$ points (a week form of Bezout’s
theorem[35]). When the cardinality of $X$ is exactly $d\cdot e$, $X$ is called
a complete intersection. How does one describe polynomials of degree at most
$k$ that vanish on a complete intersection $X$ or on its subsets? The case in
which both plane curves $\mathcal{D}$ and $\mathcal{E}$ are simply unions of
lines and the union $\mathcal{D}\cup\mathcal{E}$ is the $(d\times e)$-cage in
question.
Our main results in this paper are enlightened by studying the instability of
spline space and are proved by spline method and the “principle of duality ”
in the projective plane. This paper is organized as follows: In section 2,
some basic preliminaries of the projective geometry are given. In section 3,
some new concepts such as characteristic ratio, characteristic mapping and
characteristic number of algebraic curve are introduced by discussing the
properties of a line and a conic. Section 4 gives our main results for the
invariant of cubic and presents a generalization of the Pascal type theorem to
cubic. Moreover, some corresponding conclusions to the case of algebraic
curves of higher degrees $(n>3)$ are also stated in this section without
proofs. The basic theory of bivariate spline , a series of results on the
singularity of spline space and the proof of the main result of this paper are
given in Appendix in the end of the paper.
## 2 Preliminaries of Projective Geometry
It is well known that the “homogeneous coordinates” and the “ principle of
duality”111Poncelet claimed this principle as his own discovery, but its
nature was more clearly understood by another Franchman, J. D.
Gergonne(1771-1859)[11]. are the essential tools in the projective geometry.
A point is the set of all triads equivalent to given triad
$(x)=(x_{1},x_{2},x_{3})$, and a line is the set of all triads equivalent to
given triad $[X]=[X_{1},X_{2},X_{3}]$. By a suitable multiplication (if
necessary), any point in the projective plane can be expressed in the form
$(x_{1},x_{2},1)$, which can be shortened to $(x_{1},x_{2})$, and the two
numbers $x_{1}$ and $x_{2}$ are called the affine coordinates. In other words,
if $x_{3}\neq 0$, the point $(x_{1},x_{2},x_{3})$ in the projective plane can
be regarded as the point $(x_{1}/x_{3},x_{2}/x_{3})$ in the affine plane. The
“principle of duality” in the projective plane can be seen clearly from the
following result: ”three points $(u),(v)$ and $(w)$ in $\mathbb{P}^{2}$ are
collinear” is equivalent to ”three lines $[u],[v]$ and $[w]$ in
$\mathbb{P}^{2}$ are concurrent”. In fact, the necessary and sufficient
condition for the both statements is: there are numbers $\lambda,\mu,\nu$, not
all zero, such that $\lambda u_{i}+\mu v_{i}+\nu w_{i}=0(i=1,2,3),$ namely,
$\displaystyle\left|\begin{array}[]{ccc}u_{1}&u_{2}&u_{3}\\\
v_{1}&v_{2}&v_{3}\\\ w_{1}&w_{2}&w_{3}\\\ \end{array}\right|=0.$
If $(u),(v)$ are distinct points, $\nu\neq 0.$ Hence the general point
collinear with $(u)$ and $(v)$ can be formed a linear combination of $(u)$ and
$(v)$. In other word, a point $(u)=(u_{1},u_{2},u_{3})\in\mathbb{P}^{2}$
corresponds uniquely to a line
$[u]=[u_{1},u_{2},u_{3}]:u_{1}x+u_{2}y+u_{3}z=0$, while a line
$[u]=[u_{1},u_{2},u_{3}]:u_{1}x+u_{2}y+u_{3}z=0$ corresponds uniquely a point
$(u)=(u_{1},u_{2},u_{3})$. We say that a point $(u)$ and the corresponding
line $[u]$ are dual to each other - which is the two-dimensional “principle of
duality”. Under this duality, it follows the following definition.
###### Definition 2.1 (Duality of planar figure).
Let $\Delta$ be a planar figure consisting of lines and points in the
projective plane. A planar figure obtained by the corresponding dual lines and
points of the points and lines in $\Delta$ respectively is called the Dual
figure of $\Delta$, denotes by $\Delta^{*}$.
For instance, the dual figure of Fig. 2 is shown in Fig. 2, where $[\cdot]$
represents the corresponding dual line of the point $(\cdot)$ in Fig. 2.
Figure 1:
Figure 2:
## 3 New Definitions
In what follows, we shall use $u$ to represent a point $(u)$ or a line $[u]$
when no ambiguities exist, $u=<a,b>$ for the intersection point of lines $a$
and $b$, and $a=(u,v)$ for the line which joins the points $u$ and $v$.
Figure 3:
First, we review the following properties of a line and a conic. Suppose a
line $l$ be cut by any three lines $a,b$ and $c$ with no common zero (see Fig.
3). Let $u=<c,a>,v=<a,b>$ and $w=<b,c>$, $P=<l,a>,Q=<l,b>$ and $R=<l,c>$.
Obviously, there exist numbers $a_{i},b_{i}\ \ (i=1,2,3)$ such that
$P=a_{1}u+b_{1}v,Q=a_{2}v+b_{2}w,R=a_{3}w+b_{3}u,$ provided in turn $u,v,w$,
then we have
###### Proposition 3.1.
$\frac{b_{1}}{a_{1}}\cdot\frac{b_{2}}{a_{2}}\cdot\frac{b_{3}}{a_{3}}=-1.$
###### Proof.
Without loss of generality, we assume that $u=(1,0,0),v=(0,1,0)$ and
$w=(0,0,1)$. Since $P,Q$ and $R$ are collinear, hence
$\left|\begin{array}[]{ccc}a_{1}&b_{1}&0\\\ 0&a_{2}&b_{2}\\\
b_{3}&0&a_{3}\end{array}\right|=0.$
It follows that
$\frac{b_{1}}{a_{1}}\cdot\frac{b_{2}}{a_{2}}\cdot\frac{b_{3}}{a_{3}}=-1.$ ∎
With the same notations, it follows that the necessary and sufficient
condition for $P,Q$ and $R$ to be collinear is
$\frac{b_{1}}{a_{1}}\cdot\frac{b_{2}}{a_{2}}\cdot\frac{b_{3}}{a_{3}}=-1$.
Now let us replace the line $l$ in proposition 3.1 by a conic $\Gamma$. There
are two intersections between $\Gamma$ and each $a,b,c$. Let
$\\{p_{1},p_{2}\\}=<\Gamma,a>$, $\\{p_{3},p_{4}\\}=<\Gamma,b>$ and
$\\{p_{5},p_{6}\\}=<\Gamma,c>$. Consequently, there are real numbers
$\\{a_{i},b_{i}\\}_{i=1}^{6}$ such that
$\displaystyle\left\\{\begin{array}[]{ll}p_{1}=a_{1}u+b_{1}v\\\
p_{2}=a_{2}u+b_{2}v\end{array}\right.,\left\\{\begin{array}[]{ll}p_{3}=a_{3}v+b_{3}w\\\
p_{4}=a_{4}v+b_{4}w\end{array}\right.\mbox{and}\left\\{\begin{array}[]{ll}p_{5}=a_{5}w+b_{5}u\\\
p_{6}=a_{6}w+b_{6}u\end{array}\right..$ (3.7)
We have
###### Theorem 3.2.
Let a conic be cut by any three lines with no common zero. Under the notations
above, we have
$\displaystyle\frac{b_{1}b_{2}}{a_{1}a_{2}}\cdot\frac{b_{3}b_{4}}{a_{3}a_{4}}\cdot\frac{b_{5}b_{6}}{a_{5}a_{6}}=1.$
(3.8)
###### Proof.
Let $u=<c,a>,v=<a,b>$, $w=<b,c>$. Notice that the duality of the figure
composed of the points $\\{p_{i}\\}_{i=1}^{6}$, $u,v,w$ and the lines $a,b,c$
turns out a planar figure with a structure of Morgan-Scott triangulation with
inner edges consists of the dual lines of the points $\\{p_{i}\\}_{i=1}^{6}$,
$u,v,w$ (see Fig. 6). Note that the six points $\\{p_{i}\\}_{i=1}^{6}$ lie on
a conic, it is shown from Theorem 5.4 (see appendix 5.1) that the spline space
$S_{2}^{1}(\Delta_{MS})$ (the set of all piecewise polynomial of degree 2 with
smoothness 1 over Morgan-Scott triangulation $\Delta_{MS}$) is singular, that
is $\dim S_{2}^{1}(\Delta_{MS})=7$. Which implies from Theorem 5.5 (see
appendix 5.1) that Theorem 3.3 thus follows. ∎
On the other hand, Theorem 3.2 can be used to tell whether or not any six
points simultaneously lie on a conic. In fact, let
$p_{i}\in\mathbb{P}^{2}(i=1,2,\cdots,6)$ be any six distinct points without
any three points are collinear, $a=(p_{1},p_{2}),b=(p_{3},p_{4})$,
$c=(p_{5},p_{6})$, and $u=<c,a>,v=<a,b>$, $w=<b,c>$. Using the same notations
as in (3.2), it follows from the proof of Theorem 3.2 that
###### Proposition 3.3.
For any given six points $p_{1},p_{2},\cdots,p_{6}$ without no three points
are collinear, (3.2) is a necessary and sufficient condition for those six
points to be lying on a conic.
Actually, Theorem 3.2 is equivalent to the Pascal theorem.
[Proof of Pascal theorem.] Let $p_{i}\in\mathbb{P}^{2}(i=1,2,\cdots,6)$ be any
six distinct points without any three points are collinear. Denoted by
$a=(p_{1},p_{2}),b=(p_{3},p_{4}),c=(p_{5},p_{6})$ and $u=<c,a>,v=<a,b>$,
$w=<b,c>$. Without loss of generality, we assume $u=(1,0,0),v=(0,1,0)$ and
$w=(0,0,1)$. Since the 6 points $p_{i}\in\mathbb{P}^{2}(i=1,2,\cdots,6)$ lie
on a conic, (3.2) holds. It is clear that
$\begin{array}[]{ll}q_{1}=<(p_{1},p_{2}),(p_{4},p_{5})>=(b_{4}b_{5},-a_{4}a_{5},0)=b_{4}b_{5}u-a_{4}a_{5}w,\\\
q_{2}=<(p_{2},p_{3}),(p_{5}p_{6})>=(a_{2}a_{3},0,-b_{2}b_{3})=-b_{2}b_{3}v+a_{2}a_{3}u,\\\
q_{3}=<(p_{3},p_{4}),(p_{1},p_{6})>=(0,-b_{1}b_{6},a_{1}a_{6})=-b_{1}b_{6}w+a_{1}a_{6}v,\\\
\end{array}$
and (3.2) is equivalent to
$\displaystyle(-\frac{b_{1}b_{6}}{a_{1}a_{6}})\cdot(-\frac{b_{2}b_{3}}{a_{2}a_{3}})\cdot(-\frac{b_{4}b_{5}}{a_{4}a_{5}})=-1.$
(3.9)
By Proposition 3.1, three points $\\{q_{1},q_{2},q_{3}\\}$ must be collinear.
This is the conclusion of the Pascal theorem.
Notice that Proposition 3.1 and Theorem 3.2, -1 and 1 are invariants of line
and conic respectively. We therefore introduce the following definitions.
###### Definition 3.4 (Characteristic ratio).
Let $u,v\in\mathbb{P}^{2}$ be two distinct points (or lines),
$p_{1},p_{2},\cdots,p_{k}$ be points (or lines) on the line $(u,v)$ (or
passing through $<u,v>$), then there are numbers $a_{i},b_{i}$ such that
$p_{i}=a_{i}u+b_{i}v,i=1,2,\cdots,k$. The ratio
$[u,v;p_{1},\cdots,p_{k}]:=\frac{b_{1}b_{2}\cdots,b_{k}}{a_{1}a_{2}\cdots,a_{k}}$
is called the Characteristic ratio of $p_{1},p_{2},\cdots,p_{k}$ with respect
to the basic points (or lines) $u,v$. If there are multiple points in the
intersection points, the corresponding characteristic ratio is defined by
their limit form.
###### Remark 3.5.
For four collinear points $u,v,p_{1},p_{2}$, while the Characteristic ratio of
$p_{1},p_{2}$ with respective to $u,v$ is $\frac{b_{1}b_{2}}{a_{2}b_{2}}$, the
cross ratio in the projective geometry is defined as
$\frac{a_{1}b_{2}}{a_{2}b_{1}}$.
###### Definition 3.6 (Characteristic mapping).
Let $u$ and $v$ be two distinct points, and the line $(u,v)$ join the points
$p$ and $q$. We call $q($or $p)$ the characteristic mapping point of $p($or
$q)$ with respect to the basic points $u$ and $v$ if
$[u,v;p,q]=1,$
and denote $q=\chi_{(u,v)}(p)$ (or $p=\chi_{(u,v)}(q)$).
Apparently if $q$ is the characteristic mapping point (or line) of $p$, then
$p$ is the characteristic mapping of $q$ as well. That is, the characteristic
mapping is reflexive, i.e., $\chi_{(u,v)}\circ\chi_{(u,v)}=I$ (identity
mapping). Geometrically, $\chi_{(u,v)}(p)$ and $p$ are symmetric with respect
to the mid-point of $u$ and $v$.
From the definition of the characteristic mapping, Proposition 3.1 and Theorem
3.2, the property of the characteristic mapping can be shown in the following
corollaries.
###### Corollary 3.7.
Any three points $P,Q$ and $R$ in the projective plane $\mathbb{P}^{2}$ are
collinear if and only if their characteristic mapping points
$\chi_{(u,v)}(P),\chi_{(v,w)}(Q)$ and $\chi_{(w,u)}(R)$ are collinear.
###### Corollary 3.8.
Any six distinct points $p_{i}\in\mathbb{P}^{2}(i=1,2,\cdots,6)$ lie on a
conic if and only if the image of their characteristic mapping
$\chi_{(u,v)}(p_{1})$, $\chi_{(u,v)}(p_{2})$, $\chi_{(v,w)}(p_{3})$,
$\chi_{(v,w)}(p_{4})$, $\chi_{(w,u)}(p_{5})$ and $\chi_{(w,u)}(p_{6})$ lie on
a conic as well.
Bezout’s theorem[35] says that two algebraic curves of degree $r$ and $s$ with
no common components have exactly $r\cdot s$ points in the projective complex
plane. In particular, a line $l$ and an algebraic curve $C$ of degree $n$
without the component $l$ meet in exactly $n$ points in the projective complex
plane.
###### Definition 3.9 (Characteristic number).
Let $\Gamma_{n}$ be an algebraic curve of degree $n$, and $a,b,c$ be any three
distinct lines (without common zero) where none of them is a component of
$\Gamma_{n}$. Suppose that there exist $n$ intersections between the each line
and $\Gamma$, and denoted by $\\{p_{i}^{(a)},p_{i}^{(b)},$
$p_{i}^{(c)}\\}_{i=1}^{n}$ the intersections between $\Gamma_{n}$ and the
lines $a,b,c$, respectively. Let $u=<c,a>,v=<a,b>$, $w=<b,c>$. The number
$\mathcal{K}_{n}(\Gamma_{n}):=[u,v;p_{1}^{(a)},\cdots,p_{n}^{(a)}]\cdot[v,w;p_{1}^{(b)},\cdots,p_{n}^{(b)}]\cdot[w,u;p_{1}^{(c)},\cdots,p_{n}^{(c)}],$
independent of $a,b$ and $c$ (See Theorem 4.4 below), is called the
characteristic number of algebraic curve $\Gamma_{n}$ of degree $n$.
It is obvious from the Definition 3.9 that if $\Gamma_{n}$ is a reducible
curve of degree $n$ and has components $\Gamma_{n_{1}}$ and
$\Gamma_{n_{2}},n=n_{1}+n_{2}$ , then
$\mathcal{K}_{n}(\Gamma_{n})=\mathcal{K}_{n_{1}}(\Gamma_{n_{1}})\cdot\mathcal{K}_{n_{1}}(\Gamma_{n_{1}})$.
From the discussion below the Characteristic number is a global invariant of
algebraic curves.
By Definition 3.9, the characteristic numbers of line and conic are -1 and +1
respectively.
###### Definition 3.10 (Pascal mapping).
For any 6 points $p_{1},p_{2},\cdots,p_{6}$ without any three points are
collinear in the projective plane, first define $\Phi$ by
$\Phi(\\{p_{1},p_{2},\cdots,p_{6}\\})=\\{q_{1},q_{2},q_{3}\\},$
where
$q_{1}=<(p_{1},p_{2}),(p_{4},p_{5})>,q_{2}=<(p_{2},p_{3}),(p_{5},p_{6})>$ and
$q_{3}=<(p_{3},p_{4}),$ $(p_{6},p_{1})>$(i.e. $\\{q_{i}\\}_{i=1}^{3}$ are the
three pairs of the continuations of opposite side of the hexagon determined by
$\\{p_{i}\\}_{i=1}^{6}$). Then the Pascal mapping $\Psi$ to
$\\{p_{1},p_{2},\cdots,p_{6}\\}$ is defined by
$\Psi\\{p_{1},p_{2},\cdots,p_{6}\\}:=\chi\circ\Phi\\{p_{1},p_{2},\cdots,p_{6}\\}:=\\{\chi_{(u,v)}(q_{1}),\chi_{(w,u)}(q_{2}),\chi_{(v,w)}(q_{3})\\},$
where $u=<(p_{1},p_{2}),(p_{5},p_{6})>,v=<(p_{1},p_{2}),(p_{3},p_{4})>$ and
$w=<(p_{3},p_{4}),$ $(p_{5},p_{6})>$.
Notice that the Pascal mapping on $p_{1},p_{2},\cdots,p_{6}$ giving above
depends on the order of $\\{p_{i}\\}_{i=1}^{6}$. One can also define the
Pascal mapping on $p_{1},p_{2},\cdots,p_{6}$ by
$u=<(p_{2},p_{3}),(p_{4},p_{5})>$, $w=<(p_{4},p_{5}),(p_{1},p_{6})>$ and
$v=<(p_{2},p_{3}),$ $(p_{1},p_{6})>$ instead, which will not affect the result
of the Pascal theorem (Theorem 3.11) giving below. But for the case of higher
degrees as stated below, we must insist on $u,v$ and $w$ being defined as in
the definition above.
Figure 4: The Pascal mapping and Pascal Theorem
Fig. 4 illustrates the Pascal mapping and the Pascal Theorem that the three
points $\chi_{(u,v)}(q_{1}),\chi_{(w,u)}(q_{2}),\chi_{(v,w)}(q_{3})$ are
derived by applying the Pascal mapping to $p_{1},p_{2},\cdots,p_{6}$. From
Corollary 3.7, we state the following version of Pascal theorem in order to
generalize it to cubic.
###### Theorem 3.11 (Pascal theorem).
For given 6 points $p_{1},p_{2},\cdots,p_{6}$ on a conic section, the 3 points
of image of the Pascal mapping on these six points,
$\Psi(\\{p_{i}\\}_{i=1}^{6})$, will all lie on a single line.
## 4 Invariant and Pascal Type Theorem
In this section, we show our main results on the characteristic number to
algebraic curves. Consequently, a generalization of the Pascal theorem to
curves of higher degree is given by the “principle of duality” and the spline
method.
For cubic, we have
###### Theorem 4.1.
The characteristic number of cubic is $-1$, that is,
$\mathcal{K}_{3}(\Gamma_{3})=-1.$
###### Proof.
See Appendix 5.2. ∎
Let $a,b$ and $c$ be any three distinct lines with no common zero in the
projective plane, denoted by $u=<c,a>,v=<a,b>,w=<b,c>$. Assume that
$p_{1},p_{2},p_{3}$ are three points on $a$, $p_{4},p_{5},p_{6}$ are on $b$,
and $p_{7},p_{8},p_{9}$ are on $c$, then there exist real numbers
$a_{i},b_{i},i=1,2,\cdots,9$ such that
$\displaystyle\left\\{\begin{array}[]{ll}p_{1}=a_{1}u+b_{1}v\\\
p_{2}=a_{2}u+b_{2}v\\\ p_{3}=a_{3}u+b_{3}v\\\
\end{array}\right.,\left\\{\begin{array}[]{ll}p_{4}=a_{4}v+b_{4}w\\\
p_{5}=a_{5}v+b_{5}w\\\ p_{6}=a_{6}v+b_{6}w\\\
\end{array}\right.and\left\\{\begin{array}[]{ll}p_{7}=a_{7}w+b_{7}u\\\
p_{8}=a_{8}w+b_{8}u\\\ p_{9}=a_{9}w+b_{9}u,\\\ \end{array}\right.$ (4.10)
Similar to Proposition 3.3, one can easily show, following the proof of
Theorem 4.1, that
###### Proposition 4.2.
The nine points $p_{1},p_{2},\cdots,p_{9}$ lie on a cubic which differs from
$a\cdot b\cdot c=0$ if and only if
$\displaystyle\frac{b_{1}b_{2}b_{3}}{a_{1}a_{2}a_{3}}\cdot\frac{b_{4}b_{5}b_{6}}{a_{4}a_{5}a_{6}}\cdot\frac{b_{7}b_{8}b_{9}}{a_{7}a_{8}a_{9}}=-1,$
(4.11)
holds.
Now, from Proposition 4.2, we provide in the following a new generalization of
the Pascal theorem to cubic.
###### Theorem 4.3.
For any given 9 intersections between a cubic $\Gamma_{3}$ and any three lines
$a,b,c$ with no common zero, none of them is a component of $\Gamma_{3}$, then
the six points consisting of the three points determined by the Pascal mapping
applied to any six points (no three points of which are collinear) among those
9 intersections as well as the remaining three points of those 9 intersections
must lie on a conic.
###### Proof.
Let $\\{p_{1},p_{2},p_{7}\\}=\Gamma_{3}\bigcap a$,
$\\{p_{3},p_{4},p_{8}\\}=\Gamma_{3}\bigcap b$,
$\\{p_{5},p_{6},p_{9}\\}=\Gamma_{3}\bigcap c$, and $u=<c,a>,v=<a,b>,w=<b,c>$.
Without loss of generality, we assume that $u=(1,0,0),v=(0,1,0)$ and
$w=(0,0,1)$. It is shown in Theorem 4.1 that those 9 points
$\\{p_{i}\\}_{i=1}^{9}\in\mathbb{P}^{2}$ lying on a cubic implies
$\mathcal{K}_{3}(\Gamma_{3})=-1$, or equivalently, (4.2) holds. Notice that
$\begin{array}[]{ll}q_{1}=<(p_{1},p_{2}),(p_{4},p_{5})>=(b_{4}b_{5},-a_{4}a_{5},0)=b_{4}b_{5}u-a_{4}a_{5}v,\\\
q_{2}=<(p_{2},p_{3}),(p_{5}p_{6})>=(a_{2}a_{3},0,-b_{2}b_{3})=-b_{2}b_{3}w+a_{2}a_{3}u,\\\
q_{3}=<(p_{1},p_{6}),(p_{3},p_{4})>=(0,-b_{1}b_{6},a_{1}a_{6})=-b_{1}b_{6}v+a_{1}a_{6}w,\\\
\end{array}$
So applying the Pascal mapping on $p_{1},p_{2},p_{3},p_{4},p_{5},p_{6}$, we
have
$\\{\chi_{(u,v)}(q_{1}),\chi_{(w,u)}(q_{2}),\chi_{(v,w)}(q_{3})\\}=\\{-a_{4}a_{5}u+b_{4}b_{5}v,a_{2}a_{3}w-b_{2}b_{3}u,a_{1}a_{6}v-b_{1}b_{6}w\\}.$
Since (4.2) is equivalent to
$\displaystyle(\frac{b_{4}b_{5}}{-a_{4}a_{5}})\frac{b_{7}}{a_{7}}\cdot(\frac{-b_{1}b_{6}}{a_{1}a_{6}})\frac{b_{8}}{a_{8}}\cdot(\frac{-b_{2}b_{3}}{a_{2}a_{3}})\frac{b_{9}}{a_{9}}=1.$
(4.12)
Thus by Theorem 3.2 and Proposition 3.3, the six points
$\\{\chi_{(u,v)}(q_{1}),$ $\chi_{(w,u)}(q_{2}),$
$\chi_{(v,w)}(q_{3}),p_{7},p_{8},p_{9}\\}$ must lie on a conic. ∎
Figure 5: Generalization of Pascal Theorem
Theorem 4.3 implies that if $p_{1},p_{2},\cdots,p_{9}$ are intersection points
between a cubic and and any three distinct lines where non of them is a
component of the cubic (see Fig. 5), the three points
$\chi_{(u,v)}(q_{1}),\chi_{(w,u)}(q_{2}),\chi_{(v,w)}(q_{3})$ along with
$p_{7},p_{8},p_{9}$ will lie on a conic. Obviously, this is an intrinsic
property of cubic!
Here, let us give an example to illustrate the Pascal type theorem 4.3. Let a
cubic $\Gamma_{3}$ be given by
$\displaystyle-1120x^{3}+560x^{2}y-60xy^{2}+1008y^{3}-450xyz+1200y^{2}z+580xz^{2}-1514yz^{2}$
$\displaystyle-729z^{3}=0,$
and three lines $a:x+z=0$, $b:-y+z=0$ and $c:-x+z=0$ be given. Then the 9
intersections between $\Gamma_{3}$ and $a,b,c$ are
$\displaystyle p_{1}$ $\displaystyle=(-4,-1,4),$ $\displaystyle p_{2}$
$\displaystyle=(-1,-\frac{3}{2},1),$ $\displaystyle p_{3}$
$\displaystyle=(\frac{1}{4},1,1),$ $\displaystyle p_{4}$
$\displaystyle=(-\frac{1}{4},1,1),$ $\displaystyle p_{5}$
$\displaystyle=(1,-\frac{3}{2},1),$ $\displaystyle p_{6}$
$\displaystyle=(1,-\frac{3}{4},1),$ $\displaystyle p_{7}$
$\displaystyle=(2,-1,-2),$ $\displaystyle p_{8}$
$\displaystyle=(\frac{1}{2},1,1),$ $\displaystyle p_{9}$
$\displaystyle=(1,\frac{47}{42},1),$
and $u=(0,-1,0),v=(-1,1,1),w=(1,1,1,)$. By direct computation, we have
$\displaystyle q_{1}$
$\displaystyle=<(p_{1},p_{2}),(p_{4},p_{5})>=(-1,\frac{5}{2},1),$
$\displaystyle q_{2}$
$\displaystyle=<(p_{2},p_{3}),(p_{5},p_{6})>=(1,\frac{5}{2},1),$
$\displaystyle q_{3}$ $\displaystyle=<(p_{1},p_{6}),(p_{3},p_{4})>=(-6,1,1)$
and consequently
$\chi_{(u,v)}(q_{1})=(-1,\frac{5}{3},1),\chi_{(w,u)}(q_{2})=(1,\frac{5}{3},1),\chi_{(v,w)}(q_{3})=(6,1,1).$
It is easy to verify that the six points
$\chi_{(u,w)}(q_{1}),\chi_{(v,u)}(q_{2}),\chi_{(w,v)}(q_{3})$ as well as
$p_{7},p_{8},p_{9}$ lie on a conic:
$4x^{2}+39xy-126y^{2}-65xz+312yz-174z^{2}=0.$
In general, for algebraic curves of degree $n(n\geq 3)$, we have proved the
invariant of algebraic curves and the Pascal type theorem to higher degrees.
They are listed in the paper without proofs.
###### Theorem 4.4.
For any algebraic curve $\Gamma_{n}$ of degree $n$, its characteristic number
$\mathcal{K}_{n}(\Gamma_{n})$ is always equal to $(-1)^{n}.$
With this invariant, we may formulate a Pascal type Theorem for algebraic
curves of higher degrees:
###### Theorem 4.5 (Pascal type Theorem).
Let $a,b,c$ be any three distinct lines with no common zero in the projective
plane, and $\\{p_{i}^{(a)}\\}_{i=1}^{n}$, $\\{p_{i}^{(b)}\\}_{i=1}^{n}$,
$\\{p_{i}^{(c)}\\}_{i=1}^{n}$ be given $n$ points lying on $a,b$ and $c$,
respectively. Then those $3n$ points $\\{p_{i}^{(a)}\\}_{i=1}^{n}$,
$\\{p_{i}^{(b)}\\}_{i=1}^{n}$, $\\{p_{i}^{(c)}\\}_{i=1}^{n}$ lie on an
algebraic curve of degree $n$ if and only if the $3(n-1)$ points consisting of
the three points determined by the Pascal mapping applied to any six points
(no three points of which are collinear) among those $3n$ intersections as
well as the remaining $3(n-2)$ points of those $3n$ intersections must lie on
an algebraic curve of degree $n-1$ as well.
In view of the simplicity of the invariant, some known results of algebraic
curves (see [35], pp.123) can be easily understood from our invariant (the
characteristic number).
###### Theorem 4.6.
If a line cuts a cubic in three distinct points, the residual intersections of
the tangents at these three points are collinear.
###### Proof.
Let $l$ be a line cutting a given cubic $\Gamma_{3}$ at three points
$p_{1},p_{2},p_{3}$ and $l_{1},l_{2},l_{3}$ be the three tangents at these
points respectively. Denote by $q_{1},q_{2},q_{3}$ the residual intersections
between $\Gamma_{3}$ and $l_{1},l_{2},l_{3}$, respectively. Let
$u=<l_{2},l_{3}>,v=<l_{1},l_{2}>,w=<l_{3},l_{1}>$. Then there are real numbers
$\\{a_{i},b_{i},c_{i},d_{i}\\}_{i=1}^{3}$ such that
$p_{1}=a_{1}v+b_{1}w,p_{2}=a_{2}u+b_{2}v,p_{3}=a_{3}w+b_{3}u$ and
$q_{1}=c_{1}v+d_{1}w,q_{2}=c_{2}u+d_{2}v,q_{3}=c_{3}w+d_{3}u$. From Theorem
4.1 and Proposition 4.2, we have
$(\frac{b_{1}}{a_{1}})^{2}\frac{c_{1}}{d_{1}}\cdot(\frac{b_{2}}{a_{2}})^{2}\frac{c_{2}}{d_{2}}\cdot(\frac{b_{3}}{a_{3}})^{2}\frac{c_{3}}{d_{3}}=-1.$
Since $p_{1},p_{2},p_{3}$ are collinear, then
$\frac{b_{1}}{a_{1}}\cdot\frac{b_{2}}{a_{2}}\cdot\frac{b_{3}}{a_{3}}=-1$.
Hence, we have
$\frac{c_{1}}{d_{1}}\cdot\frac{c_{2}}{d_{2}}\cdot\frac{c_{3}}{d_{3}}=-1$, and
those three points $q_{1},q_{2},q_{3}$ are collinear. ∎
###### Theorem 4.7.
A line joining two flexes of a cubic passes through a third flexes.
###### Proof.
Let $p_{1},p_{2},p_{3}$ be three flexes of a cubic, and $l_{1},l_{2},l_{3}$ be
the three tangents at these points. Let
$u=<l_{2},l_{3}>,v=<l_{1},l_{2}>,w=<l_{3},l_{1}>$. Then there are real numbers
$\\{a_{i},b_{i}\\}_{i=1}^{3}$ such that
$p_{1}=a_{1}v+b_{1}w,p_{2}=a_{2}u+b_{2}v,p_{3}=a_{3}w+b_{3}u$. From Theorem
4.1, we have
$(\frac{b_{1}}{a_{1}})^{3}\cdot(\frac{b_{2}}{a_{2}})^{3}\cdot(\frac{b_{3}}{a_{3}})^{3}=-1,$
hence
$\frac{b_{1}}{a_{1}}\cdot\frac{b_{2}}{a_{2}}\cdot\frac{b_{3}}{a_{3}}=-1.$
Which implies $p_{1},p_{2},p_{3}$ are collinear. ∎
Similar to the proofs of Theorem 4.6 and Theorem 4.7, the following theorem
can be also easily proved by using the invariant that we found.
###### Theorem 4.8.
If a conic is tangent to a cubic at three distinct points, the residual
intersections of the tangents at these points are collinear.
## 5 Appendix
### 5.1 Bivariate Spline Space over Triangulations
It is well known that spline is an important approximation tool in
computational geometry, and it is widely used in CAGD, scientific computations
and many fields of engineering. Splines, i.e., piecewise polynomials, forms
linear spaces that have a very simple structure in univariate case. However,
it is quite complicated to determine the structure of a space of bivariate
spline over arbitrary triangulation.
Bivariate spline is defined as follows[37]:
###### Definition 5.1.
Let $\Omega$ be a given planar polygonal region and $\Delta$ be a
triangulation or partition of $\Omega$, denoted by $T_{i},i=1,2,\cdots,V$,
called cells of $\Delta$. For integer $k>\mu\geq 0$, the linear space
$S_{k}^{\mu}(\Delta):=\\{s\mid s|_{T_{i}}\in\bf{\mathbb{P}_{k}},s\in
C^{\mu}(\Omega),\forall T_{i}\in\Delta\\}$
is called the spline space of degree $k$ with smoothness $\mu$, where
$\bf{\mathbb{P}_{k}}$ is the polynomial space of total degree less than or
equal to $k$.
From the Smoothing Cofactor method[37], the fundamental theorem on bivariate
splines was established.
###### Theorem 5.2.
$s(x,y)\in S_{k}^{\mu}(\Delta)$ if and only if the following conditions are
satisfied:
1. 1.
For each interior edge of $\Delta$, which is defined by
$\Gamma_{i}:l_{i}(x,y)=0,$ there exists a so-called smoothing cofactor
$q_{i}(x,y),$ such that
$p_{i1}(x,y)-p_{i2}(x,y)=l_{i}^{\mu+1}(x,y)q_{i}(x,y),$
where the polynomials $p_{i1}(x,y)$ and $p_{i2}(x,y)$ are determined by the
restriction of $s(x,y)$ on the two cells $\Delta_{i1}$ and $\Delta_{i2}$ with
$\Gamma_{i}$ as the common edge and $q_{i}(x,y)\in\mathbb{P}_{k-(\mu+1)}$.
2. 2.
For any interior vertex $v_{j}$ of $\Delta$, the following conformality
conditions are satisfied
$\sum[l_{i}^{(j)}(x,y)]^{\mu+1}q_{i}^{(j)}(x,y)\equiv 0,$ (5.1)
where the summation is taken on all interior edges $\Gamma_{i}^{(j)}$ passing
through $v_{j}$, and the sign of the smoothing cofactors $q_{i}^{(j)}$ are
refixed in such a way that when a point crosses $\Gamma_{i}^{(j)}$ from
$\Delta_{i1}$ to $\Delta_{i2}$, it goes around $v_{j}$ counter-clockwisely.
From Theorem 5.2, the dimension of the space $S_{k}^{\mu}(\Delta)$ can be
expressed as
$\dim S_{k}^{\mu}(\Delta)=\left(\begin{array}[]{c}k+2\\\
2\end{array}\right)+\tau,$
where $\tau$ is the dimension of the linear space defined by the conformality
conditions (5.1).
However, for an arbitrary given triangulation, the dimension of these spaces
depends not only on the topology of the triangulation, but also on the
geometry of the triangulation. In general cases, no dimension formula is
known. We say that a triangulation is singular to $S_{k}^{\mu}(\Delta)$ if the
dimension of the spline space depends on, in additional to the topology of the
triangulation, the geometric position of the vertices of $\Delta$, and
$S_{k}^{\mu}(\Delta)$ is singular when its dimension increases according to
the geometric property of $\Delta$. Hence, the singularity of multivariate
spline spaces is an important object that is inevitable in the research of the
structure of multivariate spline spaces. For example, Morgan and Scott’s
triangulation $\Delta_{MS}$([27], see Fig. 6) is singular to
$S_{2}^{1}(\Delta_{MS})$. That is to say that the dimension of the space
$S_{2}^{1}(\Delta_{MS})$ is 6 in general but it increases to 7 when the
position of the inner vertices satisfy certain conditions.
Figure 6: Morgan-Scott triangulation
While the singularity of multivariate spline over any triangulation has not
been completely settled, many results on the structure of multivariate spline
space in the past 30 years can be found in many of references [1, 2, 3, 4, 5,
7, 8, 9, 10, 12, 13, 14, 29, 28, 36, 37, 23]. For Morgan-Scott’s
triangulation, Shi[31] and Diener[14] independently obtained the geometric
significance of the necessary and sufficient condition of $\dim
S_{2}^{1}(\Delta_{MS})=7$, respectively, and an equivalent geometric necessary
and sufficient condition of singularity of $S_{2}^{1}(\Delta_{MS})$ from the
viewpoint of projective geometry was obtained in [15].
Now, we take an example for $S_{2}^{1}(\Delta_{MS})$ to intuitively understand
Theorem 5.2. Let $l_{i}:\alpha_{i}x+\beta_{i}y+\gamma_{i}z=0\ \
(i=1,2,\cdots,6)$,
$u:\alpha_{u}x+\beta_{u}y+\gamma_{u}z=0$,$v:\alpha_{v}x+\beta_{v}y+\gamma_{v}z=0$
and $w:\alpha_{w}x+\beta_{w}y+\gamma_{w}z=0$ in Morgan-Scott triangulation
shown in Fig. 6. From Theorem 5.2, the global conformality condition in
$S_{2}^{1}(\Delta_{MS})$ is
$\displaystyle\left\\{\begin{array}[]{ll}\lambda_{1}l_{1}^{2}+\lambda_{2}l_{2}^{2}+\lambda_{u}u^{2}+\lambda_{v}v^{2}=0,\\\
\lambda_{3}l_{3}^{2}+\lambda_{4}l_{4}^{2}-\lambda_{v}v^{2}+\lambda_{w}w^{2}=0,\\\
\lambda_{5}l_{5}^{2}+\lambda_{6}l_{6}^{2}-\lambda_{w}w^{2}-\lambda_{u}u^{2}=0,\end{array}\right.$
(5.5)
where all letters of $\lambda^{\prime}s$ are undetermined real constants. Then
the $\dim S_{2}^{1}(\Delta_{MS})=6+\tau$, where $\tau$ is the dimension of the
linear space defined by (5.2). However, the structure of
$S_{2}^{1}(\Delta_{MS})$ depends on the geometric positions of the inner
vertices $a,b$ and $c$, which can be obviously shown from the following
conclusions.
###### Theorem 5.3 ([31]).
The spline space $S_{2}^{1}(\Delta_{MS})$ is singular (i.e. $\dim
S_{2}^{1}(\Delta_{MS})=7$) if and only if $Aa,Bb,Cc$ are concurrent, otherwise
$\dim S_{2}^{1}(\Delta_{MS})=6$(see Fig.6).
###### Theorem 5.4 ([15]).
Let $l_{i}(x,y,z)=\alpha_{i}x+\beta_{i}y+\gamma_{i}z=0$ $(i=1,2,\cdots,6)$,
then the spline space $S_{2}^{1}(\Delta_{MS})$ is singular (i.e. $\dim
S_{2}^{1}(\Delta_{MS})=7$) if and only if 6 points
$\\{(\alpha_{i},\beta_{i},\gamma_{i})\\}_{i=1}^{6}$ lie on a conic, otherwise
$\dim S_{2}^{1}(\Delta_{MS})=6$.
Using the principle of duality, an interesting fact is that the equivalent
relations in Theorem 5.3 and Theorem 5.4 hold because of the Pascal theorem!
More Precisely, for the Morgan-Scott triangulation, let
$\displaystyle\left\\{\begin{array}[]{ll}l_{1}=a_{1}u+b_{1}v\\\
l_{2}=a_{2}u+b_{2}v\\\
\end{array}\right.,\left\\{\begin{array}[]{ll}l_{3}=a_{3}v+b_{3}w\\\
l_{4}=a_{4}v+b_{4}w\\\
\end{array}\right.and\left\\{\begin{array}[]{ll}l_{5}=a_{5}w+b_{5}u\\\
l_{6}=a_{6}w+b_{6}u,\\\ \end{array}\right.$ (5.12)
where all $a_{i}^{\prime}s$ and $b_{i}^{\prime}s$ are constants, then by
solving the system of equations in (5.2), we have
###### Theorem 5.5 ([22],[31]).
The spline space $S_{2}^{1}(\Delta_{MS})$ is singular (i.e. $\dim
S_{2}^{1}(\Delta_{MS})=7$) if and only if
$\displaystyle\frac{b_{1}b_{2}}{a_{1}a_{2}}\cdot\frac{b_{3}b_{4}}{a_{3}a_{4}}\cdot\frac{b_{5}b_{6}}{a_{5}a_{6}}=1.$
(5.13)
Figure 7: Partition $\Delta$
###### Remark 5.6.
In fact, there also exists the singularity in the simplest spline space
$S_{1}^{0}(\Delta)$ consisting of continuous piecewise linear polynomials over
arbitrary partition $\Delta$. For instance, let $\Delta$ be a partition shown
in Fig. 7, the dual figure of $\Delta$ is in Fig. 3. Using the same notations
in Proposition 3.1, it is easy to verify through the duality principle that
$\dim S_{1}^{0}(\Delta)=4$ when
$\frac{b_{1}}{a_{1}}\cdot\frac{b_{2}}{a_{2}}\cdot\frac{b_{3}}{a_{3}}=-1$,
otherwise $\dim S_{1}^{0}(\Delta)=3.$
In general, for $\mu\geq 3$, Luo & Chen[24] gave an equivalent condition in an
algebraic form to the singularity of
$S_{\mu+1}^{\mu}(\Delta_{MS}^{\mu})(\mu\geq 3)$ as follows: for a given
triangulation $\Delta_{MS}^{\mu}$(see Fig. 9), suppose
$\displaystyle\left\\{\begin{array}[]{ll}l_{i}=a_{i}u+b_{i}v,\qquad
i=1,2,\ldots\ldots\mu+1,\\\ l_{j}=a_{j}v+b_{j}w,\qquad
j=\mu+2,\mu+3,\ldots\ldots 2\mu+2,\\\ l_{k}=a_{k}w+b_{k}u,\qquad
k=2\mu+3,2\mu+4,\ldots\ldots 3\mu+3,\\\ \end{array}\right.$ (5.17)
then
###### Theorem 5.7 ([24]).
The spline space $S_{\mu+1}^{\mu}(\Delta_{MS}^{\mu})$ is singular if and only
if
$\frac{a_{1}\ldots\ldots a_{\mu+1}}{b_{1}\ldots\ldots
b_{\mu+1}}\cdot\frac{a_{\mu+2}\ldots\ldots a_{2\mu+2}}{b_{\mu+2}\ldots\ldots
b_{2\mu+2}}\cdot\frac{a_{2\mu+3}\ldots\ldots
a_{3\mu+3}}{b_{2\mu+3}\ldots\ldots b_{3\mu+3}}={(-1)}^{\mu+1}.$ (5.18)
Figure 8: Morgan-Scott’s type triangulation $\Delta_{MS}^{2}$
Figure 9: Morgan-Scott’s type triangulation $\Delta_{MS}^{\mu}$
For the geometric condition of the singularity of
$S_{3}^{2}(\Delta_{MS}^{2})$, it was analyzed in [23] from projective geometry
point of view and the following result was obtained.
Let $l_{i}:\alpha_{i}x+\beta_{i}y+\gamma_{i}z=0(i=1,2,\ldots,9)$,
$a=(a_{1},a_{3},a_{3}),b=(b_{1},b_{2},b_{3})$, and $c=(c_{1},c_{2},c_{3})$ in
$\Delta_{MS}^{2}$ triangulation (see Fig.5.3). Let
$l_{a}=a_{1}x+a_{2}y+a_{3}z,l_{b}=b_{1}x+b_{2}y+b_{3}z$ and
$l_{c}=c_{1}x+c_{2}y+c_{3}z$. We define $\bar{\mathbb{P}}_{3}$ to be the cubic
polynomial subspaces spanned by any nine monomials of
$\\{x^{3},y^{3},z^{3},x^{2}y,xy^{2},$ $y^{2}z,yz^{2},x^{2}z,xz^{2},xyz\\}$ as
in [23].
###### Theorem 5.8 ([23]).
The spline space $S_{3}^{2}(\Delta_{MS}^{2})$ is singular (i.e.
$dimS_{3}^{2}(\Delta_{MS}^{2})=11$) if and only if
$p_{i}=(\alpha_{i},\beta_{i},\gamma_{i}),(i=1,2,\ldots,9)$ lie on a plane
curve, which differs from $l_{a}\cdot l_{b}\cdot l_{c}=0$, in
$\bar{\mathbb{P}}_{3}$.
###### Proof.
Embedding $\Delta_{MS}^{2}$ to $\mathbb{P}^{2}$ by the map:
$(x,y)\longmapsto[x,y,1]$. Suppose the lines $\bar{bc},\bar{ca}$ and
$\bar{ab}$ are given by $u=0,v=0$ and $w=0$, respectively. There are real
numbers $a_{i},b_{i}(i=1,2,\cdots,9)$ such that
$\displaystyle\left\\{\begin{array}[]{ll}l_{1}=a_{1}u+b_{1}v\\\
l_{2}=a_{2}u+b_{2}v\\\
l_{3}=a_{3}u+b_{3}v\end{array}\right.,\left\\{\begin{array}[]{ll}l_{4}=a_{4}v+b_{4}w\\\
l_{5}=a_{5}v+b_{5}w\\\
l_{6}=a_{6}v+b_{6}w\end{array}\right.and\left\\{\begin{array}[]{ll}l_{7}=a_{7}w+b_{7}u\\\
l_{8}=a_{8}w+b_{8}u\\\ l_{9}=a_{9}w+b_{9}u.\end{array}\right.$ (5.28)
Let $\lambda_{i}(i=1,2,\ldots,9)$ be the corresponding smoothing cofactors and
let $p_{i}=(\alpha_{i},\beta_{i},\gamma_{i}),i=1,2,\ldots,9$. Then the global
conformality conditions in $S_{3}^{2}(\Delta_{MS}^{2})$ become
$\displaystyle\left\\{\begin{array}[]{ll}\lambda_{1}l_{1}^{3}+\lambda_{2}l_{2}^{3}+\lambda_{3}l_{3}^{3}+\lambda_{u}u^{3}+\lambda_{v}v^{3}=0,\\\
\lambda_{4}l_{4}^{3}+\lambda_{5}l_{5}^{3}+\lambda_{6}l_{6}^{3}-\lambda_{v}v^{3}+\lambda_{w}w^{3}=0,\\\
\lambda_{7}l_{7}^{3}+\lambda_{8}l_{8}^{3}+\lambda_{9}l_{9}^{3}-\lambda_{w}w^{3}-\lambda_{u}u^{3}=0.\end{array}\right.$
(5.32)
Let
$\psi:(\lambda_{1},\lambda_{2},\cdots,\lambda_{9},\lambda_{u},\lambda_{v},\lambda_{w})\longmapsto(\lambda_{1},\lambda_{2},\cdots,\lambda_{9})$,
then $\psi$ is an injective linear map from the solution spaces of (5.7) to
the solution space of
$\displaystyle\sum_{i=1}^{9}\lambda_{i}l_{i}^{3}(x,y,z)=0.$ (5.33)
Similar to [15], we intend to prove that the map $\psi$ is also bijective. For
this purpose, let $\Lambda=(\lambda_{1},\lambda_{2},\cdots,\lambda_{9})$ be a
solution of (5.9). Taking $u=0$, $w=0$ and $v=0$ in (5.9) respectively, we
have
$\displaystyle\begin{array}[]{lll}\lambda_{4}l_{4}^{3}+\lambda_{5}l_{5}^{3}+\lambda_{6}l_{6}^{3}+k_{1}w^{3}+k_{2}v^{3}=0,\\\
\lambda_{1}l_{1}^{3}+\lambda_{2}l_{2}^{3}+\lambda_{3}l_{3}^{3}+k_{3}u^{3}+k_{4}v^{3}=0,\\\
\end{array}$
and
$\displaystyle\begin{array}[]{l}\lambda_{7}l_{7}^{3}+\lambda_{8}l_{8}^{3}+\lambda_{9}l_{9}^{3}+k_{5}u^{3}+k_{6}w^{3}=0,\end{array}$
where all $k_{i}(i=1,2,\cdots,6)$ are real numbers determined by $\Lambda$ and
the coefficients in (5.9). Since $\Lambda$ is a solution of (5.9), it follows
that
$k_{1}w^{3}+k_{2}v^{3}+k_{3}u^{3}+k_{4}v^{3}+k_{5}u^{3}+k_{6}w^{3}=0,$
and $k_{1}=-k_{6},k_{2}=-k_{4},k_{3}=-k_{5}$. Consequently,
$\tilde{\Lambda}:=(\lambda_{1},\cdots,\lambda_{9},k_{3},k_{2},k_{1})$ is a
solution of (5.8).
Hence, $\dim S_{3}^{2}(\Delta_{MS}^{2})=11$(or $S_{3}^{2}(\Delta_{MS}^{2})$ is
singular) if and only if there exists a nonzero solution of equation (5.9).
Now expand (5.9) with respect to $x,y,z$, will result in a system of linear
equations:
$\displaystyle\hskip
22.76228pt\mathbb{M}\Lambda:=\left(\begin{array}[]{ccccc}\alpha_{1}^{3}&\alpha_{2}^{3}&\cdots&\alpha_{8}^{3}&\alpha_{9}^{3}\\\
\alpha_{1}^{2}\beta_{1}&\alpha_{2}^{2}\beta_{2}&\cdots&\alpha_{8}^{2}\beta_{8}&\alpha_{9}^{2}\beta_{9}\\\
\cdots&\cdots&\cdots&\cdots&\cdots\\\
\alpha_{1}\beta_{1}\gamma_{1}&\alpha_{2}\beta_{2}\gamma_{2}&\cdots&\alpha_{8}\beta_{8}\gamma_{8}&\alpha_{9}\beta_{9}\gamma_{9}\\\
\cdots&\cdots&\cdots&\cdots&\cdots\\\
\beta_{1}\gamma_{1}^{2}&\beta_{2}\gamma_{2}^{2}&\cdots&\beta_{8}\gamma_{8}^{2}&\beta_{9}\gamma_{9}^{2}\\\
\gamma_{1}^{3}&\gamma_{2}^{3}&\cdots&\gamma_{8}^{3}&\gamma_{9}^{3}\\\
\end{array}\right)\cdot\left(\begin{array}[]{c}\lambda_{1}\\\ \lambda_{2}\\\
\vdots\\\ \lambda_{9}\\\ \end{array}\right)=0$ (5.47)
Notice that $p_{i}=(\alpha_{i},\beta_{i},\gamma_{i})(i=1,2,\ldots,9)$ lie on
the cubic $C_{3}:=l_{a}\cdot l_{b}\cdot l_{c}=0$, obviously the row vectors of
the coefficient matrix are linearly dependent. Since no four points in
$\\{p_{i}=(\alpha_{i},\beta_{i},\gamma_{i})\\}_{i=1}^{9}$ are collinear, it
can be shown from a classical results of algebraic geometry that
rank$(\mathbb{M})\geq 8$. Hence, (5.10) has a non-zero solution $\Lambda$ if
and only if the rank of the coefficient matrix of (5.10) is equal to 8,
implying that those nine points
$p_{i}=(\alpha_{i},\beta_{i},\gamma_{i})(i=1,2,\ldots,9)$ lie on a cubic in
$\bar{\mathbb{P}}_{3}$. Conversely, let
$\\{p_{i}=(\alpha_{i},\beta_{i},\gamma_{i})\\}_{i=1}^{9}$ lie on a cubic
$\Gamma_{3}$ in $\bar{\mathbb{P}}_{3}$, and $\Gamma_{3}$ differ from $C_{3}$.
Without loss of generality, suppose
$\Gamma_{3}:\ \
a_{1}x^{3}+a_{2}x^{2}y+a_{3}xy^{2}+a_{4}y^{3}+a_{5}xyz+a_{6}x^{2}z+a_{7}xz^{2}+a_{8}y^{2}z+a_{9}yz^{2}=0$
(no $z^{3}$ term),then we claim that $C_{3}$ must contain a $z^{3}$ term.
Otherwise, by simple computation, there exists constant $d$ such that a cubic
$\bar{\Gamma}_{3}=\Gamma_{3}+dC_{3}$, composed some 8 basis elements in
$\\{x^{3},y^{3},z^{3},x^{2}y,xy^{2},y^{2}z,yz^{2},x^{2}z,xz^{2},xyz\\}$,
passes through the nine points
$\\{p_{i}=(\alpha_{i},\beta_{i},\gamma_{i})\\}_{i=1}^{9}$. Thus the rank of
the coefficient matrix $\mathbb{M}$ must be less than 8, which is
contradictory. Since $p_{i}(i=1,2,\cdots,9)$ lie on $\Gamma_{3}$,
$\displaystyle\left(\begin{array}[]{ccccc}\alpha_{1}^{3}&\alpha_{1}^{2}\beta_{1}&\cdots&\beta_{1}^{2}\gamma_{1}&\beta_{1}\gamma_{1}^{2}\\\
\alpha_{2}^{3}&\alpha_{2}^{2}\beta_{2}&\cdots&\beta_{2}^{2}\gamma_{2}&\beta_{2}\gamma_{2}^{2}\\\
\cdots&&\cdots&&\cdots\\\
\alpha_{9}^{3}&\alpha_{9}^{2}\beta_{9}&\cdots&\beta_{9}^{2}\gamma_{9}&\beta_{9}\gamma_{9}^{2}\\\
\end{array}\right)\cdot\left(\begin{array}[]{c}a_{1}\\\ a_{2}\\\ \vdots\\\
a_{9}\\\ \end{array}\right)=0.$ (5.56)
Obviously, the system of linear equations:
$\displaystyle\left(\begin{array}[]{ccccc}\alpha_{1}^{3}&\alpha_{2}^{3}&\cdots&\alpha_{8}^{3}&\alpha_{9}^{3}\\\
\alpha_{1}^{2}\beta_{1}&\alpha_{2}^{2}\beta_{2}&\cdots&\alpha_{8}^{2}\beta_{8}&\alpha_{9}^{2}\beta_{9}\\\
\cdots&\cdots&\cdots&\cdots&\cdots\\\
\alpha_{1}\beta_{1}\gamma_{1}&\alpha_{2}\beta_{2}\gamma_{2}&\cdots&\alpha_{8}\beta_{8}\gamma_{8}&\alpha_{9}\beta_{9}\gamma_{9}\\\
\cdots&\cdots&\cdots&\cdots&\cdots\\\
\beta_{1}\gamma_{1}^{2}&\beta_{2}\gamma_{2}^{2}&\cdots&\beta_{8}\gamma_{8}^{2}&\beta_{9}\gamma_{9}^{2}\\\
\end{array}\right)\cdot\left(\begin{array}[]{c}\lambda_{1}\\\ \lambda_{2}\\\
\vdots\\\ \lambda_{9}\\\ \end{array}\right)=0$ (5.67)
has a non-zero solution. The condition of $C_{3}$ containing a term $z^{3}$
and passing through $p_{i}(i=1,2,\cdots,9)$ show that the vector
$(\gamma_{1}^{3},\gamma_{2}^{3},\cdots,\gamma_{9}^{3})$ can be expressed by
the linear combination of the 9 row vectors in (5.12). Therefore, the non-zero
solution $\Lambda$ of (5.12) is also solution of (5.9) and (5.8). This
completes the proof. ∎
###### Remark 5.9.
In fact, it can be easily seen from the process of the proof of Theorem 5.8 or
from the Chasles’s Theorem that Theorem 5.8 can be improved as:The spline
space $S_{3}^{2}(\Delta_{MS}^{2})$ is singular (i.e.
$dimS_{3}^{2}(\Delta_{MS}^{2})=11$) if and only if
$p_{i}=(\alpha_{i},\beta_{i},\gamma_{i}),(i=1,2,\ldots,9)$ lie on a cubic,
which differs from $l_{a}\cdot l_{b}\cdot l_{c}=0$.
### 5.2 Proof of Theorem 4.1
Let $a,b$ and $c$ be any three distinct lines in the projective plane
$\mathbb{P}^{2}$, denoted by $u=<c,a>,v=<a,b>$ and $w=<b,c>$, and $\Gamma_{3}$
be a cubic in $\mathbb{P}^{2}$. Assume that $p_{1},p_{2},p_{3}$ are three
intersection points of $a$ and $\Gamma_{3}$, $p_{4},p_{5},p_{6}$ are
intersection points of $b$ and $\Gamma_{3}$, and $p_{7},p_{8},p_{9}$ are
intersection points of $c$ and $\Gamma_{3}$. Then there are real numbers
$\\{a_{i},b_{i}\\}$ such that
$\displaystyle\left\\{\begin{array}[]{ll}p_{1}=a_{1}u+b_{1}v\\\
p_{2}=a_{2}u+b_{2}v\\\
p_{3}=a_{3}u+b_{3}v\end{array}\right.,\left\\{\begin{array}[]{ll}p_{4}=a_{4}v+b_{4}w\\\
p_{5}=a_{5}v+b_{5}w\\\
p_{6}=a_{6}v+b_{6}w\end{array}\right.and\left\\{\begin{array}[]{ll}p_{7}=a_{7}w+b_{7}u\\\
p_{8}=a_{8}w+b_{8}u\\\ p_{9}=a_{9}w+b_{9}u.\end{array}\right.$
Using Definition 1.3, the duality of the figure composed by the lines $a,b$
and $c$, the points $u,v$, $w$ and $\\{p_{i}\\}_{i=1}^{9}$ turns precisely out
the Morgan-Scott type partition $\Delta_{MS}^{2}$ (in which $\mu=2$) as shown
in Fig.5.3, where $l_{i}=\alpha_{i}x+\beta_{i}y+\gamma_{i}z,i=1,2,\cdots,9$.
From Theorem 5.8, we see that the spline space $S_{3}^{2}(\Delta_{MS}^{2})$ is
singular, that is, $\dim S_{3}^{2}(\Delta_{MS}^{2})=11$. Consequently, it
follows from Theorem 5.7 ($\mu=2$) that the characteristic number of a cubic
is always equal to $(-1)^{3}=-1$. Which completes the proof of our main
result.
Acknowledgement. The author would like to appreciate Prof. Prof. T.Y. Li of
Michigan State University for his valuable comments and suggestions, that
helped to improve the paper. The author also thanks Prof. R.H. Wang and J.Z.
Nan of Dalian University of Technology for their kind suggestions from the
aspect of computational geometry and projective geometry.
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|
arxiv-papers
| 2012-01-06T03:27:50 |
2024-09-04T02:49:26.012198
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhongxuan Luo",
"submitter": "Zhongxuan Luo",
"url": "https://arxiv.org/abs/1201.1344"
}
|
1201.1364
|
# Two-qubit gate operations in superconducting circuits with strong coupling
and weak anharmonicity
Xin-You Lü1,3, S. Ashhab1,2, Wei Cui1, Rebing Wu1,4, Franco Nori1,2 1Advanced
Science Institute, RIKEN, Wako-shi, Saitama 351-0198, Japan
2Physics Department, The University of Michigan, Ann Arbor, Michigan
48109-1040, USA
3School of Physics, Ludong University, Yantai 264025, P. R. China
4Department of Automation, Center for Quantum information Science and
Technology, Tsinghua University, Beijing, 100084, P.R. China
###### Abstract
We investigate theoretically the implementation of two-qubit gates in a system
of two coupled superconducting qubits. In particular, we analyze two-qubit
gate operations under the condition that the coupling strength is comparable
to or even larger than the anharmonicity of the qubits. By numerically solving
the time-dependent Schrödinger equation under the assumption of negligible
decoherence, we obtain the dependence of the two-qubit gate fidelity on the
system parameters in the case of direct and indirect qubit-qubit coupling. Our
numerical results can be used to identify the “safe” parameter regime for
experimentally implementing two-qubit gates with high fidelity in these
systems.
###### pacs:
03.67.-a; 42.50.Pq; 85.25.-j
## 1 Introduction
Superconducting (SC) circuits based on Josephson junctions are promising
candidates for the realization of scalable quantum computing on a solid-sate
platform, due to their design flexibility, large-scale integration and
controllability (see the reviews in Refs. [1, 2, 3, 4, 5, 6, 7]). SC qubits,
include the charge [8], flux [9], and phase qubits [10, 11] as well as their
variants, capacitively shunted flux qubits [12] and capacitively shunted
charge qubits (transmon) [13]. The phase qubit, the capacitively shunted flux
qubit and the transmon qubit are relatively insensitive to charge noise and
can be operated over a wide range of parameters. Single-qubit gates [14], two-
qubits gates [15, 16] and simple quantum algorithms [17] with these types of
qubits have been demonstrated experimentally in recent years. However,
comparing with the flux qubits, the common disadvantage of these types of
qubits is their weakly-anharmonic energy level structure, i.e., the detuning
between adjacent transition frequencies is very small.
Generally, the influence of the small anharmonicity (denoted by $\Delta$) on
quantum gate operations can be neglected when the qubit-field or qubit-qubit
coupling strength is very small compared with $\Delta$. However, for the
practical application of quantum computation, one wants to maximize the number
of quantum gate operations with a given coherence time. In other words, we
must implement quantum operations as fast as possible, which requires a strong
qubit-qubit or qubit-field coupling to be employed during the single- and two-
qubit gate operations [18]. The anharmonicity of SC qubits will influence the
quality of quantum gates more and more with increasing coupling strength.
Recently, there have been a number of theoretical studies analyzing the
effects of weak anharmonicity of SC qubits on the operation of single-qubit
gates and several optimization strategies have been proposed based on varying
driving pulse shapes and sequences [19, 20, 21, 22, 23]. Similar to single-
qubit gates, the weak anharmonicity of SC qubits will also influence the
implementation of two-qubit gates. Then two questions arise naturally: (1) how
much the weak anharmonicity of the qubits influence the implementation of two-
qubit gates in a system of coupled SC qubits? (2) how strong can the coupling
be while allowing a high two-qubit gate fidelity? In other words, how fast can
two-qubit gates with high fidelity be implemented, given the weak
anharmonicity of SC qubits?
Motivated by the above questions, in this paper we study the implementation of
two-qubit gates with superconducting systems in the strong coupling regime.
First, we introduce some possible methods for implementing two-qubit gates and
qualitatively discuss the effect of strong coupling (section II). Then, in
section III, we numerically simulate the influence of the coupling strength
and anharmonicity on the fidelities of two-qubit gates in different
superconducting systems, and show that the “safe” parameter regime for
implementing two-qubit gates with high fidelity can be identified, which is
useful for guiding experimental efforts based on superconducting qubits.
Finally, we conclude with a brief summary in section IV.
Figure 1: (Color online) System with direct (a) and indirect (b) qubit-qubit
coupling. Here, $g$, $G_{j}$ and $\Delta_{j}$ $(j=A,B)$ are the qubit-qubit,
qubit-cavity coupling strength and anharmonicity, respectively.
## 2 Model and qualitative discussion
As shown in Fig. 1, as model systems we consider two directly (a) or
indirectly (b) coupled SC qubits with weakly-anharmonic multilevel structure
(such as transmon or phase qubits). Here it should be pointed out the flux
qubits have a strong anharmonicity, and the problem discussed in this paper is
not a serious limitation. The two lowest levels $\\{|0\rangle_{j}$,
$|1\rangle_{j}\\}$, separated in energy by $\hbar\omega_{j}$ ($j=A,B$), are
the computational basis, and the $n$th ($n\geq 2$) higher levels are different
from $n\hbar\omega_{j}$ by $\hbar\epsilon^{j}_{n}$. Here $\epsilon^{j}_{n}$
has the standard nonlinear oscillator form
$\epsilon^{j}_{n}=\Delta_{j}(n-1)n/2$ [24] and $\Delta_{j}$ is the
anharmonicity of the qubit, and it is positive in our paper.
In the case of direct qubit-qubit coupling, two qubits are directly
(capacitively) coupled, while they are dispersively coupled to a common
transmission line resonator in the case of indirect qubit-qubit coupling. The
Hamiltonian of these two types of coupled system is given by ($\hbar=1$) [25,
26, 27, 28, 29, 30, 31, 32]
$\displaystyle H^{direct}$ $\displaystyle=$
$\displaystyle\sum^{N-1}_{n=1}\left[\left(n\omega_{A}-\epsilon^{A}_{n}\right)|n\rangle_{A}\langle
n|+\left(n\omega_{B}-\epsilon^{B}_{n}\right)|n\rangle_{B}\langle
n|\right]+gJ^{x}_{A}\otimes J^{x}_{B},$ (1a) $\displaystyle H^{indirect}$
$\displaystyle=$
$\displaystyle\omega_{c}a^{{\dagger}}a+\sum_{j=A,B}\left[\sum^{N-1}_{n=1}\left(n\omega_{j}-\epsilon^{j}_{n}\right)|n\rangle_{j}\langle
n|+G_{j}(a+a^{{\dagger}})J^{x}_{j}\right],$ (1b) $\displaystyle J^{x}_{A}$
$\displaystyle=$
$\displaystyle\sum^{N-1}_{n=1}\eta^{A}_{n-1,n}\sigma^{Ax}_{n-1,n},\;\;\;J^{x}_{B}=\sum^{N-1}_{n=1}\eta^{B}_{n-1,n}\sigma^{Bx}_{n-1,n},$
(1c)
where $H^{d}$ and $H^{id}$ denote the Hamiltonian for the system with direct
and indirect qubit-qubit coupling, $N$ is the number of levels in each SC
qubit, $\eta^{j}_{n-1,n}=\sqrt{n}$ is the level-dependent coupling matrix
element, and $\sigma^{jx}_{n-1,n}=|n-1\rangle_{j}\langle
n|+|n-1\rangle_{j}\langle n|$ is the effective Pauli spin operators for levels
$|n-1\rangle$ and $|n\rangle$. Also, $\omega_{c}$ is the frequency of the
quantized cavity mode; $g$ and $G_{j}$ denote the qubit-qubit and qubit-cavity
coupling strength.
In order to qualitatively analyze the implementation and fidelity of two-qubit
gates, we assume that each qubit has three levels. Then, the Hamiltonian of
direct qubit-qubit coupled system ($H^{direct}$), under the rotation-wave
approximation (RWA), can be reduced to
$\displaystyle\;\;\;\;\;\;H^{direct}_{I}=\sum_{j=A,B}\left[\omega_{j}|1\rangle_{j}\langle
1|+\left(2\omega_{j}-\Delta_{j}\right)|2\rangle_{j}\langle 2|\right]$
$\displaystyle+g[|01\rangle\langle 10|+\sqrt{2}|02\rangle\langle
11|+\sqrt{2}|20\rangle\langle 11|+2|12\rangle\langle 21|+h.c.],$ (1b)
where $|mn\rangle$ denotes $|m\rangle_{A}|n\rangle_{B}$.
For the system with indirect qubit-qubit coupling, under the dispersive qubit-
cavity-coupling condition, i.e.,
$\mid\delta_{j}\mid=\mid\omega_{j}-\omega_{c}\mid\gg G_{j}$ $(j=A,B)$, the
qubits will exchange energy by virtual photon processes. Then we can obtain
the Hamiltonian of the effective qubit-qubit interaction by a Fröhlich
transformation [33, 34, 35, 36],
$\displaystyle H^{indirect}_{{\rm eff},1}=\exp(-S)H^{id}\exp(S)$
$\displaystyle\;\;\;\;\;\;\;\;\;\approx\sum_{j=A,B}\left\\{\left[\left(\omega_{j}+\frac{G^{2}}{\delta_{j}}\right)|1\rangle_{j}\langle
1|+\left(2\omega_{j}-\Delta_{j}+\frac{2G^{2}}{\delta_{j}-\Delta_{j}}\right)|2\rangle_{j}\langle
2|+\frac{G^{2}}{2\delta_{j}}a^{{\dagger}}a\left(|1\rangle_{j}\langle
1|-|0\rangle_{j}\langle 0|\right)\right.\right.$
$\displaystyle\;\;\;\;\;\;\;\;\;\left.\left.+\frac{G^{2}}{\delta_{j}-\Delta_{j}}a^{{\dagger}}a\left(|2\rangle_{j}\langle
2|-|1\rangle_{j}\langle
1|\right)\right]+\left[\frac{\sqrt{2}G^{2}}{2}\left(\frac{1}{\delta_{j}-\Delta_{j}}-\frac{1}{\delta_{j}}\right)a^{2}|2\rangle_{j}\langle
0|\right.\right.$
$\displaystyle\;\;\;\;\;\;\;\;\;\left.\left.+\frac{G^{2}}{2}\left(\frac{1}{\delta_{A}}+\frac{1}{\delta_{B}}\right)|01\rangle\langle
10|+\frac{\sqrt{2}G^{2}}{2}\left(\frac{1}{\delta_{B}-\Delta_{B}}+\frac{1}{\delta_{A}}\right)|02\rangle\langle
11|\right.\right.$
$\displaystyle\;\;\;\;\;\;\;\;\;\left.\left.+\frac{\sqrt{2}G^{2}}{2}\left(\frac{1}{\delta_{A}-\Delta_{A}}+\frac{1}{\delta_{B}}\right)|20\rangle\langle
11|+G^{2}\left(\frac{1}{\delta_{A}-\Delta_{A}}+\frac{1}{\delta_{B}-\Delta_{B}}\right)|12\rangle\langle
21|+h.c.\right]\right\\},$ (1c)
where
$\displaystyle
S=\sum_{j=A,B}\left[\frac{G}{\delta_{j}}a^{{\dagger}}|0\rangle_{j}\langle
1|+\frac{\sqrt{2}G}{\delta_{A}-\Delta_{A}}a^{{\dagger}}|1\rangle_{j}\langle
2|-h.c.\right].$ (1d)
Here, we have assumed that $G_{A}=G_{B}=G$.
Figure 2: (Color online) The energy-level diagram of two-qubit product states
for the iSWAP gate (a), and the controlled-Z gate (b) in the system with
direct qubit-qubit coupling. Red levels denote the states in the computational
basis. The black dashed arrows are the resonant transitions used for realizing
the two-qubit gates and the green dotted arrows are the main $undesired$
transitions, which adversely affect the implementation of two-qubit gates. The
couplings $g$ and $\sqrt{2}g$ are indicated in blue, while the detuning
between levels is indicated in black. This figure also applies the system with
indirect qubit-qubit coupling when the corresponding couplings is replaced by
$g_{{\rm eff},m}$ $(m=1,2,3,4)$.
The terms proportional to $G^{2}$ in the first four terms of equation (3)
represent level shifts, and the fifth term describes two-photon processes.
Under the dispersive qubit-cavity-coupling condition, the cavity mode is only
virtually excited during the gate operation, and therefore the third, fourth,
and fifth terms of equation (3) vanish. Then, the Hamiltonian (3) can be
simplified further as [37, 38, 39, 40, 41]
$\displaystyle H^{indirect}_{{\rm
eff},2}=\sum_{j=A,B}\left[\tilde{\omega}_{j1}|1\rangle_{j}\langle
1|+\left(\tilde{\omega}_{j2}-\Delta_{j}\right)|2\rangle_{j}\langle 2|\right]$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;+\left[\sqrt{2}g_{{\rm
eff},1}|02\rangle\langle 11|+\sqrt{2}g_{{\rm eff},2}|20\rangle\langle
11|+g_{{\rm eff},3}|01\rangle\langle 10|+2g_{{\rm eff},4}|12\rangle\langle
21|+h.c.\right].$ (1e)
where
$\displaystyle\tilde{\omega}_{j1}=\omega_{j}+\frac{G^{2}}{\delta_{j}},$ (1fa)
$\displaystyle\tilde{\omega}_{j2}=2\omega_{j}+\frac{2G^{2}}{\delta_{j}-\Delta_{j}},$
(1fb) $\displaystyle g_{{\rm eff},1}$ $\displaystyle=$
$\displaystyle\frac{G^{2}}{2}\left(\frac{1}{\delta_{B}-\Delta_{B}}+\frac{1}{\delta_{A}}\right),$
(1fc) $\displaystyle g_{{\rm eff},2}$ $\displaystyle=$
$\displaystyle\frac{G^{2}}{2}\left(\frac{1}{\delta_{A}-\Delta_{A}}+\frac{1}{\delta_{B}}\right),$
(1fd) $\displaystyle g_{{\rm eff},3}$ $\displaystyle=$
$\displaystyle\frac{G^{2}}{2}\left(\frac{1}{\delta_{A}}+\frac{1}{\delta_{B}}\right),$
(1fe) $\displaystyle g_{{\rm eff},4}$ $\displaystyle=$
$\displaystyle\frac{G^{2}}{2}\left(\frac{1}{\delta_{A}-\Delta_{A}}+\frac{1}{\delta_{B}-\Delta_{B}}\right).$
(1ff)
Now, we obtain an effective interaction Hamiltonian similar to the Hamiltonian
(2) in the system with direct qubit-qubit coupling.
From the Hamiltonians (2) and (5), it is easily seen that various two-qubit
gates can be realized by appropriately adjusting the qubit frequencies
($\omega_{A}$, $\omega_{B}$) both in the system with direct and indirect
qubit-qubit coupling. For example, by setting $\omega_{A}=\omega_{B}$
($\omega_{B}=\omega_{A}+\Delta_{B}$), the resonant transition between state
$|01\rangle$ and $|10\rangle$ ($|11\rangle$ and $|02\rangle$) can be obtained
as shown in Fig. 2. Then the two-qubit iSWAP [15] (CZ [16, 17]) gate can be
realized after an interaction time $gt_{g}=\pi/2$ or $g_{{\rm
eff},3}t_{g}=\pi/2$ ($\sqrt{2}gt=\pi$ or $\sqrt{2}g_{{\rm eff},1}t=\pi$). Here
it should be pointed out that some undesired transitions [see the (green)
dotted arrows in Fig. 2] have been neglected in the weak-coupling regime
$g\ll|\Delta_{j}|$ or $g_{{\rm eff},m}\ll|\Delta_{j}|$ $(m=1-4;j=A,B)$.
However with increasing coupling strength $g$ or $g_{{\rm eff},m}$, the
average amplitude $g/|\Delta_{j}|$ or $g_{{\rm eff},m}/|\Delta_{j}|$ of
undesired transitions will become larger and larger, which can not be
neglected again and will reduce the fidelity of the two-qubit gate. So, the
relative value of the coupling strength $g$ or $g_{{\rm eff},m}$ and the
anharmonicity $\Delta_{j}$ is an important parameter for the quality of the
two-qubit gate. In the two-qubit gate scheme based on SC qubits, a very strong
qubit-qubit or qubit-cavity coupling strength cannot be employed due to the
weak anharmonicity of the qubits, if one wants to obtain a high fidelity. How
strong the coupling can be, while allowing high two-qubit-gate fidelities,
will be analyzed in detail in the next section.
## 3 Numerical results
In this section, we will numerically calculate the fidelity of two-qubit gates
in the circuits with either direct or indirect qubit-qubit coupling.
Importantly, the present numerical results can help identify the safe
parameter regime for implementing two-qubit gates with high fidelity. Here, we
neglect the noise and decoherence of system in order to show explicitly the
influence of coupling strength and anharmonicity on the fidelity of two-qubit
gates. Here, it should also be pointed out that the single-qubit gates are
performed using microwave pulses (with frequencies of a few of GHz), while the
frequency tuning for the two-qubit gates are implemented using trapezoidal
pulses.
Here, the fidelity of a two-qubit gate is defined as the Euclidean distance
between the target $U_{T}$ and the actual evolution $U(t_{g})$ [22],
$\displaystyle F=1-\frac{1}{16}\|U_{T}-P^{{\dagger}}U(t_{g})P\|^{2}_{2},$
(1fg)
where $U(t)$ is the usual time evolution operator obeying the Schrödinger
equation $\dot{U}(t)=-\frac{i}{\hbar}H(t)U(t)$ in the full space of the
quantum system. Here $\|X\|_{2}^{2}={\rm tr}(X^{{\dagger}}X)$ where $X$ is an
arbitrary operator. $P$ is the projection operator on the two-qubit
computational $\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$;
$U_{T}=|00\rangle\langle 00|-i|01\rangle\langle 10|-i|10\rangle\langle
01|+|11\rangle\langle 11|$
corresponds to the two-qubit iSWAP gate, and
$U_{T}=|00\rangle\langle 00|+|01\rangle\langle 01|+|10\rangle\langle
10|-|11\rangle\langle 11|$
corresponds to the two-qubit CZ gate. Here it should be pointed out that
single-qubit rotations and an overall phase factor
$U^{A}_{z}=e^{i\theta_{A}\sigma^{A}_{z}}$,
$U^{B}_{z}=e^{i\theta_{B}\sigma^{B}_{z}}$, $U_{I}=e^{i\theta I}$ are used in
the numerical calculations in order to eliminate any extra phase factors; $I$
is the unit matrix and
$\sigma^{A}_{z}=|00\rangle\langle 00|+|01\rangle\langle 01|-|10\rangle\langle
10|-|11\rangle\langle 11|,$ $\sigma^{B}_{z}=|00\rangle\langle
00|-|01\rangle\langle 01|+|10\rangle\langle 10|-|11\rangle\langle 11|.$
Specifically, in our numerical calculations, we replace the unitary operation
$U(t_{g})$ in Eq. (7) by $U^{\prime}(t_{g})=U_{I}U^{B}_{z}U^{A}_{z}U(t_{g})$
and choose $\theta_{A}$, $\theta_{B}$ and $\theta$ that maximize the fidelity.
We also note here that in our numerical calculations we do not use the RWA.
But, there is almost no difference between these results shown below and the
numerical results with the RWA (not shown in this paper). The reason is that
the parameter regime that we consider does not reach the ultrastrong coupling
regime and thus the RWA is valid here. Very recently, the influence of the
counter-rotating terms in the Hamiltonian on the two-qubit gates in the
ultrastrong coupling regime has been studied in a related system [42]. Also,
the effect of counter-rotating terms were studied in [43].
### 3.1 System with direct qubit-qubit coupling
Figure 3: (Color online) The fidelities of the two-qubit iSWAP (a) and CZ (b)
gate as functions of $g/\Delta_{B}$ in a circuit with direct qubit-qubit
coupling. Some representative dots are denoted by the dashed lines and red
circles in order to present the relationship between the gate time $t_{g}$ and
fidelity $F$. The red arrows point out the parameter regime corresponding to
two-qubit gate with high fidelity. In figure (b), the qubit frequencies are
adiabatically adjusted during the gate operation, as shown in the inset part.
The system parameters used here are: (a) $\omega_{A}/2\pi=5.5$ GHz,
$\omega_{B}=\omega_{A}$, $\Delta_{A}/2\pi=0.15$ GHz, and $\Delta_{B}/2\pi=0.1$
GHz; (b) $\omega_{A}/2\pi=7.16$ GHz, $\Delta_{A}/2\pi=0.087$ GHz,
$\Delta_{B}/2\pi=0.114$ GHz, and $\omega_{B}=\omega_{A}+\Delta_{B}$.
Figure 4: (Color online) The fidelities of the two-qubit iSWAP gate (a)
$F_{\rm iSWAP}$ and CZ gate $F_{\rm CZ}$ (b) versus $\Delta_{A}/g$ and
$\Delta_{B}/g$ in a circuit with direct qubit-qubit coupling. The dashed lines
correspond to the parameter regime for implementing a two-qubit gate with
fidelities 95$\%$ and 99$\%$. The system parameters are the same as in Fig. 3
except for $g/2\pi=0.2$ GHz. Figure 5: (Color online) The fidelities of the
two-qubit iSWAP gates versus $\Delta_{B}$ and $g$ in a circuit with direct
qubit-qubit coupling. The dashed lines correspond to the parameter regime for
implementing two-qubit gate with fidelities 95$\%$ and 99.5$\%$. The system
parameters are the same as in Fig. 3 except for $\Delta_{A}=\Delta_{B}$.
In this subsection, based on the original Hamiltonian Eq. (1a), we numerically
calculate the influence of the coupling strength $g$ and anharmonicity
$\Delta_{j}$ on the fidelities of the two-qubit iSWAP and CZ gates (see Figs.
3-5). Here we consider the two-qubit iSWAP and CZ gates implemented in
experiments [15]. In Figs. 3(a) and (b), we plot the fidelities of the two-
qubit iSWAP gate ($F_{\rm iSWAP}$) and the CZ gate ($F_{\rm CZ}$) as functions
of $g/\Delta_{B}$ in a circuit with direct qubit-qubit coupling, where we
consider each SC qubit to have three levels (same approximation will be used
in Figs. 4 and 5). From Fig. 3(a) and the (green) solid line in Fig. 3(b), it
can be seen that the fidelities of these gates decrease with increasing
$g/\Delta_{B}$, and the present numerical results can help identify the safe
parameter regime for realizing two-qubit gates with high fidelities. As shown
in Fig. 3(a), if we want to implement the two-qubit iSWAP (CZ) gate with
fidelity higher than 99$\%$ (99.2$\%$), the safe parameter regime is
$g/\Delta_{B}<0.152$ ($g/\Delta_{B}<0.24$). In other words, based on the
relationship $gt_{g}=\pi/2$ for the iSWAP gate and $\sqrt{2}gt_{g}=\pi$ for
the CZ gate, the present numerical results can also identify the time limit
for implementing two-qubit gates with high fidelity. For example, here the
shortest gate time is $t_{g}\approx 16.4$ ns ($t_{g}\approx 12.9$ ns) for
implementing a two-qubit iSWAP (CZ) gate with fidelity higher than 99$\%$
(99.2$\%$).
The (green) solid line in Fig. 3(b) shows small oscillations in the fidelity
of the two-qubit CZ gate. This result is due to the frequency mismatch between
the undesired transitions and the resonant transition [see Fig. 2(b)], and it
demonstrated that the fluctuations of the system parameters will influence the
implementation of two-qubit gates. Based on the idea of adiabatically
eliminating undesired transitions, these oscillations can be reduced by slowly
adjusting the frequencies of the qubits during the gate operation. As shown in
the inset of Fig. 3(b), the frequency of qubit B starts at 1.1$\omega_{B}$, is
first ramped down to $\omega_{B}$ in $\tau_{d}$, then ramped up to
1.1$\omega_{B}$ after an interaction time $t_{g}$ $(\sqrt{2}gt_{g}=\pi)$.
During the full gate operation time ($2\tau_{d}$+$t_{g}$), the frequency of
qubit A is fixed. Using such pulses, we numerically calculate the fidelities
of the two-qubit CZ gate for different values of $\tau_{d}$ and present the
results in Fig. 3(b) [See dashed, dotted and dot-dashed lines in Fig. 3(b)].
It can be seen that the oscillations of the fidelity can be eliminated by
adiabatically adjusting the qubit frequencies during the gate operation. This
numerical result provides a method to reduce the influence of parameter
fluctuations on the implementation of two-qubit gates.
Figure 6: (Color online) The fidelity of the two-qubit CZ gate versus
$\Delta_{A}/g_{{\rm eff},1}$, $\Delta_{B}/g_{{\rm eff},1}$ (a) and versus
$g_{{\rm eff},1}$, $\Delta_{B}$ (b) in the system with indirect qubit-qubit
coupling. The dashed lines correspond to the parameter regime for implementing
two-qubit gate with fidelities 95$\%$ and 99$\%$. The basal system parameters
are: $\omega_{c}/2\pi=6.9$ GHz, $\omega_{A}/2\pi=8.2$ GHz,
$\omega_{B}=\omega_{A}+\Delta_{B}$, $\delta_{j}=\omega_{j}-\omega_{c}$
$(j=A,B)$; And $G=0.2$ GHz for panel (a), $\Delta_{A}/2\pi=\Delta_{B}/2\pi$
GHz for panel (b).
In order to show the influence of $\Delta_{A}$ and $\Delta_{B}$ on the two-
qubit gates, we plot the fidelities of the two-qubit iSWAP and CZ gates as
functions of $\Delta_{A}/g$ and $\Delta_{B}/g$ in Fig. 4. It is easily seen
from Fig. 4(a) that the anharmonicities $\Delta_{A}$ and $\Delta_{B}$ have
equal effects on the two-qubit iSWAP gate, i.e., the larger the
anharmonicities $\Delta_{j}$ $(j=A,B)$ are, the higher the fidelity. This
symmetric property disappears in the two-qubit CZ gate due to the asymmetry in
the condition on the parameters, $\omega_{B}=\omega_{A}+\Delta_{B}$ [see Fig.
4(b)]. In other words, the influence of the anharmonicity $\Delta_{A}$ on the
two-qubit CZ gate can be neglected when $\omega_{B}=\omega_{A}+\Delta_{B}$ is
chosen. In addition, the dashed lines in Fig. 4 indicate the safe regime of
$\Delta_{j}/g$ $(j=A,B)$ for implementing two-qubit iSWAP and CZ gates with
fidelity higher than $99\%$.
In Figs. 3 and 4, either the anharmonicity $\Delta_{j}$ or the coupling
strength $g$ have been set to a fixed value. A natural question is whether the
conclusions obtained from Figs. 3 and 4 are universal. In other words, will
the properties of Figs. 3 and 4 change much when either $\Delta_{j}$ or $g$ is
changed? Thus, we now present in Fig. 5 three-dimensional (3D) plots of the
dependence of $F_{\rm iSWAP}$ on $g$ and $\Delta_{B}$. It is shown that the
fidelity of two-qubit gates are approximately determined by the ratio of the
qubit-qubit coupling strength $g$ to the anharmonicity $\Delta_{j}$ of the SC
qubits. As a result, the conclusion obtained from Fig. 3(a) [or Fig. 4(a)]
will not be changed when adjusting $\Delta_{B}$ (or $g$). A similar property
is also obtained from the two-qubit CZ gate (the corresponding figures are not
shown in this paper because are very similar to Fig. 5).
### 3.2 System with indirect qubit-qubit coupling
Figure 7: (Color online) The fidelities of the two-qubit gates as a function
of $g/\Delta_{B}$ (a) and $g_{{\rm eff},1}/\Delta_{B}$ (b) in systems with
direct (a) and indirect (b) qubit-qubit coupling, when the three, four, or
five lowest levels are considered for each qubit. The system parameters are
the same as in Fig. 3 or 6. The green, red circles in (a) and cyan circle in
(b) mark respectively the experimental parameters regime in Refs. [15], [16],
[17].
In this subsection, based on the Hamiltonian Eq. (1b), we present the results
of numerical calculations for the dependence of the fidelity of the two-qubit
gates on the effective qubit-qubit coupling $g_{\rm eff1}$ and anharmonicity
$\Delta_{j}$ of SC qubits. Here the two-qubit CZ gates are realized based on
the qubit-cavity dispersive interaction method [17], and the parameter
$\displaystyle g_{{\rm
eff},1}=\frac{G^{2}}{2}\left(\frac{1}{\delta_{B}-\Delta_{B}}+\frac{1}{\delta_{A}}\right)=\frac{G^{2}}{\delta_{A}}$
under the condition $\omega_{B}=\omega_{A}+\Delta_{B}$.
In Fig. 6, we present the 3D plots of the dependence of $F_{\rm CZ}$ on
$\Delta_{A}/g_{{\rm eff},1}$ and $\Delta_{B}/g_{{\rm eff},1}$ [panel (a)], and
$g_{{\rm eff},1}$ and $\Delta_{B}$ [panel (b)], where we consider the SC
qubits to have three levels. Using dashed lines, we have denoted the parameter
regime for implementing two-qubit CZ gate with fidelities $95\%$ and $99\%$.
It is shown from Figs. 6(a) and (b) that high-fidelity areas correspond to the
weak-coupling regime $g_{{\rm eff},1}/\Delta_{j}\ll 1$ $(j=A,B)$, while low
fidelity corresponds to the strong-coupling regime, where $g_{{\rm eff},1}$ is
comparable to or larger than $\Delta_{j}$. This property is similar as that in
the system with direct qubit-qubit coupling. The present numerical results can
be used to identify the safe parameter regime for implementing the two-qubit
CZ gate with high fidelity in the circuit with indirect qubit-qubit coupling.
### 3.3 Going beyond the three-level approximation
Until now, three-level-system approximation for qubits has been used in the
above numerical calculations. It is then natural to ask the following
question: will our conclusions, obtained from the above numerical results,
still be valid for qubits with $N$ ($N>$3) levels? To explore this, in Fig. 7,
we plot the fidelities of the two-qubit iSWAP and CZ gates as functions of
$g/\Delta_{B}$ (or $g_{{\rm eff},1}/\Delta_{B}$) in the system with direct (or
indirect) qubit-qubit coupling when each qubit has three, four or five levels.
It can be seen from Fig. 7 that there is not much difference between the
numerical results based on the three-, four- and five-level approximations for
the qubits. So, our conclusions obtained from the above numerical calculations
are still valid for $N$-level (with $N>$3) SC qubits.
### 3.4 Limits on the gate fidelities of recent experiments imposed by weak
anharmonicity
In order to serve as a guide for future experiments, we compare our numerical
results with corresponding experiments and show the limited fidelity of two-
qubit gate based on SC qubits with weak anharmonicity. Based on the
experimental parameters ($\omega_{A}/2\pi$, $\omega_{B}/2\pi$,
$\Delta_{A}/2\pi$, $\Delta_{B}/2\pi$, $g/2\pi$) equal to (5.5, 5.5, 0.15, 0.1,
0.011) GHz and (7.16, 7.274, 0.087, 0.114, 0.0091) GHz, two-qubit iSWAP [15]
and CZ [16] gates with fidelities 63% and 70% were implemented in the circuit
with direct qubit-qubit coupling. In the circuit with indirect qubit-qubit
coupling, a two-qubit gate [17] with fidelity 85% was realized with system
parameters ($\omega_{c}/2\pi$, $\omega_{A}/2\pi$, $\omega_{B}/2\pi$,
$\Delta_{A}/2\pi$, $\Delta_{B}/2\pi$, $G_{A}/2\pi=G_{B}/2\pi$) equal to (6.9,
8.2, 8.45, 0.2, 0.25, 0.199) GHz. Corresponding to the above experimental
parameters, in Fig. 7 we indicate the ideal fidelity (see the green, red and
magenta circles) based on our theoretical calculations. From the comparison
between experiments and our numerical calculations, we show that two-qubit
gates with fidelities 99.52%, 99.91%, and 99.2% can be realized, in principle,
if the influence of decoherence can be eliminated. Recently, the effects of
decoherence on quantum gates and possible optimization routes were also
studied in Ref. [44].
## 4 Conclusion
We have studied the performance of two-qubit gates in a system of two coupled
SC qubits under the condition that the coupling strength is comparable to or
larger than the anharmonicity of the qubits. First of all, by using the three-
level approximation for the qubits, we analyzed and numerically calculated the
dependence of the two-qubit gate fidelity on the qubit-qubit coupling strength
and the anharmonicity of the qubits. Based on extensive numerical results, the
safe parameter regime was identified for experimentally implementing two-qubit
gates with high fidelity. Secondly, we numerically calculated the fidelity of
the two-qubit gates in the case of four- and five-level approximations for the
qubits, and demonstrated the validity of our numerical results for $N$-level
qubits with $N>3$. Our results can serve as a guide for future experiments
based on SC qubits.
We would like to thank E. Solano for useful discussions. This work was
partially supported by ARO grant No. 0726909, JSPS-RFBR (No. 09-02- 92114),
Grant-in-Aid for Scientific Research (S), MEXT Kakenhi on Quantum Cybernetics,
the JSPS via its FIRST program. XYL was supported by the National Natural
Science Foundation of China (Grant No. 11005057).
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|
arxiv-papers
| 2012-01-06T08:17:32 |
2024-09-04T02:49:26.021763
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin-You L\\\"u, S. Ashhab, Wei Cui, Rebing Wu, and Franco Nori",
"submitter": "Xinyou Lu Dr.",
"url": "https://arxiv.org/abs/1201.1364"
}
|
1201.1375
|
# Efficient Estimation of Nonlinear Finite Population
Parameters Using Nonparametrics
Camelia Goga 1 and Anne Ruiz-Gazen2
1 IMB, Université de Bourgogne, DIJON - France
2 TSE, Université Toulouse 1, Toulouse, France.
email : $camelia.goga$@u-bourgogne.fr, $ruiz$@cict.fr
###### Abstract
Currently, the high-precision estimation of nonlinear parameters such as Gini
indices, low-income proportions or other measures of inequality is
particularly crucial. In the present paper, we propose a general class of
estimators for such parameters that take into account univariate auxiliary
information assumed to be known for every unit in the population. Through a
nonparametric model-assisted approach, we construct a unique system of survey
weights that can be used to estimate any nonlinear parameter associated with
any study variable of the survey, using a plug-in principle. Based on a
rigorous functional approach and a linearization principle, the asymptotic
variance of the proposed estimators is derived, and variance estimators are
shown to be consistent under mild assumptions. The theory is fully detailed
for penalized B-spline estimators together with suggestions for practical
implementation and guidelines for choosing the smoothing parameters. The
validity of the method is demonstrated on data extracted from the French Labor
Force Survey. Point and confidence intervals estimation for the Gini index and
the low-income proportion are derived. Theoretical and empirical results
highlight our interest in using a nonparametric approach versus a parametric
one when estimating nonlinear parameters in the presence of auxiliary
information.
Keywords auxiliary information; penalized B-splines; calibration;
concentration and inequality measures; influence function; linearization;
model-assisted approach; total variation distance.
## 1 Introduction
The estimation of nonlinear parameters in finite populations has become a
crucial problem in many recent surveys. For example, in the European
Statistics on Income and Living Conditions (EU-SILC) survey, several
indicators for studying social inequalities and poverty are considered; these
include the Gini index, the at-risk-of-poverty rate, the quintile share ratio
and the low-income proportion. Thus, deriving estimators and confidence
intervals for such indicators is particularly useful. In the present paper,
assuming that we have a single continuous auxiliary variable available for
every unit in the population, we propose a general class of estimators that
take into account the auxiliary variable, and we derive their asymptotic
properties for general survey designs. The class of estimators we propose is
based on a nonparametric model-assisted approach. Interestingly, the
estimators can be written as a weighted sum of the sampled observations,
allowing a unique weight variable that can be used to estimate any complex
parameter associated with any study variable of the survey. Having a unique
system of weights is very important in multipurpose surveys such as the EU-
SILC survey.
The estimation of nonlinear parameters is a problem that has already been
addressed in several papers such as Shao (1994) for L-estimators, Binder and
Kovacevic (1995) for the Gini index and Berger and Skinner (2003) for the low-
income proportion. We mention also the very recent work of Opsomer and Wang
(2011). Taking auxiliary information into account for estimating means or
totals is a topic that has been extensively studied in the literature; it now
encompasses the model-assisted and the calibration approaches, which coincide
in particular cases (Särndal, 2007). In a model-assisted setting, linear
models are usually used, thus leading to the well-known generalized regression
estimators (GREG). Some nonparametric models have also been considered (Breidt
and Opsomer, 2009). However, to the best of our knowledge, ratios,
distribution functions and quantiles are the only examples of nonlinear
parameters estimated using auxiliary information.
To derive our class of estimators and their asymptotic properties, we use an
approach based on the influence function developed by Deville (1999). This
approach utilizes a functional interpretation of the parameter of interest and
a linearization principle to derive asymptotic approximations of the
estimators. In general, the precision of an estimator $\widehat{\Phi}$ of a
nonlinear finite population parameter $\Phi$ is obtained by resampling
techniques or linearization approaches and in the present paper we focus on
linearization techniques. When a sample $s$ is selected from the finite
population $U$ according to a sampling design $p(\cdot)$, the linearization of
$\widehat{\Phi}$ leads under some assumptions, to the following approximation:
$\displaystyle\hat{\Phi}-\Phi\simeq\sum_{s}\frac{u_{k}}{\pi_{k}}-\sum_{U}u_{k}$
(1)
where $\pi_{k}=Pr(k\in s)>0$ denotes the first-order inclusion probability for
element $k$ under the design $p(\cdot)$. The right term of (1) is the
difference between the well-known Horvitz-Thompson estimator and the parameter
it estimates, namely the total of the variable $u_{k}$ over the population
$U$. Here, $u_{k}$ referred to as the linearized variable of $\Phi$ and the
way it is derived depends on the type of linearization method used which could
include the Taylor series (Särndal et al., 1992), estimating equations
(Binder, 1983) or influence function (Deville, 1999) approaches. The
artificial variable $u_{k}$ is used to compute the approximative variance of
$\widehat{\Phi}$ as
$\displaystyle\sum_{s}\sum_{s}(\pi_{kl}-\pi_{k}\pi_{l})\frac{u_{k}}{\pi_{k}}\frac{u_{l}}{\pi_{l}},$
(2)
with $\pi_{kl}=Pr(k\in s,l\in s)$ the joint inclusion probability for the
elements $k,l\in U.$
Roughly speaking, when examining (1) and (2), we can see that, if we estimate
in an efficient way $\sum_{U}u_{k}$, we will achieve a small approximative
variance and good precision for $\widehat{\Phi}$. As stated above, it is well
known that auxiliary information is useful for improving on the estimation of
a total in terms of efficiency and, based on a linear model, the use of a GREG
estimator is the most common alternative. When estimating a total, note that
the asymptotic variance of the GREG estimator depends on the residuals of the
study variable on the auxiliary variable. Because linearized variables may
have complicated mathematical expressions, fitting a linear model onto
linearized variables may not be the most appropriate choice. This may occur
even if the study and the auxiliary variables have a clear linear
relationship, as illustrated in the following example. Consider a data set of
size 1000 extracted from the French Labor Force Survey and consider $y_{k}$
(the wages of person $k$ in 2000) as the study variable and $x_{k}$ (the wages
of person $k$ in 1999) as the auxiliary variable. We now consider the problem
of estimating the Gini index. The expression of the linearized variable
$u_{k}$, $k\in U$ for the Gini index is given in Binder and Kovacevic (1995)
and recalled in equation (17). It is a complex function of the study variable
$y_{k}$, $k\in U$. In the left (resp. right) graphic of Figure 1, the study
variable $y_{k}$ is plotted (resp. the linearized variable $u_{k}$) on the
$y$-axis and the auxiliary variable $x_{k}$ is plotted on the $x$-axis. The
relationship between the study variable and the auxiliary variable is almost
linear; however the relationship between the linearized variable of the Gini
index and the auxiliary information is no longer linear. The consequence of
this is that we cannot increase the efficiency of estimating a Gini index if
we take the auxiliary information into account through a GREG estimator.
Therefore, nonparametric models should be preferred to estimate nonlinear
parameters $\Phi$.
Figure 1: Left plot: $y_{k}$: the wages of person $k$ in 2000 against $z_{k}$:
the wages of person $k$ in 1999. Right plot: $u_{k}$: linearized variable of
the Gini index for the wages in 2000 for person $k$ against $z_{k}$: the wages
of person $k$ in 1999.
Recent work already employs nonparametric models to estimate totals (Breidt
and Opsomer, 2000, Breidt et al., 2005 and Goga, 2005). The use of
nonparametrics prevents model failure; however the improvement over parametric
estimation for totals and means may not be significant enough to justify the
supplemental difficulties of implementing nonparametric methodology. As
illustrated above, the motivation for using nonparametrics becomes much
stronger when estimating nonlinear parameters. Note that the use of
nonparametric regression to estimate distribution functions and quantiles has
also been studied, for example in Johnson et al. (2008); however, to our
knowledge, this has not been performed for other nonlinear parameters.
We propose a novel methodology that allows for the efficient estimation of any
parameter $\Phi$ by combining the functional approach (Deville, 1999) with any
of the previously suggested nonparametric methods. One issue with the
functional approach is that several technical details are not provided in
Deville (1999); thus it is difficult to derive rigorous proof of asymptotic
results by following this approach. In the present paper, we propose to
clarify some important points and derive rigorous proofs of our asymptotic
results. Most importantly, we prove that the total variation distance between
finite measures is an adequate choice for the derivation of asymptotic
approximations in this context. Asymptotic results are detailed at length for
penalized B-spline nonparametric estimators.
The estimators under study combine two types of nonlinearity: nonlinearity due
to the expression of a complex parameter and nonlinearity due to nonparametric
estimation. We propose a two-step linearization procedure that provides an
approximation of the nonparametric estimator via a Horvitz-Thompson estimator
of a total using an artificial variable. Roughly speaking, this artificial
variable corresponds to the residuals of the linearized variable $u_{k}$ on
the fitted values under the model. Because the linearized variables depend on
the parameter of interest, the residuals will also depend on this parameter.
The consequence of this important and general property is that the
nonparametric approach helps to get a unique system of weights that may lead
to a gain in efficiency for different complex parameters.
The paper is structured as follows: the second section provides some
background information on the nonparametric estimation of a finite population
total in a general framework. In the third section, a class of nonparametric
substitution estimators based on nonparametric regression is introduced.
Variance approximations are derived using the influence function linearization
approach (Deville, 1999) in a general nonparametric setting. We propose in the
fourth section a penalized B-spline model-assisted estimator for the finite
population totals which is in fact an extension to a survey sampling framework
of the penalized B-spline estimator studied in Claeskens et al. (2009). We
prove that the estimator is asymptotically design-unbiased and consistent.
Next, we build the nonparametric penalized spline estimation for nonlinear
parameters and we assess the validity of the two-step linearization technique.
The fifth section defines a class of consistent variance estimators while
section six contains a case study. The data set is extracted from the French
Labor Force surveys of 1999 and 2000 as presented previously. Asymptotic and
finite-sample properties of the regression B-spline estimators are illustrated
for the simple random sampling without replacement and the stratified simple
random sampling. This section also includes suggestions for practical
implementation and guidelines for choosing the smoothing parameters. Finally,
section seven concludes this study and the assumptions and the technical
proofs together with some discussion are provided in the Appendix.
## 2 Nonparametric model-assisted estimation of finite population totals
We focus on the estimation of the total
$t_{y}=\sum_{k=1}^{N}y_{k}=\sum_{U}y_{k}$ of the study variable $\mathcal{Y}$
over $U$, taking into account the univariate auxiliary variable $\mathcal{Z}.$
The values $z_{1},\ldots,z_{N}$ of $\mathcal{Z}$ are assumed to be known for
the entire population.
Many approaches can be used to take into account auxiliary information
$\mathcal{Z}$ and thus improve on the Horvitz-Thompson estimator
$\hat{t}_{y,HT}=\sum_{s}y_{k}/\pi_{k}.$ The goal is to derive a weighted
linear estimator $\hat{t}_{yw}=\sum_{s}w_{ks}y_{k}$ of $t_{y},$ such that the
sample weights $w_{ks}$ do not depend on the study variable values $y_{k}$ but
include the values $z_{k},$ for all $k\in U.$ The construction of the model-
assisted (MA) class of estimators $\hat{t}_{yw}$ is based on a superpopulation
model $\xi$:
$\displaystyle\xi:\quad y_{k}=f(z_{k})+\varepsilon_{k}$ (3)
where the $\varepsilon_{k}$ are independent random variables with mean zero
and variance $v(z_{k}).$ If $f(z_{k})$ was known for all $k\in U,$ the total
$t_{y}$ may be estimated by the generalized difference estimator (Cassel et
al., 1976),
$\displaystyle\hat{t}_{y,\mbox{\tiny{diff}}}=\sum_{s}\frac{y_{k}-f(z_{k})}{\pi_{k}}+\sum_{U}f(z_{k}).$
(4)
Note that $\hat{t}_{y,\mbox{\tiny{diff}}}$ consists in the difference between
the Horvitz-Thompson estimator $\hat{t}_{y,HT}$ and its bias under the model
$\xi,$ namely $\sum_{s}f(z_{k})/\pi_{k}-\sum_{U}f(z_{k})$. As a consequence,
$\hat{t}_{y,\mbox{\tiny{diff}}}$ is unbiased under the model,
$E_{\xi}(\hat{t}_{y,\mbox{\tiny{diff}}})=t_{y}$ and moreover, it is unbiased
under the sampling design, $E_{p}(\hat{t}_{y,\mbox{\tiny{diff}}})=t_{y}.$ The
variance of $\hat{t}_{y,\mbox{\tiny{diff}}}$ under the sampling design is
given by
$\displaystyle
V_{p}(\hat{t}_{y,\mbox{\tiny{diff}}})=\sum_{U}\sum_{U}(\pi_{kl}-\pi_{k}\pi_{l})\frac{y_{k}-f(z_{k})}{\pi_{k}}\frac{y_{l}-f(z_{l})}{\pi_{l}}$
(5)
which shows clearly that the difference estimator
$\hat{t}_{y,\mbox{\tiny{diff}}}$ is more efficient than the Horvitz-Thompson
estimator $\hat{t}_{y,HT}$ if $f(z_{k})$ approximates well $y_{k}$ for all
$k\in U.$
In practice, we don’t know the true regression function $f,$ thus we use an
estimator of it. Generally, this estimator is obtained using a two-step
procedure: we estimate first $f$ by $\tilde{f}$ under the model $\xi$ and
next, we estimate $\tilde{f}$ by $\hat{f}$ using the sampling design. Plugging
$\hat{f}$ in (4), yields the final estimator of $t_{y}.$
The linear regression function
$f(z_{k})=\mathbf{z}^{\prime}_{k}\mathbf{\beta}$ yields the generalized
regression estimator (GREG) extensively studied by Särndal et al. (1992). The
GREG estimator is efficient if the model fits the data well, but if the model
is misspecified, the GREG estimator exhibits no improvement over the Horvitz-
Thompson estimator and may even lead to a loss of efficiency. One way of
guarding against model failure is to use nonparametric regression which does
not require a predefined parametric mathematical expression for $f$.
Recently, Breidt and Opsomer (2000) proposed local linear estimators and
Breidt et al. (2005) and Goga (2005) used nonparametric spline regression. The
unknown $f$ function is approximated by the projection of the population
vector $\mathbf{y}_{U}=(y_{1},\ldots,y_{N})^{\prime}$ onto different basis
functions, such as the basis of truncated $q$th degree polynomials in Breidt
et al. (2005) and the B-spline basis in Goga (2005). In the following, we
briefly recall the definition and the main asymptotic properties of
nonparametric model-assisted estimators for finite population totals (see also
Breidt and Opsomer, 2009).
Let $\tilde{f}_{y,k}$ be the estimator of $f(z_{k})$ obtained at the
population level using one of the three nonparametric methods mentioned above.
Plugging $\tilde{f}_{y,k}$ into (4) results in the following nonparametric
generalized difference pseudo-estimator of the finite population total:
$\displaystyle t^{*}_{y,\mbox{\tiny diff}}$ $\displaystyle=$
$\displaystyle\sum_{s}\frac{y_{k}-\tilde{f}_{y,k}}{\pi_{k}}+\sum_{U}\tilde{f}_{y,k}.$
(6)
Note that $t^{*}_{y,\mbox{\tiny diff}}$ is called a pseudo-estimator because
it is not feasible in practice since $\tilde{f}_{y,k}$ is unknown. This
pseudo-estimator is still design-unbiased but it is model-biased because
nonparametric estimators $\tilde{f}_{y,k}$ are biased for $f(z_{k})$ (Sarda
and Vieu, 2000). Nevertheless, under supplementary assumptions (Breidt and
Opsomer, 2000 and Goga, 2005), the bias under the model vanishes
asymptotically to zero when the population and the sample sizes go to
infinity. The unknown quantities $\tilde{f}_{y,k}$ are usually obtained by
least squares methods (ordinary, weighted or penalized) and we may write
$\displaystyle\tilde{f}_{y,k}=\mathbf{q}_{k}^{\prime}\mathbf{y}_{U},\quad\mbox{for
all }k\in U$ (7)
where the $N$ dimensional vector $\mathbf{q}_{k}$ depends on the population
values $z_{k},$ $k\in U$ as well as on the projection matrix for the
considered basis functions, but does not depend on $\mathcal{Y}.$ The
expression of $\mathbf{q}_{k}$ depends on the chosen nonparametric method, as
discussed in Breidt and Opsomer (2000), Breidt et al. (2005) and Goga (2005).
As in the parametric case, we estimate $\tilde{f}_{y,k}$ by ${\hat{f}}_{y,k}$
using the sampling design,
$\displaystyle{\hat{f}}_{y,k}=\widehat{\mathbf{q}}^{\prime}_{ks}\mathbf{y}_{s},\quad\mbox{for
all }k\in U$ (8)
where $\widehat{\mathbf{q}}^{\prime}_{ks}$ is the $n$-dimensional design-based
estimator of $\mathbf{q}^{\prime}_{k}$ and $\mathbf{y}_{s}=(y_{k})_{k\in s}$
is the sample restriction of $\mathbf{y}_{U}.$ Plugging ${\hat{f}}_{y,k}$ into
(6) yields the following nonparametric model-assisted estimator (NMA)
$\displaystyle\hat{t}_{y,np}$ $\displaystyle=$
$\displaystyle\sum_{s}\frac{y_{k}-{\hat{f}}_{y,k}}{\pi_{k}}+\sum_{U}{\hat{f}}_{y,k}.$
(9)
This estimator can be written as a weighted sum of the sampled observations
$\displaystyle\hat{t}_{y,np}=\sum_{s}w_{ks}y_{k}=\mathbf{w}^{\prime}_{s}\mathbf{y}_{s},$
(10)
where the weights $\mathbf{w}_{s}=(w_{ks})_{k\in s}$ depend only on the sample
and on the auxiliary information,
$\displaystyle\mathbf{w}_{s}=\mathbf{\Pi}_{s}^{-1}\mathbf{1}_{s}-\widehat{\mathbf{Q}}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{1}_{s}+\widehat{\mathbf{Q}}^{\prime}_{U}\mathbf{1}_{U},$
(11)
with $\mathbf{1}_{s}$ the $n$ dimensional vector of ones, $\mathbf{\Pi}_{s}$
the $n\times n$ diagonal matrix with $\pi_{k},$ $k\in s,$ along the diagonal
and $\widehat{\mathbf{Q}}_{U}$ the $N\times n$ matrix having
$\widehat{\mathbf{q}}^{\prime}_{ks}$ as rows with sample restriction
$\widehat{\mathbf{Q}}_{s}=(\widehat{\mathbf{q}}^{\prime}_{ks})_{k\in s}.$ The
estimator (10) is a nonlinear function of Horvitz-Thompson estimators, and its
asymptotic variance has been obtained on a case-by-case study. Under mild
hypothesis (Breidt and Opsomer, 2000, Breidt et al., 2005 and Goga, 2005),
$\hat{t}_{y,np}$ is asymptotically design-unbiased, namely
$\mbox{lim}_{N\rightarrow\infty}E_{p}(\hat{t}_{y,np}-t_{y})/N=0$ and design
$\sqrt{n}$-consistent in the sense that
$\displaystyle N^{-1}(\hat{t}_{y,np}-t_{y})$ $\displaystyle=$ $\displaystyle
O_{p}(n^{-1/2}).$ (12)
Moreover, it can be approximated by the nonparametric generalized difference
estimator $t^{*}_{y,\mbox{\tiny diff}},$
$\displaystyle N^{-1}(\hat{t}_{y,np}-t_{y})$ $\displaystyle=$ $\displaystyle
N^{-1}(t^{*}_{y,\mbox{\tiny diff}}-t_{y})+o_{p}(n^{-1/2}).$ (13)
Furthermore, if the asymptotic distribution of $V_{p}(t^{*}_{y,\mbox{\tiny
diff}})^{-1/2}(t^{*}_{y,\mbox{\tiny diff}}-t_{y})$ is normal
$\mathcal{N}(0,1)$, we have that the asymptotic distribution of
$V_{p}(t^{*}_{y,\mbox{\tiny diff}})^{-1/2}(\hat{t}_{y,np}-t_{y})$ is also
normal $\mathcal{N}(0,1)$ where $V_{p}(t^{*}_{y,\mbox{\tiny diff}})$ is
obtained according to formula (5) applied to residuals
$y_{k}-\tilde{f}_{y,k}.$ This means that the NMA estimators bring an
improvement over parametric methods and the Horvitz-Thompson estimator when
the relation between $\mathcal{Y}$ and $\mathcal{Z}$ is not linear. In this
case, the residuals $y_{k}-\tilde{f}_{y,k}$ will be smaller than under a
parametric smoother, which explains the diminution of the design variance of
NMA estimators. Nevertheless, nonparametric estimators require that the
auxiliary information should be known on the whole population unlike the GREG
estimator that requires only the finite population total for $\mathcal{Z}.$
The efficiency of NMA estimators depends on the choice of the smoothing
parameters. Opsomer and Miller (2005) and Harms and Duchesne (2010) derive the
optimal bandwidth for the local polynomial regression, while Breidt et al.
(2005) circumvent the issue of the number of knots by introducing a penalty
coefficient. They also give a practical method for estimating this penalty.
## 3 Nonparametric model-assisted estimation
of nonlinear finite population parameters
### 3.1 Definition of the nonparametric substitution estimator
Let us consider the estimation of some nonlinear parameters $\Phi$ by taking
into account univariate auxiliary information known for all the population
units. Examples of a nonlinear parameter of interest $\Phi$ include the ratio,
the Gini coefficient and the low-income proportion. A parameter $\Phi$ may
depend on one or several variables of interest; however, the same auxiliary
variable $\mathcal{Z}$ will be used to explain these variables of interest.
We aim to provide a general method for the estimation of $\Phi$ using
$\mathcal{Z}$ and considering the functional approach introduced by Deville
(1999). The methodology consists in considering a discrete and finite measure
$M=\sum_{U}\delta_{y_{k}}$ where $\delta_{y_{k}}$ is the Dirac measure at the
point $y_{k}$ and $M$ is such that there is unity mass on each point $y_{k}$
with $k\in U$ and zero mass elsewhere. Furthermore, we write $\Phi$ as a
functional $T$ of $M,$
$\displaystyle\Phi=T(M).$ (14)
The nonparametric weights $w_{ks}$ are provided by (11) and $M$ is estimated
by
$\displaystyle\widehat{M}_{np}=\sum_{s}w_{ks}\delta_{y_{k}}.$
Even if these weights are derived to estimate the total $t_{y},$ they do not
depend on the study variable $\mathcal{Y}$; thus they can be used to estimate
any nonlinear parameter of interest $\Phi$ when it can be expressed as a
function of $M.$ Note that $\widehat{M}_{np}$ is a random measure of total
mass equal to $\hat{N}_{np}=\sum_{s}w_{ks}.$
Plugging $\widehat{M}_{np}$ into (14) provides the following nonparametric
substitution estimator for $\Phi$,
$\displaystyle\widehat{\Phi}_{np}=T(\widehat{M}_{np}).$
We will now illustrate the computation of $\widehat{\Phi}_{np}$ using the
simple case of a ratio $R$ and subsequently the more intricate case of the
Gini index and parameters defined by implicit equations.
a. The ratio R between two finite population totals. We write
$R=\sum_{U}y_{k}/\sum_{U}x_{k}$ in a functional form as $\displaystyle
R=\frac{\int ydM(y)}{\int xdM(x)}.$ The nonparametric estimator of $R$ is
easily obtained by replacing the measure $M$ with $\hat{M}_{np},$ namely
$\hat{R}_{np}=\displaystyle\frac{\int yd\hat{M}_{np}(y)}{\int
xd\hat{M}_{np}(x)}=\frac{\sum_{s}w_{ks}y_{k}}{\sum_{s}w_{ks}x_{k}}.$ A similar
estimation of $R$ using GREG weights was previously considered by Särndal et
al. (1992).
b. The Gini index. The Gini index (Nygard and Sandström, 1985) is given by
$\mbox{G}=\frac{\sum_{U}y_{k}\left(2F(y_{k})-1\right)}{t_{y}}=\frac{\int(2F(y)-1)ydM(y)}{\int
ydM(y)}$
where $F(y)=\int\mathbf{1}_{\\{\xi\leq y\\}}dM(\xi)/\int
dM(y)=\sum_{U}\textbf{1}_{\\{y_{k}\leq y\\}}/N$ is the empirical distribution
function. Again, the nonparametric estimator for $G$ is obtained by simply
replacing $M$ with $\widehat{M}_{np}.$ Hence,
$\displaystyle\widehat{\mbox{G}}_{np}$ $\displaystyle=$
$\displaystyle\frac{\sum_{s}w_{ks}(2\hat{F}_{np}(y_{k})-1)y_{k}}{\sum_{s}w_{ks}y_{k}},$
(15)
where $\hat{F}_{np}(y)=\displaystyle\frac{\int\mathbf{1}_{\\{\xi\leq
y\\}}d\hat{M}_{np}(\xi)}{\int
d\hat{M}_{np}(y)}=\frac{\sum_{s}w_{ks}\mathbf{1}_{\\{y_{k}\leq
y\\}}}{\sum_{s}w_{ks}}.$
c. Parameters defined by an implicit equation. Let $\Phi$ be defined as the
unique solution of an implicit estimating equation $\sum_{U}\phi_{k}(\Phi)=0$
(Binder, 1983) that may be written in a functional form as
$\int\phi(\Phi)dM=0.$ We replace $M$ with $\widehat{M}_{np}$ and the
nonparametric sample-based estimator of $\Phi$ is the unique solution of the
sample-based estimating equation
$\int\phi(\Phi)d\widehat{M}_{np}=\sum_{s}w_{ks}\phi_{k}(\widehat{\Phi}_{np})=0.$
An example of such a parameter is the odds-ratio which is extensively used in
epidemiological studies. Goga and Ruiz-Gazen (2012) have studied the
estimation of the odds-ratio by taking into account auxiliary information and
nonparametric regression.
### 3.2 Asymptotic properties of the nonparametric substitution estimator
under the sampling design
In this section, we investigate the asymptotic properties of the nonparametric
estimator $\hat{\Phi}_{np}$, using the asymptotic framework suggested by Isaki
and Fuller (1982). Additionally, we make several assumptions (detailed in the
Appendix) regarding the regularity of the functional $T$ and the first order
inclusion probabilities of the sampling design.
The nonparametric estimator $\widehat{\Phi}_{np}$ is doubly nonlinear, with
nonlinearity due to the parameter $\Phi$ and nonlinearity due to the
nonparametric estimation. Our main goal is to approximate
$\widehat{\Phi}_{np}$ using a linear estimator (Horvitz-Thompson type) which
will allow to compute the asymptotic variance of $\widehat{\Phi}_{np}.$ This
approximation will be accomplished in two steps: first, we will linearize
$\Phi$ and next, we will linearize the nonparametric estimator obtained in
step one.
The first linearization step is a first-order expansion of
$\widehat{\Phi}_{np}$ with the reminder going to zero. The parameter of
interest $\Phi$ is a statistical functional $T$ defined with respect to the
measure $M$ or equivalently, with respect to the probability measure $M/N$ (by
assumption A1). Using the first-order expansion of statistical functionals $T$
as introduced by von Mises (1947) and under the assumption of Fréchet
differentiability of $T$, the reminder depends on some distance function
between $M/N$ and an estimator of this measure (Huber, 1981). Deville (1999)
uses these facts to prove the linearization of the Horvitz-Thompson
substitution estimator of $\Phi$; however, no details are given about the
considered distance, while Goga et al. (2009) provide only minimal details. In
what follows, we provide a distance between $\widehat{M}_{np}/N$ and the true
$M/N$ which goes to zero when the sample and the population sizes go to
infinity.
We consider the total variation distance for two finite and positive measures
$M_{1}$ and $M_{2}$ to be defined by
$d_{\mbox{tv}}(M_{1},M_{2})=\sup_{h\in\cal{H}}\left|\int h\,dM_{1}-\int
h\,dM_{2}\right|$
with ${\cal H}=\\{h:\mathbb{R}\rightarrow\mathbb{R}|\sup_{x}|h(x)|\leq 1\\}$.
We first prove (lemma 1 from below), that the distance $d_{\mbox{tv}}$ between
the Horvitz-Thompson estimator of $M/N$ and the true $M/N$ goes to zero. Next,
we extend the result (lemma 2 from below) to the nonparametric estimator
$\widehat{M}_{np}/N.$
Let $w_{ks}$ represent the Horvitz-Thompson weights, namely $w_{ks}=1/\pi_{k}$
for all $k\in s$ and let $\widehat{M}_{HT}=\sum_{s}\delta_{y_{k}}/\pi_{k}$ be
the estimator of $M$ using these weights. Let $h\in\mathcal{H}$ and for ease
of notation, $x_{k}=h(y_{k})$. Thus, for all $k\in U,$ $|x_{k}|\leq 1$
uniformly in $h\in\mathcal{H}$ and
$\int h\,d\widehat{M}_{HT}-\int
h\,dM=\sum_{s}\frac{h(y_{k})}{\pi_{k}}-\sum_{U}h(y_{k})=\sum_{U}\left(\frac{I_{k}}{\pi_{k}}-1\right)h(y_{k}),$
where $I_{k}=\mathbf{1}_{\\{k\in s\\}}$ is the sample membership indicator.
###### Lemma 1.
Assume (A3) and (A5) from the Appendix. Then,
$d_{\mbox{tv}}\left(\widehat{M}_{HT}/N,M/N\right)=O_{p}(n^{-1/2}).$
The proof is provided in the Appendix. We extend now lemma 1 to nonparametric
weights $w_{ks}$ given by (11). Consider again $h\in\mathcal{H}$ and let
$\displaystyle\int h\,d\widehat{M}_{np}-\int h\,dM$ $\displaystyle=$
$\displaystyle\sum_{s}w_{ks}x_{k}-\sum_{U}x_{k}$ $\displaystyle=$
$\displaystyle\sum_{s}\frac{x_{k}-\hat{f}_{x,k}}{\pi_{k}}+\sum_{U}\hat{f}_{x,k}-\sum_{U}x_{k}$
where $\hat{f}_{x,k}$ is obtained from (8) for $y_{k}$ replaced with
$x_{k}=h(y_{k}).$ Let also $\tilde{f}_{x,k}$ obtained from (7) for $y_{k}$
replaced with $x_{k}.$
###### Lemma 2.
Assume (A3) and (A5) from the Appendix. Assume in addition that:
($A^{*}$) for all $k\in U,$ $\frac{1}{N}\sum_{U}\tilde{f}^{2}_{x,k}=O(1)$
uniformly in $h$.
($A^{**}$)
$E_{p}\left|\frac{1}{N}\sum_{U}\left(\frac{I_{k}}{\pi_{k}}-1\right)(\tilde{f}_{x,k}-\hat{f}_{x,k})\right|=O(n^{-1/2})$
uniformly in $h.$
Then,
$d_{\mbox{tv}}\left(\widehat{M}_{np}/N,M/N\right)=O_{p}(n^{-1/2}).$
The proof is provided in the Appendix. In section 4, we prove that the
nonparametric estimator of $M$ constructed using B-spline estimators satisfies
the assumptions ($A^{*}$) and ($A^{**}$) from the above lemma. The results
from Breidt and Opsomer (2000) may be used to prove the assumptions for local
polynomial regression; however, this issue will not be pursued further here.
To provide the first order expansion of $\Phi=T(M),$ we must also define its
first derivative. This derivative is referred to as the influence function and
is defined as follows (Deville, 1999)
$\displaystyle IT(M,y)=\lim_{\varepsilon\rightarrow
0}\frac{T(M+\varepsilon\delta_{y})-T(M)}{\varepsilon}$
where $\delta_{y}$ is the Dirac measure at point $y$. Note that the above
definition is slightly different from the definition of the influence function
given by Hampel (1974) in robust statistics, which is based on a probability
distribution instead of a finite measure.
Let $u_{k},$ for all $k\in U$ be the influence function $IT$ computed at
$y=y_{k}$, namely
$u_{k}=IT(M,y_{k}),\quad k\in U.$
These quantities are referred to as the linearized variables of $\Phi$ and
serve as a tool for computing the approximative variance of $\hat{\Phi}_{np}.$
They depend on the parameter of interest and they are usually unknown even for
the sampled individuals. Deville (1999) provides many practical rules for
computing $u_{k}$ for rather complicated parameters $\Phi.$
Examples. The linearized variable of a ratio $R$ is
$\displaystyle u_{k}=\frac{1}{\sum_{U}x_{k}}(y_{k}-Rx_{k})$ (16)
and for the Gini index, it is given by
$\displaystyle
u_{k}=2F(y_{k})\frac{y_{k}-\overline{y}_{k,<}}{t_{y}}-y_{k}\frac{1+G}{t_{y}}+\frac{1-G}{N}$
(17)
where $\overline{y}_{k,<}$ is the mean of $y_{j}$ lower than $y_{k}.$
We now provide the main result of this paper. The following theorem is the
first linearization step of $\widehat{\Phi}_{np}$. This proves that under
broad assumptions the nonparametric estimator $\widehat{\Phi}_{np}$ is
approximated by the nonparametric estimator for the population total
$\sum_{U}u_{k}$ of the linearized variable. The proof is provided in the
Appendix.
###### Theorem 3.
(First linearization step) Assume (A1)-(A3) and (A5) from the Appendix.
Additionally assume ($A^{*}$) and $(A^{**})$ from lemma 2. Then, the
nonparametric substitution estimator $\widehat{\Phi}_{np}$ fulfills
$\displaystyle N^{-\alpha}\left(\widehat{\Phi}_{np}-\Phi\right)$
$\displaystyle=$ $\displaystyle
N^{-\alpha}\left(\sum_{s}w_{ks}u_{k}-\sum_{U}u_{k}\right)+o_{p}(n^{-1/2}).$
We can put $\sum_{s}w_{ks}u_{k}$ in the form of an NMA estimator. Let denote
$t^{*}_{u,np}=\sum_{s}w_{ks}u_{k}.$ Using (11), we can write
$\displaystyle t^{*}_{u,np}$ $\displaystyle=$
$\displaystyle\mathbf{w}^{\prime}_{s}\mathbf{u}_{s}=\sum_{s}\frac{u_{k}-g^{*}_{u,k}}{\pi_{k}}+\sum_{U}g^{*}_{u,k},$
(18)
where $g^{*}_{u,k}=\widehat{\mathbf{q}}^{\prime}_{ks}\mathbf{u}_{s}$ with
$\widehat{\mathbf{q}}_{ks}$ is given by (8) and $\mathbf{u}_{s}=(u_{k})_{k\in
s}$ is the sample restriction of $\mathbf{u}_{U}=(u_{k})_{k\in U}.$
Remark 1: A model-based interpretation of $g^{*}_{u,k}$ may be given. For the
nonparametric model $\xi^{\prime}$, the linearized variable $u_{k}$ can be
fitted using the auxiliary variable $z_{k},$
$\displaystyle\xi^{\prime}:\quad u_{k}=g(z_{k})+\eta_{k}$
where the $\eta_{k}$ are independent random variables with mean zero and
variance $\tilde{v}(z_{k}).$ The estimator of $g$ under the model
$\xi^{\prime},$ denoted by $\tilde{g}_{u,k}$, is obtained using the same
nonparametric method employed for estimating $f$ under the model $\xi.$ This
implies that $\tilde{g}_{u,k}=\mathbf{q}_{k}^{\prime}\mathbf{u}_{U}$ is the
best fit of the population vector $\mathbf{u}_{U}=(u_{k})_{k\in U}$ with
$\mathbf{q}_{k}$ given by (7). Furthermore, $\mathbf{q}_{k}$ is estimated by
$\hat{\mathbf{q}}_{ks}$ which leads to the pseudo-estimator
$g^{*}_{u,k}=\widehat{\mathbf{q}}^{\prime}_{ks}\mathbf{u}_{s}$ of
$\tilde{g}_{u,k}.$ However, unlike the linear case, $g^{*}_{u,k}$ is not an
estimate of $\tilde{g}_{u,k}$ because the sample linearized variable vector
$\mathbf{u}_{s}$ is not known and we refer to it as a pseudo-estimator. Remark
also that the estimator $\hat{\Phi}_{np}$ is efficient if the nonparametric
model $\xi^{\prime}$ holds.
The nonparametric pseudo-estimator $t^{*}_{u,np}$ given by (18) is a nonlinear
function of Horvitz-Thompson estimators; however, it estimates a linear
parameter of interest, namely the total of $u_{k},$ $t_{u}=\sum_{U}u_{k}.$
This indicates that $t^{*}_{u,np}$ is similar to estimators used by Breidt and
Opsomer (2000), Breidt et al. (2005) and Goga (2005) although it is computed
for the artificial variable $u_{k}.$ The second linearization step
approximates $t^{*}_{u,np}$ by the generalized difference estimator of
$\sum_{U}u_{k}$ given by
$\displaystyle t^{*}_{u,\mbox{\tiny diff}}$ $\displaystyle=$
$\displaystyle\sum_{s}\frac{u_{k}-\tilde{g}_{u,k}}{\pi_{k}}+\sum_{U}\tilde{g}_{u,k}.$
(19)
###### Proposition 4.
(Second linearization step) Assume that
$N^{-\alpha}(t^{*}_{u,np}-t^{*}_{u,\mbox{\tiny diff}})=o_{p}(n^{-1/2}).$ Then,
$\displaystyle
N^{-\alpha}(t^{*}_{u,np}-t_{u})=N^{-\alpha}(t^{*}_{u,\mbox{\tiny
diff}}-t_{u})+o_{p}(n^{-1/2}).$
Based on theorem 3 and proposition 4, we see that the asymptotic variance of
$\widehat{\Phi}_{np}$ is the variance of $t^{*}_{u,\mbox{\tiny diff}},$ namely
$V_{p}(t^{*}_{u,\mbox{\tiny
diff}})=\sum_{U}\sum_{U}\Delta_{kl}\frac{u_{k}-\tilde{g}_{u,k}}{\pi_{k}}\frac{u_{l}-\tilde{g}_{u,l}}{\pi_{l}}.$
Moreover, if the asymptotic distribution of $V^{-1/2}_{p}(t^{*}_{u,\mbox{\tiny
diff}})(t^{*}_{u,\mbox{\tiny diff}}-t_{u})$ is $\mathcal{N}(0,1),$ then the
asymptotic distribution of $V^{-1/2}_{p}(t^{*}_{u,\mbox{\tiny
diff}})(\widehat{\Phi}_{np}-\Phi)$ is also $\mathcal{N}(0,1).$ In section 4,
we provide the necessary assumptions for the linearized variables and the
auxiliary variable $\mathcal{Z}$ to obtain an approximation of $t^{*}_{u,np}$
by $t^{*}_{u,\mbox{\tiny diff}}$ in a B-spline estimation context.
Remark 2. When the linearized variable $u_{k}$ is a linear combination of the
study variables, the assumption from proposition 4 is reduced to assumptions
on the study variables. For example, this occurs in the case of a ratio
$R=t_{y}/t_{x},$ where the linearized variable is given by $\displaystyle
u_{k}=\frac{1}{t_{x}}(y_{k}-Rx_{k})=A_{1}y_{k}+A_{2}x_{k}.$ The error
$t^{*}_{u,\mbox{\tiny diff}}-t^{*}_{u,np}$ can be written as a linear
combination of errors between $t^{*}_{y,\mbox{\tiny diff}}-\hat{t}_{y,np}$ and
$t^{*}_{x,\mbox{\tiny diff}}-\hat{t}_{x,np}$, respectively. Using mild
regularity assumptions on $\mathcal{X},$ $\mathcal{Y}$ and on the sampling
design, $N^{-1}(\hat{t}_{y,np}-t^{*}_{y,\mbox{\tiny diff}})$ and
$N^{-1}(\hat{t}_{x,np}-t^{*}_{x,\mbox{\tiny diff}})$ are shown to be of order
$o_{p}(n^{-1/2})$ (see Fuller, 2009, for linear regression and section 4 for
B-spline estimators). Thus $t^{*}_{u,np}-t^{*}_{u,\mbox{\tiny diff}}$ is also
of order $o_{p}(n^{-1/2})$ provided that $R$ and $N^{-1}t_{x}$ are bounded.
Remark 3. The asymptotic variance $\widehat{\Phi}_{np}$ given by theorem 3 and
proposition 4 depends on the population residuals $u_{k}-\tilde{g}_{u,k}$ of
the linearized variables $u_{k}$ under the model $\xi^{\prime}$. For the
simple case of a ratio, the relationship between $u_{k}$ and the study
variables is explicit and given by $\displaystyle
u_{k}=A_{1}y_{k}+A_{2}x_{k}$. If linear models fit the data $x_{k}$ and
$y_{k}$ well, then a linear model will also fit $u_{k}$ well. Nevertheless,
for nonlinear parameters such as the Gini index, the relationship between
$u_{k}$ and the study variable is not as simple as that for the ratio. In such
situations, the use of nonparametric regression methods may provide a major
improvement with respect to variance compared to parametric regression.
## 4 Penalized B-spline estimators
Spline functions have many attractive properties, and they are often used in
practice due to their good numerical features and ease of implementation. We
suppose without loss of generality that all $z_{k}$ have been normalized and
lie in $[0,1].$ For a fixed $m>1,$ the set $S_{K,m}$ of spline functions of
order $m,$ with $K$ equidistant interiors knots
$0=\xi_{0}<\xi_{1}<\ldots<\xi_{K}<\xi_{K+1}=1$ is the set of piecewise
polynomials of degree $m-1$ that are smoothly connected at the knots (Zhou et
al., 1998),
$S_{K,m}=\\{t\in C^{m-2}[0,1]:t(z)\quad\mbox{is a polynomial of degree
}(m-1)\mbox{ on each interval}\quad[\xi_{i},\xi_{i+1}]\\}$
For $m=1,$ $S_{K,m}$ is the set of step functions with jumps at knots. For
each fixed set of knots, $S_{K,m}$ is a linear space of functions of dimension
$q=K+m$. A basis for this linear space is provided by the B-spline functions
(Schumaker, 1981, Dierckx, 1993) $B_{1},\ldots,B_{q}$ defined by
$\displaystyle
B_{j}(x)=(\xi_{j}-\xi_{j-m})\sum_{l=0}^{m}\frac{(\xi_{j-l}-x)_{+}^{m-1}}{\Pi_{r=0,r\neq
l}^{m}(\xi_{j-l}-\xi_{j-r})}$
where $(\xi_{j-l}-x)_{+}^{m-1}=(\xi_{j-l}-x)^{m-1}$ if $\xi_{j-l}\geq x$ and
zero, otherwise. For all $j=1,\ldots,q,$ each function $B_{j}$ has the knots
$\xi_{j-m},\ldots,\xi_{j}$ with $\xi_{r}=\xi_{\min(\max(r,0),K+1)}$ for
$r=j-m,\ldots,j$ (Zhou et al., 1998) which means that its support consists of
a small, fixed, finite number of intervals between knots. Moreover, B-spline
are positive functions with a total sum equal to unity:
$\displaystyle\sum_{j=1}^{q}B_{j}(x)=1\ ,\quad\quad x\in[0,1].$ (20)
For the same order $m$ and the same knot location, one can use the truncated
power basis (Ruppert and Carroll, 2000) given by
$1,z,z^{2},\ldots,z^{m-1},(z-\xi_{1})^{m-1}_{+},\ldots,(z-\xi_{K})^{m-1}_{+}$.
The B-spline and the truncated power bases are equivalent in the sense that
they span the same set of spline functions $S_{K,m}$ (Dierckx, 1993).
Nevertheless, as indicated by Rupert et al. (2003), “the truncated power bases
have the practical disadvantage that they are far from orthogonal”, which
leads to numerical instability especially if a large number of knots are used.
### 4.1 Nonparametric penalized spline estimation for finite population
totals
We now consider the superpopulation model $\xi$ given by (3). To estimate the
regression function $f,$ we use spline approximation and a penalized least
squares criterion. We define the spline basis vector of dimension $q\times 1$
as $\mathbf{b^{\prime}}(z_{k})=(B_{1}(z_{k}),\ldots,B_{q}(z_{k})),$ $k\in U.$
The penalized spline estimator $\tilde{f}_{y,k}$ of $f(z_{k})$ is given by
$\tilde{f}_{y,k}=\mathbf{b^{\prime}}(z_{k})\tilde{\boldsymbol{\theta}}_{y,\lambda}$
with $\tilde{\boldsymbol{\theta}}_{y,\lambda}$ as the least squares minimizer
of
$\displaystyle\sum_{k=1}^{N}(y_{k}-\mathbf{b^{\prime}}(z_{k})\boldsymbol{\theta})^{2}+\lambda\int_{0}^{1}[(\mathbf{b^{\prime}}(t)\boldsymbol{\theta})^{(p)}]^{2}dt,$
(21)
where (p) represents the $p$-th derivate with $p\leq m-1.$ The solution of
(21) is a ridge-type estimator,
$\displaystyle\tilde{\boldsymbol{\theta}}_{y,\lambda}=(\mathbf{B}^{\prime}_{U}\mathbf{B}_{U}+\lambda\mathbf{D}_{p})^{-1}\mathbf{B}^{\prime}_{U}\mathbf{y}_{U},$
(22)
where $\mathbf{B}_{U}$ is the $N\times q$ matrix with rows
$\mathbf{b^{\prime}}(z_{k})$ and the $q\times q$ matrix $\mathbf{D}_{p}$ is
the squared $L^{2}$ norm applied to the $p$th derivative of
$\mathbf{b^{\prime}}\boldsymbol{\theta}$. Because the derivative of a
$B$-spline function of order $m$ may be written as a linear combination of
$B$-spline functions of order $m-1$, for equidistant knots
$\mathbf{D}_{p}=K^{2p}\nabla^{\prime}_{p}\mathbf{R}\nabla_{p}$ (Claeskens et
al., 2009) where the matrix $\mathbf{R}$ has elements
$R_{ij}=\int_{0}^{1}B_{i}^{(m-p)}(t)B_{j}^{(m-p)}(t)dt$ with $B_{i}^{(m-p)}$
as the $B$-spline function of order $m-p$ and $\nabla_{p}$ as the matrix
corresponding to the $p$th order difference operator.
The amount of smoothing is controlled by $\lambda>0.$ The case $\lambda=0$
results in an unpenalized B-spline estimator the asymptotic properties of
which have been extensively studied in the literature (Agarwal and Studden,
1980, Burman, 1991, and Zhou et al., 1998, among others). The case
$\lambda\rightarrow\infty$ is equivalent to fitting a $(p-1)$th degree
polynomial. The theoretical properties of penalized splines with $\lambda>0,$
have been studied only recently by Cardot (2000), Hall and Opsomer (2005),
Kauermann et al. (2009) and Claeskens et al. (2009).
The design-based estimators of $\tilde{f}_{y,k}$ are
$\displaystyle\hat{f}_{y,k}=\mathbf{b^{\prime}}(z_{k})\hat{\boldsymbol{\theta}}_{y,\lambda}$
(23)
where
$\hat{\boldsymbol{\theta}}_{y,\lambda}=(\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{B}_{s}+\lambda\mathbf{D}_{p})^{-1}\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{y}_{s}$
is the design-based estimator of
$\mathbf{\tilde{\boldsymbol{\theta}}}_{y,\lambda}$ and $\mathbf{B}_{s}$ is the
$n\times q$ matrix given by $\mathbf{B}_{s}=(\mathbf{b^{\prime}}(z_{k}))_{k\in
s}.$ We note that $\hat{f}_{y,k}$ may be written as in formula (8) for
$\widehat{\mathbf{q}}^{\prime}_{ks}=\mathbf{b^{\prime}}(z_{k})(\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{B}_{s}+\lambda\mathbf{D}_{p})^{-1}\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}.$
Finally, the $B$-spline NMA estimator of $t_{y}$ is as follows:
$\displaystyle\hat{t}_{y,BS}$ $\displaystyle=$
$\displaystyle\sum_{s}\frac{y_{k}-\hat{f}_{y,k}}{\pi_{k}}+\sum_{U}\hat{f}_{y,k}$
(24) $\displaystyle=$
$\displaystyle\sum_{s}\frac{y_{k}}{\pi_{k}}-\left(\sum_{s}\frac{\mathbf{b}(z_{k})}{\pi_{k}}-\sum_{U}\mathbf{b}(z_{k})\right)^{\prime}\hat{\boldsymbol{\theta}}_{y,\lambda}.$
This indicates that $\hat{t}_{y,BS}$ may be written as a GREG estimator that
uses the vectors $\mathbf{b^{\prime}}(z_{k})$ as regressors of dimension
$q\times 1$ with $q$ going to infinity and a ridge-type regression coefficient
$\hat{\boldsymbol{\theta}}_{y,\lambda}.$ Furthermore, $\hat{t}_{y,BS}$ is a
weighted sum of sampled values $y_{k}$ with weights $\mathbf{w}_{s}$ expressed
as in (11),
$\displaystyle\mathbf{w}_{s}=\mathbf{\Pi}_{s}^{-1}\mathbf{1}_{s}-\mathbf{\Pi}_{s}^{-1}\mathbf{B}_{s}\left(\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{B}_{s}+\lambda\mathbf{D}_{p}\right)^{-1}\left(\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{1}_{s}-\mathbf{B}^{\prime}_{U}\mathbf{1}_{U}\right).$
(25)
#### Regression splines
For $\lambda=0,$ we obtained the unpenalized B-spline estimator studied by
Goga (2005) and called the regression splines. The B-spline property given in
(20) may be written as $\mathbf{1}^{\prime}_{q}\cdot\mathbf{b}(z_{k})=1$ with
$\mathbf{1}_{q}$ the $q$ dimensional vector of ones, implying that
$\mathbf{1}_{s}=\mathbf{B}_{s}\mathbf{1}_{q}$ and
$\mathbf{1}_{U}=\mathbf{B}_{U}\mathbf{1}_{q}.$ Using these two relations in
(25) (Goga, 2005), we observe that $\hat{t}_{y,BS}$ is equal to the finite
population total of the prediction
$\hat{f}_{y,k}=\mathbf{b}^{\prime}(z_{k})(\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{B}_{s})^{-1}\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{y}_{s},$
$\hat{t}_{y,BS}=\sum_{U}\hat{f}_{y,k}=\mathbf{w}^{\prime}_{s}\mathbf{y}_{s}$
where the weights are given by,
$\displaystyle\mathbf{w}_{s}=\mathbf{\Pi}^{-1}_{s}\mathbf{B}_{s}\left(\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{B}_{s}\right)^{-1}\mathbf{B}^{\prime}_{U}\mathbf{1}_{U}.$
(26)
Note the similarity with the GREG weights obtained in the case of a linear
model when the variance of errors is linearly related to the auxiliary
information (Särndal, 1980). We note that for a B-spline of order $m=1,$ the
estimator $\hat{t}_{y,BS}$ becomes the well-known poststratified estimator
(Särndal et al., 1992).
Based on assumptions regarding the sampling design and the variable
$\mathcal{Y},$ (assumptions (A3)-(A5) from the Appendix) and assumptions
regarding the distribution of $\mathcal{Z}$ and the knot number (assumptions
(B1)-(B2) in the Appendix), Goga (2005) proved that the B-spline estimator for
the total $t_{y}$ is asymptotically design-unbiased and consistent (equation
(12)) and may be approximated by a nonparametric generalized difference
estimator (equation (13)). These results are valid without supplementary
assumptions regarding the smoothness of the regression function $f.$
#### Penalized splines using truncated polynomial basis functions
Let
$\mathbf{c}^{\prime}(z_{k})=\\{1,z_{k},\ldots,z^{m-1}_{k},(z_{k}-\xi_{1})^{m-1}_{+},\ldots,(z_{k}-\xi_{K})^{m-1}_{+}\\}$
be the vector basis and let
$\tilde{f}_{y,k}=\mathbf{c^{\prime}}(z_{k})\mathbf{\tilde{\eta}}_{y,\rho}$
with $\mathbf{\tilde{\eta}}_{y,\rho}$ be the least squares minimizer of
$\sum_{k=1}^{N}(y_{k}-\mathbf{c^{\prime}}(z_{k})\mathbf{\eta})^{2}+\rho\sum_{j=1}^{K}\eta_{m-1+j}^{2}$
for $\mathbf{\eta}^{\prime}=(\eta_{0},\ldots,\eta_{m-1+K}).$ The solution is
given by
$\mathbf{\tilde{\eta}}_{y,\rho}=(\mathbf{C}^{\prime}_{U}\mathbf{C}_{U}+\rho\mathbf{A})^{-1}\mathbf{C}^{\prime}_{U}\mathbf{y}_{U}$
with $\mathbf{C}_{U}=(\mathbf{c^{\prime}}(z_{k}))_{k\in U}$ and the penalty
matrix $\mathbf{A}$ having $m-1$ zeros on the diagonal followed by $K$ one
values, $\mathbf{A}=\mbox{diag}(0,\ldots,0,1,\ldots,1).$ Note that for
$\rho=0,$ we obtain the same prediction $\tilde{f}_{y,k}$ as with an
unpenalized B-spline estimation. This results follows from the fact that the
two bases are equivalent, thus there exists a square and invertible transition
matrix $\mathbf{L}_{U}$ such that
$\mathbf{B}_{U}=\mathbf{C}_{U}\mathbf{L}_{U}$ (Ruppert et al., 2003). For
$\rho>0,$ we have
$\tilde{f}_{y,k}=\mathbf{B}^{\prime}_{U}(\mathbf{B}^{\prime}_{U}\mathbf{B}_{U}+\rho\mathbf{L}^{\prime}_{U}\mathbf{A}\mathbf{L}_{U})^{-1}\mathbf{B}^{\prime}_{U}\mathbf{y}_{U},$
which indicates equivalency to the estimator $\tilde{f}_{y,k}$ obtained with
penalized B-spline fitting given by (22) for
$\rho\mathbf{L}^{\prime}_{U}\mathbf{A}\mathbf{L}_{U}=\lambda\mathbf{D}_{q+1}$
(see Claeskens et al. (2009) for the expression of $\mathbf{L}_{U}$ satisfying
this equation).
In a design-based approach, Claeskens et al. (2005) proved that the NMA
estimator $\hat{t}_{y,BS}$ is the population total of the design-based
predictions
$\hat{f}_{y,k}=\mathbf{c^{\prime}}(z_{k})(\mathbf{C}^{\prime}_{s}\mathbf{\Pi}^{-1}_{s}\mathbf{C}_{s}+\rho\mathbf{A})^{-1}\mathbf{C}^{\prime}_{s}\mathbf{\Pi}^{-1}_{s}\mathbf{y}_{s}.$
They also proved that $\hat{t}_{y,BS}$ fulfils properties (12) and (13).
### 4.2 Asymptotic properties of the B-spline estimator of totals under the
sampling design
In the following, we study the asymptotic properties of $\hat{t}_{y,BS}$ under
the sampling design. We first provide a lemma concerning the convergence of
$\hat{\boldsymbol{\theta}}_{y,\lambda}.$ The proofs are based on the results
provided by Goga (2005) for the unpenalized B-spline estimator and on the fact
that the inverse of the matrix
$\frac{1}{N}\mathbf{B}^{\prime}_{U}\mathbf{B}_{U}+\frac{\lambda}{N}\mathbf{D}_{p}$
is of order $O(K)$ for the penalized B-spline estimator (lemma 1 from
Claeskens et al., 2009).
###### Lemma 5.
* (a)
Assume assumptions (A4)-(b) and (B1), (B2)-(a) and (B3) from the Appendix.
Then, $||\tilde{\boldsymbol{\theta}}_{y,\lambda}||=O(K^{1/2}).$
* (b)
Assume assumptions (A3), (A4)-(b), (A5) and (B1)-(B3) from the Appendix. Then,
$E_{p}(||\hat{\boldsymbol{\theta}}_{y,\lambda}-\tilde{\boldsymbol{\theta}}_{y,\lambda}||^{2})=O\left(\frac{K^{3}}{n}\right).$
where $||\cdot||$ is the usual euclidian norm.
The proof is provided in the Appendix. We note that for B-spline functions of
order $m=1$ and $\lambda=0,$ we obtain a poststratified estimator with a
number of poststrata going to infinity. In this context, lemma 5, (b) provides
a detailed theoretical justification for the poststratification example in
Deville (1999, p. 196). We note also that to obtain the convergence of
$\hat{t}_{y,BS},$ Claeskens et al. (2005) assume that the result from lemma 5,
(b) holds. Finally, we note that GREG estimators may be viewed as a special
case when the number of knots is fixed. Papers dealing with this issue usually
assume that the regression coefficient satisfies the results from the above
lemma (see for example Robinson and Särndal, 1983, or Isaki and Fuller, 1983).
A similar result was proved by Cardot et al. (2012).
Using lemma 5, we derive the following results.
###### Proposition 6.
Assume assumptions (A3), (A4)-(b), (A5) and (B1)-(B3) from the Appendix. Then,
* (a)
$E_{p}\left|\frac{1}{N}\left(\hat{t}_{y,BS}-t_{y}\right)\right|=O((K/n)^{1/2})).$
* (b)
$\frac{1}{N}\left(\hat{t}_{y,BS}-t_{y}\right)=\displaystyle{\frac{1}{N}\left(t^{*}_{y,diff}-t_{y}\right)+o_{p}(n^{-1/2})}$
where
$t^{*}_{y,diff}=\sum_{s}\frac{y_{k}-\tilde{f}_{y,k}}{\pi_{k}}+\sum_{U}\tilde{f}_{y,k}.$
The proof is provided in the Appendix. Using the Markov inequality, we see
from the first point of proposition 6 that $\hat{t}_{y,BS}$ is asymptotically
design-unbiased for $t_{y}$ and $\sqrt{n}$-consistent as
$(\hat{t}_{y,BS}-t_{y})/N=O_{p}(n^{-1/2}).$ The second point provides an
approximation of $\hat{t}_{y,BS}$ by the nonparametric generalized difference
estimator $t^{*}_{y,diff}.$
### 4.3 Calibration with penalized splines
The spline approach has some interesting calibration properties. Under the
unpenalized B-spline framework, the weights $w_{ks}$ given by (26) satisfy the
calibration equation for the known population total of B-spline functions,
namely
$\sum_{s}w_{ks}B_{j}(z_{k})=\sum_{U}B_{j}(z_{k}),\quad\mbox{for all
}j=1,\ldots,q.$
This relation is easily obtained using (20) (Goga, 2005). Because the spline
space $S_{K,m}$ is spanned by the B-spline functions $B_{j}$, these weights
will be calibrated to the total of any polynomial $z^{r}$ of degree $r\leq
q=K+m.$ In particular, $\sum_{s}w_{ks}=N$ and
$\sum_{s}w_{ks}z_{k}=\sum_{U}z_{k}.$ Claeskens et al. (2005) prove that using
the penalized splines and the truncated polynomial basis functions l provides
weights that are also calibrated for the finite population totals of the
polynomial basis functions $1,z,z^{2},\ldots,z^{m-1}.$
### 4.4 Nonparametric penalized spline estimation for nonlinear parameters
We now consider the nonlinear parameter $\Phi$ estimated by
$\hat{\Phi}_{BS}=T(\widehat{M}_{BS})$ with
$\widehat{M}_{BS}=\sum_{s}w_{ks}\delta_{y_{k}}$ and the weights $w_{ks}$ given
by (25). As in section 3, to linearize $\hat{\Phi}_{BS}$ we use a two-step
procedure. The first-step linearization is given in theorem 3 provided that
the assumptions $(A^{*})$ and $(A^{**})$ from lemma (2) are fulfilled. These
assumptions are crucial because they ensure the convergence of some
nonparametric estimator of $M$ to the true measure $M$ according to the
distance $d_{tv}.$ Using classical assumptions from a B-spline framework
(assumptions (B1)-(B3) from the Appendix) and mild assumptions regarding the
sampling design (assumptions (A3) and (A5) from the Appendix), we prove in
theorem 7 below that $(A^{*})$ and $(A^{**})$ are verified. The proof is
basically based on lemma 5 and the fact that the distance $d_{tv}$ is defined
for uniformly bounded functions $h\in\mathcal{H},$ ensuring that the
assumption (A4)-(b) is automatically fulfilled.
By conducting this first linearization step, we see that the nonparametric
B-spline estimator $\hat{\Phi}_{BS}$ will be approximated by the nonparametric
B-spline estimator of the total of the linearized variables $u_{k}$ given by
$t^{*}_{u,BS}=\mathbf{w}^{\prime}_{s}\mathbf{u}_{s}=\sum_{s}\frac{u_{k}-g^{*}_{u,k}}{\pi_{k}}+\sum_{U}g^{*}_{u,k},$
where
$g^{*}_{u,k}=\mathbf{b^{\prime}}(z_{k})\hat{\boldsymbol{\theta}}_{u,\lambda}$
with
$\hat{\boldsymbol{\theta}}_{u,\lambda}=(\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{B}_{s}+\lambda\mathbf{D}_{p})^{-1}\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{u_{s}}.$
The second-step linearization consists of providing an approximation of
$t^{*}_{u,BS}$ by a nonparametric generalized difference estimator,
$t^{*}_{u,\mbox{\tiny
diff}}=\sum_{s}\frac{u_{k}-\tilde{g}_{u,k}}{\pi_{k}}+\sum_{U}\tilde{g}_{u,k}.$
where
$\tilde{g}_{u,k}=\mathbf{b^{\prime}}(z_{k})(\mathbf{B}^{\prime}_{U}\mathbf{B}_{U}+\lambda\mathbf{D}_{p})^{-1}\mathbf{B}^{\prime}_{U}\mathbf{u_{U}}.$
To obtain this result, we state in theorem 7, (b) a supplementary assumption
regarding the linearized variable $u_{k}.$ Goga and Ruiz-Gazen (2012) prove
that the linearized variable $u_{k}$ of the odds-ratio satisfies this
assumption.
###### Theorem 7.
Suppose that the sampling design satisfies assumptions (A3) and (A5). In
addition, assume that (B1)-(B3) hold.
* (a)
Assumptions $(A^{*})$ and $(A^{**})$ from lemma 2 are fulfilled.
As a consequence, $d_{tv}(\widehat{M}_{BS}/N,M/N)=O_{p}(n^{-1/2}).$ Moreover,
if the functional $T$ satisfies (A1) and (A2), then
$N^{-\alpha}(\hat{\Phi}_{BS}-\Phi)=N^{-\alpha}(\hat{t}_{u,BS}-t_{u})+o_{p}(n^{-1/2}).$
* (b)
Suppose that the linearized variables are such that for all $k\in U,$
$N^{-\alpha+1}u_{k}$ satisfy (A4)-(b). Then,
$N^{-\alpha}\left(\hat{t}_{u,BS}-t_{u}\right)=N^{-\alpha}\left(\hat{t}_{u,\mbox{\tiny
diff}}-t_{u}\right)+o_{p}(n^{-1/2}).$
The proof is provided in the Appendix.
## 5 Variance estimation
In this section we undertake a detailed study of the variance estimation of
$\widehat{\Phi}_{np}.$ We first give the functional form of the variance of
$\hat{t}_{y,HT}$ as well as of its variance estimator and we propose a
variance estimator for $\widehat{\Phi}_{np}$ and assumptions under which this
estimator is consistent.
The Horvitz-Thompson variance
$V_{p}(\hat{t}_{y,HT})=\sum_{U}\sum_{U}\Delta_{kl}(y_{k}/\pi_{k})(y_{l}/\pi_{l})$
for $\Delta_{kl}=\pi_{kl}-\pi_{k}\pi_{l}$ is a quadratic form that can be
written as a functional of some finite and discrete measure. We can write the
variance as follows (Liu and Thompson, 1983),
$\displaystyle V_{p}(\hat{t}_{y,HT})=\sum_{(k,l)\in U^{*}}\psi(y_{k},y_{l})$
(27)
where $U^{*}=\\{(k,l),k,l=1,\ldots N\\}$ and
$\psi(y_{k},y_{l})=\Delta_{kl}(y_{k}/\pi_{k})(y_{l}/\pi_{l})$ is a bilinear
function of $y_{k}$ and $y_{l}.$ It follows from (27), that the Horvitz-
Thompson variance $V_{p}$ is the finite population total of
$\psi(y_{k},y_{l})$ over the derived synthetic population $U^{*}$ of size
$N^{*}=N^{2}.$ This variance can be put in a functional form as follows
$V_{p}(\hat{t}_{y,HT})=T^{*}(M^{*})=\int\psi(y,y)dM^{*}(y,y)$
where $M^{*}=\sum_{(k,l)\in U^{*}}\delta_{(y_{k},y_{l})}.$ Note that $T^{*}$
is a functional of degree $1$ with respect to $M^{*},$ namely
$T^{*}(M^{*}/N^{*})=T^{*}(M^{*})/N^{*}.$ A sample in this population $U^{*}$
is $s^{*}=\\{(k,l),k,l\in s\\}$ and has size $n^{*}=n^{2}.$ Moreover, the
first-order inclusion probabilities over the synthetic population $U^{*}$ are
$\pi^{*}_{(k,l)}=\pi_{kl}$, which are exactly the second-order inclusion
probabilities with respect to the initial sampling design $p(s).$ The measure
$M^{*}$ is estimated on $s^{*}$ by $\widehat{M}^{*}=\sum_{(k,l)\in
s^{*}}\delta_{(y_{k},y_{l})}/\pi_{kl}=\sum_{s^{*}}w^{*}_{(kl)}\delta_{(y_{k},y_{l})}$
where $w^{*}_{(kl)}=1/\pi_{kl}.$ The resulting estimator of
$V_{p}(\hat{t}_{y,HT})$ is as follows
$\widehat{V}_{p}(\hat{t}_{y,HT})=T^{*}(\widehat{M}^{*})=\int\psi(y,y)d\widehat{M}^{*}(y,y)=\sum_{(k,l)\in
s^{*}}\frac{\Delta_{kl}}{\pi_{kl}}\frac{y_{k}}{\pi_{k}}\frac{y_{l}}{\pi_{l}}.$
This is exactly the Horvitz-Thompson variance estimator, as $\sum_{(k,l)\in
s^{*}}$ is equal to $\sum_{k\in s}\sum_{l\in s}.$ Moreover, the functional
$T^{*}$ is Fréchet differentiable, with first derivative given by
$IT^{*}(M^{*},y)=\psi(y,y).$
Consider now the asymptotic variance $AV_{p}(\widehat{\Phi}_{np})$ of
$\widehat{\Phi}_{np}$ given by
$\displaystyle
AV_{p}(\widehat{\Phi}_{np})=\displaystyle\sum_{U}\sum_{U}\Delta_{kl}\frac{u_{k}-\tilde{g}_{u,k}}{\pi_{k}}\frac{u_{l}-\tilde{g}_{u,l}}{\pi_{l}}$
(28)
where $u_{k}$ is the linearized variable of $\Phi$ and
$\tilde{g}_{u,k}=\mathbf{q}^{\prime}_{k}\mathbf{u}_{U}$ for
$\mathbf{u}_{U}=(u_{k})_{k\in U}.$ We recognize the Horvitz-Thompson variance
of the total of the population residuals $e_{ks}=u_{k}-\tilde{g}_{u,k}.$ We
suggest estimating the variance of $\widehat{\Phi}_{np}$ by using the Horvitz-
Thompson variance estimator with $u_{k}$ replaced by the sample estimators
$\hat{u}_{k},$
$\displaystyle\widehat{V}_{p}(\widehat{\Phi}_{np})=\displaystyle\sum_{s}\sum_{s}\frac{\Delta_{kl}}{\pi_{kl}}\frac{\hat{u}_{k}-\hat{g}_{\hat{u},k}}{\pi_{k}}\frac{\hat{u}_{l}-\hat{g}_{\hat{u},l}}{\pi_{l}}$
(29)
where $\hat{g}_{\hat{u},k}=\hat{\mathbf{q}}^{\prime}_{ks}\mathbf{\hat{u}}_{s}$
is the sample estimate of
$\tilde{g}_{u,k}=\mathbf{q}^{\prime}_{k}\mathbf{u}_{U}.$ The Horvitz-Thompson
variance estimator with true linearized variables given by
$\displaystyle\widehat{AV}_{p}(\widehat{\Phi}_{np})=\sum_{s}\sum_{s}\frac{\Delta_{kl}}{\pi_{kl}}\frac{u_{k}-\tilde{g}_{u,k}}{\pi_{k}}\frac{u_{l}-\tilde{g}_{u,l}}{\pi_{l}}.$
(30)
The three expressions of variance above depend on the population fits
residuals $e_{ks},$ for all $k\in U.$ It follows that we may write
$AV_{p}(\widehat{\Phi}_{np})$ as a functional of $M^{*}$ depending on
parameter $\mathbf{e}_{U}=(e_{ks})_{k\in U},$
$AV_{p}(\widehat{\Phi}_{np})=T^{*}(M^{*},\mathbf{e}_{U}).$
Furthermore, the Horvitz-Thompson estimator
$\widehat{AV}_{p}(\widehat{\Phi}_{np})$ and the variance estimator
$\widehat{V}_{p}(\widehat{\Phi}_{np})$ can be treated in a functional form as
follows
$\widehat{AV}_{p}(\widehat{\Phi}_{np})=T^{*}(\widehat{M}^{*},\mathbf{e}_{U}),\quad\widehat{V}_{p}(\widehat{\Phi}_{np})=T^{*}(\widehat{M}^{*},\mathbf{\hat{e}}_{U}).$
Note that $\mathbf{\hat{e}}_{U}=(\hat{e}_{ks})_{k\in U}$ is the vector of
sample-based fit residuals with
$\hat{e}_{ks}=\hat{u}_{k}-\hat{g}_{\hat{u},k},$ for all $k\in U.$ Theorem 3
from Goga et al. (2009) allows us to establish under additional assumptions
that the variance estimator (29) is $n$-consistent for the asymptotic
variance.
###### Theorem 8.
Assume that assumptions (A3) and (A5) from the Appendix hold. Also assume that
$N^{1-\alpha}e_{ks}=O(1)$ holds uniformly in k and
$\displaystyle{nN^{-2\alpha}\sum_{U}(\hat{e}_{ks}-e_{ks})^{2}=o_{p}(1)}.$ If
the Horvitz-Thompson variance estimator
$\widehat{AV}_{p}(\widehat{\Phi}_{np})$ is $n$-consistent for
$AV_{p}(\widehat{\Phi}_{np}),$ then the variance estimator
$\widehat{V}_{p}(\widehat{\Phi}_{np})$ is also $n$-consistent for
$AV_{p}(\widehat{\Phi}_{np})$ in the sense that
$nN^{-2\alpha}(\widehat{V}_{p}(\widehat{\Phi}_{np})-AV_{p}(\widehat{\Phi}_{np}))=o_{p}(1).$
The proof is given in the Appendix. Note that because the functional $T^{*}$
is Fréchet differentiable, the $n$-consistency of the Horvitz-Thompson
estimator $\widehat{AV}_{p}(\widehat{\Phi}_{np})$ for
$AV_{p}(\widehat{\Phi}_{np})$ may also be derived with assumptions on fourth
moment of $e_{ks}$ and on fourth-order inclusion probabilities. The reader is
referred to Breidt and Opsomer (2000) for additional details.
## 6 Empirical results
Let us consider a data set from the French Labor Force surveys of 1999 and
2000 as the finite populations of interest. The data consist of the monthly
wages (in euros) of 19,378 wage-earners who were sampled in both years. The
study variable $y_{k}$ (resp. the auxiliary variable $x_{k}$) is the wage of
person $k$ in 2000 (resp. 1999). The objective of the simulation studies is to
investigate the finite-sample performance of the regression spline estimators
for two nonlinear parameters of interest and two different survey designs. We
concentrate in practice on the simple approach of regression B-splines and do
not consider the penalized B-splines with $\lambda>0$. The empirical study of
penalized splines raises the problem of estimating the parameter $\lambda$
which is beyond the scope of the present paper. We illustrate the efficiency
of the regression B-splines estimators compared to other estimators, and we
also confirm the possibility of conducting valid inference using variance
estimators as detailed in the previous section.
The parameters to estimate include the mean, the Gini index and the poverty
rate for the wages in 2000 using the wages in 1999 as auxiliary information.
The poverty rate is the proportion of individuals whose wages are below the
threshold of 60% of the median wage and correspond to the low-income
proportion studied in Berger and Skinner (2003). The Gini index and the low-
income proportion are the complex parameters to be estimated and we provide
results for the mean as a benchmark. Note that details on the low-income
proportion estimator and its associated linearized variable can be found in
Berger and Skinner (2003) and are not provided in the present paper. In
subsection 6.1, we focus on simple random sampling without replacement and in
subsection 6.2, we focus on a stratified simple random sampling without
replacement. We consider the following estimators for each parameter:
\- the Horvitz-Thompson estimator (HT), which does not incorporate any
auxiliary information,
\- poststratified estimators (POST) with a different number of strata bounded
at the empirical quantiles for 1999 wages,
\- the GREG estimator (GREG), which takes into account the 1999 wages as
auxiliary information using a simple linear model,
\- B-spline estimators (BS($m$) where $m$ denotes the spline order), which
take into account the wages from 1999 as auxiliary information by using a
nonparametric model with different numbers of knots ($K$) located at the
quantiles of the empirical distribution for wages from 1999. The $m=2$ and
$m=3$ orders are considered.
The poststratified estimator is an example of a B-spline estimator with order
$m=1$. The number of strata correspond to the number of interior knots $K$
plus one.
To use the regression B-spline estimators we propose in a complex survey, and
derive confidence intervals, the user must be able to calculate the weights
given in equation (26) and the residuals $\hat{u}_{k}-\hat{g}_{\hat{u},k}$ of
equation (29). The weights depend on a spline basis that is easy to obtain
using for instance the transreg procedure in the SAS software or the functions
spline.des or bs from the splines package in the R software. Then, it is
possible to use standard calibration algorithms by simply providing the
($m+K$) B-spline basis functions as auxiliary variables for calculating the
calibrated weights that correspond to equation (26). These weights are needed
to calculate the substitution estimator of the parameter of interest (e.g. the
expression (15) for the Gini index). To estimate the variance, the linearized
variables associated with the parameter have to be estimated. For several
inequality indicators, including the Gini index and the low-income proportion,
some SAS macro programs are freely available on the web site of Xavier
d’Haultfœuille. Similar functions are available in the R language upon request
from the authors of the present paper. Once the linearized variable is
estimated, the residuals of this variable against the auxiliary variable using
regression splines are calculated; this can be accomplished with the transreg
procedure in the SAS software. Then, by using the residuals as if they were
the study variable in standard variance estimation tools for complex surveys,
the user can obtain the estimated approximative variance and derive confidence
intervals.
For each simulation scheme, we draw $NS$ samples according to the sampling
design and compare the finite-sample properties of the HT estimator, the GREG
estimator, the POST and the BS(2) and BS(3) estimators. We set knots at the
quantiles of the empirical distribution of the auxiliary variable in the
sample. We also compared the results with knots set at the quantiles of the
empirical distribution of the auxiliary variable over the entire population.
Both results are very similar; thus, we report only on the first method. For
the POST, BS(2) and BS(3) estimators we tried different numbers of knots $K$
but only report the results for $K=2$ and $K=4$. Note that in the tables, the
results for $K=2$ and $K=4$ are reported in the same columns and separated by
a dash. For the poststratified estimator, $K=2$ (resp. $K=4$) corresponds to 3
(resp. 5) strata. To summarize, in the following, we compare eight estimators
(HT, GREG and POST, BS(2) and BS(3) with $K=2$ and $K=4$).
There are several ways to estimate the linearized variable (see section 5). In
this section, the results are almost the same, regardless of whether we use
the simple HT weights, the GREG weights or the B-spline weights for estimating
the linearized variable. We recommend using the simplest weights (that is, the
HT weights), which is what we do in the present study.
Estimators performance of $\hat{\theta}$ for a parameter $\theta$ is evaluated
using the following Monte-Carlo measures:
* •
Relative bias in percentage:
$\displaystyle\mbox{RB}=\frac{100}{NS}\times\sum_{i=1}^{NS}(\hat{\theta}_{i}-\theta)/\theta$.
* •
Ratio of root mean squared errors in percentage:
$\mbox{RRMSE}=100\times\sqrt{\sum_{i=1}^{NS}(\hat{\theta}_{i}-\theta)^{2}}/\sqrt{\sum_{i=1}^{NS}(\hat{\theta}_{i,HT}-\theta)^{2}}.$
* •
Monte-Carlo Coverage probabilities for a nominal coverage probability of 95%.
### 6.1 Simple random sampling without replacement
The first survey design we consider is simple random sampling without
replacement with three sample sizes ($n=200$, $n=500$ and $n=1000$). The
number of simulations is $NS=$3,000. The eight estimators are compared and
relative biases and ratios of the roots of the mean squared errors are
provided in Table 1 for the different parameters and sample sizes.
Not surprisingly, for complex parameters, the largest efficiency gain is
observed when the B-spline estimators are compared to the HT estimator without
auxiliary information. Because the wages from 2000 are almost linearly related
to the wages from 1999, considering the B-spline estimator instead of the GREG
estimator does not improve the performance of the mean estimation. However,
regarding the Gini index and the low-income proportion, the incorporation of
auxiliary information using GREG estimators does not improve efficiency
compared to the HT estimator while using a B-spline approach improves the
results especially for spline functions of order $m=2$. When comparing the
POST estimator with the BS(2) and BS(3) estimators we notice that there is
quite a large gain in efficiency when order $m=2$ is used instead of $m=1$,
while there is an efficiency loss when $m=3$ is used instead of $m=2$,
especially for sample sizes smaller than 1,000. Moreover, for $m=2$ and $m=3$,
the results do not depend heavily on the number of knots and are similar for
$K$ between 2 and 4 while for the poststratified estimator, there are large
variations in the results, regardless of whether we consider 3 or 5 strata.
The coverage probabilities in table 2 illustrate that valid inference can be
carried out using B-spline estimators as long as the spline order is not too
high, especially when the sample size is not very large. No problems are
detected for B-splines of order $m=1$ and order $m=2$ even when the sample
size is $n=200$; however for $m=3$ and $n=200$, the coverage probabilities for
the Gini index estimation are approximately 75% which is quite far from the
95% nominal probability. This result indicates that for a moderate sample
size, the variance may be underestimated when the order of the splines is
larger than two. The results are not given for $m=4$ but we have observed that
the problem worsens when we increase the order of the splines. This is not
really surprising due to double linearization and nonparametric estimation.
Table 1: RRMSE (RB) of HT, GREG and POST, BS(2) and BS(3) with $K=2$ \- 4 for
the mean, the Gini index and the low-income proportion
Parameter $n$ HT GREG POST BS(2) BS(3) Mean 200 100 (0) 38 (0) 71 (0) - 63 (0)
38 (0) - 37 (0) 39 (0) - 41 (0) 500 100 (0) 40 (0) 73 (0) - 65 (0) 40 (0) - 39
(0) 38 (0) - 39 (0) 1,000 100 (0) 40 (0) 73 (0) - 66 (0) 40 (0) - 40 (0) 38
(0) - 39 (0) Gini index 200 100 (1) 96 (1) 92 (1) - 80 (1) 53 (2) - 53 (2) 70
(3) - 70 (3) 500 100 (1) 93 (0) 93 (1) - 85 (1) 50 (1) - 50 (1) 59 (1) - 56
(1) 1,000 100 (0) 92 (0) 93 (0) - 86 (0) 49 (0) - 48 (0) 55 (1) - 51 (1)
Poverty rate 200 100 (2) 95 (0) 92 (0) - 80 (0) 65 (1) - 65 (1) 72 (1) - 63
(1) 500 100 (0) 95 (0) 88 (0) - 78 (0) 64 (0) - 64 (0) 68 (0) - 62 (0) 1,000
100 (1) 94 (0) 89 (0) - 78 (0) 64 (0) - 64 (0) 67 (0) - 61 (0)
Table 2: Coverage probabilities (in %) for HT, GREG and POST, BS(2) and BS(3)
with $K=2$ \- 4
Parameter $n$ HT GREG POST BSPL(2) BSPL(3) Mean 200 94 95 93 - 92 93 - 93 90 -
88 500 95 94 93 - 94 93 - 93 91 - 91 1,000 95 95 94 - 93 94 - 94 93 - 93 Gini
index 200 94 93 94 - 94 89 - 87 74 - 75 500 93 93 93 - 94 91 - 90 83 - 85
1,000 95 94 95 - 94 94 - 93 88 - 90 Poverty rate 200 94 95 95 - 95 95 - 94 94
- 94 500 93 95 95 - 94 95 - 95 96 - 95 1,000 94 95 96 - 96 95 - 96 96 - 95
### 6.2 Stratified simple random sampling without replacement
For each simulation, we draw $NS=$5,000 samples from the French Labor Force
population according to a stratified simple random sampling design without
replacement. We compare the finite-sample properties of the eight estimators
considered in the previous subsection. The strata are spatial divisions of the
French territory in six “regions” that correspond to the major socio-economic
regions of metropolitan France as defined by Eurostat. These regions are the
first level of the nomenclature of territorial units for statistics
classification (NUTS-1). For our example, we grouped the Northern and Eastern
regions together and we grouped the Mediterranean and the Southwestern regions
together. The sample size inside each stratum is 200 making the total sample
size 1200. Thus, we used an unequal probability design with a sample rate
inside the strata that varied from 5 to 9.3%.
As previously described, we set the knots at the quantiles of the empirical
distribution of the auxiliary variable in the sample and we estimate the
linearized variables using the HT weights. The simulation results are reported
in Table 3 and 4 and the conclusions are similar to those obtained from the
simple random sampling design without replacement when the size of the sample
is $n=200$ which corresponds to the sample sizes inside each stratum. It is
beneficial to use the available auxiliary information when estimating the mean
but there is no need to use nonparametric estimators because they are not more
efficient than the GREG estimator. However, for complex parameters, using a
GREG estimator to take auxiliary information into account is not worthwhile in
terms of variance while important gains can be made by using B-spline
estimators. The empirical coverage probabilities are all very good except for
the Gini index B-spline estimator of order three with values equal to 89-90%
which confirms the problem of variance underestimation for moderate sample
sizes and splines of order three.
Based on this example we do not recommend using high order values for B-spline
regression, especially when the sample sizes are smaller than 500. However,
choosing $m=2$ instead of $m=1$ (which corresponds to poststratification)
leads to a clear improvement in terms of efficiency for complex parameters
such as the Gini index or the low-income proportion, and we recommend this
choice.
Table 3: RRMSE (RB) of HT, GREG and POST, BS(2) and BS(3) with $K=2$ \- 4
Parameter HT GREG POST BS(2) BS(3) Mean 100 (0) 40 (0) 73 (0) - 66 (0) 40 (0)
- 40 (0) 40 (0) - 40 (0) Gini index 100 (0) 93 (0) 94 (0) - 88 (0) 50 (0) - 50
(0) 55 (1) - 52 (1) Poverty rate 100 (0) 93 (0) 88 (0) - 77 (0) 65 (0) - 64
(0) 68 (0) - 62 (0)
Table 4: Coverage probabilities for HT, GREG and POST, BS(2) and BS(3) with
$K=2$ \- 4
Parameter HT GREG POST BS(2) BS(3) Mean 95 95 95 - 94 94 - 94 93 - 92 Gini
index 95 95 95 - 95 93 - 93 89 - 90 Poverty rate 94 95 95 - 95 95 - 95 96 - 95
## 7 Discussion
In this paper we considered the important problem of nonlinear parameter
estimation in a finite population framework by taking into account the survey
design and a unique auxiliary variable known for all the population units.
Examples of nonlinear parameters are concentration and inequality measures,
such as the Gini index or the low-income proportion. We proposed a general
class of substitution estimators that allows us to take into account the
auxiliary information via a nonparametric model-assisted approach. The
asymptotic variance of this class of estimators was derived, based on broad
assumptions, and variance estimators were proposed. Our main result was that
the asymptotic variance depends on the extent to which the auxiliary variable
$z_{k}$ explains the variation in the linearized variable $u_{k}$. Because
linearized variables of nonlinear parameter are likely to be nonlinearly
related to auxiliary information, a nonparametric approach is recommended. The
proposed estimators are based on weights that are flexible enough to increase
the efficiency of finite population totals estimators for any study variable
and to allow the consideration of parameters that are more complex than
totals. Moreover, the penalized B-spline estimators were studied in detail,
and the theoretical results were confirmed for regression B-spline estimators
using one case study.
Our proposal can be extended in several different ways. In particular, further
research can extend this proposal to include multivariate auxiliary
information by means of additive models, as in Breidt et al. (2005), or single
index models as in Wang (2009).
Acknowledgement: we are grateful to Patrick Gabriel for his precious help for
lemma 1, to Hervé Cardot for helpful discussions and to Didier Gazen for his
assistance with the simulations.
## Appendix: assumptions and proofs
Assumptions on functional $T$ and on sampling design.
* (A1).
The functional $T$ is homogeneous, in that there exists a real number
$\alpha>0,$ dependent on $T$ such that $T(rM)=r^{\alpha}T(M)$ for any real
$r>0.$ We assume also that $\lim_{N\rightarrow\infty}N^{-\alpha}T(M)<\infty.$
* (A2).
The functional $T$ is Fréchet differentiable at $M/N$; that is, there exists a
functional $T(M/N;\Delta)$ that is linear in $\Delta$ such that
$\displaystyle\left|T\left(\frac{G}{N}\right)-T\left(\frac{M}{N}\right)-T\left(\frac{M}{N};\frac{G-M}{N}\right)\right|=o\left(d\left(\frac{G}{N},\frac{M}{N}\right)\right)$
with $d\left(\frac{G}{N},\frac{M}{N}\right)\longrightarrow 0.$
We note that the strong assumption of Fréchet differentiability can be
weakened to compact or Hadamard differentiability. However, for Hadamard
differentiability, functionals are considered with respect to the empirical
distribution function and the distance $d_{\mbox{tv}}$ should be replaced by
the sup norm. Supplementary assumptions need to be supposed in order to have
the consistency of the estimator of the empirical distribution function.
Motoyama and Takahashi (2008) study the asymptotic behavior of Hadamard
statistical functionals but only for simple random sampling without
replacement.
* (A3).
$\displaystyle\lim_{N\rightarrow\infty}\frac{n}{N}=\pi\in(0,1).$
* (A4).
* (a)
$\overline{\lim}N^{-1}\sum_{U}y_{k}^{2}<\infty$ with $\xi$-probability 1.
* (b)
$\mbox{sup}_{k\in U}y_{k}\leq C$ with $C$ a positive constant not depending on
$N.$
* (A5).
$\displaystyle\min_{k\in U}\pi_{k}\geq\lambda$, $\min_{i,k\in
U}\pi_{ik}\geq\lambda*$ with $\lambda,\lambda*$ with some positive constants
and $\displaystyle\overline{\lim}_{N\rightarrow\infty}n\ \max_{i\neq k\in
U}|\pi_{ik}-\pi_{i}\pi_{k}|<\infty.$
Assumption (A3) and (A5) deal with first and second order inclusion
probabilities and are rather classical in survey sampling theory (see also
Robinson and Särndal, 1983 and Breidt and Opsomer, 2000). They are satisfied
for many sampling designs. Assumption (A4)-(a) is a regularity condition
necessary to get the consistency results. Some results need the stronger
assumption (A4)-(b).
Assumptions on B-splines
* (B1).
There exists a distribution function $Q(z)$ with strictly positive density on
$[0,1]$ such that $\sup_{z\in[0,1]}|Q_{N}(z)-Q(z)|=o(K^{-1}),$ with $Q_{N}(z)$
the empirical distribution of $(z_{i})_{i=1}^{N}.$
* (B2).
* (a)
$K=o(N)$;
* (b)
$K=O(n^{a})$ with $a<1/3.$
* (B3).
$K_{p}=(K+m-p)(\lambda\tilde{c})^{1/(2p)}N^{-1/(2p)}<1$ where
$\tilde{c}=c(1+o(1))$ with $c$ a constant that depends only on $p$ and the
design density.
These assumptions are classical in nonparametric regression (Agarwal and
Studden, 1980, Burman, 1991, Zhou et al., 1998); (B1) means that
asymptotically, there is no sub-interval in $[0,1]$ without points $z_{i}$ and
(B2) ensures that the dimension of the B-spline basis goes to infinity but not
too fast when the population and the sample sizes go to infinity. Assumption
(B3) concerns the penalty $\lambda$ as used by Claeskens et al. (2009).
### Proofs of results from section 3
Proof of Lemma 1. Now, let $h\in{\cal H}$ and let $I_{k}=1_{\\{k\in s\\}}$ be
the sample membership. Following the same lines as in Breidt and Opsomer
(2000), we have,
$\displaystyle E_{p}\left|\int h\,d\widehat{M}_{HT}/N-\int h\,dM/N\right|^{2}$
$\displaystyle=$ $\displaystyle
N^{-2}\mbox{Var}_{p}\left(\sum_{U}\frac{I_{k}}{\pi_{k}}h(y_{k})\right)$
$\displaystyle\leq$ $\displaystyle\left(\frac{1-\lambda}{\lambda
N}+\frac{n\mbox{max}_{k\neq
l}|\Delta_{kl}|}{\lambda^{2}n}\right)\frac{1}{N}\sum_{U}h^{2}(y_{k})$
$\displaystyle\leq$ $\displaystyle\frac{1-\lambda}{\lambda
N}+\frac{n\mbox{max}_{k\neq l}|\Delta_{kl}|}{\lambda^{2}n}=O(n^{-1})$
uniformly in $h$ by assumption (A3),(A5) and using the fact that $h\in\cal H.$
Proof of Lemma 2. We have
$\displaystyle E_{p}\left|\int
h\,d\left(\frac{\widehat{M}_{np}}{N}\right)-\int
h\,d\left(\frac{M}{N}\right)\right|\leq
E_{p}\left|\frac{1}{N}\sum_{U}\left(\frac{I_{k}}{\pi_{k}}-1\right)(x_{k}-\tilde{f}_{x,k})\right|+E_{p}\left|\frac{1}{N}\sum_{U}\left(\frac{I_{k}}{\pi_{k}}-1\right)(\tilde{f}_{x,k}-\hat{f}_{x,k})\right|$
From the proof of lemma 1, we see that the first term from the right-side is
of order $O(n^{-1/2})$ uniformly in $h$ because
$(1/N)\sum_{U}(x_{k}-\tilde{f}_{x,k})^{2}\leq(2/N)\sum_{U}(x^{2}_{k}+\tilde{f}^{2}_{x,k})\leq
2(1+C)$ by construction of $x_{k}$ and assumption ($A^{*}$). The result
follows because $|\int h\,d\hat{M}_{np}/N-\int h\,dM/N|=O_{p}(n^{-1/2})$
uniformly in $h\in\mathcal{H}.$
Proof of Theorem 3 Under assumption (A2), we provide a first-order von-Mises
(1947) expansion of $T$ in $\widehat{M}_{np}/N$ around $M/N,$
$\displaystyle
T\left(\frac{\widehat{M}_{np}}{N}\right)=T\left(\frac{M}{N}\right)+\int
IT\left(\frac{M}{N},\xi\right)d\left(\frac{\widehat{M}_{np}}{N}-\frac{M}{N}\right)(\xi)+o\left(d_{\mbox{tv}}\left(\frac{\widehat{M}_{np}}{N},\frac{M}{N}\right)\right).$
Using the fact that for a functional of degree $\alpha$ (assumption A1), we
have $IT\left(\frac{M}{N},\xi\right)=N^{1-\alpha}\cdot IT\left(M,\xi\right)$
(Deville, 1999), we write
$\displaystyle N^{-\alpha}T(\widehat{M}_{np})=N^{-\alpha}T(M)+N^{-\alpha}\int
IT\left(M,\xi\right)d(\widehat{M}_{np}-M)(\xi)+o_{p}(n^{-1/2})$ (31)
since $d_{\mbox{tv}}\left(\widehat{M}_{np}/N,M/N\right)=O_{p}(n^{-1/2}).$ Now,
$u_{k}=IT(M,y_{k})$ and hence, relation (31) becomes,
$\displaystyle N^{-\alpha}\left(\widehat{\Phi}_{np}-\Phi\right)$
$\displaystyle=$ $\displaystyle
N^{-\alpha}\left(\sum_{s}w_{ks}u_{k}-\sum_{U}u_{k}\right)+o_{p}(n^{-1/2}).$
### Proofs of results from section 4
We state below several lemmas useful for the proofs of our main results. For a
matrix $\mathbf{A}=(a_{ij})_{i,j=1}^{p},$ we consider the norm defined by
$||\mathbf{A}||_{\infty}=\mbox{max}_{i=1}^{p}\sum_{j=1}^{p}|a_{ij}|$ and the
trace norm
$||\mathbf{A}||_{2}=(\mbox{trace}(\mathbf{A^{\prime}}\mathbf{A}))^{1/2}.$
We denote by
$\mathbf{H}_{\lambda}=\frac{1}{N}\mathbf{B}^{\prime}_{U}\mathbf{B}_{U}+\frac{\lambda}{N}\mathbf{D}_{p}$
and by
$\widehat{\mathbf{H}}_{\lambda}=\frac{1}{N}\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{B}_{s}+\frac{\lambda}{N}\mathbf{D}_{p}$
its estimator.
###### Lemma 9.
Assume assumptions (B1), (B2)-(a) and (B3). Then,
1. 1.
$||\frac{1}{N}(\mathbf{B}^{\prime}_{U}\mathbf{B}_{U})||_{\infty}=O(K^{-1}),$
(lemma 6.3 from Agarwal and Studden, 1980).
We also have
$||(\frac{1}{N}\mathbf{B}^{\prime}_{U}\mathbf{B}_{U})^{-1}||_{\infty}=O(K),$
(lemma 6.3 from Zhou et al., 1998).
2. 2.
$||(\frac{1}{N}\mathbf{B}^{\prime}_{U}\mathbf{B}_{U}+\frac{\lambda}{N}\mathbf{D}_{p})^{-1}||_{\infty}=||\mathbf{H}^{-1}_{\lambda}||_{\infty}=O(K)$
(lemma 1 from Claeskens et al., 2009)
###### Lemma 10.
(Goga, 2005) Assume (A3), (A4)-(a), (A5) and (B1), (B2)-(a). Then,
1. 1.
$E_{p}||\frac{1}{N}\left(\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{B}_{s}-\mathbf{B}^{\prime}_{U}\mathbf{B}_{U}\right)||_{2}^{2}=O(\frac{1}{n})$
2. 2.
$E_{p}||\frac{1}{N}\left(\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{y}_{s}-\mathbf{B}^{\prime}_{U}\mathbf{y}_{U}\right)||^{2}=O(\frac{1}{n})$
Proof of lemma 5. When $y_{k}$ is uniformly bounded (assumption A4,b), we
have, using lemma 3 (a) (Goga, 2005) that
$\displaystyle||\frac{1}{N}\mathbf{B}^{\prime}_{U}\mathbf{y}_{U}||^{2}\leq\frac{C^{2}}{N}||\sum_{U}\mathbf{b}(z_{k})||^{2}\leq\frac{1}{K}$
(32)
since for $k,l\in U$ with $|k-l|>m$ we have $B_{j}(x_{k})B_{j}(x_{l})=0.$
For (a), $\tilde{\boldsymbol{\theta}}_{y,\lambda}$ is bounded following Goga
(2005),
$\displaystyle||\tilde{\boldsymbol{\theta}}_{y,\lambda}||$ $\displaystyle\leq$
$\displaystyle||\mathbf{H}^{-1}_{\lambda}||_{\infty}\cdot||(1/N)\mathbf{B}^{\prime}_{U}\mathbf{y}_{U}||$
(33) $\displaystyle=$ $\displaystyle O(K^{-1/2})$
by lemma 9-(b) and relation (32). Furthermore, we have
$\displaystyle||\hat{\boldsymbol{\theta}}_{y,\lambda}-\tilde{\boldsymbol{\theta}}_{y,\lambda}||^{2}$
(34) $\displaystyle\leq$
$\displaystyle||\widehat{\mathbf{H}}^{-1}_{\lambda}-\mathbf{H}^{-1}_{\lambda}||_{\infty}^{2}\cdot||\frac{1}{N}\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{y}_{s}||^{2}+||\mathbf{H}^{-1}_{\lambda}||_{\infty}^{2}\cdot||\frac{1}{N}\left(\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{y}_{s}-\mathbf{B}^{\prime}_{U}\mathbf{y}_{U}\right)||^{2}$
Under the assumption (A4)-(b),
$||\frac{1}{N}\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{y}_{s}||^{2}$
is bounded by $||\frac{1}{N}\sum_{U}\mathbf{b}(z_{k})||^{2}=O(K^{-1}).$ We
have that
$\displaystyle\widehat{\mathbf{H}}^{-1}_{\lambda}-\mathbf{H}^{-1}_{\lambda}$
$\displaystyle=$
$\displaystyle-\mathbf{H}^{-1}_{\lambda}\left(\frac{1}{N}\left(\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{B}_{s}-\mathbf{B}^{\prime}_{U}\mathbf{B}_{U}\right)\right)\left(\mathbf{H}^{-1}_{\lambda}\frac{1}{N}\left(\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{B}_{s}-\mathbf{B}^{\prime}_{U}\mathbf{B}_{U}\right)+\mathbf{I}_{q}\right)^{-1}\mathbf{H}^{-1}_{\lambda}$
and
$||\mathbf{H}^{-1}_{\lambda}\frac{1}{N}\left(\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{B}_{s}-\mathbf{B}^{\prime}_{U}\mathbf{B}_{U}\right)||_{\infty}=o_{p}(1)$
for $K=O(n^{a})$ with $a<1/3,$ implying that
$||\left(\mathbf{H}^{-1}_{\lambda}\frac{1}{N}\left(\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{B}_{s}-\mathbf{B}^{\prime}_{U}\mathbf{B}_{U}\right)+\mathbf{I}_{q}\right)^{-1}||_{\infty}\leq
1.$ Using lemma 10-(a), we obtain that
$\displaystyle
E_{p}||\widehat{\mathbf{H}}^{-1}_{\lambda}-\mathbf{H}^{-1}_{\lambda}||_{\infty}^{2}$
$\displaystyle=O\left(\frac{K^{4}}{n}\right)$
From lemmas 9 and 10, we obtain that
$\displaystyle
E_{p}\left(||\mathbf{H}^{-1}_{\lambda}||_{\infty}^{2}\cdot||\frac{1}{N}\left(\mathbf{B}^{\prime}_{s}\mathbf{\Pi}_{s}^{-1}\mathbf{y}_{s}-\mathbf{B}^{\prime}_{U}\mathbf{y}_{U}\right)||^{2}\right)=O\left(\frac{K^{2}}{n}\right).$
Finally, we have that
$E_{p}||\hat{\boldsymbol{\theta}}_{y,\lambda}-\tilde{\boldsymbol{\theta}}_{y,\lambda}||^{2}=O\left(\frac{K^{3}}{n}\right).$
Proof of proposition 6. Consider first $(b).$ Using the same lines as in the
proof of lemma 1 and the fact that $||\mathbf{b}(z_{k})||\leq 1$ for all $k\in
U$ (Burman, 1991), we obtain that
$\displaystyle
E_{p}\left|\left|\frac{1}{N}\sum_{U}\left(\frac{I_{k}}{\pi_{k}}-1\right)\mathbf{b}^{\prime}(z_{k})\right|\right|=O(n^{-1/2}).$
(35)
Furthermore,
$\displaystyle
E_{p}\left|\frac{1}{N}\left(\hat{t}_{y,BS}-t^{*}_{y,diff}\right)\right|$
$\displaystyle\leq$ $\displaystyle
E_{p}\left(\left|\left|\frac{1}{N}\sum_{U}\left(\frac{I_{k}}{\pi_{k}}-1\right)\mathbf{b}^{\prime}(z_{k})\right|\right|\cdot||\hat{\boldsymbol{\theta}}_{y,\lambda}-\tilde{\boldsymbol{\theta}}_{y,\lambda}||\right)$
$\displaystyle\leq$
$\displaystyle\sqrt{E_{p}\left|\left|\frac{1}{N}\sum_{U}\left(\frac{I_{k}}{\pi_{k}}-1\right)\mathbf{b}^{\prime}(z_{k})\right|\right|^{2}\cdot
E_{p}||\hat{\boldsymbol{\theta}}_{y,\lambda}-\tilde{\boldsymbol{\theta}}_{y,\lambda}||^{2}}$
$\displaystyle=$ $\displaystyle
O\left(\frac{K^{3/2}}{n}\right)=O\left(\frac{1}{\sqrt{n}}\right)\cdot
O((K^{3}/n)^{1/2})=O\left(\frac{1}{\sqrt{n}}\right)\cdot O((n^{3a-1})^{1/2})$
$\displaystyle=$ $\displaystyle o(n^{-1/2})$
by (35) and lemma 5-(b). Then, the result follows by using the Markov
inequality.
$(a)$ Now, we consider the error $\hat{t}_{y,BS}-t_{y}.$ We write
$\displaystyle\frac{1}{N}\left(\hat{t}_{y,BS}-t_{y}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{N}\left(\hat{t}_{y,HT}-t_{y}\right)$ $\displaystyle-$
$\displaystyle\frac{1}{N}\sum_{U}\mathbf{b}^{\prime}(z_{k})\left(\frac{I_{k}}{\pi_{k}}-1\right)(\hat{\mathbf{\theta}}_{y,\lambda}-\tilde{\boldsymbol{\theta}}_{y,\lambda})-\frac{1}{N}\sum_{U}\mathbf{b}^{\prime}(z_{k})\left(\frac{I_{k}}{\pi_{k}}-1\right)\tilde{\boldsymbol{\theta}}_{y,\lambda}$
By assumptions (A3), (A4-b) and (A5), we have that
$E_{p}\left|\frac{1}{N}\left(\hat{t}_{y,HT}-t_{y}\right)\right|=O(n^{-1/2}).$
Moreover, using relation (35) and lemma 5, (a) we have
$E_{p}\left|\frac{1}{N}\sum_{U}\left(\frac{I_{k}}{\pi_{k}}-1\right)\mathbf{b}^{\prime}(z_{k})\tilde{\boldsymbol{\theta}}_{y,\lambda}\right|=O((K/n)^{1/2})$
which implies that
$E_{p}\left|\frac{1}{N}\left(\hat{t}_{y,BS}-t_{y}\right)\right|\leq
O(n^{-1/2})+O(K^{3/2}n^{-1})+O(K^{1/2}n^{-1/2})=O((K/n)^{1/2})$
by the fact that $(K/n)^{1/2}>n^{-1/2}>K^{3/2}n^{-1}$ using assumption (B2).
Proof of Theorem 7. (a) We check that assumptions ($A^{*}$) and ($A^{**}$) are
fulfilled. We have
$\tilde{\boldsymbol{\theta}}_{x,\lambda}=\mathbf{H}^{-1}_{\lambda}(\sum_{U}\mathbf{b}(z_{k})x_{k}/N)$
with $|x_{k}|=|h(y_{k})|\leq 1$ for all $k\in U.$ Following (32) and (33), we
obtain that $||\tilde{\boldsymbol{\theta}}_{x,\lambda}||=O(K^{1/2})$ uniformly
in $h$ and
$\displaystyle\frac{1}{N}\sum_{U}\tilde{f}^{2}_{x,k}=\frac{1}{N}\tilde{\boldsymbol{\theta}}^{\prime}_{x,\lambda}\mathbf{B}^{\prime}_{U}\mathbf{B}_{U}\tilde{\boldsymbol{\theta}}_{x,\lambda}\leq||\tilde{\boldsymbol{\theta}}_{x,\lambda}||^{2}||\frac{1}{N}\mathbf{B}^{\prime}_{U}\mathbf{B}_{U}||_{\infty}=O(1),$
(36)
uniformly in $h.$ Now, we check the assumption ($A^{**}$), namely
$E_{p}\left|\frac{1}{N}\sum_{U}\left(\frac{I_{k}}{\pi_{k}}-1\right)(\tilde{f}_{x,k}-\hat{f}_{x,k})\right|=O(n^{-1/2})$
uniformly in $h.$
We have
$\displaystyle
E_{p}\left|\frac{1}{N}\sum_{U}\left(\frac{I_{k}}{\pi_{k}}-1\right)(\tilde{f}_{x,k}-\hat{f}_{x,k})\right|$
$\displaystyle\leq$ $\displaystyle
E_{p}\left(\left|\left|\frac{1}{N}\sum_{U}\left(\frac{I_{k}}{\pi_{k}}-1\right)\mathbf{b}^{\prime}(z_{k})\right|\right|\cdot||\tilde{\boldsymbol{\theta}}_{x,\lambda}-\hat{\boldsymbol{\theta}}_{x,\lambda}||\right)$
$\displaystyle\leq$
$\displaystyle\sqrt{E_{p}\left|\left|\frac{1}{N}\sum_{U}\left(\frac{I_{k}}{\pi_{k}}-1\right)\mathbf{b}^{\prime}(z_{k})\right|\right|^{2}\cdot
E_{p}||\tilde{\boldsymbol{\theta}}_{x,\lambda}-\hat{\boldsymbol{\theta}}_{x,\lambda}||^{2}}$
The first term from the right-side does not depend on $h$ and is of order
$O(n^{-1})$ (equation (35)). For the second term from the right-side, we can
use the proof of lemma (5), more exactly the equation (34), and the fact that
$\mbox{sup}_{k\in U}|h(y_{k})|\leq 1$ to obtain
$E_{p}||\tilde{\boldsymbol{\theta}}_{x,\lambda}-\hat{\boldsymbol{\theta}}_{x,\lambda}||^{2}=O\left(\frac{K^{3}}{n}\right)\quad\mbox{uniformly
in h}.$
Finally, we obtain that
$E_{p}\left|\frac{1}{N}\sum_{U}\left(\frac{I_{k}}{\pi_{k}}-1\right)(\tilde{f}_{x,k}-\hat{f}_{x,k})\right|=o(n^{-1/2})$
for $K=O(n^{a})$ with $a<1/3.$
(b) We write equation ((b)) as follows:
$\displaystyle N^{-\alpha}(t^{*}_{u,BS}-t^{*}_{u,\mbox{\tiny
diff}})=N^{-\alpha}\sum_{U}\left(\frac{I_{k}}{\pi_{k}}-1\right)\mathbf{b^{\prime}}(z_{k})(\hat{\boldsymbol{\theta}}_{u,\lambda}-\tilde{\boldsymbol{\theta}}_{u,\lambda})=o_{p}(n^{-1/2})$
because
$N^{-1}\sum_{U}\left(\displaystyle\frac{I_{k}}{\pi_{k}}-1\right)\mathbf{b^{\prime}}(z_{k})=O_{p}(n^{-1/2})$
(equation (35)) and
$N^{-\alpha+1}(\hat{\boldsymbol{\theta}}_{u}-\tilde{\boldsymbol{\theta}}_{u})=O_{p}(K^{3/2}n^{-1/2})$
by lemma 5.
Proof of Theorem 8. The proof follows the same basic steps as in Theorem 3
from Goga et al. (2009) and result 4 from Chaouch and Goga (2010). Let
$A_{N}=\widehat{V}_{p}(\widehat{\Phi}_{np})-\widehat{AV}_{p}(\widehat{\Phi}_{np}),\quad
B_{N}=\widehat{AV}_{p}(\widehat{\Phi}_{np})-AV_{p}(\widehat{\Phi}_{np})$
with $\widehat{AV}_{p}(\widehat{\Phi}_{np})$ given by (30) and let also
$c_{kl}=\displaystyle\frac{\Delta_{kl}}{\pi_{kl}}\frac{I_{k}}{\pi_{k}}\frac{I_{l}}{\pi_{l}}$.
Furthermore, the quantity $A_{N}$ can be written as
$\displaystyle A_{N}$ $\displaystyle=$
$\displaystyle\sum_{U}\sum_{U}c_{kl}(\hat{e}_{ks}\hat{e}_{ls}-e_{ks}e_{ls})$
$\displaystyle=$
$\displaystyle\sum_{U}\sum_{U}c_{kl}(\hat{e}_{ks}-e_{ks})(\hat{e}_{ls}-e_{ls})+2\sum_{U}\sum_{U}c_{kl}(\hat{e}_{ks}-e_{ks})e_{ls}$
$\displaystyle=$ $\displaystyle A_{1N}+A_{2N}$
Now,
$\displaystyle\frac{n}{N^{2\alpha}}|A_{1N}|\leq\frac{1-\lambda}{\lambda^{2}}\frac{n}{N^{2\alpha}}\sum_{U}(\hat{e}_{ks}-e_{ks})^{2}+\frac{n\max|\Delta_{kl}|}{\lambda^{2}\lambda^{*}N^{2\alpha-1}}\sum_{U}(\hat{e}_{ks}-e_{ks})^{2}=o_{p}(1)$
by assumptions (A3) and (A5). Using the same arguments as above, we obtain
$nN^{-2\alpha}|A_{2N}|=o_{p}(n^{-1}).$ Hence,
$nN^{-2\alpha}|A_{N}|=o_{p}(n^{-1})$ and the result then follows because
$nN^{-2\alpha}B_{N}=o_{p}(1)$
$\left|\frac{n}{N^{2\alpha}}(\widehat{V}_{p}(\widehat{\Phi}_{np})-AV_{p}(\widehat{\Phi}_{np}))\right|\leq\frac{n}{N^{2\alpha}}(|A_{N}|+|B_{N}|).$
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|
arxiv-papers
| 2012-01-06T09:09:32 |
2024-09-04T02:49:26.030382
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Camelia Goga and Anne Ruiz-Gazen",
"submitter": "Camelia Goga",
"url": "https://arxiv.org/abs/1201.1375"
}
|
1201.1552
|
# GRB 090618: different pulse temporal and spectral characteristics within a
burst
Fu-Wen Zhang College of Science, Guilin University of Technology, Guilin,
Guangxi 541004, China fwzhang@pmo.ac.cn
###### Abstract
GRB 090618 was simultaneously detected by Swift-BAT and Fermi-GBM. Its light
curve shows two emission episodes consisting of four prominent pulses. The
pulse in the first episode (episode A) has a smoother morphology than the
three pulses in the second episode (episode B). Using the pulse peak-fit
method, we have performed a detailed analysis of the temporal and spectral
characteristics of these four pulses and found out that the first pulse (pulse
A) exhibits distinctly different properties than the others in episode B
(pulses B1, B2 and B3) in the following aspects. (i) Both the pulse width
($w$) and the rise-to-decay ratio of pulse ($r/d$, pulse asymmetry) in GRB
090618 are found to be energy-dependent. The indices of the power-law
correlation between $w$ and $E$ for the pulses in episode B however are larger
than that in episode A. Moreover the pulses B1, B2 and B3 tend to be more
symmetric at the higher energy bands while the pulse A displays a reverse
trend. (ii) Pulse A shows a hard-to-soft spectral evolution pattern, while the
three pulses in the episode B follow the light curve trend. (iii) Pulse A has
a longer lag than the pulses B1, B2 and B3. The mechanism which causes the
different pulse characteristics within one single GRB is unclear.
gamma-ray bursts; statistical
## 1 Introduction
Gamma-ray bursts (GRB) have remained enigmatic since their discovery in the
late 1960s (for reviews, see Piran 2004; Zhang 2007). Although in the last ten
years our understanding of GRBs has been advanced significantly, due mainly to
the study of GRB afterglows (e.g., Sari et al. 1998; Fan & Wei 2005; Zhang et
al. 2006), the exact mechanism which produces the prompt gamma-ray emission
has not been definitively established (e.g., Fan 2010; Ghisellini 2010). The
temporal structures of the prompt emission are very complicated, consisting of
many overlapping pulses. Pulses are the basic, central building blocks of the
prompt emission, and their correlative properties imply that the pulses are
responsible for many luminosity-related characteristics. Recent studies showed
that the lag vs. luminosity relation (Norris et al. 2000), the variability vs.
luminosity relation (Reichart et al. 2001), the $E_{\rm peak}$ vs. $E_{\rm
iso}$ relation (Amati et al. 2002) and the $E_{\rm peak}$ vs. $L_{\rm iso}$
relation (Wei & Gao 2003; Yonetoku et al. 2004) all seem to be better
explained by pulse rather than bulk emission properties (see, Hakkila et al.
2008; Hakkila & Cumbee 2009; Krimm et al. 2009; Firmani et al. 2009; Ohno et
al. 2009; Ghirlanda, Nava & Ghisellini 2010; Arimoto et al. 2010). In
principle, the bulk characteristics of the prompt emission can be derived from
our knowledge of the decomposition of the burst in pulses and their individual
properties. Therefore, it is essential to our understanding of the physics of
the bulk prompt emission of GRBs, that we properly measure and understand the
properties of the individual pulses.
Hakkila et al. (2008) isolated and delineated pulse spectral properties of
GRBs detected by BATSE with known redshifts, and found that pulse lag, pulse
luminosity, and pulse duration are strongly correlated. They also found that
pulse peak lag, pulse asymmetry, and pulse hardness are correlated for a large
number of pulses of long GRBs (Hakkila & Cumbee 2009). These results indicate
that most pulses of long GRBs within a given burst as well as when comparing
different bursts might have similar physical origins.
However, in some cases, which show two or more separated distinct emission
episodes, and each emission episode consists of one or more pulses, their
pulse properties and origins are likely complicated. For example, Hakkila &
Giblin (2004) identified two cases (GRBs 960530 and 980125), consisting of two
separated emission episodes, and found that the pulses in the second emission
episodes of these two GRBs have longer lags, smoother morphologies, and softer
spectral evolution than those in the first episodes. It has been suggested
that internal- and external-shock emission might overlap in these two cases
(Hakkila & Giblin 2004).
Recently, the Swift Burst Alert Telescope (BAT) detected a burst, GRB 090618,
which shows two emission episodes with four prominent pulses (Baumgartner et
al. 2009). It is obvious that the pulse in the first episode has a smoother
morphology than the three pulses in the second episode. We wonder whether the
pulses in the two emission episodes within this burst have different
properties and/or origins. To this end, we have performed a detailed analysis
of the pulse temporal and spectral characteristics of GRB 090618 (preliminary
results are reported in Zhang 2011).
## 2 Observations
GRB 090618 was detected by Swift-BAT at 08:28:29 UT on 2009 June 18 (this time
is used as $T_{0}$ throughout the paper, Schady et al. 2009). The burst was
also observed by Fermi-GBM (McBreen et al. 2009), AGILE (Longo et al. 2009),
Suzaku WAM (Kono et al. 2009), KONUS-WIND and KONUS-RF (Golenetskii et al.
2009). The Swift X-ray telescope (XRT) began follow up observations of its
X-ray light curve 124 s after the BAT trigger and its UVO telescope detected
its optical afterglow 129 s after the trigger (Schady et al. 2009). Absorption
features which were detected in its bright optical afterglow with the 3m Shane
telescope at Lick observatory yielded a redshift of $z=0.54$ (Cenko et al.
2009).
The BAT burst light curve shows a smooth multipeak structure with 4 prominent
pulses. Significant spectral evolution was observed during the burst. The
spectrum at the maximum count rate, measured from $T_{0}$+62.720 to
$T_{0}+64.0$ s, was well fitted (Golenetskii et al. 2009) in the 20 keV$-$2
MeV range by the Band function (Band et al. 1993) with a low-energy photon
index $-0.99(-0.06,+0.07)$, a high energy photon index $-2.29(-0.5,+0.23)$,
and peak energy $E_{p}=440\pm 70$ keV, while the time integrated spectrum had
a low-energy photon index $-1.28\pm 0.02$, a high energy photon index
$-2.66(-0.2,+0.14)$, and a peak energy $E_{p}=186\pm 8$ keV (Golenetskii et
al. 2009). The isotropic equivalent energy in the 8$-$1000 keV band was
$E_{iso}=2.0\times 10^{53}$ erg (standard cosmology, McBreen et al. 2009).
## 3 Pulse Temporal Properties
Figure 1 shows the BAT and GBM light curves over the standard energy bins
(BAT: 15$-$25, 25$-$50, 50$-$100 and 100$-$350 keV; GBM: 8$-$1000 keV (NaI)
and 0.2$-$30 MeV (BGO)). The first episode (episode A) is a smooth 50 s pulse
starting at $T_{0}-5$ s, and ends at $T_{0}+45$ s (pulse A). The second
episode (episode B) starts at $\sim T_{0}+45$ s and is about 275 s long,
consisting of three overlapping pulses. The first pulse peak at $\sim
T_{0}+62$ s (pulse B1), the second peak is at $\sim T_{0}+80$ s (pulse B2),
and the third peak is at $T_{0}+112$ s, finally ending at $T_{0}+320$ s (pulse
B3). $T_{90}$ (15$-$350 keV) is $113.2\pm 0.6$ s (estimated error including
systematics, Baumgartner et al. 2009). We focus attention on how the pulse
width and pulse width ratio depend on energy in the two emission episodes,
while checking if that dependence is maintained during this burst.
Figure 1: Broadband light curves of GRB 090618 observed by Swift and Fermi.
The fitting curves with eq. (1) are plotted.
Kocevski et al. (2003) developed an empirical expression, which can be used to
fit the pulses of GRBs. This function can be written as,
$F(t)=F_{m}(\frac{t+t_{0}}{t_{m}+t_{0}})^{r}[\frac{d}{d+r}+\frac{r}{d+r}(\frac{t+t_{0}}{t_{m}+t_{0}})^{(r+1)}]^{-\frac{r+d}{r+1}},$
(1)
where $t_{m}$ is the time of the maximum flux ($F_{m}$) of the pulse, $t_{0}$
is the offset time, $r$ and $d$ are the rising and decaying power-law indices,
respectively. Because the prompt emission of GRB 090618 is concentrated mainly
in the Swift-BAT energy range, only the BAT light curves are considered. We
fit all the light curves (see Figure 1) of the burst in the different BAT
energy bands with equation (1) and then measure the values of pulse-width
($w$) and the rise-to-decay ratio of pulse ($r/d$, pulse asymmetry). The
errors of $w$ and $r/d$ are derived from simulations by assuming a normal
distribution of the errors of the fitting parameters. The reported errors are
at $1\sigma$ confidence level. The results are listed in Table 1.
Table 1: Pulse temporal characteristics of GRB 090618. | Pulse A | Pulse B1 | Pulse B2 | Pulse B3
---|---|---|---|---
Band | $w$ | $r/d$ | $w$ | $r/d$ | $w$ | $r/d$ | $w$ | $r/d$
(keV) | (s) | | (s) | | (s) | | (s) |
(1) 15-25 | 32.3$\pm$4.6 | 0.68$\pm 0.11$ | 13.2$\pm 1.4$ | 1.44$\pm 0.29$ | 16.1$\pm 1.4$ | 0.55$\pm 0.11$ | 15.5$\pm 3.5$ | 0.48$\pm 0.13$
(2) 25-50 | 28.4$\pm$3.2 | 0.63$\pm 0.09$ | 12.6$\pm 1.1$ | 1.36$\pm 0.29$ | 15.1$\pm 1.5$ | 0.59$\pm 0.13$ | 15.0$\pm 4.2$ | 0.52$\pm 0.16$
(3) 50-100 | 24.9$\pm$2.4 | 0.57$\pm 0.10$ | 11.9$\pm 1.1$ | 1.29$\pm 0.27$ | 14.1$\pm 2.1$ | 0.65$\pm 0.15$ | 14.3$\pm 7.2$ | 0.59$\pm 0.24$
(4) 100-350 | 20.3$\pm$5.1 | 0.37$\pm 0.10$ | 11.1$\pm 1.8$ | 1.31$\pm 0.35$ | 13.3$\pm 3.1$ | 0.71$\pm 0.19$ | 13.4$\pm 6.9$ | 0.62$\pm 0.26$
From Table 1, we find a significant trend: all the pulses tend to be narrower
at higher energies. However, the pulse asymmetry dependence on the energy are
different for the two emission episodes. The pulses B2 and B3 tend to be more
symmetric at higher energies while the pulse A follows a reverse trend. To
further study how the pulse width depends on energy in detail, we show $w$ and
$r/d$ as functions of energy ($E$) in Figure 2, where $E$ is the geometric
mean of the lower and upper boundaries of the corresponding energy band (this
is adopted throughout this paper unless otherwise noted). Apparently both $w$
and $r/d$ are correlated with $E$. The correlation analysis yields $w\propto
E^{-0.20\pm 0.01}$ and $r/d\propto E^{-0.24\pm 0.06}$ for the pulse A,
$w\propto E^{-0.07\pm 0.01}$ and $r/d\propto E^{-0.05\pm 0.04}$ for the pulse
B1, $w\propto E^{-0.09\pm 0.01}$ and $r/d\propto E^{0.12\pm 0.01}$ for the
pulse B2, and $w\propto E^{-0.06\pm 0.01}$ and $r/d\propto E^{0.12\pm 0.02}$
for the pulse B3. It is found that the $w-E$ relations of GRB 090618 are well
consistent with those observed in the majority of long GRBs (e.g., Norris et
al. 1996; 2005; Peng et al. 2006), but the power-law indices of the $w-E$
relations within this event are larger than those previously observed in
typical GRBs (e.g., Fenimore et al. 1995; Norris et al. 1996; 2005), and the
indices in the episode B are larger than that in the episode A. The large
power-law indices of the $w-E$ relations in GRB 090618 can be explained from
the fact that the distribution of power-law index of the $w-E$ relation has a
large dispersion (see, Jia & Qin 2005; Peng et al. 2006; Zhang et al. 2007,
Zhang & Qin 2008; Zhang 2008). In addition, we also find that the energy
dependence of $r/d$ is different for the 4 pulses in the burst. The power-law
indices of $r/d-E$ relation for the pulses A and B1 are negative111For the
pulse B1, the power-law anti-correlation between $r/d$ and $E$ is not very
robust, this is so because the pulse rising phase is likely affected by
overlapping mini-pulses (see Figure 1)., while the power-law indices of the
relation for the pulses B2 and B3 are positive. The two different energy
dependence correlations of $r/d$ were observed previously within different
bursts for a large set of GRBs in the BATSE database (see, Peng et al. 2006).
The power-law correlation between $r/d$ and $E$ has been predicted
theoretically by Qin et al (2004; 2005), who suggested that the emission
associated with the shocks occurs on a relativistically expanding fireball
surface, where the curvature effect must be important. However, it is unclear
which mechanism is responsible of the power-law anti-correlation between $r/d$
and $E$. As proposed by Peng et al. (2006), a varying synchrotron or
comptonized radiation or a different pattern of the spectral evolution should
be considered. Furthermore, the different dependence on energy of pulse
asymmetries in one single GRB is reported firstly, this indicates that the
evolution and/or nature of pulses might different in some GRBs and the
different emission episodes are likely to originate from different physical
mechanisms (e.g., Hakkila & Giblin 2004).
Figure 2: Dependence of the pulse width ($w$, $top$ $panels$) and pulse rise-
to-decay ratio ($r/d$, $bottom$ $pannels$) on energy in GRB 090618. The solid
lines in the plots represent the best fits.
## 4 Pulse Spectral characteristics
To further check if the pulses in the two emission episodes of GRB 090618 have
different properties and/or different physical origins, we have performed a
detailed pulse spectral analysis.
### 4.1 Pulse spectral evolution
Pulse spectral evolution is very important to understand the physics of GRB
pulses (and thus of GRB prompt emission). Page et al. (2011) performed 14
time-slices spectra for GRB 090618, and found that the peak energy initially
decreases with time, then moves to higher energies during flaring activity. In
general, there is a positive trend between peak energy and flux. In order to
perform a more detailed study of the individual pulse spectral evolution in
GRB 090618, 23 time-sliced spectra from both the Swift-BAT and Fermi-GBM (NaI
and BGO) detectors, covering $-5-$150 s after the trigger, were extracted with
single power-law (BAT) and cutoff power-law (CPL, joint BAT-GBM) models222The
BAT and GBM data are publicly available at http://swift.gsfc.nasa.gov/ and
http://fermi.gsfc.nasa.gov/.. In general, 5 s time interval is selected to
perform time-resolved spectral analysis. For the weak emission in the begin
and end stage of pulses, 10 s or 30 s time interval is adopted (see Table 2).
The standard data analysis methods according to the BAT Analysis
Threads333http://heasarc.gsfc.nasa.gov/docs/swift/analysis/threads/
bat_threads.html and the GBM Analysis
Threads444http://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/
gbm_grb_analysis.html are used. The useful energy ranges for the BAT, NaI and
BGO spectral fitting are 15$-$150, 8$-$1000 and 200$-$30 000 keV,
respectively. Spectra were analyzed with Xspec(v12) software. Note that the
Band model (Band et al. 1993) is extensively used to fit the GRB spectra. For
GRB 090618, the high energy index in the Band model cannot be well constrained
in most of the time slices (also see Page et al. 2011). For the purpose of
comparing the spectral evolution of the different pulses of GRB 090618 under
one spectral model, we choose the minimal simplest model, i.e. the CPL. The
power-law index ($\Gamma_{\rm PL}$) from the BAT fit with the single PL model,
and the peak energy $E_{\rm peak}$ and low-energy index ($\Gamma_{\rm CPL}$)
from the joint BAT-GBM fits with the CPL model are shown in Figure 3 and Table
2.
Table 2: Spectral results of the time resolved analysis in GRB 090618. | | PL (BAT) | CPL (BAT+GBM)
---|---|---|---
t1 | t2 | $\Gamma_{\rm PL}$ | $\chi^{2}$/dof | $\Gamma_{\rm CPL}$ | Epeak | $\chi^{2}$/dof
s | s | | | | keV |
-5 | 5 | 1.01 $\pm$ 0.02 | 38/56 | 0.71 $\pm$ 0.03 | 235 $\pm$ 15 | 432/410
5 | 10 | 1.26 $\pm$ 0.03 | 62/56 | 0.89 $\pm$ 0.04 | 193 $\pm$ 18 | 381/410
10 | 15 | 1.38 $\pm$ 0.03 | 63/56 | 0.97 $\pm$ 0.06 | 156 $\pm$ 19 | 418/410
15 | 20 | 1.51 $\pm$ 0.03 | 64/56 | 1.11 $\pm$ 0.07 | 155 $\pm$ 24 | 467/410
20 | 25 | 1.66 $\pm$ 0.03 | 77/56 | 1.3 $\pm$ 0.09 | 162 $\pm$ 39 | 472/410
25 | 35 | 1.87 $\pm$ 0.04 | 77/56 | 1.05 $\pm$ 0.16 | 64 $\pm$ 13 | 454/410
35 | 45 | 2.16 $\pm$ 0.08 | 61/56 | 1.53 $\pm$ 0.22 | 98 $\pm$ 55 | 434/410
45 | 50 | 1.69 $\pm$ 0.03 | 43/56 | 1.37 $\pm$ 0.04 | 317 $\pm$ 50 | 463/410
50 | 55 | 1.52 $\pm$ 0.02 | 38/56 | 1.2 $\pm$ 0.03 | 313 $\pm$ 28 | 524/410
55 | 60 | 1.41 $\pm$ 0.02 | 39/56 | 1.06 $\pm$ 0.1 | 500 $\pm$ 26 | 922/410
60 | 65 | 1.38 $\pm$ 0.01 | 44/56 | 1.14 $\pm$ 0.1 | 389 $\pm$ 13 | 721/410
65 | 70 | 1.51 $\pm$ 0.01 | 53/56 | 1.23 $\pm$ 0.02 | 234 $\pm$ 12 | 508/410
70 | 75 | 1.74 $\pm$ 0.02 | 56/56 | 1.37 $\pm$ 0.02 | 245 $\pm$ 17 | 604/410
75 | 80 | 1.66 $\pm$ 0.02 | 50/56 | 1.31 $\pm$ 0.02 | 278 $\pm$ 14 | 642/410
80 | 85 | 1.6 $\pm$ 0.02 | 57/56 | 1.3 $\pm$ 0.02 | 250 $\pm$ 14 | 577/410
85 | 90 | 1.71 $\pm$ 0.02 | 60/56 | 1.33 $\pm$ 0.04 | 153 $\pm$ 14 | 455/410
90 | 95 | 1.83 $\pm$ 0.02 | 63/56 | 1.39 $\pm$ 0.07 | 116 $\pm$ 15 | 421/410
95 | 100 | 1.99 $\pm$ 0.03 | 60/56 | 1.39 $\pm$ 0.09 | 81 $\pm$ 12 | 415/410
100 | 105 | 1.97 $\pm$ 0.03 | 78/56 | 1.4 $\pm$ 0.07 | 103 $\pm$ 13 | 449/410
105 | 110 | 1.97 $\pm$ 0.02 | 73/56 | 1.48 $\pm$ 0.05 | 111 $\pm$ 10 | 409/410
110 | 115 | 2.13 $\pm$ 0.02 | 63/56 | 1.6 $\pm$ 0.06 | 97 $\pm$ 11 | 437/410
115 | 120 | 2.22 $\pm$ 0.03 | 70/56 | 1.56 $\pm$ 0.07 | 74 $\pm$ 10 | 395/410
120 | 150 | 2.39 $\pm$ 0.03 | 61/56 | 1.81 $\pm$ 0.1 | 80 $\pm$ 14 | 447/410
Figure 3: Spectral evolution of GRB 090618, where the values of $E_{\rm peak}$
are obtained from the joint GBM-BAT fits with the cutoff power-law model, and
the values of $\Gamma_{\rm PL}$ are measured from the Swift-BAT fit with the
single power-law model. The light curve in the BAT band ($15-350$ keV) is also
displayed.
From Figure 3, we conclude that GRB 090618 exhibits significant spectral
evolution and the pulses in the different episodes have different spectral
evolution trends555Here we only show the whole spectral evolution of GRB
090618 to depict individual pulse spectral evolution. It is known that the
individual pulse spectrum cannot be divided from a GRB which have several
overlapping pulses. The three pulses in the episode B of GRB 090618 are
overlapping, but they can be identified well (see Figure 1). Therefore, the
individual pulse spectral evolution trend in the episode B cannot be
significantly affected by the overlapping effect.. $E_{\rm peak}$ of the pulse
A shows a hard-to-soft evolutionary pattern, decreasing monotonically while
the flux rises and falls, $\Gamma$ shows an opposite trend. In the three
pulses of the episode B, there is a positive trend between $E_{\rm peak}$ and
flux, while $\Gamma$ follows an opposite trend. The two types of spectral
evolution patterns have been previously observed in pulses from different GRBs
(e.g., Golenetskii et al.1983; Norris et al. 1986; Preece et al. 1998; Kaneko
et al. 2006), but the phenomenon that the two types of spectral evolution
patterns exist simultaneously in one single GRB is very infrequent. GRB 921207
is another case following such spectral evolution trend (see, Figure 4 of Ford
et al. 1995 and Figure 2 of Lu et al. 2010). It is difficult to accommodate
the two different spectral evolution trends under one mechanism. Lu et al.
(2010) argued that it could be explained in terms of the viewing angle and jet
structure effects.
### 4.2 Pulse spectral lag
Table 3: Pulse peak lags of GRB 090618. The numbering represent the energy bands used to calculated the pulse peak lags listed in the column 1 of Table 1 (e.g., Lag 21 represent the lag is measured between (2)25-50 keV and (1)15-25 keV energy bands.). Pulse | Lag 21 | Lag 31 | Lag 41 | Lag 32 | Lag 42 | Lag 43
---|---|---|---|---|---|---
| (s) | (s) | (s) | (s) | (s) | (s)
Pulse A | 2.19$\pm 0.36$ | 6.01$\pm 0.37$ | 10.88$\pm 0.36$ | 3.82$\pm 0.12$ | 8.69$\pm 0.15$ | 4.87$\pm 0.20$
Pulse B1 | 0.53$\pm 0.20$ | 0.98$\pm 0.21$ | 1.34$\pm 0.08$ | 0.45$\pm 0.09$ | 0.81$\pm 0.10$ | 0.36$\pm 0.20$
Pulse B2 | 0.32$\pm 0.20$ | -0.20$\pm 0.20$ | 0.38$\pm 0.22$ | -0.52$\pm 0.19$ | 0.06$\pm 0.13$ | 0.58$\pm 0.14$
Pulse B3 | 0.49$\pm 0.56$ | 1.06$\pm 0.56$ | 1.70$\pm 0.75$ | 0.57$\pm 0.21$ | 1.21$\pm 0.54$ | 0.64$\pm 0.54$
Another observed effect of the spectral evolution in GRB data is spectral lag.
Spectral lags are energy-dependent delays in the GRB temporal structure. Pulse
peak lags are defined as the differences between the pulse peak times in
different energy channels, which can be obtained for any pulse between two
energy channels (e.g., Norris et al. 2005; Liang et al. 2006; Hakkila et al.
2008; Zhang et al. 2007; Zhang 2008). In general, soft pulses lag behind hard
pulses. The pulse peak-fit method gives a simple straightforward way for
extracting lags ( Norris et al. 2005; Hakkila et al. 2008). The pulse spectral
lags between the four standard BAT energy bands (see Table 1) are displayed in
Table 3. We find that the pulse A of GRB 090618 has a very longer lag (in all
energy channel combinations) than all three pulses in the episode B. Using the
cross-correlation function (CCF) analysis method, Page et al. (2011) analyzed
the whole spectral lags in the two episodes of GRB 090618 and found that the
episode A have a lag about a factor of 6 longer than for the episode B. Their
result is consistent with our finding, although the episode B of GRB 090618
comprises three pulses. A similar phenomenon was also obtained by Hakkila &
Giblin (2004). The early studies of burst spectral lags show that they vary
within a given burst as well as from burst to burst (e.g., Norris 2002; Ryde
et al. 2005; Chen et al. 2005). Multi-lag GRBs are ubiquitous. Therefore, we
can not differentiate between their physical origins by only taking into
account the spectral lags.
## 5 Conclusions and Discussion
Figure 4: _Left_ : Rest frame pulse duration $w_{0}$ vs. pulse peak lag
$\tau_{0}$ for fit pulses of BATSE GRBs having known redshifts (the data are
taken from Hakkila et al. 2008, GRB 980425 is excluded) as well as GRB 090618.
_Right_ : Isotropic pulse peak luminosity $L$ vs. pulse peak lag $\tau_{0}$
for the pulses shown in the left panel. The open circles represent the pulses
from GRB 971214, GRB 980703, GRB 970508, GRB 990510, GRB 991216 and GRB
990123, and the filled square represent GRB 090618. The solid lines are the
best fits obtained by Hakkila et al. (2008).
In this work we have used the pulse peak-fit method to analyze the pulse
temporal and spectral characteristics of GRB 090618. We find that the pulses
in the two emission episodes have different properties, including the energy
dependence of pulse widths and the pulse asymmetries, the pulse spectral
evolution patterns as well as the pulse lags. The different pulse temporal and
spectral characteristics exhibit simultaneously in one single GRB, indicating
there might be different origins in the different emission episodes of some
GRBs. None of the mechanisms proposed so far can be used to account for this
fact. Recently, Hakkila et al. (2008) and Hakkila and Cumbee (2009) found that
isotropic pulse peak luminosity ($L$), rest frame pulse peak lag ($\tau_{0}$),
and pulse duration ($w_{0}$) are correlated intrinsic properties of most GRB
pulses, and argued that most pulses might result from variations on a single
pulse type. To further understand the different pulse properties, we also
calculated the values of $L$, $\tau_{0}$ and $w_{0}$ for all pulses in GRB
090618 and compare their relations with the Hakkila et al. (2008) result
(Figure 4). We find that the distributions of $L$, $\tau_{0}$ and $w_{0}$ for
the four pulses basically comply with the relations found by Hakkila et al.
(2008). Such a result renders the interpretation of the different pulse
temporal and spectral properties found in our earlier analysis much more
challenging. The first episode is dimmer than the second episode and may be
identified as a precursor of the burst. A precursor could either have the same
origin as the main emission episode or it could be due to a different
mechanism (see, Koshut et al. 1995; Lazzati 2005; Burlon et al. 2008, 2009).
Recently, Penacchioni et al. (2011) proposed that GRB 090618 might be a
members of a specific new family of GRBs presenting a double astrophysical
component. A first one, related to the proto-black hole, prior to the process
of gravitational collapse (episode A) and a second one which is the canonical
GRB (episode B) emitted during the formation of the black-hole.
Better measurements are needed in order to improve our understanding of GRB
pulse properties. Description and analysis of pulse properties can help to
constrain physical models. The similar time evolution of pulse structures,
combined with the fact that their measurable properties correlate strongly,
suggests that one physical mechanism produces the observed array of pulse
characteristics (see Hakkila et al. 2008; Hakkila & Cumbee 2009). There is
strong evidence that the majority of GRB pulses results from internal shocks
in relativistic winds (e.g. Sari & Piran 1997; Kobayashi et al. 1997; Daigne &
Mochkovitch 1998; Ramirez-Ruiz & Fenimore 2000; Nakar & Piran 2002). Katz
(1994) suggested that GRB pulse shapes originate from time delays inherent in
the geometry of spherically expanding emission fronts. Liang et al. (1997)
argued that saturated Compton up-scattering of softer photons may be the
dominant physical mechanism that creates the shape of GRB pulses. According to
Ryde and Petrosian (2002), the simplest scenario accounting for the observed
GRB pulses is to assume an impulsive heating of the leptons and their
subsequent cooling and emission. In addition, in the impulsive external shock
model, a single relativistic wave of plasma interacts with inhomogeneities in
the surrounding medium and form external shocks that accelerate particles
which can also contribute to the formation of GRB pulses (Dermer et al. 1999).
Although many studies have been performed to interpret the pulse
characteristics, their nature is still unclear.
Acknowledgments We acknowledge the use of the public data from the Swift and
Fermi data archive. We thank the anonymous referee for very detailed comments
which helps in improving the quality of the paper. We also thank Yi-Zhong Fan
for insightful comments. This work was supported by the National Natural
Science Foundation of China (grants 11163003) and Guangxi Natural Science
Foundation (2010GXNSFB013050).
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|
arxiv-papers
| 2012-01-07T12:50:53 |
2024-09-04T02:49:26.050783
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fu-Wen Zhang",
"submitter": "Fu-Wen Zhang",
"url": "https://arxiv.org/abs/1201.1552"
}
|
1201.1630
|
11institutetext: Nonlinear Physics Centre and Centre for Ultra-high bandwidth
Devices for Optical Systems (CUDOS),
Australian National University, Canberra ACT 0200, Australia
Laboratory of Photonic Information Technology, School for Information and
Optoelectronic Science and Engineering,
South China Normal University, Guangzhou 510006, P.R. China
Wave propagation, transmission and absorption Optical bistability,
multistability, and switching, including local field effects Nonlinear
waveguides
# Nonlinear Mach-Zehnder-Fano interferometer
Yi Xu 1122 Andrey E. Miroshnichenko E-mail: 11 aem124@physics.anu.edu.au 1122
###### Abstract
We demonstrate that the interaction of loop and nonlinear Fano resonances
results in a formation of hybrid resonant states in Mach-Zehnder type
interferometers, providing with opportunities for an advanced phase
manipulation. The nonlinear response of such structures can be greatly
enhanced, leading to a low threshold 100$\%$ switching operation. We further
propose one of the possible realizations based on nonlinear photonic crystal
circuits, suitable for optimal all-optical switching.
###### pacs:
42.25.Bs
###### pacs:
42.65.Pc
###### pacs:
42.65.Wi
## 1 Introduction
Mach-Zehnder interferometer (MZI) is a key component in many branches of
physics because of its ability to manipulate a coherent signal [1]. By
coupling a resonator to the MZI can further increase the phase sensitivity of
the coherent manipulation [2, 3]. The enhanced all-optical switching [2] and
the bistability [4] have been demonstrated in a coupled ring-resonator Mach-
Zehnder interferometer, which provides the possibility for the effective and
coherent control by using a nonlinear resonator.
Figure 1: (color online)(a) and (b) are the generic discrete models for the
system exhibiting Fano resonance. $A$ and $B$ are the incoming waves, $T$ and
$R$ are scattered amplitudes. $C_{i}$ and $D_{i}$ are the forward and backward
scattered waves. $V$ is the coupling strength between the nonlinear Fano
defect(located by $M$) and the site $A_{M}$ in the linear chain. $L$ and $N$
specify the number of sites. Without loss of generality, we choose L=1 which
provides the phase reference while $N$ represents the length of MZFI arm
including two junction sites.
Recently, we have introduced the concept of Mach-Zehnder-Fano interferometer
(MZFI) [5] providing with unique physical property that can not be found in a
macroscopic resonator enhanced MZI [2, 3, 4, 6]. The MZFI allows us to
manipulate the interaction of different types of resonances which leads to the
formation of a novel hybrid Fano-like resonant states [7]. Furthermore, the
counterpart of the ring-resonator-coupled Mach-Zehnder interferometer in the
microscopic scale, i.e. MZFI based on a photonics crystal (PhC) platform seems
to be more promising for future application owing to small volume compared
with the macroscopic resonators. Recent advantages in PhCs fabrication
technology [8], allow us to achieve ultra high-Q cavities facilitating low
threshold nonlinear bistability [9, 10, 11, 12, 13]. Indeed, the manifested
optical bistable state is the nonlinear Fano resonance [15]. As a result, the
systems supporting Fano resonances [14, 15, 16], associated with an asymmetric
scattering profile, attract a significant attention recently. It’s their
unique property that allows us to achieve an optimal high extinction ratio,
large modulation depth and lowest threshold nonlinear switching [17, 6].
The aim of this Letter is to introduce and demonstrate unique properties of
the nonlinear MZFI, which originate from the excitation of the nonlinear
hybrid Fano resonances. Such resonant states appear because of the interaction
between MZI loop’s resonance and nonlinear Fano resonances of the side-coupled
defect. It provides with an enhanced nonlinear response and optimal conditions
for low threshold dynamic bistability. As a particular realisation of our
model we provide with a PhC circuit example, which supports our results.
Figure 2: (color online)(a) and (b) are the transmission of the linear MZFI
and the intensity of the defect, respectively. Here, $L=1$ and $N=7$. The
circles in (b) represent the normalized intensity in the nonlinear Fano defect
when the input power $I=0.01$ and $\chi^{(3)}=1$. (c) nonlinear response of
different resonances marked $\omega_{a-d}$ at Fig. 2 (a). Solid pink line
stands for the hybrid Fano resonance marked $\omega_{a}$ in Fig. 2 (a), dotted
red line presents the resonance of the MZI’s loop marked $\omega_{d}$ while
dashed green line and dashed-dotted blue line represents the resonance similar
with the one in resonator enhanced MZI [2, 3, 4, 6](marked $\omega_{c}$) and
conventional one(marked $\omega_{b}$), respectively. The shaded pink region
represents the area of the dynamic modulation instability initialled by a
pulse of $t_{0}=1.3\times 10^{4}$, $W=5\times 10^{3}$ and $I_{0}=0.25$. (d)
Double bistability at the frequency $\omega=0.43$ with detuning
$\Delta\omega=0.11$ respecting to $\omega_{a}$. Here, $t_{0}=1.3\times
10^{4}$, $W=5\times 10^{3}$ and $I_{0}=0.025$. The arrows indicate the upward
and downward nonlinear Fano bistable operation. The responses of
$\omega_{b-d}$ for the same detuning value are also presented.
## 2 Discrete nonlinear MZFI model
We start our analysis from a generic discrete model [15], which can describe
the dynamics of MZFI like structures. By using the modified Fano-Anderson
model, dynamical equations describing the nonlinear MZFI system shown in Fig.
1 (b) can be written as follows:
$\begin{array}[]{rcl}i\dot{\psi}_{n}=\sum_{k}\psi_{k}+\delta_{n,M}V\varphi_{d}\;,\\\
i\dot{\varphi}_{d}=E_{d}\varphi_{d}+\chi^{(3)}|\varphi_{d}|^{2}\varphi_{d}+V\psi_{M}\;,\end{array}$
(1)
where $\psi_{n}$ represents the linear chain with complex field amplitude,
$\varphi_{d}$ stands for the Fano defect, $M$ gives the location of the Fano
defect in the arm, $k$ is the total number of the nearest neighbour sites in
the chain($k=3$ in the Y-junction while $k=2$ at the others) and we neglect
the nonlocal interactions in this paper, $\chi^{(3)}$ is the cubic nonlinear
parameter, $V$ is the coupling strength between the chain and the Fano defect,
$E_{d}$ is the eigenfrequency of the Fano defect.
As it was demonstrated in Ref. [18], the side-coupled Fano defect acts as an
effective scattering potential, whose strength depends on the input frequency
$\omega$. In the presence of nonlinearity the effective scattering potential
of the Fano defect becomes input intensity dependent, which brings extra
flexibility to tune the Fano resonances of the MZFI.
To find static solutions of system (1) we employed S-matrix approach. The
S-matrices of the Y-junction can be obtained by applying scattering boundary
condition at three branches respectively [19]. A set of nonlinear equations
which describes the stationary nonlinear response of the system is as follow:
$\displaystyle\left(\begin{array}[]{c}R\\\ C_{1}\\\ C_{2}\end{array}\right)$
$\displaystyle={\mathbf{S}}^{Y}_{L}\left(\begin{array}[]{c}A\\\ D_{1}\\\
D_{2}\end{array}\right)\;,\;\left(\begin{array}[]{c}T\\\ D_{1}\\\
D_{3}\end{array}\right)={\mathbf{S}}^{Y}_{-N}\left(\begin{array}[]{c}B\\\
C_{1}\\\ C_{3}\end{array}\right),$ (14)
$\displaystyle\left(\begin{array}[]{c}D_{2}\\\ C_{3}\end{array}\right)$
$\displaystyle={\mathbf{S}}_{M}\left(\begin{array}[]{c}C_{2}\\\
D_{3}\end{array}\right),$ $\displaystyle X_{d}$
$\displaystyle=\frac{A_{M}V}{(\omega-E_{d}-\lambda|X_{d}|^{2})}$
where
${\mathbf{S}}^{Y}_{K}=\left(\begin{array}[]{ccc}e^{-2iKq}r_{Y}&t_{Y}&t_{Y}\\\
t_{Y}&e^{2iKq}r_{Y}&e^{2iKq}t_{Y}\\\
t_{Y}&e^{2iKq}t_{Y}&e^{2iKq}r_{Y}\end{array}\right),\\\
{\mathbf{S}}_{M}=\frac{1}{\varepsilon-2i\sin
q}\left(\begin{array}[]{cl}-\varepsilon e^{-2iMq}&-2i\sin q\\\ -2i\sin
q&-\varepsilon e^{2iMq}\end{array}\right),\\\
\varepsilon=\frac{V^{2}}{(\omega-E_{d}-\lambda|X_{d}|^{2})}$
As can be seen from the complex defect field $X_{d}$, the nonlinearity has an
effect on the phase and amplitude of the Fano defect and the corresponding
strengths in turn depend on the power of the Fano defect. At the same time,
the nonliear response of the Fano defect would give a feedback to the
scattering waves in the arms of MZFI (read as the nonlinear scattering
potential $\varepsilon$) of which forms a complex nonlinear resonance system.
Then, such nonlinear MZFI, whose nonlinear response can not be mapped from the
standard nonlinear Fano resonant system [18], is sophisticated even in the
linear region [7].
We study two cases, where the Fano defect is placed symmetrically and
asymmetrically. The detail parameters can be found in the caption of Fig. 2.
These two cases allow us to excite different sets of MZI loop’s modes, which
have nonzero overlap with the coupled site [7]. The scattering of the
conventional Fano resonance geometry [see Fig. 1(a)] is shown by dotted line
in Fig. 2(a) as a reference. In the symmetric case, the transmission in the
centre of the propagation band resembles a step-function. In such a case, the
eigenfrequency of the Fano defect is in the vicinity of MZI loop’s resonance,
where a hybrid Fano resonance is formed. At the same time, such resonance has
the highest intensity at the defect site among other resonances because of the
interference of two Fano-like resonances [7]. Intuitively, the high contrast
step function like transmission together with high resonant intensity of the
nonlinear defect are positive factors for further enhancment of the nonlinear
behaviour [20]. We, thus, investigate the nonlinear response of the system (1)
at these specific resonances.
The stationary nonlinear switching of four resonances marked
$\omega_{a-d}$(corresponding to transmission minima) in Fig. 2 (a) are shown
in Fig. 2 (c). It should be pointed out that the maximum of the Fano defect’s
intensity is in between the transmission dip and tip because of the sharp
asymmetric line shapes [15]. We emphasize that the resonance $\omega_{d}$ is
excited due to a symmetry breaking by side-coupled Fano defect [7] and it is
not the eigenfrequency of the defect. At the same time, the nonlinear response
at $\omega_{a}$ is greatly reduced by the step-function like linear
transmission compared to resonance $\omega_{b}$ and $\omega_{c}$. If we define
a figure of merit as $FoM=\Delta T/P_{th}$, where $\Delta
T=\max\limits_{P_{in}}[T(P_{in})-T(0)]$ refers to the maximum contrast of the
transmission and $P_{th}$ represents the necessary input power to pull the
transmission of the system up to $90\%$ of the $\Delta T$. The FoM of
resonance $\omega_{a}$, which describes both the enhanced transmission
contrast in the linear case and the reduction of the switching power, can be
enhanced more than 60 times compared to a given defect supporting conventional
Fano resonance $\omega_{b}$ [18]. The nonlinear responses at $\omega_{b}$ and
$\omega_{c}$ are similar because they are originated from the eigenfrequency
of the Fano defect.
To study the dynamical switching we employed high accuracy Crank-Nicolson
method [22] with suitable absorption boundary condition [23]. The dynamic
nonlinear response of the system at specified resonances is obtained by a long
pulse with $I=I_{0}\exp(-(t-t_{0})^{2}/W^{2})\sin(\omega t)$, where $W$ is the
pulse width, $\omega$ is set at the same frequencies as $\omega_{a-d}$.
Compared to the stationary one, the dynamic solution predicts similar
nonlinear response except for the case of $\omega_{a}$. One can see that
transmission cannot be well defined at frequency $\omega_{a}$ above a certain
threshold of the input power, indicated by the shaded region. This is caused
by the modulational instability of the Fano resonances [21], which also
indicates an enhanced nonlinear response. It is similar to the modulational
instability of waves scattered by a nonlinear centre [24]. Usually, it happens
in the finite interval of frequencies . The stability analysis of our system
with the nonlinear hybridization between Fano resonances gives similar results
to the conventional nonlinear Fano resonance with the same instability regions
[21]. Note here, that dynamical instability occurs independently from
bistability conditions.
Figure 3: (color online) (a) Typical perfect bistability at the frequency
$\omega=-0.2$ with respect to $\omega_{a^{\prime}}$. Here, $t_{0}=1.3\times
10^{4}$, $W=5\times 10^{3}$ and $I_{0}=0.025$. (b) Intensity of the Fano
defect with detuning $\Delta\omega=0.11$ with respect to
$\omega_{a,a^{\prime}}$.
Figure 2 (d) demonstrates a butterfly like double bistability obtained by
dynamic pulse feeding, where the operation frequency is referred to
$\omega_{a}=0.43$. Such a hysteresis loop at $\omega_{a}$ shows interesting
properties that one loop is clockwise while the other is anti-clockwise. The
non-uniform shift of the normalized intensity in the nonlinear Fano defect
[see balls in Fig. 2 (b)] leads to the double bistable operation. As can be
seen from this figure, that such a detuning is not enough to initiate a
bistable response for other case of $\omega_{b}$, $\omega_{c}$, and
$\omega_{d}$. Thus, the nonlinear hybridization of various resonant states
offers a unique opportunity to manipulate the nonlinear Fano resonances.
According to the stationary normalized intensity of the defect [see Fig. 2
(b)], we can obtain further nonlinear enhancement by working with
$\omega_{a^{\prime}}$. Figure 3 (a) shows an enhanced bistability when the
operation frequency is set at $\omega=-0.21$. This kind of perfect bistable
state benefits from the sharp step function like linear transmission and the
further enhanced intensity of the nonlinear defect. The intensities of the
Fano defect [see Fig. 3(b)] which demonstrate the further enhancement of the
intensity in the Fano defect are in accord with the static result [see Fig.
2(b)]. The oscillation between two bistable state is the process that one
bistable state transfers to the other one and it is the properties of the
dynamical bistability [21].
Figure 4: (color online)(a) Bistable operation obtain by pulse excitation in
nonlinear FDTD experiment; (b) electric field distribution ($|E|^{2}$) at the
upward dynamic switching point.
## 3 Photonic Crystal based nonlinear MZF interferometer
Using the modified Fano-Anderson model, we have proved that the nonlinear MZFI
can be considered as suitable candidate to realize the enhanced nonlinear
response and manipulate the dynamical bistability. We thus suggest a PhCs
platform as one of the possibilities to realize our idea, while the results
above can be applied to other varieties of nonlinear discrete system. The PhC
structure is shown in Fig. 4 (b) and It is formed by dielectric rods embedded
in air(the radius of dielectric rods is $r=0.19a$ where $a$ is the lattice
constant and refractive index $n=3.14$) except for the nonlinear Fano defect
(polymer rod $n=1.59$ and the third-order nonlinearity susceptibility of the
polymer is $\chi^{(3)}=1.14\times 10^{-12}cm^{2}/W$). We use the nonlinear
FDTD method to solve the Maxwell’s equations. The bistable response shown in
Fig. 4 (a) is obtained by the response of a Gaussian pulse with input
frequency $f=0.373$ $2\pi c/a$ and duration $30$ picoseconds. The profile is
similar to the theoretical result in Fig. 3. Figure 4 (b) shows the transient
electric field distribution ($|E|^{2}$) exactly at the downward switching
point (from on to off-state). Nearly perfect blocking of the input pulse
demonstrates the dynamical shutting down operation by a pulse and successful
suppression of the modulation instability. Both high intensities of the Fano
defect and the loop are the signature of nonlinear switching involving unique
hybrid Fano resonances in the MZFI [7]. The transmission contrast can be
further enhanced by careful engineering the interaction between the eigen-Fano
resonance and the MZI loop’s resonance which takes place exactly at the
propagation band center.
## 4 Conclusions
In conclusion, we have investigated the interaction of loop and side-coupled
Fano resonances in nonlinear MZFI type structures. By introducing nonlinear
Fano defects we are able to excite ”dark” states of pure MZI, which results in
the formation of very narrow hybrid resonant states. Both stationary and
dynamic studies convince the superiority of enhanced nonlinear response and
dynamical bistability in nonlinear MZFI. Our direct numerical simulations of
two-dimensional PhCs confirm the theoretical predictions of the dynamic
characteristics. We anticipate that such structures can be realized on
different platforms for ultra-high sensitive sensing operations. The idea of
nonlinear Fano resonances hybridization can also be generalized to the other
nanoscale structure. Particularly, in the new emerging field of plasmonics and
metamaterial where the electromagnetic wave is confined in subwavelength
scale, such nonlinear resonance interaction could be a basic physical idea for
designing the logic devices.
###### Acknowledgements.
The authors thank Prof. Yuri Kivshar and Dr. Anton Desyatnikov for useful
discussions. Y. Xu acknowledges the support from the China Scholarship Council
and the Nonlinear Research Centre at ANU for their hospitality. The work of A.
E. Miroshnichenko was supported by the Australian Research Council through
Future Fellowship program.
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|
arxiv-papers
| 2012-01-08T13:10:04 |
2024-09-04T02:49:26.059992
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yi Xu and Andrey E. Miroshnichenko",
"submitter": "Yi Xu Mr",
"url": "https://arxiv.org/abs/1201.1630"
}
|
1201.1632
|
# Metric tensors for the interpolation error and its gradient in $L^{p}$ norm
Hehu Xie111LSEC, ICMSEC, Academy of Mathematics and Systems Science, CAS,
Beijing 100080, China email: hhxie@lsec.cc.ac.cn Xiaobo Yin222Department of
Mathematics, Central China Normal University, Wuhan 430079, China email:
yinxb@lsec.cc.ac.cn
> Abstract. A uniform strategy to derive metric tensors in two spatial
> dimension for interpolation errors and their gradients in $L^{p}$ norm is
> presented. It generates anisotropic adaptive meshes as quasi-uniform ones in
> corresponding metric space, with the metric tensor being computed based on a
> posteriori error estimates in different norms. Numerical results show that
> the corresponding convergence rates are always optimal.
>
> Keywords. metric tensor, interpolation; gradient; anisotropic.
>
> AMS subject classification. 65N30, 65N50
## 1 Introduction
Generation of adaptive meshes is now the standard option in most software
packages. Traditionally, isotropic mesh adaptation has received much
attention, where regular mesh elements are only adjusted in size based on an
error estimate. However, for problems with anisotropic solutions (with, say,
sharp boundary or internal layers), the shape of elements can be further
optimized and an equidistribution of a scalar error density is not sufficient
to ensure that a mesh is optimally efficient [14]. Indeed anisotropic meshes
have been used successfully in many areas, for example in singular
perturbation and flow problems [4, 5, 6, 21, 22, 35, 43] and in adaptive
procedures [2, 7, 8, 10, 11, 23, 35, 36]. For anisotropic mesh adaptation, the
common practice is to generate the needed anisotropic mesh as a quasi-uniform
one in the metric space determined by a tensor (or a matrix-valued function),
always called monitor function or metric tensor. Both the monitor function
(denoted by the letter $M$) and metric tensor (denoted by the calligraphy
letter $\mathcal{M}$) play the same role in mesh generation, i.e., they are
used to specify the size, shape, and orientation of mesh elements throughout
the physical domain. The only difference lies in the way they specify the size
of elements. Indeed, the former specifies the element size through the
equidistribution condition, while the latter determines the element size
through the unitary volume requirement. Readers could regard the metric tensor
as normalization for the monitor function. Examples of anisotropic meshing
strategies include blue refinement [29, 30], directional refinement [36],
Delaunay-type triangulation method [7, 8, 11, 35], advancing front method
[19], bubble packing method [41], local refinement and modification [21, 37],
variational methods [9, 17, 24, 27, 28, 31], and so on. Readers are referred
to [18] and [34] for an overview.
Among these meshing strategies, the definition of the metric tensor (or
monitor function) based on the Hessian of the solution seems widespread in the
meshing community[1, 11, 12, 13, 14, 15, 20, 21, 23, 25, 26, 24, 37, 39].
Especially, Huang and Russell [26] propose the monitor function
$\displaystyle
M=\det{\Big{(}}I+\frac{1}{\alpha}|H(u)|{\Big{)}}^{-\frac{1}{d+p(2-m)}}{\Big{\|}}I+\frac{1}{\alpha}|H(u)|{\Big{\|}}^{\frac{mp}{d+p(2-m)}}{\Big{[}}I+\frac{1}{\alpha}|H(u)|{\Big{]}},$
(1.1)
for the interpolation error in $W^{m,p}$ norm ($m=0,1$, $p\in[1,+\infty)$),
where $d$ stands for the spatial dimension. Set
$\mathcal{H}=I+\frac{1}{\alpha}|H(u)|$, when $d=2$,
$\displaystyle
M_{m,p}=\det(\mathcal{H})^{-\frac{1}{2+p(2-m)}}\|\mathcal{H}\|^{\frac{mp}{2+p(2-m)}}\mathcal{H}.$
(1.2)
Separately, it becomes
$\displaystyle M_{0,p}=\det(\mathcal{H})^{-\frac{1}{2(p+1)}}\mathcal{H},$
(1.3)
for the interpolation error in $L^{p}$ norm and
$\displaystyle
M_{1,p}=\det(\mathcal{H})^{-\frac{1}{p+2}}\|\mathcal{H}\|^{\frac{p}{p+2}}\mathcal{H}.$
(1.4)
for the gradient of interpolation error in $L^{p}$ norm.
The objective of this paper is to give a unified strategy deriving metric
tensors in two spatial dimension for interpolation error and its gradients in
$L^{p}$ norm. The development begin with the error estimates [32] for $L^{2}$
norm and our recent work [42] for $H^{1}$ norm on linear interpolation for
quadratic functions on triangles. These estimates are anisotropic in the sense
that they allow a full control of the shape of elements when used within a
mesh generation strategy. Using the relationship between different norms, a
posterior error estimates for other norms ($W^{m,p},m=0,1$, $p\neq 2$) can be
gained. We will apply these error estimates to formulate corresponding metric
tensors in a unified way. The procedure is based on two considerations: on the
one hand the anisotropic mesh is generated as a quasi-uniform mesh in the
metric tensor. On the other hand, the anisotropic mesh is required to minimize
the error for a given number of triangles. To compare with those existing
methods, we list our main results using monitor function style, that is
$\displaystyle M_{0,p}^{n}({\bf
x})=\det(\mathcal{H})^{-\frac{1}{2(p+1)}}\mathcal{H},$ (1.5)
for interpolation errors in $L^{p}$ norm and
$\displaystyle M_{1,p}^{n}({\bf
x})=\det(\mathcal{H})^{-\frac{1}{p+2}}\mbox{tr}(\mathcal{H})^{\frac{p}{p+2}}\mathcal{H},$
(1.6)
for gradient of interpolation errors in $L^{p}$ norm. To sum up, the metric
tensor can be expressed by
$\displaystyle M_{m,p}^{n}({\bf
x})=\det(\mathcal{H})^{-\frac{1}{2+p(2-m)}}\mbox{tr}(\mathcal{H})^{\frac{mp}{2+p(2-m)}}\mathcal{H},$
(1.7)
for the $W^{m,p}$ norm ($m=0,1$, $p\in(0,+\infty]$) of the interpolation
error.
The paper is organized as follows. In Section 2, we describe the anisotropic
error estimates on linear interpolation for quadratic functions on triangles
obtained in our recent work [42]. The formulation of the monitor function and
metric tensor is developed in Section 3. Numerical results are presented in
Section 4 to illustrating our analysis. Finally, conclusions are drawn in
Section 5.
## 2 Estimates for interpolation error and its gradient
As we know, the interpolation error depends on the solution, the size and
shape of the elements in the mesh. Understanding this relation is crucial for
the generating efficient meshes for the finite element method. In the mesh
generation community, this relation is studied more closely for the model
problem of interpolating quadratic functions. This treatment yields a reliable
and efficient estimator of the interpolation error for general functions
provided a saturation assumption is valid [3, 16]. For instance, Nadler [32]
derived an exact expression for the $L^{2}$-norm of the linear interpolation
error in terms of the three sides ${\bf\ell}_{1}$, ${\bf\ell}_{2}$, and
${\bf\ell}_{3}$ of the triangle $K$,
$\displaystyle\|u-u_{I}\|^{2}_{L^{2}(K)}=\frac{|K|}{180}{\Big{[}}{\Big{(}}d_{1}+d_{2}+d_{3}{\Big{)}}^{2}+d_{1}d_{2}+d_{2}d_{3}+d_{1}d_{3}{\Big{]}},$
(2.1)
where $|K|$ is the area of the triangle, $d_{i}={\bf\ell}_{i}\cdot
H{\bf\ell}_{i}$ with $H$ being the Hessian of $u$. Assuming
$u=\lambda_{1}x^{2}+\lambda_{2}y^{2}$, D’Azevedo and Simpson [13] derived the
exact formula for the maximum norm of the interpolation error
$\displaystyle\|(u-u_{I})\|^{2}_{L^{\infty}(K)}=\frac{D_{12}D_{23}D_{31}}{16\lambda_{1}\lambda_{2}|K|^{2}},$
(2.2)
where
$D_{ij}={\bf\ell}_{i}\cdot\mbox{diag}(\lambda_{1},\lambda_{2}){\bf\ell}_{j}$.
Based on the geometric interpretation of this formula, they proved that for a
fixed area the optimal triangle, which produces the smallest maximum
interpolation error, is the one obtained by compressing an equilateral
triangle by factors $\sqrt{\lambda_{1}}$ and $\sqrt{\lambda_{2}}$ along the
two eigenvectors of the Hessian of $u$. Furthermore, the optimal incidence for
a given set of interpolation points is the Delaunay triangulation based on the
stretching map (by factors $\sqrt{\lambda_{1}}$ and $\sqrt{\lambda_{2}}$ along
the two eigenvector directions) of the grid points. Rippa [38] showed that the
mesh obtained in this way is also optimal for the $L^{p}$-norm of the error
for any $1\leq p\leq\infty$.
The element-wise error estimates in the following theorem are developed in
[42] using the theory of interpolation and proper numerical quadrature
formula.
###### Theorem 2.1.
Let $u$ be a quadratic function and $u_{I}$ is the Lagrangian linear finite
element interpolation of $u$. The following relationship holds:
$\displaystyle\|\nabla(u-u_{I})\|^{2}_{L^{2}(K)}=\frac{1}{48|K|}\sum_{i=1}^{3}({\bf\ell}_{i+1}\cdot
H{\bf\ell}_{i+2})^{2}|{\bf\ell}_{i}|^{2},$ (2.3)
where we prescribe $i+3=i,i-3=i$.
To get the a posteriori error estimate of the interpolation error in $L^{p}$
and $W^{1,p}$ norms for $p\neq 2$, we need some lemmas below.
###### Lemma 2.1.
For any $d$ positive numbers $a_{1},\cdots,a_{d}$, the inequalities
$\displaystyle{\Big{(}}\sum\limits_{j=1}^{d}a_{j}^{2}{\Big{)}}^{\frac{1}{2}}\leq{\Big{(}}\sum\limits_{j=1}^{d}a_{j}^{p}{\Big{)}}^{\frac{1}{p}}\leq
d^{\frac{1}{p}-\frac{1}{2}}{\Big{(}}\sum\limits_{j=1}^{d}a_{j}^{2}{\Big{)}}^{\frac{1}{2}},$
(2.4)
and
$\displaystyle
d^{\frac{1}{p}-\frac{1}{2}}{\Big{(}}\sum\limits_{j=1}^{d}a_{j}^{2}{\Big{)}}^{\frac{1}{2}}\leq{\Big{(}}\sum\limits_{j=1}^{d}a_{j}^{p}{\Big{)}}^{\frac{1}{p}}\leq{\Big{(}}\sum\limits_{j=1}^{d}a_{j}^{2}{\Big{)}}^{\frac{1}{2}}$
(2.5)
hold for numbers $0<p<2$ and $p>2$, respectively.
###### Proof.
We just give the proof for the case $0<p<2$, it is similar for the case $p>2$.
For any number $0<p<2$,
$\displaystyle{\Big{(}}\sum\limits_{j=1}^{d}a_{j}^{2}{\Big{)}}^{\frac{1}{2}}\leq{\Big{(}}\sum\limits_{j=1}^{d}a_{j}^{p}{\Big{)}}^{\frac{1}{p}}$
holds due to the Jensen’s inequality. From the generalized arithmetic-mean
geometric-mean inequality, for any positive numbers $a_{1},\cdots,a_{d}$,
$\displaystyle{\Big{(}}\sum\limits_{j=1}^{d}\frac{1}{d}a_{j}^{p}{\Big{)}}^{\frac{1}{p}}\leq{\Big{(}}\sum\limits_{j=1}^{d}\frac{1}{d}a_{j}^{2}{\Big{)}}^{\frac{1}{2}}.$
Then
$\displaystyle{\Big{(}}\sum\limits_{j=1}^{d}a_{j}^{p}{\Big{)}}^{\frac{1}{p}}\leq
d^{\frac{1}{p}-\frac{1}{2}}{\Big{(}}\sum\limits_{j=1}^{d}a_{j}^{2}{\Big{)}}^{\frac{1}{2}}.$
∎
To sum up, for any $d$ positive numbers $a_{1},\cdots,a_{d}$, the inequalities
$\displaystyle\underline{C}_{p}{\Big{(}}\sum\limits_{j=1}^{d}a_{j}^{2}{\Big{)}}^{\frac{1}{2}}\leq{\Big{(}}\sum\limits_{j=1}^{d}a_{j}^{p}{\Big{)}}^{\frac{1}{p}}\leq\overline{C}_{p}{\Big{(}}\sum\limits_{j=1}^{d}a_{j}^{2}{\Big{)}}^{\frac{1}{2}}$
(2.6)
holds for any numbers $p>0$, where $\underline{C}_{p}=1$ for $0<p<2$ and
$d^{\frac{1}{p}-\frac{1}{2}}$ for $p>2$,
$\overline{C}_{p}=d^{\frac{1}{p}-\frac{1}{2}}$ for $0<p<2$ and $1$ for $p>2$.
###### Lemma 2.2.
[3] For any $p\in(0,+\infty]$ and any non-negative $v\in P_{2}(K)$ it holds
$\displaystyle
C_{1/p}^{-\frac{1}{p}}|K|^{\frac{1}{p}-1}\|v\|_{L^{1}(K)}\leq\|v\|_{L^{p}(K)}\leq
C_{p}|K|^{\frac{1}{p}-1}\|v\|_{L^{1}(K)}$ (2.7)
with
$\left\\{\begin{array}[]{lll}C_{p}=1&\mbox{if}\,\,0<p\leq 1,&\\\
C_{p}=(d+1)(d+2)(d!)^{\frac{1}{p}}{\Big{(}}\prod\limits_{j=1}^{d}(p+j){\Big{)}}^{-\frac{1}{p}}&\mbox{if}\,\,1<p<+\infty,&\\\
C_{\infty}=\lim\limits_{p\rightarrow+\infty}C_{p}=(d+1)(d+2),\\\
C_{1/\infty}=\lim\limits_{p\rightarrow+\infty}C_{1/p}=1.\end{array}\right.$
### 2.1 Estimates for interpolation errors in $L^{p}$ norm
We consider the error of linear interpolation $e=u-u_{I}$ for a quadratic
function $u$ on $K$. Since the function $e$ is quadratic on $K$, we can apply
Lemma 2.2 to obtain
$\displaystyle
C_{1/p}^{-1/p}|K|^{\frac{1}{p}-1}\|e\|_{L^{1}(K)}\leq\|e\|_{L^{p}(K)}\leq
C_{p}|K|^{\frac{1}{p}-1}\|e\|_{L^{1}(K)}.$ (2.8)
Set $p=2$,
$\displaystyle|K|^{-\frac{1}{2}}\|e\|_{L^{1}(K)}\leq\|e\|_{L^{2}(K)}\leq
C_{2}|K|^{-\frac{1}{2}}\|e\|_{L^{1}(K)},$
or
$\displaystyle
C_{2}^{-1}|K|^{\frac{1}{2}}\|e\|_{L^{2}(K)}\leq\|e\|_{L^{1}(K)}\leq|K|^{\frac{1}{2}}\|e\|_{L^{2}(K)}.$
(2.9)
Combine (2.8) and (2.9), we get
$\displaystyle
C_{1/p}^{-1/p}C_{2}^{-1}|K|^{\frac{1}{p}-\frac{1}{2}}\|e\|_{L^{2}(K)}\leq\|e\|_{L^{p}(K)}\leq
C_{p}|K|^{\frac{1}{p}-\frac{1}{2}}\|e\|_{L^{2}(K)}.$ (2.10)
In this article, $A\sim B$ stands for that there exist two constants
$\underline{C}$ and $\overline{C}$ such that
$\displaystyle\underline{C}A\leq B\leq\overline{C}A,$
where the two constants $\underline{C}$ and $\overline{C}$ may depend on the
prescribed error, the index $p$, the dimension $d$, and the numbers of
elements $N$, however are independent of function at hand. So (2.10) can be
rewritten as
$\displaystyle\|e\|_{L^{p}(K)}\sim|K|^{\frac{1}{p}-\frac{1}{2}}\|e\|_{L^{2}(K)}.$
Together with the expression (2.1) for the $L^{2}$ norm of the linear
interpolation error derived by Nadler[32], we have the a posteriori error
estimate in $L^{p}$ norms as follows:
$\displaystyle\|e\|^{2}_{L^{p}(K)}\sim|K|^{\frac{2}{p}-1}\|e\|^{2}_{L^{2}(K)}$
$\displaystyle=$
$\displaystyle\frac{|K|^{\frac{2}{p}}}{180}{\Big{[}}{\Big{(}}\sum_{i=1}^{3}d_{i}{\Big{)}}^{2}+d_{1}d_{2}+d_{2}d_{3}+d_{1}d_{3}{\Big{]}}.$
(2.11)
### 2.2 Estimates for gradient of interpolation errors in $L^{p}$ norm
Now we consider the gradient of linear interpolation error $\nabla
e=\nabla(u-u_{I})$ for a quadratic function $u$. Since the function
$\displaystyle v_{j}(x)={\Big{(}}\frac{\partial e}{\partial
x_{j}}{\Big{)}}^{2}$
is quadratic on $K$, we can apply Lemma 2.2 to obtain
$\displaystyle\|v_{j}\|_{L^{p/2}(K)}^{1/2}\geq
C_{2/p}^{-1/p}|K|^{\frac{1}{p}-\frac{1}{2}}\|v_{j}\|_{L^{1}(K)}^{1/2}=C_{2/p}^{-1/p}|K|^{\frac{1}{p}-\frac{1}{2}}{\Big{\|}}\frac{\partial
e}{\partial x_{j}}{\Big{\|}}_{L^{2}(K)},$ (2.12)
and
$\displaystyle\|v_{j}\|_{L^{p/2}(K)}^{1/2}\leq
C_{p/2}^{1/2}|K|^{\frac{1}{p}-\frac{1}{2}}\|v_{j}\|_{L^{1}(K)}^{1/2}=C_{p/2}^{1/2}|K|^{\frac{1}{p}-\frac{1}{2}}{\Big{\|}}\frac{\partial
e}{\partial x_{j}}{\Big{\|}}_{L^{2}(K)}.$ (2.13)
Since
$\displaystyle\|\nabla
e\|_{L^{p}(K)}^{p}=\sum\limits_{j=1}^{d}\int_{K}{\Big{|}}\frac{\partial
e}{\partial x_{j}}{\Big{|}}^{p}d{\bf
x}=\sum\limits_{j=1}^{d}\|v_{j}\|_{L^{p/2}(K)}^{p/2},$
then together with (2.12) and (2.13), we have
$\displaystyle
C_{2/p}^{-1/p}|K|^{\frac{1}{p}-\frac{1}{2}}{\Big{(}}\sum\limits_{j=1}^{d}{\Big{\|}}\frac{\partial
e}{\partial x_{j}}{\Big{\|}}_{L^{2}(K)}^{p}{\Big{)}}^{\frac{1}{p}}\leq\|\nabla
e\|_{L^{p}(K)}\leq
C_{p/2}^{1/2}|K|^{\frac{1}{p}-\frac{1}{2}}{\Big{(}}\sum\limits_{j=1}^{d}{\Big{\|}}\frac{\partial
e}{\partial x_{j}}{\Big{\|}}_{L^{2}(K)}^{p}{\Big{)}}^{\frac{1}{p}}.$
From (2.6), the inequality
$\displaystyle\underline{C}_{p}C_{2/p}^{-1/p}|K|^{\frac{1}{p}-\frac{1}{2}}\|\nabla
e\|_{L^{2}(K)}\leq\|\nabla
e\|_{L^{p}(K)}\leq\overline{C}_{p}C_{p/2}^{1/2}|K|^{\frac{1}{p}-\frac{1}{2}}\|\nabla
e\|_{L^{2}(K)},$
holds, or simply
$\displaystyle\|\nabla e\|_{L^{p}(K)}\sim|K|^{\frac{1}{p}-\frac{1}{2}}\|\nabla
e\|_{L^{2}(K)}.$
Together with the a posteriori error estimate (2.3) of the interpolation error
in $H^{1}$($=W^{1,2}$) norm, we have the a posteriori error estimate in
$W^{1,p}$ norms as follows:
$\displaystyle\|\nabla e\|^{2}_{L^{p}(K)}$ $\displaystyle\sim$
$\displaystyle|K|^{\frac{2}{p}-1}\|\nabla e\|_{L^{2}(K)}^{2}$ (2.14)
$\displaystyle=$
$\displaystyle|K|^{\frac{2}{p}-1}\frac{1}{48|K|}\sum_{i=1}^{3}({\bf\ell}_{i+1}\cdot
H_{K}{\bf\ell}_{i+2})^{2}|{\bf\ell}_{i}|^{2}$ $\displaystyle=$
$\displaystyle\frac{|K|^{\frac{2}{p}-2}}{48}\sum_{i=1}^{3}({\bf\ell}_{i+1}\cdot
H_{K}{\bf\ell}_{i+2})^{2}|{\bf\ell}_{i}|^{2}.$
## 3 Metric tensors for anisotropic mesh adaptation
We now use the results of Section 2 to develop metric tensors for
interpolation errors and their gradients in $L^{p}$ norm in a unified way. As
a common practice in anisotropic mesh generation, the metric tensor,
$\mathcal{M}({\bf x})$, is used in a meshing strategy in such a way that an
anisotropic mesh is generated as a quasi-uniform mesh in the metric space
determined by $\mathcal{M}({\bf x})$. Mathematically, this can be interpreted
as the shape, size and equidistribution requirements as follows.
The shape requirement. The elements of the new mesh, $\mathcal{T}_{h}$, are
(or are close to being) equilateral in the metric.
The size requirement. The elements of the new mesh $\mathcal{T}_{h}$ have a
unitary volume in the metric, i.e.,
$\displaystyle\int_{K}\sqrt{\det(\mathcal{M}({\bf x}))}d{\bf x}=1,\quad\forall
K\in\mathcal{T}_{h}.$ (3.1)
The equidistribution requirement. The anisotropic mesh is required to
minimize the error for a given number of mesh points (or equidistribute the
error on every element).
Notice that to derive the monitor function, we just need the shape and
equidistribution requirements.
### 3.1 Metric tensors for gradients of interpolation errors in $L^{p}$ norm
$\mathcal{F}_{K}$$\bf\ell_{1}$$\bf\ell_{2}$$\bf\ell_{3}$$\theta_{1}$$\theta_{2}$$\theta_{3}$$\hat{\theta}_{1}$$\hat{\theta}_{2}$$\hat{\theta}_{3}$$\hat{\ell}_{1}$$\hat{\ell}_{2}$$\hat{\ell}_{3}$
Figure 1: Affine map ${\bf\hat{x}}=\mathcal{F}_{K}{\bf x}$ from $K$ to the
reference triangle $\hat{K}$.
We derive the monitor function $M({\bf x})$ first. Assume $H(u)$ be a
symmetric positive definite matrix on every point ${\bf x}$, this assumption
will be dropped later. Set $M({\bf x})=C({\bf x})H(u)$. Consider the $L^{2}$
projection of $H(u)$ on $K$, denoted by $H_{K}$, then so does $M_{K}$. Since
$H_{K}$ is a symmetric positive definite matrix, we consider the singular
value decomposition $H_{K}=R^{T}\Lambda R$, where
$\Lambda=\mbox{diag}(\lambda_{1},\lambda_{2})$ is the diagonal matrix of the
corresponding eigenvalues ($\lambda_{1},\lambda_{2}>0$) and $R$ is the
orthogonal matrix having as rows the eigenvectors of $H_{K}$. Denote by
$F_{K}$ and ${\bf t}_{K}$ the matrix and the vector defining the invertible
affine map $\hat{\bf x}=\mathcal{F}_{K}({\bf x})=F_{K}{\bf x}+{\bf t}_{K}$
from the generic element $K$ to the reference triangle $\hat{K}$ (see Figure
1).
Obviously, $M_{K}=C_{K}H_{K}$. Let $M_{K}=F_{K}^{T}F_{K}$, then
$F_{K}=C_{K}^{\frac{1}{2}}\Lambda^{\frac{1}{2}}R$. Mathematically, the shape
requirement can be expressed as
$\displaystyle|\hat{\ell}_{i}|=L\,\,\mbox{and}\,\,\cos\hat{\theta}_{i}=\frac{\hat{\ell}_{i+1}\cdot\hat{\ell}_{i+2}}{L^{2}}=\frac{1}{2},\,i=1,2,3,$
(3.2)
where $L$ is a constant for every element $K$. Enforcing the shape
requirement, we get
$\displaystyle\|\nabla e\|^{2}_{L^{p}(K)}$ $\displaystyle\sim$
$\displaystyle\frac{|K|^{\frac{2}{p}-2}}{48}\sum_{i=1}^{3}({\bf\ell}_{i+1}\cdot
H_{K}{\bf\ell}_{i+2})^{2}|{\bf\ell}_{i}|^{2}$ $\displaystyle=$
$\displaystyle\frac{|K|^{\frac{2}{p}-2}}{48C_{K}^{2}}\sum_{i=1}^{3}({\bf\ell}_{i+1}\cdot
M_{K}{\bf\ell}_{i+2})^{2}|{\bf\ell}_{i}|^{2}$ $\displaystyle=$
$\displaystyle\frac{L^{4}|K|^{\frac{2}{p}-2}}{48C_{K}^{2}}\sum_{i=1}^{3}(\cos\hat{\theta}_{i})^{2}|{\bf\ell}_{i}|^{2}=\frac{L^{4}|K|^{\frac{2}{p}-2}}{192C_{K}^{2}}\sum_{i=1}^{3}|{\bf\ell}_{i}|^{2}.$
Notice that,
$\displaystyle|K|=\frac{|\hat{K}|}{C_{K}\sqrt{\det(H_{K})}},$
we have
$\displaystyle\|\nabla e\|^{2}_{L^{p}(K)}$ $\displaystyle\sim$
$\displaystyle\frac{L^{4}|\hat{K}|^{\frac{2}{p}-2}C_{K}^{2-\frac{2}{p}}\det(H_{K})^{1-\frac{1}{p}}}{192C_{K}^{2}}\sum_{i=1}^{3}{\Big{|}}C_{K}^{-\frac{1}{2}}R^{-1}\Lambda^{-\frac{1}{2}}\hat{\ell}_{i}{\Big{|}}^{2}$
$\displaystyle=$
$\displaystyle\frac{L^{4}|\hat{K}|^{\frac{2}{p}-2}\det(H_{K})^{1-\frac{1}{p}}}{192C_{K}^{1+\frac{2}{p}}}\sum_{i=1}^{3}{\Big{|}}\Lambda^{-\frac{1}{2}}\hat{\ell}_{i}{\Big{|}}^{2}$
$\displaystyle=$
$\displaystyle\frac{L^{4}|\hat{K}|^{\frac{2}{p}-2}\det(H_{K})^{1-\frac{1}{p}}}{192C_{K}^{1+\frac{2}{p}}}\frac{\mbox{tr}(H_{K})}{\det(H_{K})}$
$\displaystyle=$
$\displaystyle\frac{L^{4}|\hat{K}|^{\frac{2}{p}-2}\det(H_{K})^{-\frac{1}{p}}\mbox{tr}(H_{K})}{192C_{K}^{1+\frac{2}{p}}}$
$\displaystyle\sim$
$\displaystyle\frac{\det(H_{K})^{-\frac{1}{p}}\mbox{tr}(H_{K})}{C_{K}^{1+\frac{2}{p}}},$
then
$\displaystyle\|\nabla e\|^{p}_{L^{p}(K)}=(\|\nabla
e\|^{2}_{L^{p}(K)})^{p/2}\sim\frac{\det(H_{K})^{-\frac{1}{2}}\mbox{tr}(H_{K})^{\frac{p}{2}}}{C_{K}^{1+\frac{p}{2}}}.$
To satisfy the equidistribution requirement, let
$\displaystyle\|\nabla
e\|^{p}_{L^{p}(K)}={\Big{(}}\sum\limits_{K\in\mathcal{T}_{h}}e_{K}^{p}{\Big{)}}/N=\epsilon^{p}/N,$
where $N$ is the number of elements of $\mathcal{T}_{h}$. Then
$\displaystyle
C_{K}\sim\det(H_{K})^{-\frac{1}{p+2}}\mbox{tr}(H_{K})^{\frac{p}{p+2}}.$
So $M({\bf x})$ could be the form
$\displaystyle M({\bf
x})=\det(H)^{-\frac{1}{p+2}}\mbox{tr}(H)^{\frac{p}{p+2}}H(u),$
since $M({\bf x})$ can be modified by multiplying a constant. Since it
corresponds the gradient of interpolation errors in $L^{p}$ norm, we denote it
by $M_{1,p}^{n}({\bf x})$.
To establish the metric tensor $\mathcal{M}_{1,p}^{n}({\bf x})$, set
$\mathcal{M}_{1,p}^{n}({\bf x})=\theta_{1,p}M_{1,p}^{n}({\bf x})$, at this
time, the size requirement (3.1) should be used, which leads to
$\displaystyle\theta_{1,p}\int_{K}\rho_{1,p}({\bf x})d{\bf x}=1,$
where
$\displaystyle\rho_{1,p}({\bf x})=\sqrt{\det(M_{1,p}^{n}({\bf x}))}.$
Summing the above equation over all the elements of $\mathcal{T}_{h}$, one
gets
$\displaystyle\theta_{1,p}\sigma_{1,p}=N,$
where
$\displaystyle\sigma_{1,p}=\int_{\Omega}\rho_{1,p}({\bf x})d{\bf x}.$
Thus, we get
$\displaystyle\theta_{1,p}=\frac{N}{\sigma_{1,p}},$
and as a consequence,
$\displaystyle\mathcal{M}_{1,p}^{n}({\bf
x})=\frac{N}{\sigma_{1,p}}\det(H)^{-\frac{1}{p+2}}\mbox{tr}(H)^{\frac{p}{p+2}}H(u).$
### 3.2 Metric tensor for the interpolation errors in $L^{p}$ norm
Using the error estimates (2.11) for interpolation errors in $L^{p}$ norm and
the shape requirement (3.2), we have
$\displaystyle\|e\|^{2}_{L^{p}(K)}$ $\displaystyle\sim$
$\displaystyle\frac{|K|^{\frac{2}{p}}}{180}{\Big{[}}{\Big{(}}\sum_{i=1}^{3}d_{i}{\Big{)}}^{2}+d_{1}d_{2}+d_{2}d_{3}+d_{1}d_{3}{\Big{]}}$
$\displaystyle=$
$\displaystyle\frac{|K|^{\frac{2}{p}}}{180C_{K}^{2}}{\Big{[}}{\Big{(}}\sum_{i=1}^{3}|\hat{\ell}_{i}|^{2}{\Big{)}}^{2}+\sum_{i=1}^{3}{\Big{(}}|\hat{\ell}_{i+1}||\hat{\ell}_{i+2}|{\Big{)}}^{2}{\Big{]}}$
$\displaystyle=$
$\displaystyle\frac{L^{4}|K|^{\frac{2}{p}}}{15C_{K}^{2}}=\frac{L^{4}|\hat{K}|^{\frac{2}{p}}}{15C_{K}^{2+\frac{2}{p}}\det(H)^{\frac{1}{p}}}\sim\frac{1}{C_{K}^{2+\frac{2}{p}}\det(H)^{\frac{1}{p}}}.$
Then,
$\displaystyle\|e\|^{p}_{L^{p}(K)}=(\|e\|^{2}_{L^{p}(K)})^{p/2}\sim\frac{1}{C_{K}^{p+1}\det(H)^{\frac{1}{2}}}.$
To satisfy the equidistribution requirement, let
$\displaystyle\|e\|^{p}_{L^{p}(K)}={\Big{(}}\sum\limits_{K\in\mathcal{T}_{h}}e_{K}^{p}{\Big{)}}/N=\epsilon^{p}/N.$
Using similar argument in last subsection, we easily get monitor functions
$\displaystyle M_{0,p}^{n}({\bf x})=\det(H)^{-\frac{1}{2(p+1)}}H(u),$
and metric tensors
$\displaystyle\mathcal{M}_{0,p}^{n}({\bf
x})=\frac{N}{\sigma_{0,p}}\det(H)^{-\frac{1}{2(p+1)}}H(u),$
for the interpolation errors in $L^{p}$ norm.
### 3.3 Practice use of metric tensor
So far we assume that $H(u)$ is a symmetric positive definite matrix at every
point. However this assumption doesn’t hold in many cases. In order to obtain
a symmetric positive definite matrix, the following procedure are often
implemented. First, the Hessian $H$ is modified into
$|H|=R^{T}\,\,\mbox{diag}(|\lambda_{1}|,|\lambda_{2}|)R$ by taking the
absolute value of its eigenvalues ([22]). Since $|H|$ is only semi-positive
definite, $\mathcal{M}_{m,p}^{n}$ cannot be directly applied to generate the
anisotropic meshes. To avoid this difficulty, we regularize the expression
with the flooring parameter $\alpha_{m,p}>0$ (see, e.g., [24]). Replacing
$|H|$ with
$\displaystyle\mathcal{H}=\alpha_{m,p}I+|H|,$
we get the modified metric tensors, also denoted by $\mathcal{M}_{m,p}^{n}$,
that is
$\displaystyle\mathcal{M}_{m,p}^{n}({\bf
x})=\frac{N}{\sigma_{m,p}}\det(\mathcal{H})^{-\frac{1}{2+p(2-m)}}\mbox{tr}(\mathcal{H})^{\frac{mp}{2+p(2-m)}}\mathcal{H},$
(3.3)
which are suitable for practical mesh generation.
### 3.4 Comparison with existing methods using monitor function style
When $m=0$, the new monitor function $M_{0,p}^{n}$ (1.5) is in fact the same
with (1.3) in [26, 25]. Chen, Sun and Xu [12] proved that under suitable
conditions, the error estimate
$\displaystyle\|u-u_{I}\|_{L^{p}(\Omega)}\leq CN^{-\frac{2}{d}}\|\sqrt[d]{\det
H}\|_{L^{\frac{pd}{2p+d}}(\Omega)},1\leq p\leq\infty,$
holds on the quasi-uniform mesh determined by the metric
$(\det{H})^{-\frac{1}{2p+d}}H$, where $H$ is a majorant of the Hessian matrix,
$N$ is the number of elements in the triangulation and the constant $C$ does
not depend on $u$ and $N$. This estimate is optimal in the sense that it is a
lower bound if $u$ is strictly convex or concave. Note that $\mathcal{H}$ can
be chosen as a majorant of the Hessian matrix.
When $m=1$, the new monitor function $M_{1,p}^{n}$ (1.6) is different with
(1.4) [26] that the former refers to $\mbox{tr}(\mathcal{H})$ and the latter
involves $\|\mathcal{H}\|$. In some cases, the two monitor functions are
pretty much alike. However, in other cases, the effect of the former is
superior to the latter for mesh generation. Numerical results in [40] have
shown our approach’s superiority for the error in $H^{1}$ norm.
Figure 2: Example 1: Interpolation error and its gradient in $L^{p}$ norm
## 4 Numerical experiments
In this section, we present some numerical results for three problems with
given analytical solutions. The numerical results are performed by using the
BAMG software [23]. Given a background mesh and an approximation solution,
BAMG generates the mesh according to the metric tensor. The code allows the
user to supply his/her own metric tensor defined on a background mesh. In our
computation, the background mesh has been taken as the most recent mesh
available.
Denote by $nbt$ the number of triangles in the current mesh. The number of
triangles is adjusted when necessary by trial and errors through the
modification of the multiplicative coefficient of the metric tensors.
Figure 3: Example 1: plots of the solution (a) and corresponding mesh (b)
using $\mathcal{M}_{1,1}$
Example 1 This example is to generate adaptive meshes for
$\displaystyle u({\bf
x})=\frac{1}{1+e^{-200(\sqrt{x_{1}^{2}+x_{2}^{2}}-0.8)}},\quad{\bf
x}\in(0.1,1)\times(0.1,1).$ (4.1)
This function is anisotropic along the quarter circle
$x_{1}^{2}+x_{2}^{2}=0.8^{2}$ and changes sharply in the direction normal to
this curve. A similar example was presented in [33] where the region is
$(0,1)\times(0,1)$. In the current computation, each run is stopped after 15
iterations to guarantee that the adaptive procedure tends towards stability.
We show in Figure 2 the $L^{p}$ norms of the interpolation error and its
gradient using corresponding metric tensors, for $p=1,2,4,\infty$. For
example, the curve $p=2$ in (a) stands for the interpolation error using the
metric tensor $\mathcal{M}_{0,2}$, while $p=\infty$ in (b) stands for the
gradient interpolation error using the metric tensor $\mathcal{M}_{1,\infty}$.
We see that the convergence rates for the interpolation error and its gradient
are always nearly optimal, i.e. $\|e\|_{L^{p}}\sim N^{-1}$ and $\|\nabla
e\|_{L^{p}}\sim N^{-0.5}$. We also show in Figure 3 plots of the solution and
corresponding mesh using the metric tensor $\mathcal{M}_{1,1}$.
Figure 4: Example 2: Interpolation error and its gradient in $L^{p}$ norm
Example 2 This example is to generate adaptive meshes for
$\displaystyle u({\bf
x})=x_{1}^{2}x_{2}+x_{2}^{3}+\tanh(10(\sin(5x_{2})-2x_{1})),\quad{\bf
x}\in(-1,1)\times(-1,1).$ (4.2)
This function is anisotropic along the zigzag curve $\sin(5x_{2})-2x_{1}=0$
and changes sharply in the direction normal to this curve (taken from [3]). In
the current computation, each run is stopped after 20 iterations to guarantee
that the adaptive procedure tends towards stability. We show in Figure 4 the
$L^{p}$ norms of the interpolation error and its gradient using corresponding
metric tensors, for $p=1,2,4,\infty$. As in Example 1, the convergence rates
for the interpolation error and its gradient here are always nearly optimal.
In Figure 5 we select 6 meshes with 4000 triangles generated by corresponding
metric tensors. We can learn that the optimal meshes in different norms are
different. For example, the mesh generated by the metric tensor
$\mathcal{M}_{1,\infty}$ concentrates more triangles and nodes along the
zigzag line.
Figure 5: Example 2: Meshes generated by the metric tensor $\mathcal{M}_{m,p}$
for (a)$m=0,p=1$, (b)$m=0,p=2$, (c)$m=0,p=\infty$, (d)$m=1,p=1$, (e)$m=1,p=2$,
(f)$m=1,p=\infty$.
Example 3 (Taken from [40]) This example is to solve the boundary value
problem of Poisson’s equation
$\displaystyle-\triangle u$ $\displaystyle=$ $\displaystyle f,\quad{\bf
x}\in\Omega\equiv(-1.2,1.2)\times(-1.2,1.2),$ (4.3)
with the Dirichlet boundary condition and the right-hand side term being
chosen such that the exact solution is given by
$\displaystyle u({\bf
x})=\sum_{i=1}^{5}\big{[}(1+e^{\frac{x+y-c_{i}}{2\epsilon}})^{-1}+(1+e^{\frac{x-y-
d_{i}}{2\epsilon}})^{-1}\big{]},$ (4.4)
where $c_{i}=0,-0.6,0.6,-1.2,1.2;\,d_{i}=0,-0.6,0.6,-1.2,1.2.$ The solution
exhibits ten sharp layers on lines $x+y-c_{i}=0$ and $x-y-d_{i}=0$,
$i=1,2,\cdots,5$, when $\epsilon$ is small. In our computations, $\epsilon$ is
taken as 0.01. Numerical results in [40] have shown that our approach’s
superiority for the error in $H^{1}$ norm. In the current computation, each
run is stopped after 20 iterations to guarantee that the adaptive procedure
tends towards stability, except that governed by $\mathcal{M}_{1,\infty}$,
which need 30 iterations. We show in Figure 6 the $L^{p}$ norms of the
interpolation error and its gradient using corresponding metric tensors, for
$p=1,2,4,\infty$. As in Example 1 and Example 2, the convergence rates for the
interpolation error and its gradient here are always nearly optimal. Another
purpose to select this example is to describe the difference of finding layers
using different norms. In Figure 7 we list meshes in different stage during
one selected run governed by corresponding metric tensors. While in Figure 8
convergence history is shown. From the three figures we can learn that most of
the metric tensors can quickly find the layers except the metric tensor
$\mathcal{M}_{1,\infty}$ when dealing with the complex problems, e.g., with
multiple layers.
Figure 6: Example 3: Interpolation error and its gradient in $L^{p}$ norm
Figure 7: Example 3: Meshes generated by the metric tensors
$\mathcal{M}_{1,2}$ after (a) 5 step, (b) 10 step, (c) 15 step, and
$\mathcal{M}_{1,\infty}$ after (d) 5 step, (e) 10 step, (f) 15 step, (g) 20
step.
Figure 8: Example 3: Convergence history versus number of triangles by the
metric tensor $\mathcal{M}_{0,p}$ and $\mathcal{M}_{1,p}$ for
$p=1,2,4,\infty$.
## 5 Conclusions
In the previous sections we have developed a uniform strategy to derive metric
tensors in two spatial dimension for interpolation errors and their gradients
in $L^{p}$ norm. The metric tensor $\mathcal{M}_{0,p}^{n}$ for the $L^{p}$
norm of the interpolation error is similar to some existing methods. However,
the metric tensor $\mathcal{M}_{1,p}^{n}$ is essentially different with those
metric tensors existed. There is a fine distinction between the new metric
tensor $\mathcal{M}_{1,p}^{n}$ and $\mathcal{M}_{1,p}^{h}$ proposed by Huang
and Russell[26] that the former refers to $\mbox{tr}(\mathcal{H})$ and the
latter involves $\|\mathcal{H}\|$. In some cases, the two metric tensors are
pretty much alike. However, when dealing with the complex problems, e.g., with
multiple layers, the effect of the former is superior to the latter for mesh
generation. Numerical results show that the corresponding convergent rates are
always almost optimal.
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|
arxiv-papers
| 2012-01-08T13:29:17 |
2024-09-04T02:49:26.066351
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaobo Yin, Hehu Xie",
"submitter": "Xiaobo Yin",
"url": "https://arxiv.org/abs/1201.1632"
}
|
1201.1686
|
# Reversed Drifting Quasi-periodic Pulsating Structure in an X1.3 Solar Flare
on 2005 July 30
Rui Wang Key Laboratory of Solar Activity, National Astronomical
Observatories,
Chinese Academy of Sciences, Beijing, 100012, China.
Email: Ray@nao.cas.cn Baolin Tan Key Laboratory of Solar Activity, National
Astronomical Observatories,
Chinese Academy of Sciences, Beijing, 100012, China.
Email: Ray@nao.cas.cn Chengming Tan Key Laboratory of Solar Activity, National
Astronomical Observatories,
Chinese Academy of Sciences, Beijing, 100012, China.
Email: Ray@nao.cas.cn Yihua Yan Key Laboratory of Solar Activity, National
Astronomical Observatories,
Chinese Academy of Sciences, Beijing, 100012, China.
Email: Ray@nao.cas.cn
(Received: ****; Accepted: 7 January 2012)
###### Abstract
Based on the analysis of the microwave observations at frequency of 2.60 –
3.80 GHz in a solar X1.3 flare event observed at Solar Broadband
RadioSpectrometer in Huairou (SBRS/Huairou) on 2005 July 30, an interesting
reversed drifting quasi-periodic pulsating structure (R-DPS) is confirmed. The
R-DPS is mainly composed of two drifting pulsating components: one is a
relatively slow very short-period pulsation (VSP) with period of about 130 –
170 ms, the other is a relatively fast VSP with period of about 70 – 80 ms.
The R-DPS has a weak left-handed circular polarization. Based on the synthetic
investigations of Reuven Ramaty High Energy Solar Spectroscopic Imaging
(RHESSI) hard X-ray, Geostationary Operational Environmental Satellite (GOES)
soft X-ray observation, and magnetic field extrapolation, we suggest the R-DPS
possibly reflects flaring dynamic processes of the emission source regions.
###### keywords:
Sun: quasi-periodic pulsation — Sun: microwave burst — Sun: flares
## 1 Introduction
Quasi-periodic pulsations (QPPs) associated with solar flares are observed
frequently in optical, EUV, soft X-ray, hard X-ray, and radio emissions (see
the recent review of Nakariakov and Melnikov, 2009). For pulsation events,
Aschwanden (1987, 2004) presented an extensive review about the models, and
classified them mainly into three groups: (1) magnetohydrodynamic (MHD) flux
tube oscillations (eigenmodes); (2) Cyclic self-organizing systems of plasma
instabilities; (3) Modulation of periodic electron acceleration. Based on the
radio observations and the period of pulsation (P), QPPs can be classified
into three types (Wang and Xie, 2000): (1) long period pulsation (LPP), P
$\sim$ tens of seconds; (2) short period pulsation (SPP), P $\sim$ several
seconds; and (3) very short period pulsation (VSP), P $\sim$ subseconds.
Recently, some supplements and extensions had made the classification more
comprehensive and detailed (Tan et al, 2010). The very long period pulsation
(VLP) was added, whose period is in the hectosecond or several minutes range.
Generally it is defined as P $>$ 100 s. On the other hand, the VSP was divided
into two sub-classes: slow-VSP, where the period is in the decisecond, 0.1 $<$
P $<$ 1.0 s and the other is fast-VSP, P $<$ 0.1 s. The different QPPs should
be corresponding to different generation mechanisms, and might reveal
different physical conditions in the source region. The flare-associated QPP
can provide information of solar flaring regions, and give some prospective
insight into coronal plasma dynamic processes, providing remote diagnostics of
the microphysics of energy release sites. The understanding of flaring QPP in
the solar corona will open up very interesting perspectives for the
diagnostics of stellar coronae (Mathioudakis, et al., 2003).
Usually, some QPPs frequently have another important feature: frequency-time
drift, recognized as drifting quasi-periodic pulsating structures (DPS)
(Kliem, Karlický, and Benz, 2000). The analysis of the frequency drift rate in
DPS may provide information not only about the dynamical processes of the
source region but it can also reveal atmospheric properties. Kliem, Karlický,
and Benz (2000) proposed a model in which the decimetric DPS is caused by
quasi-periodic particle acceleration episodes that result from a highly
dynamic regime of magnetic reconnection in an extended large-scale current
sheet above the soft X-ray flare loop, where reconnection is dominated by
repeated formation and subsequent coalescence of magnetic islands, known as
secondary tearing modes. With this model, they explained the global frequency
drifting pulsating structure as a motion of the plasmoid in the solar
atmosphere with density gradient. Here, particles are accelerated near the
magnetic X-points in the DC electric field associated with magnetic
reconnection. The strongest electric fields occur at the main magnetic
X-points adjacent to the plasmoid, and a large fraction of the accelerated
particles may be temporarily trapped in the plasmoid; the accelerated process
itself may form an anisotropic velocity distribution, which excites the
observed radio emission. In fact, there are a series of works to explain DPSs
as the radio emission being generated during multi-scale tearing and
coalescence processes in the extended current sheet of a flare (Karlicky,
2004; Karlicky et al., 2005). Based on particle-in-cell simulation, Karlicky
and Barta (2007) found that electrons are accelerated most efficiently around
the X-point of the magnetic configuration at the end of the tearing process
and the beginning of plasmoid coalescence. The most energetic electrons are
mainly localized along the X-lines of the magnetic configuration.
However, so far, from the observations, we only obtained DPS with single-
directional frequency drifting rate, i.e., drift from high frequency to lower
frequency, or from low frequency to higher frequency in the single DPS event.
A DPS with double-directional frequency drifting rate, i.e., the emission
drifts from higher frequency to lower and then reversed, namely from lower
frequency to higher may be called as reversed drifting quasi-periodic
pulsating structure (R-DPS). By scrutinizing the microwave observation data
obtained in Chinese Solar Broadband RadioSpectrometer (SBRS/Huairou), we find
a particular example of R-DPS in the flare on 2005 July 30.
This paper is arranged as follows. Section 2 presents the observational data
and the data analysis. Section 3 gives some discussions on physical processes
related to the R-DPS. Finally, Section 4 draws our conclusions.
## 2 Observations and Data Analysis
### 2.1 Observations
On 2005 July 30 an X1.3 flare/CME event occurred from 06:10 UT to 07:00 UT,
with the peak at 06:35 UT in AR 10792 at N11∘, E52∘, near the east edge of the
solar disk. During this flare event, several solar telescopes got the perfect
observational data, such as the solar microwave (SBRS/Huairou), Reuven Ramaty
High Energy Solar Spectroscopic Imaging (RHESSI) hard X-ray, Geostationary
Operational Environmental Satellite (GOES) soft X-ray, optical Michelson
Doppler Imager on Solar and Heliospheric Observatory (MDI/SOHO), and Big Bear
Solar Observatory (BBSO), etc. In this work, our focus is on the microwave
observations. We mainly use the observation of SBRS/Huairou to investigate the
properties of QPP. SBRS/Huairou includes three parts: 1.10 – 2.06 GHz, 2.60 –
3.80 GHZ and 5.20 – 7.60 GHz (Fu et al., 1995; Fu et al., 2004; Yan et al.,
2002). R-DPS appeared in the frequency band of 2.60 – 3.80 GHz and the time
range from 06:24:15 – 06:24:21 UT, and the duration lasts for about 6 s
(Figure 1).
Figure 1: The spectrum of the solar microwave drifting quasi-periodic
pulsating structure at 06:24:12 UT – 06:24:24 UT, on 2005 July 30, observed at
SBRS/Huairou with the spectrometer of 2.60 – 3.80 GHZ. Two yellow arrows
indicate frequency drifting directions.
The antenna diameter of the SBRS/Huairou at frequency of 2.60 – 3.80 GHz is
3.2 m. It is controlled by a computer to automatically trace the solar disk
center and can receive the total flux of solar radio emission with dual
circular polarization. The dynamic range of this instrument is 10dB above
quiet solar background emission and the observation sensitivity is $\triangle
S/S_{\odot}\leq 2\%$, where S⊙ is the quiet solar background emission. The
data processing used the software in IDL language and data calibration
followed the method proposed by Tanaka et al. (1973). The standard flux values
of the quiet Sun are adopted from the data published by the Solar Geophysical
Data (SGD). For strong bursts, the receiver may work beyond its linear range
and a nonlinear calibration method will be used instead (Yan et al., 2002).
In order to make our data more convincible, the other instruments were also
utilized to support the radio emission data. The soft X-ray data from GOES was
used to make a comparison. Also hard X-ray observations with different energy
ranges from RHESSI were adopted. In addition, the photospheric magnetograph of
the line-of-sight magnetic component obtained from MDI/SOHO was adopted to
extrapolate and model the coronal magnetic field.
### 2.2 Data Analysis
Figure 1 presents the QPP event, which occurred at 06:24:15 – 06:24:21 UT, on
2005 July 30 in the frequency range of 2.60 – 3.50 GHz. The upper and lower
panels give the left- and right-handed circular polarization, respectively.
From this figure, we can see that the QPP had a negative frequency drift rate
(drift from high frequency to the lower frequency) during 06:24:15 – 06:24:18
UT (named left wing, hereafter), and then the frequency drift rate became
positive (drift from low frequency to the higher frequency) during 06:24:18 –
06:24:21 UT (right wing), with the inflexion occurring around 06:24:18 UT. The
two yellow arrows indicate the frequency drifting directions. With a linear
fit we find that the frequency drift rates at each wing of the QPP are -285
MHz ${s}^{-1}$ and 186 MHz ${s}^{-1}$, respectively.
In order to make sure that the QPP signals originate from the flare bursts and
they are not simply noise, Figure 2 presents three profiles at the frequencies
of 2.80, 3.00, and 3.20 GHz, respectively. From the SGD database we may
extrapolate that the radio mean flux at frequency 2.60 – 3.80 GHz of the quiet
Sun on 2005 July 30 is about 100 – 135 sfu. So the instrument sensitivity is
about $\triangle S/S_{\odot}\leq 2\%\simeq 2-2.7$ sfu. Figure 2 indicates that
there are enhancements of more than 15 sfu in the left and right wings of the
QPP with respect to the background emission. Moreover the enhancements around
the QPP exceed the instrument sensitivity greatly, so we may confirm that the
QPP is real, this dynamic spectrum is clear and reliable.
Figure 2: The profiles of radio emission at frequencies 2.80 GHz, 3.00 GHz and
3.20 GHz, respectively. The dashed lines mark the positions of the maximum
flux intensity at 2.80 GHz. The relative positions to the dashed lines of the
maximum flux intensity at 3.00 GHz and 3.20 GHz reflect the frequency drift
rates at the left and right part of the R-DPS.
Figure 1 shows that the patterns or intensities of the QPP are almost the same
in the left-handed or right-handed polarization spectrogram, indicating that
the polarization of the QPP is not obvious. Calculation indicates that the
total polarization degree ($(R-L)/(R+L)$) is around -0.04%, the polarization
degree of left wing is around -2.33% and the right wing is around -3.22%,
where R and L are the intensities of the right- and left- handed circular
polarization emission which subtract the background components, respectively.
From the bright lines of the left and right wings of the structure in Figure
1, we find that it is quasi-periodic, maybe it is hybrid of more periodic
components than one. The best way to analyze this kind of structures is by
using wavelet analysis, which can get information on both the amplitude of any
periodic component within the series, and the temporal evolution of the QPP.
Figure 3: The bottom two panels show the wavelet power spectrum at the left
and right part of R-DPS, using the Morlet wavelet. The black contours are the
95% confidence regions and anything ”below” this line is dubious. The region
below the parabolic curve indicates the ”cone of influence”, where edges
influence is important. The A1 $\scriptsize{\sim}$ 80 ms, B1
$\scriptsize{\sim}$ 170 ms, A2 $\scriptsize{\sim}$ 70 ms, B2
$\scriptsize{\sim}$ 130 ms. The top two panels give corresponding radio fluxes
for time comparison.
Figure 3 presents the wavelet spectrum at frequency of 3.00 GHz during
06:24:14 UT to 06:24:22 UT which just contains the time interval of the QPP.
The black contours plot the confident region with 95% confidence level. In the
left part of the figure, there are two obvious spectrum peaks corresponding to
the left wing of the QPP in the confident region, the periods are about 80 ms
(marked as A1) and 170 ms (marked as B1), respectively. This implies that
there are two pulsating components overlapped around 06:24:16 UT. On the right
wing, the analogous structures appear between 06:24:20 UT and 06:24:21 UT and
the periods are about 70 ms (marked as A2) and 130 ms (marked as B2),
respectively, which are slightly shorter than that in the left wing. Both of
them are VSPs.
According to the analysis above, we could find some significant relations
between the left wing and right wings of this QPP. Firstly, the gap between
the two parts is only about 1 second, which is much shorter than the duration
of each part in the QPP; secondly, the periods are very close in each part (80
ms at A1 to 70 ms at A2, 170 ms at B1 to 130 ms at B2, respectively) of the
emission frequency band; thirdly, both of the degrees of polarization at each
part of the QPP are not obvious. Therefore, the name R-DPS should be more
appropriate to describe such kind of structures.
Figure 4: The relative position of the reversed drifting quasi-periodic
pulsation (R-DPS) in the profile of the X1.3 flare event on July 30, 2005. The
thick curve presents the profile of solar radio emission observed at frequency
2.80 GHz with SBRS/Huairou spectrometer. The thin curve and dot-dashed curve
are the RHESSI hard X-ray curves in 25.0 – 50.0 kev and 50.0 – 800.0 kev,
respectively. The dotted line shows soft X-rays from GOES satellite. The flare
peak of this radio emission occurred at 06:35 UT.
Figure 4 presents context data to the microwave at 2.80 GHz (which has a
similar overall profile with that in the other frequencies), GOES soft X-rays,
and RHESSI hard X-rays associated with the X1.3 flare/CME event from 06:10 UT
to 07:00 UT. Here, the R-DPS is marked with a black arrow, which indicates
that the R-DPS occurred in the flare ascending phase, just after the onset of
the flare. It is associated with a time just when the gradient of the soft
X-ray reaches to its maximum. RHESSI also observed this flare event, however,
we have obtained RHESSI hard X-ray data only in the time interval from 06:28
UT to 06:36 UT, and no valid data during the R-DPS. Anyway, we still present
the hard X-ray emission curves in 25 – 50 keV and 50 – 800 keV as reference.
The magnetic field configuration has importance for understanding the physical
processes of the QPP. Figure 5 gives the magnetic topology of the flare active
region AR 10792 obtained from potential extrapolation computed from the
observed line-of-sight magnetic field using a Green’s function. The initial
version of this technique is implemented by T. Metcalf and G. Barnes on 2005
October 25 and this program can be found in the SolarSoftWare (SSW). The
background of the magnetic extrapolation in the bottom panel is the line-of-
sight magnetogram observed by MDI/SOHO. Red lines present closed magnetic
field lines, while blue lines are open field lines. Through the magnetic
model, we can make rough scale estimations of the coronal loops. However, as
there is no microwave imaging observation at the corresponding frequency, we
do not know which loop is associated with the R-DPS exactly. For comparison,
we present the H$\alpha$ image in the same area (the upper panel in Figure 5).
By reason of lacking data in the same time range, here we just got a image of
AR 10792 at 06:35:49 UT on 2005 July 30, observed by BBSO, while it is still
valuable for comparison with the extrapolated model. It is obvious that there
is a two-ribbon flare in this image. This structure often indicates that
magnetic reconnection has allowed the coronal magnetic field to relax into a
lower energy state. Practically, it is natural to assume that only the coronal
loops which are adjacent to the flare ribbons are related to the microwave
bursts. From Figure 5 we may obtain the lengths of these coronal loops are
about 2” – 100”. Suppose the coronal loops are semicircles, then the lengths
of the coronal loops are about $2.3\times 10^{3}$ – $1.14\times 10^{5}$ km.
Figure 5: Potential extrapolation of magnetic field lines in the lower panel
using the observed line-of-sight field of the MDI/SOHO during 06:24:12 –
06:24:24 UT, on July 30, 2005. Coordinates are in arcseconds. The coordinate
of the center of the Sun is (0,0). The image in the upper panel is from the
H$\alpha$ data of the Big Bear Solar Observatory (BBSO) at 06:35:49 UT. There
is an obvious two-ribbon flare in this image.
## 3 Discussion on the possible process of R-DPS
According to the work of Tan et al. (2007), Tan (2008) and Tan et al. (2010),
VSP can be explained as a result of modulations of the resistive tearing-mode
oscillation in some electric current-carrying flare loops. The pulsating
emission is possibly plasma emission. As we know that the plasma emission is
always generated at the plasma fundamental frequency ($\omega_{pe}$) or at the
second harmonic frequency ($\sim 2\omega_{pe}$). The degree of polarization of
fundamental plasma emission is very strong and usually in the sense of O-mode,
while the second harmonic plasma emission is always a weak circular
polarization. As the R-DPS is weakly polarized, we may suppose that it is
possibly related to the second harmonic plasma emission. The central frequency
of the R-DPS is about 3.00 GHz, and implies that the plasma density is about
$2.78\times 10^{10}$cm-3. Plasma with such high density is probably very close
to the flare core.
Based on the plasma emission mechanism, we have the emission frequency:
$f=sf_{pe}\simeq 9sn_{e}^{1/2}$, we may obtain the frequency drift rate as:
$\frac{df}{dt}\simeq\frac{f}{2H}v$ (1)
Here, $H=\mid n_{e}/\frac{dn_{e}}{dr}\mid$ is the inhomogeneous scale length
of the plasma in the source, $v=\frac{dr}{dt}$ is the moving velocity of the
emission source region. Then we may get the moving velocity:
$v=\frac{2H}{f}\frac{df}{dt}=2H\varepsilon$,
$\varepsilon=\frac{1}{f}\frac{df}{dt}$ is the relative frequency drift rate.
From here we know that the moving velocity is only proportional to the
relative frequency drift rate. Usually, the inhomogeneous scale length $H$
should be induced from the solar active region atmospheric model. For
simplicity, we may assume that $H\sim 10^{4}$ km. Then we may estimate that
the source moving velocity associated with the left wing of the R-DPS is about
1900 km s-1 , and 1240 km s-1 in opposite direction with the right wing. This
may imply that the R-DPS reflects a following process: during the rising phase
of the X1.3 flare, the closed flaring coronal loop has an upthrust in velocity
of 1900 km s-1, and then falls down slowly in velocity of 1240 km s-1.
To the explanation of the loop upward and downward movings, a two-dimensional
(2D) resistive-MHD numerical simulation of the reconnection starting from the
Harris-type current sheet has been done (Bárta, Všnak, and Karlický, 2008).
The result of simulation indicated that if the reconnection rate v$\times$B at
the X-point below the plasmoid is higher than the one at the X-point above the
plasmoid, the plasmoid moves upward since the net tension causes an upward
electron acceleration and then excites the plasma emission in the upper source
region. If the magnetic flux is reconnected in the upper diffusion region is
higher than in the lower one, the plasmoid moves downward and the high energy
electron flow excites plasma emission from the lower source region. However as
it is stated above, if the source regions are located at different altitudes,
the density of the source regions would be very different, depending on the
altitudes, and then different waveband signals from the right and left wing of
the R-DPS would be received. This does not agree with our observation, that
the frequency range of the R-DPS event from 2.60 GHz to 3.80 GHz. Therefore we
take another way to interpret our observations.
If the emission source region could be located within a loop with up-and-down
motions, it would be more consistent with the observation. We assume that the
up and down motions corresponded to the expansion and shrinkage of the loops.
These processes should have a relation with intense energy injection (Li and
Gan, 2005). During the shrinkage of the loops, there were few intense energy
injections, since the chromospheric evaporation needed several minutes to fill
the loops, and during this time the density of loops was rather low while it
was opposite around the loops. Afterward, the injection process completed, the
density of the region above the loop top was lower corresponding to the loop
system, then the loop began to expand.
We may assume that the flaring loop is current-carrying plasma loop, having an
up-and-down motion, a resistive tearing-mode instability will be triggered in
the flaring loop and a series of multi-scale magnetic islands would form.
Electron acceleration will occur at X-points between every two adjacent
magnetic islands. Then the energetic electrons will excite some Langmuir
turbulence in the flare plasma loop and make the plasma emission enhanced.
Modulated by the resistive tearing-mode oscillation, the emission will behave
as pulsating structure in the spectrogram.
At the same time, we know there are two pulsating components in both the left
and right wings of the R-DPS, and this may indicate that there are two
different flaring plasma loops in the same oscillating source region. The
difference may be in loop radius, or electric current, etc. (Tan, 2008).
However, as we have no corresponding imaging observations, we could not
confirm which factor is the real candidate.
## 4 Conclusions
In this work we present detailed observations of a particular reversed
drifting quasi-periodic pulsation (R-DPS) associated with the rising phase of
an X1.3 flare event. From the above data analysis and discussions, we may
reach the following conclusions:
(1) It is observationally confirmed that the theoretically predicted reversed
direction frequency drift structures in microwave emission indeed exist.
(2) The R-DPS is mainly composed of two pulsating components: one is a slow-
VSP with period of about 130 – 170 ms, the other is a fast-VSP with period of
about 70 – 80 ms.
(3) The frequency drift rate in the left wing of the R-DPS is about -285 MHz
${s}^{-1}$, and in the right wing about 186 MHz ${s}^{-1}$.
(4) The polarization of the R-DPS is a weak left-handed circular polarization.
Based on the assumption of plasma emission mechanism that the tearing mode
oscillation modulates the plasma emission in current-carrying plasma loops,
the R-DPS may reflect the dynamic processes of the emission source regions.
From the frequency drift rates we make an estimation of the source up-and-down
motion velocity being about 1900 km s-1 up and then 1240 km s-1 down. The
variations of the plasma density in the loop with respect to the background
during the up-and-down motion result the reversed drifting quasi-periodic
pulsations. In order to confirm this deduction, some microwave imaging
observations at the corresponding frequency is necessary. The constructing
Chinese Spectral Radioheliograph (0.4 – 15 GHz) will satisfy this need (Yan et
al, 2009).
###### Acknowledgements.
The authors would like to thank the referee for the helpful and valuable
comments on this paper. We would also thank the the GOES, RHESSI, MDI/SOHO,
BBSO and SBRS/Huairou teams for providing observation data. This work was
supported by NSFC Grant No. 10873021, 10921303, 10903013, 11103044, 11103039,
MOST Grant No. 2011CB811401, and the National Major Scientific Equipment R&D
Project ZDYZ2009-3.
## References
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* [Aschwanden(2004)] Aschwanden, M.J.: 2004, _Physics of the Solar Corona: An introduction, Springer, Berlin._
* [Barta(2008)] Bárta, M., Vr$\breve{s}$nak, B., Karlický, M.: 2008, _Astron. Astrophys._ 477, 649.
* [Fu et al(1995)] Fu, Q.J., Qin, Z.H., Ji, H.R., Pei, L.B.: 1995, _Solar Phys._ 160, 97.
* [Fu et al(2004)] Fu, Q.J., Ji, H.R., Qin, Z.H., Xu, J.C., Xia, Z.G., Wu, H.A., Liu, Y.Y., Yan, Y.H., Huang, G.L., Chen, Z.J., Jin, Z.Y., Yao, Q.J., Cheng, C.L., Xu, F.Y., Wang, M., Pei, L.B., Chen, S.H., Yang, G., Tan, C.M., Shi, S.B.: 2004, _Solar Phys._ 222, 167.
* [Li et al(1995)] Li, Y.P., Gan, W.Q.: 2005, _Astrophys. J._ 629, 137.
* [Karlický(2004)] Karlický, M.: 2004, _Astron. Astrophys._ 417, 325.
* [Karlický & Bárta(2007)] Karlický, M., Bárta, M.: 2007, _Astron. Astrophys._ 464, 735.
* [Karlický et al(2005)] Karlický, M., Bárta, M., Mészárosová, H., Zlobec, P.: 2005, _Astron. Astrophys._ 432, 705.
* [Kliem et al(2000)] Kliem, B., Karlický, M., Benz, A.O.: 2000, _Astron. Astrophys._ 360, 715.
* [Mathioudakis et al(2003)] Mathioudakis, M. Seiradakis, J.H., Williams, D.R., Avgoloupis, S., Bloomfield, D.S., McAteer, R.T.J.: 2003, _Astron. Astrophys._ 403, 1101.
* [Nakariakov et al(2009)] Nakariakov, V.M., Melnikov, V.F.: 2009, _Space Sci. Rev._ 149, 119.
* [Tan(2008)] Tan, B.L.: 2008, _Solar Phys._ 253, 117.
* [Tan et al(2007)] Tan, B.L., Yan, Y.H., Tan, C.M., Liu, Y.Y.: 2007, _Astrophys. J._ 671, 964.
* [Tan et al(2010)] Tan, B.L., Zhang, Y., Tan, C.M., Liu, Y.Y.: 2010, _Astrophys. J._ 723, 25.
* [Tanaka et al(1973)] Tanaka, H., Castelli, J.P., Covington, A.E., Krüger, A., Landecker, T.L., Tlamcha, A.: 1973, _Solar Phys._ 29, 243.
* [Wang(2000)] Wang, M., Xie, R.X.: 2000, _China. J. Astron. Astrophys._ 24, 95.
* [Yan et al(2002)] Yan, Y.H., Tan, C.M., Xu, L., Ji, H.R., Fu, Q.J., Song, G.X.: 2002, _Science in China Ser. A_ 45, 89.
* [Yan et al(2009)] Yan, Y.H., Zhang, J., Wang, W., Liu, F., Chen, Z.J., Ji, G.S: 2009, _Earth Moon Planet_ 104, 97.
|
arxiv-papers
| 2012-01-09T04:31:37 |
2024-09-04T02:49:26.074578
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rui Wang, Baolin Tan, Chengming Tan and Yihua Yan",
"submitter": "Wang Rui",
"url": "https://arxiv.org/abs/1201.1686"
}
|
1201.1749
|
# Operator Covariant Transform and Local Principle
Vladimir V. Kisil School of Mathematics, University of Leeds, Leeds LS2 9JT,
UK
On leave from Odessa University kisilv@maths.leeds.ac.uk
(Date: 27th August 2024)
###### Abstract.
We describe connections between the localization technique introduced by I.B.
Simonenko and operator covariant transform produced by nilpotent Lie groups.
###### Contents
1. 1 Introduction
2. 2 Preliminaries
1. 2.1 Classic Localization Technique
2. 2.2 Covariant Transform
3. 2.3 Inverse Covariant Transform
3. 3 Semidirect Products and Localization
4. 4 Localization and Invariance
5. 5 Closing Remarks
## 1\. Introduction
In 1965 I.B. Simonenko pioneered [Simonenko65a, Simonenko65b] localization
technique in the theory of operators. It still remains an important tool in
this area, see for example [Vasilevski08a, BoetcherKarlovichSpitkovsky02a,
DuduchavaSaginashviliShargorodsky97a, KarlovichSpitkovsky95a,
RabinovichSamko11a, KarlovichSilbermann04a]. Many questions addressed by this
technique, e.g. boundary value problems, are rooted in mathematical physics.
We also discuss connections with quantum mechanics in the closing section of
this paper.
The localisation method was developed in various directions and there is no
possibility to mention all works based on numerous existing variants and
modifications of the localization technique. Several generalizations, e.g.
within $C^{*}$-algebras setup [Douglas72]*Prop. 4.5, capture the abstract
skeleton of the localization technique. However the idea of “localization” has
an explicit geometrical meaning, which often escapes those general schemes.
We present here a different point of view on the original works of Simonenko,
which highlights the rôle of groups in the constructions. Thus it is not a
generalization but rather an attempt to link certain geometrical meaning of
locality with homogeneous structure of nilpotent Lie groups. This paper grown
up from our earlier works [Kisil94a, Kisil96e, Kisil92, Kisil93e, Kisil93b,
Kisil94f, Kisil98a] revised in the light of recent research [Kisil09d,
Kisil10c].
The paper outline is as follows: Section 2 collects preliminary information
from other works, which will be used here. In Section 3 we use homogeneous
structure of nilpotent Lie groups to define basic elements of localization.
Operators which are invariant under certain group action are main building
blocks for localization, we demonstrate this in Section 4. The final Section 5
offers summary of our observations which lead to new directions for further
research.
## 2\. Preliminaries
### 2.1. Classic Localization Technique
We present here the fundamental definitions from the work of I.B. Simonenko
[Simonenko65a, Simonenko65b] formulated for operators on
$L_{p}{}(\mathbb{R}^{n}{})$. Essential norm of an operator is defined by
$\left|\left|\left|A\right|\right|\right|=\inf_{K}\left\|A-K\right\|,$
where the infimum is taken over all compact operators $K$. For a measurable
set $F\subset\mathbb{R}^{n}{}$ we define the projection operator
$P_{F}:L_{p}{}(\mathbb{R}^{n}{})\rightarrow L_{p}{}(\mathbb{R}^{n}{})$ by:
(1) $[P_{F}f](x)=\left\\{\begin{array}[]{ll}f(x),&\text{ if }x\in F;\\\
0,&\text{ otherwise}.\end{array}\right.$
The operators, most suitable for the localization method, are defined as
follows.
###### Definition 1.
[Simonenko65a]*§ I.1 An operator $A$ is of _local type_ if for any two closed
disjoint sets $F_{1}$ and $F_{2}$ the operator $P_{F_{1}}AP_{F_{2}}$ is
compact.
The cornerstone definition for the whole theory is
###### Definition 2.
[Simonenko65a]*§ I.2 Operators $A$, $B:L_{p}{}(\mathbb{R}^{n}{})\rightarrow
L_{p}{}(\mathbb{R}^{n}{})$ are called _equivalent_ at a point $x_{0}$ if for
any $\varepsilon>0$ there is a neighborhood $u$ of $x_{0}$ such that
$\left|\left|\left|AP_{u}-BP_{u}\right|\right|\right|<\varepsilon$ and
$\left|\left|\left|P_{u}A-P_{u}B\right|\right|\right|<\varepsilon$. This is
denoted $A\stackrel{{\scriptstyle x_{0}}}{{\sim}}B$.
As usual there are two stages in this method: analysis and synthesis. Local
equivalence decomposes operators into families of local representatives. Now
we define the opposite process of a reconstruction.
###### Definition 3.
[Simonenko65a]*§ I.5 Let $A_{x}$ be a family of operators
$L_{p}{}(X)\rightarrow L_{p}{}(X)$ depending from $x\in X$. An operator
$A:L_{p}{}(X)\rightarrow L_{p}{}(X)$ is an _envelope_ of $A_{x}$ if for every
$x$ we have $A\stackrel{{\scriptstyle x}}{{\sim}}A_{x}$.
An envelope can be build [Simonenko65a]*§ I.5 as the limit $A$ of a sequence
$A_{n}$ which is defined by the expression:
(2) $A_{n}=\sum_{j=1}^{n}P_{u_{j}}A_{x_{j}}P_{u_{j}},$
where sets $u_{n}$ make a decomposition of $X$ and $x_{n}\in u_{n}$.
### 2.2. Covariant Transform
The following concept is a natural development of the coherent states
(wavelets) based on group representations.
###### Definition 4.
[Kisil09d, Kisil10c] Let ${\rho}$ be a representation of a group $G$ in a
space $V$ and $F$ be an operator from $V$ to a space $U$. We define a
_covariant transform_ $\mathcal{W}$ from $V$ to the space $L{}(G,U)$ of
$U$-valued functions on $G$ by the formula:
(3) $\mathcal{W}:v\mapsto\hat{v}(g)=F({\rho}(g^{-1})v),\qquad v\in V,\ g\in
G.$
Operator $F$ will be called _fiducial operator_ in this context.
We borrow the name for operator $F$ from fiducial vectors of Klauder and
Skagerstam [KlaSkag85]. The wavelet transform, which is a particular case of
the covariant transform, corresponds to the fiducial operator which is a
linear functional. Thus its image consists scalar-valued functions. It seems
to be most favorable situation, cf. [Kisil10c]*Rem. 3, and was believed to be
the only possible one for a long time. A moral of the present work is that the
covariant transform can be useful even in the other extreme limit: if the
range of the fiducial operator is the entire space $V$.
By the way, we do not require that the fiducial operator $F$ shall be linear
in general, however it will be always linear in the present work. Sometimes
the positive homogeneity, i.e. $F(tv)=tF(v)$ for $t>0$, alone can be already
sufficient, see [Kisil10c, Kisil11c].
The following property is inherited by the coherent transform from the wavelet
one.
###### Theorem 5.
[Kisil09d, Kisil10c] The covariant transform (3) intertwines ${\rho}$ and the
left regular representation $\Lambda$ on $L{}(G,U)$:
$\mathcal{W}{\rho}(g)=\Lambda(g)\mathcal{W}.$
Here $\Lambda$ is defined as usual by:
(4) $\Lambda(g):f(h)\mapsto f(g^{-1}h).$
The next result follows immediately.
###### Corollary 6.
The image space $\mathcal{W}(V)$ is invariant under the left shifts on $G$.
### 2.3. Inverse Covariant Transform
An object invariant under the left action $\Lambda$ (4) is called _left
invariant_. For example, let $L$ and $L^{\prime}$ be two left invariant spaces
of functions on $G$. We say that a pairing
$\left\langle\cdot,\cdot\right\rangle:L\times
L^{\prime}\rightarrow\mathbb{C}{}$ is _left invariant_ if
(5) $\left\langle\Lambda(g)f,\Lambda(g)f^{\prime}\right\rangle=\left\langle
f,f^{\prime}\right\rangle,\quad\textrm{ for all }\quad f\in L,\ f^{\prime}\in
L^{\prime}.$
###### Remark 7.
1. (1)
We do not require the pairing to be linear in general.
2. (2)
If the pairing is invariant on space $L\times L^{\prime}$ it is not
necessarily invariant (or even defined) on the whole $C{}(G)\times C{}(G)$.
3. (3)
An invariant pairing on $G$ can be obtained from an invariant functional $l$
by the formula $\left\langle f_{1},f_{2}\right\rangle=l(f_{1}\bar{f}_{2})$.
Such a functional are often associated to the (quasi-) invariant measures.
###### Example 8.
Let $G$ be the $ax+b$ group, cf. Ex. 12 below. There are essentially two non-
trivial invariant pairings for it. The first one is based on the left Haar
measure $\frac{da\,db}{a^{2}}$ and integration over the entire group:
(6) $\left\langle
f_{1},f_{2}\right\rangle=\int\limits_{-\infty}^{\infty}\int\limits_{0}^{\infty}f_{1}(a,b)\,\bar{f}_{2}(a,b)\,\frac{da\,db}{a^{2}}.$
Another invariant pairing on $G$, which is not generated by the Haar measure,
is:
(7) $\left\langle f_{1},f_{2}\right\rangle=\lim_{a\rightarrow
0}\int\limits_{-\infty}^{\infty}f_{1}(a,b)\,\bar{f}_{2}(a,b)\,db.$
This pairing participates in the definition of the inner product on the Hardy
space, thus we call it _Hardy-type pairing_ [Kisil10c].
For a representation ${\rho}$ of $G$ in $V$ and $v_{0}\in V$ we fix a function
$w(g)={\rho}(g)v_{0}$. We assume that the pairing can be extended in its
second component to this $V$-valued functions, say, in the weak sense.
###### Definition 9.
Let $\left\langle\cdot,\cdot\right\rangle$ be a left invariant pairing on
$L\times L^{\prime}$ as above, let ${\rho}$ be a representation of $G$ in a
space $V$, we define the function $w(g)={\rho}(g)v_{0}$ for $v_{0}\in V$. The
_inverse covariant transform_ $\mathcal{M}$ is a map $L\rightarrow V$ defined
by the pairing:
(8) $\mathcal{M}:f\mapsto\left\langle f,w\right\rangle,\qquad\text{ where
}f\in L.$
There is an easy consequence of this definition.
###### Proposition 10.
The inverse wavelet transform intertwines the left regular representation and
${\rho}(g)$.
## 3\. Semidirect Products and Localization
Let $G$ be an $m$-dimensional exponential nilpotent Lie group of the length
$k$. That means that
* •
we can identify $G$ with its Lie algebra $\mathfrak{g}\sim\mathbb{R}^{m}{}$
through the exponential map;
* •
there is a linear space decomposition
(9) $\mathfrak{g}=\oplus_{j=1}^{k}V_{j},\qquad\text{ such that
}\quad[V_{i},V_{j}]\in V_{i+j},$
where $[V_{i},V_{j}]$ denotes the space of all commutators $[x,y]=xy-yx$ with
$x\in V_{i}$, $y\in V_{j}$ and $V_{l}=\\{0\\}$ for all $l>k$.
###### Example 11.
Here are two most fundamental examples.
1. (1)
The group of Euclidean shifts in $\mathbb{R}^{n}{}$—a nilpotent group of the
length $1$.
2. (2)
The Heisenberg group $\mathbb{H}^{n}{}$ [Folland89, Howe80b]—a nilpotent group
of dimensionality $m=2n+1$ and the length $2$. Its element is $(s,x,y)$, where
$x$, $y\in\mathbb{R}^{n}{}$ and $s\in\mathbb{R}{}$. The group law on
$\mathbb{H}^{n}{}$ is given as follows:
(10)
$\textstyle(s,x,y)\cdot(s^{\prime},x^{\prime},y^{\prime})=(s+s^{\prime}+\frac{1}{2}(xy^{\prime}-x^{\prime}y),x+x^{\prime},y+y^{\prime}).$
For a generic group $G$ described above there is a one-parameter group of
automorphisms of $\mathfrak{g}$ defined in terms of decomposition (9):
$\tau_{t}(v_{j})=t^{j}v_{j},\qquad\text{ for }\quad v_{j}\in V_{j},\
t\in\mathbb{R}_{+}{}.$
The exponential map sends $\tau_{t}$ to automorphisms of the group $G$ by the
Baker–Campbell–Hausdorff formula. Thus we consider the semidirect product
$\bar{G}=G\rtimes\mathbb{R}_{+}{}$ of the group $G$ and positive reals with
the group law:
$(t,g)\cdot(t^{\prime},g^{\prime})=(tt^{\prime},g\cdot\tau_{t}(g^{\prime})),\qquad\text{
where }t,t^{\prime}\in\mathbb{R}_{+}{},\ g,g^{\prime}\in G.$
The unit in $\bar{G}$ is $(1,e)$ and
$(t,g)^{-1}=(t^{-1},\tau_{t^{-1}}(g^{-1}))$.
###### Example 12.
Returning to groups introduced in Example 11:
1. (1)
If $G$ is the group of shifts on the real line $\mathbb{R}{}$ then the above
semidirect $\bar{G}$ product is the $ax+b$ group (or _affine_ group). The
group $\bar{G}$ is isomorphic to $\mathbb{R}_{+}{}\times\mathbb{R}{}$ with the
group law:
$(a,b)\cdot(a^{\prime},b^{\prime})=(aa^{\prime},ab^{\prime}+b),\quad\text{
where }a,a^{\prime}\in\mathbb{R}_{+}{},\ b,b^{\prime}\in\mathbb{R}{}.$
2. (2)
For the Heisenberg group $\mathbb{H}^{n}{}$ the above automorphisms is
$\tau_{t}(s,x,y)=(t^{2}s,tx,ty)$ [Dynin75], thus the respective group law on
${\bar{\mathbb{H}}}^{n}$ is:
(11)
$(t,s,x,y)\cdot(t^{\prime},s^{\prime},x^{\prime},y^{\prime})=(tt^{\prime},s+t^{2}s^{\prime}+\frac{t}{2}(xy^{\prime}-x^{\prime}y),x+tx^{\prime},y+ty^{\prime}).$
There is a linear action of $\bar{G}$ on functions over $G$ cooked by the
“$ax+b$-recipe”:
(12) $[{\rho}(t,g)f](g^{\prime})=t^{\frac{k}{p}}\,f(\tau_{t^{-1}}(g^{-1}\cdot
g^{\prime})),$
where $k=\sum_{j}j\cdot\dim V_{j}$. This action is an isometry of
$L_{p}{}(G)=L_{p}{}(G,d\mu)$, where $d\mu$ is the Haar measure on $G$ (recall
that it is unimodular as a nilpotent one). Then we can define the respective
representation ${\rho_{d}}$ of $\bar{G}\times\bar{G}$ on operators [Kisil98a,
Kisil10c, Kisil11c]:
(13)
${\rho_{d}}(t,g;t^{\prime},g^{\prime}):A\mapsto{\rho}(t^{-1},\tau_{t^{-1}}(g^{-1}))A{\rho}(t^{\prime},g^{\prime}),$
for a linear operator $A:L_{p}{}(G)\rightarrow L_{p}{}(G)$.
Let $F_{e}\subset G$ be a bounded closed subset, which contains a
neighbourhood of the unit $e\in G$. We will denote by $F_{(t,g)}=(t,g)\cdot
F_{e}$ for $(t,g)\in\bar{G}$, its image under the left action of $\bar{G}$ on
$G$. Define the associated projection $P_{e}=P_{F_{e}}$ by (1). It is a
straightforward verification that
(14) ${\rho_{d}}(t,g;t,g)P_{e}=P_{F_{(t,g)}},\qquad\text{ where
}F_{(t,g)}=(t,g)\cdot F_{e}.$
We shall use a simpler notation $P_{(t,g)}=P_{F_{(t,g)}}$ again. The exact
form of $F_{e}$ is not crucial for the following construction, but the
following property simplifies technical issues:
###### Definition 13.
We say that $F_{e}$ is _$r$ -self-covering_ if for any two intersecting sets
$F_{(1,g_{1})}$ and $F_{(1,g_{2})}$ there is such $g\in G$ that $F_{(r,g)}$
covers the union of $F_{(1,g_{1})}$ and $F_{(1,g_{2})}$ .
For example, the closed unit ball in $\mathbb{R}^{n}{}$ is $2$-self-covering
with no other $F_{e}$ having a smaller value of $r$ for the self-covering
property.
For a Banach space $V$, we denote by $B(V)$ the collection of all bounded
linear operators $V\rightarrow V$.
###### Definition 14.
We select a fiducial operator $F:B(L_{p}{}(G))\rightarrow B(L_{p}{}(G))$ by
the identity
(15) $F(A)=P_{e}AP_{e},\qquad\text{ where }A\in B(L_{p}{}(G)).$
Then _Simonenko presymbol_ $\hat{S}_{A}(t,g;t^{\prime},g^{\prime})$ of an
operator $A$ is the covariant transform (3) generated by the representation
${\rho_{d}}$ (13) and the fiducial operator $F$ (15):
$\displaystyle\hat{S}_{A}(t,g;t^{\prime},g^{\prime})$ $\displaystyle=$
$\displaystyle F({\rho_{d}}(t,g;t^{\prime},g^{\prime})A)$ $\displaystyle=$
$\displaystyle
P_{e}\,{\rho}(t^{-1},\tau_{t^{-1}}(g^{-1}))\,A\,{\rho}(t^{\prime},g^{\prime})\,P_{e}.$
Thus the Simonenko presymbol is $B(L_{p}{}(G))$-valued function on
$\bar{G}\times\bar{G}$. We can consider a definition of the alternative
presymbol:
(16)
$\tilde{S}_{A}(t,g;t^{\prime},g^{\prime})=P_{(t,g)}\,A\,P_{(t^{\prime},g^{\prime})},$
which is closer to the original geometrical spirit of Simonenko’s works
[Simonenko65a, Simonenko65b]. However there is an easy explicit connection
between them:
$\hat{S}_{A}((t,g)^{-1};(t^{\prime},g^{\prime})^{-1})={\rho_{d}}(t,g;t^{\prime},g^{\prime})\,\tilde{S}_{A}\,(t,g;t^{\prime},g^{\prime}),$
which is a local transformation of the function value at every point. Thus
both symbols shall bring equivalent theories, although each of them seems to
be more suitable for particular purposes.
For operators of local type the whole presymbol is excessive due to the
following result.
###### Proposition 15.
Let $F_{e}$ be $r$-self-similar and $A$ be an operator of local type. Then for
any reals $t>t^{\prime}>0$ and $g\in G$ the operator
$\hat{S}_{A}(t_{1},g_{1};t_{2},g_{2})$ with $t_{i}>t$, $i=1,2$ can be
expressed as a finite sum
(17)
$\hat{S}_{A}(t_{1},g_{1};t_{2},g_{2})=\sum_{k=1}^{n}B_{k}\hat{S}_{A}(t^{\prime},h_{k};t^{\prime},h_{k})C_{k},$
for some $h_{k}\in F_{(t_{1},g_{1})}\cup F_{(t_{2},g_{2})}$ and constant
operator coefficients $B_{k}$ and $C_{k}$, which do not depend on $A$.
###### Proof.
We will proceed in terms of the equivalent presymbol $\tilde{S}_{A}$ (16)
since it better reflects geometrical aspects. We also note that if we obtain
the decomposition
$\tilde{S}_{A}(t_{1},g_{1};t_{2},g_{2})=\sum_{k=1}^{n}B_{k}\tilde{S}_{A}(t_{k},h_{k};t_{k},h_{k})C_{k},$
with all $t_{k}\leq t^{\prime}$ then we will be able to replace $t_{k}$ by
$t^{\prime}$ with the simultaneous change of coefficient $B_{k}$ and $C_{k}$
in order to get the required identity (17).
Now we put $t^{\prime\prime}=t^{\prime}/r$ and find a finite covering of the
compact sets $F_{(t_{1},g_{1})}\cup F_{(t_{2},g_{2})}$ by the interiors of
sets $F_{(t^{\prime\prime},h_{k})}$ with $h_{k}\in F_{(t_{1},g_{1})}\cup
F_{(t_{2},g_{2})}$. Using the inclusion-exclusion principle we can write:
$\displaystyle P_{(t_{i},g_{i})}$ $\displaystyle=$
$\displaystyle\sum_{k}P_{(t^{\prime\prime},h_{k})}-\sum_{k,l}P_{(t^{\prime\prime},h_{k})}P_{(t^{\prime\prime},h_{l})}+\ldots$
$\displaystyle{}-\sum_{k}P_{(t^{\prime\prime},h_{k})}{P}^{\perp}_{(t_{i},g_{i})}+\sum_{k,l}P_{(t^{\prime\prime},h_{k})}P_{(t^{\prime\prime},h_{l})}{P}^{\perp}_{(t_{i},g_{i})}-\ldots,$
where all sums are finite and the number of sums is finite as well. Moreover
each term in the summation contains at least one projection
$P_{(t^{\prime\prime},h_{k})}$. We use this decomposition for the presymbol
$P_{(t_{1},g_{1})}AP_{(t_{2},g_{2})}$ of an operator $A$ of local type. Then
we need to take care only on the terms
$P_{(t^{\prime\prime},h_{k})}AP_{(t^{\prime\prime},h_{l})}$ where
$F_{(t^{\prime\prime},h_{k})}$ and $F_{(t^{\prime\prime},h_{n})}$ intersect.
Due to the $r$-self-covering property each such term can be represented as
$B_{m}P_{(t^{\prime},h_{m})}AP_{(t^{\prime},h_{m})}C_{m}$ for some $h_{m}\in
F_{(t_{1},g_{1})}\cup F_{(t_{2},g_{2})}$ with $B_{m}$ and $C_{m}$ depending on
the geometry of sets only. ∎
Thus for the operators of local type we give the following definition.
###### Definition 16.
For an operator $A$ of local type we define _Simonenko symbol_
$S_{A}(t,g)=\hat{S}_{A}(t,g;t,g)$, that is:
$\displaystyle
S_{A}(t,g)=P_{e}\,{\rho}(t^{-1},\tau_{t^{-1}}(g^{-1}))\,A\,{\rho}(t,g)\,P_{e}.$
###### Corollary 17.
For an operator $A$ of local type the value of the presymbol
$\hat{S}_{A}(t^{\prime},g^{\prime};t^{\prime\prime},g^{\prime\prime})$ at a
point
$(t^{\prime},g^{\prime};t^{\prime\prime},g^{\prime\prime})\in\bar{G}\times\bar{G}$
is completely determined by the values of symbol $S_{A}(t,g)$, $g\in G$ for an
arbitrary fixed $t$ such that $t\leq\min(t^{\prime},t^{\prime\prime})$.
###### Corollary 18.
Tho operators $A$ and $B$ of local type are equal if and only if for any
$\varepsilon>0$ there is a positive $t<\varepsilon$ such that
$S_{A}(t,g)=S_{B}(t,g)$ for all $g\in G$.
In other words even the symbol $S_{A}(t,g)$ contains an excessive information:
in a sense we shall look for values of $\lim_{t\rightarrow 0}S_{A}(t,g)$ only.
We conclude this section by the restatement of the Definition 2.
###### Definition 19.
Two operators $A$ and $B$ of local type are _equivalent_ at a point $g\in G$,
denoted by $A\stackrel{{\scriptstyle g}}{{\sim}}B$, if
$\lim_{t\rightarrow 0}\left|\left|\left|S_{A-B}(t,g)\right|\right|\right|=0.$
## 4\. Localization and Invariance
The paper of Simonenko [Simonenko65b] already contains results which can be
easily adopted to covariant transform setup. This was already used in our
previous work [Kisil94a, Kisil96e, Kisil92, Kisil93e, Kisil93b, Kisil94f] to
study singular integral operators on the Heisenberg group. In this section we
provide such restatements of results in term of the representation from (12).
Proofs will be omitted since they are easy modifications of the original ones
[Simonenko65b].
###### Definition 20.
An operator is called _homogeneous_ if it commutes with all transformations
${\rho}(t,e)$, $t\in\mathbb{R}_{+}{}$ (12). If an operator commutes with
${\rho}(1,g)$, $g\in G$ (12) then it is called _shift-invariant_.
There is an immediate consequence of Thm. 5.
###### Corollary 21.
The symbols of a homogeneous (or shift-invariant) operator is a function on
$\bar{G}$, which is invariant under the action of the subgroup
$\mathbb{R}_{+}{}\subset\bar{G}$ (or $G\subset\bar{G}$ respectively).
Thus homogeneous shift-invariant operators have constant symbols. Tame
behavior of operators from those classes is described by the following
statements, cf. [Simonenko65b]*§ II.2.
###### Lemma 22.
For two homogeneous operators $A$ and $B$ the following are equivalent:
1. (1)
$A\stackrel{{\scriptstyle e}}{{\sim}}B$, where $e\in G$ is the unit;
2. (2)
$S_{A}(t,e)=S_{B}(t,e)$ for certain $t\in\mathbb{R}_{+}{}$;
3. (3)
$A=B$.
###### Lemma 23.
[Simonenko65b]*§ II.2 For two homogeneous shift-invariant operators $A$ and
$B$ the following are equivalent:
1. (1)
$A\stackrel{{\scriptstyle g}}{{\sim}}B$ for certain $g\in G$;
2. (2)
$S_{A}(t,g)=S_{B}(t,g)$ for certain $(t,g)\in\bar{G}$;
3. (3)
$A=B$.
A shift-invariant operator on $G$ can be associated to a convolution. A
convolution, which is also a homogeneous operator, shall have singular
kernels. A study of such convolutions can be carried out by means of (non-
commutative) harmonic analysis on $G$. For the (commutative) Euclidean group
this was illustrated in [Simonenko65a, Simonenko65b]. A non-commutative
example of the Heisenberg group can be found in [Kisil94a, Kisil96e, Kisil92,
Kisil93e, Kisil93b, Kisil94f]. It is also possible to study this operators
through further versions of wavelets (coherent) transform, e.g. the Berezin-
type symbols [Kisil98a]. In the common case boundedness of the Berezin symbols
corresponds to the boundedness of the operator, and if the symbol vanishes at
the infinity then the operator is compact.
Once a good description of singular convolutions is obtained (through
covariant transform or several such transforms applied in a sequence) we can
consider the class of operators which can be reduced to them.
###### Definition 24.
[Simonenko65b]*§ III.1 A linear operator $A$ of local type is called a
_generalized singular integral_ if $A$ is equivalent at every point of $G$ to
a some homogeneous shift-invariant operator.
The final step of the construction is synthesis of an operator from the field
of local representatives using the inverse covariant transform from Subsection
2.3. To this end we need to chose an invariant pairing on the group $\bar{G}$,
keeping the $ax+b$ group as an archetypal example. For operators of local type
the whole information is concentrated in the arbitrary small neighborhood of
the subgroup $G\subset\bar{G}$, cf. Cor. 18. Thus we select the Hardy-type
functional (7) instead of the Haar one (6). Let $d\mu$ be the Haar measure on
the group $G$. Then the following integral
(18) $\left\langle f_{1},f_{2}\right\rangle=\lim_{t\rightarrow
0}\int_{G}f_{1}(t,g)f_{2}(t,g)\,d\mu(g),$
defines an invariant pairing on the group $\bar{G}$.
We again make use of the fiducial operator $F(A)=P_{e}AP_{e}$ (15). In the
language of wavelet theory we may say that analyzing and reconstructing
vectors are the same. The respective transformation ${\rho_{f}}(t,g)F$ by an
element of the group $\bar{G}$ is defined through the identity
$[{\rho_{f}}(t,g)F](A)=P_{(t,g)}AP_{(t,g)}$ for an arbitrary $A$. Consequently
the inverse covariant transform (8) sends an operator valued function $A(t,g)$
to an operator through the invariant pairing:
$\mathcal{M}:A(t,g)\mapsto A=\lim_{t\rightarrow
0}\int_{G}P_{(t,g)}A(t,g)P_{(t,g)}\,d\mu(g).$
The last integral may be realized through Riemann-type sums which are lead to
the approximation (2) of an envelope of $A(t,g)$.
## 5\. Closing Remarks
In this work we outlined an interpretation of the classical Simonenko’s
localization method [Simonenko65a, Simonenko65b] in the context of recently
formulated covariant transform [Kisil09d, Kisil10c]. The original localization
was used to study singular integral operators, which are convolutions on the
Euclidean group. Our interpretation allows to make a straightforward
modification of localization technique for non-commutative nilpotent Lie
groups. The crucial role is played by the one-parameter group of automorphism
realized as dilations.
Once local representatives are obtained they can be studied further by other
forms of wavelet (covariant) transform. The Berezin symbol seems to be very
suitable for this task. Such a chain (Simonenko–Berezin–…) of covariant
transforms shall lead to the full dissection of initial operator into a very
detailed symbol, which may be even scalar valued. The opposite process,
reconstruction of an operator from its symbol or local representatives, can be
done by the inverse covariant transform, which uses the same group structure.
The original coherent states in quantum mechanics are obtained from the ground
state of the harmonic oscillator by a unitary action of the Weyl–Heisenberg
group [AliAntGaz00]*Ch. 1. The next standard move is a decomposition of an
arbitrary state into a linear superposition of coherent states, which form an
overcomplete set. Consequently, observables can be investigated through such
decompositions of states.
However, observables are primary notions of quantum theory, thus direct
techniques, which circumvent decomposition of states, look more preferable.
Classical coherent states have the best possible (within the Heisenberg
uncertainty relations) localisation in the phase space. Thus our localisation
on nilpotent Lie groups, in particular the Heisenberg group, has a particular
significance for quantum theory. Any observable corresponding to an operator
of local type can be represented as a compact operator and a continuous field
of local representatives. Compact operators have a discrete spectrum with a
complete set of eigenvectors each having at most a finite degeneracy. Local
representatives corresponds to observables which are highly localised on the
phase space. Thus operators of local type is a large set of quantum
observables admitting efficient calculations of their spectrum.
It would be interesting to look for a similar construction in other classes of
Lie groups. For example, Toeplitz operators on the Bergman space
[Vasilevski08a] may be treated through the group
$\mathrm{SL}_{2}(\mathbb{R}{})$ [Kisil11c], which is semisimple. Such groups
do not admit a group of dilation-type global automorphisms, thus some
adjustments to the scheme are required at this point.
Another interesting direction of development is operators of non-local type.
They may look very different from the view-point of geometrical localization,
however it terms of covariant transform the distinction is not so huge. For
operators of local type their Simonenko presymbol over $\bar{G}\times\bar{G}$
is excessive and we can consider only the symbol in a small vicinity of the
boundary $G$ of the diagonal in $\bar{G}\times\bar{G}$. For operators of non-
local type the presymbol on the whole group $\bar{G}\times\bar{G}$ shall be
used. This topic deserves a further consideration.
Acknowledgements: I am grateful to anonymous referees for useful comments and
suggestions, which helped to improve the paper.
## References
|
arxiv-papers
| 2012-01-09T13:09:59 |
2024-09-04T02:49:26.082463
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vladimir V. Kisil",
"submitter": "Vladimir V Kisil",
"url": "https://arxiv.org/abs/1201.1749"
}
|
1201.1866
|
# Universal response of optimal granular damping devices
Martín Sánchez sanchez.martin@frlp.utn.edu.ar Gustavo Rosenthal
guser@frlp.utn.edu.ar Luis A. Pugnaloni luis@iflysib.unlp.edu.ar Instituto de
Física de Líquidos y Sistemas Biológicos (CONICET La Plata, UNLP), Calle 59
Nro 789, 1900 La Plata, Argentina. Centro de Ensayos Estructurales, Facultad
Regional Delta, Universidad Tecnológica Nacional, Av. San Martín 1171,
B2804GBW Campana, Argentina. Departamento de Ingeniería Mecánica, Facultad
Regional La Plata, Universidad Tecnológica Nacional, 60 esq. 124 S/N, 1900 La
Plata, Argentina.
###### Abstract
Granular damping devices constitute an emerging technology for the attenuation
of vibrations based on the dissipative nature of particle collisions. We show
that the performance of such devices is independent of the material properties
of the particles for working conditions where damping is optimal. Even the
suppression of a dissipation mode (collisional or frictional) is unable to
alter the response. We explain this phenomenon in terms of the inelastic
collapse of granular materials. These findings provide a crucial standpoint
for the design of such devices in order to achieve the desired low maintenance
feature that makes particle dampers particularly suitable to harsh
environments.
###### keywords:
Particle dampers , Granular materials , Vibration attenuation
††journal: Journal of Sound and Vibration
## 1 Introduction
Granular dampers (or particle dampers, PDs) are devices aimed at the
attenuation of mechanical vibrations by exploiting the dissipative character
of the interaction between macroscopic particles. A PD consists in a number of
particles (grains) enclosed in a receptacle that is attached or embedded in a
vibrating structure (see Fig. 1). The motion of the grains inside the
enclosure, as the structure vibrates, is able to dissipate part of the energy
through the non-conservative collisions, so reducing the vibration amplitude.
This emerging technology can replace the widely used viscous and viscoelastic
dampers in particular applications where extreme temperatures (either low or
high) are involved or where low maintenance is required. The leading sector in
this regard is the aerospace industry [1, 2, 3]. However, the automotive [4]
and oil and gas [5] industries are catching up in recent years. PDs are the
descendant of the older impact dampers designed to operate by the use of a
single body inside an enclosure [6, 7]. PDs are now preferred over impact
dampers due to the lower noise level they produce. The performance of a PD
depends on a number of design characteristics such as the relative size and
shape of the particles and the enclosure, the total weight $m_{p}$ of the
particles, the number $N$ of grains, the working vibration intensity and
frequency, etc. These have been studied to some extent in the last two decades
[8, 9, 10]. However, less attention has been paid to the role that the grain-
grain interaction plays in these systems.
In this paper, we show that the response of a PD is independent of the
particle-particle interaction to the extent that even friction or inelastic
collisions can be suppressed without altering the vibration attenuation. This
effect is explained in terms of the effective zero restitution of the granular
system caused by the effective _inelastic collapse_ [11]. The inelastic
collapse in dense granular materials refers to the effect by which the system
can dissipate its entire kinetic energy in a short finite time even if
collisions have a very high restitution coefficient. In dense systems, the
number of collisions grows so rapidly that even a minute dissipation in each
collision suffices to make the system as a whole fully dissipative. We show
that this interpretation allows us to set the limits to the universal
response. Crucial implications for the design and maintenance of PDs are
discussed.
Figure 1: (Color online) Schematic view of a particle damper. ($M$) the
primary mass of the structure, ($m_{p}$) the total mass of the particles in
the enclosure, ($K$) the spring constant, ($C$) the viscous structural
damping, ($B$) the vibrating base where the displacement is imposed.
## 2 DEM Simulations
We have carried out simulations of a PD by means of a Discrete Element Method
(DEM). A number N (with $5<N<250$) of spherical particles of mass
$m=9.08\times 10^{-4}$ kg are deposited in a prismatic enclosure (see Fig. 1)
of mass $M=2.37$ kg, base $L\times L$ ($L=0.03675$ m) and height $L_{z}$ (with
$0.057$ m$<L_{z}<0.282$ m). The enclosure is attached to a vibrating base by
means of a Hookean spring (spring constant $K=21500$ Nm-1) and a viscous
damper of low dissipation constant ($C=7.6$ Nsm-1). The particles interact
through a Hertz-Kuwabara-Kono [12] normal contact force
($F_{\mathrm{n}}=-k_{\mathrm{n}}\alpha^{3/2}-\gamma_{\mathrm{n}}\upsilon_{\mathrm{n}}\sqrt{\alpha}$)
plus a tangential force
($F_{\mathrm{t}}=-\min\left(\left|\gamma_{\mathrm{t}}\upsilon_{\mathrm{t}}\sqrt{\alpha}\right|,\left|\mu_{\mathrm{d}}F_{\mathrm{n}}\right|\right)\rm{sgn}\left(\upsilon_{\mathrm{t}}\right)$)
that implements the frictional property of the grain surfaces [13]. Here,
$\upsilon_{\mathrm{n}}$ and $\upsilon_{\mathrm{t}}$ are the relative normal
and tangential velocities and $\alpha=r_{ij}-d$ the virtual overlap between
the interacting particles $i$ and $j$. The diameter of the particles is
$d=0.006$m. The parameters that set the grain-grain interactions are:
$k_{n}=4.02\times 10^{9}$ Nm-3/2 (which corresponds to the Young modulus
$E=2.03\times 10^{11}$ Nm-2 and Poison ratio $\nu=0.28$ for steel),
$0<\gamma_{n}<1\times 10^{4}$ kgm-1/2s-1, $0<\gamma_{t}<1\times 10^{4}$
kgm-1/2s-1 and $\mu_{\mathrm{d}}=0.3$. The grain-wall interaction is taken
equal to the grain-grain interaction. The vibrating base and the enclosure are
constrained to move only in the vertical $z$-direction, whereas the particles
can move in the three-dimensional volume of the receptacle subjected to the
action of gravity. The motion $z_{base}(t)$ of the base is set to a harmonic
function [$z_{base}(t)=0.0045\cos(\omega t)$ m] whose frequency $\omega=2\pi
f$ is varied in the range $0.5\mathrm{Hz}<f<30\mathrm{Hz}$. We monitor the
amplitude of the oscillation $z_{max}$ of the enclosure in response to the
base vibration, the motion of the grains inside and the energy dissipated in
each cycle. All analysis is done over the stationary state of the system for
each given $f$ after any transient has died out.
Figure 2: (Color online) (a) Frequency response function of the system with
$N=250$, $\gamma_{n}=3660.0$ kgm-1/2s-1 and $\gamma_{t}=10980.0$ kgm-1/2s-1.
We plot the maximum amplitude of the oscillation $z_{max}$ of the primary mass
as a function of the excitation frequency $f$ imposed to the base for
different heights $L_{z}$ of the enclosure (see legend) and compare with the
theoretical FRF for an empty enclosure (black solid line). Lines joining
symbols are only to guide the eye. In panels (b), (c) and (d), we plot the
trajectory of the enclosure (black solid lines define the floor and ceiling)
and the motion of the granular sample inside (colored area defined by the
position of the uppermost and lowermost particle at each time; the dotted red
line is the position of the center of mass) for the optimum height
$L_{z}=0.12255$m: (b) $f=5.5$Hz, (c) $14.5$Hz and (d) $f=21.0$Hz.
Figure 2(a) shows the frequency response function (FRF) —i.e., $z_{max}$ as a
function of $f$— of the system with $N=250$ for a given choice of the granular
interaction and three different height $L_{z}$ of the enclosure in comparison
with the response obtained when the enclosure is empty. As discussed in
previous studies [9, 14, 15], there exists an optimum enclosure height (in
this case $L_{z}=0.12255$m) for which the maximum attenuation is achieved. The
resonant frequency is shifted due to the added mass (the granular mass), but
in a non-trivial way, with overshoot and undershoot effective masses [16]. The
motion of the granular bed inside the enclosure for the optimum enclosure
height can be seen in Fig. 2(b), (c) and (d) for different values of $f$. The
grains behave as a more or less dense lumped mass for a wide range of
frequencies ($0<f<18$Hz). However, for high frequencies and tall enclosures,
the grains enter a gas-like state [17].
Figure 3: (Color online) (a) The FRF for a PD of optimum height
($L_{z}=0.12255$m) for $N=250$ particles with different interaction parameters
$\gamma_{n}$ and $\gamma_{t}$ (see legend for the values used in units of
kgm-1/2s-1). The dashed lines correspond to the analytical solution for: an
empty enclosure (orange), and an empty enclosure with primary mass corrected
to $M+m_{p}$ (brown). The black solid line corresponds to the zero restitution
single particle model. (b) The energy dissipated per cycle as a function of
$f$ for the frictional (squares) and collisional (triangles) modes for three
different particle–particle interactions (see legend). The total energy
dissipated (circles) is independent of $\gamma_{n}$ and $\gamma_{t}$. The
black solid line corresponds to the zero restitution single particle model.
(c) Energy dissipated as a function of the collisional dissipation
$\gamma_{n}$ at the resonant frequency for $\gamma_{t}=10980.0$ kgm-1/2s-1.
(d) Energy dissipated as a function of the frictional dissipation $\gamma_{t}$
at the resonant frequency for $\gamma_{n}=3660.0$ kgm-1/2s-1.
## 3 Effect of particle–particle interactions
In order to asses the effect of the particle–particle interactions, we plot in
Fig. 3(a) the FRF for the optimum enclosure height for different values of
$\gamma_{n}$ and $\gamma_{t}$. The effective normal and tangential restitution
coefficient is an exponentially decaying function of these parameters and
depend on the relative velocity at impact [13]. As we can see, different
interaction parameters yield the same FRF, suggesting a universal response.
Notice that even eliminating the frictional character of the particles or,
alternatively, eliminating the dissipative nature of the normal interaction is
not sufficient to induce a change in the FRF over a wide range of frequencies.
As it is to be expected, eliminating both, the normal and tangential
dissipative part of the interaction converts the system into a conservative
molecular-like system which yields no attenuation of the response [see
diamonds in Fig. 3(a)]. In this case only a shift of the resonant frequency is
observed due to the added mass $m_{p}$ [see brown dashed line in Fig. 3(a) for
the analytical solution]. We have also considered different material
properties such as Young modulus and dynamic friction coefficient, but have
observed no change in the FRF.
The energy dissipated in each oscillation cycle is shown in Fig. 3(b) as a
function of the excitation frequency. Notice that the frictional dissipation
and the collisional dissipation are higher around the resonant frequency. The
proportion of energy dissipated through one mode or the other (collisional or
frictional) depends on the actual interaction parameters. However, the total
energy dissipated is independent of the friction and restitution
characteristics of the particles.
To further explore the extent of the apparent universal response of the PD, we
have considered a wide range of the dissipative interaction parameters
$\gamma_{n}$ and $\gamma_{t}$. As it can be seen in Figs. 3(c), where the
response of the system with $N=250$ at the resonant frequency is considered,
increasing $\gamma_{n}$ at constant $\gamma_{t}$ leads to an increase in the
energy dissipated though the collisional mode. The converse is true if the
frictional parameter $\gamma_{t}$ is increased [see Fig. 3(d)]. However, the
total dissipated energy remains constant even if $\gamma_{n}$ or $\gamma_{t}$
drops to zero. Hence, the system is able to dissipate the same amount of
energy irrespective of the dissipation modes available to the particles.
## 4 Origin of the universal response
We speculate that this universal response is found whenever a large number of
particles is used and the motion of the granular bed is set into a more or
less dense lumped mass, as oppose to a gas-like state. For a dense granular
layer, the number of collisions per unit time as the bed collides with the
boundaries increases dramatically due to an effective inelastic collapse [18,
11]. Although _inelastic collapse_ refers to a mathematical divergence of the
number of collisions per unit time when instantaneous interactions are
considered, real systems do also exhibit a remarkable increases in the
collision rate [11, 19]. The behavior of such granular systems has been
recently proven to be well modeled by a single mass $m$ with an effective zero
restitution coefficient and no frictional properties [20]. Also, recent
simulations with no frictional components where able to fit experimental data
on PDs in microgravity [21]. This fact —that friction can be neglected— is
evidenced in Fig. 3(d). However, according to Fig. 3(c), restitution can also
be set to unity and any non-vanishing friction will suffice to render the
universal FRF shown in Fig. 3(a). This indicates that a much wider set of
interactions can lead to the effective inelastic collapse than previously
shown.
We have solved a simple one-dimensional model where a single particle of mass
$m_{p}$ and zero restitution coefficient moves between the floor an ceiling.
The model is adapted from [8]. The results are shown in Figs. 3(a) and 3(b)
with black solid lines. It is clear that such simple model provides the
essence to describe the PD. Hence, design can be based in this simple model
without worrying about a careful selection of the particle properties.
Previous workers have used slightly more complicated models where the
restitution coefficient was used as a fitting parameter [8, 21]. Our results
indicate that this complication may be unnecessary.
## 5 Limits to the universal response
We now turn into surveying the limits of this universal response. Inelastic
collapse is known to take place only if a large number of grains at high
densities are considered. Therefore, we can set the system into a regime where
this universal response does not apply by reducing $N$ or by promoting a
diluted granular state in the enclosure.
Figure 4: (Color online) (a) The vibration amplitude $z_{r}$ at resonance as a
function of the collisional dissipation $\gamma_{n}$ for $\gamma_{t}=10980.0$
kgm-1/2s-1 for different number $N$ of grains in the enclosure (see legend).
The particle size is chosen to yield a total particle mass $m_{p}=0.227$ kg.
(b) Same as (a) but the dependence on $\gamma_{t}$ is considered for
$\gamma_{n}=3660.0$ kgm-1/2s-1.
In Fig. 4, we present the variation of the amplitude of vibration $z_{r}$ at
the resonant frequency as a function of the normal [Fig. 4(a)] and tangential
[Fig. 4(b)] dissipative parameters for different $N$. For these simulations,
we have changed the diameter of the particles in order to keep the total mass
$m_{p}$ of the grains constant as $N$ is changed. For $N>100$, a constant
(_universal_) response is recovered, whereas smaller systems present a better
attenuation as any of the two dissipative properties ($\gamma_{n}$ or
$\gamma_{t}$) is increased, in accord with intuition. These results confirm
the speculation that the universal response only applies if a relatively large
number of particles is involved. Of course, the values of $N$ at which this
universal response is reached will depend on the horizontal size of the
enclosure. We estimate from our simulations that whenever the enclosure is
filled with three or more layers of particles the system response near
resonance becomes independent of the particle–particle interaction.
A way to induce a gas-like behavior of the granular sample in the enclosure
—so as to create a dilute regime where an effective inelastic collapse is not
expected— is to increase the height of the cavity. In Fig. 5(a) and (b) we
plot the energy dissipated per cycle as a function of $\gamma_{n}$ and
$\gamma_{t}$ for an enclosure with $L_{z}=0.282$m. For this large $L_{z}$, the
granular sample expands significantly and does not move as a lumped mass, so
reducing dramatically the number of collisions per unit time. The final
results is an effective dissipation that depends on the particle–particle
dissipative interaction. Interestingly, in this regime, increasing
$\gamma_{n}$ or $\gamma_{t}$ leads to a decrease of the total energy
dissipated. Since PDs are designed to optimize attenuation in a number of
applications, the size of the enclosure generally promotes the dense lumped
mass motion of the grains inside [1]. Therefore, in many working conditions of
interest, the system will be in a regime where the universal FRF is obtained.
Figure 5: (Color online) (a) The energy dissipated per cycle at resonance as a
function of the collisional dissipation $\gamma_{n}$ for a large enclosure
($L_{z}=0.282$m) with $N=250$ particles and $\gamma_{t}=10980.0$ kgm-1/2s-1.
(b) Same as (a) but the effect of the frictional dissipation $\gamma_{t}$ is
considered for $\gamma_{n}=3660.0$ kgm-1/2s-1.
## 6 Conclusions
We have shown that a basic phenomenon (inelastic collapse) leads to a
universal response of a PD —in the sense that the particle–particle
interaction becomes irrelevant. This allowed us to determine the limits of
this universality: relatively large numbers of particles must be used and the
system has to be set in a state of dense lumped mass.
It is worth mentioning that the particular contact force model used for the
simulations is of little relevance to the results presented here. We have
shown that a much simplistic model (the zero-restitution-single-particle
model) also shows the same response. Indeed, the proposed universality implies
that details of the interactions are not relevant. Moreover, despite our
studies being done on a system vibrating in the direction of gravity, we
expect results will apply to horizontally vibrating PDs.
This universal response is consistent with some observations in experiments
and simulations where a few values of the material properties where tested
[26, 22, 23, 24, 25] and can explain the unexpected agreement between
simplified models and complex experiments [21]. It is worth mentioning that
powders, as opposed to granulars, may not follow this universal response even
at resonance. Powders are fine graded particles and the effects of the
hydrodynamics of the surrounding air affects the motion of the particles to a
large extent. Fine powders will expand due to air-particle interactions and
the inelastic collapse will be unlikely. Some preliminary experiments with PDs
using fine powders seem to confirm this [26].
The suitability of PDs to work in harsh environments can be understood as a
consequence of this phenomenon. Extreme temperatures and pressures may induce
mild changes in frictional properties, but these will not alter the PD
response. More importantly, degradation of the particles during operation due
to wear, deformation and fragmentation are not likely to compromise the PD
performance. Changes in friction or restitution are unimportant. Although we
have not studied particles of different shapes, we speculate that fragments of
particles may be as effective as the original particles as long as they are
not fine graded. Moreover, fragmentation can only increase $N$, which should
not take the system out of the universal FRF. This is the underlying
phenomenon that explains the characteristic low maintenance required for this
devices. Notice however that very high temperatures may weld particles
together inducing an effective reduction of $N$ which can reduce vibration
attenuation.
Design of PDs can be greatly simplified by choosing to work with large $N$ and
using a simple model such as the zero restitution single mass used here. Under
these conditions, the selection of the particle material properties is
unimportant for the PD performance and one can focus, for example, on cost
effectiveness.
## Acknowledgements
LAP acknowledges support from CONICET (Argentina).
## References
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* [2] H. V. Panossian, Non-obstructive particle damping experience and capabilities. Proceedings of SPIE 4753 (2002) 936-941.
* [3] R. Ehrgott, H. V. Panossian and G. Davis, Modeling techniques for evaluating the effectiveness of particle damping in turbomachinery. 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Material Conference, Palm Springs (2009).
* [4] Z. Xia, X. Liu and Y. Shan, Application of particle damping for vibration attenuation in brake drum. Int. J. of Vehicle Noise and Vibration 7 (2011) 178-194. doi:10.1504/IJVNV.2011.040573
* [5] A. S. Velichkovich and S. V. Velichkovich, Vibration-Impact Damper for Controlling the Dynamic Drillstring Conditions. Chem. Petroleum Eng. 37 (2001) 213-215. doi:10.1023/A:1017650519261
* [6] P. Lieber and D. P. Jensen, An acceleration damper: development, design and some applications. Trans. ASME 67 (1945) 523-530.
* [7] M. R. Duncan, C. R. Wassgren and C. M. Krousgrill, The damping performance of a single particle impact damper. J. Sound Vib. 286 (2005) 123-144. doi:10.1016/j.jsv.2004.09.028
* [8] R. D. Friend and V. K. Kinra, Particle impact damping. J. Sound Vib. 233 (2000) 93-118. doi:10.1006/jsvi.1999.2795
* [9] M. Saeki, Impact damping with granular materials in a horizontally vibrating system. J. Sound Vib. 251 (2002) 153-161. doi:10.1006/jsvi.2001.3985
* [10] W. Liu, G. R. Tomlinson and J. A. Rongong, The dynamic characterisation of disk geometry particle dampers. J. Sound Vib. 280 (2005) 849-861. doi:10.1016/j.jsv.2003.12.047
* [11] L. P. Kadanoff, Built upon sand: Theoretical ideas inspired by granular flows. Rev. Mod. Phys. 71 (1999) 435-444. doi:10.1103/RevModPhys.71.435
* [12] N. V. Brilliantov, F. Spahn, J. M. Hertzsch and T. Pöschel, Model for collisions in granular gases. Phys. Rev. E 53 (1996) 5382–5392. doi:10.1103/PhysRevE.53.5382
* [13] J. Schäfer, S. Dippel and D. E. Wolf, Force Schemes in simulations of granular materials. J. Phys. I France 6 (1996) 5-20.
* [14] A. Papalou and S. F. Masri, An experimental investigation of particle dampers under harmonic excitation. J. Vibr. Control 4 (1998) 361-379. doi:10.1177/107754639800400402
* [15] M. Sánchez and L. A. Pugnaloni, Modelling of a granular damper. Mecánica Computacional XXIX (2010) 1849-1859.
* [16] M. Sánchez and L. A. Pugnaloni. Effective mass overshoot in single degree of freedom mechanical systems with a particle damper. J. Sound Vib. 330 (2011) 5812-5819. doi:10.1016/j.jsv.2011.07.016
* [17] C. Salueña, T. Pöschel and S. E. Espiov, Dissipative properties of vibrated granular materials. Phys. Rev. E 59 (1999) 4422-4425. doi:10.1103/PhysRevE.59.4422
* [18] S. McNamara and W. R. Young, Inelastic collapse and dumping in a one-dimensional granular medium. Phys. Fluids A 4 (1992) 496-504. doi:10.1063/1.858323
* [19] T. Pöschel and T. Schwager, Computational Granular Dynamics. Models and Algorithms, Springer, Berlin (2005).
* [20] F. F. Chung, S. S. Liaw and W. C. Chang, Collective motion of inelastic particles between two oscillating walls. Granular Matter 13 (2011) 787-794. doi:10.1007/s10035-011-0291-2
* [21] M. N. Bannerman, J. E. Kollmer, A. Sack, M. Heckel, P. Müller and T. Pöschel, Movers and shakers: Granular damping in microgravity. Phys. Rev. E 84 (2011) 011301. doi:10.1103/PhysRevE.84.011301
* [22] T. Chen, K. Mao, X. Huang and M. Y. Wang, Dissipation mechanisms of non-obstructive particle damping using discrete element method. Proceedings of SPIE 4331 (2001) 294-301.
* [23] X. M. Bai, L. M. Keer, Q. J. Wang and R. Q. Snurr, Investigation of particle damping mechanism via particle dynamics simulations. Granular Matter 11 (2009) 417-429. doi:10.1007/s10035-009-0150-6
* [24] Z. Lu, S. F. Masri and X. Lu, Parametric studies of the performance of particle dampers under harmonic excitation. Struct. Control Health Monitoring 18 (2011) 79-98. doi:10.1002/stc.359
* [25] Y. Duan and Q. Chen, Simulation and experimental investigation on dissipative properties of particle dampers. J. Vib. Control 17 (2011) 777-788. doi:10.1177/1077546309356183
* [26] K. S. Marhadi and V. K. Kinra, Particle impact damping: effect of mass ratio, material,and shape. J. Sound Vib. 283 (2005) 433-448. doi:10.1016/j.jsv.2004.04.013
Figure Captions
Figure 1: (Color online) Schematic view of a particle damper. ($M$) the
primary mass of the structure, ($m_{p}$) the total mass of the particles in
the enclosure, ($K$) the spring constant, ($C$) the viscous structural
damping, ($B$) the vibrating base where the displacement is imposed.
Figure 2: (Color online) (a) Frequency response function of the system with
$N=250$, $\gamma_{n}=3660.0$ kgm-1/2s-1 and $\gamma_{t}=10980.0$ kgm-1/2s-1.
We plot the maximum amplitude of the oscillation $z_{max}$ of the primary mass
as a function of the excitation frequency $f$ imposed to the base for
different heights $L_{z}$ of the enclosure (see legend) and compare with the
theoretical FRF for an empty enclosure (black solid line). Lines joining
symbols are only to guide the eye. In panels (b), (c) and (d), we plot the
trajectory of the enclosure (black solid lines define the floor and ceiling)
and the motion of the granular sample inside (colored area defined by the
position of the uppermost and lowermost particle at each time; the dotted red
line is the position of the center of mass) for the optimum height
$L_{z}=0.12255$m: (b) $f=5.5$Hz, (c) $14.5$Hz and (d) $f=21.0$Hz.
Figure 3: (Color online) (a) The FRF for a PD of optimum height
($L_{z}=0.12255$m) for $N=250$ particles with different interaction parameters
$\gamma_{n}$ and $\gamma_{t}$ (see legend for the values used in units of
kgm-1/2s-1). The dashed lines correspond to the analytical solution for: an
empty enclosure (orange), and an empty enclosure with primary mass corrected
to $M+m_{p}$ (brown). The black solid line corresponds to the zero restitution
single particle model. (b) The energy dissipated per cycle as a function of
$f$ for the frictional (squares) and collisional (triangles) modes for three
different particle–particle interactions (see legend). The total energy
dissipated (circles) is independent of $\gamma_{n}$ and $\gamma_{t}$. The
black solid line corresponds to the zero restitution single particle model.
(c) Energy dissipated as a function of the collisional dissipation
$\gamma_{n}$ at the resonant frequency for $\gamma_{t}=10980.0$ kgm-1/2s-1.
(d) Energy dissipated as a function of the frictional dissipation $\gamma_{t}$
at the resonant frequency for $\gamma_{n}=3660.0$ kgm-1/2s-1.
Figure 4: (Color online) (a) The vibration amplitude $z_{r}$ at resonance as a
function of the collisional dissipation $\gamma_{n}$ for $\gamma_{t}=10980.0$
kgm-1/2s-1 for different number $N$ of grains in the enclosure (see legend).
The particle size is chosen to yield a total particle mass $m_{p}=0.227$ kg.
(b) Same as (a) but the dependence on $\gamma_{t}$ is considered for
$\gamma_{n}=3660.0$ kgm-1/2s-1.
Figure 5: (Color online) (a) The energy dissipated per cycle at resonance as a
function of the collisional dissipation $\gamma_{n}$ for a large enclosure
($L_{z}=0.282$m) with $N=250$ particles and $\gamma_{t}=10980.0$ kgm-1/2s-1.
(b) Same as (a) but the effect of the frictional dissipation $\gamma_{t}$ is
considered for $\gamma_{n}=3660.0$ kgm-1/2s-1.
|
arxiv-papers
| 2012-01-09T18:39:02 |
2024-09-04T02:49:26.091409
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mart\\'in S\\'anchez, Gustavo Rosenthal and Luis A. Pugnaloni",
"submitter": "Luis Ariel Pugnaloni",
"url": "https://arxiv.org/abs/1201.1866"
}
|
1201.2035
|
# On the Characterization of the Duhem Hysteresis Operator with Clockwise
Input-Output Dynamics
Ruiyue Ouyang r.ouyang@rug.nl Vincent Andrieu vincent.andrieu@gmail.com Bayu
Jayawardhana† bayujw@ieee.org (Corresponding author) Dept. Discrete Technology
and Production Automation, University of Groningen, Groningen 9747AG, The
Netherlands Université Lyon 1, Villeurbanne; CNRS, UMR 5007, LAGEP. 43 bd du
11 novembre, 69100 Villeurbanne, France Dept. Discrete Technology and
Production Automation, University of Groningen, Groningen 9747AG, The
Netherlands
###### Abstract
In this paper we investigate the dissipativity property of a certain class of
Duhem hysteresis operator, which has clockwise (CW) input-output (I/O)
behavior. In particular, we provide sufficient conditions on the Duhem
operator such that it is CW and propose an explicit construction of the
corresponding function satisfying dissipation inequality of CW systems. The
result is used to analyze the stability of a second order system with
hysteretic friction which is described by a Dahl model.
###### keywords:
Hysteresis, clockwise I/O dynamics , dissipative systems
## 1 Introduction
Hysteresis is a common nonlinear phenomena that is present in diverse physical
systems, such as piezo-actuator, ferromagnetic material and mechanical
systems. From the perspective of input-output behavior, the hysteretic
phenomena can be characterized into counterclockwise (CCW) input-output (I/O)
dynamics [1], clockwise (CW) I/O dynamics [21], or even more complex I/O map
(such as, butterfly map [3]). For example, backlash operator generates CCW I/O
dynamics; elastic-plastic operator generates CW I/O dynamics and Preisach
operator can have either CCW or CW I/O dynamics depending on the weight of the
hysterons which are used in the Preisach model [5, 18, 16].
In the recent work by Angeli [1], the counterclockwise (CCW) I/O dynamics of a
single-input single-output system is characterized by the following inequality
$\liminf_{T\rightarrow\infty}\int^{T}_{0}{\dot{y}(t)u(t)dt>-\infty},$ (1)
where $u$ is the input signal and $y$ is the corresponding output signal. It
is assumed that $u\in U$ where $U$ is the set of input signals for which $y$
exists and is well defined for all positive time. Compare with the classical
definition of passivity [30], it can be interpreted as the system is passive
from the input $u$ to the time derivative of the corresponding output $y$. In
particular, (1) holds if there exists a function
$H:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}_{+}$ such that
$\frac{{\rm d}\hbox{\hskip 0.5pt}H(y(t),u(t))}{{\rm d}\hbox{\hskip
0.5pt}t}\leq\dot{y}(t)u(t).$ (2)
Indeed, integrating (2) from $0$ to $\infty$ we obtain (1).
Correspondingly, clockwise (CW) I/O dynamics can be described by the following
dissipation inequality
$\liminf_{T\rightarrow\infty}\int^{T}_{0}{\dot{u}(t)y(t)dt>-\infty}.$ (3)
The notions of counterclockwise (CCW) I/O and clockwise (CW) I/O are also
discussed in [20].
In our previous results in [13], we show that for a certain class of Duhem
hysteresis operator $\Phi:u\mapsto\Phi(u,y_{0}):=y$, we can construct a
function $H_{\circlearrowleft}:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}_{+}$
which satisfies
$\frac{{\rm d}\hbox{\hskip 0.5pt}H_{\circlearrowleft}(y(t),u(t))}{{\rm
d}\hbox{\hskip 0.5pt}t}\leq\dot{y}(t)u(t).$ (4)
This inequality immediately implies that such Duhem hysteresis operator is
dissipative with respect to the supply rate $\dot{y}(t)u(t)$ and has CCW
input-output dynamics. The symbol $\circlearrowleft$ in $H_{\circlearrowleft}$
indicates the counterclockwise behavior of $\Phi$.
In this paper, as a dual extension to [13], we focus on the clockwise (CW)
hysteresis operator where the supply rate is given by $\dot{u}y$ which is dual
to the supply rate $u\dot{y}$ considered in [13]. This is motivated by the
friction induced hysteresis phenomenon in the mechanical system which has CW
I/O behavior from the input relative displacement to the output friction
force. One may intuitively consider to reverse the input-output relation of
the CW hysteresis operator for getting the CCW I/O behavior in the reverse I/O
setting. However, this consideration has two drawbacks: 1). the reverse input-
output pair may not be physically realizable (this is related to the causality
problem in the port-based modeling, such as, the bond graph modeling framework
[4]); 2). the operator itself may not be invertible (for example, if the
output of the hysteresis operator can be saturated).
In Theorem 1, we provide sufficient conditions on the underlying functions
$f_{1}$ and $f_{2}$ of the Duhem operator, such that it has CW I/O dynamics.
Roughly speaking, the functions $f_{1}$ and $f_{2}$ (as defined later in
Section 2) determine two possible different directions $(y,u)$ depending on
whether the input $u$ is increasing or decreasing. By evaluating these two
functions on two disjoint domains (which are separated by an anhysteresis
curve), we can determine whether it has CW I/O dynamics using Theorem 1. This
is shown by constructing a function
$H_{\circlearrowright}:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}_{+}$ such that
the following inequality
$\frac{{\rm d}\hbox{\hskip 0.5pt}H_{\circlearrowright}(y(t),u(t))}{{\rm
d}\hbox{\hskip 0.5pt}t}\leq y(t)\dot{u}(t).$ (5)
holds. The function $H_{\circlearrowright}$ can also be related to the concept
of available storage function from [30] where, instead of using the standard
supply rate $yu$, we use the CW supply rate $y\dot{u}$ as shown in Proposition
1 in this paper.
The dissipativity property (5) can be further used in the stability analysis
of the systems with CW hysteresis, such as, a second-order mechanical system
with hysteretic friction as discussed in Section 4.2. As an illustrative
example on the application of (5), let us consider a mechanical system
described by
$\left.\begin{array}[]{rl}m\ddot{x}&=F-F_{{\rm friction}},\\\ F_{{\rm
friction}}&=\Phi(x,y_{0}),\end{array}\right.$
with the hysteresis operator $\Phi$ satisfying the Dahl model as follows
$\dot{F}_{\rm friction}=\rho\left(1-\frac{F_{\rm
friction}}{F_{C}}\right)\max\\{0,\dot{x}\\}+\rho\left(1+\frac{F_{\rm
friction}}{F_{C}}\right)\min\\{0,\dot{x}\\},$
where $m$ refers to the mass, $x$ refers to the displacement, $F$ is the
applied force, $\rho>0$ describes the stiffness constant, $F_{C}>0$ represents
the Coulomb friction constant and $y_{0}$ is the initial condition of the Dahl
model (see, for example, [20]). By taking $x_{1}=x,x_{2}=\dot{x}$ and
$x_{3}=F_{friction}$ as the state variables, we can rewrite this hysteretic
system into state-space form as follows
$\left.\begin{array}[]{rl}\dot{x}_{1}&=x_{2},\\\
\dot{x}_{2}&=\frac{F}{m}-\frac{x_{3}}{m},\\\
\dot{x}_{3}&=\rho\left(1-\frac{x_{3}}{F_{C}}\right)\max\\{0,x_{2}\\}+\rho\left(1+\frac{x_{3}}{F_{C}}\right)\min\\{0,x_{2}\\}.\end{array}\right.$
In Section 4.1, we obtain the function $H_{\circlearrowright}$ satisfying (5)
explicitly and it is parameterized by $\rho$ and $F_{C}$. Using
$V(x_{1},x_{2},x_{3})=\frac{1}{2}mx_{2}^{2}+H_{\circlearrowright}(x_{3},x_{1})$
as a Lyapunov function we have
$\displaystyle\dot{V}$ $\displaystyle=m\dot{x}_{2}x_{2}+\frac{{\rm
d}\hbox{\hskip 0.5pt}H_{\circlearrowright}(x_{3},x_{1})}{{\rm d}\hbox{\hskip
0.5pt}t}$ $\displaystyle=-x_{3}x_{2}+Fx_{2}+\frac{{\rm d}\hbox{\hskip
0.5pt}H_{\circlearrowright}(x_{3},x_{1})}{{\rm d}\hbox{\hskip 0.5pt}t}$
$\displaystyle\leq Fx_{2}.$
This inequality establishes that the closed loop system is passive from the
applied force $F$ to the velocity $x_{2}$. Thus a simple propositional
feedback $F=-dx_{2}$, where $d>0$, can guarantee the asymptotic convergence of
the velocity $x_{2}$ to zero without having to know precisely the parameters
$\rho$ and $F_{C}$.
## 2 Duhem operator and clockwise hysteresis operators
Denote $C^{1}({\mathbb{R}}_{+})$ the space of continuously differentiable
functions $f:{\mathbb{R}}_{+}\to{\mathbb{R}}$ and $AC({\mathbb{R}}_{+})$ the
space of absolutely continuous functions
$f:{\mathbb{R}}_{+}\rightarrow{\mathbb{R}}$. Define $\frac{{\rm d}\hbox{\hskip
0.5pt}z(t)}{{\rm d}\hbox{\hskip 0.5pt}t}:=\lim_{h\searrow
0^{+}}\frac{z(t+h)-z(t)}{h}$.
The Duhem operator $\Phi:AC({\mathbb{R}}_{+})\times{\mathbb{R}}\to
AC({\mathbb{R}}_{+}),(u,y_{0})\mapsto\Phi(u,y_{0})=:y$ is described by [18,
20, 28]
$\dot{y}(t)=f_{1}(y(t),u(t))\dot{u}_{+}(t)+f_{2}(y(t),u(t))\dot{u}_{-}(t),\
y(0)=y_{0},$ (6)
where $\dot{u}_{+}(t):=\max\\{0,\dot{u}(t)\\}$,
$\dot{u}_{-}(t):=\min\\{0,\dot{u}(t)\\}$. The functions $f_{1}$ and $f_{2}$
are assumed to be $C^{1}$.
The existence of solutions to (6) has been reviewed in [18]. In particular, if
for every $\xi\in{\mathbb{R}}$, $f_{1}$ and $f_{2}$ satisfy
$\displaystyle(\sigma_{1}-\sigma_{2})[f_{1}(\sigma_{1},\xi)-f_{1}(\sigma_{2},\xi)]$
$\displaystyle\leq\lambda_{1}(\xi)(\sigma_{1}-\sigma_{2})^{2},$ (7)
$\displaystyle(\sigma_{1}-\sigma_{2})[f_{2}(\sigma_{1},\xi)-f_{2}(\sigma_{2},\xi)]$
$\displaystyle\geq-\lambda_{2}(\xi)(\sigma_{1}-\sigma_{2})^{2},$
for all $\sigma_{1}$, $\sigma_{2}\in{\mathbb{R}}$, where $\lambda_{1}$ and
$\lambda_{2}$ are nonnegative, then the solution to (6) exist and $\Phi$ maps
$AC({\mathbb{R}}_{+})\times{\mathbb{R}}\rightarrow AC({\mathbb{R}}_{+})$. We
will assume throughout the paper that the solution to (6) exists for all $u\in
AC({\mathbb{R}}_{+})$ and $y_{0}\in{\mathbb{R}}$.
As a dual definition to counterclockwise (CCW) I/O behavior [1], we define the
clockwise (CW) I/O dynamics as follows
###### Definition 1
An operator $Q$ is clockwise (CW) if for every $u\in U$ with the corresponding
output map $y:=Qu$, where $U$ is the space of input signals such that $y$ is
well-defined for all positive time, the following inequality holds
$\liminf_{T\rightarrow\infty}\int^{T}_{0}{y(t)\dot{u}(t){\rm d}\hbox{\hskip
0.5pt}t>-\infty}.$ (8)
For the Duhem operator $\Phi$, inequality (8) holds if there exists a function
$H_{\circlearrowright}:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}_{+}$ such that
for every $u\in AC({\mathbb{R}}_{+})$ and $y_{0}\in{\mathbb{R}}$, the
inequality
$\frac{{\rm d}\hbox{\hskip 0.5pt}H_{\circlearrowright}(y(t),u(t))}{{\rm
d}\hbox{\hskip 0.5pt}t}\leq y(t)\dot{u}(t),$ (9)
holds for all $t$ where $y:=\Phi(u,y_{0})$.
In the following subsections, we describe several well-known hysteresis
operators which generate clockwise I/O dynamics and we recast these operators
into the Duhem operator as in (6).
### 2.1 Dahl model
The Dahl model [7, 22] is commonly used in mechanical systems, which
represents the friction force with respect to the relative displacement
between two surfaces in contact. The general representation of the Dahl model
is given by
$\dot{y}(t)=\rho\left|1-\frac{y(t)}{F_{c}}\textrm{sgn}(\dot{u}(t))\right|^{r}\textrm{sgn}\left(1-\frac{y(t)}{F_{c}}\textrm{sgn}(\dot{u}(t))\right)\dot{u}(t),$
(10)
where $y$ denotes the friction force, $u$ denotes the relative displacement,
$F_{c}>0$ denotes the Coulomb friction force, $\rho>0$ denotes the rest
stiffness and $r\geq 1$ is a parameter that determines the shape of the
hysteresis loops.
The Dahl model can be described by the Duhem hysteresis operator (6) with
$f_{1}(\sigma,\xi)=\rho\left|1-\frac{\sigma}{F_{c}}\right|^{r}\textrm{sgn}\left(1-\frac{\sigma}{F_{c}}\right),$
(11)
$f_{2}(\sigma,\xi)=\rho\left|1+\frac{\sigma}{F_{c}}\right|^{r}\textrm{sgn}\left(1+\frac{\sigma}{F_{c}}\right).$
(12)
In Figure 1, we illustrate the behavior of the Dahl model where $F_{c}=0.75$,
$\rho=1.5$ and $r=3$.
(a) (b)
Figure 1: The input-output dynamcis of the Dahl model with $F_{c}=0.75$,
$\rho=1.5$ and $r=3$.
### 2.2 Bouc-Wen model
The Bouc-Wen model [25, 29] is commonly used to model the elastic stress-
strain relationships in structures. Moreover, it is also used to represent the
magnetorheological behavior in the MR damper [8]. The general representation
of the Bouc-Wen model is given by
$\dot{y}(t)=\alpha\dot{u}(t)-\beta\dot{u}(t)|y(t)|^{n}-\gamma|\dot{u}(t)|y(t)|y(t)|^{n-1},$
where $u$ denotes the displacement, $y$ denotes the elastic strain, $n\geq 1$
and $\beta,\zeta$ are the parameters determine the shape of the hysteresis
curve.
The Bouc-Wen model can be described by the Duhem hysteresis operator (6) with
$f_{1}(\sigma,\xi)=\alpha-\beta|\sigma|^{n}-\zeta\sigma|\sigma|^{n-1},$ (13)
$f_{2}(\sigma,\xi)=\alpha-\beta|\sigma|^{n}+\zeta\sigma|\sigma|^{n-1}.$ (14)
In Figure 2, we illustrate the behavior of the Bouc-Wen model where
$\alpha=1$, $\beta=1$, $\zeta=1$ and $n=3$.
(a) (b)
Figure 2: The input-output dynamcis of the Bouc-Wen model with $\alpha=1$,
$\beta=1$, $\zeta=1$ and $n=3$.
## 3 Main result
Before stating our main contribution, we need to introduce three functions in
the following subsections: an anhysteresis function $f_{an}$, a traversing
function $\omega_{\Phi}$ and an intersecting function $\Lambda$; these
functions will play an important role in the characterization of dissipativity
and in the construction of the storage function. These three functions are
defined based on the knowledge of $f_{1}$ and $f_{2}$. Generally speaking, the
anhysteresis function $f_{an}$ defines the curve where $f_{1}=f_{2}$, the
function $\omega_{\Phi}$ describes the trajectory of $\Phi$ when a monotone
increasing $u$ or a monotone decreasing $u$ is applied from a given point in
the hysteresis phase plot, and the intersecting function $\Lambda$ defines the
intersection of the anhysteresis function $f_{an}$ and function
$\omega_{\Phi}$ from a given point. The anhysteresis function $f_{an}$ and the
traversing function $\omega_{\Phi}$ have the same definitions as given in our
previous results in [12].
### 3.1 Anhysteresis function
In order to define the anhysteresis function, we rewrite $f_{1}$ and $f_{2}$
as follows
$\left.\begin{array}[]{ll}f_{1}(y(t),u(t))&=F(y(t),u(t))+G(y(t),u(t)),\\\
f_{2}(y(t),u(t))&=-F(y(t),u(t))+G(y(t),u(t)),\end{array}\right\\}$ (15)
where $F,G:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$. We assume that the
implicit function $F(\sigma,\xi)=0$ can be represented by an explicit function
$\sigma=f_{an}(\xi)$ or $\xi=g_{an}(\sigma)$. Such function $f_{an}$ (or
$g_{an}$) is called an anhysteresis function and the corresponding graph
$\\{(\xi,f_{an}(\xi))|\xi\in{\mathbb{R}}\\}$ is called an anhysteresis curve.
Using $f_{an}$, it can be checked that
$f_{1}(f_{an}(\xi),\xi)=f_{2}(f_{an}(\xi),\xi)$ holds. Note also that the
functions $F$ and $G$ in (15) are defined by
$F=\frac{f_{1}-f_{2}}{2}\qquad G=\frac{f_{1}+f_{2}}{2}.$
### 3.2 Traversing function $\omega_{\Phi}$
For every given point $(\sigma,\xi)\in\mathbb{R}^{2}$ in the hysteresis phase
plot, let $\omega_{\Phi,1}(\cdot,\sigma,\xi):[\xi,\infty)\to\mathbb{R}$ be the
solution $x$ of
$x(\tau)-x(\xi)=\int^{\tau}_{\xi}{f_{1}(x(\lambda),\lambda)\ {\rm
d}\hbox{\hskip 0.5pt}\lambda}\ \quad
x(\xi)=\sigma\quad\forall\tau\in[\xi,\infty),$
and let $\omega_{\Phi,2}(\cdot,\sigma,\xi):(-\infty,\xi]\to{\mathbb{R}}$ be
the solution $x$ of
$x(\tau)-x(\xi)=\int_{\xi}^{\tau}{f_{2}(x(\lambda),\lambda)\ {\rm
d}\hbox{\hskip 0.5pt}\lambda}\ \quad
x(\xi)=\sigma\quad\forall\tau\in(-\infty,\xi].$
Using the above definitions, for every point $(\sigma,\xi)\in\mathbb{R}^{2}$
in the hysteresis phase plot, the traversing function
$\omega_{\Phi}(\cdot,\sigma,\xi):\mathbb{R}\to\mathbb{R}$ is defined by the
concatenation of $\omega_{\Phi,2}(\cdot,\sigma,\xi)$ and
$\omega_{\Phi,1}(\cdot,\sigma,\xi)$:
$\omega_{\Phi}(\tau,\sigma,\xi)=\left\\{\begin{array}[]{ll}\omega_{\Phi,2}(\tau,\sigma,\xi)&\forall\tau\in(-\infty,\xi),\\\
\omega_{\Phi,1}(\tau,\sigma,\xi)&\forall\tau\in[\xi,\infty).\end{array}\right.$
(16)
We remark that the function $\omega_{\Phi}(\cdot,\sigma,\xi)$ defines the
(unique) hysteresis curve where the curve
$\\{(\tau,\omega_{\Phi}(\tau,\sigma,\xi))\,|\,\tau\in(-\infty,\xi]\\}$ is
obtained by applying a monotone decreasing $u$ to $\Phi(\cdot,\sigma)$ with
$u(0)=\xi$, $\lim_{t\to\infty}u(t)=-\infty$ and, similarly, the curve
$\\{(\tau,\omega_{\Phi}(\tau,\sigma,\xi))\,|\,\tau\in[\xi,\infty)\\}$ is
obtained by introducing a monotone increasing $u$ to $\Phi(\cdot,\sigma)$ with
$u(0)=\xi$ and $\lim_{t\to\infty}u(t)=\infty$.
### 3.3 Intersecting function $\Lambda$
The intersecting function $\Lambda$ describes the intersection between the
anhysteresis curve $f_{an}$ and the curve $\omega_{\Phi}$. The function
$\Lambda:{\mathbb{R}}^{2}\to{\mathbb{R}}$ is an intersecting function
(corresponding to $\omega_{\Phi}$ and $f_{an}$) if: i)
$\omega_{\Phi}(\Lambda(\sigma,\xi),\sigma,\xi)=f_{an}(\Lambda(\sigma,\xi))$
for all $(\sigma,\xi)\in{\mathbb{R}}^{2}$ and; ii)
$\Lambda(\sigma,\xi)\leq\xi$ whenever $\sigma\geq f_{an}(\xi)$ and
$\Lambda(\sigma,\xi)>\xi$ otherwise. This implies that the two functions
$\omega_{\Phi}(\cdot,\sigma,\xi)$ and $f_{an}(\cdot)$ intersect at a unique
point larger or smaller than $\xi$ depending on the sign of $\sigma-
f_{an}(\xi)$. In our main result, we also need that $\frac{{\rm d}\hbox{\hskip
0.5pt}\Lambda(y(t),u(t))}{{\rm d}\hbox{\hskip 0.5pt}t}$ exists for every
solutions $(y,u)$ of (6).
In the following lemma we give sufficient conditions for the existence of such
intersecting function $\Lambda$.
###### Lemma 1
Assume that $f_{1}$ and $f_{2}$ in (15) be such that $f_{1}$, $f_{2}$ are
$C^{1}$. Moreover, assume that $f_{an}$ is strictly increasing and there
exists a positive real constant $\epsilon>0$ such that for all
$(\sigma,\xi)\in{\mathbb{R}}^{2}$ the following inequality holds
$\displaystyle f_{1}(\sigma,\xi)>\frac{{\rm d}\hbox{\hskip
0.5pt}f_{an}(\xi)}{{\rm d}\hbox{\hskip 0.5pt}\xi}+\epsilon$ whenever
$\displaystyle\sigma>f_{an}(\xi)\ ,$ (17) $\displaystyle
f_{2}(\sigma,\xi)>\frac{{\rm d}\hbox{\hskip 0.5pt}f_{an}(\xi)}{{\rm
d}\hbox{\hskip 0.5pt}\xi}+\epsilon$ whenever $\displaystyle\sigma<f_{an}(\xi)\
.$ (18)
Then there exists an intersecting function $\Lambda\in
C^{1}({\mathbb{R}}^{2},{\mathbb{R}})$ such that
(1)
$\Lambda(\sigma,\xi)\leq\xi$ whenever $\sigma\geq f_{an}(\xi)$ and
$\Lambda(\sigma,\xi)>\xi$ otherwise.
(2)
$\displaystyle\omega_{\Phi}(\Lambda(\sigma,\xi),\sigma,\xi)=f_{an}(\Lambda(\sigma,\xi)).$
($19$)
(3)
Moreover, for all $u\in C^{1}$, $y:=\Phi(u,y_{0})$, we have that $\frac{{\rm
d}\hbox{\hskip 0.5pt}}{{\rm d}\hbox{\hskip 0.5pt}t}\Lambda(y(t),u(t))$ exists.
The proof of Lemma 1 is given in the A.
###### Example 1
In order to illustrate these functions, let us consider the Duhem operator
$\Phi$ with $f_{1}(\sigma,\xi)=e^{0.5(-1.2\sigma+\xi)}+0.83$ and
$f_{2}(\sigma,\xi)=e^{0.5(1.2\sigma-\xi)}+0.83$ as shown in Figure 3. It can
be checked that the anhysteresis function of the operator is
$f_{an}(\xi)=0.83\xi$ and the functions $f_{1}$ and $f_{2}$ satisfy the
hypotheses in Lemma 1. With a reference to Figure 3, let the current state of
$\Phi$ be given by $(y(t),u(t))$. In this figure, the traversing function
$\omega_{\phi}(\cdot,y(t),u(t))$ is depicted by the dashed-line and the
anhysteresis function $f_{an}$ is shown by the thick solid-line. The point
$(y(t),u(t))$ is located above the anhysteresis curve, i.e.,
$y(t)>f_{an}(u(t))$. It can be seen from the figure that the intersecting
point $\Lambda(y(t),u(t))$ (which is shown by the solid circle) is less than
$u(t)$, i.e., $\Lambda(y(t),u(t))\leq u(t)$. This shows that the property (1)
in Lemma 1 holds.
Figure 3: Illustration of a Duhem operator with
$f_{1}(\sigma,\xi)=e^{0.5(-1.2\sigma+\xi)}+0.83$ and
$f_{2}(\sigma,\xi)=e^{0.5(1.2\sigma-\xi)}+0.83$ for all
$(\sigma,\xi)\in{\mathbb{R}}^{2}$. The anhysteresis curve
$f_{an}(\xi)=0.83\xi$ is shown by the thick solid-line. If the current state
be given by $(y(t),u(t))$, the traversing function
$\omega_{\phi}(\cdot,y(t),u(t))$ is depicted by the dashed-line and the
intersecting point $\Lambda(y(t),u(t))$ is shown by the solid circle.
### 3.4 Duhem operator with clockwise hysteresis
Based on the three functions $\omega_{\Phi}$, $f_{an}$ and $\Lambda$, we
define $H_{\circlearrowright}:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}_{+}$ as
follows
$H_{\circlearrowright}(\sigma,\xi)=\int_{0}^{\Lambda(\sigma,\xi)}{f_{an}(\tau){\rm
d}\hbox{\hskip
0.5pt}\tau}-\int_{\xi}^{\Lambda(\sigma,\xi)}{\omega_{\Phi}(\tau,\sigma,\xi){\rm
d}\hbox{\hskip 0.5pt}\tau}.$ (20)
###### Theorem 1
Consider the Duhem hysteresis operator $\Phi$ defined in (6) and (15) with
$C^{1}$ functions $F,G:{\mathbb{R}}^{2}\to{\mathbb{R}}$ and with the
traversing function $\omega_{\Phi}$ and the anhysteresis function $f_{an}$.
Suppose that there exists an intersecting function $\Lambda$ (e.g. the
hypotheses in Lemma 1 hold). Let the following condition holds for all
$(\sigma,\xi)$ in ${\mathbb{R}}^{2}$
(A)
$F(\sigma,\xi)\geq 0$ whenever $\sigma\leq f_{an}(\xi)$, and $F(\sigma,\xi)<0$
otherwise.
Then for every $u\in AC({\mathbb{R}}_{+})$ and for every
$y_{0}\in{\mathbb{R}}$, the function $t\rightarrow
H_{\circlearrowright}(y(t),u(t))$ with $H_{\circlearrowright}$ as in (20) and
$y:=\Phi(u,y_{0})$, is right differentiable and satisfies (5). Moreover, if
the anhysteresis function $f_{an}$ satisfies $f_{an}(0)=0$, then
$H_{\circlearrowright}\geq 0$ and the Duhem operator is clockwise (CW).
The proof of Theorem 1 is given in the B.
###### Remark 1
In addition to the result in Theorem 1, if $f_{1}$ and $f_{2}$ satisfy the
hypotheses given in Theorem 1, then for every $u\in AC({\mathbb{R}}_{+})$ and
$y_{0}\in{\mathbb{R}}$, the function $t\rightarrow
H_{\circlearrowright}(y(t),u(t))$ with $H_{\circlearrowright}$ as in (20) is
left-differentiable and satisfies
$\lim_{h\nearrow
0^{-}}\frac{H_{\circlearrowright}(y(t+h),u(t+h))-H_{\circlearrowright}(y(t),u(t))}{h}\leq
y(t)\dot{u}(t).$
The proof of this claim follows a similar line as that of Theorem 1.
In order to depict the storage function $H_{\circlearrowright}$ that is
constructed in Theorem 1, we recall again the example of the Duhem operator
$\Phi$ in Example 1 where $f_{1}=e^{0.5(-1.2y+u)}+0.83$ and
$f_{2}=e^{0.5(1.2y-u)}+0.83$, and it is shown in Figure 3. Based on the
functions $f_{an}$ and $\omega_{\Phi}(\cdot,y(t),u(t))$ as shown in Figure 3
and following the construction of the storage function $H_{\circlearrowright}$
as in (20), the first component on the RHS of (20) corresponds to the light
grey area in Figure 3. Correspondingly, the second component on the RHS of
(20) refers to the dark grey area in Figure 3. The summation of these two
areas gives the storage function $H_{\circlearrowright}$ for a given state
$(y(t),u(t))$ satisfying (5) according to Theorem 1.
The principle of the construction of $H_{\circlearrowright}$ in (20) can be
described in words as follows. From a given state $(y(0),u(0))$, let us define
the trajectory that crosses the anhysteresis curve at a given time $T$ by
applying either a monotonically increasing input signal $u(t)=u(0)+t$ or a
monotonically decreasing input signal $u(t)=u(0)-t$. Denote this trajectory by
$y$ and the intersecting point by $(y(T),u(T))$. Then the storage function
$H_{\circlearrowright}$ is given by the integral of the anhysteresis function
from $0$ to $u(T)$ minus the integral of $y$ from $0$ to $T$.
###### Proposition 1
Consider the Duhem operator $\Phi$ satisfying the hypotheses in Theorem 1.
Moreover, we assume that the anhysteresis function $f_{an}$ satisfies
$f_{an}(\xi)=0$ for all $\xi\in{\mathbb{R}}$. Then for every
$y_{0},u_{0}\in{\mathbb{R}}$, the function $H_{\circlearrowright}$ as in (20)
satisfies
$H_{\circlearrowright}(y_{0},u_{0})=\sup\limits_{\begin{subarray}{c}u\in
AC({\mathbb{R}}_{+})\\\
u(0)=u_{0}\end{subarray}}-\int_{0}^{T}{y(\tau)\dot{u}(\tau){\rm d}\hbox{\hskip
0.5pt}\tau},$
where $y:=\Phi(u,y_{0})$. In other words, $H_{\circlearrowright}$ defines the
available storage function (as discussed in [30]) where the supply rate is
given by $y\dot{u}$ (instead of $yu$ as in [30]).
* Proof.
As given in the first part of the proof of Theorem 1, we have
$\frac{{\rm d}\hbox{\hskip 0.5pt}}{{\rm d}\hbox{\hskip
0.5pt}t}H_{\circlearrowright}(y(t),u(t))=\dot{u}(t)y(t)-\int_{u(t)}^{u^{*}(t)}{\frac{{\rm
d}\hbox{\hskip 0.5pt}}{{\rm d}\hbox{\hskip
0.5pt}t}\omega_{\Phi}(\tau,y(t),u(t)){\rm d}\hbox{\hskip 0.5pt}\tau}.$ (21)
Integrating (21) from $t=0$ to $T$, we obtain
$H_{\circlearrowright}(T)-H_{\circlearrowright}(0)=\int_{0}^{T}{y(\tau)\dot{u}(\tau){\rm
d}\hbox{\hskip 0.5pt}\tau}-\int_{0}^{T}{\int_{u(t)}^{u^{*}}{\frac{{\rm
d}\hbox{\hskip 0.5pt}}{{\rm d}\hbox{\hskip
0.5pt}t}\omega_{\Phi}(\tau,y(t),u(t){\rm d}\hbox{\hskip 0.5pt}\tau{\rm
d}\hbox{\hskip 0.5pt}t,}}$
where $u^{*}=\Lambda(y(t),u(t))$ and we have used the shorthand notation of
$H_{\circlearrowright}(t):=H_{\circlearrowright}(y(t),u(t))$.
By rearranging the terms in this equation, we arrive at
$-\int_{0}^{T}{y(\tau)\dot{u}(\tau){\rm d}\hbox{\hskip
0.5pt}\tau}=H_{\circlearrowright}(0)-H_{\circlearrowright}(T)-\int_{0}^{T}{\int_{u(t)}^{u^{*}}{\frac{{\rm
d}\hbox{\hskip 0.5pt}}{{\rm d}\hbox{\hskip
0.5pt}t}\omega_{\Phi}(\tau,y(t),u(t){\rm d}\hbox{\hskip 0.5pt}\tau{\rm
d}\hbox{\hskip 0.5pt}t}}.$ (22)
The supremum of the LHS of (22) over all possible $u$ and $T$ defines the
available storage function where the supply rate is $y\dot{u}$. Note that this
supply rate is a particular class of the general supply rate as studied in
[27, 30]. Since the last two terms on the RHS of (22) is non-positive, we will
show that we can define $u$ and $T$ such that these two terms are equal to
zero, and thus the supremum of the LHS of (22) is equal to
$H_{\circlearrowright}(y(0),u(0))$, which is equivalent to
$H_{\circlearrowright}(y_{0},u_{0})$, i.e., $H_{\circlearrowright}$ is the
available storage function.
From a given initial condition $(y_{0},u_{0})$, let us introduce an input
signal $u(t)=u_{0}(T-t)+t\Lambda(y_{0},u_{0})$ for all $t\in[0,T]$ and
$u(t)=\Lambda(y_{0},u_{0})$ otherwise. This means that we have an input signal
$u$ which starts from $u_{0}$, ends at $\Lambda(y_{0},u_{0})$ at $t=T$ and
remains there for all $t>T$. Together with the corresponding signal
$y=\Phi(u,y_{0})$, we have $\Lambda(y(t),u(t))=\Lambda(y_{0},u_{0})$ for all
$t$, i.e. the intersecting point is always the same. Indeed, this follows from
the fact that $\Lambda(y(t),u(t))$ remains the same along the trajectories
that converge to the intersection point
$(\omega_{\Phi}(u^{*},y_{0},u_{0}),u^{*})$ where $u^{*}=\Lambda(y_{0},u_{0})$.
Following the same arguments as in the proof of Theorem 1 (c.f., the arguments
that lead to Eq. (40)), this input signal ensures that the last term on the
RHS of (22) is equal to zero. Since $u(T)=\Lambda(y_{0},u_{0})$ for all $t>T$,
we also have that $H_{\circlearrowright}(y(t),u(t))=0$ for all $t>T$, i.e. the
second term on the RHS of (22) is zero using such an input signal. Hence
$H_{\circlearrowright}$ as in (20) is an available storage function.
$\Box\Box\Box$
The results given in Theorem 1 can be slightly generalized in order to
incorporate the case when the Duhem hysteresis operator $\Phi$ has saturated
output. Consider the set $D\subset{\mathbb{R}}^{2}$ which contains all
relations of $\Phi$, i.e., ${\cal
R}_{y_{0},u}:=\\{(y(t),u(t))\in{\mathbb{R}}^{2}|y=\Phi(u,y_{0}),t\in{\mathbb{R}}_{+}\\}\subset
D$ holds for all $u\in AC({\mathbb{R}}_{+})$ and $(y_{0},u(0))\in D$. For
example, the set $D$ for the Dahl model in Section 2 is given by
$D=(-F_{C},F_{C})\times{\mathbb{R}}$. Using $D$, we can generalize Theorem 1
as follows.
###### Proposition 2
Consider the Duhem hysteresis operator $\Phi$ defined in (6) and (15) with
$C^{1}$ functions $F,G:D\to{\mathbb{R}}$ and with the traversing function
$\omega_{\Phi}$ and the anhysteresis function $f_{an}$. Assume that the
anhysteresis curve is in $D$ and there exists an intersecting function
$\Lambda$ (e.g., the hypotheses in Lemma 1 hold). Assume further that the
Assumption (A) holds for all $(\sigma,\xi)$ in $D$. Then for every $u\in
AC({\mathbb{R}}_{+})$ and $(y_{0},u(0))\in D$, the function $t\rightarrow
H_{\circlearrowright}(y(t),u(t))$ with $H_{\circlearrowright}$ as in (20) and
$y:=\Phi(u,y_{0})$ is right differentiable and satisfies (5). Moreover, if the
anhysteresis function $f_{an}$ satisfies $f_{an}(0)=0$, then
$H_{\circlearrowright}\geq 0$ and the Duhem operator is clockwise (CW).
The proof follows the same arguments as that of Theorem 1.
## 4 Examples
### 4.1 The function $H_{\circlearrowright}$ for the Dahl model
Recall the Dahl model as defined in Section 2.1 and consider the case when
$r=1$. In this case, the Dahl model can be reformulated into the Duhem
operator as in (6) with
$f_{1}(\sigma,\xi)=\rho\left(1-\frac{\sigma}{F_{c}}\right),\
f_{2}(\sigma,\xi)=\rho\left(1+\frac{\sigma}{F_{c}}\right),$ (23)
where $\rho>0$ and $F_{c}>0$. It is immediate to check that the conditions as
given in (7) are satisfied, which means there exists solution for this Duhem
operator for all positive time. The anhysteresis function of the Dahl model is
$f_{an}(\xi)=0$.
Calculating the curve $\omega_{\Phi}$, we have
$\omega_{\Phi}(\tau,y(t),u(t))=\left\\{\begin{array}[]{ll}F_{c}+(y(t)-F_{c})e^{\frac{\rho}{F_{c}}(u(t)-\tau)}&\
\tau\in[u(t),\ \infty),\\\
-F_{c}+(y(t)+F_{c})e^{\frac{\rho}{F_{c}}(\tau-u(t))}&\ \tau\in(-\infty,\
u(t)].\end{array}\right.$ (24)
From (23) and (24), it is immediate to see that the pair $(y,u)$ is well-
defined in $D=(-F_{C},F_{C})\times{\mathbb{R}}$. The intersecting function
$\Lambda(y(t),u(t))$ is given as follows
$\Lambda(y(t),u(t))=\left\\{\begin{array}[]{ll}u(t)+\frac{F_{c}}{\rho}\ln{\frac{F_{c}}{y(t)+F_{c}}}&\
y(t)\geq 0,\\\ u(t)-\frac{F_{c}}{\rho}\ln{\frac{-F_{c}}{y(t)-F_{c}}}&\
y(t)<0.\end{array}\right.$ (25)
Since $f_{1}$ and $f_{2}$ in (23) satisfy the Assumption (A) for all
$(\sigma,\xi)\in D$, the result in Proposition 2 holds.
By denoting $u^{*}(t)=\Lambda(y(t),u(t))$, we can compute explicitly the
function $H_{\circlearrowright}$ as follows
$\displaystyle H_{\circlearrowright}(y(t),u(t))$
$\displaystyle=\left\\{\begin{array}[]{ll}-F_{c}(u(t)-u^{*}(t))+\frac{F_{c}}{\rho}(y(t)+F_{c})(1-e^{\frac{\rho}{F_{c}}(u^{*}(t)-u(t))})&\
y(t)\geq 0\\\
F_{c}(u(t)-u^{*}(t))+\frac{F_{c}}{\rho}(y(t)-F_{c})(e^{\frac{\rho}{F_{c}}(u(t)-u^{*}(t))}-1)&\
y(t)<0\end{array}\right.$ (28)
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{F_{C}^{2}}{\rho}\ln{\frac{F_{c}}{y(t)+F_{c}}}+\frac{F_{C}}{\rho}y(t)&\
y(t)\geq 0\\\
\frac{F_{C}^{2}}{\rho}\ln{\frac{-F_{c}}{y(t)-F_{c}}}-\frac{F_{C}}{\rho}y(t)&\
y(t)<0.\end{array}\right.$ (31)
Indeed, it can be checked that
$\dot{H}_{\circlearrowright}\leq\dot{u}(t)y(t)$.
### 4.2 Stability analysis of a second-order mechanical system with
hysteretic friction
Now, let us consider an example of a mechanical system with the Dahl friction
model given by $m\ddot{x}+d\dot{x}+kx+\Phi(x)=0$, where $m>0$, $d>0$, $k>0$,
the hysteresis operator $\Phi$ is given as in (23) with $\rho>0$ and
$F_{c}>0$. As discussed before, the functions $f_{1}$ and $f_{2}$ satisfy the
hypotheses of Proposition 2.
The state space representation of the system is given as follows
$\left.\begin{array}[]{rl}\dot{x}_{1}&=x_{2},\\\
\dot{x}_{2}&=-\frac{k}{m}x_{1}-\frac{d}{m}x_{2}-\frac{x_{3}}{m},\\\
\dot{x}_{3}&=\rho\left(1-\frac{x_{3}}{F_{C}}\right)x_{2+}+\rho\left(1+\frac{x_{3}}{F_{C}}\right)x_{2-}.\end{array}\right.$
Using
$V(x_{1},x_{2},x_{3})=\frac{1}{2}kx_{1}^{2}+\frac{1}{2}mx_{2}^{2}+H_{\circlearrowright}(x_{3},x_{1})$,
where $H_{\circlearrowright}$ is as in (31) and satisfies (5), a routine
calculation shows that
$\displaystyle\dot{V}$ $\displaystyle\leq-x_{2}x_{3}-dx_{2}^{2}+x_{3}x_{2}$
$\displaystyle=-dx_{2}^{2}.$
Since the relations of the corresponding Dahl operator (i.e. the set ${\cal
R}_{y_{0},u}:=\\{(y(t),u(t))|y=\Phi(u,y_{0})\\}$) is contained in
$(-F_{C},F_{C})\times{\mathbb{R}}$ for all $y_{0}\in(-F_{C},F_{C})$ and $u\in
AC({\mathbb{R}}_{+})$, then it implies that $x_{3}$ (which is the output of
the Dahl operator) is bounded and lies in the interval $(-F_{C},F_{C})$.
Additionally, we have $V$ which is lower bounded and radially unbounded in the
first and second arguments, i.e. $V(x_{1},x_{2},x_{3})\rightarrow\infty$ as
$\left\|\left.\begin{smallmatrix}x_{1}\\\
x_{2}\end{smallmatrix}\right.\right\|\rightarrow\infty$. Thus $\dot{V}\leq-
dx_{2}^{2}$ implies that the state trajectory $(x_{1},x_{2})$ is bounded.
Moreover, using the boundedness of $(x_{1},x_{2})$ and the boundedness of
$x_{3}$, an application of the Lasalle’s invariance principle shows that
$(x_{1},x_{2},x_{3})$ converges to the largest invariant set where $x_{2}=0$.
By analyzing the corresponding state equations, this invariant set is given by
$\\{(x_{1},x_{2},x_{3})|kx_{1}=-x_{3},\ x_{2}=0\\}$.
## 5 Conclusion
In this paper, we have investigated the clockwise I/O dynamics of a class of
Duhem hysteresis operator by obtaining sufficient conditions for the Duhem
operators to be CW. The CW property is obtained via the construction of a
suitable function satisfying the CW dissipation inequality which can be useful
for studying stability of systems having CW hysteretic element, such as,
mechanical systems with hysteretic friction. The sufficient conditions for CW
I/O dynamics incorporates also the knowledge of anhysteresis function which is
commonly neglected in the literature of hysteretic systems. For systems
identification of hysteresis systems, the results provide additional
characterization of the Duhem operators that can be used to restrict the class
of the Duhem operators which will be fitted with the measurement data.
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## Appendix A Proof of Lemma 1
* Proof.
The proof is similar to the proof of [12, Lemma 3.1]. Consider the continuous
function $\varphi:{\mathbb{R}}^{3}\rightarrow{\mathbb{R}}$ defined as
$\varphi(\xi,y_{0},u_{0})=\omega_{\Phi}(\xi,y_{0},u_{0})-f_{an}(\xi)$.
Consider also $A_{0}$ and $A_{1}$ the two subsets of ${\mathbb{R}}^{3}$
defined as,
$\displaystyle A_{0}=\\{(\xi,y_{0},u_{0})\in{\mathbb{R}}^{3},\
y_{0}>f_{an}(u_{0})\ ,\ \xi<u_{0}\\}\ ,$ $\displaystyle
A_{1}=\\{(\xi,y_{0},u_{0})\in{\mathbb{R}}^{3},\ y_{0}<f_{an}(u_{0})\ ,\
\xi>u_{0}\\}\ .$
Note that the function $f_{an}$ being strictly increasing by assumption,
implies that these sets are open sets. Also, the function $\omega_{\Phi}$
satisfies
$\displaystyle\frac{\partial\omega_{\Phi}}{\partial\xi}(\xi,y_{0},u_{0})$
$\displaystyle=f_{2}(\omega_{\Phi}(\xi,y_{0},u_{0}),\xi)\qquad\forall(\xi,y_{0},u_{0})\
\in\ A_{0}\ ,$
$\displaystyle\frac{\partial\omega_{\Phi}}{\partial\xi}(\xi,y_{0},u_{0})$
$\displaystyle=f_{1}(\omega_{\Phi}(\xi,y_{0},u_{0}),\xi)\qquad\forall(\xi,y_{0},u_{0})\
\in\ A_{1}\ .$
Consequently, $\omega_{\Phi}(\xi,y_{0},u_{0})$ is the solution of ordinary
differential equations computed from $C^{1}$ vector field. With [10, Theorem
V.3.1], it implies that $\omega_{\Phi}$ is a $C^{1}$ function in $A_{0}\cup
A_{1}$. Moreover, the function $f_{an}$ being $C^{1}$ implies that the
function $\varphi$ is $C^{1}$ in $A_{0}\cup A_{1}$. With (17) and (18), the
function $\varphi$ satisfies,
$\displaystyle\frac{\partial\varphi}{\partial\xi}(\xi,y_{0},u_{0})$
$\displaystyle>$ $\displaystyle\epsilon\neq 0\ ,\ \forall(\xi,y_{0},u_{0})\
\in\ A_{0}\cup A_{1}\ .$
Consequently, $\varphi$ is a strictly increasing function in its first
argument in the set $A_{0}\cup A_{1}$. This also implies that,
$\displaystyle\varphi(\xi,y_{0},u_{0})$
$\displaystyle<\varphi(u_{0},y_{0},u_{0})+\epsilon(\xi-u_{0})\ $
$\displaystyle\qquad\forall(\xi,y_{0},u_{0})\ \in\ A_{0}\ ,$
$\displaystyle\varphi(\xi,y_{0},u_{0})$
$\displaystyle>\varphi(u_{0},y_{0},u_{0})+\epsilon(\xi-u_{0})\ $
$\displaystyle\qquad\forall(\xi,y_{0},u_{0})\ \in\ A_{1}\ .$
Note that if $y_{0}>f_{an}(u_{0})$, then $\varphi(u_{0},y_{0},u_{0})>0$ and
consequently there exists a unique real number $u^{*}$ such that
$\varphi(u^{*},y_{0},u_{0})=0$ and $(u^{*},y_{0},u_{0})\in A_{0}$. On the
other hand, if $y_{0}<f_{an}(u_{0})$, then $\varphi(u_{0},y_{0},u_{0})<0$ and
consequently there exists a unique real number $u^{*}$ such that
$\varphi(u^{*},y_{0},u_{0})=0$ and $(u^{*},y_{0},u_{0})\in A_{1}$. Therefore,
by denoting $\Lambda(y_{0},u_{0})=u^{*}$, by employing the implicit function
theorem and using the fact that $\varphi$ is $C^{1}$, it can be shown that
$\Lambda$ is $C^{1}$.
$\Box\Box\Box$
## Appendix B Proof of Theorem 1
* Proof.
The proof of Theorem 1 follows the same line as in our previous work [12]. In
the first part of the proof we will prove that for all $t\in{\mathbb{R}}_{+}$,
$\dot{H}_{\circlearrowright}\big{(}y(t),u(t)\big{)}$ exists and satisfies (5).
In the second part we show the non-negativeness of
$H_{\circlearrowright}\big{(}y(t),u(t)\big{)}$.
To show that $H_{\circlearrowright}$ exists, let us denote
$u^{*}:=\Lambda(y,u)$. Using the Leibniz derivative rule, we have
$\displaystyle\frac{{\rm d}\hbox{\hskip 0.5pt}}{{\rm d}\hbox{\hskip
0.5pt}t}H_{\circlearrowright}(y(t),u(t))$ $\displaystyle=\frac{{\rm
d}\hbox{\hskip 0.5pt}}{{\rm d}\hbox{\hskip
0.5pt}t}\left[\int_{0}^{\Lambda(y(t),u(t))}{f_{an}(\tau){\rm d}\hbox{\hskip
0.5pt}\tau}-\int_{u(t)}^{\Lambda(y(t),u(t))}{\omega_{\Phi}(\tau,y(t),u(t)){\rm
d}\hbox{\hskip 0.5pt}\tau}\right]$
$\displaystyle=\dot{\Lambda}(y(t),u(t))f_{an}(\Lambda(y(t),u(t)))-\dot{\Lambda}(y(t),u(t))\omega_{\Phi}(\Lambda(y(t),u(t)),y(t),u(t))$
$\displaystyle+\dot{u}(t)\omega_{\Phi}(u(t),y(t),u(t))-\int_{u(t)}^{\Lambda(y(t),u(t))}{\frac{{\rm
d}\hbox{\hskip 0.5pt}\omega_{\Phi}(\tau,y(t),u(t))}{{\rm d}\hbox{\hskip
0.5pt}t}{\rm d}\hbox{\hskip 0.5pt}\tau}$
$\displaystyle=\dot{u}(t)y(t)-\int_{u(t)}^{u^{*}(t)}{\frac{{\rm d}\hbox{\hskip
0.5pt}}{{\rm d}\hbox{\hskip 0.5pt}t}\omega_{\Phi}(\tau,y(t),u(t)){\rm
d}\hbox{\hskip 0.5pt}\tau},$ (32)
where $u^{*}(t)=\Lambda(y(t),u(t))$ and the last equation is due to
$\omega_{\Phi}(u(t),y(t),u(t))=y(t)$ and by the hypothesis given in Lemma 1.
The first term in the RHS of (32) exists for all $t\geq 0$ since $u(t)$
satisfies (6). In order to get (5), it remains to check whether the last term
of (32) exists, is finite and satisfies
$\int_{u(t)}^{u^{*}(t)}{\frac{{\rm d}\hbox{\hskip 0.5pt}}{{\rm d}\hbox{\hskip
0.5pt}t}\omega_{\Phi}(\tau,y(t),u(t)){\rm d}\hbox{\hskip 0.5pt}\tau}\geq 0.$
(33)
It suffices to show that, for every $\tau\in[u(t),u^{*}(t)]$, the following
limit
$\lim_{\epsilon\searrow
0^{+}}\frac{1}{\epsilon}[\omega_{\Phi}(\tau,y(t+\epsilon),u(t+\epsilon))-\omega_{\Phi}(\tau,y(t),u(t))]$
(34)
exist and the limit of (34) is greater or equal to zero when $u^{*}(t)>u(t)$
and the limit is less or equal to zero elsewhere.
For any $\epsilon\geq 0$, let us introduce the continuous function
$\omega_{\epsilon}:{\mathbb{R}}\rightarrow{\mathbb{R}}$ by
$\omega_{\epsilon}(\tau)=\omega_{\Phi}(\tau,y(t+\epsilon),u(t+\epsilon)).$
(35)
More precisely, using (16), $\omega_{\epsilon}$ is the unique solution of
$\omega_{\epsilon}(\tau)=\left\\{\begin{array}[]{l}y(t+\epsilon)+\displaystyle\int_{u(t+\epsilon)}^{\tau}{f_{1}(\omega_{\epsilon}(s),s){\rm
d}\hbox{\hskip 0.5pt}s}\quad\forall\tau\geq u(t+\epsilon)\\\\[15.00002pt]
y(t+\epsilon)+\displaystyle\int_{u(t+\epsilon)}^{\tau}{f_{2}(\omega_{\epsilon}(s),s){\rm
d}\hbox{\hskip 0.5pt}s}\quad\forall\tau\leq u(t+\epsilon).\end{array}\right.$
(36)
Note that $\omega_{0}(\tau)=\omega_{\Phi}(\tau,y(t),u(t))$ as in (16) for all
$\tau\in{\mathbb{R}}$ and
$\omega_{\epsilon}(u(t+\epsilon))=y(t+\epsilon)\qquad\forall\;\epsilon\;\in\;{\mathbb{R}}_{+}\
.$ (37)
In order to show the existence of (34) and the validity of (33), we consider
several cases depending on the sign of $\dot{u}(t)$ and $F(y(t),u(t))$. It can
be checked that the hypothesis (A) on $F$ implies that $f_{1}(y(t),u(t))\geq
f_{2}(y(t),u(t))$ whenever $y(t)\leq f_{an}(u)$, and
$f_{1}(y(t),u(t))<f_{2}(y(t),u(t))$ otherwise.
First, we assume that $\dot{u}(t)>0$ and $y(t)\geq f_{an}(u(t))$. In this
case, according to Lemma 1, we have $u^{*}(t)<u(t)$. Since $\dot{u}(t)>0$,
there exists $\gamma>0$ such that $\tau\leq u(t)<u(s)$ for all $s$ in
$(t,t+\gamma)$. It follows from (36) and assumption (A) that for every
$\epsilon\in(0,\gamma)$:
$\displaystyle\frac{{\rm d}\hbox{\hskip 0.5pt}\omega_{\epsilon}(u(s))}{{\rm
d}\hbox{\hskip 0.5pt}s}$
$\displaystyle=f_{2}(\omega_{\epsilon}(u(s)),u(s))\;\dot{u}(s)$
$\displaystyle\geq f_{1}(\omega_{\epsilon}(u(s)),u(s))\;\dot{u}(s)\quad\forall
s\in[t,t+\epsilon],$
and the function $\omega_{0}$ satisfies
$\frac{{\rm d}\hbox{\hskip 0.5pt}\omega_{0}(u(s))}{{\rm d}\hbox{\hskip
0.5pt}s}\;=\;f_{1}(y(s),u(s))\;\dot{u}(s)\qquad\forall
s\;\in\;[t,t+\epsilon].$
Since the functions $\epsilon\mapsto w_{0}(u(t+\epsilon))$ and
$\epsilon\mapsto y(t+\epsilon)$ with $\epsilon\in(0,\gamma]$ are two $C^{1}$
functions which are solutions of the same locally Lipschitz ODE and with the
same initial value. By uniqueness of solution, we get
$\omega_{0}(u(t+\epsilon))=y(t+\epsilon)$.
This together with the fact that
$\omega_{\epsilon}(u(t+\epsilon))=y(t+\epsilon)$ and using the comparison
principle (in reverse direction), we get that for every
$\epsilon\in[0,\gamma)$:
$\omega_{\epsilon}(u(s))\;\leq\;\omega_{0}(u(s))\qquad\forall\;s\;\in\;[t,t+\epsilon].$
Since the two functions $\omega_{\epsilon}(\tau)$ and $\omega_{0}(\tau)$ for
$\tau\in[u^{*}(t),u(t)]$ are two solutions of the same ODE, it follows that
111Otherwise there exist $\tau_{1}<\tau_{2}$ such that
$\omega_{\epsilon}(\tau_{1})\;=\;\omega_{0}(u(\tau_{1}))$ and
$\omega_{\epsilon}(\tau_{2})\;>\;\omega_{0}(u(\tau_{2}))$ which contradict the
uniqueness of the solution of the locally Lipschitz ODE.
$\omega_{\epsilon}(\tau)\;\geq\;\omega_{0}(\tau)$ and we get that if it
exists:
$\lim_{\epsilon\searrow
0^{+}}\frac{1}{\epsilon}[\omega_{\epsilon}(\tau)-\omega_{0}(\tau)]\;\leq\;0\qquad\forall\tau\in[u^{*}(t),u(t)].$
(38)
Then it is clear that
$\lim_{\epsilon\searrow
0^{+}}\frac{1}{\epsilon}[\omega_{\epsilon}(\tau)-\omega_{0}(\tau)]\;\geq\;0\qquad\forall\tau\in[u(t),u^{*}(t)].$
(39)
In the following, we show the existence of the limit given in (38) by
computing a bound on the function
$\epsilon\mapsto\frac{1}{\epsilon}[\omega_{\epsilon}(\tau)-\omega_{0}(\tau)]$.
Note that for every $\epsilon\in[0,\gamma]$,
$\displaystyle|\omega_{\epsilon}(\tau)-\omega_{0}(\tau)|\leq|y(t+\epsilon)-y(t)|+\left|\int_{u(t+\epsilon)}^{u(t)}f_{2}(\omega_{\epsilon}(s),s)\,{\rm
d}\hbox{\hskip 0.5pt}s\right|$
$\displaystyle\qquad\qquad+\left|\int_{u(t)}^{\tau}f_{2}(\omega_{\epsilon}(s),s)-f_{2}(\omega_{0}(s),s)\,{\rm
d}\hbox{\hskip 0.5pt}s\right|$
$\displaystyle\leq|y(t+\epsilon)-y(t)|+\int_{u(t)}^{u(t+\epsilon)}|f_{2}(\omega_{\epsilon}(s),s)|\,{\rm
d}\hbox{\hskip 0.5pt}s$
$\displaystyle\qquad\qquad+\int_{\tau}^{u(t)}|f_{2}(\omega_{\epsilon}(s),s)-f_{2}(\omega_{0}(s),s)|\,{\rm
d}\hbox{\hskip 0.5pt}s,$
for all $\tau\in[u^{*}(t),u(t)]$. By the locally Lipschitz property of $f_{2}$
and by the boundedness of $\omega_{\epsilon}$ on $[\tau,u(t)]$ for all
$\epsilon\in[0,\gamma]$, it can be shown that there exists $\alpha$, such that
$\alpha$ is a bound of $f_{2}$ on a compact set. Then
$|\omega_{\epsilon}(\tau)-\omega_{0}(\tau)|\leq|y(t+\epsilon)-y(t)|\\\
+\int_{\tau}^{u(t)}\,L\,|\omega_{\epsilon}(s)-\omega_{0}(s)|\,{\rm
d}\hbox{\hskip 0.5pt}s+\alpha|u(t+\epsilon)-u(t)|\ ,$
where $L$ is the Lipschitz constant of $f_{2}$ on
$[\omega_{\min{}},\omega_{\max{}}]\times[\tau,u(t)]$ with
$\displaystyle\omega_{\min{}}$
$\displaystyle=\min_{(c,s)\in[0,\gamma]\times[\tau,u(t)]}\omega_{c}(s),$
$\displaystyle\omega_{\max{}}$
$\displaystyle=\max_{(c,s)\in[0,\gamma]\times[\tau,u(t)]}\omega_{c}(s)\ .$
With Gronwall’s lemma, this implies that for every $\epsilon\in[0,\gamma]$
$|\omega_{\epsilon}(\tau)-\omega_{0}(\tau)|\leq\exp((u(t)-\tau)L)\Big{[}|y(t+\epsilon)-y(t)|+\alpha|u(t+\epsilon)-u(t)|\Big{]},$
for all $\tau\in[u^{*}(t),u(t)]$. Hence
$\lim_{\epsilon\searrow
0^{+}}\frac{1}{\epsilon}|\omega_{\epsilon}(\tau)-\omega_{0}(\tau)|\leq\exp((u(t)-\tau)L)\Big{[}|f_{1}(y(t),u(t))|+\alpha\Big{]}\,\dot{u}(t),$
for all $\tau\in[u^{*}(t),u(t)]$. Consequently the limit given in (38) exists.
It implies that the inequality (33) holds when $\dot{u}(t)>0$ and $y(t)\geq
f_{an}(u(t))$.
For the next case, we assume that $\dot{u}(t)>0$ and $y(t)<f_{an}(u(t))$.
Again, according to Lemma 1, we have $u^{*}(t)>u(t)$. Since for every
$\epsilon\in(0,\gamma]$ the two functions $\omega_{\epsilon}(\tau)$ and
$\omega_{0}(\tau)$ satisfy the same ODE for222we have for all
$\tau\in[u(t+\epsilon),u^{*}(t)]$ : $\frac{{\rm d}\hbox{\hskip
0.5pt}\omega_{\epsilon}(\tau)}{{\rm d}\hbox{\hskip
0.5pt}\tau}\;=\;f_{1}(\omega_{\epsilon}(\tau),\tau)\quad,\qquad\frac{{\rm
d}\hbox{\hskip 0.5pt}\omega_{0}(\tau)}{{\rm d}\hbox{\hskip
0.5pt}\tau}\;=\;f_{1}(\omega_{0}(\tau),\tau)$
$\tau\in[u(t+\epsilon),u^{*}(t)]$, we have
$\omega_{\epsilon}(\tau)=\omega_{0}(\tau)\qquad\forall\tau\in[u(t+\epsilon),u^{*}(t)],$
for all $\epsilon\in[0,\gamma]$. This implies that
$\lim_{\epsilon\searrow
0^{+}}\frac{1}{\epsilon}[\omega_{\epsilon}(\tau)-\omega_{0}(\tau)]\;=\;0.$
(40)
We can use similar arguments to prove that (33) is satisfied when
$\dot{u}(t)<0$.
Finally, when $\dot{u}(t)=0$, we simply get
$\lim_{\epsilon\searrow
0^{+}}\frac{1}{\epsilon}|\omega_{\epsilon}(\tau)-\omega_{0}(\tau)|=0,$
by continuity of the above bound.
For the second step, we need to show that $H_{\circlearrowright}$ is non-
negative. Consider the case when $y(t)\geq f_{an}(u(t))$, we have
$u^{*}(t)<u(t)$ and $\omega_{\Phi}(\tau)\geq f_{an}(\tau)$ for all
$\tau\in[u^{*}(t),u(t)]$ by Lemma 1. Since $f_{an}(\tau)$ belongs to the
sector $[0,\ \infty)$ for all $\tau\in{\mathbb{R}}$, we have
$H_{\circlearrowright}(y(t),u(t))=\int_{0}^{u(t)}{f_{an}(\tau){\rm
d}\hbox{\hskip
0.5pt}\tau}+\int_{u(t)}^{u^{*}(t)}{f_{an}(\tau)-\omega_{\Phi}(\tau,y(t),u(t)){\rm
d}\hbox{\hskip 0.5pt}\tau}\geq 0.$
In case when $y(t)<f_{an}(u(t))$, we can show the non-negativeness of $H$ by
using similar arguments.
$\Box\Box\Box$
|
arxiv-papers
| 2012-01-10T12:00:24 |
2024-09-04T02:49:26.106607
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ruiyue Ouyang, Vincent Andrieu, Bayu Jayawardhana",
"submitter": "Bayu Jayawardhana",
"url": "https://arxiv.org/abs/1201.2035"
}
|
1201.2139
|
# The Electronic Correlation Strength of Pu
A. Svane svane@phys.au.dk Department of Physics and Astronomy, Aarhus
University, DK 8000 Aarhus C, Denmark R. C. Albers Theoretical Division, Los
Alamos National Laboratory, Los Alamos, NM 87545 USA N. E. Christensen
Department of Physics and Astronomy, Aarhus University, DK 8000 Aarhus C,
Denmark M. van Schilfgaarde Department of Physics, King’s College London,
The Strand, London WC2R 2LS, UK A. N. Chantis American Physical Society, 1
Research Road, Ridge, New York 11961, USA Jian-Xin Zhu Theoretical Division,
Los Alamos National Laboratory, Los Alamos, NM 87545 USA
###### Abstract
An electronic quantity, the correlation strength, is defined as a necessary
step for understanding the properties and trends in strongly correlated
electronic materials. As a test case, this is applied to the different phases
of elemental Pu. Within the $GW$ approximation we have surprisingly found a
“universal” scaling relationship, where the f-electron bandwidth reduction due
to correlation effects is shown to depend only on the local density
approximation bandwidth and is otherwise independent of crystal structure and
lattice constant.
###### pacs:
71.10.-w, 71.27.+a
## I Introduction
Many technologically important materials have strong electron-electron
correlation effects. They exhibit large anomalies in their physical properties
when compared with materials that are weakly correlated, and have significant
deviations in their electronic-structure from that predicted by conventional
band-structure theory based on the local-density approximation (LDA). Because
the anomalies and deviations are caused by electronic correlation effects,
which often dominate the physics of these materials, in this paper we define a
quantity that we call the “correlation strength,” or $C$, as a necessary step
in order to be able to describe trends and bring order into our understanding
of correlated materials. We emphasize the word “quantity” since a quantitative
measure is needed to answer the question, “How strong are the electronic
correlations?” Without some understanding of how big this is, it is not
possible to make sense of the properties of these materials. In this context,
“correlation” is defined in a way somewhat different from how it is sometimes
used (e.g., in the term “exchange-correlation potential”). By “correlation” we
specifically mean “correlation beyond LDA theory.” This usage reflects the way
the term is often loosely used in common terminology in the area of strongly
correlated electronic systems.
To create a new quantity requires determining a “scale” by which to measure
its size. In principle, any experimental or theoretical property (e.g.,
specific heat) that monotonically increases or decreases over the full range
of correlation effects, where we define correlation strength to lie between
zero for none and one for full correlation, can be used as a measure of this
quantity. Hence correlation strength is an indeterminant quantity and depends
on the property used to define it. However, this does not matter since only
relative rather than any absolute strength is important for characterizing
these materials and for predicting trends in their properties. Any measure
based on one property can easily be converted to that based on another
property. In this paper we develop a theoretical correlation strength based on
the $GW$ approximation Hedin (1965); Hedin and Lundqvist (1969); Hedin (1999);
Aryasetiawan and Gunnarsson (1998) to electronic-structure theory and apply it
to plutonium, Chantis _et al._ (2009); Kutepov _et al._ (2012) which is
known to have significant correlation effects. The $GW$ approximation is named
for the correction term in this theory, which is a Green’s function G times a
screened Coulomb interaction W. We also demonstrate a scaling relationship
that is universal in that it is independent of crystal structure and atomic
volume. The ideas in this paper could certainly be modified and generalized to
be able to treat other types of correlated materials (e.g., spin-fluctuation
or high-temperature superconducting materials) by using other electronic
properties to determine a correlation strength and by using more sophisticated
theoretical techniques than are considered here.
Of course, there is a long history in physics and chemistry of using various
quantities to predict materials trends. For example, with respect to the
actinides, in 1970 HillHill (1970) plotted the magnetic and superconducting
transition temperatures of actinide compounds as a function of the actinide-
actinide nearest-neighbor distance. These “Hill plots” brought some sensible
order into what had previously been seen as a somewhat random occurrence of
these various ground states, and also provided some degree of predictability,
in that superconducting compounds tended to occur for short actinide spacings
and magnetic compounds at large spacings.Boring and Smith (2000) The plots
were intuitively based on the idea that $f$-wave-function overlap was the key
factor determining the stability of the relative ground states. These plots
failed for heavy-fermion compoundsBoring and Smith (2000) and our
understanding of electronic structure has now advanced to the point where we
realize that at large actinide nearest-neighbor distances the $f$ electrons
tend to hop predominantly through hybridizations with other orbitals on nearby
atoms rather than through a direct $f$-$f$ hybridization.
Another important actinide trend was developed by Smith and Kmetko.Smith and
Kmetko (1983) They showed that the crystal structures of the actinides can be
plotted as a continuous function of atomic number ($Z$), with alloys filling
in between the atomic numbers of the pure elements. When plotted in this way,
one obtains “connected binary alloy phase diagrams for the light actinides,”
which provide a clear picture of the trends and relationships between the
crystal structures of all the light actinides “at a glance.”
More generally, in materials science, many different variables have been used
in an attempt to understand systematic trends in crystal structures among
classes of different compounds. Such variables have included electronegativity
differences, covalent and ionic contribution to the average spectroscopic
energy gap, and various types of core, ionic, and metallic radii. These have
been reviewed in a review article on “Structure Mapping” by Pettifor;Pettifor
(2000) see also Refs. Villars, 1983; Pettifor, 1986; Christensen _et al._ ,
1987; Pettifor, 1988; Fischer _et al._ , 2006. However, these methods are not
relevant for our purposes, since, as we shall show below, correlation effects
are more important than crystal structure for determining the properties of
many actinide metals.
Among different classes of correlated materials, superconducting transition
pressures have often been plotted versus either specific structural properties
or some characteristic correlated quantity. These are too numerous to report
in full. A typical example are trends in superconducting transition
temperaturesFischer _et al._ (2007); Takahashi _et al._ (2008) with numbers
of planar (layered or two-dimensional) structural units (e.g., CuO2 or FeAs
planes), and similarly for representative classes of some heavy-fermion
superconductors (e.g., CeMIn5 and PuMGa5 for M=Co, Rh, Ir, also including c/a
structural anisotropiesBauer _et al._ (2004)). Closer in spirit to this paper
are trends in superconducting transition temperature versus characteristic
spin-fluctuation energies, except that the trends were all based on
experimental measurements rather than theoretical input.Bauer _et al._
(2004); Sarrao and Thompson (2007); Pfleiderer (2009)
Perhaps the closest analog to the ideas of our paper is the correlation
between crystal structure and $d$-occupation numbers in rare-earth systems
(including under pressure).Duthie and Pettifor (1977); Skriver (1985) In this
case theoretical calculations are required to determine the number of occupied
$d$ electrons as a function of $d$ element and volume per atom (which can be
equated to pressure). Given this input, however, the correct crystal structure
can then usually be predicted. What is different about our approach is that we
believe that not just one property such as crystal structure or transition
temperature, but many properties of actinide metals will follow trends based
on our correlation scale (see below).
The outline of the paper is as follows: In Sec. II, a theoretical definition
of the correlation scale is presented. It is expressed in terms of the
effective band width based on the parameter-free LDA and $GW$ approaches. In
Sec. III, we apply the scenario to determine the correlation strength in
elemental Pu solids. A universal scaling relationship is obtained, where the
$f$-electron bandwidth reduction due to correlation effects is shown to depend
only upon the LDA bandwidth and is otherwise independent of crystal structure
and lattice constant. The same type of trend is also found for the
$d$-electron systems. A concluding summary is given in Sec. IV.
## II Theoretical Method
Our meaning of correlation makes it necessary to use a theory that includes
correlation effects that go beyond those included by the LDA approximation in
order to determine a theoretical correlation strength. This is challenging,
since the most sophisticated treatments of correlation effects have
historically been mainly confined to abstract theoretical models, and have
parameterized the electronic structure in such an oversimplified manner that
the connection with actual materials examined experimentally was often
somewhat vague.Albers _et al._ (2009) In the last decade, however, great
progress has been made in this area, especially those involving dynamical
mean-field theory (DMFT) Georges _et al._ (1996); Kotliar _et al._ (2006);
Held _et al._ (2008); Kuneš _et al._ (2010) techniques, and strong
correlation effects are beginning to be integrated into true first principles
methods. To achieve this, instead of using ad hoc Hubbard Hamiltonians that
were essentially added without derivation to local density approximation
calculations, more recent methods have been attempting to explicitly calculate
screened Coulomb interactions directly in the random phase approximation (RPA)
and related approximations. These techniques have been recently reviewed by
Imada and Miyake.Imada and Miyake (2010) One direction that has been
particularly fruitful recently is the construction of low-energy effective
models involving a downfolding of the electronic states and using localized
Wannier orbitals and ab initio real-space tight-binding models. States far
from the Fermi energy can be treated with conventional LDA-like techniques,
while correlation effects are taken explicitly into account for the important
states around the Fermi energy. Usually constrained RPA (or cRPA) methods are
used to screen the Coulomb interactions. Such methods have achieved a fair
degree of success for semiconductors, 3$d$ transition-metal oxides, iron-based
superconductors, and organic superconductors.
However, these methods rely upon being able to separate the electronic
structure into some electrons belonging to fairly isolated bands near the
Fermi level and the rest to band degrees of freedom far from the Fermi level.
For metals, as we are considering, such methods therefore appear to be
unlikely to be successful. Another approach,Sun and Kotliar (2002); Biermann
_et al._ (2003) which seems more suitable to our case, is GW+DMFT. This has
also been reviewed in Ref. Imada and Miyake, 2010. Such a method involves $GW$
(or RPA-like) methods for calculating the Coulomb interactions that are then
integrated with DMFT techniques. In the full implementation the entire scheme
would be made self-consistent and would be independent of the initial $GW$
calculations used to initiate the method. In the initial description of the
methodBiermann _et al._ (2003) only a simplified one-shot approach was
applied to nickel. Since the initial papers outlining the methodology, almost
no progress has been made, perhaps indicating the difficulty of this approach.
Very recently, however, a more sophisticated implementationTomczak _et al._
(2012) has been applied to SrVO3. While these calculations are not yet fully
self-consistent, they may stimulate more interest in pushing through the
technical issues involved in implementing this method.
Since there is not yet widely available a suitable code that involves these
more sophisticated treatments of correlation for the metallic systems that we
are interested in, we have used the $GW$ methodHedin (1965); Aryasetiawan and
Gunnarsson (1998); Hedin (1999) as a theoretical method for estimating
correlation effects. Although this is a low-order approximation that
definitely fails for very strong correlation effects, it is sufficient for our
purposes as a way to estimate correlation deviations from LDA band-structure
theory, and in particular for the main purpose of our work, which is to show
that it is possible and useful to define a new quantity, which we call
correlation strength, in order to be able to place new materials in their
proper physics context and hence to be able to observe important trends in
their properties.
Among the available $GW$ codes, we have used the quasiparticle self-consistent
$GW$ approximation (QSGW).van Schilfgaarde _et al._ (2006a, b); Kotani _et
al._ (2007) The $GW$ approximation, itself, can be viewed as the first term in
the expansion of the nonlocal energy-dependent self-energy
$\Sigma(\bf{r},\bf{r}^{\prime},\omega)$ in the screened Coulomb interaction
$W$. From a more physical point of view it can also be interpreted as a
dynamically screened Hartree-Fock approximation plus a Coulomb hole
contribution.Hedin (1999); Aryasetiawan and Gunnarsson (1998) Therefore, $GW$
is a well defined perturbation theory. In its usual implemention, sometimes
called the “one-shot” approximation, it depends on the one-electron Green’s
functions which use LDA eigenvalues and eigenfunctions, and hence the results
can depend on this choice. Unfortunately, as correlations become stronger
serious practical and formal problems can arise in this approximation. van
Schilfgaarde _et al._ (2006b) However, Kotani et al. Kotani _et al._ (2007)
have provided a way to surmount this difficulty, by using a self-consistent
one-electron Green’s function that is derived from the self-energy (the quasi-
particle eigenvalues and eigenfunctions) instead of LDA as the starting point.
In the literature, it has been demonstrated that the QSGW form of $GW$ theory
reliably describes a wide range of semiconductors, van Schilfgaarde _et al._
(2006a); Svane _et al._ (2010a, b, 2011) $spd$, van Schilfgaarde _et al._
(2006a); Faleev _et al._ (2004); Chantis _et al._ (2006) and rare-earth
systems. Chantis _et al._ (2007) It should be noted that the energy
eigenvalues of the QSGW method are the same as the quasiparticle spectra of
the $GW$ method. This captures the many-body shifts in the quasiparticle
energies. However, when presenting the quasiparticle DOS, this ignores the
smearing by the imaginary part of the self-energy of the spectra due to
quasiparticle lifetime effects, which should increase as quasiparticle
energies become farther away from the Fermi energy.
To define a theoretical correlation strength some electronic-structure
quantity that scales with an intuitive notion of correlation strength is
needed. In our application to Pu, we propose to consider the $f$ bandwidth,
$W_{f}$, and use the relative bandwidth reduction in QSGW compared to LDA,
${\textrm{w}_{rel}}=W_{f}(\text{GW})/W_{f}(\text{LDA}),$ (1)
as the key quantity, where $W_{f}(\text{GW})$ and $W_{f}(\text{LDA})$ are the
$f$ bandwidths as obtained from QSGW and LDA calculations, respectively. This
is consistent with the correlation-induced QSGW $f$-bandwidth reduction in Pu
that was demonstrated in Ref. Chantis _et al._ , 2009.
Using a quasiparticle calculation is important since lifetime effects, which
are absent in the LDA calculations, would obscure the band narrowing in $GW$
relative to LDA. We also need a measure that is robust at the high
temperatures of the strongly correlated phases of Pu, where any low-energy
features in the electronic structure are likely to be thermally averaged away.
111The most strongly correlated phase of Pu is $\delta$ Pu, which has a
temperature between about 600 and 700 K. As noted in Ref. Boivineau, 2001,
“the delocalization process of the $5f$ electrons,” i.e., the correlation
effects, continues to produce anomalies as high as 2000 K in temperature in
many of these properties, well into the high-temperature liquid phase of Pu.
Also, see Ref. Boivineau, 2009. In this regard, it should be noted that
although temperature certainly plays an important role in predicting the
correct equilibrium crystal structure, we believe that it is the resulting
volume per atom of any Pu phase that determines the amount of correlation,
since this is an electronic property. In particular, we do not expect that the
bandwidth predicted by our zero-temperature $GW$ calculations will be
sensitive to any temperature in the range set by the Pu solid phases.
The choice of bandwidth narrowing as a measure of correlation strength is
consistent with ideas of correlation going back almost to the beginning of
modern electronic structure theory. Quasiparticle descriptions of electronic
structure have been standard since Landau developed Fermi liquid theory and
have been derived from standard many-body approaches (see, for example, the
discussion in Refs. Abrikosov _et al._ , 1963; Nozieres, 1964; Hedin and
Lundqvist, 1969). They have since been extended to strongly correlated
electronic materials (see, for example, the review in Ref. Hewson, 1993). Much
of our modern understanding of correlation effects has been developed using
simple model Hamiltonians, especially the Hubbard model.Hubbard (1963) For
metals, most of these approaches for strong correlations have focused on low-
temperatures,Hewson (1993) where the electronic structure at the Fermi energy
can yield a rich and diverse set of phenomena at low-energy scales. In such a
case, for example, specific heat or effective mass enhancements at the Fermi
energy have often been used to characterize the strength of correlations. As
we describe below, pure elemental plutonium forms correlated states at very
high temperatures, and therefore electronic states are sampled that are far
from the Fermi energy. Although it is an interesting question how far away
from the Fermi energy correlations effects extend (see, e.g., Ref. Byczuk _et
al._ , 2007), it is nonetheless important to include correlation effects for
all the quasiparticle states of the $f$ electrons in Pu. By including the real
part of the self-energy for all of these states, which are involved in the
band narrowing, our $GW$ approach is thus more relevant for these high-
temperature correlated phases than more traditional measures of correlation
that focus exclusively on effects at or near the Fermi energy.
To set an appropriate correlation scale, we define our theoretical $C$ by
$C=1-{\textrm{w}_{rel}},$ (2)
which ranges from $C=0$ (no bandwidth reduction) in the LDA limit to $C=1$ in
the fully localized or atomic limit (the bandwidth becomes zero).
As mentioned above, our test case for correlation is elemental Pu, an actinide
metal, which exhibits large volume changes compared to predictions from band
structure theory that are clearly due to correlation effects.Wick (1967);
Hecker (2001, 2004); Hecker _et al._ (2004); Albers (2001) The large
variation in volumes is controlled by the amount of strong $f$-bonding, which
is due to direct $f$-$f$ wave-function overlap. The $f$ bonding for many of
the different phases is greatly reduced leading to anomalous volume expansions
due to the narrowing of the $f$ bands that results from correlation effects.
Albers (2001) If no correlation were present, the $f$ bonds would have their
full strength and a relatively small volume per atom for all phases would be
accurately predicted by LDA band-structure methods. In the limit of extremely
strong correlation the bands would have narrowed so much that the $f$
electrons would be fully localized, and they would not contribute to the
bonding. The volume per atom would then be much larger and close to that of
Am, which has fully localized $f$ electrons that do not extend outside the
atomic core.
Using the QSGW approximation we have calculated 222We have not included spin-
orbit effects, which can be safely ignored for the purposes of this paper. The
Pu $f$ DOS splits into a pair of clearly separated $j=5/2$ and $7/2$ peaks. To
include spin-orbit, we would need to calculate the bandwidth of each peak
separately and use that corresponding to $j=5/2$. By ignoring spin-orbit
coupling, we are saved from this additional trouble, which is not expected to
change the effective $f$ bandwidths. Recent spin-orbit $GW$ calculations have
been calculated in Pu (Ref. Kutepov _et al._ , 2012). However these have been
done in the fully self-consistent $GW$ method, which usually is a poor
approximation in solids due to an incorrect treatment of plasmon effects.
Since the DOS in this paper includes broadening effects due to the imaginary
part of the self-energy in all of the different approximation that were used,
it is also unclear how bandwidth narrowing would separately be affected by
spin-orbit effects. the quasiparticle band structures of the fcc, bcc, simple
cubic (sc), $\gamma$, and pseudo-$\alpha$ phases of Pu as a function of
volume. The pseudo-$\alpha$ is a two-atom per unit cell approximation Bouchet
_et al._ (2004) to the true $\alpha$ structure of Pu that preserves the
approximate nearest-neighbor distances and other essential features needed for
the electronic-structure. In this way we avoid performing an extremely large
and expensive 16-atom per unit cell calculation for the $\alpha$ structure. We
are unfortunately unable to present $GW$ results for the $\beta$ structure,
which is even more complex than the $\alpha$ structure, since no
pseudostructure for this crystal structure is available and a QSGW calculation
is presently not feasible for so many atoms per unit cell.
To calculate the $f$-electron bandwidths from the $f$-electron projected
density of states (DOS), Df(E), an algorithm is needed to determine the width
of the main peak in this DOS. A simple first guess is to choose a rectangular
DOS and to use a least-squares fit to the $GW$ or LDA $f$-DOS to determine the
best height and width of the rectangle. A drawback of this method is that an
artificial broadening of the effective $f$ bandwidth appears, which is due to
a significant $d$-$f$ hybridization at the bottom of the $f$-DOS that creates
an extra peak at low energies. This masks the correlation-induced band
narrowing. Since this peak has relatively lower height than the main $f$ peak,
we may avoid this complication by generating an algorithm that emphasizes the
“high-peak” part of the $f$-DOS. The algorithm we have used is therefore the
second moment of the $f$ DOS
$W=2(\langle E^{2}\rangle-\langle E\rangle^{2})^{1/2}.$ (3)
The factor of two is needed because the bandwidth extends above and below the
mean energy and is not just the average deviation from the mean energy. To
emphasize the main part of the $f$-DOS peak, the square of the $f$ DOS is used
as weight function:333Choosing instead for example $D_{f}(E)$ as the weight
function does not serve the purpose of emphasizing the central over the
“hybridization wings” in the $f$ DOS.
$\langle f(E)\rangle\equiv\int dEf(E)D_{f}^{2}(E)/\int dED_{f}^{2}(E).$ (4)
## III Numerical Results and Discussion
In Fig. 1 we illustrate how wrel varies with volume for the five different
phases considered here. 444For the atomic volumes we have ignored any thermal
volume expansion. Each phase is represented by a volume corresponding to a
fixed temperature within that phase. We have used the original data of
Zachariasen and Ellinger (Refs. Zachariasen and Ellinger, 1963a, b, 1955;
Ellinger, 1956) corresponding to the volumes at the temperatures 21, 190, 235,
320, 477, and 490 ∘C, for the $\alpha$, $\beta$, $\gamma$, $\delta$,
$\delta^{\prime}$, and $\epsilon$ phases, respectively. Large volume
variations ranging between about 14–28 Å3 per atom are considered, with
bandwidths that span almost an order of magnitude, from about 0.5 eV to 2.5
eV. Although the LDA bandwidth decreases with increased volume due to
reduction in $f$-$f$ overlap of the wavefunctions, the QSGW bandwidth
decreases even faster illustrating increased correlation effects with lattice
expansion. The bandwidth at a specific volume depends on crystal structure
(due to differences in coordination and bond lengths), as does also the
correlation strength.
Figure 1: (Color online) Plot of wrel= $W_{f}$(GW)/$W_{f}$(LDA) versus
volume, $V$, per atom, for the $\gamma$, fcc, bcc, sc, and ps-$\alpha$
[pseudo-$\alpha$, an approximate $\alpha$-phasen (Ref. Bouchet _et al._ ,
2004)] crystal phases of Pu. Note that the sc (simple cubic) is a hypothetical
structure for Pu. The small, vertical bars at the top of the figure mark the
experimentally observed atomic volumes (Ref. 54).
Although we expect electronic-structure calculations to strongly depend on the
crystal structure and lattice constant, we surprisingly found that correlation
effects were approximately independent of these. Indeed, Fig. 2 shows that all
of our different calculations for our measure of correlation strength, the
reduced bandwidth, collapse to a single “universal” curve when plotted as a
function of the LDA bandwidth. In making this plot, it is likely that the
effective screened Coulomb interaction between the $5f$ electrons is
approximately constant and that the correlation effects are being tuned by the
effective average kinetic energy of these electrons as reflected in their LDA
bandwidth. In the range of $W_{f}$ values considered here the curve is
approximately quadratic, i.e.,
${\textrm{w}_{rel}}(x)=0.15+0.43x-0.07x^{2},$ (5)
where $x$ = $W_{f}$(LDA) in eV. From Eq. (2) we can use these results to
determine a correlation strength $C$. It is remarkable that the many-body
properties of a strongly correlated system can be tuned with what is normally
considered to be a one-electron property.
Figure 2: (Color online) Plot of wrel= $W_{f}$(GW)/$W_{f}$(LDA) versus
$W_{f}$(LDA) for the $\gamma$, fcc, bcc, sc, and ps-$\alpha$. The dashed red
line represents the fit of Eq. (5) The small, vertical bars at the top of the
figure mark the values of $W_{f}$(LDA) calculated at the experimental volumes
of the five Pu phases (Ref. 54).
In Fig. 3 we show 555For the volumes of the different phases of Pu, we have
followed the same method used to generate Fig. 1. We have also used the same
volumes of the different phases for the sound velocity and resistivity needed
to determine the correlation strength from the $GW$ calculations plotted in
Fig. 1. Note that, since we have not directly calculated the value of
${\textrm{w}_{rel}}$ for the $\beta$ phase, we instead used the availability
of the bandwidth reduction of Eq. (5) together with the calculated LDA
bandwidth for the correct crystal structure of $\beta$ Pu to determine
${w_{rel}}(\beta)$ = 0.55. that our definition of theoretical correlation
strength does indeed fulfill our expectations and can be used to bring order
into the trends for various experimental properties, including volume, sound
velocity, and resistivity. These properties exhibit an approximately 25%, 50%,
and 35% change over the correlation range (about 0.2 to 0.6) between the
$\alpha$ and $\delta$ phases of Pu and, with some scatter that might partially
depend on sample quality, fall on smooth curves when plotted as a function of
our theoretical correlation strength. It is remarkable that all of these data
should collapse to a single curve for each property that is independent of any
explicit consideration of temperature, crystal structure, or other variable.
However, more generally, we would only expect this to be true for a property
that was predominantly affected by correlation effects.
Figure 3: (Color online) Trends in Pu properties as a function of correlation
strength $C$, including (a) volume per atom (Ref. 54), (b) sound velocity
(Ref. Boivineau, 2001), and (c) resistivity (Ref. Boivineau, 2001).
In terms of theoretical trends, various theories have often attempted to
estimate the amount of correlation in terms of the $Z$-factor,
$Z_{n{\mathbf{k}}}=\left(1-\langle\Psi_{n{\mathbf{k}}}|\frac{\partial\Sigma(\epsilon_{n{\mathbf{k}}})}{\partial\omega}|\Psi_{n{\mathbf{k}}}\rangle\right)^{-1},$
(6)
where $\Psi_{n{\mathbf{k}}}$ are the (LDA) electronic eigenfunctions with
energies $\epsilon_{n{\mathbf{k}}}$, and $\Sigma$ denotes the self-energy. We
have found that the volume dependence of the $Z$-factors follows the trend of
the $f$-bandwidth reduction in Fig. 1, i.e., our measure of correlation
strength, albeit with variations due to ${\mathbf{k}}$\- and hybridization-
dependence. However, it should be noted that the relation between $Z$ and
bandwidth reduction is not the same in all materials, especially for weakly
correlated broad-band systems, which seem very different from strongly
correlated materials such as Pu.
The simplest Hubbard-like HamiltonianHubbard (1963) to describe strongly
correlated electron systems has a form
$H=\sum_{ij,\sigma}t_{ij}c^{\dagger}_{i\sigma}c_{j\sigma}+U\sum_{i}n_{i\uparrow}n_{i\downarrow}.$
(7)
with two parameters: the Hubbard parameter $U$ which induces correlation, and
an effective $t$, which can be related to the uncorrelated bandwidth $W$. When
$W$ dominates, the system is in a weakly correlated limit and, when $U$
dominates, the system is in a strongly correlated regime. Hence, one can study
the solutions as a function of $U/W$ to go from one limit to another. In more
realistic electronic-structure calculations, the same physics is intuitively
expected to carry over. The Hubbard $U$ can then be thought of as a screened
on-site Coulomb interaction and the bandwidth as due to the normal band-
structure hybridization. In our context, this suggests that the correlation
strength $C$ should also be a function of $U/W$. To test this, in Fig. 4 we
plot $C$ versus $1/W_{f}(\text{LDA})$. If the effective $U$ were approximately
constant, we had hoped to observe some approximate linear behavior at weak
correlations, but any such behavior is unclear in Fig. 4. To show what might
happen at weaker correlation strengths we have also included in Fig. 4 the
equilibrium-volume results for Co, Rh, and Ir for the $d$-electron projected
DOS. Interestingly enough, the $d$-electron results seem to follow the same
overall trend to large bandwidths (small correlation). Among the transition
metals included in the plot, Co (3$d$) has the most narrow $d$ band, and the
correlation value is close to the lowest values for Pu in the figure.
Figure 4: (Color online) $C$ from $GW$ theory versus 1/W${}_{f}(LDA)$. The
data for Co, Rh and Ir are for the $3d$, $4d$, and $5d$ bandwidths,
respectively. The small, vertical bars at the top of the figure mark the
values of Wf(LDA)-1 calculated at the experimental volumes of the five Pu
phases (Ref. 54).
## IV Conclusion
In summary, we have introduced the idea of a “correlation strength” quantity
$C$, which must be taken into account in order to explain the properties of
strongly correlated electronic materials. As an example, we have shown how to
use the $GW$ method to define a theoretical $C$ for metallic Pu, and that
various experimental physical properties, including anomalous volume
expansion, sound velocity, and resistivity, for the different phases of Pu
follow well-defined trends when plotted versus our theoretical correlation
strength. We have also demonstrated a universal scaling relationship for the
correlation-reduced bandwidth as a function of the LDA bandwidth.
###### Acknowledgements.
This work was carried out under the auspices of the National Nuclear Security
Administration of the U.S. Department of Energy at Los Alamos National
Laboratory under Contract No. DE-AC52-06NA25396, the Los Alamos LDRD Program,
and the Research Foundation of Aarhus University. The calculations were
carried out at the Centre for Scientific Computing in Aarhus (CSC-AA),
financed by the Danish Centre for Scientific Computing (DCSC) and the Faculty
of Science and Technology, Aarhus University.
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* Chantis _et al._ (2006) A. N. Chantis, M. van Schilfgaarde, and T. Kotani, Phys. Rev. Lett. 96, 086405 (2006).
* Chantis _et al._ (2007) A. N. Chantis, M. van Schilfgaarde, and T. Kotani, Phys. Rev. B 76, 165126 (2007).
* Note (1) The most strongly correlated phase of Pu is $\delta$ Pu, which has a temperature between about 600 and 700 K. As noted in Ref. boivineau01, “the delocalization process of the $5f$ electrons,” i.e., the correlation effects, continues to produce anomalies as high as 2000 K in temperature in many of these properties, well into the high-temperature liquid phase of Pu. Also, see Ref. boivineau09.
* Abrikosov _et al._ (1963) A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, _Methods of Quantum Field Theory in Statistical Physics_ (Prentice-Hall, Englewood Cliffs, N.J., 1963).
* Nozieres (1964) P. Nozieres, _Theory of Interacting Fermi Systems_ (W. A. Benjamin, New York, 1964).
* Hewson (1993) A. C. Hewson, _The Kondo Problem to Heavy Fermions_ (Cambridge University Press, Cambridge, 1993).
* Hubbard (1963) J. Hubbard, Proc. Roy. Soc A276, 38 (1963).
* Byczuk _et al._ (2007) K. Byczuk, M. Kollar, K. Held, Y.-F. Yang, I. A. Nekrasov, T. Pruschke, and D. Vollhardt, Nat. Phys. 3, 168 (2007).
* Wick (1967) O. J. Wick, _Plutonium Handbook: A Guide to the Technology_ (Gordon and Breach, New York, 1967).
* Hecker (2001) S. S. Hecker, MRS Bull. 26, 672 (2001).
* Hecker (2004) S. S. Hecker, Met. Mat. Trans. A 35A, 2207 (2004).
* Hecker _et al._ (2004) S. S. Hecker, D. R. Harbur, and T. G. Zocco, Prog. Mat. Sci. 49, 429 (2004).
* Albers (2001) R. C. Albers, Nature (London) 410, 759 (2001).
* Note (2) We have not included spin-orbit effects, which can be safely ignored for the purposes of this paper. The Pu $f$ DOS splits into a pair of clearly separated $j=5/2$ and $7/2$ peaks. To include spin-orbit, we would need to calculate the bandwidth of each peak separately and use that corresponding to $j=5/2$. By ignoring spin-orbit coupling, we are saved from this additional trouble, which is not expected to change the effective $f$ bandwidths. Recent spin-orbit $GW$ calculations have been calculated in Pu (Ref. kutepov12). However these have been done in the fully self-consistent $GW$ method, which usually is a poor approximation in solids due to an incorrect treatment of plasmon effects. Since the DOS in this paper includes broadening effects due to the imaginary part of the self-energy in all of the different approximation that were used, it is also unclear how bandwidth narrowing would separately be affected by spin-orbit effects.
* Bouchet _et al._ (2004) J. Bouchet, R. C. Albers, M. D. Jones, and G. Jomard, Phys. Rev. Lett. 92, 095503 (2004).
* Note (3) Choosing instead for example $D_{f}(E)$ as the weight function does not serve the purpose of emphasizing the central over the “hybridization wings” in the $f$ DOS.
* Note (4) For the atomic volumes we have ignored any thermal volume expansion. Each phase is represented by a volume corresponding to a fixed temperature within that phase. We have used the original data of Zachariasen and Ellinger (Refs. zachariasen63b,zachariasen63a,zachariasen55,ellinger56) corresponding to the volumes at the temperatures 21, 190, 235, 320, 477, and 490 ∘C, for the $\alpha$, $\beta$, $\gamma$, $\delta$, $\delta^{\prime}$, and $\epsilon$ phases, respectively.
* Note (5) For the volumes of the different phases of Pu, we have followed the same method used to generate Fig. 1. We have also used the same volumes of the different phases for the sound velocity and resistivity needed to determine the correlation strength from the $GW$ calculations plotted in Fig. 1. Note that, since we have not directly calculated the value of ${\textrm{w}_{rel}}$ for the $\beta$ phase, we instead used the availability of the bandwidth reduction of Eq. (5) together with the calculated LDA bandwidth for the correct crystal structure of $\beta$ Pu to determine ${w_{rel}}(\beta)$ = 0.55.
* Boivineau (2001) M. Boivineau, J. Nuc. Mater. 297, 97 (2001).
* Boivineau (2009) M. Boivineau, J. Nuc. Mater. 392, 568 (2009).
* Zachariasen and Ellinger (1963a) W. H. Zachariasen and F. H. Ellinger, Acta Cryst. 16, 777 (1963a).
* Zachariasen and Ellinger (1963b) W. H. Zachariasen and F. H. Ellinger, Acta Cryst. 16, 395 (1963b).
* Zachariasen and Ellinger (1955) W. H. Zachariasen and F. H. Ellinger, Acta Cryst. 8, 431 (1955).
* Ellinger (1956) F. H. Ellinger, J. of Metals 8, 1256 (1956).
|
arxiv-papers
| 2012-01-10T19:02:11 |
2024-09-04T02:49:26.118142
|
{
"license": "Public Domain",
"authors": "A. Svane, R. C. Albers, N. E. Christensen, M. van Schilfgaarde, A. N.\n Chantis, Jian-Xin Zhu",
"submitter": "Robert Albers",
"url": "https://arxiv.org/abs/1201.2139"
}
|
1201.2215
|
# Existence and concentration of semiclassical states for nonlinear
Schrödinger equations
Shaowei Chen
School of Mathematical Sciences, Capital Normal University,
Beijing 100048, P. R. China
E-mail adress: chensw@amss.ac.cn (S. Chen),
Abstract: In this paper, we study the following semilinear Schrödinger
equation
$-\epsilon^{2}\triangle u+u+V(x)u=f(u),\ u\in H^{1}(\mathbb{R}^{N}),$
where $N\geq 2$ and $\epsilon>0$ is a small parameter. The function $V$ is
bounded in $\mathbb{R}^{N},$ $\inf_{\mathbb{R}^{N}}(1+V(x))>0$ and it has a
possibly degenerate isolated critical point. Under some conditions on $f,$ we
prove that as $\epsilon\rightarrow 0,$ this equation has a solution which
concentrates at the critical point of $V$.
Key words: semilinear Schrödinger equation, variational reduction method.
2000 Mathematics Subject Classification: 35J20, 35J70
††††
## 1 Introduction and main result
In this paper, we are concerned with the following semilinear Schrödinger
equation
$-\epsilon^{2}\triangle u+u+V(x)u=f(u),\ u\in H^{1}(\mathbb{R}^{N}),$ (1.1)
where $N\geq 2$ and $\epsilon>0$ is a small parameter. The function
$f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies
$(\bf{F_{1}}).$ $f\in C^{1}(\mathbb{R})$ and there exist $q\in(2,2^{*}),$
$2<p_{1}<p_{2}<2^{*}$ and a constant $C>0$ such that
$|f^{\prime}(t)|\leq C(|t|^{p_{1}-2}+|t|^{p_{2}-2}),\ t\in\mathbb{R}$
and for any $L>0$,
$\displaystyle\sup\\{|f^{\prime}(t)-f^{\prime}(s)|/|t-s|^{q-2}\ |\
t,s\in[-L,L],\ t\neq s\\}<\infty,$ (1.2)
where $2^{*}=2N/(N-2)$ if $N\geq 3$ and $2^{*}=\infty$ if $N=2$;
$(\bf{F_{2}}).$ there exists $\mu>2$ such that $f(t)t\geq\mu F(t)>0,$ $t\neq
0$, where $F(t)=\int^{t}_{0}f(s)ds$;
$(\bf{F_{3}}).$ $f(t)/|t|$ is an increasing function on
$\mathbb{R}\setminus\\{0\\}$;
###### Remark 1.1.
A typical function which satisfies $\bf(F_{1})-(F_{3})$ is
$f(t)=\sum^{m}_{i=1}a_{i}|t|^{\beta_{i}-2}t$
with $2<\beta_{1}<\cdots<\beta_{m}<2^{*}$ and $a_{i}>0,$ $1\leq i\leq m.$
The potential function $V$ satisfies the following conditions:
$(\bf{V_{0}}).$ $\inf_{x\in\mathbb{R}^{N}}(1+V(x))>0$ and
$\max_{x\in\mathbb{R}^{N}}|V(x)|<\infty;$
$(\bf{V_{1}}).$ $V\in C^{2}(\mathbb{R}^{N})$ has an isolated critical point
$x_{0}$ such that
$V(x)=Q_{n^{*}}(x-x_{0})+o(|x-x_{0}|^{n^{*}})$
in some neighborhood of $x_{0},$ where $n^{*}\geq 2$ is an even integer and
$Q_{n^{*}}$ is an $n^{*}$\- homogeneous polynomial in $\mathbb{R}^{N}$ which
satisfies that $\triangle Q_{n^{*}}\geq 0$ in $\mathbb{R}^{N}$ or $\triangle
Q_{n^{*}}\leq 0$ in $\mathbb{R}^{N}$ and $\triangle Q_{n^{*}}\not\equiv 0$ in
$\mathbb{R}^{N}$.
###### Remark 1.2.
Without loss of generality, in what follows, we always assume that $x_{0}=0.$
Typical examples for $Q_{n^{*}}$ are $\pm|x|^{n^{*}}$ $(n^{*}\geq 2).$
Our main result of this paper is the following theorem
###### Theorem 1.3.
Suppose that $f$ satisfies $\bf(F_{1})-\bf(F_{4})$ and $V$ satisfies
$\bf(V_{0})$ and $\bf(V_{1})$. Then there exist $\epsilon_{0}>0$ and a set
$\mathcal{K}$ whose elements are radially symmetric solutions of equation
$-\triangle u+u=f(u),\ u\in H^{1}(\mathbb{R}^{N})$ (1.3)
such that if $0<\epsilon<\epsilon_{0}$, then equation (1.1) has a solution
$u_{\epsilon}$ satisfying that
$\lim_{\epsilon\rightarrow
0}\mbox{dist}_{{}_{Y}}(v_{\epsilon},\mathcal{K})=0,$
where $v_{\epsilon}(x)=u_{\epsilon}(\epsilon x),$ $x\in\mathbb{R}^{N}$ and
$Y=H^{1}(\mathbb{R}^{N}).$
The analysis of the semilinear Schrödinger equation (1.1) has recently
attracted a lot of attention due to its many applications in mathematical
physics.
If $v$ is a solution of equation (1.1), then $v(\epsilon x)$ is a solution of
the following equation
$-\triangle u+u+V(\epsilon x)u=f(u),\ u\in H^{1}(\mathbb{R}^{N}).$ (1.4)
Equation (1.4) is a perturbation of the limit equation (1.3). If equation
(1.3) has a solution $w\in C^{2}(\mathbb{R}^{N})$ satisfying the non-
degeneracy condition:
$\ker L_{0}=\mbox{span}\left\\{\frac{\partial\omega}{\partial x_{i}}\ |\ 1\leq
i\leq N\right\\},$
where $L_{0}v=-\triangle v+v-f^{\prime}(\omega)v$, then in the celebrated
paper [1] (see also [2]), Ambrosetti, Badiale and Cingolani developed a kind
of variational reduction method and showed that if the potential function $V$
has a strictly local minimizer or maximizer $x_{0}$, then equation (1.4)
admits a solution $u_{\epsilon}$ which converges to $\omega(\cdot-x_{0})$ in
$H^{1}(\mathbb{R}^{N})$ as $\epsilon\rightarrow 0$. In their argument, the
non-degeneracy property of $\omega$ plays essential role. Using the non-
degeneracy condition and the reduction method, it was shown by Kang and Wei
[20] that, at a strict local maximum point $x_{0}$ of $V$ and for any positive
integer $k$, (1.1) has a positive solution with $k$ interacting bumps
concentrating near $x_{0}$, while at a non-degenerate local minimum point of
$V(x)$ such solutions do not exist. Moreover, under the assumption of the non-
degeneracy condition, multiplicity of solutions with one bump has also been
considered by Grossi [16].
However, for a general nonlinearity $f$, it is very difficult to verify the
non-degeneracy condition for a solution of (1.3). An effective method to
attack problem (1.1) without using the non-degeneracy condition is variational
method. In [21], Rabinowitz used a global variational method to show the
existence of least energy solutions for (1.1) when $\epsilon>0$ is small, and
the condition imposed on $V$ is a global one, namely
$0<\inf_{x\in\mathbb{R}^{N}}(1+V(x))<\liminf_{|x|\rightarrow\infty}(1+V(x)).$
In [12], [13], [14], [15] and [17], Del Pino, Felmer and Gui used different
variational methods to obtain nontrivial solution of (1.1) for small
$\epsilon>0$ under local conditions which can be roughly described as follows:
$V$ is local Hölder continuous on $\mathbb{R}^{N},$
$\displaystyle\inf_{x\in\mathbb{R}^{N}}(1+V(x))>0$ (1.5)
and there exists $k$ disjoint bounded regions $\Omega_{1},\cdots,\Omega_{k}$
in $\mathbb{R}^{N}$ such that
$\displaystyle\inf_{x\in\partial\Omega_{i}}V(x)>\inf_{x\in\Omega_{i}}V(x).$
(1.6)
Their methods involve the deformation of nonlinearity $f$ and some prior
estimates. Recently, Byeon, Jeanjean and Tanaka [5] [6] developed the
variational methods and made great advance in problem (1.1). Byeon and
Jeanjean showed in [5] that if $N\geq 3$, $V$ satisfies (1.5) and (1.6) with
$k=1$ and $f$ satisfies
$\bf(f_{1}).$ $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and
$\lim_{t\rightarrow 0+}f(t)/t=0;$
$\bf(f_{2}).$ there exists some $p\in(1,2^{*}-1)$ such that
$\lim_{t\rightarrow\infty}f(t)/t^{p}<\infty;$
$\bf(f_{3}).$ there exists $T>0$ such that $\frac{1}{2}mT^{2}<F(T)$, where
$F(t)=\int^{t}_{0}f(s)ds$ and $m=\inf_{x\in\Omega_{1}}V(x),$
then (1.1) exists positive solution $v_{\epsilon}$ concentrating in the
minimizers of $V$ in $\Omega_{1}$ as $\epsilon\rightarrow 0.$ And in [6],
Byeon, Jeanjean and Tanaka considered the case $N=1,2$ and obtained similar
results. Their conditions on the nonlinearity $f$ are almost optimal.
Moreover, when $V$ satisfies (1.5) and (1.6) with $k>1$ and $f$ satisfies
$\bf(f_{1})-(f_{3})$, in [10], Cingolani, Jeanjean and Secchi constructed
multi-bump solutions for magnetic nonlinear Schödinger equations which contain
equation (1.1) as a special case.
Comparing to the variational methods mentioned above, the Lyapunov reduction
method of Ambrosetti and Badiale, although it need the non-degeneracy
condition, has its advantages that their method can be used to deal with
elliptic equations involving critical Sobolev exponent (see, for example, [3])
and other problems involving concentration compactness (see, for example,
[18]).
In this paper, we indent to attack the problem (1.1) though a Lyapunov
reduction method, but avoiding the non-degeneracy condition for the solutions
of limit equation (1.3). In this paper, we develop a new reduction method for
an isolated critical set $\mathcal{K}$ of the functional corresponding to
(1.3). This method can be regarded as a generalization of Ambrosetti and
Badiale’s method. The non-degeneracy conditions for the solutions in this
critical set are no longer necessary and it does not involve the deformation
of nonlinearity. By combination of the new reduction method and Conley index
theory which was developed by Chang and Ghoussoub in [9](see also [8]), we
obtain a solution of (1.4) in a neighborhood of $\mathcal{K}$ for sufficiently
small $\epsilon>0.$ Our method is new and it can be used to other problems
which involve concentration compactness. In contrast with the results of
Byeon, Jeanjean and Tanaka, although the assumptions we imposed on the
nonlinearity $f$ are much stronger, the assumptions we made on $V$ seem weaker
in a sense, because by the assumption $\bf(V_{1})$, $x_{0}$ can be a local
maximum point of $V.$
This paper is organized as follows: In section 2, we obtain a critical set of
the functional corresponding to (1.3) with nontrivial Topology. In section 3
and section 4, a reduction for the function corresponding to (1.4) is
developed. In section 5, we give the proof of Theorem 1.3. Section 6 and 7 are
appendixes.
Notations. $\mathbb{R},\ \mathbb{Z}$ and $\mathbb{N}$ denote the sets of real
number, integer and positive integer respectively. Let $E$ be a metric space.
$B_{E}(a,\rho)$ denotes the open ball in $E$ centered at $a$ and having radius
$\rho$. The closure of a set $A\subset E$ is denoted by $\overline{A}$ or
$cl_{E}(A).$ $\mbox{dist}_{E}(a,A)$ denotes the distance from the point $a$ to
the set $A\subset E$. By $\rightarrow$ we denote the strong and by
$\rightharpoonup$ the weak convergence. By $\ker A$ denotes the null space of
the operator $A.$ If $g$ is a $C^{2}$ functional defined on a Hilbert space
$H$, $\nabla g$ (or $Dg$) and $\nabla^{2}g$ (or $D^{2}g$) denote the gradient
of $g$ and the second derivative of $g$ respectively. And for
$a,b\in\mathbb{R},$ we denote $g^{a}:=\\{u\in H\ |\ g(u)\leq a\\}$ and
$g_{b}:=\\{u\in H\ |\ g(u)\geq b\\}$ the sub- and super-level sets of the
functional $g,$ moreover, $g^{a}_{b}:=\\{u\in H\ |\ b\leq g(u)\leq a\\}.$
$\delta_{i,j}$ denotes the Kronecker notation, i.e., $\delta_{i,j}=1$ if $i=j$
and $0$ if $i\neq j.$ For a Banach space $E,$ denote $\mathcal{L}(E)$ the
Banach space consisting of all bounded linear operator from $E$ to $E.$ If $H$
is a Hilbert space and $W$ is a closed subspace of $H,$ we denote the
orthogonal complement space of $W$ in $H$ by $W^{\bot}.$ For a subset
$A\subset H,$ $\mbox{span}\\{A\\}$ denotes the subspace of $H$ generated by
$A.$ For a topology pair $(A,B)$ in metric space, $\check{H}^{*}(A,B)$ denotes
the $\check{\mbox{C}}\mbox{ech}$-Alexander-Spanier cohomology with coefficient
group $\mathbb{Z}_{2}$ (see [23]).
## 2 Critical sets of limit functional with nontrivial Topology
Throughout this paper, we denote the Sobolev space $H^{1}(\mathbb{R}^{N})$ and
the radially symmetric function space
$H^{1}_{r}(\mathbb{R}^{N}):=\\{u\in H^{1}(\mathbb{R}^{N})\ |\ u\ \mbox{is
radially symmetric}\\}$
by $Y$ and $X$ respectively. The inner product of $Y$ is
$\langle u,v\rangle=\int_{\mathbb{R}^{N}}(\nabla u\nabla v+uv)dx,$
and we use $||\cdot||$ to denote the norm of $Y$ corresponding to this inner
product. Define
$I(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}(|\nabla
u|^{2}+|u|^{2})dx-\int_{\mathbb{R}^{N}}F(u)dx,\ u\in X.$
$J(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}(|\nabla
u|^{2}+|u|^{2})dx-\int_{\mathbb{R}^{N}}F(u)dx,\ u\in Y,$
$E_{\epsilon}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}(|\nabla
u|^{2}+|u|^{2}+V(\epsilon x)|u|^{2})dx-\int_{\mathbb{R}^{N}}F(u)dx,\ u\in Y.$
For $h\in H^{-1}(\mathbb{R}^{N})$, let $(-\triangle+1)^{-1}h$ and
$(-\triangle+1+V(\epsilon x))^{-1}h$ be the solutions of
$\displaystyle-\triangle u+u=h,\ u\in H^{1}(\mathbb{R}^{N})$ (2.1)
and
$\displaystyle-\triangle u+u+V(\epsilon x)u=h,\ u\in H^{1}(\mathbb{R}^{N})$
(2.2)
respectively.
Under conditions $\bf(F_{1})-(F_{3})$, $I$ satisfies Palais-Smale condition
(see, for example, [24]) and has a mountain pass geometry, that is,
* (i)
$I(0)=0$,
* (ii)
there exist $\rho_{0}>0$ and $\delta_{0}>0$ such that $I(u)\geq\delta_{0}$ for
all $||u||=\rho_{0},$
* (iii)
there exists $u_{0}\in X$ such that $||u_{0}||>\rho_{0}$ and $I(u_{0})<0.$
Thus the following minimax value is well defined and is larger than
$\delta_{0},$
$\displaystyle c=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}I(\gamma(t))$ (2.3)
where
$\displaystyle\Gamma=\\{\gamma\in C([0,1],X)\ |\ \gamma(0)=0,\
I(\gamma(1))<0\\}.$ (2.4)
###### Lemma 2.1.
For any $\sigma\in(0,\delta_{0}),$ if $a\in(c-\sigma,c)$ and
$b\in(c,c+\sigma)$ are regular values of $I$, then
$\check{H}^{1}(I^{b},I^{a})\neq 0.$
Proof. Since $b>c,$ by the definition of minimax value $c,$ there exists
$\gamma\in\Gamma$ such that
$\displaystyle\max_{t\in[0,1]}I(\gamma(t))<b.$ (2.5)
Let $u_{0}=\gamma(1).$ We infer that $0$ and $u_{0}$ lie in different
connected component of $I^{a}.$ It follows that the homomorphism
$\iota^{*}:\check{H}^{0}(I^{a})\rightarrow\check{H}^{0}(\\{0,u_{0}\\})\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$
which is induced by the inclusion mapping $\iota:\\{0,u_{0}\\}\hookrightarrow
I^{a}$ is a surjection. Consider the following homomorphism which is induced
by the inclusion mapping $j:\\{0,u_{0}\\}\hookrightarrow I^{b},$
$j^{*}:\check{H}^{0}(I^{b})\rightarrow\check{H}^{0}(\\{0,u_{0}\\}).$
By (2.5), $0$ and $u_{0}$ lie in the same connected component of $I^{b}.$ It
follows that $j^{*}$ is not a surjection.
Consider the following communicative diagram
$\check{H}^{0}(I^{b})$$\check{H}^{1}(I^{b},I^{a})$$\check{H}^{0}(I^{a})$$\check{H}^{0}(\\{0,u_{0}\\})$$i^{*}$$\iota^{*}$$j^{*}$$\alpha^{*}$
Since $j^{*}$ is not a surjection and $\iota^{*}$ is a surjection, by this
communicative diagram, we deduce that
$\mbox{Image}(i^{*})\neq\check{H}^{0}(I^{a}).$ Moreover, by the property of
exact sequence, we have $\mbox{Image}(i^{*})=\ker\alpha^{*}.$ Thus
$\ker\alpha^{*}\neq\check{H}^{1}(I^{a})$. It follows that $\alpha^{*}\neq 0$.
Therefore, $\check{H}^{1}(I^{b},I^{a})\neq 0.$ $\Box$
From Chapter 4 of [24], we have the following lemma
###### Lemma 2.2.
If $\nabla I(u)=0$ and $I(u)<2c$, then $u$ does not change sign in
$\mathbb{R}^{N}$.
Let $\mathcal{F}$ be a $C^{1}$ functional defined on a Hilbert space $M$ with
critical set $K_{\mathcal{F}}$. And let $V$ be a pesudo-gradient vector field
with respect to $D\mathcal{F}$ on $M$. A pesudo-gradient flow associated with
$V$ is the unique solution of the following ordinary differential equation in
$M:$
$\dot{\eta}=-V(\eta(x,t)),\ \eta(x,0)=x.$
A subset $W$ of $M$ is said to have the mean value property (for short (MVP))
if for any $x\in M$ and any $t_{0}<t_{1}$ we have
$\eta(x,[t_{0},t_{1}])\subset W$ whenever $\eta(x,t_{i})\in W,\ i=1,2.$
###### Definition 2.3.
(Definition I.10 of [9]) Let $\mathcal{F}$ be a $C^{1}$ functional on a
Hilbert space $M$. A subset $S$ of the critical set $K$ of $\mathcal{F}$ is
said to be a dynamically isolated critical set if there exist a closed
neighborhood $\mathcal{O}$ of $S$ and regular values $a<b$ of $\mathcal{F}$
such that
$\mathcal{O}\subset\mathcal{F}^{-1}[a,b]$ (2.6)
and
$cl(\widetilde{\mathcal{O}})\cap K\cap\mathcal{F}^{-1}[a,b]=S,$ (2.7)
where $\widetilde{\mathcal{O}}=\bigcup_{t\in\mathbb{R}}\eta(\mathcal{O},t)$.
$(\mathcal{O},a,b)$ is called an isolating triplet for $S.$
###### Definition 2.4.
(Definition III.1 of [9]) Let $\mathcal{F}$ be a $C^{1}$ functional on a
Hilbet space $M$ and let $S$ be a subset of the critical set $K_{\mathcal{F}}$
for $\mathcal{F}$. A pair $(W,W_{-})$ of subset is said to be a GM pair for
$S$ associated with a pesudo-gradient vector field $V$, if the following
conditions hold:
(1). $W$ is a closed (MVP) neighborhood of $S$ satisfying $W\cap K=S$ and
$W\cap\mathcal{F}_{\alpha}=\emptyset$ for some $\alpha.$
(2). $W_{-}$ is an exit set for $W,$ i.e., for each $x_{0}\in W$ and $t_{1}>0$
such that $\eta(x_{0},t_{1})\not\in W,$ there exists $t_{0}\in[0,t_{1})$ such
that $\eta(x_{0},[0,t_{0}])\subset W$ and $\eta(x_{0},t_{0})\in W_{-}.$
(3). $W_{-}$ is closed and is a union of a finite number of sub-manifolds that
transversal to the flow $\eta.$
For $\alpha,\beta\in\mathbb{R},$ define
$\mathcal{K}^{\beta}_{\alpha}:=\\{u\in X\ |\ \nabla I(u)=0,\ \alpha\leq
I(u)\leq\beta\\}.$
Let $a$ and $b$ are the regular values which come from Lemma 2.1. Then by
Definition 2.4, $\mathcal{K}^{b}_{a}$ is a dynamically isolated critical set
of $I$. By Lemma 2.1 and Theorem III.3 of [9], we have the following lemma
###### Lemma 2.5.
Let $\sigma>0$ be sufficiently small and $a\in(c-\sigma,c)$,
$b\in(c,c+\sigma)$ be regular values of $I$. If $(W,W_{-})$ is a GM pair of
$\mathcal{K}^{b}_{a}$ associated with some pseudo-gradient vector field of
$I$, then
$\check{H}^{1}(W,W_{-})\neq 0.$
###### Remark 2.6.
In this remark, we shall show that the set of regular values of $I$ is dense
in $\mathbb{R}.$ Therefore, for any $\sigma>0$, there always exist regular
values of $I$ in $(c-\sigma,c)$ and $(c,c+\sigma).$ In fact, we shall show
that $I(C)$ is of first category, where $C$ is the set of critical points of
$I$. It suffices to prove that for any $u\in C$, there exists $\delta_{u}>0$
such that $\overline{I(C\cap B_{X}(u,\delta_{u}))}$ does not contain interior
points.
Let $u\in C$. Since $u$ is radially symmetric, the dimension of the kernel
space of the following operator is at most one
$\displaystyle\nabla^{2}I(u):X\rightarrow X,\ h\in X\mapsto
h-(-\triangle+1)^{-1}f^{\prime}(u)h.$
If $\dim\nabla^{2}I(u)=0,$ then by Morse Lemma (see, e.g., Lemma 4.1 of [7]),
there exists $\delta_{u}>0$ such that $u$ is the unique critical point of $I$
in $B_{X}(u,\delta_{u}).$ Thus, in this case, $I(C\cap
B_{X}(u,\delta_{u}))=\\{I(u)\\}.$
If $\dim\nabla^{2}I(u)=1,$ let $N=\mbox{ker}\nabla^{2}I(u)$ and note that $I$
is a $C^{2}$ functional, then by Lemma 1 of [19] (see also Theorem 5.1 of
[7]), there exist an origin preserving $C^{1}$ diffeomorphism $\Phi$ of some
$B_{X}(0,\delta_{u})$ into $X$ and an an origin preserving $C^{1}$ map $h$
defined in $N\cap B_{X}(0,\delta_{u})$ into $X$ such that
$I\circ\Phi(z,y)=I(u)+||Pz||^{2}-||(\mbox{id}-P)z||^{2}+I(h(y)+y)$
where $P:N^{\bot}\rightarrow N^{\bot}$ is an orthogonal projection and
$N^{\bot}$ is the orthogonal complement of $N$ in $X$. Let $U=\\{y\in N\cap
B_{X}(0,\delta_{u})\ |\ h(y)+y\\}.$ Then $U$ is a $C^{1}$ one-dimensional
manifold. Let us restrict $I$ to $U$. Then $I:U\rightarrow\mathbb{R}$ is
$C^{1}$. Moreover, $C\cap B_{X}(0,\delta_{u})=C\cap U,$ so $I(C\cap
B_{X}(0,\delta_{u}))=I(C\cap U)$. Therefore, by classical Sard theorem,
$\overline{I(C\cap B_{X}(0,\delta_{u}))}$ does not contain interior points.
For $r>0,$ $A\subset X,$ let
$\displaystyle N_{r}(A):=\\{v\in X\ |\ \mbox{dist}_{X}(v,A)<r\\}.$ (2.8)
###### Lemma 2.7.
Let $c$ be the mountain pass value coming from Lemma 2.1. For any $r>0,$ there
exists $\sigma_{r}>0$ such that if $a\in(c-\sigma_{r},c)$ and
$b\in(c,c+\sigma_{r})$ are regular values of $I$, then there exists a GM pair
$(W,W_{-})$ of the critical set $\mathcal{K}^{b}_{a}$ of the functional $I$
associated with the negative gradient vector field of $I$ such that $W\subset
N_{r}(\mathcal{K}^{b}_{a}).$
Proof. By $\bf(F_{1})-(F_{3}),$ we know that $I$ satisfies the Palais-Smale
condition (see [24]). Therefore, for any $r>0,$ there exists $\kappa_{r}>0$
such that if $a\in(c-1,c)$ and $b\in(c,c+1)$, then
$\displaystyle||\nabla I(v)||\geq\kappa_{r},\ \forall v\in
I^{-1}[a,b]\setminus N_{r/3}(\mathcal{K}^{b}_{a}).$ (2.9)
Let
$\displaystyle 0<\sigma_{r}<\min\\{r\kappa_{r}/6,1\\}$ (2.10)
and $a\in(c-\sigma_{r},c)$ and $b\in(c,c+\sigma_{r})$ be regular values of
$I$. For
$\displaystyle u\in I^{-1}[a,b]\cap N_{r/3}(\mathcal{K}^{b}_{a}),$ (2.11)
consider the negative gradient flow:
$\displaystyle\dot{\eta}(t)=-\nabla I(\eta(t)),\ \eta(0)=u.$ (2.12)
Let
$T^{+}_{u}=\sup\\{t\geq 0\ |\ \mbox{for every}\ s\in[0,t],\ I(\eta(s))\geq
a\\}$
and
$T^{-}_{u}=\inf\\{t\leq 0\ |\ \mbox{for every}\ s\in[t,0],\ I(\eta(s))\leq
b\\}.$
Let
$U=\bigcup_{t\in[T^{-}_{u},T^{+}_{u}]}\\{\eta(t,u)\ |\ u\in I^{-1}[a,b]\cap
N_{r/3}(\mathcal{K}^{b}_{a})\\}.$
Then
$[\mathcal{K}^{b}_{a}]\subset U,$
where
$[\mathcal{K}^{b}_{a}]=\\{v\in X\ |\
\omega(v)\cup\omega^{*}(v)\in\mathcal{K}^{b}_{a}\\},$
$\omega(v)=\cap_{t>0}\overline{\eta(v,[t,+\infty))}$ is the $\omega-$limit set
of $v$ and $\omega^{*}(v)=\cap_{t>0}\overline{\eta(v,(-\infty,-t])}$ is the
$\omega^{*}-$limit set of $v$.
By [9, Proposition III.2], we deduce that there exists a GM pair $(W,W_{-})$
of $\mathcal{K}^{b}_{a}$ such that $W\subset U$. Thus, to prove this Lemma, it
suffices to prove that if $\sigma_{r}>0$ is small enough, then for $u$ which
satisfies (2.11),
$\displaystyle\sup_{t\in(T^{-}_{u},T^{+}_{u})}||\eta(t)-u||\leq\frac{2}{3}r.$
(2.13)
Since their arguments are similar, we only give the proof for
$\displaystyle\sup_{t\in[0,T^{+}_{u})}||\eta(t)-u||\leq\frac{2}{3}r.$ (2.14)
If (2.14) were not true, then there exist $0\leq t_{1}<t_{2}<T^{+}_{u}$ such
that
$r/3\leq||\eta(t)-u||\leq 2r/3,\ \forall t\in[t_{1},t_{2}]$
$\displaystyle||\eta(t_{1})-u||=r/3,\ ||\eta(t_{2})-u||=2r/3.$ (2.15)
According to (2.9), we have
$\displaystyle b-a\geq I(\eta(t_{1}))-I(\eta(t_{2}))$
$\displaystyle=\int^{t_{1}}_{t_{2}}\langle\nabla
I(\eta(t)),\dot{\eta}(t)\rangle dt=\int^{t_{2}}_{t_{1}}||\nabla
I(\eta(t))||^{2}dt\geq\kappa^{2}_{r}(t_{2}-t_{1}).$
It follows that
$\displaystyle t_{2}-t_{1}\leq(b-a)/\kappa^{2}_{r}.$ (2.16)
Combining (2.15) and (2.16) leads to
$\displaystyle\frac{r}{3}$ $\displaystyle\leq$
$\displaystyle||\eta(t_{2})-\eta(t_{1})||\leq\int^{t_{2}}_{t_{1}}||\dot{\eta}(t)||dt$
$\displaystyle\leq$
$\displaystyle(t_{2}-t_{1})^{1/2}(\int^{t_{2}}_{t_{1}}||\dot{\eta}(t)||^{2})^{1/2}=(t_{2}-t_{1})^{1/2}(\int^{t_{2}}_{t_{1}}||\nabla
I(\eta(t))||^{2})^{1/2}$ $\displaystyle\leq$
$\displaystyle(t_{2}-t_{1})^{1/2}(b-a)^{1/2}\leq(b-a)/\kappa_{r}<2\sigma_{r}/\kappa_{r}.$
It contradicts (2.10). Thus, (2.14) holds. $\Box$
## 3 A variational reduction for the limiting functional $I$
Let $\sigma>0$ be sufficiently small and $a\in(c-\sigma,c)$,
$b\in(c,c+\sigma)$ be regular values of $I$, where $c$ is defined by (2.3). In
what follows, for the sake of simplicity, we denote the critical set
$\mathcal{K}^{b}_{a}$ by $\mathcal{K}.$
By [4], if $u\in Y$ is a weak solution of
$-\triangle u+u=f(u),$ (3.1)
then $u$ and $\frac{\partial u}{\partial x_{i}},$ $1\leq i\leq N$ satisfy
exponential decay at infinity. As a consequence, $\mathcal{K}$ is a compact
subset of $W^{2,2}(\mathbb{R}^{N}).$ If $u\in Y$ is a solution of equation
(3.1), then $\frac{\partial u}{\partial x_{i}},$ $i=1,\cdots,N$ are the
eigenfunctions for the eigenvalue problem
$-\triangle h+h=f^{\prime}(u)h.$ (3.2)
###### Remark 3.1.
By [22, Theorem C. 3.4]), any eigenfunction of the eigenvalue problem (3.2)
satisfies exponential decay at infinity.
The argument in [11, Page 970-971] implies the following Lemma.
###### Lemma 3.2.
Suppose that $u\in X$ is a solution of equation (3.1) and it does not change
sign in $\mathbb{R}^{N}$. If $v\in Y$ is a solution of (3.2) and satisfies
$\left\langle v,\frac{\partial u}{\partial x_{i}}\right\rangle=0,\
i=1,\cdots,N,$
then $v\in X.$
###### Remark 3.3.
By Lemma 2.2, we infer that if $u\in\mathcal{K}$, then $u$ does not change
sign in $\mathbb{R}^{N}$.
As it has been mentioned above, $\mathcal{K}$ is a compact subset in
$W^{2,2}(\mathbb{R}^{N})$. Thus for any $u\in\mathcal{K}$ and any
$\varsigma>0$, there exists $\tau_{u}>0$ such that
$\displaystyle\sum^{N}_{j=1}\left|\left|\frac{\partial v}{\partial
x_{j}}-\frac{\partial u}{\partial x_{j}}\right|\right|<\varsigma,\ \forall
v\in\mathcal{K}\cap B_{X}(u,2\tau_{u}).$ (3.3)
Therefore, we can choose a finite open sub-covering of $\mathcal{K}$
$\displaystyle\mathcal{A}=\\{B_{X}(u_{i},\tau_{u_{i}})\ |\ i=1,\cdots,s\\}$
(3.4)
from the open covering $\\{B_{X}(u,\tau_{u})\ |\ u\in\mathcal{K}\\}$. Let
$\zeta\in C^{\infty}([0,+\infty))$ be such that $0\leq\zeta(t)\leq 1$ for all
$t,$ $\zeta(t)=1$ for $t\in[0,1/2]$ and $\zeta(t)=0$ for $t\in[1,\infty).$ Let
$\xi_{i}(u)=\frac{\zeta(||u-u_{i}||/\tau_{u_{i}})}{\sum^{s}_{i=1}\zeta(||u-u_{i}||/\tau_{u_{i}})},\
1\leq i\leq s.$
Then $\\{\xi_{i}\ |\ 1\leq i\leq s\\}$ is a $C^{\infty}$ partition of unity
corresponding to the covering $\mathcal{A}$.
For $u\in\mathcal{K},$ let
$Y_{u}:=\\{h\in X\ |\ \nabla^{2}I(u)h=0\\},\
Z_{u}:=\mbox{span}\\{\frac{\partial u}{\partial x_{i}}\ |\ 1\leq i\leq N\\}.$
Let
$\mathcal{Y}=\mbox{span}\\{\cup^{s}_{i=1}Y_{u_{i}}\\}.$ (3.5)
Let
$q=\dim\mathcal{Y}.$ (3.6)
Let $\\{e_{1},e_{2},\cdots,e_{q}\\}$ be an orthogonal normal base of
$\mathcal{Y}$. As mentioned in Remark 3.1, for every $1\leq n\leq q,$
$e_{n}\in W^{2,2}_{r}(\mathbb{R}^{N})$ and $e_{n}$ satisfies exponential decay
at infinity.
Let $\\{e^{\prime}_{1},e^{\prime}_{2}\cdots\\}$ be an orthogonal normal base
of $\mathcal{Y}^{\bot},$ where $\mathcal{Y}^{\bot}$ is the orthogonal
complement space of $\mathcal{Y}$ in $X$. From the appendix A of this paper,
for every $k\in\mathbb{N},$ there exists
$\displaystyle E_{k}:=\\{\tilde{e}_{j,k}\ |\ 1\leq j\leq k\\},$ (3.7)
such that
* (i)
For every $k,$ $E_{k}\subset X\cap W^{2,2}_{r}(\mathbb{R}^{N})$ and
$E_{k}\bot\mathcal{Y}$;
* (ii)
Every $\tilde{e}_{j,k}$ satisfies exponential decay at infinity,
$\langle\tilde{e}_{j,k},\tilde{e}_{j^{\prime},k}\rangle=\delta_{j,j^{\prime}}$
and
$\displaystyle\sup_{1\leq j\leq k}||\tilde{e}_{j,k}-e^{\prime}_{j}||\leq
1/2^{k}.$
For every $k,$ denote
$\displaystyle X_{k}:=\mbox{span}\\{E_{k}\\}\oplus\mathcal{Y}.$
Let $P_{k}:X\rightarrow X_{k}$ and $P^{\bot}_{k}:X\rightarrow X^{\bot}_{k}$ be
the orthogonal projections, where $X^{\bot}_{k}$ is the orthogonal complement
space of $X_{k}$ in $X.$ By the definition of $X_{k}$ and the properties
$\bf(i)$ and $\bf(ii)$ mentioned above, we have the following Lemma which is
easy to prove.
###### Lemma 3.4.
For every $h\in X,$
$\lim_{k\rightarrow\infty}||h-P_{k}h||=\lim_{k\rightarrow\infty}||P_{k}^{\bot}h||=0.$
###### Lemma 3.5.
For any $r>0,$ there exists $l_{r}\in\mathbb{N}$ such that if $k\geq l_{r}$,
then for every $v\in N_{r}(\mathcal{K})$,
$P^{\bot}_{k}\nabla^{2}I(v)|_{X^{\bot}_{k}}$ is invertible and
$\displaystyle||(P^{\bot}_{k}\nabla^{2}I(v)|_{X^{\bot}_{k}})^{-1}||_{\mathcal{L}(X^{\bot}_{k})}\leq
2.$
Proof. For $w\in X^{\bot}_{k}$,
$\displaystyle
P^{\bot}_{k}\nabla^{2}I(v)w=w-P^{\bot}_{k}(-\triangle+1)^{-1}f^{\prime}(v)w.$
Denote the operator $w\mapsto P^{\bot}_{k}(-\triangle+1)^{-1}f^{\prime}(v)w$
by $A_{v,k}.$ If we can prove that
$\limsup_{k\rightarrow\infty}\sup\\{||A_{v,k}||_{\mathcal{L}(X^{\bot}_{k})}\
|\ v\in N_{r}(\mathcal{K})\\}=0,$ (3.8)
then the conclusion of this Lemma follows. If (3.8) were not true, we can
choose $v_{k}\in N_{r}(\mathcal{K})$ and $w_{k}\in X_{k}^{\bot}$ with
$||w_{k}||=1$, $k=1,2,\cdots,$ such that
$\displaystyle\limsup_{k\rightarrow\infty}||A_{v_{k},k}w_{k}||>0.$ (3.9)
Without loss of generality, we assume that $v_{k}\rightharpoonup v_{0}$ in $X$
and $w_{k}\rightharpoonup w_{0}$ in $X$ as $k\rightarrow\infty.$ Since for any
$2\leq p<2^{*}$, $X$ can be compactly embedded into the radially symmetric
$L^{p}$ space (see, for example, [24, Corollary 1.26])
$L^{p}_{r}(\mathbb{R}^{N}):=\\{u\in L^{p}(\mathbb{R}^{N})\ |\ u\ \mbox{ is
radially symmetric}\\},$
combining the condition $\bf(F_{1})$, we can get that
$\lim_{k\rightarrow\infty}\sup\\{\int_{\mathbb{R}^{N}}|f^{\prime}(v_{k})w_{k}h-f^{\prime}(v_{0})w_{0}h|\
|\ h\in X,\ ||h||\leq 1\\}=0.$
It follows that
$\displaystyle\lim_{k\rightarrow\infty}||(-\triangle+1)^{-1}(f^{\prime}(v_{k})w_{k}-f^{\prime}(v_{0})w_{0})||=0.$
(3.10)
By (3.10) and Lemma 3.4, we deduce that
$\lim_{k\rightarrow\infty}||A_{v_{k},k}w_{k}||=0.$ But this contradicts (3.9).
$\Box$
For $u\in\mathcal{K}$, denote $X_{k}\oplus Z_{u}$ by $W_{u,k}$ and let
$W_{u,k}^{\bot}$ be the orthogonal complement space of $W_{u,k}$ in $Y.$ Let
$P_{W_{u_{i},k}}:Y\rightarrow W_{u_{i},k}$ and
$P_{W^{\bot}_{u_{i},k}}:Y\rightarrow W^{\bot}_{u_{i},k}$ be the orthogonal
projections.
###### Lemma 3.6.
Suppose that $\kappa:=\max\\{\tau_{u_{i}}\ |\ 1\leq i\leq s\\}$ is
sufficiently small, where $\tau_{u_{i}}$ comes from (3.4). Then there exist
$C>0$ and $l_{\kappa}\in\mathbb{N}$ such that if $k\geq l_{\kappa}$ and $v\in
B_{X}(u_{i},\tau_{u_{i}})$ for some $1\leq i\leq s$, then
$P_{W^{\bot}_{u_{i},k}}\nabla^{2}J(v)|_{W^{\bot}_{u_{i},k}}$ is invertible and
$\displaystyle||(P_{W^{\bot}_{u_{i},k}}\nabla^{2}J(v)|_{W^{\bot}_{u_{i},k}})^{-1}||_{\mathcal{L}(W^{\bot}_{u_{i},k})}\leq
C.$ (3.11)
Proof. We note that for $w\in W^{\bot}_{u_{i},k}$,
$\displaystyle
P_{W^{\bot}_{u_{i},k}}\nabla^{2}J(v)w=w-P_{W^{\bot}_{u_{i},k}}(-\triangle+1)^{-1}f^{\prime}(u)w.$
Since for any $p\in[2,2^{*})$, $X$ can be compactly embedded into the radially
symmetric $L^{p}$ space, by the condition $\bf(F_{1})$, we deduce that
$w\mapsto P_{W^{\bot}_{u_{i},k}}(-\triangle+1)^{-1}f^{\prime}(v)w$ is a
compact operator. It follows that
$P_{W^{\bot}_{u_{i},k}}\nabla^{2}J(v)|_{W^{\bot}_{u_{i},k}}$ is a Fredholm
operator with index zero. Therefore, if we can prove that there exists $C>0$
which is independent of $k$ such that, for sufficiently large $k,$
$\displaystyle||P_{W^{\bot}_{u_{i},k}}\nabla^{2}J(v)w||_{\mathcal{L}(W^{\bot}_{u_{i},k})}\geq\frac{1}{C}||w||,\
\forall w\in W^{\bot}_{u_{i},k},\ \forall v\in B_{X}(u_{i},\tau_{u_{i}})$
then the conclusion of this Lemma follows.
Without loss of generality, we assume that $u_{i}\equiv u_{1}$ and for the
sake of simplicity, we denote the operator
$P_{W^{\bot}_{u_{1},k}}\nabla^{2}J(v)|_{W^{\bot}_{u_{1},k}}$ by $H_{v,k}.$ If
such $C>0$ does not exist, then there exist sequences
$\\{\tau^{k}_{u_{1}}\\}$, $\\{v_{k}\\}\subset X$ and $\\{w_{k}\\}\subset Y$
such that $\tau^{k}_{u_{1}}\rightarrow 0$ as $k\rightarrow\infty,$ $v_{k}\in
B_{X}(u_{1},\tau^{k}_{u_{1}})$, $w_{k}\in W^{\bot}_{u_{1},k}$, $||w_{k}||=1$,
$k=1,2,\cdots$ and
$\displaystyle\lim_{k\rightarrow\infty}||H_{v_{k},k}w_{k}||=0.$ (3.12)
Passing to a subsequence, we may assume that $w_{k}\rightharpoonup w_{0}$ in
$Y$ as $k\rightarrow\infty.$ By $\tau^{k}_{u_{1}}\rightarrow 0$ as
$k\rightarrow\infty$ and the assumption that $\\{v_{k}\\}\subset
B_{X}(u_{1},\tau^{k}_{u_{1}})$, we get that
$\displaystyle\lim_{k\rightarrow\infty}||v_{k}-u_{1}||=0.$ (3.13)
By $w_{k}\in W^{\bot}_{u_{1},k}$ and $w_{k}\rightharpoonup w_{0}$ in $Y$, we
get that $w_{0}\bot X\oplus Z_{u_{1}}$. Combining the condition $\bf(F_{1})$,
(3.13) and the fact that $w_{k}\rightharpoonup w_{0}$ in $Y$ leads to
$\displaystyle\lim_{k\rightarrow\infty}||(-\triangle+1)^{-1}(f^{\prime}(v_{k})w_{k}-f^{\prime}(u_{1})w_{k})||=0$
(3.14)
and
$\displaystyle\lim_{k\rightarrow\infty}||(-\triangle+1)^{-1}(f^{\prime}(u_{1})w_{k}-f^{\prime}(u_{1})w_{0})||=0.$
(3.15)
By (3.15) and (3.14), we get that
$\displaystyle\lim_{k\rightarrow\infty}||(-\triangle+1)^{-1}(f^{\prime}(v_{k})w_{k}-f^{\prime}(u_{1})w_{0})||=0.$
(3.16)
By Lemma 3.4, we deduce that
$\displaystyle\lim_{k\rightarrow\infty}||P_{W^{\bot}_{u_{1},k}}h-P_{(X\oplus
Z_{u_{{}_{1}}})^{\bot}}h||=0,\ \forall h\in Y,$ (3.17)
where $P_{(X\oplus Z_{u_{1}})^{\bot}}:Y\rightarrow(X\oplus
Z_{u_{{}_{1}}})^{\bot}$ is the orthogonal projection. By (3.16) and (3.17), we
get that
$\displaystyle\lim_{k\rightarrow\infty}||P_{W^{\bot}_{u_{1},k}}((-\triangle+1)^{-1}f^{\prime}(v_{k})w_{k})-P_{(X\oplus
Z_{u_{{}_{1}}})^{\bot}}((-\triangle+1)^{-1}f^{\prime}(u_{1})w_{0})||=0.$
(3.18)
By definition,
$\displaystyle
H_{v_{k},k}w_{k}=w_{k}-P_{W^{\bot}_{u_{1},k}}(-\triangle+1)^{-1}f^{\prime}(v_{k})w_{k}.$
(3.19)
By (3.18) and the assumption
$\lim_{k\rightarrow\infty}||H_{v_{k},k}w_{k}||=0$, we deduce that
$\\{w_{k}\\}$ is compact in $Y.$ Therefore, $||w_{k}-w_{0}||\rightarrow 0$ as
$k\rightarrow\infty.$ It follows that $||w_{0}||=1,$ since $||w_{k}||=1$ for
every $k.$
Sending $k$ into infinity in the equality (3.19), by $w_{0}\in(X\oplus
Z_{u_{{}_{1}}})^{\bot},$ (3.12) and (3.18), we get that
$\displaystyle P_{(X\oplus
Z_{u_{{}_{1}}})^{\bot}}(w_{0}-(-\triangle+1)^{-1}f^{\prime}(u_{1})w_{0})=0.$
(3.20)
By $w_{0}\bot X$ and $u_{1}\in X,$ we have
$\displaystyle\langle
w_{0}-(-\triangle+1)^{-1}f^{\prime}(u_{1})w_{0},h\rangle$ (3.21)
$\displaystyle=$ $\displaystyle\langle
w_{0},h\rangle-\langle(-\triangle+1)^{-1}f^{\prime}(u_{1})h,w_{0}\rangle=0,\
\forall h\in X.$
Since for any $h\in Z_{u_{1}},$
$h-(-\triangle+1)^{-1}f^{\prime}(u_{1})h=0,$
we get that
$\displaystyle\langle
w_{0}-(-\triangle+1)^{-1}f^{\prime}(u_{1})w_{0},h\rangle$ (3.22)
$\displaystyle=$ $\displaystyle\langle
h-(-\triangle+1)^{-1}f^{\prime}(u_{1})h,w_{0}\rangle=0,\ \forall h\in
Z_{u_{1}}.$
By (3.21) and (3.22), we get that
$\displaystyle P_{X\oplus
Z_{u_{{}_{1}}}}(w_{0}-(-\triangle+1)^{-1}f^{\prime}(u_{1})w_{0})=0.$ (3.23)
By (3.20) and (3.23), we obtain
$w_{0}-(-\triangle+1)^{-1}f^{\prime}(u_{1})w_{0}=0,$
that is, $w_{0}$ is an eigenfunction of (3.2) with $u=u_{1}\in\mathcal{K}.$
But $w_{0}$ satisfies $w_{0}\bot X\oplus Z_{u_{1}}$ and $||w_{0}||=1.$ This
contradicts Lemma 3.2. $\Box$
For $v\in\cup^{s}_{i=1}B_{X}(u_{i},\tau_{u_{i}}),$ let
$\displaystyle\mathcal{T}_{v}=\mbox{span}\\{\sum^{s}_{i=1}\xi_{i}(v)\frac{\partial
u_{i}}{\partial x_{j}}\ |\ 1\leq j\leq N\\}.$ (3.24)
The space $X_{k}\oplus\mathcal{T}_{v}$ is denoted by $E_{v,k}$. Let
$P_{E^{\bot}_{v,k}}:Y\rightarrow E^{\bot}_{v,k}$ be the orthogonal projection.
###### Lemma 3.7.
Suppose that $\kappa=\max\\{\tau_{u_{i}}\ |\ 1\leq i\leq s\\}$ is sufficiently
small. Then there exist $C^{\prime}>0$ and $l_{\kappa}\in\mathbb{N}$ such that
if $k\geq l_{\kappa}$, then for every
$v\in\cup^{s}_{i=1}B_{X}(u_{i},\tau_{u_{i}})$, the operator
$P_{E^{\bot}_{v,k}}\nabla^{2}J(v)|_{E^{\bot}_{v,k}}$ is invertible and
$\displaystyle||(P_{E^{\bot}_{v,k}}\nabla^{2}J(v)|_{E^{\bot}_{v,k}})^{-1}||_{\mathcal{L}(E^{\bot}_{v,k})}\leq
C^{\prime}.$ (3.25)
Proof. As the proof of Lemma 3.6, it suffices to prove that there exists
$C^{\prime}>0$ which is independent of $k$ such that, for sufficiently large
$k,$
$\displaystyle||P_{E^{\bot}_{v,k}}\nabla^{2}J(v)w||_{\mathcal{L}(E^{\bot}_{v,k})}\geq\frac{1}{C^{\prime}}||w||,\
\forall w\in E^{\bot}_{v,k},\ \forall
v\in\cup^{s}_{i=1}B_{X}(u_{i},\tau_{u_{i}}).$ (3.26)
Without loss of generality, we assume that $v\in B(u_{1},\tau_{u_{{}_{1}}})$.
Let $P_{X_{k}}:Y\rightarrow X_{k}$ and
$P_{\mathcal{T}_{v}}:Y\rightarrow\mathcal{T}_{v}$ be orthogonal projections.
For $h\in Y,$
$\displaystyle P_{E^{\bot}_{v,k}}h=h-P_{X_{k}}h-P_{\mathcal{T}_{v}}h,$ (3.27)
and
$\displaystyle
P_{\mathcal{T}_{v}}h=\sum^{N}_{j=1}\Big{\langle}h,\sum^{s}_{i=1}\xi_{i}(v)\frac{\partial
u_{i}}{\partial
x_{j}}\Big{\rangle}\frac{\sum^{s}_{i=1}\xi_{i}(v)\frac{\partial
u_{i}}{\partial x_{j}}}{||\sum^{s}_{i=1}\xi_{i}(v)\frac{\partial
u_{i}}{\partial x_{j}}||^{2}}.$ (3.28)
Since $\\{\xi_{i}\ |\ 1\leq i\leq s\\}$ is a partition of unity, we get that
for every $1\leq j\leq N,$
$\displaystyle||\frac{\partial u_{1}}{\partial
x_{j}}-\sum^{s}_{i=1}\xi_{i}(v)\frac{\partial u_{i}}{\partial x_{j}}||$
$\displaystyle=$ $\displaystyle||\sum^{s}_{i=1}\xi_{i}(v)\frac{\partial
u_{1}}{\partial x_{j}}-\sum^{s}_{i=1}\xi_{i}(v)\frac{\partial u_{i}}{\partial
x_{j}}||$ (3.29) $\displaystyle\leq$
$\displaystyle\sum^{s}_{i=1}\xi_{i}(v)||\frac{\partial u_{1}}{\partial
x_{j}}-\frac{\partial u_{i}}{\partial x_{j}}||.$
If $\xi_{i}(v)\neq 0,$ then $v\in B_{X}(u_{i},\tau_{u_{i}})$. Combining the
assumption $v\in B_{X}(u_{1},\tau_{u_{1}})$, we get that $u_{1}\in
B_{X}(u_{i},2\tau_{u_{i}})\cap\mathcal{K}$. Therefore, by (3.3), we deduce
that
$\displaystyle\sum^{s}_{i=1}||\frac{\partial u_{1}}{\partial
x_{j}}-\frac{\partial u_{i}}{\partial x_{j}}||<\varsigma,\ \mbox{if}\
\xi_{i}(v)\neq 0.$ (3.30)
Combining (3.29) and (3.30) leads to
$\displaystyle||\frac{\partial u_{1}}{\partial
x_{j}}-\sum^{s}_{i=1}\xi_{i}(v)\frac{\partial u_{i}}{\partial
x_{j}}||<\varsigma,\ \mbox{for every }\ 1\leq j\leq N.$ (3.31)
Thus, there exists $C>0$ which is independent of $k$ such that
$\displaystyle||P_{\mathcal{T}_{v}}h-P_{Z_{u_{1}}}h||\leq C\varsigma||h||,\
\forall h\in Y,$ (3.32)
where
$P_{Z_{u_{1}}}:Y\rightarrow Z_{u_{1}},\
h\mapsto\sum^{N}_{j=1}\Big{\langle}h,\frac{\partial u_{1}}{\partial
x_{j}}\Big{\rangle}\frac{\frac{\partial u_{1}}{\partial
x_{j}}}{||\frac{\partial u_{1}}{\partial x_{j}}||^{2}}$
is orthogonal projection. By (3.27) and (3.32), we have
$\displaystyle||P_{E^{\bot}_{v,k}}h-P_{W^{\bot}_{u_{{}_{1}},k}}h||\leq
C\varsigma||h||,\ \forall h\in Y.$ (3.33)
For $w\in E^{\bot}_{v,k},$ we have
$\displaystyle||P_{E^{\bot}_{v,k}}\nabla^{2}J(v)w||$ $\displaystyle\geq$
$\displaystyle||P_{W^{\bot}_{u_{1},k}}\nabla^{2}J(v)w||-||(P_{E^{\bot}_{v,k}}-P_{W^{\bot}_{u_{1},k}})\nabla^{2}J(v)w||$
$\displaystyle\geq$
$\displaystyle||P_{W^{\bot}_{u_{1},k}}\nabla^{2}J(v)w||-C\varsigma||\nabla^{2}J(v)||_{\mathcal{L}(Y)}||w||\
(\mbox{by}\ (\ref{gdvc66dtdfaa}))$ $\displaystyle\geq$
$\displaystyle||P_{W^{\bot}_{u_{1},k}}\nabla^{2}J(v)(w-P_{Z_{u_{1}}}w)||-||P_{W^{\bot}_{u_{1},k}}\nabla^{2}J(v)(P_{Z_{u_{1}}}w)||-C\varsigma||\nabla^{2}J(v)||_{\mathcal{L}(Y)}||w||$
$\displaystyle\geq$ $\displaystyle
C||w-P_{Z_{u_{1}}}w||-||\nabla^{2}J(v)||_{\mathcal{L}(Y)}||P_{Z_{u_{1}}}w||$
$\displaystyle-C\varsigma||\nabla^{2}J(v)||_{\mathcal{L}(Y)}||w||\ (\mbox{by}\
w-P_{Z_{u_{1}}}w\in W^{\bot}_{u_{1},k}\ \mbox{and}\ (\ref{gdg5w5qdaxz}))$
$\displaystyle\geq$ $\displaystyle
C||w||-(C+||\nabla^{2}J(v)||_{\mathcal{L}(Y)})||P_{Z_{u_{1}}}w||-C\varsigma||\nabla^{2}J(v)||_{\mathcal{L}(Y)}||w||$
$\displaystyle=$ $\displaystyle
C||w||-(C+||\nabla^{2}J(v)||_{\mathcal{L}(Y)})||P_{\mathcal{T}_{v}}w-P_{Z_{u_{1}}}w||$
$\displaystyle-C\varsigma||\nabla^{2}J(v)||_{\mathcal{L}(Y)}||w||\
(\mbox{since}\ P_{\mathcal{T}_{v}}w=0)$ $\displaystyle\geq$ $\displaystyle
C||w||-\varsigma
C(C+||\nabla^{2}J(v)||_{\mathcal{L}(Y)})||w||-C\varsigma||\nabla^{2}J(v)||_{\mathcal{L}(Y)}||w||.\
(\mbox{by}\ (\ref{4gdvc66dtdfaa}))$
It follows that if $\kappa>0$ is sufficiently small, then there exist
$l_{\kappa}\in\mathbb{N}$ and $C^{\prime}>0$ such that for every $k\geq
l_{\kappa},$ (3.26) holds. $\Box$
Recall that $X^{\bot}_{k}$ is the orthogonal complement space of $X_{k}$ in
$X$ and $P_{k}:X\rightarrow X_{k}$, $P^{\bot}_{k}:X\rightarrow X^{\bot}_{k}$
are orthogonal projections. Let
$\mathcal{N}_{\delta,\tau,k}:=\\{u+v\in X\ |\ u\in X_{k},\
\mbox{dist}_{X}(u,P_{k}\mathcal{K})<\delta,\ v\in X^{\bot}_{k},\
||v||<\tau\\},$
where $P_{k}\mathcal{K}=\\{P_{k}v\ |\ v\in\mathcal{K}\\}.$ By Lemma 3.4 and
the fact that $\mathcal{K}$ is a compact subset of $X,$ we get that as
$k\rightarrow\infty$, the Hausdorff distance of $\mathcal{K}$ and
$P_{k}\mathcal{K}$,
$\displaystyle\sup_{v\in
P_{k}\mathcal{K}}\mbox{dist}_{X}(v,\mathcal{K})+\sup_{u\in\mathcal{K}}\mbox{dist}_{X}(u,P_{k}\mathcal{K})\rightarrow
0.$ (3.35)
Thus, for any $\delta>0$, $\tau>0$ and $0<r<\min\\{\delta,\tau\\}$, if $k$ is
sufficiently large, then
$N_{r}(\mathcal{K})\subset\mathcal{N}_{\delta,\tau,k},$ (3.36)
where $N_{r}(\mathcal{K})$ comes from (2.8). And for any $r>0$, if
$\delta,\tau\in(0,r/2)$, then for sufficiently large $k$,
$\mathcal{N}_{\delta,\tau,k}\subset N_{r}(\mathcal{K}).$ (3.37)
Let
$\displaystyle\mathcal{N}_{\delta,k}:=\\{u\in X_{k}\ |\
\mbox{dist}_{X}(u,P_{k}\mathcal{K})<\delta\\}.$ (3.38)
###### Lemma 3.8.
If $\delta>0$ is sufficient small and $k$ is sufficiently large, then there
exists a $C^{1}-$mapping
$\pi_{k}:\mathcal{N}_{\delta,k}\rightarrow X^{\bot}_{k},$
satisfying
* (i)
$\langle\nabla I(v+\pi_{k}(v)),\phi\rangle=0,$ $\forall\phi\in X^{\bot}_{k};$
* (ii)
$\lim_{k\rightarrow\infty}\sup\\{||\pi_{k}(v)||\ |\
v\in\mathcal{N}_{\delta,k}\\}=0;$
* (iii)
$\lim_{k\rightarrow\infty}\sup\\{||D\pi_{k}(v)h||\ |\
v\in\mathcal{N}_{\delta,k},\ h\in X_{k},\ ||h||=1\\}=0;$
* (iv)
If $v$ is a critical point of $I(v+\pi_{k}(v))$, then $v+\pi_{k}(v)$ is a
critical point of $I.$
Proof. By Lemma 3.5, if $r>0$ is small enough, then the operator
$L_{v,k}:=P^{\bot}_{k}\nabla^{2}I(v)|_{X^{\bot}_{k}}:X^{\bot}_{k}\rightarrow
X^{\bot}_{k}$
is invertible and if $k\geq l_{\kappa}$,
$\displaystyle||L_{v,k}^{-1}||_{\mathcal{L}(X_{k}^{\bot})}\leq 2,\ \forall
v\in N_{r}(\mathcal{K}).$ (3.39)
Assume that $0<\delta<r$, by (3.37), if $k$ is large enough, then
$\mathcal{N}_{\delta,k}\subset N_{r}(\mathcal{K})$.
For $\rho>0$ and $v\in\mathcal{N}_{\delta,k},$ define
$\Psi_{v,k}:\overline{B_{X^{\bot}_{k}}(0,\rho)}\rightarrow X^{\bot}_{k},\
w\mapsto w-L^{-1}_{v,k}P^{\bot}_{k}\nabla I(v+w).$
For any $w_{i}\in\overline{B_{X^{\bot}_{k}}(0,\rho)},$ $i=1,2,$ by the
definition of $L_{v,k}$, we have
$w_{2}-w_{1}-L^{-1}_{v,k}P^{\bot}_{k}\nabla^{2}I(v)(w_{2}-w_{1})=0.$
Therefore,
$\displaystyle||\Psi_{v,k}(w_{2})-\Psi_{v,k}(w_{1})||$ (3.40) $\displaystyle=$
$\displaystyle||w_{2}-w_{1}-L^{-1}_{v,k}P^{\bot}_{k}\nabla^{2}I(v+\theta
w_{2}+(1-\theta)w_{1})(w_{2}-w_{1})||$ $\displaystyle(\mbox{by the mean value
theorem},\ 0<\theta=\theta(x)<1)$ $\displaystyle\leq$
$\displaystyle||w_{2}-w_{1}-L^{-1}_{v,k}P^{\bot}_{k}\nabla^{2}I(v)(w_{2}-w_{1})||$
$\displaystyle+||L^{-1}_{v,k}P^{\bot}_{k}(\nabla^{2}I(v+\theta
w_{2}+(1-\theta)w_{1})-\nabla^{2}I(v))(w_{2}-w_{1})||$ $\displaystyle=$
$\displaystyle||L^{-1}_{v,k}P^{\bot}_{k}(\nabla^{2}I(v+\theta
w_{2}+(1-\theta)w_{1})-\nabla^{2}I(v))(w_{2}-w_{1})||$ $\displaystyle\leq$
$\displaystyle 2||(\nabla^{2}I(v+\theta
w_{2}+(1-\theta)w_{1})-\nabla^{2}I(v))(w_{2}-w_{1})||\ (\mbox{by}\
(\ref{nnb99ifufjjj})).$
Since $I\in C^{2}(X,\mathbb{R})$ and $\mathcal{K}$ is compact in $X$, if
$\delta$ and $\rho$ are small enough, then for any
$v\in\mathcal{N}_{\delta,k}$ and $w\in\overline{B_{X^{\bot}_{k}}(0,\rho)},$
$||\nabla^{2}I(v+w)-\nabla^{2}I(v)||_{\mathcal{L}(X)}<1/4.$
Thus, by (3.40), we get that for any
$w_{i}\in\overline{B_{X^{\bot}_{k}}(0,\rho)},$ $i=1,2,$
$\displaystyle||\Psi_{v,k}(w_{2})-\Psi_{v,k}(w_{1})||\leq\frac{1}{2}||w_{2}-w_{1}||.$
(3.41)
If $\delta>0$ is small enough and $k$ is large enough, then for every
$v\in\mathcal{N}_{\delta,k}$,
$||\Psi_{v,k}(0)||\leq\rho/2.$
Then by (3.41), we get that for every
$w\in\overline{B_{X^{\bot}_{k}}(0,\rho)}$,
$\displaystyle||\Psi_{v,k}(w)||\leq||\Psi_{v,k}(w)-\Psi_{v,k}(0)||+||\Psi_{v,k}(0)||\leq\rho.$
(3.42)
By (3.41) and (3.42), $\Psi_{v,k}$ is a contractive mapping in
$\overline{B_{X^{\bot}_{k}}(0,\rho)}$ if $\delta$ and $\rho$ are small enough
and $k$ is large enough. Thus, by Banach fixed point theorem, there exists
unique fixed point $\pi_{k}(v)\in\overline{B_{X^{\bot}_{k}}(0,\rho)}.$ It is
easy to verify that $\pi_{k}$ is a $C^{1}-$mapping and it satisfies the result
$\bf(i)$.
Now, we give the proof of $\bf(ii).$ By $P_{k}^{\bot}\nabla I(v+\pi_{k}(v))=0$
and $\pi_{k}(v)\in X^{\bot}_{k}$, we get that
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\langle\nabla
I(v+\pi_{k}(v)),\pi_{k}(v)\rangle$ (3.43) $\displaystyle=$
$\displaystyle||\pi_{k}(v)||^{2}-\int_{\mathbb{R}^{N}}f(v+\pi_{k}(v))\cdot\pi_{k}(v).$
By Lemma 3.4, we deduce that for any sequence $\\{v_{k}\\}$ with
$v_{k}\in\mathcal{N}_{\delta,k}$, $\pi_{k}(v_{k})\rightharpoonup 0$ in $X$ as
$k\rightarrow\infty$. Combining the compact embedding $X\hookrightarrow
L^{p}_{r}(\mathbb{R}^{N})$, we obtain
$\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}|f(v_{k}+\pi_{k}(v_{k}))|\cdot|\pi_{k}(v_{k})|=0.$
It follows that
$\displaystyle\lim_{k\rightarrow\infty}\sup\\{\int_{\mathbb{R}^{N}}f(v+\pi_{k}(v))\cdot\pi_{k}(v)\
|\ v\in\mathcal{N}_{\delta,k}\\}=0.$ (3.44)
The conclusion $\bf(ii)$ follows from (3.43) and (3.44).
Differentiating equation $P^{\bot}_{k}\nabla I(v+\pi_{k}(v))=0$ for the
variable $v$ in the direction $h\in X_{k}$, we get that
$\displaystyle
D\pi_{k}(v)h-P^{\bot}_{k}(-\triangle+1)^{-1}f^{\prime}(v+\pi_{k}(v))(h+D\pi_{k}(v)h)=0.$
(3.45)
Note that $D\pi_{k}(v)h\in X^{\bot}_{k}$. By (3.39), (3.45) and
$\lim_{k\rightarrow\infty}||\pi_{k}(v)||=0$, we get that if $k$ is large
enough, then
$\displaystyle\frac{1}{2}||D\pi_{k}(v)h||$ $\displaystyle\leq$
$\displaystyle||D\pi_{k}(v)h-P^{\bot}_{k}(-\triangle+1)^{-1}f^{\prime}(v+\pi_{k}(v))D\pi_{k}(v)h||$
$\displaystyle=$
$\displaystyle||P^{\bot}_{k}(-\triangle+1)^{-1}f^{\prime}(v+\pi_{k}(v))h||$
It follows that for sufficiently large $k,$
$\displaystyle\sup\\{||D\pi_{k}(v)h||\ |\ v\in\mathcal{N}_{\delta,k},\ h\in
X_{k},\ ||h||\leq 1\\}<\infty.$ (3.47)
By (3.45), we get that
$\displaystyle||D\pi_{k}(v)h||^{2}=\int_{\mathbb{R}^{N}}f^{\prime}(v+\pi_{k}(v))\cdot(h+D\pi_{k}(v)h)\cdot
D\pi_{k}(v)h.$ (3.48)
(3.47) and the same argument as (3.44) yield
$\lim_{k\rightarrow\infty}\sup\\{\int_{\mathbb{R}^{N}}f^{\prime}(v+\pi_{k}(v))\cdot(h+D\pi_{k}(v)h)\cdot
D\pi_{k}(v)h\ |\ v\in\mathcal{N}_{\delta,k},\ h\in X_{k},\ ||h||\leq 1\\}=0.$
Combining (3.48), we get the conclusion $\bf(iii).$
By $\bf(iii),$ if $k$ is sufficiently large, then
$\\{h+D\pi_{k}(v)h\ |\ h\in X_{k}\\}+X^{\bot}_{k}=X.$
Combining the result $\bf(i)$, we get that if $v_{0}$ is a critical point of
$I(v+\pi_{k}(v))$, then $v_{0}+\pi_{k}(v_{0})$ is a critical point of $I$.
$\Box$
###### Remark 3.9.
By $\bf(ii)$ and $\bf(iv)$ of Lemma 3.8, $\mathcal{N}_{\delta,\tau,k}$ is a
neighborhood of $\mathcal{K}$ if
$\displaystyle\tau>\sup\\{||\pi_{k}(v)||\ |\ v\in\mathcal{N}_{\delta,k}\\}.$
(3.49)
###### Lemma 3.10.
Let
$\mathcal{I}_{k}(u)=\frac{1}{2}||P^{\bot}_{k}u||^{2}+I(P_{k}u+\pi_{k}(P_{k}u)).$
Then
$\lim_{k\rightarrow\infty}||\mathcal{I}_{k}-I||_{C^{1}(\overline{\mathcal{N}_{\delta,\tau,k}})}=0.$
Proof. By definition, we have
$\mathcal{I}_{k}(u)=\frac{1}{2}||u||^{2}+\frac{1}{2}||\pi_{k}(P_{k}u)||^{2}-\int_{\mathbb{R}^{N}}F(P_{k}u+\pi_{k}(P_{k}u)).$
For any sequence $\\{u_{k}\\}$ with
$u_{k}\in\overline{\mathcal{N}_{\delta,\tau,k}}$, by the mean value theorem,
we get that
$\displaystyle F(P_{k}u_{k}+\pi_{k}(P_{k}u_{k}))-F(u_{k})$ $\displaystyle=$
$\displaystyle\zeta(u_{k},\theta)(P_{k}u_{k}+\pi_{k}(P_{k}u_{k})-u_{k})$
$\displaystyle=$
$\displaystyle\zeta(u_{k},\theta)(\pi_{k}(P_{k}u_{k})-P_{k}^{\bot}u_{k})$
where
$\displaystyle\zeta(u_{k},\theta)=f^{\prime}(\theta
P_{k}u_{k}+\theta\pi_{k}(P_{k}u_{k})+(1-\theta)u_{k})$
with $0<\theta(x)<1,$ $x\in\mathbb{R}^{N}$. Then we have
$\displaystyle\int_{\mathbb{R}^{N}}\Big{|}F(P_{k}u_{k}+\pi_{k}(P_{k}u_{k}))-F(u_{k})\Big{|}=\int_{\mathbb{R}^{N}}|\zeta(u_{k},\theta)|\cdot|\pi_{k}(P_{k}u_{k})-P_{k}^{\bot}u_{k}|.$
(3.50)
By $\bf(ii)$ of Lemma 3.8, we get that for every $2\leq p<2^{*},$
$\displaystyle\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}|\pi_{k}(P_{k}u_{k})|^{p}=0.$
(3.51)
By Lemma 3.4,we have
$\displaystyle P^{\bot}_{k}u_{k}\rightharpoonup 0\ \mbox{in}\ X.$ (3.52)
Since $X$ can be compactly embedded into $L^{p}_{r}(\mathbb{R}^{N})$, by
(3.52), we get that for every $2\leq p<2^{*},$
$\displaystyle\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}|P^{\bot}_{k}u_{k}|^{p}=0.$
(3.53)
By (3.50), (3.51), (3.53) and the condition $\bf(F_{1})$, we obtain
$\displaystyle\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}\Big{|}F(P_{k}u_{k}+\pi_{k}(P_{k}u_{k}))-F(u_{k})\Big{|}=0.$
Thus
$\displaystyle\lim_{k\rightarrow\infty}\sup\\{\int_{\mathbb{R}^{N}}\Big{|}F(P_{k}u+\pi_{k}(P_{k}u))-F(u)\Big{|}\
|\ u\in\overline{\mathcal{N}_{\delta,\tau,k}}\\}=0.$ (3.54)
By $\bf(ii)$ of Lemma 3.8 and (3.54), we get that
$\displaystyle\lim_{k\rightarrow\infty}||\mathcal{I}_{k}-I||_{C^{0}(\overline{\mathcal{N}_{\delta,\tau,k}})}=0.$
(3.55)
For $h\in X,$
$\displaystyle\langle\nabla\mathcal{I}_{k}(u),h\rangle$ $\displaystyle=$
$\displaystyle\langle
u,h\rangle+\langle\pi_{k}(P_{k}u),D\pi_{k}(P_{k}u)(P_{k}h)\rangle$
$\displaystyle-\int_{\mathbb{R}^{N}}f(P_{k}u+\pi_{k}(P_{k}u))\cdot(P_{k}h+D\pi_{k}(P_{k}u)(P_{k}h)).$
By $\bf(iii)$ of Lemma 3.8 and the same argument as above, we can get that
$\displaystyle\lim_{k\rightarrow\infty}\sup\\{\langle\nabla\mathcal{I}_{k}(u)-\nabla
I(u),h\rangle\ |\ u\in\overline{\mathcal{N}_{\delta,\tau,k}},\ ||h||\leq
1\\}=0.$ (3.56)
The result of this Lemma follows from (3.55) and (3.56). $\Box$
###### Remark 3.11.
For $r>0,$ let $\sigma\in(0,\sigma_{r/2})$, where $\sigma_{r/2}$ comes from
Lemma 2.7, and let $a\in(c-\sigma,c)$, $b\in(c,c+\sigma)$ be regular values of
$I$, where $c$ comes from (2.3). By Lemma 2.7, there exists a GM pair
$(W,W_{-})$ of $\mathcal{K}^{b}_{a}$ associated with some pseudo-gradient
vector field of $I$ such that $W\subset N_{r/2}(\mathcal{K}^{b}_{a}).$ By
(3.36), if $0<r<\min\\{\delta,\tau\\}$, then
$N_{r}(\mathcal{K})\subset\mathcal{N}_{\delta,\tau,k}$ if $k$ is sufficiently
large. Denote the critical set of $\mathcal{I}_{k}$ in
$\mathcal{N}_{\delta,\tau,k}$ by $\widehat{\mathcal{K}}_{k}$. By $\bf(i)$ and
$\bf(iv)$ of Lemma 3.8, we deduce that
$\widehat{\mathcal{K}}_{k}=P_{k}\mathcal{K}^{b}_{a}$. Then by (3.35),
$\widehat{\mathcal{K}}_{k}\subset\mbox{int}\ W$ if $k$ is large enough. By [9,
Theorem III.4] and Lemma 3.10, we infer that for sufficiently large $k$,
$(W,W_{-})$ is also a GM pair of $\mathcal{I}_{k}$ for
$\widehat{\mathcal{K}}_{k}$ associated with some pseudo-gradient vector filed
of $\mathcal{I}_{k}.$
For $v\in\mathcal{N}_{\delta,k},$ denote $I(v+\pi_{k}(v))$ by $g_{k}(v).$ And
denote the critical set of $g_{k}$ in $W$ by $\mathcal{K}_{k}$. By $\bf(i)$
and $\bf(iv)$ of Lemma 3.8, we deduce that
$\mathcal{K}_{k}=P_{k}\mathcal{K}^{b}_{a}=\widehat{\mathcal{K}}_{k}$. Let
$(W_{k},W^{-}_{k})$ be a GM pair of $g_{k}$ for $\mathcal{K}_{k}$. Note that
for $u=w+v\in\mathcal{N}_{\delta,\tau,k}$ with $w\in X^{\bot}_{k},$ $v\in
X_{k},$ $\mathcal{I}_{k}(u)=\frac{1}{2}||w||^{2}+g_{k}(v)$. By shifting
theorem (see Lemma 5.1 of [7]), we have
$\displaystyle\check{H}^{q}(W_{k},W^{-}_{k})=\check{H}^{q}(W,W^{-}),\
q=0,1,2,\cdots.$
Combining Lemma 2.5, we get that, for sufficiently large $k,$
$\displaystyle\check{H}^{1}(W_{k},W^{-}_{k})=\check{H}^{1}(W,W^{-})\neq 0.$
(3.57)
## 4 A variational reduction for the functional $E_{\epsilon}$
For $v\in\cup^{s}_{i=1}B_{X}(u_{i},\tau_{u_{i}})$ and $y\in\mathbb{R}^{N}$,
denote the space
$\\{\zeta(\cdot-y)\ |\ \zeta\in X_{k}\\}\oplus\mathcal{T}_{v}(\cdot-y)$
by $T_{v,y,k}$, where $\mathcal{T}_{v}$ comes from (3.24). Denote the
orthogonal complemental space of $T_{v,y,k}$ in $Y$ by $T_{v,y,k}^{\bot}$.
Recall that (see (3.38))
$\displaystyle\mathcal{N}_{\delta,k}=\\{u\in X_{k}\ |\
\mbox{dist}_{X}(u,P_{k}\mathcal{K})<\delta\\}.$
For $v\in\mathcal{N}_{\delta,k},$ define
$L_{v,y,\epsilon,k}:T_{v,y,k}^{\bot}\rightarrow T_{v,y,k}^{\bot}$
by
$\displaystyle w\in T_{v,y,k}^{\bot}\mapsto
w-S_{v,y,k}(-\triangle+1+V(\epsilon x))^{-1}(f^{\prime}(v(\cdot-y))w)$ (4.1)
where $S_{v,y,k}:Y\rightarrow T_{v,y,k}^{\bot}$ is orthogonal projection and
the operator $(-\triangle+1+V(\epsilon x))^{-1}$ is defined by (2.2).
###### Lemma 4.1.
Given $R>0,$ there exist $\delta_{0}>0$, $\epsilon_{0}>0$, $l^{*}>0$ and $C>0$
which are independent of $k,$ such that if $k\geq l^{*},$
$0<\delta\leq\delta_{0}$ and $0\leq\epsilon\leq\epsilon_{0},$ then for any
$v\in\overline{\mathcal{N}_{\delta,k}}$ and
$y\in\overline{B_{\mathbb{R}^{N}}(0,R)}$, $L_{v,y,\epsilon,k}$ is invertible
and
$\displaystyle||L_{v,y,\epsilon,k}w||\geq C||w||,\ \forall|y|\leq R,\ \forall
w\in T_{v,y,k}^{\bot}.$ (4.2)
Proof. Suppose $\kappa=\max\\{\tau_{u_{i}}\ |\ 1\leq i\leq s\\}$ is small
enough such that Lemma 3.7 holds. By (3.37), for sufficiently small
$\delta_{0}>0,$ there exists $l^{\prime}_{\kappa}>0$ such that
$\mathcal{N}_{\delta_{0},k}\subset\cup^{s}_{i=1}B_{X}(u_{i},\tau_{u_{i}})$ if
$k\geq l^{\prime}_{\kappa}.$ Note that $L_{v,0,0,k}$ is exactly the operator
$P_{E^{\bot}_{v,k}}\nabla^{2}J(v)|_{E^{\bot}_{v,k}}$ which has been defined in
Lemma 3.7 and for every $w\in T^{\bot}_{v,y,k}$,
$L_{v,y,0,k}w=L_{v,0,0,k}w(\cdot-y).$
Thus, by Lemma 3.7, there exists $C^{\prime}>0$ such that if $k\geq
l^{*}:=\max\\{l_{\kappa},l^{\prime}_{\kappa}\\}$, then for any
$v\in\mathcal{N}_{\delta_{0},k}$,
$\displaystyle||L_{v,y,0,k}w||\geq C^{\prime}||w||,\ \forall|y|\leq R,\
\forall w\in T_{v,y,k}^{\bot},$
where $l_{\kappa}$ is the constant comes from Lemma 3.7. Therefore, to prove
(4.2), it suffices to prove that
$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\Big{\\{}||L_{v,y,\epsilon,k}w-L_{v,y,0,k}w||\ |\ w\in
T_{v,y,k}^{\bot},\ ||w||\leq 1,$ (4.3) $\displaystyle\quad\quad\quad\quad\quad
v\in\overline{\mathcal{N}_{\delta_{0},k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\ k\geq l^{*}\Big{\\}}=0.$
If we can prove that for any given sequences $\\{k_{n}\\}\subset\mathbb{N},$
$\\{\epsilon_{n}\\}\subset(0,+\infty),$
$\\{y_{n}\\}\subset\overline{B_{\mathbb{R}^{N}}(0,R)}$, $\\{v_{n}\\}$ and
$\\{w_{n}\\}$ which satisfy that $\epsilon_{n}\rightarrow 0$ as
$n\rightarrow\infty$, $v_{n}\in\overline{\mathcal{N}_{\delta_{0},k_{n}}}$,
$w_{n}\in T^{\bot}_{v_{n},y_{n},k_{n}}$ and $||w_{n}||\leq 1$, $n=1,2,\cdots$,
$\displaystyle\lim_{n\rightarrow\infty}||L_{v_{n},y_{n},\epsilon_{n},k_{n}}w_{n}-L_{v_{n},y_{n},0,k_{n}}w_{n}||=0,$
(4.4)
then (4.3) holds. We only give the proof of (4.4) in the case
$k_{n}\rightarrow\infty,$ $n\rightarrow\infty$, since the proofs in other
cases are similar. Without loss of generality, we assume that $\\{k_{n}\\}$ is
exactly the sequence $\\{k\\}$ and we shall denote $\epsilon_{n}$, $y_{n},$
$v_{n}$ and $w_{n}$ by $\epsilon_{k}$, $y_{k},$ $v_{k}$ and $w_{k}$
respectively, $k=1,2,\cdots.$
Passing to a subsequence, we may assume that as $k\rightarrow\infty$,
$y_{k}\rightarrow y_{0}$, $v_{k}\rightharpoonup v_{0}$ in $X$ and
$w_{k}\rightharpoonup w_{0}$ in $Y$.
Let
$\eta_{k}=(-\triangle+1+V(\epsilon_{k}x))^{-1}(f^{\prime}(v_{k}(\cdot-
y_{k}))w_{k}).$
It is easy to verify that $\\{\eta_{k}\\}$ is bounded in $Y$ and
$\displaystyle\eta_{k}=(-\triangle+1)^{-1}(f^{\prime}(v_{k}(\cdot-
y_{k}))w_{k})-(-\triangle+1)^{-1}V(\epsilon_{k})\eta_{k}.$ (4.5)
Passing to a subsequence, we may assume that $\eta_{k}\rightharpoonup\eta_{0}$
in $Y$ as $k\rightarrow\infty.$
By definition of $L_{v,y,\epsilon,k}$ and (4.5), we get that
$\displaystyle
L_{v_{k},y_{k},\epsilon,k}w-L_{v_{k},y_{k},0,k}w=S_{v_{k},y_{k},k}(-\triangle+1)^{-1}V(\epsilon_{k}x)\eta_{k}.$
(4.6)
The condition $\bf(V_{1})$ implies that $V(0)=0$. It follows that for any
$h\in Y,$
$\displaystyle\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}V(\epsilon_{k}x)\eta_{k}h=0.$
(4.7)
Since $\eta_{k}$ is a weak solution of the equation:
$\displaystyle-\triangle\eta_{k}+\eta_{k}+V(\epsilon_{k}x)\eta_{k}=f^{\prime}(v_{k}(\cdot-
y_{k}))w_{k},$ (4.8)
by (4.7), $y_{k}\rightarrow y_{0},$ $\eta_{k}\rightharpoonup\eta_{0}$ and
$w_{k}\rightharpoonup w_{0}$ in $Y$, we get that $\eta_{0}$ is a weak solution
of the equation:
$\displaystyle-\triangle\eta_{0}+\eta_{0}=f^{\prime}(v_{0}(\cdot-
y_{0}))w_{0}.$ (4.9)
From (4.8) and (4.9), we obtain
$\displaystyle-\triangle(\eta_{k}-\eta_{0})+(\eta_{k}-\eta_{0})+V(\epsilon_{k}x)(\eta_{k}-\eta_{0})$
$\displaystyle=$ $\displaystyle(f^{\prime}(v_{k}(\cdot-
y_{k}))w_{k}-f^{\prime}(v_{0}(\cdot-y_{0}))w_{0})-V(\epsilon_{k}x)\eta_{0}.$
Multiplying the above equation by $\eta_{k}-\eta_{0}$ and integrating, we get
that there exists a constant $C>0$ such that
$\displaystyle C||\eta_{k}-\eta_{0}||^{2}$ (4.10) $\displaystyle\leq$
$\displaystyle||\eta_{k}-\eta_{0}||^{2}+\int_{\mathbb{R}^{N}}V(\epsilon_{k}x)(\eta_{k}-\eta_{0})^{2}\
(\mbox{by the condition }\ {\bf(V_{0})})$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}\Big{(}f^{\prime}(v_{k}(\cdot-
y_{k}))w_{k}-f^{\prime}(v_{0}(\cdot-
y_{0}))w_{0}-V(\epsilon_{k}x)\eta_{0}\Big{)}\cdot(\eta_{k}-\eta_{0})$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{N}}\Big{|}f^{\prime}(v_{k}(\cdot-
y_{k}))w_{k}-f^{\prime}(v_{0}(\cdot-
y_{0}))w_{0}\Big{|}\cdot|\eta_{k}-\eta_{0}|$
$\displaystyle+(\int_{\mathbb{R}^{N}}V^{2}(\epsilon_{k}x)\eta^{2}_{0})^{\frac{1}{2}}\cdot||\eta_{k}-\eta_{0}||_{L^{2}(\mathbb{R}^{N})}.$
Since $v_{k}\rightharpoonup v_{0}$ in $X$ and $y_{k}\rightarrow y_{0}$ as
$k\rightarrow\infty$, by the fact that $X$ can be compactly embedding into
$L^{p}_{r}(\mathbb{R}^{N})$ ($\forall p\in[2,2^{*})$), we get that
$\displaystyle\lim_{k\rightarrow\infty}||v_{k}(\cdot-y_{k})-v_{0}(\cdot-
y_{0})||_{L^{p}(\mathbb{R}^{N})}=0,\ \forall p\in[2,2^{*}).$ (4.11)
By (4.11) and the condition $\bf(F_{1})$, we get that
$\displaystyle\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}\Big{|}f^{\prime}(v_{k}(\cdot-
y_{k}))w_{k}-f^{\prime}(v_{0}(\cdot-
y_{0}))w_{0}\Big{|}\cdot|\eta_{k}-\eta_{0}|=0.$ (4.12)
By (4.10), (4.12) and
$\displaystyle\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}V^{2}(\epsilon_{k}x)\eta^{2}_{0}=0,$
(4.13)
we get that
$\lim_{k\rightarrow\infty}||\eta_{k}-\eta_{0}||=0.$ (4.14)
(4.13) and (4.14) yield
$\displaystyle\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}V^{2}(\epsilon_{k}x)\eta^{2}_{k}=0.$
(4.15)
It follows that
$\displaystyle\lim_{k\rightarrow\infty}||(-\triangle+1)^{-1}V(\epsilon_{k}x)\eta_{k}||=0.$
(4.16)
Combining (4.16) and (4.6) leads to (4.4).
Finally, by definition, $L_{v,y,\epsilon,k}$ is a Fredholm operator with index
zero and by (4.2), it is an injection. Therefore, it is invertible. $\Box$
###### Theorem 4.2.
Given $R>0.$ There exist $\delta^{*}>0$ and $\epsilon^{*}>0$ such that if
$0<\delta\leq\delta^{*}$ and $0\leq\epsilon\leq\epsilon^{*}$, then there exist
$k(\delta)$ and a $C^{1}-$mapping
$w_{\delta,k}(\cdot,\cdot,\epsilon):\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\rightarrow
Y,\ (u,y)\mapsto w_{\delta,k}(u,y,\epsilon)$
for $k\geq k(\delta)$, satisfying
* (i)
$w_{\delta,k}(u,y,\epsilon)\in T_{u,y,k}^{\bot},$
$\forall(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)};$
* (ii)
$\langle\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)),\phi\rangle=0,$
$\forall\phi\in T_{u,y,k}^{\bot};$
* (iii)
$w_{\delta,k}(u,y,0)=(\pi_{k}(u))(\cdot-y),$
$\forall(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}$;
* (iv)
for any $r>0,$ there exists $\delta_{r}>0$ such that if
$0<\delta\leq\delta_{r}$, $u\in\overline{\mathcal{N}_{\delta,k}}$,
$y\in\overline{B_{\mathbb{R}^{N}}(0,R)}$ and $k\geq k(\delta)$, then
$||w_{\delta,k}(u,y,\epsilon)||\leq r;$
* (v)
for any $n>0$,
$\displaystyle\sup\\{||(1+|x|)^{n}w_{\delta,k}(u,y,\epsilon)||_{L^{\infty}(\mathbb{R}^{N})}\
|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},\
0\leq\epsilon\leq\epsilon^{*}\\}$ $\displaystyle<\infty.$ (4.17)
Proof. By Lemma 4.1, we know that for any $R>0$, $L_{u,y,\epsilon,k}$ is
invertible if $0<\delta\leq\delta_{0}$, $0\leq\epsilon\leq\epsilon_{0}$ and
$k\geq l^{*}$. Moreover, the upper bound of $||L^{-1}_{u,y,\epsilon,k}||$ is
independent of $u,$ $y$, $\epsilon$ and $k.$ For
$u\in\overline{\mathcal{N}_{\delta,k}}$ and $r>0$, let
$\Phi_{u,y,\epsilon,k}:\overline{B_{T_{u,y,k}^{\bot}}(0,r)}\rightarrow
T_{u,y,k}^{\bot},$ $w\mapsto w-L^{-1}_{u,y,\epsilon,k}S_{u,y,k}\nabla
E_{\epsilon}(u(\cdot-y)+w).$
Now, we show that if $r$, $\delta$ and $\epsilon$ are small enough and $k$ is
large enough, then for any $u\in\overline{\mathcal{N}_{\delta,k}}$,
$\Phi_{u,y,\epsilon,k}$ is a contractive mapping in
$\overline{B_{T_{u,y,k}^{\bot}}(0,r)}.$
Using
$\displaystyle\nabla E_{\epsilon}(u(\cdot-y)+w)$ $\displaystyle=$
$\displaystyle u(\cdot-y)+w-(-\triangle+1+V(\epsilon x))^{-1}f(u(\cdot-y)+w)$
and the mean value theorem, we get that for any
$w_{1},w_{2}\in\overline{B_{T_{u,y,k}^{\bot}}(0,r)}$,
$\Phi_{u,y,\epsilon,k}(w_{1})-\Phi_{u,y,\epsilon,k}(w_{2})$ equals
$\displaystyle(w_{1}-w_{2})-L^{-1}_{u,y,\epsilon,k}S_{u,y,k}\Big{\\{}(w_{1}-w_{2})$
(4.18) $\displaystyle-(-\triangle+1+V(\epsilon
x))^{-1}(f^{\prime}(u(\cdot-y)+\tilde{w})\cdot(w_{1}-w_{2}))\Big{\\}}$
$\displaystyle=$
$\displaystyle(w_{1}-w_{2})-L^{-1}_{u,y,\epsilon,k}S_{u,y,k}\Big{\\{}(w_{1}-w_{2})$
$\displaystyle-(-\triangle+1+V(\epsilon
x))^{-1}f^{\prime}(u(\cdot-y))(w_{1}-w_{2})$
$\displaystyle-(-\triangle+1+V(\epsilon
x))^{-1}(f^{\prime}(u(\cdot-y)+\tilde{w})-f^{\prime}(u(\cdot-y)))(w_{1}-w_{2})\Big{\\}}$
where $\tilde{w}=\theta w_{1}+(1-\theta)w_{2}$ for some $0<\theta<1.$ By the
condition $\bf(F_{1})$, we can prove that
$\displaystyle\lim_{r\rightarrow 0}\sup\\{||(-\triangle+1+V(\epsilon
x))^{-1}(f^{\prime}(u(\cdot-y)+\tilde{w})-f^{\prime}(u(\cdot-y)))\varphi||$
(4.19) $\displaystyle\quad\quad\quad\quad\ |\
u\in\overline{\mathcal{N}_{\delta,k}},\ |y|\leq R,\ \varphi\in Y,\
||\varphi||\leq 1,\ 0\leq\epsilon\leq\epsilon_{0}\\}=0.$
By $||L^{-1}_{u,y,\epsilon,k}||_{\mathcal{L}(Y)}\leq 1/C$ (see Lemma 4.1 ),
$||S_{u,y,k}||_{\mathcal{L}(Y)}\leq 1$ and (4.19), we deduce that if $r$ is
small enough, then
$\displaystyle||L^{-1}_{u,y,\epsilon,k}S_{u,y,k}(-\triangle+1+V(\epsilon
x))^{-1}(f^{\prime}(u(\cdot-y)+\tilde{w})-f^{\prime}(u(\cdot-y)))(w_{1}-w_{2})||$
(4.20) $\displaystyle\leq\frac{1}{C}||(-\triangle+1+V(\epsilon
x))^{-1}(f^{\prime}(u(\cdot-y)+\tilde{w})-f^{\prime}(u(\cdot-y)))(w_{1}-w_{2})||$
$\displaystyle\leq\frac{1}{2}||w_{1}-w_{2}||.$
By the definition of $L_{u,y,\epsilon,k}$,
$\displaystyle
L^{-1}_{u,y,\epsilon,k}S_{u,y,k}\Big{\\{}(w_{1}-w_{2})-(-\triangle+1+V(\epsilon
x))^{-1}(f^{\prime}(u(\cdot-y))(w_{1}-w_{2}))\Big{\\}}$ (4.21)
$\displaystyle=(w_{1}-w_{2}).$
Combining (4.20), (4.21) and (4.18), we deduce that there exists $r_{0}>0$
such that if $0<r\leq r_{0}$, $0<\delta\leq\delta_{0}$,
$0\leq\epsilon\leq\epsilon_{0}$ and $k\geq l^{*}$, then for any
$(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}$
and $w_{1},w_{2}\in\overline{B_{T_{u,y,k}^{\bot}}(0,r)},$
$||\Phi_{u,y,\epsilon,k}(w_{1})-\Phi_{u,y,\epsilon,k}(w_{2})||\leq\frac{1}{2}||w_{1}-w_{2}||.$
(4.22)
Claim: For any $0<r\leq r_{0},$ there exist $\epsilon_{r}$, $\delta_{r}$ and
$k(\delta,r)$ such that if $0<\delta\leq\delta_{r}$,
$0\leq\epsilon\leq\epsilon_{r}$ and $k\geq k(\delta,r)$, then
$\displaystyle||\Phi_{u,y,\epsilon,k}(0)||\leq r/2,\
\forall(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}.$
(4.23)
Let $h_{u,y,\epsilon}=(-\triangle+1+V(\epsilon x))^{-1}f(u(\cdot-y)).$ It is
easy to verify
$\displaystyle
h_{u,y,\epsilon}=(-\triangle+1)^{-1}f(u(\cdot-y))-(-\triangle+1)^{-1}V(\epsilon
x)h_{u,y,\epsilon}.$ (4.24)
The same argument as (4.15) yields
$\lim_{\epsilon\rightarrow 0}\sup\\{\int_{\mathbb{R}^{N}}V^{2}(\epsilon
x)h^{2}_{u,y,\epsilon}\ |\ \ u\in\overline{\mathcal{N}_{\delta_{{}_{0}},k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\ k\geq l^{*}\\}=0.$
Thus, by (4.24), as $\epsilon\rightarrow 0,$
$\displaystyle\sup\\{||(-\triangle+1+V(\epsilon x))^{-1}f(u(\cdot-y))$
$\displaystyle\quad\quad\quad-(-\triangle+1)^{-1}f(u(\cdot-y))||\ |\
u\in\overline{\mathcal{N}_{\delta_{{}_{0}},k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\ k\geq l^{*}\\}$
$\displaystyle\rightarrow 0.$
It follows that as $\epsilon\rightarrow 0,$
$\displaystyle\sup\\{||\nabla E_{\epsilon}(u(\cdot-y))-\nabla J(u(\cdot-y))||\
|\ u\in\overline{\mathcal{N}_{\delta_{{}_{0}},k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\ k\geq l^{*}\\}$ (4.25)
$\displaystyle\rightarrow 0.$
Therefore, for $0<r\leq r_{0}$, there exists $\epsilon_{r}>0$ such that for
any $u\in\overline{\mathcal{N}_{\delta_{{}_{0}},k}},$
$y\in\overline{B_{\mathbb{R}^{N}}(0,R)}$ and $k\geq l^{*}$,
$\displaystyle||\nabla E_{\epsilon}(u(\cdot-y))-\nabla
J(u(\cdot-y))||<\frac{C}{4}r\ \ \mbox{if}\ 0\leq\epsilon\leq\epsilon_{r},$
(4.26)
where the constant $C$ comes from Lemma 4.1. Since $\nabla
J(v(\cdot-y))=\nabla J(v)=0,$ $\forall v\in\mathcal{K},$ we get that for any
$0<r\leq r_{0}$, there exists $\delta_{r}$ such that for any
$0<\delta\leq\delta_{r}$ and any $u\in N_{2\delta}(\mathcal{K})$,
$\displaystyle||\nabla J(u(\cdot-y))||<\frac{C}{4}r.$ (4.27)
By (4.27) and the fact that (see (3.35))
$\lim_{k\rightarrow\infty}\overline{\mathcal{N}_{\delta,k}}\subset
N_{2\delta}(\mathcal{K}),$
we deduce that there exists $k(\delta,r)$ such that if $k\geq k(\delta,r)$,
then for any $0<\delta\leq\delta_{r}$ and any
$u\in\overline{\mathcal{N}_{\delta,k}}$,
$\displaystyle||\nabla J(u(\cdot-y))||<\frac{C}{4}r.$ (4.28)
Thus, the claim follows from (4.26), (4.28) and the fact that
$||\Phi_{u,y,\epsilon,k}(0)||\leq\frac{1}{C}||\nabla
E_{\epsilon}(u(\cdot-y))||.$
Combining (4.22) and (4.23) leads to
$||\Phi_{u,y,\epsilon,k}(w)||\leq r$
for every $w\in\overline{B_{T_{u,y,k}^{\bot}}(0,r)}$. Therefore,
$\Phi_{u,y,\epsilon,k}$ is a contractive mapping in
$\overline{B_{T_{u,y,k}^{\bot}}(0,r)}.$ By Banach fixed point theorem, there
exists unique fixed point $w_{\delta,k}(u,y,\epsilon)$ of
$\Phi_{u,y,\epsilon,k}$. Denote $\delta_{r_{{}_{0}}}$ by $\delta^{*}$,
$\epsilon_{r_{{}_{0}}}$ by $\epsilon^{*}$ and $k(\delta,r_{0})$ by
$k(\delta)$. It is easy to verify that the conclusions $\bf(i)-(iv)$ hold for
$w_{\delta,k}(u,y,\epsilon)$.
Now, we prove that
$w_{\delta,k}:\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\rightarrow
Y$ is $C^{1}$. For any
$(u_{0},y_{0})\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}$
and $(u,y)$ close to $(u_{0},y_{0})$, both
$S_{u_{0},y_{0},k}|_{T^{\perp}_{u,y,k}}:T^{\perp}_{u,y,k}\to
T^{\perp}_{u_{0},y_{0},k}$ and
$S_{u,y,k}|_{T^{\perp}_{u_{0},y_{0},k}}:T^{\perp}_{u_{0},y_{0},k}\to
T^{\perp}_{u,y,k}$ are isomorphisms, and finding a solution $w\in
T^{\perp}_{u,y,k}$ to the equation $S_{u,y,k}\nabla
E_{\epsilon}(u(\cdot-y)+w)=0$ is equivalent to finding a solution $w\in
T^{\bot}_{u_{0},y_{0},k}$ to the equation $S_{u_{0},y_{0},k}S_{u,y,k}\nabla
E_{\epsilon}(u(\cdot-y)+S_{u,y,k}w)=0$. Note that
$S_{u_{0},y_{0},k}S_{u,y,k}\nabla E_{\epsilon}(u(\cdot-y)+S_{u,y,k}w)$ is
$C^{1}$ near
$(u_{0},y_{0},w_{0})\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\times
T^{\perp}_{u_{0},y_{0},k}$ and the Fréchet partial derivative of
$S_{u_{0},y_{0},k}S_{u,y,k}\nabla E_{\epsilon}(u(\cdot-y)+S_{u,y,k}w)$ at
$(u_{0},y_{0},w_{0})$ with respect to $w$ is $L_{u_{0},y_{0},\epsilon,k}$
which is invertible. Therefore, the implicit functional theorem implies that
$w_{\delta,k}(\cdot,\cdot,\epsilon):\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\rightarrow
Y$
is $C^{1}$.
Finally, we give the proof of $\bf(v).$ Let
$\displaystyle\varphi_{u,y,\epsilon,k}=u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)-P_{T_{u,y,k}}(\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))),$ (4.29)
where $P_{T_{u,y,k}}:Y\rightarrow T_{u,y,k}$ is orthogonal projection. By the
conclusion $\bf(ii)$ of this Theorem, we get that
$\displaystyle P_{T_{u,y,k}}(\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)))=\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)).$ (4.30)
Thus, by (4.29) and (4.30), $\varphi_{u,y,\epsilon,k}$ satisfies
$\displaystyle-\triangle\varphi_{u,y,\epsilon,k}+\varphi_{u,y,\epsilon,k}+V(\epsilon
x)\varphi_{u,y,\epsilon,k}=f(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)).$ (4.31)
By the definition of $T_{u,y,k}$, we have
$\displaystyle P_{T_{u,y,k}}(\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)))$ (4.32) $\displaystyle=$
$\displaystyle\sum^{N}_{j=1}\Big{\langle}\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)),\sum^{s}_{i=1}\xi_{i}(u)\frac{u_{i}(\cdot-y)}{\partial
x_{j}}\Big{\rangle}\frac{\sum^{s}_{i=1}\xi_{i}(u)\frac{u_{i}(\cdot-y)}{\partial
x_{j}}}{||\sum^{s}_{i=1}\xi_{i}(u)\frac{u_{i}(\cdot-y)}{\partial
x_{j}}||^{2}}$ $\displaystyle+\sum^{k}_{i=1}\langle\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)),\tilde{e}_{i,k}(\cdot-y)\rangle\tilde{e}_{i,k}(\cdot-y)$
$\displaystyle+\sum^{q}_{i=1}\langle\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)),e_{i}(\cdot-y)\rangle
e_{i}(\cdot-y).$
Since $\tilde{e}_{i,k}$, $e_{i}$, $u$ and $\frac{\partial u_{i}}{\partial
x_{j}}$ satisfy exponential decay at infinity, by (4.32), for any given $k\geq
k(\delta)$ and $n\geq 0$, there exists $C^{\prime}_{n,k}>0$ such that
$\displaystyle\sup\\{||(1+|x|)^{n}(P_{T_{u,y,k}}(\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))))||_{L^{\infty}(\mathbb{R}^{N})}$
$\displaystyle\quad\quad|\
u\in\overline{\mathcal{N}_{\delta,k}},y\in\overline{B_{\mathbb{R}^{N}}(0,R)},0\leq\epsilon\leq\epsilon^{*}\\}\leq
C^{\prime}_{k,n}$ (4.33)
and
$\displaystyle\sup_{u\in\overline{\mathcal{N}_{\delta,k}},y\in\overline{B_{\mathbb{R}^{N}}(0,R)}}||(1+|x|)^{n}u(\cdot-y)||_{L^{\infty}(\mathbb{R}^{N})}\leq
C^{\prime}_{k,n}.$ (4.34)
Note that $\varphi_{u,y,\epsilon,k}$ satisfies the elliptic equation (4.31).
Therefore, by the bootstrap argument and the fact that
$\\{w_{\delta,k}(u,y,\epsilon))\ |\ u\in\overline{\mathcal{N}_{\delta,k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\ 0\leq\epsilon\leq\epsilon^{*}\\}$
is compact in $Y$ (because for fixed $k$, $\overline{\mathcal{N}_{\delta,k}}$
is compact), we get that
$\displaystyle\sup\\{||\varphi_{u,y,\epsilon,k}||_{L^{\infty}(\mathbb{R}^{N})}\
|\ u\in\overline{\mathcal{N}_{\delta,k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\
0\leq\epsilon\leq\epsilon^{*}\\}<\infty$ (4.35)
and
$\displaystyle\lim_{\rho\rightarrow\infty}\sup\\{||\varphi_{u,y,\epsilon,k}||_{L^{\infty}(\mathbb{R}^{N}\setminus\overline{B_{\mathbb{R}^{N}}(0,\rho)})}\
|\ u\in\overline{\mathcal{N}_{\delta,k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\ 0\leq\epsilon\leq\epsilon^{*}\\}=0.$
(4.36)
By (4.35), (4.36) and (4.29), we get that
$\displaystyle\sup\\{||w_{\delta,k}(u,y,\epsilon)||_{L^{\infty}(\mathbb{R}^{N})}\
|\ u\in\overline{\mathcal{N}_{\delta,k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\
0\leq\epsilon\leq\epsilon^{*}\\}<\infty.$ (4.37)
and
$\displaystyle\lim_{\rho\rightarrow\infty}\sup\\{||w_{\delta,k}(u,y,\epsilon)||_{L^{\infty}(\mathbb{R}^{N}\setminus\overline{B_{\mathbb{R}^{N}}(0,\rho)})}\
|\ u\in\overline{\mathcal{N}_{\delta,k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\ 0\leq\epsilon\leq\epsilon^{*}\\}$
$\displaystyle=0.$
Let $d(t)=f(t)/t,$ $t\in\mathbb{R}.$ Then by (4.37), (4.34) and the condition
$\bf(F_{1})$, we have
$\displaystyle\sup\\{||d(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))||_{L^{\infty}(\mathbb{R}^{N})}\
|\ u\in\overline{\mathcal{N}_{\delta,k}},\
y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\ 0\leq\epsilon\leq\epsilon^{*}\\}$
(4.39) $\displaystyle<\infty.$
By the condition $\bf(V_{0})$, the condition $\bf(F_{1})$ and (4), we deduce
that there exists $\rho_{0}$ such that
$\displaystyle\inf\\{1+V(\epsilon x)-d(u(x-y)+w_{\delta,k}(u,y,\epsilon))\ |\
|x|>\rho_{0},\ u\in\overline{\mathcal{N}_{\delta,k}},$
$\displaystyle\quad\quad\ y\in\overline{B_{\mathbb{R}^{N}}(0,R)},\
0\leq\epsilon\leq\epsilon^{*}\\}>0.$ (4.40)
Let $\eta$ be a cut-off function which satisfies that $\eta\equiv 1$ in
$B_{\mathbb{R}^{N}}(0,\rho_{0})$ and $\eta\equiv 0$ in
$\mathbb{R}^{N}\setminus\overline{B_{\mathbb{R}^{N}}(0,\rho_{0}+1)}$. We can
rewrite equation (4.31) as
$\displaystyle-\triangle\varphi_{u,y,\epsilon,k}+(1+V(\epsilon
x)-(1-\eta(x))d(u(x-y)+w_{\delta,k}(u,y,\epsilon)))\varphi_{u,y,\epsilon,k}$
$\displaystyle=$ $\displaystyle f_{u,y,\epsilon,k}$
with
$\displaystyle f_{u,y,\epsilon,k}$ $\displaystyle=$ $\displaystyle
d(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))\cdot u(\cdot-y)$ (4.42)
$\displaystyle+\eta(x)\cdot d(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))\cdot
w_{\delta,k}(u,y,\epsilon)$ $\displaystyle-(1-\eta(x))\cdot
d(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))$
$\displaystyle\quad\quad\quad\times(u(\cdot-y)-P_{T_{u,y,k}}(\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))).$
By (4.34), (4), (4.39) and the fact that
$\eta(x)d(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))\cdot
w_{\delta,k}(u,y,\epsilon)$
has compact support, we deduce that there exists
$C^{\prime\prime\prime}_{n,k}>0$ such that
$\displaystyle\sup_{u\in\overline{\mathcal{N}_{\delta,k}},y\in\overline{B_{\mathbb{R}^{N}}(0,R)}}||(1+|x|)^{n}f_{u,y,\epsilon,k}||_{L^{\infty}(\mathbb{R}^{N})}\leq
C^{\prime\prime\prime}_{k,n}.$ (4.43)
By (4.43), (4), (4) and [25, Proposition 4.2], we get that there exists
$C^{\prime\prime}_{n,k}>0$ such that
$\displaystyle\sup_{u\in\overline{\mathcal{N}_{\delta,k}},y\in\overline{B_{\mathbb{R}^{N}}(0,R)}}||(1+|x|)^{n}\varphi_{u,y,\epsilon,k}||_{L^{\infty}(\mathbb{R}^{N})}\leq
C^{\prime\prime}_{k,n}.$ (4.44)
Then the conclusion $\bf(v)$ follows from (4.29), (4.44), (4) and (4.34).
$\Box$
By the conclusion $\bf(iii)$ of Theorem 4.2, we get that
$\displaystyle J(u(\cdot-y)+w_{\delta,k}(u,y,0))\equiv I(u+\pi_{k}(u)),\
\forall(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}.$
(4.45)
In what follows, for a $C^{1}$ mapping $f$ defined in
$\mathcal{N}_{\delta,k}\times B_{\mathbb{R}^{N}}(0,R)$, we use the the
notations $Df$, $D_{u}f$ and $D_{y}f$ to denote the derivatives of $f$ with
respect to $(u,y)$ variable, $u$ variable and $y$ variable respectively and
use $Df(u,y)[\bar{u},\bar{y}]$ to denote the derivative of $f$ at the point
$(u,y)$ along the vector $(\bar{u},\bar{y})\in X_{k}\times\mathbb{R}^{N}.$
Furthermore, we use $D_{u}f(u,y)[\bar{u}]$ and $D_{y}f(u,y)[\bar{y}]$ to
denote the Fréchet partial derivatives with respect to the $u$ and $y$
variables along the vectors $\bar{u}$ and $\bar{y}$ respectively.
The condition $\bf(V_{1})$ for the potential $V$ yields
$\displaystyle\lim_{\epsilon\rightarrow 0}\frac{V(\epsilon
x)}{\epsilon^{n^{*}}}=Q_{n^{*}}(x).$ (4.46)
The proof of the following proposition will be given in appendix.
###### Proposition 4.3.
Let $\delta>0$ be sufficiently small and $k\geq k(\delta)$. If $\iota<n^{*},$
then
$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\\{\frac{1}{\epsilon^{\iota}}\Lambda_{k}(u,y,\epsilon)\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\\}=0$
where
$\displaystyle\Lambda_{k}(u,y,\epsilon)$ $\displaystyle=$
$\displaystyle||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||$
$\displaystyle+\sup_{\bar{y}\in\mathbb{R}^{N},|\bar{y}|\leq
1}||Dw_{\delta,k}(u,y,\epsilon)[0,\bar{y}]-D(\pi_{k}(u)(\cdot-y))[0,\bar{y}]||$
$\displaystyle+\sup_{v\in X_{k},||v||\leq
1}||Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]||.$
Moreover, there exists a constant $M>0$ which is independent of $(u,y)$ and
$\epsilon$ such that for every
$(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}$
and $0\leq\epsilon\leq\epsilon^{*},$
$\displaystyle\Lambda_{k}(u,y,\epsilon)\leq M\epsilon^{n^{*}}.$
For $0<\delta\leq\delta^{*}$ and $0\leq\epsilon\leq\epsilon^{*},$ denote the
functional
$E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)),\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}$
(4.47)
by $\Psi_{k}(u,y,\epsilon)$.
###### Theorem 4.4.
Suppose that $0<\delta\leq\delta^{*}$ and $k\geq k(\delta)$. Then there exists
$\epsilon_{k}>0$ such that if $0\leq\epsilon\leq\epsilon_{k}$ and
$(u_{\epsilon},y_{\epsilon})\in\mathcal{N}_{\delta,k}\times
B_{\mathbb{R}^{N}}(0,R)$ is a critical point of the functional
$\Psi_{k}(u,y,\epsilon)$, that is,
$\displaystyle D\Psi_{k}(u_{\epsilon},y_{\epsilon},\epsilon)[v,\bar{y}]=0,\
\forall(v,\bar{y})\in X_{k}\times\mathbb{R}^{N},$ (4.48)
then $u_{\epsilon}(\cdot-
y_{\epsilon})+w_{\delta,k}(u_{\epsilon},y_{\epsilon},\epsilon)$ is a critical
point of $E_{\epsilon}$.
Proof. By the conclusion $\bf(ii)$ of Theorem 4.2 and hypothesis (4.48), we
deduce that to prove $u_{\epsilon}(\cdot-
y_{\epsilon})+w_{\delta,k}(u_{\epsilon},y_{\epsilon},\epsilon)$ is a critical
point of $E_{\epsilon},$ it suffices to prove that for sufficiently small
$\epsilon>0,$
$\displaystyle\\{v(\cdot-
y_{\epsilon})-(\bar{y}\cdot\nabla_{x}u_{\epsilon})(\cdot-
y_{\epsilon})+Dw_{\delta,k}(u_{\epsilon},y_{\epsilon},\epsilon)[v,\bar{y}]\ |\
v\in X_{k},\ \bar{y}\in\mathbb{R}^{N}\\}$
$\displaystyle+T^{\perp}_{u_{\epsilon},y_{\epsilon},k}=Y.$ (4.49)
If (4) were not true, then there exist $\epsilon_{n}\rightarrow 0$ as
$n\rightarrow\infty$ such that $Y_{n}\neq Y,$ where $Y_{n}$ denotes the space
appeared in the left side of (4) with $\epsilon=\epsilon_{n}.$ Passing to a
subsequence, we may assume that $y_{\epsilon_{n}}\rightarrow y_{k}$ and
$u_{\epsilon_{{}_{n}}}\rightarrow u_{k}$ in $Y$ as $n\rightarrow\infty,$ since
$\\{(u_{\epsilon_{n}},y_{\epsilon_{n}})\\}$ is a bounded sequence in the
finite dimensional space $X_{k}\times\mathbb{R}^{N}.$ By the hypothesis (4.48)
and Proposition 4.3, we deduce that $u_{k}$ is a critical point of
$I(v+\pi_{k}(v))$. Then by the conclusion $\bf(iv)$ of Lemma 3.8,
$u_{k}+\pi_{k}(u_{k})$ is a critical point of $I$. We denote it by
$\tilde{u}_{k}$. Since $D\pi_{k}(u_{k})v\in X$ and $\mathcal{T}_{u_{k}}\subset
X^{\bot}$, we get $D\pi_{k}(u_{k})v\bot\mathcal{T}_{u_{k}}$, where
$\mathcal{T}_{u_{k}}$ comes from (3.24). Moreover, by Lemma 3.8, we get that
$D\pi_{k}(u_{k})v\in X^{\bot}_{k}$. Thus,
$D\pi_{k}(u_{k})v\bot X_{k}\oplus\mathcal{T}_{u_{k}}=T_{u_{k},0,k}.$
It follows that the following subspace of $Y:$
$\\{v-\bar{y}\nabla_{x}u_{k}-\bar{y}\nabla_{x}\pi_{k}(u_{k})+D\pi_{k}(u_{k})v\
|\ v\in X_{k},\ \bar{y}\in\mathbb{R}^{N}\\}+T^{\bot}_{u_{k},0,k}$ (4.50)
is equal to
$\displaystyle\\{v-\bar{y}\nabla_{x}u_{k}-\bar{y}\nabla_{x}\pi_{k}(u_{k})\ |\
v\in X_{k},\ \bar{y}\in\mathbb{R}^{N}\\}+T^{\bot}_{u_{k},0,k}$ (4.51)
$\displaystyle=$ $\displaystyle\\{v-\bar{y}\nabla_{x}\tilde{u}_{k}\ |\ v\in
X_{k},\ \bar{y}\in\mathbb{R}^{N}\\}+T^{\bot}_{u_{k},0,k}.$
As it has been mentioned above,
$\tilde{u}_{k}=u_{k}+\pi_{k}(u_{k})\in\mathcal{K}.$ Therefore, by (3.3), we
get that for every $1\leq j\leq N,$
$\displaystyle||\frac{\partial\tilde{u}_{k}}{\partial
x_{j}}-\sum^{s}_{i=1}\xi_{i}(\tilde{u}_{k})\frac{\partial u_{i}}{\partial
x_{j}}||\leq\sum^{s}_{i=1}\xi_{i}(\tilde{u}_{k})||\frac{\partial\tilde{u}_{k}}{\partial
x_{j}}-\frac{\partial u_{i}}{\partial x_{j}}||\leq\varsigma.$ (4.52)
By $\bf(ii)$ of Lemma 3.8 and the fact that every $\xi_{i}$ is a Lipschitz
function, we deduce that for every $1\leq j\leq N,$ as $k\rightarrow\infty$,
$\displaystyle||\sum^{s}_{i=1}\xi_{i}(\tilde{u}_{k})\frac{\partial
u_{i}}{\partial x_{j}}-\sum^{s}_{i=1}\xi_{i}(u_{k})\frac{\partial
u_{i}}{\partial x_{j}}||$ (4.53) $\displaystyle\leq$
$\displaystyle\sum^{s}_{i=1}|\xi_{i}(\tilde{u}_{k})-\xi_{i}(u_{k})|\cdot||\frac{\partial
u_{i}}{\partial x_{j}}||\leq
C\sum^{s}_{i=1}||\tilde{u}_{k}-u_{k}||\cdot||\frac{\partial u_{i}}{\partial
x_{j}}||\rightarrow 0,$
where $C$ is the the Lipschitz constant of $\xi_{i}.$ By (4.52) and (4.53), we
obtain that for every $1\leq j\leq N,$
$\limsup_{k\rightarrow\infty}||\frac{\partial\tilde{u}_{k}}{\partial
x_{j}}-\sum^{s}_{i=1}\xi_{i}(u_{k})\frac{\partial u_{i}}{\partial
x_{j}}||\leq\varsigma.$
It follows that
$\displaystyle\limsup_{k\rightarrow\infty}\sup_{|\bar{y}|\leq
1}||\bar{y}\nabla_{x}\tilde{u}_{k}-\sum^{N}_{j=1}\bar{y}_{j}\sum^{s}_{i=1}\xi_{i}(u_{k})\frac{\partial
u_{i}}{\partial x_{j}}||\leq\varsigma.$
Thus, when $\varsigma$ is sufficiently small and $k$ is sufficiently large,
the space defined by (4.51) is equal to $Y$. As a consequence, when
$\varsigma$ is sufficiently small and $k$ is sufficiently large, the space
defined by (4.50) is also $Y$. Therefore, the space
$\displaystyle\\{v(\cdot-y_{k})-(\bar{y}\nabla_{x}u_{k})(\cdot-
y_{k})-(\bar{y}\nabla_{x}\pi_{k}(u_{k}))(\cdot-
y_{k})+(D\pi_{k}(u_{k})v)(\cdot-y_{k})$ $\displaystyle\quad\quad|\ v\in
X_{k},\ \bar{y}\in\mathbb{R}^{N}\\}+T^{\bot}_{u_{k},y,k}$ (4.54)
is equal to $Y$. Then we can define a bounded linear operator
$\displaystyle H_{n}:Y\rightarrow Y,$ $\displaystyle w=v(\cdot-
y_{k})-(\bar{y}\nabla_{x}u_{k})(\cdot-
y_{k})-(\bar{y}\nabla_{x}\pi_{k}(u_{k}))(\cdot-
y_{k})+(D\pi_{k}(u_{k})v)(\cdot-y_{k})+\phi$ $\displaystyle\mapsto
H_{n}(w)=v(\cdot-
y_{\epsilon_{{}_{n}}})-(\bar{y}\nabla_{x}u_{\epsilon_{{}_{n}}})(\cdot-
y_{\epsilon_{{}_{n}}})+Dw_{\delta,k}(u_{\epsilon_{{}_{n}}},y_{\epsilon_{{}_{n}}},\epsilon_{n})[v,\bar{y}]+\phi,$
where $\phi\in T^{\bot}_{u_{k},y,k}.$ It satisfies $Y_{n}=H_{n}(Y)$, where
$Y_{n}$ denotes the space appeared in the left side of (4) with
$\epsilon=\epsilon_{n}.$ By $u_{\epsilon_{n}}\rightarrow u_{k}$,
$y_{\epsilon_{n}}\rightarrow y_{k}$ and Proposition 4.3, we get that as
$n\rightarrow\infty$,
$||H_{n}-id||_{\mathcal{L}(Y)}\rightarrow 0.$
Therefore, when $n$ is large enough, $H_{n}(Y)=Y$. It follows that $Y_{n}=Y,$
which contradicts the assumption. Thus, when $k(\delta)$ is large enough and
$k\geq k(\delta)$, there exists $\epsilon_{k}>0$ such that if
$0\leq\epsilon\leq\epsilon_{k},$ then (4) holds. $\Box$
## 5 Proof of Theorem 1.3
By the conclusions $\bf(iii)$ and $\bf(v)$ of Theorem 4.2, if
$u\in\overline{\mathcal{N}_{\delta,k}}$, then $\pi_{k}(u)$ decays
exponentially at infinity. Therefore, for
$u\in\overline{\mathcal{N}_{\delta,k}}$ and $y\in\mathbb{R}^{N}$, we can
define
$\Gamma_{k}(u,y)=\int_{\mathbb{R}^{N}}Q_{n^{*}}(x+y)(u+\pi_{k}(u))^{2}dx.$
By the same argument as Lemma 3.2 of [1] and by (4.46), (4.34) and the
Lebesgue Convergence Theorem, we can get the following Lemma:
###### Lemma 5.1.
For any given $k\geq k(\delta)$, as $\epsilon\rightarrow 0,$
$\sup\Big{\\{}\Big{|}\frac{1}{\epsilon^{n^{*}}}\int_{\mathbb{R}^{N}}V(\epsilon(x+y))(u+\pi_{k}(u))^{2}dx-\Gamma_{k}(u,y)\Big{|}\
|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\Big{\\}}\rightarrow
0$
and
$\displaystyle\sup\Big{\\{}\Big{|}D\Big{(}\frac{1}{\epsilon^{n^{*}}}\int_{\mathbb{R}^{N}}V(\epsilon(x+y))(u+\pi_{k}(u))^{2}dx-\Gamma_{k}(u,y)\Big{)}[v,\bar{y}]\Big{|}\
|\ v\in X_{k},\ ||v||\leq 1,$
$\displaystyle\quad\quad\quad\bar{y}\in\mathbb{R}^{N},\ |\bar{y}|\leq 1,\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\Big{\\}}\rightarrow
0.$
From now on, for the condition $\bf(V_{1})$, we always assume that $\triangle
Q_{n^{*}}\geq 0$ and $\triangle Q_{n^{*}}\not\equiv 0$ in $\mathbb{R}^{N}$,
since the proof for the other case is similar.
###### Lemma 5.2.
If $\delta>0$ is small enough, then for any
$u\in\overline{\mathcal{N}_{\delta,k}}$, $\Gamma_{k}(u,\cdot)$ has a strict
local minimum at $y=0$ and $D^{2}_{y}\Gamma_{k}(u,0)$ is a positive-definite
matrix. More precisely, there exists a constant $A_{k}>0$ such that
$\displaystyle D^{2}_{y}\Gamma_{k}(u,0)y\cdot y\geq A_{k}|y|^{2},\ \forall
u\in\overline{\mathcal{N}_{\delta,k}},\ \forall y\in\mathbb{R}^{N}.$ (5.1)
Proof. By Lemma 4.1 of [1], we know that $y=0$ is a critical point of
$\Gamma_{k}(u,\cdot)$ for every $u\in\overline{\mathcal{N}_{\delta,k}}$. If
(5.1) were not true, then there exist $\delta_{n}>0$,
$u_{n}\subset\overline{\mathcal{N}_{\delta_{n},k}},$ $n=1,2,\cdots$ and
$\\{y_{n}\\}\subset S^{N-1}$ such that $\delta_{n}\rightarrow 0$ as
$n\rightarrow\infty$ and
$\displaystyle\lim_{n\rightarrow\infty}|D^{2}_{y}\Gamma_{k}(u_{n},0)y_{n}\cdot
y_{n}|=0.$ (5.2)
Since $(u_{n},y_{n})$ is bounded in the finite dimensional space
$X_{k}\times\mathbb{R}^{N}$, passing to a subsequence, we may assume that
$u_{n}\rightarrow u_{0}$ in $X_{k}$, and $y_{n}\rightarrow y_{0}\in S^{N-1}$
as $n\rightarrow\infty$. Let $D_{ii}\Gamma_{k}(u_{n},y)$ be the second
derivative of $\Gamma_{k}(u_{n},y)$ with respect to the variable $y_{i}$ and
$\mbox{diag}\\{D_{11}\Gamma_{k}(u_{n},0),\cdots,D_{NN}\Gamma_{k}(u_{n},0)\\}$
be diagonal matrix with diagonal elements $D_{11}\Gamma_{k}(u_{n},0),$
$\cdots,$ $D_{NN}\Gamma_{k}(u_{n},0)$. By the appendix of [1], we get that
$\displaystyle
D_{ii}\Gamma_{k}(u_{n},0)=-\frac{2}{N}\int_{\mathbb{R}^{N}}(u_{n}+\pi_{k}(u_{n}))\nabla
Q_{n^{*}}(x)\cdot\nabla(u_{n}+\pi_{k}(u_{n}))dx,\ 1\leq i\leq N.$ (5.3)
Therefore,
$\displaystyle D^{2}_{y}\Gamma_{k}(u_{n},0)y_{n}\cdot y_{n}$ $\displaystyle=$
$\displaystyle
y^{T}_{n}\cdot\mbox{diag}\\{D_{11}\Gamma_{k}(u_{n},0),\cdots,D_{NN}\Gamma_{k}(u_{n},0)\\}\cdot
y_{n}$ (5.4) $\displaystyle=$
$\displaystyle-\frac{2}{N}|y_{n}|^{2}\int_{\mathbb{R}^{N}}(u_{n}+\pi_{k}(u_{n}))\nabla
Q_{n^{*}}(x)\cdot\nabla(u_{n}+\pi_{k}(u_{n}))dx$ $\displaystyle=$
$\displaystyle-\frac{1}{N}|y_{n}|^{2}\int_{\mathbb{R}^{N}}\nabla
Q_{n^{*}}(x)\cdot\nabla(u_{n}+\pi_{k}(u_{n}))^{2}dx$ $\displaystyle=$
$\displaystyle\frac{1}{N}|y_{n}|^{2}\int_{\mathbb{R}^{N}}\triangle
Q_{n^{*}}(x)\cdot(u_{n}+\pi_{k}(u_{n}))^{2}dx$
By (5.2) and (5.4), we infer that
$\lim_{n\rightarrow\infty}D^{2}_{y}\Gamma_{k}(u_{n},0)y_{n}\cdot
y_{n}=\frac{1}{N}|y_{0}|^{2}\int_{\mathbb{R}^{N}}\triangle
Q_{n^{*}}(x)\cdot(u_{0}+\pi_{k}(u_{0}))^{2}dx=0.$
It is a contradiction, since we have assumed that $\triangle Q_{n^{*}}(x)\geq
0$ and $\triangle Q_{n^{*}}\not\equiv 0$ in $\mathbb{R}^{N}$. $\Box$
In the rest of this section, we assume that $\delta>0$ is sufficiently small
and $k\geq k(\delta)$ is sufficiently large such that (3.57) holds, where the
constant $k(\delta)$ comes from Theorem 4.2.
Proof of Theorem 1.3:
By definition of $\Psi_{k}(u,y,\epsilon)$ (see (4.47)), for
$(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},$
$\displaystyle\Psi_{k}(u,y,\epsilon)$ (5.5) $\displaystyle=$
$\displaystyle\frac{1}{2}||u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)||^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}}V(\epsilon
x)|u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)|^{2}dx$
$\displaystyle-\int_{\mathbb{R}^{N}}F(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))dx$
$\displaystyle=$
$\displaystyle\frac{1}{2}||u(\cdot-y)+w_{\delta,k}(u,y,0)||^{2}+\frac{1}{2}||w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0)||^{2}$
$\displaystyle+\langle
u(\cdot-y)+w_{\delta,k}(u,y,0),w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0)\rangle$
$\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{N}}V(\epsilon
x)|u(\cdot-y)+w_{\delta,k}(u,y,0)|^{2}dx$
$\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{N}}V(\epsilon
x)|w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0)|^{2}dx$
$\displaystyle+\int_{\mathbb{R}^{N}}V(\epsilon
x)(u(\cdot-y)+w_{\delta,k}(u,y,0))\cdot(w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0))dx$
$\displaystyle-\int_{\mathbb{R}^{N}}F(u(\cdot-y)+w_{\delta,k}(u,y,0))dx$
$\displaystyle-\int_{\mathbb{R}^{N}}f(u(\cdot-y)+w_{\delta,k}(u,y,0))\cdot(w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0))dx$
$\displaystyle-\eta_{1}(u,y,\epsilon),$
where
$\displaystyle\eta_{1}(u,y,\epsilon)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}F(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))dx-\int_{\mathbb{R}^{N}}F(u(\cdot-y)+w_{\delta,k}(u,y,0))dx$
$\displaystyle-\int_{\mathbb{R}^{N}}f(u(\cdot-y)+w_{\delta,k}(u,y,0))\cdot(w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0))dx.$
By Taylor expansion, we deduce that there exists $0<\theta=\theta(x)<1,$
$\forall x\in\mathbb{R}^{N}$ such that
$\displaystyle\eta_{1}(u,y,\epsilon)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}f^{\prime}(u(\cdot-y)+\theta
w_{\delta,k}(u,y,0)+(1-\theta)w_{\delta,k}(u,y,\epsilon))$
$\displaystyle\quad\quad\quad\times(w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0))^{2}dx$
By the condition $\bf(F_{1})$, Proposition 4.3 and (5), we deduce that
$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\\{\frac{1}{\epsilon^{n^{*}}}|\eta_{1}(u,y,\epsilon)|\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\\}=0.$
(5.7)
Note that for $v\in X_{k},$ $\bar{y}\in\mathbb{R}^{N}$,
$\displaystyle D\eta_{1}(u,y,\epsilon)[v,\bar{y}]$ (5.8) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}f(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))$
$\displaystyle\quad\quad\times(v(\cdot-y)-\bar{y}(\nabla_{x}u)(\cdot-y)+Dw_{\delta,k}(u,y,\epsilon)[v,\bar{y}])dx$
$\displaystyle-\int_{\mathbb{R}^{N}}f(u(\cdot-y)+w_{\delta,k}(u,y,0))$
$\displaystyle\quad\quad\times(v(\cdot-y)-\bar{y}(\nabla_{x}u)(\cdot-y)+Dw_{\delta,k}(u,y,0)[v,\bar{y}])dx$
$\displaystyle-\int_{\mathbb{R}^{N}}f^{\prime}(u(\cdot-y)+w_{\delta,k}(u,y,0))\cdot(w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0))$
$\displaystyle\quad\quad\quad\quad\times(v(\cdot-y)-\bar{y}(\nabla_{x}u)(\cdot-y)+Dw_{\delta,k}(u,y,0)[v,\bar{y}])dx$
$\displaystyle-\int_{\mathbb{R}^{N}}f(u(\cdot-y)+w_{\delta,k}(u,y,0))\cdot(Dw_{\delta,k}(u,y,\epsilon)[v,\bar{y}]-Dw_{\delta,k}(u,y,0)[v,\bar{y}])$
Then by the conclusion $\bf(iii)$ of Theorem 4.2, Proposition 4.3 and the
condition $\bf(F_{1})$, we deduce that
$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\\{\frac{1}{\epsilon^{n^{*}}}||D\eta_{1}(u,y,\epsilon)||\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\\}=0.$
(5.9)
Combining (5.7) and (5.9) yields
$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\\{\frac{1}{\epsilon^{n^{*}}}(|\eta_{1}(u,y,\epsilon)|+||D\eta_{1}(u,y,\epsilon)||)\
|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\\}=0.$
(5.10)
By the conclusion $\bf(ii)$ of Theorem 4.2 and the fact that
$w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0)\in T^{\bot}_{u,y,k},$
we get
$\displaystyle\langle
u(\cdot-y)+w_{\delta,k}(u,y,0),w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0)\rangle$
(5.11) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}f(u(\cdot-y)+w_{\delta,k}(u,y,0))\cdot(w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0))dx.$
By Proposition 4.3, we deduce that
$\displaystyle\eta_{2}(u,y,\epsilon)$ $\displaystyle:=$
$\displaystyle\frac{1}{2}||w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0)||^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}}V(\epsilon
x)|w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0)|^{2}dx$
$\displaystyle+\int_{\mathbb{R}^{N}}V(\epsilon
x)(u(\cdot-y)+w_{\delta,k}(u,y,0))(w_{\delta,k}(u,y,\epsilon)-w_{\delta,k}(u,y,0))dx$
also satisfies (5.10). By the conclusion $\bf(iii)$ of Theorem 4.2, we infer
that
$\displaystyle
J(u(\cdot-y)+w_{\delta,k}(u,y,0))=J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))=I(u+\pi_{k}(u)).$
(5.12)
Finally, by the conclusions $\bf(iii)$ and $\bf(v)$ of Theorem 4.2 and (4.34),
we have
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}V(\epsilon
x)|u(\cdot-y)+w_{\delta,k}(u,y,0)|^{2}dx$ (5.13) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}V(\epsilon
x)(u(\cdot-y)+\pi_{k}(u)(\cdot-y))^{2}dx$ $\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon^{n^{*}}\Gamma_{k}(u,y)+\eta_{3}(u,y,\epsilon),$
where
$\displaystyle\Gamma_{k}(u,y)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}Q_{n^{*}}(x)(u(\cdot-y)+\pi_{k}(u)(\cdot-y))^{2}dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}Q_{n^{*}}(x+y)(u+\pi_{k}(u))^{2}dx.$
By Lemma 5.1, the conclusion $\bf(v)$ of Theorem 4.2 and (4.34), we deduce
that $\eta_{3}$ satisfies (5.10). By
$(\ref{vxvvdffrrfw44e})-(\ref{nv888rtr664})$, we get that
$\displaystyle\Psi_{k}(u,y,\epsilon)=I(u+\pi_{k}(u))+\frac{1}{2}\epsilon^{n^{*}}\Gamma_{k}(u,y)+\eta(u,y,\epsilon),$
(5.14)
where $\eta=\eta_{1}+\eta_{2}+\eta_{3}$ satisfies (5.10).
By Lemma 5.2, for every $u\in\overline{\mathcal{N}_{\delta,k}}$,
$\Gamma_{k}(u,y)$ has a strict local minimum at $y=0$ and there is a constant
$A_{k}>0$ such that
$\displaystyle D^{2}_{y}\Gamma_{k}(u,0)\geq A_{k}\mbox{Id}$ (5.15)
where Id denotes the $N\times N$ identity matrix. By (5.15) and (5.14), we
deduce that there exists $\epsilon^{\prime}_{k}>0$ such that if
$0\leq\epsilon\leq\epsilon^{\prime}_{k}$, then for every
$u\in\overline{\mathcal{N}_{\delta,k}}$, there exists $y_{\epsilon}(u)\in
B_{\mathbb{R}^{N}}(0,R/2)$ such that $y_{\epsilon}(u)$ is the unique minimizer
of $\Psi_{k}(u,\cdot,\epsilon)$ in $B_{\mathbb{R}^{N}}(0,R)$. Moreover, by
implicit functional theorem, $y_{\epsilon}(\cdot)\in
C^{1}(\overline{\mathcal{N}_{\delta,k}})$. By (5.14), we get that
$\lim_{\epsilon\rightarrow
0}||\Psi_{k}(u,y_{\epsilon}(u),\epsilon)-I(u+\pi_{k}(u))||_{C^{1}(\overline{\mathcal{N}_{\delta,k}})}=0.$
(5.16)
By [9, Theorem IV.3], a GM pair is a special kind of Conley index pair which
is associated with some pseudo-gradient flow of a functional. Therefore, the
GM pair $(W_{k},W_{k}^{-})$ which was defined in Remark 3.11 is a Conley index
pair associated with some pseudo-gradient flow of the functional
$g_{k}(u)=I(u+\pi_{k}(u)).$ Then by (5.16) and Theorem III.4 of [9], we deduce
that if $\epsilon$ is small enough, then $(W_{k},W_{k}^{-})$ is also a Conley
index pair associated with some pseudo-gradient flow of the functional
$\Psi_{k}(\cdot,y_{\epsilon}(\cdot),\epsilon).$ By (3.57) and Theorem 5.5.18
of [8], we infer that if $\epsilon$ is sufficiently small, then
$\Psi_{k}(\cdot,y_{\epsilon}(\cdot),\epsilon)$ has at least a critical point
$u_{\epsilon}\in\mathcal{N}_{\delta,k}$. Then by Theorem 4.4,
$\tilde{u}_{\epsilon}:=u_{\epsilon}(\cdot-
y_{\epsilon}(u_{\epsilon}))+w_{\delta,k}(u_{\epsilon},y_{\epsilon}(u_{\epsilon}),\epsilon)$
is a critical point of $E_{\epsilon}$. Moreover, by (5.16), we have
$\lim_{\epsilon\rightarrow
0}\mbox{dist}_{{}_{Y}}(\tilde{u}_{\epsilon},\mathcal{K})=0$
with $\mathcal{K}=\mathcal{K}^{b}_{a}$. This finishes the proof of Theorem
1.3. $\Box$
## 6 Appendix A
In this appendix, we shall give the proof of the existence of
$\\{\tilde{e}_{j,k}\\}$ which satisfies the conditions $\bf(i)$ and $\bf(ii)$
in Section 3.
Since $X\cap C^{\infty}_{0}(\mathbb{R}^{N})$ is dense in $X,$ for any
$\mu_{k}>0,$ we can choose $\\{\bar{e}_{j,k}\\}\subset X\cap
C^{\infty}_{0}(\mathbb{R}^{N})$ such that
$\displaystyle\sup_{1\leq j\leq k}||\bar{e}_{j,k}-e^{\prime}_{j}||\leq\mu_{k}\
\mbox{and}\ ||\bar{e}_{j,k}||=1,\ 1\leq j\leq k.$ (6.1)
We show that if $\mu_{k}$ is small enough, then $\\{\bar{e}_{j,k}\ |\ 1\leq
j\leq k\\}\cup\\{e_{j}\ |\ 1\leq j\leq q\\}$ is linearly independent. If it
were not true, without loss of generality, we may assume that
$\displaystyle\bar{e}_{k,k}=\sum^{k-1}_{j=1}\alpha_{j}\bar{e}_{j,k}+\sum^{q}_{j=1}\beta_{j}e_{j},$
(6.2)
then
$\bar{e}_{k,k}=\sum^{k-1}_{j=1}\alpha_{j}e^{\prime}_{j}+\sum^{k-1}_{j=1}\alpha_{j}(\bar{e}_{j,k}-e^{\prime}_{j})+\sum^{q}_{j=1}\beta_{j}e_{j}.$
It follows that if $\mu_{k}<1/4\sqrt{2}$, then
$\displaystyle 1=||\bar{e}_{k,k}||^{2}$ $\displaystyle=$
$\displaystyle\sum^{k-1}_{j=1}\alpha^{2}_{j}+||\sum^{k-1}_{j=1}\alpha_{j}(\bar{e}_{j,k}-e^{\prime}_{j})||^{2}+2\langle\sum^{k-1}_{j=1}\alpha_{j}e^{\prime}_{j},\sum^{k-1}_{j=1}\alpha_{j}(\bar{e}_{j,k}-e^{\prime}_{j})\rangle$
(6.3)
$\displaystyle+\sum^{q}_{j=1}\beta^{2}_{j}+2\langle\sum^{q}_{j=1}\beta_{j}e_{j},\sum^{k-1}_{j=1}\alpha_{j}(\bar{e}_{j,k}-e^{\prime}_{j})\rangle$
$\displaystyle\geq$
$\displaystyle\frac{3}{4}\sum^{k-1}_{j=1}\alpha^{2}_{j}+\frac{3}{4}\sum^{q}_{j=1}\beta^{2}_{j}+||\sum^{k-1}_{j=1}\alpha_{j}(\bar{e}_{j,k}-e^{\prime}_{j})||^{2}-8\sum^{k-1}_{j=1}\alpha^{2}_{j}||\bar{e}_{j,k}-e^{\prime}_{j}||^{2}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\sum^{k-1}_{j=1}\alpha^{2}_{j}+\frac{1}{2}\sum^{q}_{j=1}\beta^{2}_{j}.$
By (6.2),
$e^{\prime}_{k}=\sum^{k-1}_{j=1}\alpha_{j}e^{\prime}_{j}+\sum^{k-1}_{j=1}\alpha_{j}(\bar{e}_{j,k}-e^{\prime}_{j})+\sum^{q}_{j=1}\beta_{j}e_{j}+(e^{\prime}_{k}-\bar{e}_{k,k}),$
combining (6.3), we get that
$\displaystyle 1=||e^{\prime}_{k}||^{2}$ $\displaystyle=$
$\displaystyle\sum^{k-1}_{j=1}\alpha_{j}\langle\bar{e}_{j,k}-e^{\prime}_{j},e^{\prime}_{k}\rangle+\langle
e^{\prime}_{k}-\bar{e}_{k,k},e^{\prime}_{k}\rangle\leq\mu_{k}\sum^{k-1}_{j=1}|\alpha_{j}|+\mu_{k}$
$\displaystyle\leq$ $\displaystyle(\sqrt{2k}+1)\mu_{k}.$
This induces a contradiction if we assume $(\sqrt{2k}+1)\mu_{k}<1$. Thus,
$\\{\bar{e}_{j,k}\ |\ 1\leq j\leq k\\}\cup\\{e_{j}\ |\ 1\leq j\leq k\\}$ is
linearly independent if $\mu_{k}<\min\\{1/(\sqrt{2k}+1),1/4\sqrt{2}\\}$.
By (6.1) and
$\langle\bar{e}_{j,k},\bar{e}_{j^{\prime},k}\rangle=\langle
e^{\prime}_{j}+(\bar{e}_{j,k}-e^{\prime}_{j}),e^{\prime}_{j^{\prime}}+(\bar{e}_{j^{\prime},k}-e^{\prime}_{j^{\prime}})\rangle,\
\langle\bar{e}_{j,k},e_{j^{\prime}}\rangle=\langle
e^{\prime}_{j}+(\bar{e}_{j,k}-e^{\prime}_{j}),e_{j^{\prime}}\rangle,$
we get that
$\displaystyle\sup_{1\leq j,j^{\prime}\leq k,j\neq
j^{\prime}}|\langle\bar{e}_{j,k},\bar{e}_{j^{\prime},k}\rangle|\leq
2\mu_{k}+\mu^{2}_{k},\ \sup_{j\neq
j^{\prime}}|\langle\bar{e}_{j,k},e_{j^{\prime}}\rangle|\leq\mu_{k}.$ (6.4)
Therefore, if $\mu_{k}$ is sufficiently small, using Gram-Schmidt
orthogonalizing process to $\\{e_{j}\ |\ 1\leq j\leq q\\}\cup\\{\bar{e}_{j,k}\
|\ 1\leq j\leq k\\}$, we get $\\{\tilde{e}_{j,k}\ |\ 1\leq j\leq k\\}$ which
satisfies the conditions $\bf(i)$ and $\bf(ii)$ in Section 3.
## 7 Appendix B
In this appendix, we give the proof of Proposition 4.3.
Let
$\displaystyle\eta_{u,y,k}=(-\triangle+1)^{-1}f(u(\cdot-y)+\pi_{k}(u)(\cdot-y)).$
Then
$\displaystyle\eta_{u,y,k}$ $\displaystyle=$
$\displaystyle(-\triangle+1+V(\epsilon
x))^{-1}f(u(\cdot-y)+\pi_{k}(u)(\cdot-y)))$ (7.1)
$\displaystyle+(-\triangle+1+V(\epsilon x))^{-1}V(\epsilon x)\eta_{u,y,k}.$
Subtracting equation
$S_{u,y,k}\nabla E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))=0$
from equation
$S_{u,y,k}\nabla J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))=0,$
by (7.1) and the mean value theorem, we get that
$\displaystyle
L_{u,y,\epsilon,k}\Big{(}w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)\Big{)}$
(7.2) $\displaystyle=$ $\displaystyle-S_{u,y,k}(-\triangle+1+V(\epsilon
x))^{-1}V(\epsilon x)\eta_{u,y,k}$
$\displaystyle+S_{u,y,k}(-\triangle+1+V(\epsilon
x))^{-1}\Big{(}(f^{\prime}(u(\cdot-y)+\tilde{w})-f^{\prime}(u(\cdot-y)))$
$\displaystyle\quad\quad\quad\times(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))\Big{)}$
where $\tilde{w}$ lies between $w_{\delta,k}(u,y,\epsilon)$ and
$\pi_{k}(u)(\cdot-y)$. By the conclusion $\bf(iv)$ of Theorem 4.2, we get that
$||w_{\delta,k}(u,y,\epsilon)||\leq r$ if $0<\delta\leq\delta_{r}$ and $k\geq
k(\delta)$. And by $\bf(ii)$ of Lemma 3.8, we deduce that if $k(\delta)$ is
large enough and $k\geq k(\delta),$ then $||\pi_{k}(u)(\cdot-y)||\leq r.$
Therefore, $||\tilde{w}||\leq r$ if $0<\delta\leq\delta_{r}$ and $k\geq
k(\delta)$. Moreover, by (4.19), we deduce that if $r$ is small enough,
$0<\delta\leq\delta_{r}$ and $k\geq k(\delta)$, then
$\displaystyle\Big{|}\Big{|}(-\triangle+1+V(\epsilon
x))^{-1}\Big{(}(f^{\prime}(u(\cdot-y)+\tilde{w})-f^{\prime}(u(\cdot-y))\cdot(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))\Big{)}\Big{|}\Big{|}$
$\displaystyle\leq\frac{C}{2}||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||,$
(7.3)
where $C$ is the constant appeared in Lemma 4.1. By (7), (7.2) and Lemma 4.1,
we get that
$\displaystyle C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||\leq
2||(-\triangle+1)^{-1}V(\epsilon x)\eta_{u,y,k}||.$ (7.4)
By (4.34), the conclusion $\bf(v)$ of Theorem 4.2 and [25, Proposition 4.2],
we get that for any $n>0$,
$\displaystyle\sup\\{||(1+|x|)^{n}\eta_{u,y,k}||_{L^{\infty}(\mathbb{R}^{N})}\
|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\\}<\infty.$
(7.5)
By (7.5), using the same argument as Lemma 3.2 of [1], we can get that if
$\iota<n^{*},$
$\displaystyle\lim_{\epsilon\rightarrow
0}\\{\int_{\mathbb{R}^{N}}\frac{V^{2}(\epsilon
x)}{\epsilon^{2\iota}}\eta^{2}_{u,y,k}\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\\}=0$
(7.6)
and
$\sup\\{\int_{\mathbb{R}^{N}}\frac{V^{2}(\epsilon
x)}{\epsilon^{2n^{*}}}\eta^{2}_{u,y,k}\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},\
0\leq\epsilon\leq\epsilon^{*}\\}<\infty.$
Thus, for $\iota<n^{*},$
$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\\{\frac{1}{\epsilon^{\iota}}||(-\triangle+1)^{-1}V(\epsilon
x)\eta_{u,y,k}||\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\\}=0$
(7.7)
and
$\displaystyle\sup\\{\frac{1}{\epsilon^{n^{*}}}||(-\triangle+1)^{-1}V(\epsilon
x)\eta_{u,y,k}||\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},\
0\leq\epsilon\leq\epsilon^{*}\\}<\infty.$ (7.8)
Combining (7.4), (7.7) and (7.8) yields that for $\iota<n^{*},$ if $\delta>0$
is small enough and $k\geq k(\delta),$ then
$\displaystyle\lim_{\epsilon\rightarrow
0}\\{\frac{1}{\epsilon^{\iota}}||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||\
|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\\}=0$
(7.9)
and
$\displaystyle\sup\\{\frac{1}{\epsilon^{n^{*}}}||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||\
|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},\
0\leq\epsilon\leq\epsilon^{*}\\}$ (7.10) $\displaystyle<\infty.$
Recall that $S_{u,y,k}:Y\rightarrow T^{\bot}_{u,y,k}$ is orthogonal
projection. Therefore, for $h\in Y,$
$\displaystyle S_{u,y,k}h$ $\displaystyle=$ $\displaystyle
h-\sum^{q}_{j=1}\langle h,e_{j}(\cdot-y)\rangle
e_{j}(\cdot-y)-\sum^{k}_{j=1}\langle
h,\tilde{e}_{j,k}(\cdot-y)\rangle\tilde{e}_{j,k}(\cdot-y)$
$\displaystyle-\sum^{N}_{j=1}\Big{\langle}h,\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial
x_{j}}(\cdot-y)\Big{\rangle}\frac{\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}(\cdot-y)}{||\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}||^{2}}.$
Thus, the Fréchet partial derivative of $S_{u,y,k}h$ with respect to $u$ along
the vector $v\in X_{k}$ is
$\displaystyle D_{u}(S_{u,y,k}h)[v]$ (7.11) $\displaystyle=$
$\displaystyle-\sum^{N}_{j=1}\Big{\langle}h,\sum^{s}_{i=1}D\xi_{i}(u)[v]\cdot\frac{\partial
u_{i}}{\partial
x_{j}}(\cdot-y)\Big{\rangle}\frac{\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}(\cdot-y)}{||\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}||^{2}}$
$\displaystyle-\sum^{N}_{j=1}\Big{\langle}h,\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial
x_{j}}(\cdot-y)\Big{\rangle}\frac{\sum^{s}_{i=1}(D\xi_{i}(u)[v])\cdot\frac{\partial
u_{i}}{\partial x_{j}}(\cdot-y)}{||\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}||^{2}}$
$\displaystyle+2\sum^{N}_{j=1}\Big{(}\Big{\langle}h,\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial
x_{j}}(\cdot-y)\Big{\rangle}\frac{\langle\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}},\sum^{s}_{i=1}(D\xi_{i}(u)[v])\frac{\partial
u_{i}}{\partial x_{j}}\rangle}{||\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}||^{4}}$
$\displaystyle\quad\quad\quad\times\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}(\cdot-y)\Big{)}$
and the Fréchet partial derivative of $S_{u,y,k}h$ with respect to $y$ along
the vector $\bar{y}\in\mathbb{R}^{N}$ is
$\displaystyle D_{y}(S_{u,y,k}h)[\bar{y}]$ (7.12) $\displaystyle=$
$\displaystyle\sum^{q}_{j=1}\langle h,(\bar{y}\nabla_{x}e_{j})(\cdot-y)\rangle
e_{j}(\cdot-y)+\sum^{k}_{j=1}\langle
h,(\bar{y}\nabla_{x}\tilde{e}_{j,k})(\cdot-y)\rangle\tilde{e}_{j,k}(\cdot-y)$
$\displaystyle+\sum^{q}_{j=1}\langle
h,e_{j}(\cdot-y)\rangle(\bar{y}\nabla_{x}e_{j})(\cdot-y)+\sum^{k}_{j=1}\langle
h,\tilde{e}_{j,k}(\cdot-y)\rangle(\bar{y}\nabla_{x}\tilde{e}_{j,k})(\cdot-y)$
$\displaystyle+\sum^{N}_{j=1}\Big{\langle}h,\sum^{s}_{i=1}\xi_{i}(u)\cdot(\bar{y}\nabla_{x}(\frac{\partial
u_{i}}{\partial
x_{j}}))(\cdot-y)\Big{\rangle}\frac{\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}(\cdot-y)}{||\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}||^{2}}$
$\displaystyle+\sum^{N}_{j=1}\Big{\langle}h,\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial
x_{j}}(\cdot-y)\Big{\rangle}\frac{\sum^{s}_{i=1}\xi_{i}(u)\cdot(\bar{y}\nabla_{x}(\frac{\partial
u_{i}}{\partial x_{j}}))(\cdot-y)}{||\sum^{s}_{i=1}\xi_{i}(u)\frac{\partial
u_{i}}{\partial x_{j}}||^{2}}.$
Differentiating equations $S_{u,y,k}(\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)))=0$ and
$S_{u,y,k}(\nabla J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))=0$ with respect to the
variable $u$ along the vector $v\in X_{k}$, we get that
$\displaystyle
S_{u,y,k}(\nabla^{2}E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))(v(\cdot-y)+Dw_{\delta,k}(u,y,\epsilon)[v,0]))$
$\displaystyle+D_{u}(S_{u,y,k}h_{1})[v]=0$ (7.13)
and
$\displaystyle
S_{u,y,k}(\nabla^{2}J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))(v(\cdot-y)+D\pi_{k}(u)(\cdot-y)[v,0]))$
$\displaystyle+D_{u}(S_{u,y,k}h_{2})[v]=0,$ (7.14)
where $h_{1}=\nabla E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))$ and
$h_{2}=\nabla J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))$. By (7.1) and (7), it is easy
to verify that there exists a constant $C>0$ such that
$\displaystyle||h_{1}-h_{2}||\leq
C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||+C||(-\triangle+1)^{-1}V(\epsilon
x)\eta_{u,y,k}||.$ (7.15)
By (7.15) and (7.11), we get that for $||v||\leq 1,$ there exists a constant
$C>0$ such that
$\displaystyle||D_{u}(S_{u,y,k}h_{2})[v]-D_{u}(S_{u,y,k}h_{1})[v]||$ (7.16)
$\displaystyle\leq$ $\displaystyle
C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||+C||(-\triangle+1)^{-1}V(\epsilon
x)\eta_{u,y,k}||.$
A direct computation shows that
$\displaystyle
S_{u,y,k}(\nabla^{2}E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))(v(\cdot-y)+Dw_{\delta,k}(u,y,\epsilon)[v,0]))$
(7.17) $\displaystyle-
S_{u,y,k}(\nabla^{2}J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))(v(\cdot-y)+D\pi_{k}(u)(\cdot-y)[v,0]))$
$\displaystyle=$ $\displaystyle
S_{u,y,k}(\nabla^{2}J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]))$
$\displaystyle-
S_{u,y,k}(-\triangle+1)^{-1}\Big{\\{}\Big{(}f^{\prime}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))$
$\displaystyle-f^{\prime}(u(\cdot-y)+\pi_{k}(u)(\cdot-y))\Big{)}\times(v(\cdot-y)+Dw_{\delta,k}(u,y,\epsilon)[v,0])\Big{\\}}$
$\displaystyle+S_{u,y,k}(-\triangle+1)^{-1}V(\epsilon
x)\bar{\eta}_{u,y,\epsilon,k}(v)$
where
$\bar{\eta}_{u,y,\epsilon,k}(v)=(-\triangle+1+V(\epsilon
x))^{-1}(f^{\prime}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)))\cdot(v(\cdot-y)+Dw_{\delta,k}(u,y,\epsilon)[v,0])).$
By (4.34), the conclusion $\bf(v)$ of Theorem 4.2 and (1.2) in $\bf(F_{1})$,
we get that for any $v,h\in Y,$ $||v||=||h||=1,$
$\displaystyle\int_{\mathbb{R}^{N}}\Big{|}f^{\prime}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))-f^{\prime}(u(\cdot-y)+\pi_{k}(u)(\cdot-y))\Big{|}$
$\displaystyle\quad\quad\quad\times|v(\cdot-y)+Dw_{\delta,k}(u,y,\epsilon)[v,0]|\cdot|h|dx$
$\displaystyle\leq$ $\displaystyle
C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||.$
It follows that
$\displaystyle\Big{|}\Big{|}(-\triangle+1)^{-1}\Big{\\{}\Big{(}f^{\prime}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))$
(7.18)
$\displaystyle\quad\quad-f^{\prime}(u(\cdot-y)+\pi_{k}(u)(\cdot-y))\Big{)}\times(v(\cdot-y)+Dw_{\delta,k}(u,y,\epsilon)[v,0])\Big{\\}}\Big{|}\Big{|}$
$\displaystyle\leq$ $\displaystyle
C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||.$
By (7), (7) and $(\ref{gdttter00oqppp})-(\ref{gdhf66ey161994})$, we deduce
that
$\displaystyle||S_{u,y,k}(\nabla^{2}J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]))||$
(7.19) $\displaystyle\leq$ $\displaystyle
C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||+C||(-\triangle+1)^{-1}V(\epsilon
x)\eta_{u,y,k}||$ $\displaystyle+C||(-\triangle+1)^{-1}V(\epsilon
x)\bar{\eta}_{u,y,\epsilon,k}(v)||.$
By the conclusion $\bf(ii)$ of Lemma 3.8 and (4.19), we deduce that
$\displaystyle\lim_{k\rightarrow\infty}\sup\Big{\\{}||\nabla^{2}J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))-\nabla^{2}J(u(\cdot-y))||_{\mathcal{L}(Y)}$
$\displaystyle\quad\quad\quad\quad\quad|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)}\Big{\\}}=0.$
Therefore, as $k\rightarrow\infty$,
$\displaystyle||S_{u,y,k}(\nabla^{2}J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]))$
$\displaystyle-
S_{u,y,k}(\nabla^{2}J(u(\cdot-y))(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]))||$
$\displaystyle=o(1)||Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]||.$
(7.20)
By (7.19) and (7), we get that as $k\rightarrow\infty$,
$\displaystyle||S_{u,y,k}(\nabla^{2}J(u(\cdot-y))(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]))||$
(7.21) $\displaystyle\leq$ $\displaystyle
C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||+C||(-\triangle+1)^{-1}V(\epsilon
x)\eta_{u,y,k}||$ $\displaystyle+C||(-\triangle+1)^{-1}V(\epsilon
x)\bar{\eta}_{u,y,\epsilon,k}(v)||$
$\displaystyle+o(1)||Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]||.$
Let $\mathcal{T}_{u}(\cdot-y)=\\{h(\cdot-y)\ |\ h\in\mathcal{T}_{u}\\}$ and
$\mathcal{T}^{\bot}_{u}(\cdot-y)$ be the orthogonal complement space in $Y$,
where $\mathcal{T}_{u}$ is defined in (3.24). Let
$P_{\mathcal{T}^{\bot}_{u}(\cdot-y)}:Y\rightarrow\mathcal{T}^{\bot}_{u}(\cdot-y)$
and $P_{\mathcal{T}_{u}(\cdot-y)}:Y\rightarrow\mathcal{T}_{u}(\cdot-y)$ be
orthogonal projections. Since $Dw_{\delta,k}(u,y,\epsilon)[v,0]\bot
X_{k}(\cdot-y)$ and $D(\pi_{k}(u)(\cdot-y))[v,0]\bot X_{k}(\cdot-y),$ where
$X_{k}(\cdot-y)=\\{v(\cdot-y)\ |\ v\in X_{k}\\}$, we deduce that
$P_{\mathcal{T}^{\bot}_{u}(\cdot-y)}(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0])\in
T^{\bot}_{u,y,k}.$
Therefore, by Lemma 4.1, we have
$\displaystyle||S_{u,y,k}(\nabla^{2}J(u(\cdot-y))P_{\mathcal{T}^{\bot}_{u}(\cdot-y)}(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]))||$
(7.22) $\displaystyle=$
$\displaystyle||L_{u,y,0,k}P_{\mathcal{T}^{\bot}_{u}(\cdot-y)}(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D\pi_{k}(u)(\cdot-y))[v,0])||$
$\displaystyle\geq$ $\displaystyle
C||P_{\mathcal{T}^{\bot}_{u}(\cdot-y)}(Dw_{\delta,k}(u,y,\epsilon)[v,0]-D(\pi_{k}(u)(\cdot-y))[v,0]||.$
Differentiating the following equation with respect to variable $u$ along the
vector $v,$
$\Big{\langle}w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y),\sum^{s}_{i=1}\xi_{i}(u)\frac{u_{i}(\cdot-y)}{\partial
x_{j}}\Big{\rangle}=0$
we get that
$\displaystyle\Big{\langle}D(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))[v,0],\sum^{s}_{i=1}\xi_{i}(u)\frac{u_{i}(\cdot-y)}{\partial
x_{j}}\Big{\rangle}$ $\displaystyle=$
$\displaystyle-\Big{\langle}w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y),\sum^{s}_{i=1}(D\xi_{i}(u)[v])\frac{u_{i}(\cdot-y)}{\partial
x_{j}}\Big{\rangle}.$
It follows that there exists a constant $C>0$ such that
$\displaystyle||P_{\mathcal{T}_{u}(\cdot-y)}(D(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))[v,0])||$
(7.23) $\displaystyle\leq$ $\displaystyle
C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||.$
By $(\ref{vvcnbd7duqeeadyydtdtdt})-(\ref{gfbbv8uufy5534})$, we deduce that
when $k$ is large enough, then there exists a constant $C>0$ such that
$\displaystyle||D(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))[v,0]||$
$\displaystyle\leq$ $\displaystyle
C||w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y)||+C||(-\triangle+1)^{-1}V(\epsilon
x)\eta_{u,y,\epsilon,k}||.$ $\displaystyle+C||(-\triangle+1)^{-1}V(\epsilon
x)\bar{\eta}_{u,y,\epsilon,k}(v)||.$
Then by $(\ref{baobaoainiiii})-(\ref{ncbcjdgftr745545})$ and the fact that for
$\iota<m,$
$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\Big{\\{}\frac{1}{\epsilon^{\iota}}||(-\triangle+1)^{-1}V(\epsilon
x)\bar{\eta}_{u,y,\epsilon,k}(v)||$ $\displaystyle\quad\quad\quad\quad\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},\
v\in X_{k},\ ||v||\leq 1\Big{\\}}=0$
and
$\displaystyle\sup\\{\frac{1}{\epsilon^{n^{*}}}||(-\triangle+1)^{-1}V(\epsilon
x)\bar{\eta}_{u,y,\epsilon,k}(v)||\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},$
$\displaystyle\quad\quad\ v\in X_{k},\ ||v||\leq 1,\
0\leq\epsilon\leq\epsilon^{*}\\}<\infty,$
we get that for $\iota<n^{*},$
$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\Big{\\{}\frac{1}{\epsilon^{\iota}}||D(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))[v,0]||$
$\displaystyle\quad\quad\quad\quad|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},\
v\in X_{k},\ ||v||\leq 1\Big{\\}}=0$ (7.24)
and
$\displaystyle\sup\\{\frac{1}{\epsilon^{n^{*}}}||D(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))[v,0]||\
|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},$
$\displaystyle\quad\quad\quad v\in X_{k},\ ||v||\leq 1,\
0\leq\epsilon\leq\epsilon^{*}\\}<\infty.$ (7.25)
Differentiating the two equations $S_{u,y,k}(\nabla
E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon)))=0$ and
$S_{u,y,k}(\nabla J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))=0$ with respect to the
variable $y$ along the vector $\bar{y}\in\mathbb{R}^{N}$, we get that
$\displaystyle
S_{u,y,k}(\nabla^{2}E_{\epsilon}(u(\cdot-y)+w_{\delta,k}(u,y,\epsilon))(-\bar{y}\nabla_{x}u(\cdot-y)+Dw_{\delta,k}(u,y,\epsilon)[0,\bar{y}]))$
$\displaystyle+D_{y}(S_{u,y,k}h_{1})[\bar{y}]=0$
and
$\displaystyle
S_{u,y,k}(\nabla^{2}J(u(\cdot-y)+\pi_{k}(u)(\cdot-y))(-\bar{y}\nabla_{x}u(\cdot-y)+D(\pi_{k}(u)(\cdot-y))[0,\bar{y}]))$
$\displaystyle+D_{y}(S_{u,y,k}h_{2})[\bar{y}]=0.$
The same arguments as (7) and (7) yield that for $\iota<n^{*},$
$\displaystyle\lim_{\epsilon\rightarrow
0}\sup\Big{\\{}\frac{1}{\epsilon^{\iota}}||D(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))[0,\bar{y}]||$
$\displaystyle\quad\quad\quad\quad\ |\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},\
\bar{y}\in\mathbb{R}^{N},\ |\bar{y}|\leq 1\Big{\\}}=0$
and
$\displaystyle\sup\\{\frac{1}{\epsilon^{n^{*}}}||D(w_{\delta,k}(u,y,\epsilon)-\pi_{k}(u)(\cdot-y))[0,\bar{y}]||\
|\
(u,y)\in\overline{\mathcal{N}_{\delta,k}}\times\overline{B_{\mathbb{R}^{N}}(0,R)},$
$\displaystyle\quad\quad\quad\bar{y}\in\mathbb{R}^{N},\ |\bar{y}|\leq 1,\
0\leq\epsilon\leq\epsilon^{*}\\}<\infty.$
Acknowledgements The author would like to thank the referee for her or his
comments and suggestions on the manuscript. This work was supported by NSFC
(10901112) and BNSF (1102013).
## References
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* [6] J. Byeon, L. Jeanjean, K. Tanaka, Standing waves for nonlinear Schröinger equations with a general nonlinearity: One and two dimensional cases, Comm. Partial Differential Equations 33 (2008), 1113-1136.
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* [18] V. Felli, M. Schneider, Perturbation results of critical elliptic equations of Caffarelli-Kohn-Nirenberg type, J. Differential Equations 191 (2003) 121-142.
* [19] D. Gromoll, W. Meyer, On differentiable functions with isolated critical points, Topology 8 (1969) 361-369.
* [20] X. Kang, J. C. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations 5 (2000) 899-928.
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* [22] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982) 447-526.
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|
arxiv-papers
| 2012-01-11T02:03:33 |
2024-09-04T02:49:26.128462
|
{
"license": "Public Domain",
"authors": "Shaowei Chen, Lishan Lin",
"submitter": "Shaowei Chen",
"url": "https://arxiv.org/abs/1201.2215"
}
|
1201.2229
|
# Method for classifying multi-qubit states via the rank of coefficient matrix
and its application to four-qubit states
Xiangrong Li1, Dafa Li2,3 1 Department of Mathematics, University of
California, Irvine, CA 92697-3875, USA
2 Department of mathematical sciences, Tsinghua University, Beijing 100084
CHINA
3 Center for Quantum Information Science and Technology, Tsinghua National
Laboratory for information science and technology (TNList), Beijing 100084,
CHINA
###### Abstract
We construct coefficient matrices of size $2^{\ell}$ by $2^{n-\ell}$
associated with pure $n$-qubit states and prove the invariance of the ranks of
the coefficient matrices under stochastic local operations and classical
communication (SLOCC). The ranks give rise to a simple way of partitioning
pure $n$-qubit states into inequivalent families and distinguishing degenerate
families from one another under SLOCC. Moreover, the classification scheme via
the ranks of coefficient matrices can be combined with other schemes to build
a more refined classification scheme. To exemplify we classify the nine
families of four qubits introduced by Verstraete _et al._ [Phys. Rev. A 65,
052112 (2002)] further into inequivalent subfamilies via the ranks of
coefficient matrices, and as a result, we find 28 genuinely entangled families
and all the degenerate classes can be distinguished up to permutations of the
four qubits. We also discuss the completeness of the classification of four
qubits into nine families.
## I I. Introduction
Quantum entanglement plays a crucial role in quantum information theory, with
applications to quantum teleportation, quantum cryptography, and quantum
computation Nielsen . The equivalence under stochastic local operations and
classical communication (SLOCC) induces a natural partition of quantum states.
The central task of SLOCC classification is to classify quantum states
according to a criterion that is invariant under SLOCC.
SLOCC entanglement classification has been the subject of intensive study
during the last decade Dur ; Verstraete ; Miyake ; Piza ; Chterental ; Cao ;
LDF07b ; Lamata ; LDFEPL ; LDFQIC09 ; Bastin ; Borsten ; Ribeiro ; LDFQIC11 ;
LDFJPA ; Viehmann ; LDF12b ; Sharma ; Buniy . For three qubits, there are six
SLOCC equivalence classes of which two are genuinely entanglement classes: GHZ
and $W$ Dur and four degenerate classes can be distinguished by the local
ranks (i.e., ranks of single-qubit reduced density matrices obtained by
tracing out all but one qubit Dur ). For four or more qubits, there are
infinite SLOCC classes and it is highly desirable to partition the infinite
classes into a finite number of families. The key lies in finding criteria to
determine which family an arbitrary quantum state belongs to. In a pioneering
work, Verstraete _et al._ Verstraete obtained nine SLOCC inequivalent
families of four qubits using Lie group theory: $G_{abcd}$, $L_{abc_{2}}$,
$L_{a_{2}b_{2}}$, $L_{ab_{3}}$, $L_{a_{4}}$, $L_{a_{2}0_{3\oplus{\bar{1}}}}$,
$L_{0_{5\oplus{\bar{3}}}}$, $L_{0_{7\oplus{\bar{1}}}}$, and
$L_{0_{3\oplus{\bar{1}}}0_{3\oplus{\bar{1}}}}$. It is clear that, some
families obtained by Verstraete _et al._ Verstraete contain an infinite
number of SLOCC classes and some contain both degenerate classes and genuinely
entangled classes. It is of great importance to find a more refined partition
of four-qubit states such that the degenerate classes are distinguished from
the genuinely entangled families. Many other efforts have been devoted to the
SLOCC entanglement classification of four qubits Miyake ; Chterental ; Cao ;
LDF07b ; Lamata ; LDFQIC09 ; Borsten ; Viehmann ; Buniy . More recently, a few
attempts have been made toward the generalization to higher number of qubits,
including odd $n$ qubits LDFQIC11 , even $n$ qubits LDFJPA , symmetric $n$
qubits LDFEPL ; Bastin ; Ribeiro , and general $n$ qubits LDF12b ; Sharma .
This paper is organized as follows. We first construct coefficient matrices of
size $2^{\ell}$ by $2^{n-\ell}$ associated to pure $n$-qubit states and prove
the invariance of the ranks of coefficient matrices under SLOCC in Section II.
In Section III, we present a recursive formula which allows us to easily
calculate the ranks of coefficient matrices of $n$-qubit biseparable states.
We next show that the degenerate families of general $n$ qubits are
inequivalent to one another under SLOCC in Section IV. Section V is devoted to
the classification of four qubits via the ranks of coefficient matrices.
Section VI provides the discussion of the completeness of the nine families
obtained by Verstraete _et al._ Verstraete . We finally conclude this paper in
Section VII.
## II II. The invariance of the ranks of coefficient matrices
Let $|\psi\rangle_{1\cdots n}=\sum_{i=0}^{2^{n}-1}a_{i}|i\rangle$ be an
$n$-qubit pure state. We associate with the state $|\psi\rangle_{1\cdots n}$ a
$2^{\ell}$ by the $2^{n-\ell}$ coefficient matrix
$C_{1\cdots\ell,(\ell+1)\cdots n}(|\psi\rangle_{1\cdots n})$ whose entries are
the coefficients $a_{0},a_{1},\cdots,a_{2^{n}-1}$ of the state
$|\psi\rangle_{1\cdots n}$ arranged in ascending lexicographical order. To
illustrate, we list $C_{1\cdots\ell,(\ell+1)\cdots n}(|\psi\rangle_{1\cdots
n})$ below as:
$\left(\begin{array}[]{cccc}a_{\underbrace{0\cdots
0}_{\ell}\underbrace{0\cdots 0}_{n-\ell}}&\cdots&a_{\underbrace{0\cdots
0}_{\ell}\underbrace{1\cdots 1}_{n-\ell}}&\\\ a_{\underbrace{0\cdots
1}_{\ell}\underbrace{0\cdots 0}_{n-\ell}}&\cdots&a_{\underbrace{0\cdots
1}_{\ell}\underbrace{1\cdots 1}_{n-\ell}}&\\\ \vdots&\vdots&\vdots&\\\
a_{\underbrace{1\cdots 1}_{\ell}\underbrace{0\cdots
0}_{n-\ell}}&\cdots&a_{\underbrace{1\cdots 1}_{\ell}\underbrace{1\cdots
1}_{n-\ell}}&\end{array}\right).$ (1)
In the binary form of the coefficient matrix in Eq. (1), bits $1$ to ${\ell}$
and $\ell+1$ to $n$ are referred to as the row bits and column bits,
respectively. If $\ell=0$, $C_{\emptyset,1\cdots n}(|\psi\rangle_{1\cdots n})$
reduces to the row vector $(a_{0},\cdots,a_{2^{n}-1})$, and if $\ell=n$,
$C_{1\cdots n,\emptyset}(|\psi\rangle_{1\cdots n})$ reduces to the column
vector $(a_{0},\cdots,a_{2^{n}-1})^{T}$.
Let $\\{q_{1},q_{2},\cdots,q_{n}\\}$ be a permutation of $\\{1,2,\cdots,n\\}$.
Let $C_{q_{1}\cdots q_{\ell},q_{\ell+1}\cdots q_{n}}(|\psi\rangle_{1\cdots
n})$ be the $2^{\ell}\times 2^{n-\ell}$ coefficient matrix of the state
$|\psi\rangle_{1\cdots n}$, which is constructed from the coefficient matrix
$C_{12\cdots\ell,\ell+1\cdots n}$ in Eq. (1) by taking the corresponding
permutation. Here $q_{1},\cdots,q_{\ell}$ are the row bits and
$q_{\ell+1},\cdots,q_{n}$ are the column bits. Indeed, we only need to specify
the row bits, as the column bits would simply be the rest of the bits. In the
sequel, we will omit the subscripts $q_{\ell+1},\cdots,q_{n}$ and simply write
$C_{q_{1}\cdots q_{\ell}}$, whenever the column bits are clear from the
context.
It is known that two $n$-qubit pure states $|\psi\rangle_{1\cdots n}$ and
$|\psi^{\prime}\rangle_{1\cdots n}$ are equivalent to each other under SLOCC
if and only if there are local invertible operators $\mathcal{A}_{1}$,
$\mathcal{A}_{2},\cdots$, and $\mathcal{A}_{n}$ such that Dur
$|\psi^{\prime}\rangle_{1\cdots
n}=\mathcal{A}_{1}\otimes\mathcal{A}_{2}\otimes\cdots\otimes\mathcal{A}_{n}|\psi\rangle_{1\cdots
n}.$ (2)
In terms of coefficient matrices, it can be verified that the following result
holds: For any two SLOCC equivalent $n$-qubit pure states
$|\psi\rangle_{1\cdots n}$ and $|\psi^{\prime}\rangle_{1\cdots n}$, their
coefficient matrices $C_{q_{1}\cdots q_{\ell}}$ satisfy the equation:
$\displaystyle C_{q_{1}\cdots q_{\ell}}(|\psi^{\prime}\rangle_{1\cdots n})=$
$\displaystyle(\mathcal{A}_{q_{1}}\otimes\cdots\otimes\mathcal{A}_{q_{\ell}})C_{q_{1}\cdots
q_{\ell}}(|\psi\rangle_{1\cdots
n})(\mathcal{A}_{q_{\ell+1}}\otimes\cdots\otimes\mathcal{A}_{q_{n}})^{T},$ (3)
where $\mathcal{A}_{1},\mathcal{A}_{2},\cdots$, and $\mathcal{A}_{n}$ are the
local operators in Eq. (2). Conversely, if there are local invertible
operators $\mathcal{A}_{1},\mathcal{A}_{2},\cdots$, and $\mathcal{A}_{n}$ such
that Eq. (3) holds true for some $C_{q_{1}\cdots q_{\ell}}$, then
$|\psi\rangle_{1\cdots n}$ and $|\psi^{\prime}\rangle_{1\cdots n}$ are
equivalent under SLOCC.
It immediately follows from Eq. (3) that the rank of any coefficient matrix of
an $n$-qubit pure state is invariant under SLOCC. This leads to the following
theorem.
Theorem 1. If two $n$-qubit pure states are SLOCC equivalent then their
coefficient matrices $C_{q_{1}\cdots q_{\ell}}$ given above have the same
rank.
Restated in the contrapositive the theorem reads: If two coefficient matrices
$C_{q_{1}\cdots q_{\ell}}$ associated with two $n$-qubit pure states differ in
their ranks, then the two states belong necessarily to different SLOCC
classes.
Coefficient matrices constructed above turn out to be closely related to
reduced density matrices. We let $\rho_{12\cdots n}(|\psi\rangle_{1\cdots
n})=|\psi\rangle_{1\cdots n}{}_{1\cdots n}\langle\psi|$ be the density matrix
of an $n$-qubit pure state $|\psi\rangle_{1\cdots n}$, and we let
$\rho_{q_{1}\cdots q_{\ell}}$ be the $\ell$-qubit reduced density matrix
obtained from $\rho_{12\cdots n}$ by tracing out $n-\ell$ qubits. As has been
previously noted for bipartite systems of dimensions $d\times d$, a reduced
density matrix has a full rank factorization in terms of the corresponding
coefficient matrix and its conjugate transpose LDFCTP . This factorization
also holds for $n$-qubit states Huang2 :
$\rho_{q_{1}\cdots q_{\ell}}(|\psi\rangle_{1\cdots n})=C_{q_{1}\cdots
q_{\ell}}(|\psi\rangle_{1\cdots n})C_{q_{1}\cdots
q_{\ell}}^{\dagger}(|\psi\rangle_{1\cdots n}),$ (4)
where $C^{\dagger}$ is the conjugate transpose of $C$. An important
relationship between reduced density matrices and SLOCC polynomial invariants
can be obtained by taking the determinants of both sides of Eq. (4) for even
$n$ and for $\ell=n/2$, yielding:
$\det\rho_{q_{1}\cdots q_{n/2}}(|\psi\rangle_{1\cdots n})=\bigl{|}\det
C_{q_{1}\cdots q_{n/2}}(|\psi\rangle_{1\cdots n})\bigr{|}^{2}.$ (5)
Here $\det C_{q_{1}\cdots q_{n/2}}(|\psi\rangle_{1\cdots n})$ is a SLOCC
polynomial invariant of degree $2^{n/2}$ for even $n$ qubits and its absolute
value can be used as an entanglement measure LDFJPA12 . Thus we have the
following:
Theorem 2. For even $n$-qubit pure states, the determinants of $n/2$-qubit
reduced density matrices are the squares of the SLOCC polynomial invariants of
degree $2^{n/2}$, with the absolute values of the latter quantifying
$n/2$-qubit entanglement of the even $n$-qubit states after tracing out the
other $n/2$ qubits.
As an example, when $n=4$ we have $\det\rho_{12}=|L|^{2}$,
$\det\rho_{13}=|M|^{2}$, and $\det\rho_{14}=|N|^{2}$, where $L$, $M$, and $N$
are polynomial invariants of degree 4 Luque . When $n=6$, there are 10 three-
qubit reduced density matrices and 10 polynomial invariants of degree 8:
$D_{6}^{1},\cdots,D_{6}^{10}$ LDFJPA12 . For reduced density matrix
$\rho_{123}$ and polynomial invariant $D_{6}^{1}$, we have
$\det\rho_{123}=|D_{6}^{1}|^{2}$. Similar equations hold for other reduced
density matrices and polynomial invariants with appropriate permutations of
qubits.
Remark 1. (i). The determinants of reduced density matrices are invariant
under SLOCC. (ii). It is worth noting that Eq. (5) holds for bipartite systems
of dimensions $d\times d$ as well LDFCTP .
As a particular case of Eq. (4), when $q_{i}=i$ we have $\rho_{1\cdots
n}(|\psi\rangle_{1\cdots n})=C_{1\cdots n}(|\psi\rangle_{1\cdots n})C_{1\cdots
n}^{\dagger}(|\psi\rangle_{1\cdots n})$. By virtue of Eq. (4), the rank of the
$\ell$-qubit reduced density matrix and the rank of the corresponding
coefficient matrix are the same. In light of Theorem 1, we have the following
result.
Corollary. The ranks of $\ell$-qubit reduced density matrices obtained by
tracing out $n-\ell$ qubits are invariant under SLOCC.
This is particularly true for the local ranks Dur . Note also that any complex
matrix has a singular value decomposition, with the number of nonzero singular
values equal to the rank of the matrix. This means that the number of nonzero
singular values of any coefficient matrix of an $n$-qubit pure state is
invariant under SLOCC.
## III III. A recursive formula for the ranks of $N$-qubit biseparable states
In principle, we can calculate the ranks of coefficient matrices for $n$-qubit
biseparable pure states by direct calculations. However, in practice, this is
rather cumbersome from the computational point of view, and as $n$ becomes
large, this might pose a serious problem. In order to avoid this difficulty,
we propose a simple recursive formula for the ranks of $n$-qubit biseparable
states.
Suppose that a biseparable $n$-qubit pure state $|\psi\rangle_{1\cdots n}$ is
of the form $|\psi\rangle_{1\cdots n}=|\phi\rangle_{j_{1}\cdots
j_{k}}\otimes|\varphi\rangle_{j_{k+1}\cdots j_{n}}$ with
$|\phi\rangle_{j_{1}\cdots j_{k}}$ being a $k$-qubit state and
$|\varphi\rangle_{j_{k+1}\cdots j_{n}}$ being an $(n-k)$-qubit state. We let
$C_{q_{1}\cdots q_{\ell}}(|\psi\rangle_{1\cdots n})$ be the coefficient matrix
associated with the state $|\psi\rangle_{1\cdots n}$. We let
$C_{q_{1}^{\ast}\cdots q_{s}^{\ast}}(|\phi\rangle_{j_{1}\cdots j_{k}})$ be the
$2^{s}$ by $2^{k-s}$ coefficient matrix associated with the $k$-qubit state
$|\phi\rangle_{j_{1}\cdots j_{k}}$. Here
$\\{q_{1}^{\ast},\cdots,q_{s}^{\ast}\\}=\\{q_{1},\cdots,q_{\ell}\\}\cap\\{j_{1},\cdots,j_{k}\\}$
are the row bits, and by convention, the rest $k-s$ bits are the column bits.
Moreover, we let $C_{q_{1}^{\prime}\cdots
q_{t}^{\prime}}(|\varphi\rangle_{j_{k+1}\cdots j_{n}})$ be the $2^{t}$ by
$2^{n-k-t}$ coefficient matrix associated with the $(n-k)$-qubit state
$|\varphi\rangle_{j_{k+1}\cdots j_{n}}$. Here
$\\{q_{1}^{\prime},\cdots,q_{t}^{\prime}\\}=\\{q_{1},\cdots,q_{\ell}\\}\cap\\{j_{k+1},\cdots,j_{n}\\}$
are the row bits, and by convention, the rest $n-k-t$ bits are the column
bits. It can be verified that
$\displaystyle C_{q_{1}\cdots q_{\ell}}(|\phi\rangle_{j_{1}\cdots
j_{k}}\otimes|\varphi\rangle_{j_{k+1}\cdots j_{n}})$
$\displaystyle=C_{q_{1}^{\ast}\cdots q_{s}^{\ast}}(|\phi\rangle_{j_{1}\cdots
j_{k}})\otimes C_{q_{1}^{\prime}\cdots
q_{t}^{\prime}}(|\varphi\rangle_{j_{k+1}\cdots j_{n}}).$ (6)
In view of the fact that the rank of the Kronecker product of two matrices is
the product of their ranks, we arrive at the following recursive formula for
the ranks of coefficient matrices of an $n$-qubit biseparable state:
$\displaystyle\mbox{rank}(C_{q_{1}\cdots q_{\ell}}(|\phi\rangle_{j_{1}\cdots
j_{k}}\otimes|\varphi\rangle_{j_{k+1}\cdots j_{n}}))$
$\displaystyle=\mbox{rank}(C_{q_{1}^{\ast}\cdots
q_{s}^{\ast}}(|\phi\rangle_{j_{1}\cdots
j_{k}}))\mbox{rank}(C_{q_{1}^{\prime}\cdots
q_{t}^{\prime}}(|\varphi\rangle_{j_{k+1}\cdots j_{n}})).$ (7)
The formula above allows us to calculate recursively the ranks of coefficient
matrices of $n$-qubit biseparable states in terms of the ranks of coefficient
matrices of $k$-qubit states and $(n-k)$-qubit states. To illustrate the use
of the recursive formula, we start with the initial values
$\mbox{rank}(C_{A}(|\phi\rangle_{A}))=1$ and
$\mbox{rank}(C_{\emptyset}(|\phi\rangle_{A}))=1$. It is known that a two-qubit
pure state can be either of the form $A$–$B$ (separable) or the form $AB$
(EPR). Using the recursive formula, we find
$\mbox{rank}(C_{A}(|\phi\rangle_{A}|\varphi\rangle_{B}))=\mbox{rank}(C_{A}(|\phi\rangle_{A}))\times\mbox{rank}(C_{\emptyset}(|\varphi\rangle_{B}))=1$.
On the other hand, a direct calculation shows that
$\mbox{rank}(C_{A}(|\varphi\rangle_{AB}))=2$. Using the results obtained
above, we can find the ranks of coefficient matrices of three-qubit pure
states. Consider, for example,
$\mbox{rank}(C_{C}(|\phi\rangle_{B}|\varphi\rangle_{AC}))$ for biseparable
states being of the form $B$–$AC$. Using the recursive formula, we have
$\mbox{rank}(C_{C}(|\phi\rangle_{B}|\varphi\rangle_{AC}))=\mbox{rank}(C_{\emptyset}(|\phi\rangle_{B}))\times\mbox{rank}(C_{C}(\varphi\rangle_{AC}))=2$.
In a similar fashion, we can fill in the rest of the entries in Table 1,
except those in the last row which can be obtained by direct calculations.
Proceeding in this way, we can construct Tables 2 and 3 for the ranks of
coefficient matrices for four and five qubits.
Note that in Tables 1 and 2 the ranks of only $2^{n-1}-1$ coefficient matrices
are shown. This is due to the fact that interchanging two row (resp. column)
bits or exchanging the row and column bits of a coefficient matrix does not
alter the rank of the matrix, since the former is equivalent to interchanging
two rows (resp. columns) of the matrix and the latter is equivalent to
transposing the matrix. Ignoring $C_{\emptyset}$ and $C_{1\cdots n}$ which
always have rank 1, this amounts to totally $2^{n-1}-1$ potentially different
coefficient matrices. For example, the ranks of $C_{BA}$ and $C_{BC}$ are not
shown in Table 2, since $C_{AB}$ and $C_{BA}$ differ by the interchange of two
rows, and $C_{BC}$ is the transpose of $C_{AD}$. As illustrated in Tables 1,
2, and 3, the ranks of coefficient matrices permit the partitioning of the
space of the pure states into inequivalent families under SLOCC (i.e., two
states belong to the same family if and only if the ranks of coefficient
matrices are all equal). In particular, degenerate families of three, four,
and five qubits are inequivalent from one another under SLOCC.
Table 1: Ranks of coefficient matrices of three-qubit pure states. Families Ranks of | $C_{A}$ | $C_{B}$ | $C_{C}$
---|---|---|---
$A$–$B$–$C$ | $1$ | $1$ | $1$
$A$–$BC$ | $1$ | $2$ | $2$
$B$–$AC$ | $2$ | $1$ | $2$
$C$–$AB$ | $2$ | $2$ | $1$
$ABC$ | $2$ | $2$ | $2$
Table 2: Ranks of coefficient matrices of four-qubit pure states.
Families Ranks of | $C_{A}$ | $C_{B}$ | $C_{C}$ | $C_{D}$ | $C_{AB}$ | $C_{AC}$ | $C_{AD}$
---|---|---|---|---|---|---|---
$A$–$B$–$C$–$D$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$A$–$B$–$CD$ | $1$ | $1$ | $2$ | $2$ | $1$ | $2$ | $2$
$A$–$C$–$BD$ | $1$ | $2$ | $1$ | $2$ | $2$ | $1$ | $2$
$A$–$D$–$BC$ | $1$ | $2$ | $2$ | $1$ | $2$ | $2$ | $1$
$B$–$C$–$AD$ | $2$ | $1$ | $1$ | $2$ | $2$ | $2$ | $1$
$B$–$D$–$AC$ | $2$ | $1$ | $2$ | $1$ | $2$ | $1$ | $2$
$C$–$D$–$AB$ | $2$ | $2$ | $1$ | $1$ | $1$ | $2$ | $2$
$A$–$BCD$ | $1$ | $2$ | $2$ | $2$ | $2$ | $2$ | $2$
$B$–$ACD$ | $2$ | $1$ | $2$ | $2$ | $2$ | $2$ | $2$
$C$–$ABD$ | $2$ | $2$ | $1$ | $2$ | $2$ | $2$ | $2$
$D$–$ABC$ | $2$ | $2$ | $2$ | $1$ | $2$ | $2$ | $2$
$AB$–$CD$ | $2$ | $2$ | $2$ | $2$ | $1$ | $4$ | $4$
$AC$–$BD$ | $2$ | $2$ | $2$ | $2$ | $4$ | $1$ | $4$
$AD$–$BC$ | $2$ | $2$ | $2$ | $2$ | $4$ | $4$ | $1$
$ABCD^{\mbox{a}}$00footnotemark: 0 | $2$ | $2$ | $2$ | $2$ | $\geq 2$ | $\geq 2$ | $\geq 2$
00footnotetext: ${}^{\mbox{a}}$ $ABCD$ can be further partitioned under SLOCC
in terms of the ranks of $C_{AB}$, $C_{AC}$ and $C_{AD}$.
Table 3: Ranks of coefficient matrices of five-qubit pure states.
Families Ranks of | $C_{\alpha}$ | $C_{\beta\gamma(\beta\neq\gamma)}$
---|---|---
$i$–$j$–$k$–$\ell$–$m$ | 1${}^{\mbox{b}}$ | 1${}^{\mbox{c}}$
$i$–$j$–$k$–$\ell m$ | 1, if $\alpha=i,j,k$ | 1, if $\beta,\gamma=i,j,k$
| 2, otherwise | or $\beta,\gamma=\ell,m$
| | 2, otherwise
$i$–$jk$–$\ell m$ | 1, if $\alpha=i$ | 1, if $\beta,\gamma=j,k$
| 2, otherwise | or $\beta,\gamma=\ell,m$
| | 2, if $\beta=i$ or $\gamma=i$
| | 4, otherwise
$i$–$j$–$k\ell m$ | 1, if $\alpha=i$ or $j$ | 1, if $\beta,\gamma=i,j$
| 2, otherwise | 2, otherwise
$i$–$jk\ell m$ | 1, if $\alpha=i$ | 2, if $\beta=i$ or $\gamma=i$
| 2, otherwise | 2, 3, or 4, otherwise
$ij$–$k\ell m$ | 2${}^{\mbox{b}}$ | 1, if $\beta,\gamma=i,j$
| | 2, if $\beta,\gamma=k,\ell,m$
| | 4, otherwise
$ijk\ell m$ | 2${}^{\mbox{b}}$ | 2, 3, or 4${}^{\mbox{c}}$
00footnotetext: ${}^{\mbox{a}}$ $\\{i,j,k,\ell,m\\}$ is any permutation of
$\\{A,B,C,D,E\\}$.
${}^{\mbox{b}}$ $\alpha=i,j,k,\ell,m$.
${}^{\mbox{c}}$ $\beta,\gamma=i,j,k,\ell,m$.
## IV IV. Degenerate families of general $N$ qubits are SLOCC inequivalent to
one another
The recursive formula above further gives rise to a criterion for
biseparability of an $n$-qubit pure state. Indeed, we note that Eq. (III)
holds particularly true for
$\\{q_{1},\cdots,q_{\ell}\\}=\\{j_{1},\cdots,j_{k}\\}$. In this case, the
coefficient matrices $C_{q_{1}^{\ast}\cdots q_{s}^{\ast}}$ and
$C_{q_{1}^{\prime}\cdots q_{t}^{\prime}}$ reduce to a column vector and a row
vector respectively, and therefore both of them have rank 1. It follows that
$\mbox{rank}(C_{q_{1}\cdots q_{\ell}}(|\phi\rangle_{q_{1}\cdots
q_{\ell}}\otimes|\varphi\rangle_{q_{\ell+1}\cdots q_{n}}))=1$. Conversely, if
$\mbox{rank}(C_{q_{1}\cdots q_{\ell}}(|\psi\rangle_{1\cdots n}))=1$ for an
$n$-qubit pure state $|\psi\rangle_{1\cdots n}$, then $|\psi\rangle_{1\cdots
n}$ is biseparable, being of the form $|\psi\rangle_{1\cdots
n}=|\phi\rangle_{q_{1}\cdots q_{\ell}}\otimes|\varphi\rangle_{q_{\ell+1}\cdots
q_{n}}$. This can be seen as follows. For simplicity, we assume $q_{i}=i$ with
$i=1,\cdots,n$. If $\mbox{rank}(C_{12\cdots\ell}(|\psi\rangle_{1\cdots
n}))=1$, then all columns of $C_{12\cdots\ell}$ are proportional to each other
and each column can be written into the form
$(a_{0}b_{j},a_{1}b_{j},\cdots,a_{2^{\ell}-1}b_{j})^{T}$. Hence,
$|\psi\rangle_{1\cdots n}$ can be written as $|\psi\rangle_{1\cdots
n}=|\phi\rangle_{1\cdots\ell}\otimes|\varphi\rangle_{(\ell+1)\cdots n}$ with
$|\phi\rangle_{1\cdots\ell}=\sum_{i=0}^{2^{\ell}-1}a_{i}|i\rangle_{1\cdots\ell}$
and $|\varphi\rangle_{(\ell+1)\cdots
n}=\sum_{j=0}^{2^{n-\ell}-1}b_{j}|j\rangle_{(\ell+1)\cdots n}$. This leads to
the following biseparability criterion for $n$-qubit pure states.
Biseparability criterion for $n$-qubit pure states. For any coefficient matrix
$C_{q_{1}\cdots q_{\ell}}$ associated with an $n$-qubit pure state
$|\psi\rangle_{1\cdots n}$, $\mbox{rank}(C_{q_{1}\cdots
q_{\ell}}(|\psi\rangle_{1\cdots n}))=1$ if and only if $|\psi\rangle$ is
biseparable, being of the form $|\psi\rangle_{1\cdots
n}=|\phi\rangle_{q_{1}\cdots q_{\ell}}\otimes|\varphi\rangle_{q_{\ell+1}\cdots
q_{n}}$ (see also Iwai ; Huang2 ).
Invoking the fact that an $n$-qubit pure state is entangled if it is not full
separable, we have the following criterion to identify $n$-qubit entangled
(respectively, genuinely entangled) pure states: An $n$-qubit pure state is
entangled (respectively, genuinely entangled) if and only if the rank of at
least one of its coefficient matrices is (respectively, the ranks of its all
coefficient matrices are) greater than 1.
Note that all the above criteria can be rephrased in terms of the ranks of
$\ell$-qubit reduced density matrices obtained by tracing out $n-\ell$ qubits
Chong or the number of nonzero singular values of coefficient matrices.
Theorem 1 together with the biseparability criterion above yield the following
theorem.
Theorem 3. Degenerate families of general $n$ qubits are inequivalent to one
another under SLOCC and they can be distinguished in terms of the ranks of
coefficient matrices (or in terms of the ranks of $\ell$-qubit reduced density
matrices obtained by tracing out $n-\ell$ qubits).
The validity of Theorem 3 can be seen as follows. Given an $n$-qubit pure
state, a partition $P$ of the $n$ particles is a collection of disjoint sets
in such a way that the particles within any one set are entangled and any two
particles from different sets are not entangled. Suppose $F_{1}$ and $F_{2}$
are two different degenerate families with partitions $P_{1}$ and $P_{2}$
respectively. Without loss of generality, we assume that there exists a set
$S$ such that $S\in P_{1}$ and $S\not\in P_{2}$. Then the states in $F_{1}$
can be written in the biseparable form
$|\phi\rangle_{S}|\varphi\rangle_{\bar{S}}$, where ${\bar{S}}$ is the set of
all particles except those in $S$. According to the biseparability criterion
above, $\mbox{rank}(C_{S})=1$ for states in $F_{1}$. Since the states in
$F_{2}$ cannot be written in the above biseparable form,
$\mbox{rank}(C_{S})>1$ for states in $F_{2}$. In light of Theorem 1, the two
degenerate families are inequivalent to each other under SLOCC.
In addition, we remark that degenerate families of general $n$ qubits can also
be distinguished from one another under SLOCC in terms of the ranks of
$\ell$-qubit reduced density matrices obtained by tracing out $n-\ell$ qubits
or the number of nonzero singular values of coefficient matrices.
## V V. SLOCC classification of four qubits via the ranks of coefficient
matrices
Suppose that the states $|\psi\rangle$ and $|\psi^{\prime}\rangle$ of four
qubits are SLOCC equivalent to each other, then there are local invertible
operators $\mathcal{A}_{1}$, $\mathcal{A}_{2}$, $\mathcal{A}_{3}$, and
$\mathcal{A}_{4}$ such that Dur
$|\psi^{\prime}\rangle=\mathcal{A}_{1}\otimes\mathcal{A}_{2}\otimes\mathcal{A}_{3}\otimes\mathcal{A}_{4}|\psi\rangle.$
(8)
For a four-qubit state $|\psi\rangle=\sum_{i=0}^{15}a_{i}|i\rangle$, we
consider three coefficient matrices $C_{AB}$, $C_{AC}$, and $C_{AD}$ as
follows:
$\displaystyle C_{AB}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}a_{0}&a_{1}&a_{2}&a_{3}\\\
a_{4}&a_{5}&a_{6}&a_{7}\\\ a_{8}&a_{9}&a_{10}&a_{11}\\\
a_{12}&a_{13}&a_{14}&a_{15}\end{array}\right),$ (13) $\displaystyle C_{AC}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}a_{0}&a_{1}&a_{4}&a_{5}\\\
a_{2}&a_{3}&a_{6}&a_{7}\\\ a_{8}&a_{9}&a_{12}&a_{13}\\\
a_{10}&a_{11}&a_{14}&a_{15}\end{array}\right),$ (18) $\displaystyle C_{AD}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}a_{0}&a_{4}&a_{2}&a_{6}\\\
a_{1}&a_{5}&a_{3}&a_{7}\\\ a_{8}&a_{12}&a_{10}&a_{14}\\\
a_{9}&a_{13}&a_{11}&a_{15}\end{array}\right).$ (23)
The coefficient matrices above satisfy the following equations:
$\displaystyle C_{AB}(|\psi^{\prime}\rangle)$ $\displaystyle=$
$\displaystyle\mathcal{A}_{1}\otimes\mathcal{A}_{2}C_{AB}(|\psi\rangle)(\mathcal{A}_{3}\otimes\mathcal{A}_{4})^{T},$
(24) $\displaystyle C_{AC}(|\psi^{\prime}\rangle)$ $\displaystyle=$
$\displaystyle\mathcal{A}_{1}\otimes\mathcal{A}_{3}C_{AC}(|\psi\rangle)(\mathcal{A}_{2}\otimes\mathcal{A}_{4})^{T},$
(25) $\displaystyle C_{AD}(|\psi^{\prime}\rangle)$ $\displaystyle=$
$\displaystyle\mathcal{A}_{1}\otimes\mathcal{A}_{4}C_{AD}(|\psi\rangle)(\mathcal{A}_{3}\otimes\mathcal{A}_{2})^{T}.$
(26)
It follows from Eqs. (24)-(26) that if two four-qubit states are SLOCC
equivalent then their coefficient matrices $C_{AB}$ (and also $C_{AC}$ and
$C_{AD}$) have the same rank. Conversely, if one of the coefficient matrices
$C_{AB}$, $C_{AC}$, and $C_{AD}$ differ in the ranks, then the two four-qubit
states are SLOCC inequivalent. Let family $F_{r_{AB}}^{C_{AB}}$ be the set of
all four-qubit states with the same rank $r_{AB}$ of the coefficient matrix
$C_{AB}$. Here $r_{AB}$ ranges over the values 1, 2, 3, and 4. Clearly, each
one of the nine families introduced by Verstraete _et al._ Verstraete can be
further divided into four SLOCC inequivalent subfamilies corresponding to the
four possible values of $r_{AB}$. In a similar manner, we can define the
families $F_{r_{AC}}^{C_{AC}}$ and $F_{r_{AD}}^{C_{AD}}$. One can obtain a
more refined partition by further dividing the families $F_{r_{AB}}^{C_{AB}}$,
$F_{r_{AC}}^{C_{AC}}$, and $F_{r_{AD}}^{C_{AD}}$ into subfamilies
$F_{r_{AB}r_{AC}r_{AD}}^{C_{AB}C_{AC}C_{AD}}=F_{r_{AB}}^{C_{AB}}\cap
F_{r_{AC}}^{C_{AC}}\cap F_{r_{AD}}^{C_{AD}}$. Clearly, the subfamilies
$F_{r_{AB}r_{AC}r_{AD}}^{C_{AB}C_{AC}C_{AD}}$ and
$F_{r_{AB}^{\prime}r_{AC}^{\prime}r_{AD}^{\prime}}^{C_{AB}C_{AC}C_{AD}}$ are
SLOCC inequivalent when $r_{AB}r_{AC}r_{AD}\neq
r_{AB}^{\prime}r_{AC}^{\prime}r_{AD}^{\prime}$.
We now further partition the nine families introduced by Verstraete _et al._
Verstraete into SLOCC inequivalent subfamilies via the rank of coefficient
matrix. For convenience, we rewrite the families $G_{abcd}$ and $L_{abc_{2}}$
as:
$\displaystyle G_{abcd}$ $\displaystyle=$
$\displaystyle\alpha(|0\rangle+|15\rangle)+\beta(|3\rangle+|12\rangle)+\gamma(|5\rangle+|10\rangle)$
(27) $\displaystyle+\delta(|6\rangle+|9\rangle),$ $\displaystyle L_{abc_{2}}$
$\displaystyle=$
$\displaystyle\alpha^{\prime}(|0\rangle+|15\rangle)+\beta^{\prime}(|3\rangle+|12\rangle)+\gamma^{\prime}(|5\rangle+|10\rangle)$
(28) $\displaystyle+|6\rangle.$
In Table 4, we show the subfamilies $F_{r_{AB}}^{C_{AB}}$,
$F_{r_{AC}}^{C_{AC}}$, and $F_{r_{AD}}^{C_{AD}}$ of $G_{abcd}$. As illustrated
in Table 5, $G_{abcd}$ can be further partitioned into nine genuinely
entangled subfamilies and three biseparable subfamilies (marked with “*”) via
$r_{AB}$, $r_{AC}$, and $r_{AD}$ (subfamilies not listed in the table are
empty). For simplicity, the detailed descriptions of the subfamilies are not
shown as they can be easily obtained by taking the intersections of the
corresponding descriptions in Table 4. Tables 6 and 7 illustrate the
partitions of the other eight families introduced by Verstraete _et al._ into
inequivalent subfamilies. In total, we find 28 genuinely entangled subfamilies
and all the degenerate classes can be distinguished up to permutations of the
four qubits (i.e., $A$-$B$-$C$-$D$, $A$-$B$-$CD$, $AB$-$CD$,
$|0\rangle_{A}|W\rangle_{BCD}$, and $|0\rangle_{A}|\mbox{GHZ}\rangle_{BCD}$).
Table 4: The subfamilies $F_{r_{AB}}^{C_{AB}}$, $F_{r_{AC}}^{C_{AC}}$, and
$F_{r_{AD}}^{C_{AD}}$ of $G_{abcd}$.
. Subfamily Description $F_{1}^{C_{AB}}$ $\alpha=\beta=0\ \&\
\gamma=\pm\delta\neq 0\ |\ \alpha=\pm\beta\neq 0\ \&\ \gamma=\delta=0$
$F_{2}^{C_{AB}}$ $\alpha=\beta=0\ \&\ \gamma\neq\pm\delta\ |\ \gamma=\delta=0\
\&\ \alpha\neq\pm\beta\ |\ \alpha=\pm\beta\neq 0\ \&\ \gamma=\pm\delta\neq 0$
$F_{3}^{C_{AB}}$ $\alpha=\pm\beta\neq 0\ \&\ \gamma\neq\pm\delta\ |\
\gamma=\pm\delta\neq 0\ \&\ \alpha\neq\pm\beta$ $F_{4}^{C_{AB}}$
$\alpha\neq\pm\beta\ \&\ \gamma\neq\pm\delta$ $F_{1}^{C_{AC}}$
$\alpha=\gamma=0\ \&\ \beta=\pm\delta\neq 0\ |\ \alpha=\pm\gamma\neq 0\ \&\
\beta=\delta=0$ $F_{2}^{C_{AC}}$ $\alpha=\gamma=0\ \&\ \beta\neq\pm\delta\ |\
\beta=\delta=0\ \&\ \alpha\neq\pm\gamma\ |\ \alpha=\pm\gamma\neq 0\ \&\
\beta=\pm\delta\neq 0$ $F_{3}^{C_{AC}}$ $\alpha=\pm\gamma\neq 0\ \&\
\beta\neq\pm\delta\ |\ \beta=\pm\delta\neq 0\ \&\ \alpha\neq\pm\gamma$
$F_{4}^{C_{AC}}$ $\alpha\neq\pm\gamma\ \&\ \beta\neq\pm\delta$
$F_{1}^{C_{AD}}$ $\alpha=\delta=0\ \&\ \beta=\pm\gamma\neq 0\ |\
\alpha=\pm\delta\neq 0\ \&\ \beta=\gamma=0$ $F_{2}^{C_{AD}}$ $\alpha=\delta=0\
\&\ \beta\neq\pm\gamma\ |\ \beta=\gamma=0\ \&\ \alpha\neq\pm\delta\ |\
\alpha=\pm\delta\neq 0\ \&\ \beta=\pm\gamma\neq 0$ $F_{3}^{C_{AD}}$
$\alpha=\pm\delta\neq 0\ \&\ \beta\neq\pm\gamma\ |\ \beta=\pm\gamma\neq 0\ \&\
\alpha\neq\pm\delta$ $F_{4}^{C_{AD}}$ $\alpha\neq\pm\delta\ \&\
\beta\neq\pm\gamma$
Table 5: SLOCC classification of $G_{abcd}$ via $r_{AB}$, $r_{AC}$, and
$r_{AD}$. The subfamilies marked with “*” are biseparable.
. $r_{AB}$ $r_{AC}$ $r_{AD}$ Subfamily description 222 $F_{2}^{C_{AB}}\cap
F_{2}^{C_{AC}}\cap F_{2}^{C_{AD}}$ 244 $F_{2}^{C_{AB}}\cap F_{4}^{C_{AC}}\cap
F_{4}^{C_{AD}}$ 333 $F_{3}^{C_{AB}}\cap F_{3}^{C_{AC}}\cap F_{3}^{C_{AD}}$ 344
$F_{3}^{C_{AB}}\cap F_{4}^{C_{AC}}\cap F_{4}^{C_{AD}}$ 424 $F_{4}^{C_{AB}}\cap
F_{2}^{C_{AC}}\cap F_{4}^{C_{AD}}$ 434 $F_{4}^{C_{AB}}\cap F_{3}^{C_{AC}}\cap
F_{4}^{C_{AD}}$ 442 $F_{4}^{C_{AB}}\cap F_{4}^{C_{AC}}\cap F_{2}^{C_{AD}}$ 443
$F_{4}^{C_{AB}}\cap F_{4}^{C_{AC}}\cap F_{3}^{C_{AD}}$ 444 $F_{4}^{C_{AB}}\cap
F_{4}^{C_{AC}}\cap F_{4}^{C_{AD}}$ 144∗ $F_{1}^{C_{AB}}$ (i.e., $AB$-$CD$)
414∗ $F_{1}^{C_{AC}}$ (i.e., $AC$-$BD$) 441∗ $F_{1}^{C_{AD}}$ (i.e.,
$AD$-$BC$)
Table 6: SLOCC classification of $L_{abc_{2}}$ via $r_{AB}$, $r_{AC}$, and
$r_{AD}$. The subfamilies marked with “*” are biseparable.
. $r_{AB}$ $r_{AC}$ $r_{AD}$ Subfamily description 233
$\alpha^{\prime}=\beta^{\prime}=0\ \&\ \gamma^{\prime}\neq 0$ 244
$\alpha^{\prime}=\pm\beta^{\prime}\neq 0\ \&\ \gamma^{\prime}=0$ 323
$\alpha^{\prime}=\gamma^{\prime}=0\ \&\ \beta^{\prime}\neq 0$ 332
$\alpha^{\prime}\neq 0\ \&\ \beta^{\prime}=\gamma^{\prime}=0$ 333
$\alpha^{\prime}=\pm\beta^{\prime}=\pm\gamma^{\prime}\neq 0$ 344
$\gamma^{\prime}=0\ \&\ \alpha^{\prime}\beta^{\prime}\neq 0\ \&\
\alpha^{\prime}\neq\pm\beta^{\prime}\ |\ \gamma^{\prime}\neq 0\ \&\
\alpha^{\prime}=\pm\beta^{\prime}\neq 0\ \&\
\alpha^{\prime}\neq\pm\gamma^{\prime}$ 424 $\beta^{\prime}=0\ \&\
\alpha^{\prime}=\pm\gamma^{\prime}\neq 0$ 434 $\beta^{\prime}=0\ \&\
\alpha^{\prime}\gamma^{\prime}\neq 0\ \&\
\alpha^{\prime}\neq\pm\gamma^{\prime}\ |\ \beta^{\prime}\neq 0\ \&\
\alpha^{\prime}=\pm\gamma^{\prime}\neq 0\ \&\
\alpha^{\prime}\neq\pm\beta^{\prime}$ 442 $\alpha^{\prime}=0\ \&\
\beta^{\prime}=\pm\gamma^{\prime}\neq 0$ 443 $\alpha^{\prime}=0\ \&\
\beta^{\prime}\neq\pm\gamma^{\prime}\ \&\ \beta^{\prime}\gamma^{\prime}\neq 0\
|\ \alpha^{\prime}\neq 0\ \&\ \beta^{\prime}=\pm\gamma^{\prime}\neq 0\ \&\
\alpha^{\prime}\neq\pm\beta^{\prime}$ 444 $\gamma^{\prime}\neq 0\ \&\
\alpha^{\prime}\neq\pm\beta^{\prime}\ \&\ \beta^{\prime}\neq 0\ \&\
\alpha^{\prime}\neq\pm\gamma^{\prime}\ \&\ \alpha^{\prime}\neq 0\ \&\
\beta^{\prime}\neq\pm\gamma^{\prime}$ 111∗
$\alpha^{\prime}=\beta^{\prime}=\gamma^{\prime}=0$, (i.e., $A$-$B$-$C$-$D$)
Table 7: SLOCC classifications of $L_{ab_{3}}$, $L_{a_{2}b_{2}}$, $L_{a_{4}}$,
$L_{a_{2}0_{3\oplus{\bar{1}}}}$, $L_{0_{5\oplus{\bar{3}}}}$,
$L_{0_{7\oplus{\bar{1}}}}$, and $L_{0_{3\oplus{\bar{1}}}0_{3\oplus{\bar{1}}}}$
via $r_{AB}$, $r_{AC}$, and $r_{AD}$. The subfamilies marked with “*” are
biseparable.
. Family $r_{AB}$ $r_{AC}$ $r_{AD}$ Subfamily description Family $r_{AB}$
$r_{AC}$ $r_{AD}$ Subfamily description $L_{a_{2}b_{2}}$ 333 $ab=0\ \&\ a\neq
b$ $L_{ab_{3}}$ 222 $a=b=0$ (i.e., $|W\rangle_{ABCD}$) 424 $a=\pm b\neq 0$ 344
$ab=0\ \&\ a\neq b$ 434 $ab\neq 0\ \&\ a\neq\pm b$ 424 $a=b\neq 0$ 212∗
$a=b=0$ (i.e., $A$-$C$-$BD$) 434 $b=-3a\neq 0$ $L_{a_{4}}$ 323
$L_{a_{4}}(a=0)$ 442 $a=-b\neq 0$ 434 $L_{a_{4}}(a\neq 0)$ 443 $b=3a\neq 0$
$L_{a_{2}0_{3\oplus{\bar{1}}}}$ 333 $L_{a_{2}0_{3\oplus{\bar{1}}}}(a\neq 0)$
444 $ab\neq 0\ \&\ b\neq\pm a\ \&\ b\neq\pm 3a$ 222∗ $a=0$ (i.e.,
$|0\rangle_{A}|W\rangle_{BCD}$) $L_{0_{5\oplus{\bar{3}}}}$ 333
$L_{0_{5\oplus{\bar{3}}}}$ $L_{0_{7\oplus{\bar{1}}}}$ 333
$L_{0_{7\oplus{\bar{1}}}}$ $L_{0_{3\oplus{\bar{1}}}0_{3\oplus{\bar{1}}}}$ 222∗
$|0\rangle_{A}|\mbox{GHZ}\rangle_{BCD}$
## VI VI. Discussion of the completeness of the nine families obtained by
Verstraete _et al._
The family $L_{ab_{3}}$ in Ref. Verstraete was defined as
$\displaystyle L_{ab_{3}}$ $\displaystyle=$ $\displaystyle
a(|0000\rangle+|1111\rangle)+\frac{a+b}{2}(|0101\rangle+|1010\rangle)$ (29)
$\displaystyle+\frac{a-b}{2}(|0110\rangle+|1001\rangle)$
$\displaystyle+\frac{i}{\sqrt{2}}(|0001\rangle+|0010\rangle+|0111\rangle+|1011\rangle).$
In later work, Chterental _et al._ Verstraete obtained nine SLOCC
inequivalent families of four qubits using invariant theory. Let
$L_{ab_{3}}^{\prime}$ be defined by
$\displaystyle L_{ab_{3}}^{\prime}$ $\displaystyle=$ $\displaystyle
a(|0000\rangle+|1111\rangle)+\frac{a+b}{2}(|0101\rangle+|1010\rangle)$ (30)
$\displaystyle+\frac{a-b}{2}(|0110\rangle+|1001\rangle)$
$\displaystyle+\frac{i}{\sqrt{2}}(|0001\rangle+|0010\rangle-|0111\rangle-|1011\rangle),$
that is, $L_{ab_{3}}^{\prime}$ is obtained by replacing the two “+” signs of
the last two terms in the formula of $L_{ab_{3}}$ by “-” signs Chterental . It
is claimed that there is a perfect correspondence between the nine families
obtained by Verstraete _et al._ (with $L_{ab_{3}}$ replaced by
$L_{ab_{3}}^{\prime}$) and the nine families obtained by Chterental _et al._
Chterental . Note that the formula of $L_{ab_{3}}^{\prime}$ has also been
adopted in Ref. Borsten . Since both Verstraete _et al._ and Chterental _et
al._ claimed that the nine families obtained in their work are inequivalent to
each other, a detailed study of the relation between $L_{ab_{3}}$ and
$L_{ab_{3}}^{\prime}$ can provide insights into the completeness of their
classifications.
### VI.1 A. $L_{ab_{3}}(a=0)$ is SLOCC equivalent to
$L_{ab_{3}}^{\prime}(a=0)$
It is readily verified that the following equation holds between
$L_{ab_{3}}^{\prime}(a=0)$ and $L_{ab_{3}}(a=0)$:
$L_{ab_{3}}^{\prime}(a=0)=I\otimes I\otimes i\sigma_{z}\otimes
i\sigma_{z}L_{ab_{3}}(a=0),$ (31)
where $I$ is the identity and $\sigma_{z}=\mbox{diag}\\{1,-1\\}$.
It follows from Eq. (31) that $L_{ab_{3}}(a=0)$ and $L_{ab_{3}}^{\prime}(a=0)$
are SLOCC equivalent. In particular, setting $b=0$ yields that the states
$\frac{i}{\sqrt{2}}(|0001\rangle+|0010\rangle-|0111\rangle-|1011\rangle)$ and
$\frac{i}{\sqrt{2}}(|0001\rangle+|0010\rangle+|0111\rangle+|1011\rangle)$ are
equivalent under SLOCC.
Table 8: SLOCC classification of $L_{ab_{3}}^{\prime}$ via $r_{AB}$, $r_{AC}$,
and $r_{AD}$.
. $r_{AB}$ $r_{AC}$ $r_{AD}$ Subfamily description 222 $a=b=0$ (i.e.,
$|W\rangle_{ABCD}$) 344 $ab=0\ \&\ a\neq b$ 424 $\emptyset$ 434 $a=b\neq 0\ |\
b=-3a\neq 0$ 442 $\emptyset$ 443 $a=-b\neq 0\ |\ b=3a\neq 0$ 444 $ab\neq 0\
\&\ b\neq\pm a\ \&\ b\neq\pm 3a$
### VI.2 B. $L_{ab_{3}}^{\prime}(a\neq 0)$ [respectively, $L_{ab_{3}}(a\neq
0)$] is SLOCC inequivalent to $L_{ab_{3}}$ (respectively,
$L_{ab_{3}}^{\prime}$)
We first show that the family $L_{ab_{3}}^{\prime}(a\neq 0)$ is SLOCC
inequivalent to the family $L_{ab_{3}}$. In Table 8 we show the partition of
$L_{ab_{3}}^{\prime}$ into SLOCC inequivalent subfamilies via $r_{AB}$,
$r_{AC}$, and $r_{AD}$. Consulting Tables 7 and 8, and using the fact that the
subfamilies with different ranks of coefficient matrices are SLOCC
inequivalent to each other, it suffices to consider the following six cases.
Case 1. $L_{ab_{3}}^{\prime}(a=b\neq 0)$ is SLOCC inequivalent to
$L_{ab_{3}}(b=-3a\neq 0)$.
In this case, we can resort to $D_{xy}$, a degree 6 polynomial invariant of
four qubits Luque (see the Appendix for the expression of $D_{xy}$). Indeed,
it can be verified that if $|\psi\rangle$ and $|\psi^{\prime}\rangle$ are any
two SLOCC equivalent states, that is, they satisfy Eq. (2), then the following
equation holds:
$D_{xy}(|\psi^{\prime}\rangle)=D_{xy}(|\psi\rangle)\big{[}\Pi_{i=1}^{4}\det\mathcal{A}_{i}\bigr{]}^{3}.$
(32)
It follows from Eq. (32) that for any two SLOCC equivalent states
$|\psi\rangle$ and $|\psi^{\prime}\rangle$, either
$D_{xy}(|\psi^{\prime}\rangle)$ and $D_{xy}(|\psi\rangle)$ both vanish or
neither vanishes.
A direct calculation shows that
$D_{xy}=-\frac{1}{32}\left(a-b\right)^{3}\left(a+b\right)^{3}$ (33)
for both $L_{ab_{3}}$ and $L_{ab_{3}}^{\prime}$. The desired result then
follows by noting that $D_{xy}=16a^{6}\neq 0$ for $L_{ab_{3}}(b=-3a\neq 0)$
whereas $D_{xy}=0$ for $L_{ab_{3}}^{\prime}(a=b\neq 0)$.
Case 2. $L_{ab_{3}}^{\prime}(a=-b\neq 0)$ is SLOCC inequivalent to
$L_{ab_{3}}(b=3a\neq 0)$.
This case can be dealt with similarly as case 1 by noting that
$D_{xy}=16a^{6}\neq 0$ for $L_{ab_{3}}(b=3a\neq 0)$ whereas $D_{xy}=0$ for
$L_{ab_{3}}^{\prime}(a=-b\neq 0)$.
Case 3. $L_{ab_{3}}^{\prime}(b=-3a\neq 0)$ is SLOCC inequivalent to
$L_{ab_{3}}(b=-3a\neq 0)$.
In this case, the semi-invariants defined in Ref. LDF07b turn out to be
useful. More specifically, for any four-qubit state
$|\psi\rangle=\sum_{i=0}^{15}c_{i}|i\rangle$, the semi-invariants $F_{1}$ and
$F_{2}$ are defined in Ref. LDF07b as
$\displaystyle F_{1}(\psi)$ $\displaystyle=$
$\displaystyle(c_{0}c_{7}-c_{2}c_{5}+c_{1}c_{6}-c_{3}c_{4})^{2}$ (34)
$\displaystyle-4(c_{2}c_{4}-c_{0}c_{6})(c_{3}c_{5}-c_{1}c_{7}),$
$\displaystyle F_{2}(\psi)$ $\displaystyle=$
$\displaystyle(c_{8}c_{15}-c_{11}c_{12}+c_{9}c_{14}-c_{10}c_{13})^{2}$ (35)
$\displaystyle-4(c_{11}c_{13}-c_{9}c_{15})(c_{10}c_{12}-c_{8}c_{14}).$
Let $|\phi\rangle$ be any four-qubit state SLOCC equivalent to $L_{ab_{3}}$
[i.e., they satisfy Eq. (2)]. Let
$\mathcal{A}_{1}=\left(\begin{array}[]{cc}\alpha_{1}&\alpha_{2}\\\
\alpha_{3}&\alpha_{4}\end{array}\right).$ (36)
A tedious but straightforward calculation yields
$\displaystyle F_{1}(\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(a^{2}-b^{2})\alpha_{1}^{4}\biggl{[}\prod_{i=2}^{4}\det\mathcal{A}_{i}\biggr{]}^{2},$
(37) $\displaystyle F_{2}(\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(a^{2}-b^{2})\alpha_{3}^{4}\biggl{[}\prod_{i=2}^{4}\det\mathcal{A}_{i}\biggr{]}^{2}.$
(38)
In view of Eqs. (37) and (38) and the fact that $\mathcal{A}_{1}$ is
invertible, it follows at once that if $|\phi\rangle$ is SLOCC equivalent to
$L_{ab_{3}}(a\neq\pm b)$, then the following equation holds:
$\left|F_{1}(\phi)\right|+\left|F_{2}(\phi)\right|\neq 0.$ (39)
Let $|\varphi\rangle$ be any state SLOCC equivalent to $L_{ab_{3}}^{\prime}$
[i.e., they satisfy Eq. (2)]. Again, a tedious but straightforward calculation
yields
$\displaystyle F_{1}(\varphi)$ $\displaystyle=$
$\displaystyle\frac{-1}{2\sqrt{2}}i\alpha_{1}^{3}\bigl{(}-i\sqrt{2}(3a^{2}+b^{2})\alpha_{1}+8a(a^{2}-b^{2})\alpha_{2}\bigr{)}$
(40)
$\displaystyle\times\biggl{[}\prod_{i=2}^{4}\det\mathcal{A}_{i}\biggr{]}^{2},$
$\displaystyle F_{2}(\varphi)$ $\displaystyle=$
$\displaystyle\frac{-1}{2\sqrt{2}}i\alpha_{3}^{3}\bigl{(}-i\sqrt{2}(3a^{2}+b^{2})\alpha_{3}+8a(a^{2}-b^{2})\alpha_{4}\bigr{)}$
(41)
$\displaystyle\times\biggl{[}\prod_{i=2}^{4}\det\mathcal{A}_{i}\biggr{]}^{2}.$
When $a(a^{2}-b^{2})\neq 0$, consider the operator
$\mathcal{A}_{1}^{\ast}=\left(\begin{array}[]{cc}\alpha_{1}&\frac{i\sqrt{2}(3a^{2}+b^{2})}{8a(a^{2}-b^{2})}\alpha_{1}\\\
0&\alpha_{4}\end{array}\right),$ (42)
where $\alpha_{1}\alpha_{4}\neq 0$. Clearly, $\mathcal{A}_{1}^{\ast}$ is
invertible. In view of Eqs. (40)-(42), it follows that there exists a state
$|\varphi^{\ast}\rangle$ equivalent to $L_{ab_{3}}^{\prime}(a(a^{2}-b^{2})\neq
0)$ under local invertible operators $\mathcal{A}_{1}^{\ast}$,
$\mathcal{A}_{2}$, $\mathcal{A}_{3}$, and $\mathcal{A}_{4}$, such that
$\left|F_{1}(\varphi^{\ast})\right|+\left|F_{2}(\varphi^{\ast})\right|=0.$
(43)
From Eqs. (39) and (43), $|\varphi^{\ast}\rangle$ is SLOCC inequivalent to the
state $L_{ab_{3}}(a\neq\pm b)$. Therefore,
$L_{ab_{3}}^{\prime}(a(a^{2}-b^{2})\neq 0)$ is SLOCC inequivalent to
$L_{ab_{3}}(a\neq\pm b)$. In particular, $L_{ab_{3}}^{\prime}(b=-3a\neq 0)$ is
SLOCC inequivalent to $L_{ab_{3}}(b=-3a\neq 0)$.
Case 4. $L_{ab_{3}}^{\prime}(b=3a\neq 0)$ is SLOCC inequivalent to
$L_{ab_{3}}(b=3a\neq 0)$.
This case can be treated analogously to case 3.
Case 5. $L_{ab_{3}}^{\prime}(a\neq 0\ \&\ b=0)$ is SLOCC inequivalent to
$L_{ab_{3}}(ab=0\ \&\ a\neq b)$.
In Ref. LDFQIC09 , we proved that $L_{ab_{3}}(a=0\ \&\ b\neq 0)$ and
$L_{ab_{3}}(a\neq 0\ \&\ b=0)$ are SLOCC inequivalent. A proof analogous to
that of Ref. LDFQIC09 shows that $L_{ab_{3}}^{\prime}(a=0\ \&\ b\neq 0)$ and
$L_{ab_{3}}^{\prime}(a\neq 0\ \&\ b=0)$ are SLOCC inequivalent. Using the fact
that $L_{ab_{3}}(a=0\ \&\ b\neq 0)$ is SLOCC equivalent to
$L_{ab_{3}}^{\prime}(a=0\ \&\ b\neq 0)$ [see Eq. (31)] yields that
$L_{ab_{3}}^{\prime}(a\neq 0\ \&\ b=0)$ is SLOCC inequivalent to
$L_{ab_{3}}(a=0\ \&\ b\neq 0)$. Furthermore, an argument analogous to case 3
shows that $L_{ab_{3}}^{\prime}(a\neq 0\ \&\ b=0)$ is inequivalent to
$L_{ab_{3}}(a\neq 0\ \&\ b=0)$.
Indeed, we can further conclude that $L_{ab_{3}}(a=0)$ and $L_{ab_{3}}(a\neq
0)$ are SLOCC inequivalent and $L_{ab_{3}}^{\prime}(a=0)$ and
$L_{ab_{3}}^{\prime}(a\neq 0)$ are SLOCC inequivalent.
Case 6. $L_{ab_{3}}^{\prime}(ab\neq 0\ \&\ a\neq\pm b\ \&\ b\neq\pm 3a)$ is
SLOCC inequivalent to $L_{ab_{3}}(ab\neq 0\ \&\ a\neq\pm b\ \&\ b\neq\pm 3a)$.
This case can be treated analogously to case 3.
As a consequence, $L_{ab_{3}}^{\prime}(a\neq 0)$ is SLOCC inequivalent to
$L_{ab_{3}}$. An analogous argument shows that $L_{ab_{3}}(a\neq 0)$ is SLOCC
inequivalent to $L_{ab_{3}}^{\prime}$.
### VI.3 C. The relation between $L_{ab_{3}}^{\prime}\ $and $L_{ab_{3}}$
under permutations
Let $|\gamma\rangle$ be the state of the subfamily $L_{ab_{3}}^{\prime}(a\neq
0\ \&\ b=0)$, $|\eta\rangle$ be the state of the subfamily
$L_{ab_{3}}^{\prime}(b=3a\neq 0)$, $|\vartheta\rangle$ be the state of the
subfamily $L_{ab_{3}}^{\prime}(b=-3a\neq 0)$, and $|\nu\rangle$ be the state
of the subfamily $L_{ab_{3}}^{\prime}(ab\neq 0\ \&\ a\neq\pm b\ \&\ b\neq\pm
3a)$. We argue that the above four subfamilies are SLOCC inequivalent to
$L_{ab_{3}}$ under any permutation of qubits. This can be seen as follows. Let
$(i,j)$ be the transposition of qubits $i$ and $j$. A tedious calculation
shows that the permutations giving rise to different $|\gamma\rangle$ are
$\kappa_{1}=I$, $\kappa_{2}=(1,3)$, $\kappa_{3}=(1,4)$,
$\kappa_{4}=(1,2)(1,3)$, $\kappa_{5}=(1,2)(1,4)$, and
$\kappa_{6}=(1,4)(1,2)(1,3)$. Similarly, the permutations giving rise to
different $|\eta\rangle$, $|\vartheta\rangle$, and $|\nu\rangle$ are
$\pi_{1}=I$, $\pi_{2}=(1,2)$, $\pi_{3}=(1,3)$, $\pi_{4}=(1,4)$,
$\pi_{5}=(1,3)(1,2)$, $\pi_{6}=(1,4)(1,2)$, $\pi_{7}=(1,2)(1,3)$,
$\pi_{8}=(1,2)(1,4)$, $\pi_{9}=(1,2)(1,3)(1,2)$, $\pi_{10}=(1,2)(1,4)(1,2)$,
$\pi_{11}=(1,4)(1,2)(1,3)$, and $\pi_{12}=(1,4)(1,2)(1,3)(1,2)$. The result
that $\kappa_{i}|\gamma\rangle(i=1,\cdots,6)$, $\pi_{j}|\eta\rangle$,
$\pi_{j}|\vartheta\rangle$, and $\pi_{j}|\nu\rangle(j=1,\cdots,12)$ are all
SLOCC inequivalent to $L_{ab_{3}}$ then follows by calculating the ranks
$r_{AB}$, $r_{AC}$, and $r_{AD}$ of $\kappa_{i}|\gamma\rangle$,
$\pi_{j}|\eta\rangle$, $\pi_{j}|\vartheta\rangle$ and $\pi_{j}|\nu\rangle$,
and using an argument analogous to that of case 3 in the previous section.
Remark 2. By using Tables 7 and 8, one can verify that
$(1,4)L_{ab_{3}}^{\prime}(a=b\neq 0)$ is SLOCC equivalent to $L_{ab_{3}}(a=0\
\&\ b\neq 0)$ under the invertible local operator
$\sigma_{x}\otimes\sigma_{z}\otimes iI\otimes\sigma_{y}$, and
$(1,3)L_{ab_{3}}^{\prime}(a=-b\neq 0)$ is SLOCC equivalent to $L_{ab_{3}}(a=0\
\&\ b\neq 0)$ under the invertible local operator
$\sigma_{x}\otimes\sigma_{z}\otimes\sigma_{y}\otimes iI$.
### VI.4 D. $L_{ab_{3}}^{\prime}(a\neq 0)$ is SLOCC inequivalent to the other
eight families by Verstraete _et al._
Here we show that $L_{ab_{3}}^{\prime}(a\neq 0)$ is not only SLOCC
inequivalent to $L_{ab_{3}}$ but also SLOCC inequivalent to the other eight
families by Verstraete _et al._ For simplicity, we only show that
$L_{ab_{3}}^{\prime}(a=-b\neq 0)$ is SLOCC inequivalent to the other eight
families obtained by Verstraete _et al._ From Table 8,
$r_{AB}r_{AC}r_{AD}=443$ for $L_{ab_{3}}^{\prime}(a=-b\neq 0)$. Consulting
Tables 5, 6, and 7, and using the fact that the subfamilies with different
ranks of coefficient matrices are SLOCC inequivalent to each other, it
suffices to show that $L_{ab_{3}}^{\prime}(a=-b\neq 0)$ is SLOCC inequivalent
to the subfamilies with $r_{AB}r_{AC}r_{AD}=443$ of $G_{abcd}$ and
$L_{abc_{2}}$.
To show that $L_{ab_{3}}^{\prime}(a=-b\neq 0)$ is SLOCC inequivalent to the
subfamily with $r_{AB}r_{AC}r_{AD}=443$ of $G_{abcd}$, we use the degree 6
polynomial invariant $D_{xy}$ given in Eq. (32). It is readily seen from Eq.
(33) that $D_{xy}=0$ for $L_{ab_{3}}^{\prime}(a=-b\neq 0)$. A simple
calculation shows that
$D_{xy}=(\alpha\beta-\gamma\delta)(\alpha\beta+\gamma\delta)(\alpha^{2}+\beta^{2}-\gamma^{2}-\delta^{2})$
(44)
for $G_{abcd}$ [as defined in Eq. (27)]. It is readily seen from Eq. (44) that
$D_{xy}\neq 0$ for the subfamily with $r_{AB}r_{AC}r_{AD}=443$ of $G_{abcd}$
and then the desired result follows.
Next we show that $L_{ab_{3}}^{\prime}(a=-b\neq 0)$ is SLOCC inequivalent to
the subfamily with $r_{AB}r_{AC}r_{AD}=443$ of $L_{abc_{2}}$ [as defined in
Eq. (28)]. A calculation shows that
$D_{xy}=(\alpha^{\prime}\beta^{\prime})^{2}(\alpha^{\prime 2}-\gamma^{\prime
2}+\beta^{\prime 2})$ (45)
for $L_{abc_{2}}$. From Table 6, we distinguish the following two cases.
Case 1. $\alpha^{\prime}\neq 0\ \&\ \beta^{\prime}=\pm\gamma^{\prime}\neq 0\
\&\ \alpha^{\prime}\neq\pm\beta^{\prime}$.
In this case $D_{xy}\neq 0$ and then the desired result follows.
Case 2. $\alpha^{\prime}=0\ \&\ \beta^{\prime}\neq\pm\gamma^{\prime}\ \&\
\beta^{\prime}\gamma^{\prime}\neq 0$.
In this case $D_{xy}=0$. We can resort to the semi-invariants given in Eqs.
(34) and (35). Let $|\varphi\rangle$ be any state SLOCC equivalent to
$L_{ab_{3}}^{\prime}(a=-b\neq 0)$ with $\mathcal{A}_{1}$ given by Eq. (36). A
tedious but straightforward calculation yields
$\displaystyle F_{1}(|\varphi\rangle)$ $\displaystyle=$
$\displaystyle-2a^{2}\alpha_{1}^{4}\biggl{[}\prod_{i=2}^{4}\det\mathcal{A}_{i}\biggr{]}^{2},$
(46) $\displaystyle F_{2}(|\varphi\rangle)$ $\displaystyle=$
$\displaystyle-2a^{2}\alpha_{3}^{4}\biggl{[}\prod_{i=2}^{4}\det\mathcal{A}_{i}\biggr{]}^{2}.$
(47)
In view of Eqs. (46) and (47) and the fact that $\mathcal{A}_{1}$ is
invertible, it follows at once that if $|\varphi\rangle$ is SLOCC equivalent
to $L_{ab_{3}}^{\prime}(a=-b\neq 0)$, then the following equation holds:
$\left|F_{1}(\varphi)\right|+\left|F_{2}(\varphi)\right|\neq 0.$ (48)
The desired result then follows by noting that $F_{1}=F_{2}=0$ for
$L_{abc_{2}}$ with $\alpha^{\prime}=0\ \&\
\beta^{\prime}\neq\pm\gamma^{\prime}\ \&\ \beta^{\prime}\gamma^{\prime}\neq
0$.
As a consequence, $L_{ab_{3}}^{\prime}(a=-b\neq 0)$ is SLOCC inequivalent to
the nine families obtained by Verstraete _et al._ Verstraete .
The discussion suggests that the partition in Ref. Verstraete is incomplete.
For completeness, one may add the family $L_{ab_{3}}^{\prime}$ to the family
$L_{ab_{3}}$ in Ref. Verstraete . An analogous argument shows that the
partition in Ref. Chterental is incomplete as well, and for completeness, one
may add the family $L_{ab_{3}}$ to the family 6 in Ref. Chterental .
## VII VII. Conclusion
We have recast the necessary and sufficient condition for two $n$-qubit states
to be equivalent under SLOCC into an equivalent form in terms of the
coefficient matrices associated with the states. As a direct consequence of
the new necessary and sufficient condition, we have showed that the rank of
the coefficient matrix as well as the rank of the $\ell$-qubit reduced density
matrix is invariant under SLOCC. We have also presented a recursive formula
for the calculation of the rank of coefficient matrix of an $n$-qubit
biseparable state. The recursive formula further gives rise to a
biseparability criterion in terms of the rank of coefficient matrix to
determine if an arbitrary $n$-qubit pure state is biseparable. The invariance
of the rank of coefficient matrix together with the biseparability criterion
reveals that all the degenerate families of general $n$ qubits are
inequivalent under SLOCC.
We have then classified four-qubit states under SLOCC via the ranks of
coefficient matrices and the nine families introduced by Verstraete _et al._
were further partitioned into inequivalent subfamilies. In particular, we have
found 28 genuinely entangled families and all the degenerate classes can be
distinguished up to permutations of the four qubits. We have performed a
detailed study of the relation between the family $L_{ab_{3}}$ and the family
$L^{\prime}_{ab_{3}}$ with corrections to the signs of the last two terms in
the formula of $L_{ab_{3}}$ via the ranks of coefficient matrices. By using a
degree 6 polynomial invariant and two semi-invariants of four qubits, we have
found that $L^{\prime}_{ab_{3}}(a=0)$ is SLOCC equivalent to
$L^{\prime}_{ab_{3}}(a=0)$ whereas $L^{\prime}_{ab_{3}}(a\not=0)$ is SLOCC
inequivalent to $L_{ab_{3}}(a\not=0)$. We have also demonstrated that
$L_{ab_{3}}^{\prime}(a\neq 0\ \&\ b=0)$, $L_{ab_{3}}^{\prime}(b=\pm 3a\neq
0)$, and $L_{ab_{3}}^{\prime}(ab\neq 0\ \&\ a\neq\pm b\ \&\ b\neq\pm 3a)$ are
SLOCC inequivalent to $L_{ab_{3}}$ under any permutation of qubits, whereas
$L_{ab_{3}}^{\prime}(a=\pm b\neq 0)$ are SLOCC equivalent to $L_{ab_{3}}(a=0\
\&\ b\neq 0)$ under some permutations. This suggests that the partition of
four-qubit states into the nine families by Verstraete _et al._ is incomplete,
and for completeness, one may simply add the family $L_{ab_{3}}^{\prime}$ to
the family $L_{ab_{3}}$.
## VIII Acknowledgement
This work was supported by NSFC Grant No. 10875061 and Tsinghua National
Laboratory for Information Science and Technology.
## IX Appendix
Following Luque , $D_{xy}$ can be constructed as
$D_{xy}=\left|\begin{array}[]{ccc}d_{11}&d_{12}&d_{13}\\\
d_{21}&d_{22}&d_{23}\\\ d_{31}&d_{32}&d_{33}\end{array}\right|,$ (49)
where the entries of $D_{xy}$ are given by:
$\displaystyle d_{11}$ $\displaystyle=$ $\displaystyle a_{0}a_{3}-a_{1}a_{2},$
$\displaystyle d_{12}$ $\displaystyle=$ $\displaystyle
a_{0}a_{7}-a_{1}a_{6}-a_{2}a_{5}+a_{3}a_{4},$ $\displaystyle d_{13}$
$\displaystyle=$ $\displaystyle a_{4}a_{7}-a_{5}a_{6},$ $\displaystyle d_{21}$
$\displaystyle=$ $\displaystyle
a_{0}a_{11}-a_{1}a_{10}-a_{2}a_{9}+a_{3}a_{8},$ $\displaystyle d_{22}$
$\displaystyle=$ $\displaystyle
a_{0}a_{15}-a_{1}a_{14}-a_{2}a_{13}+a_{3}a_{12}$ (50)
$\displaystyle+a_{4}a_{11}-a_{5}a_{10}-a_{6}a_{9}+a_{7}a_{8},$ $\displaystyle
d_{23}$ $\displaystyle=$ $\displaystyle
a_{4}a_{15}-a_{5}a_{14}-a_{6}a_{13}+a_{7}a_{12},$ $\displaystyle d_{31}$
$\displaystyle=$ $\displaystyle a_{8}a_{11}-a_{9}a_{10},$ $\displaystyle
d_{32}$ $\displaystyle=$ $\displaystyle
a_{8}a_{15}-a_{9}a_{14}-a_{10}a_{13}+a_{11}a_{12},$ $\displaystyle d_{33}$
$\displaystyle=$ $\displaystyle a_{12}a_{15}-a_{13}a_{14}.$
## References
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* (2) W. Dür, G. Vidal, and J.I. Cirac, Phys. Rev. A 62, 062314 (2000).
* (3) F. Verstraete, J. Dehaene, B. De Moor, and H. Verschelde, Phys. Rev. A 65, 052112 (2002).
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* (12) O. Viehmann, C. Eltschka, and J. Siewert, Phys. Rev. A 83, 052330 (2011).
* (13) R.V. Buniy and T.W. Kephart, J. Phys. A: Math. Theor. 45, 185304 (2012).
* (14) D. Li, X. Li, H. Huang, and X. Li, Europhys. Lett. 87, 20006 (2009).
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* (16) P. Ribeiro and R. Mosseri, Phys. Rev. Lett. 106, 180502 (2011).
* (17) X. Li and D. Li, Quantum Inf. Comput. 11, 0695 (2011).
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* (19) X. Li and D. Li, Phys. Rev. Lett. 108, 180502 (2012).
* (20) S.S. Sharma and N.K. Sharma, Phys. Rev. A 85, 042315 (2012).
* (21) T. Iwai, J. Phys. A: Math. Theor. 40, 12161 (2007).
* (22) D. Li, X. Li, H. Huang, and X. Li, Commun. Theor. Phys. 49, 1211 (2008).
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|
arxiv-papers
| 2012-01-11T03:07:20 |
2024-09-04T02:49:26.142953
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiangrong Li and Dafa Li",
"submitter": "Dafa Li",
"url": "https://arxiv.org/abs/1201.2229"
}
|
1201.2245
|
# Loose Legendrian Embeddings in High Dimensional Contact Manifolds
Emmy Murphy
###### Abstract.
We give an h-principle type result for a class of Legendrian embeddings in
contact manifolds of dimension at least $5$. These Legendrians, referred to as
loose, have trivial pseudo-holomorphic invariants. We demonstrate they are
classified up to ambient contact isotopy by smooth embedding class equipped
with an almost complex framing. This result is inherently high dimensional:
analogous results in dimension $3$ are false.
## 1\. Introduction
Let $(Y^{2n+1},\xi)$ be a contact manifold. A _Legendrian knot_ is defined to
be a closed, connected, embedded submanifold $L^{n}\to Y$ so that
$TL\subseteq\xi$. Though we abuse notation and say $L\subseteq Y$, we study
parametrized embeddings everywhere in this paper. Legendrian knots of
particular interest include knots with topology $S^{n}$, and/or knots embedded
in $(\mathbb{R}^{2n+1},\xi_{std}=\ker(dz-\sum y_{i}dx_{i}))$.
###### Definition 1.1.
Let $f:L^{n}\hookrightarrow(Y^{2n+1},\xi)$ be a smooth embedding, and let
$F_{s}:TL\to TY|_{L}$ be a homotopy of bundle monomorphisms, covering $f$ for
all $s$, so that $F_{0}=df$ and $F_{1}(TL)$ is a Lagrangian subspace of $\xi$.
(Recall $\xi$ has a canonical conformal symplectic structure.) The pair
$(f,F_{s})$ is called a _formal Legendrian knot_.
A Legendrian knot can be thought of as a formal Legendrian by letting
$F_{s}=df$ for all $s$. In particular, we say that two Legendrian knots are
_formally isotopic_ if there exists a smooth isotopy $f_{t}:L\to Y$ between
them, and $df_{t}$ is homotopic through paths of monomorphisms, fixed at the
endpoints, to a path of Lagrangian monomorphisms. Notice that classifying
formal Legendrian knots up to formal isotopy is a question purely about smooth
toplogy and bundle theory, we do this for the case
$(Y,\xi)=\mathbb{R}^{2n+1}_{std}$ in Appendix A. There are many infinite
familes of distinct Legendrian knots which are formally isotopic which can be
distinguished with pseudo-holomorphic curve invariants [9].
Informally, we call a Legendrian knot with $n>1$ _loose_ if it contains a
sufficiently thick Weinstein neighborhood of a stabilized Legendrian curve; we
give a precise definition in the following section. The principal purpose of
this paper is a proof of
###### Theorem 1.2.
Suppose $n>1$ and fix a contact manifold $(Y^{2n+1},\xi)$. Then for each
formal Legendrian isotopy class there is a loose Legendrian knot in that
class, unique up to ambient contact isotopy.
We will assume that the reader is familiar with the general philosophy of the
h-principle; theorems from [19], [15], and [16] are cited explicitly in the
paper. A brief outline of the paper follows. In Section 2 we cover a number of
definitions from contact topology, including a precise definition of loose
knots. We demonstrate an h-principle for $\epsilon$-Legendrian knots in
Section 3; this allows us to set up controllably transverse local charts.
Section 4 is a review of [16]; there we define the concept of wrinkled
embeddings and state an h-principle they satisfy. In the following section we
adapt this concept to the Legendrian context, and prove an h-principle for
Legendrians with prescribed singularities. Section 6 then describes a method
to resolve these singularities. The main theorem is proved in Section 7 using
the tools from the previous sections.
In the Conclusion 8 we discuss corollaries of Theorem 1.2, and compare the
result to other concepts in contact topology. Finally, we classify formal
isotopy classes of Legendrian knots in $(\mathbb{R}^{2n+1},\xi_{std})$ in the
Appendix A.
The author would like to thank her advisor, Yasha Eliashberg, for discussions
and support. His influence on this paper is more than manifest. She would also
like to thank Elizabeth Goodman and Patrick Massot for valuable comments on
early drafts of the paper.
## 2\. Definitions from Contact Topology
In this section we give some definitions and general facts about Legendrian
knots. By _Darboux neighborhood_ in $(Y,\xi)$ we mean an open set $U\subseteq
Y$, together with a contactomorphism to a (geometrically) convex subset of
$(\mathbb{R}^{2n+1},\xi_{std})$. Given a Darboux neighborhood we can define
two projections, the _Lagrangian projection_
$(x_{i},y_{i},z)\mapsto(x_{i},y_{i})$ and the _front projection_
$(x_{i},y_{i},z)\mapsto(x_{i},z)$. For the former a Legendrian will project to
an exact Lagrangian immersion in $\mathbb{R}^{2n}_{std}$, and the $z$
coordinate can be recovered (up to a constant) by integrating $\sum
y_{i}dx_{i}$. A self intersection in this projection is called a _Reeb chord_.
In the front projection a Legendrian projects to a (highly) singular
hypersurface, which nevertheless has well defined tangent fibers everywhere.
These tangent planes are nowhere vertical and the coordinate slopes recover
the $y_{i}$ coordinates of the Legendrian. A Legendrian has a Reeb chord
wherever its front is self-tangent after a local vertical translation; in
particular a Legendrian immersion has a self intersection exactly where its
front is self tangent. The kernel of the differential of the front projection
is a Legendrian foliation $\mathcal{F}$ whose leaves are the Legendrians
$\\{(x,z)=\text{constant}\\}$. A Legendrian thus has singularities in the
front projection exactly where it intersects $\mathcal{F}$ non-transversely.
In this paper we only make use of cusp singularities in the front, locally
given by the equation $z^{2}=x_{1}^{3}$.
###### Definition 2.1.
In $\mathbb{R}^{3}_{std}$, a _stabilization_ is a Legendrian curve which has
the properties depicted in Figure 1. Specifically, it is required to have a
unique transverse self-intersection and a single cusp in the front projection,
and a single Reeb chord. The length of this Reeb chord is called the _action_
of the stabilization. (We do not distinguish an orientation of a
stabilization.)
Figure 1. The front projection of a stabilization.
###### Remark.
Inside a contact $3$-manifold $(Y,\xi)$, a _bypass_ is defined to be an
embedded topological $2$-gon $D$ whose characteristic foliation $TD\cap\xi$
has no singularities on the interior, a negative elliptic singularity on one
edge, positive elliptic singularities at the two vertices, and a positive
hyperbolic singularity on the remaining edge [21]. See Figure 2. Let $L$ be a
Legendrian arc in $Y$. Then there is a Darboux chart $U\subseteq Y$ so that
$L\subseteq U$ is a stabilization if and only if there is a Legendrian arc
$\alpha\subseteq Y$ connecting the endpoints of $L$, so that $L\cup\alpha$ is
the boundary of a bypass, where $L$ (respectively $\alpha$) contains the
negative elliptic (positive hyperbolic) singularity. In the coordinate system
on $U$, the arc $\alpha$ is parallel to the $y$-axis, defining the self-
intersection in the in the front projection. The equivalence of these two
definitions is a simple application of the Weinstein neighborhood theorem, but
we will not use this alternative interpretation in this paper.
Figure 2. The characteristic foliation on a bypass $D$.
###### Definition 2.2.
Suppose $n>1$. Let $B\subseteq\mathbb{R}^{3}_{std}$ be an open ball containing
a stabilization of action $a$, and let
$V_{\rho}=\\{|p|\leqslant\rho,|q|\leqslant\rho\\}\subseteq
T^{*}\mathbb{R}^{n-1}$. Note that $B\times V_{\rho}$ is an open convex set in
$\mathbb{R}^{2n+1}_{std}$. Let $\Lambda$ be the cartesian product of the
stabilization and the zero section, which is Legendrian in $B\times V_{\rho}$.
We call the pair $(B\times V_{\rho},\Lambda)$ a _Legendrian twist_. A
Legendrian twist satisfying $\tfrac{a}{\rho^{2}}<2$ is called a _loose chart_.
Finally, let $L$ be a Legendrian knot in a contact manifold $(Y,\xi)$. If
there is a Darboux chart $U\subseteq Y$ so that $(U,U\cap L)$ is a loose chart
then $L$ is called _loose_.
For any constant $c>0$, we may change coordinates under the contactomorphism
$(x_{i},y_{i},z)\mapsto(cx_{i},cy_{i},c^{2}z)$. This can make either $a$ or
$\rho$ any given size, but not simultaneously. The requirement
$\tfrac{a}{\rho^{2}}<2$ is the essential condition in the above definition; we
claim that every Legendrian $L$ contains a Legendrian twist. To show this it
suffices to find a contact $3$-ball $B\subseteq Y$ so that $B\cap L$ is a
stabilization. Let $B^{3}$ be any small $3$-ball intersecting $L$ in a single
arc. Since $n>1$, the h-principle for isocontact embeddings of positive
codimension [19] implies that a $C^{0}$ perturbation of $B^{3}$ (fixed near
$L$) has the necessary properties.
###### Proposition 2.3.
Inside a loose chart, there is another Legendrian twist with parameters
$a,\rho$ so that $\tfrac{a}{\rho^{2}}$ is arbitarily small. A loose chart
contains two disjointly embedded loose charts.
_Proof:_ The first statement implies the second. The proof is essentially
contained in Figure 3. We first rescale coordinates on $B\times V_{\rho}$ so
that $\rho$ is normalized to $1$. Fix a small $\delta>0$, and let
$\rho^{\prime}=1-\tfrac{a}{2}-\delta>0$. Inside $B\times V_{\rho}$ we are able
to isotope $\Lambda$ to $\Lambda^{\prime}$, so that $\Lambda^{\prime}\cap
B\times V_{\rho^{\prime}}$ is a Legendrian twist with action $\delta$ (in the
given coordinates). Note that $\tfrac{\delta}{(1-^{a}/_{2}-\delta)^{2}}$ can
be made arbitrarily small by choosing sufficiently small $\delta$.
$\hfill\square$
Figure 3. We can isotope a loose chart in a neighborhood of itself, so that it
contains a Legendrian twist with arbitrarily small action as a subset. This
picture is in the front projection, note that all coordinates
$y_{i}=\tfrac{dz}{dx_{i}}$ are bounded by $\rho$
We now define an operation that alters any Legendrian knot so that it becomes
loose. This construction is unecessary for the purpose of constructing a
Legendrian isotopy between two loose knots, but we will need it to show the
existence portion of Theorem 1.2. This operation was first defined in [12];
there it was introduced (without a name) as an operation to alter Legendrian
framings in order to construct Stein manifolds. See Proposition A.3. It was
later considered in [9] where it was shown that this operation causes pseudo-
holomorphic invariants to become trivial.
In the front, consider a small neighborhood of a cusp singularity. After
flattening things out, we can say the neighborhood consists of two horizontal
open disks $\\{z=0\\}$ and $\\{z=1\\}$, connected by a strip with a single
cusp. By choosing a smaller neighborhood and rescaling coordinates we can
assume this model is arbitrarily large in all $x$ and $y$ directions. Of
course any point on a Legendrian admits local coordinates so that the given
point is on a cusp in the front, thus a small neighborhood of any point on a
Legendrian admits these coordinates.
###### Definition 2.4.
Let $L^{n}$ be a Legendrian knot in $(Y,\xi)$. Let $M\subseteq D^{n}$ be a
compact, codimension $0$ manifold, so that $M\cap\partial D^{n}=\varnothing$.
Choose a Morse function $h:M\to[0,2]$, which is identically zero near
$\partial M$ and has all critical values larger than $1$. Choose a point on
$L$ and local coordinates as above, suitably large to accomodate $h$. On the
compactly supported set they disagree, replace the disk $\\{z=0\\}$ with the
set $\\{z=h(x)\\}$. This Legendrian knot is called the _$M$ -stabilization_ of
$L$, denoted $s_{M}(L)$. See Figure 4.
This construction does not depend on the choice of neighborhood, since any
small disk in $L$ can be taken to any other by ambient contactomorphism. A
priori, $s_{M}(L)$ may depend on the isotopy class of embeddings $M\subseteq
D^{n}$; we assume this data is included in order to define $s_{M}(L)$. In
fact, Theorem 1.2 implies $s_{M}(L)$ is determined up to Legendrian isotopy by
only $\chi(M)$ and the formal Legendrian isotopy class of $L$ when $n>1$. For
the case $n=1$ the reader can check that $D^{1}$-stabilizing a knot is
equivalent to stabilizating a curve twice, once with each orientation.
Figure 4. An $M$-stabilization of a small neighborhood. Here $M$ is the
annulus. Any $M$-stabilized Legendrian contains a loose chart, shown here as
the region between the thin curves.
###### Proposition 2.5.
For any Legendrian knot of dimension $n>1$ and any $M\subseteq D^{n}$,
$s_{M}(L)$ is loose.
_Proof:_ In the coordinates defined above, there is visibly a Legendrian twist
with action $1$, see Figure 4. The radius of the neighborhood in the $x$
directions is determined by the topology of the embedding $M\subseteq D^{n}$,
but the radius in the $y$ directions may be taken to be arbitrarily large, as
discussed above. By rescaling the $x$ and $y$ coordinates in inverse
proportion (keeping the contact form fixed), we exhibit a loose chart.
$\hfill\square$
###### Proposition 2.6.
Let $L$ be a Legendrian knot, and suppose $\chi(M)=0$. Then $s_{M}(L)$ is
formally isotopic to $L$.
_Proof:_ Identify $M\subseteq L$ as the set $\\{h(x)>1\\}$, as in Definition
2.4. We first describe a smooth isotopy, undoing the $M$-stabilization. The
$y$ coordinates of our knot are given by the gradient of $h$. Fixing this near
$\partial M$, we can homotope the gradient to a nonzero vector field, since
$\chi(M)=0$. We interpret this as an isotopy which alters the $y$ coordinates
but has a fixed front projection. We can then push $M$ down through the
$\\{z=1\\}$ plane without the knot self-intersecting.
It remains to show that this smooth isotopy, $f_{t}$ is actually the base of a
formal Legendrian isotopy, that is, we need to homotope $df_{t}$ through
bundle monomorphisms to a Lagrangian monomorphism. Since we avoid the singular
set, the obvious straight line path through bundle maps projects non-
singularly to the $x$ coordinate plane. It follows that this path is in fact
through monomorphisms. $\hfill\square$
## 3\. $\epsilon$-Legendrian Knots
We demonstrate an h-principle for $\epsilon$-Legendrian knots in this section.
The advantage of working with $\epsilon$-knots rather than formal knots is
that it gives us a set of Darboux coordinates around every point, so that $L$
has a smooth front projection. For the purposes of this paper
$\epsilon=\tfrac{\pi}{3}$ is sufficiently small. First, we define a
_Legendrian plane field_ to be a Lagrangian subfield of the distribution
$\xi$.
###### Definition 3.1.
An embedded submanifold $L^{n}\subseteq(Y,\xi)$ is called _$\epsilon$
-Legendrian_ if there is a Legendrian plane field along $L$, $\lambda$, which
is $\epsilon$-close to $TL$. Here, two $n$-planes are said to be
$\epsilon$-close if the projection from one plane to the other is an
isomorphism and the angle between any vector and its projection is less than
$\epsilon$ (in some fixed metric).
We use this opportunity to discuss the general problem of $A$-directed
embeddings, which we will discuss in other contexts throughout the paper. Let
$L$ be an $n$-manifold, and $Y$ a manifold of larger dimension. Let
$A\subseteq Gr_{n}(Y)$, where $Gr_{n}(Y)$ denotes the _bundle_ of $n$-planes
in $TY$, with fiber $Gr_{\operatorname{dim}(Y),n}$. An _$A$ -directed
embedding_ is an embedding $L\to Y$ so that $TL\subseteq A$. A _formal
$A$-directed embedding_ is a smooth embedding $f:L\to Y$, together with a path
of bundle monomorphisms $F_{s}:TL\to TY$ covering $f$, so that $F_{0}=df$ and
$\operatorname{Im}(F_{1})\subseteq A$. To say _an h-principle holds_ for
$A$-directed embeddings is to say the inclusion of $A$-directed embeddings
into formal $A$-directed embeddings is a weak homotopy equivalence (with the
$C^{\infty}$ topologies). In particular, it induces a bijection on $\pi_{0}$
of these spaces: for every formal $A$-directed isotopy class, there is exactly
one $A$-directed embedding up to $A$-directed isotopy.
Even under the assumption that $A$ is open, an h-principle for $A$-directed
embeddings is not generally true. For example if $L=S^{2}$ and
$Y=\mathbb{R}^{3}$, the h-principle for $A$-directed embeddings fails for any
proper subset $A\subseteq Gr_{3,2}$. In [19], it is shown that an h-principle
holds for all open $A$, if $L$ is an open manifold. Furthermore, the concept
of convex integration is used there to prove an h-principle holds for
$A$-directed embeddings of closed manifolds, under the assumption $A$ is open
and _ample_. Rather than stating the original definition, we give the
ampleness criterion 19.1.1 from [15].
###### Proposition 3.2.
Let $A\subseteq Gr_{n}(Y)$, fix $p\in Y$, and let $S\in Gr_{n-1}(Y)_{p}$ be a
$(n-1)$-plane contained inside an element of $A$. Let $\Omega_{p,S}=\\{v\in
T_{p}Y;\,\operatorname{Span}\\{S,v\\}\in A_{p}\\}$. Assume for every choice of
$p$ and $S$, the convex hull of each connected component of $\Omega_{p,S}$ is
equal to $T_{p}Y$. Then $A$ is ample.
Let $(Y,\xi)$ be contact, and let $A\subseteq Gr_{n}(Y)$ be the subset of
$n$-planes which deviate from a Lagrangian plane in $\xi$ by angle less than
$\epsilon$. In these terms an embedding $L\to Y$ is $\epsilon$-Legendrian if
and only if it is $A$-directed. Assume that $S$ is an $(n-1)$-plane which
makes an angle less than $\epsilon$ with some Legendrian plane. Then
$\Omega_{p,S}$ is connected, open, and scalar invariant. This implies the
convex hull of $\Omega_{p,S}$ is all of $T_{p}Y$, and thus $A$ is ample by
Proposition 3.2.
Convex integration implies an h-principle for $\epsilon$-Legendrian knots;
this means the space of $\epsilon$-Legendrian knots is weakly homotopy
equivalent to formal $\epsilon$-Legendrian knots. If furthermore
$\epsilon<\tfrac{\pi}{2}$ then the space of formal $\epsilon$-Legendrian knots
is weakly homotopy equivalent to the space of formal Legendrian knots, simply
because this is true in each fiber.
###### Proposition 3.3.
Let $\epsilon<\tfrac{\pi}{2}$. Then the natural inclusion of
$\epsilon$-Legendrian knots into formal Legendrians knots is a weak homotopy
equivalence. In particular, every formal Legendrian is formally homotopic to
an $\epsilon$-Legendrian, and any formal isotopy between two
$\epsilon$-Legendrians can be $C^{0}$ perturbed (rel endpoints) to an
$\epsilon$-Legendrian isotopy.
## 4\. Review of Wrinkled Embeddings
In this section, we review concepts from [16] needed for the proof of Theorem
1.2. While attempting to be minimally complete, it would be to the reader’s
advantage to understand the constructions there more thoroughly. Theorem 1.2
can be thought of as an application of Eliashberg/Mishachev’s ideas to contact
topology.
As discussed in the previous section, an h-principle for $A$-directed
embeddings of a closed manifold $L$ is not generally true, even if we assume
$A$ is open. The motivation of the definitions in [16] is to prove an
h-principle for all open $A$, by relaxing the notion of embedding.
Specifically, a wrinkled embedding is a smooth map which is a topological
embedding, but is allowed to have prescribed singularities. These
singularities have well defined tangent fibers, allowing us to define
$A$-directed wrinkled embeddings. The main theorem from [16] is an h-principle
for $A$-directed wrinkled embeddings, for any open $A$. We now make these
statements precise, which we will adapt to a local, codimension $1$ situation.
###### Definition 4.1.
Let $W:\mathbb{R}^{n}\to\mathbb{R}^{n+1}$ be a smooth, proper map, which is a
topological embedding. Suppose $W$ is a smooth embedding away from a finite
collection of spheres, $\\{S^{n-1}_{j}\\}$. Suppose, in some coordinates near
these spheres, $W$ can be parametrized by
$W(u,\vec{v})=\left(\vec{v},u^{3}-3u(1-|\vec{v}|^{2}),\tfrac{1}{5}u^{5}-\tfrac{2}{3}u^{3}(1-|\vec{v}|^{2})+u(1-|\vec{v}|^{2})^{2}\right),$
where our domain coordinates lie in a small neighborhood of the sphere
$\\{|\vec{v}|^{2}+u^{2}=1\\}\subseteq\mathbb{R}^{n}$. Then $W$ is called a
_wrinkled embedding_ , and the spheres $S^{n-1}_{j}$ are called the
_wrinkles_.
Figure 5. The curve $\psi$.
Let $\psi:\mathbb{R}\to\mathbb{R}^{2}$ be the plane curve, defined by
$\psi(u)=(\psi^{1}(u),\psi^{2}(u))=(u^{3}-3u,\tfrac{1}{5}u^{5}-\tfrac{2}{3}u^{3}+u)$,
shown in Figure 5 (we assume $\psi$ is horizontal outside a compact subset).
Let $\psi_{\delta}$ be a rescaling of this function, defined by
$\psi_{\delta}(u)=(\delta^{{}^{3}/_{2}}\psi^{1}(\tfrac{u}{\sqrt{\delta}}),\delta^{{}^{5}/_{2}}\psi^{2}(\tfrac{u}{\sqrt{\delta}}))$.
This is well defined even when $\delta<0$, in this case $\psi_{\delta}$ is
smooth and graphical. We define $\psi_{0}(u)=(u^{3},\tfrac{1}{5}u^{5})$, which
makes $\psi_{\delta}$ a continuous family of plane curves.
In these terms, wrinkled embeddings are locally modeled by
$W(u,\vec{v})=(\vec{v},\psi_{1-|\vec{v}|^{2}}(u))$. Therefore wrinkles have
two kinds of singularities: on the singular sphere
$\\{|\vec{v}|^{2}+u^{2}=1\\}$, there are cusp singularities everywhere on the
lower and upper hemisphere. Along the equator $\\{u=0\\}$, we see “unfurled
swallowtail” singularities. See Figure 6.
Figure 6. An unfurled swallowtail singularity.
An _embryo_ of a wrinkle is defined to be the isolated singularity with a
local model given by
$(u,\vec{v})\mapsto(u^{3}+3u|\vec{v}|^{2},\vec{v},\tfrac{1}{5}u^{5}+\tfrac{2}{3}u^{3}|\vec{v}|^{2}+u|\vec{v}|^{4})$
with $(u,\vec{v})$ in a neighborhood of the origin. For
$t\in(-\epsilon,\epsilon)$, let
$W_{t}(\vec{v},u)=(\vec{v},\psi_{t-|\vec{v}|^{2}}(u))$. Then $W_{t}$ is smooth
for $t<0$ and has a single wrinkle when $t>0$. At $t=0$, there is an embryo
singularity at $(u,\vec{v})=0$. We allow embryo singularities whenever we
discuss parametric families of wrinkled embeddings. Generically, these occur
with codimension $1$ in parameter space, and are isolated points in the
embedding. We do not distinguish a time orientation, so an embryo can either
create a wrinkle in forward time, or allow one to disappear.
Even though a wrinkled embedding is singular, it does have well defined
tangent fibers of dimension $n$ everywhere. For example, let $p$ be a cusp
singularity point given in coordinates by $f(u)=(u^{2},u^{3})$. Even though
$df$ is trivial at the point $u=0$, small neighborhoods of this point are
$C^{1}$ close to uniformly horizontal. Therefore we define the tangent fiber
to be horizontal at that point. One can similarly check that the tagent fibers
near an unfurled swallowtail or embryo singularity uniformly approach
horizontal in the coordinates given above.
Therefore, given a wrinkled embedding $W:\mathbb{R}^{n}\to\mathbb{R}^{n+1}$,
we have defined the hyperplane bundle $TW\subseteq
T\mathbb{R}^{n+1}|_{\operatorname{Im}(W)}$. This allows us to define _$A$
-directed wrinkled embeddings_: wrinkled embeddings with $TW\subseteq A$. We
quote the h-principle from [16], again specialized for our purposes:
###### Theorem 4.2.
Let $f:\mathbb{R}^{n}\to\mathbb{R}^{n+1}$ be a graphical smooth map, and let
$\nu^{n}\subset T\mathbb{R}^{n+1}|_{\operatorname{Im}(f)}$ be a nowhere
vertical hyperplane distribution so that $\nu=\operatorname{Im}(df)$ outside
of a compact set, $C$. Then there is a wrinkled embedding
$W:\mathbb{R}^{n}\to\mathbb{R}^{n+1}$ so that $TW$ is $C^{0}$ close to $\nu$
and $W$ is $C^{0}$ close to $f$, equal outside $C$. This also holds
parametrically: for families $f_{t}$ and $\nu_{t}$ with $t\in D^{m}$ and
$\nu_{t}=\operatorname{Im}(df_{t})$ at $\partial D^{m}$, we can find a family
of wrinkled embeddings $W_{t}$, $C^{0}$ close to $f_{t}$, so that $TW_{t}$ is
$C^{0}$ close to $\nu_{t}$ and $W_{t}=f_{t}$ at $\partial D^{m}$ and outside
$C$.
## 5\. Wrinkled Legendrians
Say we wanted to prove that embedded Legendrian knots satisfy an h-principle,
despite knowing this is false. We will see this reduces simply to solving the
local extension problem: given a formal Legendrian
$f:\mathbb{R}^{n}\to\mathbb{R}^{2n+1}_{std}$ which is Legendrian outside a
compact set, we need to show we can find a Legendrian embedding $C^{0}$ close
to $f$, and equal to it outside the compact set. The $C^{0}$ close condition
is essential: we have no lower bounds on the size of our local charts and we
need to avoid self intersections.
The set of Legendrian planes in $Gr_{n}(\mathbb{R}^{2n+1}_{std})$ is not open,
so none of our theorems about directed embeddings apply immediately. The
advantage of the local picture is it allows us to re-interpret the geometry in
the front projection. Any smooth embedding
$W:\mathbb{R}^{n}\to\mathbb{R}^{n+1}$ which is never vertical defines a
Legendrian $L\subseteq\mathbb{R}^{2n+1}_{std}$. Assume $f$ projects to a
smooth hypersurface $H\subseteq\mathbb{R}^{n+1}$. The $y$ coordinates of $f$
define a hyperplane field $\nu=\ker(dz-\sum_{i}y_{i}dx_{i})\subseteq
T\mathbb{R}^{n+1}|_{H}$. Then $L$ is $C^{0}$ close to $f$ if $W$ is $C^{0}$
close to $H$ and $TW$ is $C^{0}$ close to $\nu$.
In fact $W$ need not be smooth, since a smooth Legendrian need not have a
smooth front projection. At this point we would like to use Theorem 4.2, but
first we need to study wrinkled singularities to determine if they have smooth
Legendrian lifts. Wrinkled embeddings have a natural tangent bundle. More
precisely, given a non-vertical wrinkled embedding
$W:\mathbb{R}^{n}\to\mathbb{R}^{n+1}$ there are unique smooth functions
$y_{i}(\vec{v},u)$ so that $dz=y_{i}dx_{i}$ everywhere. At a cusp singularity
$(x,z)=(u^{2},u^{3})$, this function is given by $y=\tfrac{3}{2}u$. In this
case, the triple $(x,y,z)$ is a smooth embedding, therefore the cusp
singularity is the front projection of a smooth Legendrian curve.
For unfurled swallowtails (as well as embryos) the functions $y_{i}$ are
uniquely defined, but the induced map
$L:\mathbb{R}^{n}\to\mathbb{R}^{2n+1}_{std}$ is not an embedding: $dL$ has
rank $n-1$ at these singularities. For now we “define away” this problem.
###### Definition 5.1.
Let $L$ be closed and connected. A _wrinkled Legendrian_ is a smooth map
$f:L\to(Y,\xi)$, which is a topological embedding, satisfying the following
properties. The image of $df$ is contained in $\xi$ everywhere, and $df$ is
full rank outside a subset of codimension $2$. This singular set is required
to be diffeomorphic to a disjoint union of $(n-2)$-spheres
$\\{S_{j}^{n-2}\\}$, called _Legendrian wrinkles_. We assume each
$S^{n-2}_{j}$ is contained in a Darboux chart $U_{j}$, so that the front
projection of $L\cap U_{j}$ is a wrinkled embedding, smooth outside of a
compact set. (In particular, the front projection of each $S^{n-2}_{j}$ is the
unfurled swallowtail singularities of a single wrinkle in the front.)
A wrinkle Legendrian is therefore a smooth Legendrian embedding outside a set
of codimension $2$, however it is permitted to contain the singularity defined
as the Legendrian lift of the unfurled swallowtail. Our definition is slightly
stronger than this: we also require a “global trivialization” of each
Legendrian wrinkle given by the Darboux charts $U_{j}$. We emphasize that
$\\{U_{j}\\}$ is considered part of the data of a wrinkled Legendrian: for a
given map $f:L\to(Y,\xi)$, different choices of $\\{U_{j}\\}$ are considered
to be different as wrinkled Legendrians. However notice there is no
requirement for these Darboux charts to be disjoint, and in fact we often take
them to be equal when multiple Legendrian wrinkles are contained in a single
Darboux chart.
Before defining a topology on the space of wrinkled Legendrians, we first
define an additional singularity that allows Legendrian wrinkles to appear and
disappear in families. Again this is completely analogous to wrinkled
embeddings. We define a _Legendrian embryo_ to be the singularity given by the
Legendrian lift of the embryo singularity defined in the previous section for
maps $\mathbb{R}^{n}\to\mathbb{R}^{n+1}$. Again these singularities are
generically codimension $1$ in parameter space and isolated points in the
domain $L$. For reference, a Legendrian embryo is given in Darboux coordinates
by
$\displaystyle x_{1}$ $\displaystyle=u^{3}+3u|\vec{v}|^{2}$
$\displaystyle(x_{2},\ldots,x_{n})$ $\displaystyle=\vec{v}$ $\displaystyle
y_{1}$ $\displaystyle=\tfrac{1}{3}(u^{2}+|\vec{v}|^{2})$
$\displaystyle(y_{2},\ldots,y_{n})$
$\displaystyle=2u(|\vec{v}|^{2}-\tfrac{1}{3}u^{2})\vec{v}$ $\displaystyle z$
$\displaystyle=\tfrac{1}{5}u^{5}+\tfrac{2}{3}u^{3}|\vec{v}|^{2}+u|\vec{v}|^{4}$
with domain coordinates $(u,\vec{v})$ in a neighborhood of the origin in
$\mathbb{R}^{n}$. As before, we allow Legendrian embryos whenever we discuss
parametric families of wrinkled Legendrians. When a Legendrian wrinkle is born
we add a new $U_{j}$ to the collection of Darboux charts which contains the
Legendrian embryo, and it is required to contain the created wrinkle
throughout its entire “lifetime”. To topologize the space of wrinkled
Legendrians we use the $C^{\infty}$ topology on the space of maps, together
with independent $C^{\infty}$ topologies for the Darboux charts $U_{j}$.
We claim that Legendrian wrinkles have a cannonical coorientation in $L$. A
point on a Legendrian wrinkle is is given in Darboux coordinates by
$\displaystyle x_{1}$ $\displaystyle=u^{3}+3ux_{2}$ $\displaystyle y_{1}$
$\displaystyle=\tfrac{1}{3}(u^{2}-x_{2})$ $\displaystyle y_{2}$
$\displaystyle=ux_{2}+\tfrac{1}{3}u^{3}$ $\displaystyle y_{i}$
$\displaystyle=0\text{ for }i>2$ $\displaystyle z$
$\displaystyle=\tfrac{1}{5}u^{5}-\tfrac{2}{3}u^{3}x_{2}+ux_{2}^{2}.$
We see $\\{u=0,\,x_{2}=0\\}$ is the singular set. $df$ has rank $n-1$ on this
set, and its kernel is spanned by $\partial_{u}$. Let
$\beta:(-\epsilon,\epsilon)\to L$ be a path with $\beta(0)=0$, so that
$\dot{\beta}$ is in the kernel of $df$ at that point. Then the second
derivative of $f\circ\beta$ at $s=0$ defines a nonzero vector $v\in TY$. For
another choice of path $\beta_{1}$, $v$ is scaled by
$\left(\tfrac{|\dot{\beta}_{1}(0)|}{|\dot{\beta}(0)|}\right)^{2}$ and added to
$df(\ddot{\beta}_{1}(0))$. Thus at the wrinkle, there is a canonical $n$-plane
containing the $\operatorname{Im}(df)$, and it is canonically cooriented.
Also notice that $\operatorname{Im}(df)$ contains the tangent space of the
singular set, and thus a Legendrian wrinkle has a canonical normal framing in
$L$.
If our goal is to prove an h-principle for wrinkled Legendrian knots, first we
must define a map from wrinkled Legendrians to formal Legendrian knots. Given
a wrinkled Legendrian $f:L\to(Y,\xi)$, we can $C^{\infty}$ perturb the map $f$
near the Legendrian wrinkles so that the perturbation $\tilde{f}$ is a smooth
embedding. This is a contractable choice and can be made completely canonical
by choosing a fixed perturbation of the model in Definition 5.1. To define the
structure map $F_{s}$ we need to find a homotopy through bundle monomorphisms
from $d\tilde{f}$ to a Lagrangian map $F_{1}:TL\to\xi$; we use the Darboux
charts $\\{U_{j}\\}$ to define this. Consider the $(n+1)$-plane bundle given
by $P=\operatorname{span}(\partial_{y_{i}},\partial_{z})$. Because $f$ is a
perturbation of a wrinkled embedding which is nowhere vertical, $d\tilde{f}$
is homotopic to a map which is everywhere transverse to $P$. The set of bundle
maps $TL\to TU_{j}$ which are transverse to $P$ forms a contractable space,
since they can be identified with the graphs of all $(n+1)\times n$ matrices.
Similarly the set of Lagrangian planes transverse to $P$ is contractable,
being equivalent to the space of skew symmetric $n\times n$ matrices.
Therefore we can homotope $d\tilde{f}$ through monomorphisms to a Lagrangian
map $F_{1}$ transverse to $P$ everywhere, and this homotopy is canonical up to
a contractable choice.
###### Theorem 5.2.
The map defined above from the space of wrinkled Legendrians to the space of
formal Legendrian knots is a weak homotopy equivalence. In particular if two
Legendrian knots are formally isotopic, then there is an isotopy through
wrinkled Legendrians between them.
This theorem is essentially a combination of the h-principles discussed so
far. We first prove a lemma to reduce the problem to Darboux charts, which is
a citation of the Holonomic Approximation Theorem (Theorem 3.1.2 in [15]). As
explained in Section 3, $\epsilon$ should be thought of as being roughly
$\tfrac{\pi}{3}$; we reserve the words “close” and “small” for an arbitrarily
small size, which may depend on $\epsilon$.
###### Lemma 5.3.
Let $L_{t}:L\to(Y,\xi)$ be a family of $\epsilon$-Legendrian knots with $t\in
D^{m}$, which are Legendrian at $\partial D^{m}$. Then, we can perturb $L_{t}$
through $\epsilon$-Legendrians ($C^{0}$ small, fixed at $\partial D^{m}$) to
$\tilde{L}_{t}$, and find a finite collection of (continuous families of)
Darboux balls $\\{B_{k}^{t}\\}$ which are disjoint for all $t$, so that
$\tilde{L}_{t}$ is Legendrian outside $\bigcup_{k}B^{t}_{k}$. Furthermore, we
can arrange that $\tilde{L}_{t}\cap B_{k}^{t}$ has a graphical front
projection for all $t$ and $k$.
_Proof:_ Choose a fixed point $t_{0}\in D^{m}$, and let $\theta_{t}:Y\to Y$ be
a family of ambient diffeomorphisms extending $L_{t}$, so that
$\theta_{t_{0}}$ is the identity. For each $p\in L_{t_{0}}$, $t\in D^{m}$, let
$U_{p,t}:B^{2n+1}_{std}\to Y$ be a small Darboux neighborhood around
$\theta_{t}(p)$, so that $U_{p,t}(0)=p$ and
$(U_{p,t})_{*}(\operatorname{span}_{i}\\{\partial_{x_{i}}\\})=T_{p}L_{t}$ at
this point. We choose $U_{p,t}$ small enough so that $\lambda_{t}$ and
$(U_{p,t})_{*}\operatorname{span}_{i}\\{\partial_{x_{i}}+y_{i}\partial_{z}\\}$
are $\epsilon$-close, and also $(U_{p,t})_{*}(\partial_{z})\notin\lambda_{t}$.
The set $\\{\bigcap_{t}\theta_{t}^{-1}(U_{p,t}(B^{2n+1}_{std})),p\in
L_{t_{0}}\\}$ is an open cover of $L_{t_{0}}$; we choose a finite subcover,
indexed by the points $p_{k}$. The cover defines a triangulation of
$L_{t_{0}}$, so that each $(n-i)$-simplex is contained in $i+1$ of these
charts, let $K$ be the codimension $1$ skeleton. Note that for all $t$,
$\\{U_{p_{k},t}\\}$ is a finite covering of $L_{t}$ by Darboux balls.
In coordinates, $L_{t}\cap U_{k}=\\{z=z(x),y_{i}=y_{i}(x)\\}$, and
$\lambda=\operatorname{span}_{i}\\{\partial_{x_{i}}+y_{i}(x)\partial_{z}+\sum_{j}g_{i}^{j}\partial_{y_{j}}\\}$,
for some functions $g_{i}^{j}(x)$. The Lagrangian condition on $\lambda$ is
equivalent to $g_{i}^{j}=g_{j}^{i}$, therefore the functions
$z(x),y_{i}(x),g_{i}^{j}(x)$ together define a section of the second jet space
of the $x$ coordinates, $J^{2}(B^{n})$.
To construct $\tilde{L}_{t}$, we perturb $L_{t}$ inductively on each Darboux
chart. Let $L^{k}_{t}=U^{-1}_{p_{k},t}(L_{t})\subseteq B^{2n+1}_{std}$. Then
the family of $\epsilon$-Legendrians $(L^{k}_{t},\lambda_{t})$ defines a
family of non-holonomic sections of $J^{2}(B^{n})$. We cite the $m$-parametric
Holonomic Approximation Theorem [15]. It states that, on some open
neighborhood $V$ of some $C^{0}$ perturbation of $K\cap L^{k}_{t}$, we may
$C^{0}$ approximate the family of sections $(z,y_{i},g_{i}^{j})_{t}$ with
sections $(\tilde{z},\tilde{y}_{i},\tilde{g}_{i}^{j})_{t}$ which are holonomic
on $V$. Over the intersection with previous charts, we fix our section where
it is already holonomic and use the extension form of Holonomic Approximation.
This defines a $\epsilon$-Legendrian family $\tilde{L}_{t}$ which is $C^{0}$
close to $L_{t}$. $\tilde{L}_{t}$ is Legendrian on $V$, since the section
defining it is holonomic there. Furthermore $T\tilde{L}_{t}$ is $C^{0}$ close
to $\lambda_{t}$ over $V$, thus $\tilde{L}_{t}$ renains graphical on $V$, in
every chart $U_{p_{k},t}$. To find $B^{t}_{k}$, take an open subset of
$U_{p_{k},t}$ which does not intersect $K$, but whose boundary is contained in
$V$. $\hfill\square$
_Proof of Theorem 5.2:_ Suppose $L_{t}$ is a family of formal Legendrian
knots, Legendrian at $t\in\partial D^{m}$. First, Proposition 3.3 states we
can first make $L_{t}$ $\epsilon$-Legendrian for all $t$. The previous lemma
then constructs a moving collection of Darboux balls, so that $L_{t}$ is
Legendrian outside. Therefore in each Darboux ball we have an
$\epsilon$-Legendrian family, which is Legendrian outside of a compact region.
Furthermore the front projection of $L_{t}$ is a smooth graphical isotopy of a
hypersurface, the $y$-coordinates determine a nowhere vertical hyperplane
field along this front. We then apply Theorem 4.2 to find a family of wrinkled
embeddings $W_{s,t}$ ($s\in[0,1]$), so that $W_{0,t}$ is the front of $L_{t}$,
$W_{s,t}$ is always $C^{0}$ close to the front of $L_{t}$, and $TW_{1,t}$ is
$C^{0}$ close to the given hyperplane field. This ensures the wrinkled
Legendrian lift of $W_{1,t}$ is $C^{0}$ close to $L_{t}$, and therefore it
remains embedded. $\hfill\square$
This theorem holds in all dimensions. In the case $n=1$, a _generic_ wrinkled
Legendrian will be smooth, since Legendrian wrinkles are codimension $2$
submanifolds. However, an isotopy will contain Legendrian embryos, so we
cannot conclude two Legendrian embeddings which are wrinkled isotopic are in
fact Legendrian isotopic. In fact, crossing through a Legendrian embryo in
this dimension is equivalent to a Legendrian curve stabilization (or
destabilization). This implies a theorem originally proved in [18]: if two
Legendrian $1$-knots are formally isotopic, they are Legendrian isotopic after
a finite number of stabilizations. It’s also true that two smoothly isotopic
Legendrian $1$-knots are formally isotopic after some number of
stabilizations, this is a simple calculation in the algebraic topology of
frame bundles (see Appendix A).
## 6\. Twist Markings
Theorem 5.2 is a major portion of the work in proving Theorem 1.2. In this
section we systemitize a method to resolve the singularities of wrinkled
Legendrians.
###### Definition 6.1.
Let $L\subseteq(Y,\xi)$ be a wrinkled Legendrian with $k$ wrinkles, and let
$\Phi\subseteq L$ be an embedded codimension $1$ smooth compact submanifold
with boundary. Assume $\Phi$ has the topology of a sphere with $k$ open disks
removed. Then $\Phi$ is called a _twist marking_ if the singular set of $L$ is
equal to $\partial\Phi$, and there is a small collar neighborhood of the
singular set so that $\Phi=\\{u=0,\,x_{2}\leqslant 0\\}\subseteq L$ in terms
of the coordinates following Definition 5.1. If $L$ has a Legendrian embryo,
we require that it is contained in the interior of $\Phi$ with local model
given by $\Phi=\\{u=0\\}$.
###### Definition 6.2.
We use the following topology on the space of wrinkled Legendrians with twist
markings. We put the $C^{\infty}$ topology on both the space of wrinkled
Legendrians as well as the space of embedded submanifolds. We also specify a
relation to accomodate discrete chages in the topology of $\Phi$: if $L_{t}$
is a path of wrinkled Legendrians containing a Legendrian embryo, and
$\Phi_{t}$ is a path of twist markings on $L_{t}$ so that $\Phi_{t}$ acquires
another puncture at the embryo, the path $(L_{t},\Phi_{t})$ is defined to be
continuous.
Recall Legendrian wrinkles have neighborhoods with front projection given by
$\\{(x_{1},\ldots,x_{n},z);\,(x_{1},z)=\psi_{x_{2}}(u),\,u\in(-\epsilon,\epsilon)\\}$,
where $\\{(x_{2},u)=(0,0)\\}$ is the singular set. Let $\delta>0$ be some
small number. $\Phi$ should be thought of as a formal representation of the
neighborhood $\\{(x_{1},z)=\psi_{\delta}(u);\,u\in(-\epsilon,\epsilon)\\}$,
where the $x_{1}$ direction is transverse to $\Phi$. This interpretation
allows us to resolve all singularities at $\partial\Phi$.
###### Proposition 6.3.
In $(Y,\xi)$, let $(L_{t},\Phi_{t})$ be a family of wrinkled Legendrians with
twist markings. Then we can construct an isotopy of Legendrian knots
$\tilde{L}_{t}:L\to(Y,\xi)$, so that $\tilde{L}_{t}$ identical to $L_{t}$
outside any small neighborhood of $\Phi_{t}$.
Figure 7. A local model describing how a twist marking resolves Legendrian
wrinkles.
_Proof:_ We only need to check things for our given models; it suffices to
work in the front projection. Note that two (possibly wrinkled) Legendrian
embeddings are $C^{0}$ close if their fronts are $C^{1}$ close. Check that
when $\delta>0$ is small, $\psi_{\delta}(u)$ is $C^{1}$ close to the
horizontal axis, and identical to it outside a small neighborhood of the
origin. On the interior of $\Phi$, we find coordinates so that
$(L,\Phi)=(\\{z=0\\},\\{(x_{1},z)=(0,0)\\})$, and replace this by
$\tilde{L}=\\{(x_{1},z)=\psi_{\delta}(u);\,u\in\mathbb{R}\\}$. This alteration
is $C^{1}$ small in the front projection and is contained in a neighborhood of
$\Phi$; it remains to describe the behavior near $\partial\Phi$.
Let $m_{\delta}:\mathbb{R}\to\mathbb{R}$ be a smoothing of the function
$\max(\delta,\cdot)$. Near points on $\partial\Phi$, we have coordinates so
that $(L,\Phi)=(\\{(x_{1},z)=\psi_{x_{2}}(u)\\},\\{u=0,\,x_{2}\leqslant 0\\}$,
which we replace with $\tilde{L}=\\{(x_{1},z)=\psi_{m_{\delta}(x_{2})}(u)\\}$.
This is nonsingular and compatible with our definition of the interior of
$\Phi$. Since both $L$ and $\tilde{L}$ are $C^{1}$ close to horizontal near
the singular set this alteration is $C^{1}$ small. See Figure 7.
Finally, we check this construction near a Legendrian embryo singularity. Let
$L_{t}$ be a path with a unique embryo, we may choose coordinates so that the
front of $L_{t}$ is given by
$\\{(x_{1},z)=\psi_{t-r^{2}}(u);\,r^{2}=x_{2}^{2}+\ldots+x_{n}^{2},\,u\in\mathbb{R}\\}$,
and $\Phi_{t}=\\{x_{1}=0,\,x_{2}^{2}+\ldots+x_{n}^{2}\geqslant t\\}$, here
$t\in(-\epsilon,\epsilon)$. We then replace this path of wrinkled Legendrians
with the path $\tilde{L}_{t}$, with front
$\\{(x_{1},z)=\psi_{m_{\delta}(t-r^{2})}(u)\\}$. $\hfill\square$
We now prove a proposition that relates resolutions of Legendrian wrinkle
singularities with loose charts. We say two properly embedded Legendrians
$(U_{0},L_{0})$ and $(U_{1},L_{1})$ in the Darboux charts $U_{i}$ are
_equivalent_ if we can find contact inclusions $\iota_{0}:U_{0}\to U_{1}$ and
$\iota_{1}:U_{1}\to U_{0}$, so that $L_{0}=\iota^{-1}_{0}(L_{1})$ and
$L_{1}=\iota^{-1}(L_{0})$. As our main example, notice the Legendrian in
$\mathbb{R}^{3}_{std}$ with front projection $\\{(x,z)=\psi(\mathbb{R})\\}$
(Figure 5) is equivalent to any stabilization (Definition 2.1, Figure 1).
Consider the properly embedded wrinkled Legendrian $\Lambda:B^{n}\to
B^{2n+1}_{std}$ with front given by
$\\{(x_{1},z)=\psi_{r^{2}-1}(u);\,r^{2}=x_{2}^{2}+\ldots+x_{n}^{2},\,u\in\mathbb{R}\\}$;
we refer to this as an _inside-out wrinkle_. Recall that Legendrian wrinkles
are given by $\\{(x_{1},z)=\psi_{1-r^{2}}(u)\\}$. An inside-out wrinkle
contains a single Legendrian wrinkle at $\\{r^{2}=1,\,x_{1}=0\\}$, however the
embedding is not compactly supported. Notice the front projection is singular
on the set $\\{|\vec{v}|^{2}-\tfrac{x_{1}^{2}}{4}=1\\}$.
An inside-out wrinkle is not technically a wrinkled Legendrian as described in
Definition 5.1, because there is no Darboux chart containing the Legendrian
wrinkle so that the front is graphical outside of a compact subset. Let
$D^{n-1}_{0}=\\{r\leqslant 1,\,u=0\\}$. Then $D^{n-1}_{0}$ is easily checked
to be a twist marking on $\Lambda$; we now show that resolving $\Lambda$ along
this twist marking is equivalent to a loose chart.
###### Proposition 6.4.
Let $(\Lambda,D^{n-1}_{0})$ be an inside out wrinkle with the twist marking
defined above. Let $\tilde{\Lambda}$ be the resolution of $D^{n-1}_{0}$ at any
scale, and let $B$ be any Darboux ball containing $D^{n-1}_{0}$. Let $(U,L)$
be any loose chart. Then $(B,\tilde{\Lambda})$ and $(U,L)$ are equivalent
models, in terms of mutual pairwise inclusion by contactomorphism.
_Proof:_ Let $\delta$ be the scale of the resolution along $D^{n-1}_{0}$, as
in of Proposition 6.3. $\tilde{\Lambda}$ is a product neighborhood of the
curve $\psi_{\delta}$, and smaller scale resolution can be realized by a
compactly supported isotopy. Therefore $\tilde{\Lambda}$ contains a subset
which is an arbitrarily wide product neighborhood of the curve $\psi$. Since a
loose chart is an arbitrarily wide product neighborhood of a stabilization
(Proposition 2.3), the propoisition follows from the observation that $\psi$
is equivalent to a stabilization. $\hfill\square$
## 7\. Completing the Main Proof
From now on, we will be interested in wrinkled Legendrians which are extended
from an inside-out wrinkle. That is, we consider singular Legendrians $L$ with
only Legendrian wrinkle singularities, so that on a fixed set $V$, $V\cap L$
is an inside-out wrinkle; and outside $V$, $L$ is a wrinkled Legendrian
(meaning the Legendrian wrinkles outside $V$ are required to be contained in
Darboux charts so that $L$ is graphical outside a compact subset). We call
such an object a _prepared wrinkled Legendrian_.
Notice that the definition of a twist marking is equally valid for prepared
wrinkled Legendrians: a twist marking is required to have one boundary
component on each Legendrian wrinkle, _including the inside out wrinkle_.
Given a twist marking on a prepared Legendrian wrinkle, we may associate to it
an element of $\pi_{n-1}L$ as follows. A twist marking is an embedded
$S^{n-1}$ with a number of punctures. Besides the inside out wrinkle, each
boundary component is contained in a Darboux neighborhood $U_{j}$ (as in
Definition 5.1). Because $U_{j}\cap L$ is contractable, we can cap off this
boundary component in a unique canonical way by choosing a cap in $U_{j}\cap
L$. After repeating this process we are left with an $(n-1)$-disk whose
boundary is contained in $V$. This gives a well defined element of
$\pi_{n-1}(L,V)$ which is isomorphic to $\pi_{n-1}L$ since $V$ is
contractable.
Given a prepared wrinkled Legendrian, we say a twist marking is _nulhomotopic_
if the element described above is $0\in\pi_{n-1}L$. Note that Proposition 6.3
only relies on the local model near the Legendrian wrinkle singularities, thus
this proposition continues to hold for prepared wrinkled Legendrians.
###### Proposition 7.1.
Let $L$ be a prepared wrinkled Legendrian with a nulhomotopic twist marking
$\Phi$. Then the smooth Legendrian obtained from resolving $L$ along $\Phi$ is
canonically formally isotopic to the wrinkled Legendrian obtained by resolving
only the inside-out wrinkle along $D^{n-1}_{0}$.
_Proof:_ Let $\tilde{L}$ be the smooth Legendrian resolved along $\Phi$, and
let $L_{0}$ be the wrinkled Legendrian obtained from resolving the inside-out
wrinkle along $D^{n-1}_{0}$. Clearly they are equal outside of $\bigcup
U_{j}\cup D^{n-1}_{0}\cup\Phi$. By our definition of how a wrinkled Legendrian
is interpreted as a formal Legendrian, we simply ignore the Legendrian
wrinkles and treat the chart $U_{j}$ as if it were graphical. Therefore,
compared to $\tilde{L}$, $L_{0}$ lacks a Legendrian twist along $\Phi$, and
also along disks in each $U_{j}$. Instead it has an additional Legendrian
twist along $D^{n-1}_{0}$. But these two sets can be homotoped from one to the
other, using the assumption that $\Phi$ is nulhomotopic. $\hfill\square$
The next proposition is a topological observation without much depth. However,
it is the only place the assumption $n>1$ is essentially used. It is also the
only step in the proof that does not obviously extend to families $L_{t}$ with
$t\in D^{m}$ for $m>1$.
###### Proposition 7.2.
Let $n>1$. Suppose $L_{t}:L\to(Y,\xi)$ is an isotopy of prepared wrinkled
Legendrian. Then there is a nulhomotopic twist marking $\Phi_{t}\subseteq
L_{t}$.
Figure 8. Building a Legendrian twist inside $L$. Pictured here is the case
$n=3$.
_Proof:_ For each $t$ and $j$, we may choose $D^{n-1}_{j}\subseteq U_{j}$ so
that it approaches the Legendrian wrinkle as specified in Definition 6.1.
For all $j$ pick points $p_{j}^{t}\in D^{n-1}_{j}$, including a point
$p_{0}\in D^{n-1}_{0}$ for the inside out wrinkle. Because each wrinkle
$S^{n-2}_{j}$ is canonically cooriented, this induces a coorientation on
$D^{n-1}_{j}$. First we describe $\Phi_{0}$. If there are no Legendrian
wrinkles besides the inside-out wrinkle, we simply let $\Phi_{0}=D^{n-1}_{0}$.
Otherwise, for each $j$, we find a curve $\alpha_{j}$, connecting $p_{0}$ to
$p_{j}^{0}$. We require that the $\alpha_{j}$ are mutually disjoint, and do
not intersect any $D^{n-1}_{j}$ on their interior. Furthermore we ask that
$\alpha_{j}$ is transverse to $D^{n-1}_{j}$ at $p^{0}_{j}$, and the outward
tangent to $\alpha_{j}$ matches the coorientation on $D^{n-1}_{j}$. Let $S$ be
the boundary of a small neighborhood of $\alpha=\bigcup_{j}\alpha_{j}$. For
any $j$, $S\cap D^{n-1}_{j}$ is a small $(n-2)$-sphere which bounds a small
disk in both $S$ and $D^{n-1}_{j}$. Discard these disks, and smooth corners to
get a connected smooth manifold (we also do this for $D^{n-1}_{0}$). After
doing this for all $j$ we obtain a manifold $\Phi_{0}$, satisfying all the
conditions in Definition 6.1.
We now construct $\Phi_{t}$. An isotopy of wrinkled Legendrian embeddings has
embryo singularities at points isolated in both space and time. On any
subinterval of time not containing an embryo, the isotopy is induced by an
ambient contact isotopy of $(Y,\xi)$. On such intervals we can simply let the
ambient isotopy act on $\alpha$, which naturally gives us an isotopy of twist
markings. Thus it suffices to describe $\Phi_{t}$ in a small time interval
around an embryo singularity at time $t_{0}$.
We first consider an embryo singularity where a wrinkle $S^{n-2}_{j}$
disappears in forward time. At the embryo
$T\Phi_{t_{0}}=\operatorname{Im}(dL_{t_{0}})$, since this equation is
satisfied for all points on $\partial\Phi_{t}$ when $t<t_{0}$. When $t>t_{0}$,
$\Phi_{t}$ has one less puncture, and has a long “tentacle” with no boundary
compenents. This can be retracted inside a neighborhood of $p_{0}$, and the
isotopy can be continued. For wrinkle creation, notice this process can be
reversed. Immediately before an embryo occurs we can extend a tentacle out
from $p_{0}$ to contain it. Furthermore while keeping everything embedded, we
can do this so that $T\Phi_{t_{0}}$ is tangent to
$\operatorname{Im}(dL_{t_{0}})$, with given orientation. $\hfill\square$
We now complete the proof of Theorem 1.2. We start by proving the existence
portion.
###### Proposition 7.3.
Let $n>1$, and suppose $(f,F_{s})$ is a formal Legendrian knot in $(Y,\xi)$.
Then there is a Legendrian knot which is formally isotopic to $(f,F_{s})$.
For any Legendrian $L$, $s_{S^{1}\times D^{n-1}}(L)$ is a loose knot in the
same formal isotopy class as $L$, so this proposition implies the existence
theorem for loose Legendrian knots. This proposition is essentially proved in
[12], let us first outline their proof here. Since immersed Legendrians
satisfy an h-principle, we focus on the set of formal Legendrian isotopy
classes in a fixed regular Legendrian homotopy class. A simple calculation
shows this set of formal isotopy classes admits a transitive $\mathbb{Z}$
action, and we then show that $M$-stabilization generates this action by
$\chi(M)$. Thus given any formal Legendrian isotopy class we can first find a
Legendrian immersion $L$ in the correct regular homotopy class, which will
generically be embedded. Then we just pick an $M$ so that $s_{M}(L)$ is in the
correct formal isotopy class; notice we can realize any integer by $\chi(M)$
since $n>1$. Much of this proof is explained in the Appendix A, though there
are a number of gaps. To the author’s knowledge a complete proof does not
exist in the literature, though it has been a “known theorem” since [12].
Here we give a different proof of the statement using our h-principle method.
It is a distinct proof in that it does not require any knowledge about the set
of all formal isotopy classes. In particular, the case for general $Y$ and $L$
is no more difficult that the case of spheres in $\mathbb{R}^{2n+1}_{std}$;
the above proof is more difficult to extend to cases where $\pi_{1}Y\neq 0$ or
when $L$ is not nulhomologous.
_Proof:_ Let $L_{0}\subseteq(Y,\xi)$ be a formal Legendrian knot. By Theorem
5.2, there is a wrinkled Legendrian $L_{1}$ which is formally isotopic to
$L_{0}$. Choose a small neighborhood of $L_{1}$ disjoint from all Legendrian
wrinkles, and let $L_{2}$ be the $S^{1}\times D^{n-1}$-stabilization of
$L_{1}$. Then $L_{2}$ is formally Legendrian isotopic to $L_{1}$ by
Proposition 2.6. Proposition 2.5 implies $L_{2}$ is loose, thus Proposition
6.4 implies there is a Darboux chart $V$, so that $(V,V\cap L_{2})$ is
contactomorphic to the standard resolution of an inside-out wrinkle.
In $L_{2}$, replace $V\cap L_{2}$ with an _unresolved_ inside-out wrinkle;
this defines a prepared wrinkled Legendrian, $L_{3}$. Proposition 7.2 implies
there is a nulhomotopic twist marking $\Phi$ on $L_{3}$. By resolving $L_{3}$
along $\Phi$ as in Proposition 6.3 we get a smooth Legendrian $\tilde{L}$, and
Proposition 7.1 says that $\tilde{L}$ is formal Legendrian isotopic to
$L_{2}$. $\hfill\square$
It remains to prove that any two loose Legendrians which are formally isotopic
are Legendrian isotopic. The proof is nearly identical to the proof of their
existence. We will in fact prove a slightly stronger statement:
###### Theorem 7.4.
Let $n>1$. Suppose $L_{0}$, $L_{1}\subseteq(Y,\xi)$ are two loose Legendrian
knots, with a formal Legendrian isotopy between them. Then they are Legendrian
isotopic, and further the Legendrian isotopy can be chosen to be formally
isotopic (rel endpoints) to the given formal isotopy.
_Proof:_ Suppose $L_{0}$, $L_{1}$ are loose Legendrian knots with a formal
isotopy $L_{t}$ between them. Let $B$ be a Darboux ball so that $B\cap L_{0}$
is a loose chart. We then can pick an ambient smooth isotopy $\zeta_{t}^{s}$
so that $B\cap\zeta_{t}^{1}(L_{t})=B\cap L_{0}$, which is the identity on
$\\{t=0\\}\cup\\{s=0\\}$ and $\zeta_{1}^{s}$ is a contact isotopy. Though
$\zeta_{t}^{s}$ cannot be made into a contact isotopy, $\zeta_{t}^{1}(L_{t})$
is a formal Legendrian isotopy with bundle homtoopy $d\zeta_{1-s}^{t}\circ
F^{t}_{s}$, where $F^{t}_{s}:TL\to TY$ is the bundle homtoopy for $L_{t}$.
Thus $\zeta_{t}^{1}(L_{t})$ is a formal isotopy between $L_{0}$ and $L_{1}$,
we relabel it $L_{t}$.
Now we apply Theorem 5.2 to find a wrinkled Legendrian isotopy between $L_{0}$
and $L_{1}$, which we denote $L^{\prime}_{t}$. We apply the theorem as an
extension from the closed set $\bar{B}$, so $L^{\prime}_{t}$ retains the
property that $B\cap L^{\prime}_{t}$ is a fixed loose chart. By Proposition
6.4 we can find a smaller ball $\tilde{B}\subseteq B$ so that $\tilde{B}\cap
L_{t}$ is isotopic to the standard resolution of an inside-out wrinkle. Let
$\Lambda_{t}$ be the prepared wrinkled Legendrian isotopy which is equal to
$L_{t}$ outside $\tilde{B}$, and $\tilde{B}\cap\Lambda_{t}$ is that inside-out
wrinkle, unresolved. We then apply Proposition 7.2 to get a path of
nulhomotopic twist markings $\Phi_{t}\subseteq\Lambda_{t}$. Since
$\Lambda_{0}$ is smooth outside $\tilde{B}$, $\Phi_{0}$ is a disk with
boundary in $\tilde{B}$; since $\Phi_{0}$ is nulhomotopic we can assume
$\Phi_{0}\subseteq B$, and similarly for $\Phi_{1}$ in $\Lambda_{1}$. Resolve
the prepared wrinkled isotopy $\Lambda_{t}$ along $\Phi_{t}$ using Proposition
6.3, this defines a genuine Legendrian isotopy $\tilde{\Lambda}_{t}$. By
definition $\tilde{\Lambda}_{0}=L_{0}$ and $\tilde{\Lambda}_{1}=L_{1}$.
Furthermore, Proposition 7.1 implies the path $\tilde{\Lambda}_{t}$ is
formally Legendrian isotopic to $L_{t}$ $\hfill\square$
## 8\. Conclusion
We take some time to discuss how Theorem 1.2 relates to other results in the
field. The term “loose Legendrian knot” comes from $3$-dimensional contact
topology [11], there it means a Legendrian knot whose complement is
overtwisted. Our concept of loose knots is significantly different: loose
knots exist in high dimensional Darboux charts. In both cases looseness is a
hypothesis about the global contact topology of the knot complement which
allows us to apply h-principle results to Legendrians. Unlike the overtwisted
complement case, the geometric model defined here is not contained in any
compact region of the complement; instead it is required to intersect the
(standard) tubular neighborhood of the knot in a prescribed way.
Presently, overtwistedness in high dimensions is not understood, if such a
concept exists at all. To justify our use of terminology, we argue that
generally there can only be one flexible class of Legendrian knots. Let
$(Y,\xi)$ be a high dimensional contact manifold which is “overtwisted”. Let
$L$ be a loose Legendrian knot, and suppose $L_{OT}$ is in the same formal
isotopy class, and has overtwisted complement. Since an $M$-stabilization
takes place in a small neighborhood of a point, $L^{\prime}=s_{S^{1}\times
D^{n-1}}(L_{OT})$ is a loose knot, whose complement remains overtwisted. By
Theorem 1.2, $L$ is Legendrian isotopic to $L^{\prime}$. If knots with
overtwisted complement have flexibility properties (as in [13] or [7] in the
$n=1$ case), $L_{OT}$ should be ambient isotopic (or at least contactomorphic)
to $L^{\prime}$. Thus in a hypothetical high dimensional overtwisted manifold,
we expect a Legendrian knot is loose exactly when it has overtwisted
complement.
We can extend our result to a h-principle for loose Legendrian links, where
the definition requires each component of the Legendrian to have a loose
chart, mutually disjoint and also disjoint from the other components of the
Legendrian. Note that a union of loose knots is not necessarily a loose link,
a fact also true of loose links in overtwisted $3$-manifolds.
In [9], Legendrian contact homology is defined (for a certain class of
$(Y,\xi)$), which is a pseudo-holomorphic curve invariant of Legendrian knots.
There it is shown that $LCH(s_{D^{n}}(L))=0$ for any knot $L$. Theorem 1.2
implies every loose knot is the $D^{n}$-stabilization of another knot, thus
every loose Legendrian has trivial $LCH$. Suppose $X$ is any exact symplectic
filling of $Y$, and $\Gamma$ is an exact Lagrangian with Legendrian boundary,
$L$. Then $\Gamma$ induces an augmentation of $LCH(L)$ [8]. The trivial
algebra admits no augmentation, thus it follows that loose knots are not
fillable by exact Lagrangians. The converse is false: a non-loose Legendrian
$T^{2}$ is described in [10] which admits no exact Lagrangian filling (though
it admits a totally real filling).
If a Weinstein manifold admits a handle decomposition so that the top
dimensional handles are attached along loose knots, it is expected to inherit
flexible properties. This topic is explored in much more depth in [5].
For tight contact structures on $3$-dimensional manifolds, one might hope to
find a similar flexible class of Legendrians. However, it is known that no
suitable one exists: the standard unknot is the unique Legendrian knot in its
formal isotopy class [13]. It is further shown there that the only
topologically local structure of Legendrian curves is the number of
stabilizations, and a result from [17] states that for any fixed $k$, there
are formally isotopic knots which are distinct even after $k$ stabilizations
of any type. Together these results imply there is no flexible class of
Legendrian $1$-knots, which can be defined independent of the global topology
of the knot.
Theorem 1.2 is for parametrized Legendrians, so it implies that loose knots
have maximal $\pi_{0}\operatorname{Diff}(L)$ symmetry: any diffeomorphism of
$L$ fixing the classical invariants can be realized by an ambient contact
isotopy. Of course any exotic diffeomorphism homotopic to the identity is an
example of such a symmetry. In contrast, there exists (in particular) an
exotic diffeomorphism of $S^{8}$ which cannot be realized by any contact
isotopy of the standard Legendrian unknot [1].
Given a Legendrian knot $L\subseteq(Y,\xi)$, we define the _twist capacity_ of
$L$ to be the non-negative real number $c(L)=\sup\\{\tfrac{\rho^{2}}{a};\text{
there exists a Legendrian twist }(U,U\cap L)\subseteq(Y,L)$ with parameters
$a,\rho\\}$. The paragraph preceding Proposition 2.3 explains why $c(L)$ is
always strictly positive; the proposition itself states that $c(L)=\infty$
whenever $c(L)>\tfrac{1}{2}$ (which occurs exactly when $L$ is loose). For
spheres in $\mathbb{R}^{2n+1}_{std}$ notice that
$c(L_{1}\\#L_{2})\geqslant\max(c(L_{1}),c(L_{2}))$, where $L_{1}\\#L_{2}$
denotes the connect sum of the knots $L_{1}$ and $L_{2}$. In particular, the
standard Legendrian unknot has smallest capacity among all Legendrian spheres.
Looking to results unique to high dimensional contact topology, we see that a
“sufficiently thick” condition is often crucial. In [22] it is shown that any
contact manifold containing a sufficiently thick Weinstein neighborhood of an
overtwisted contact submanifold is not fillable by a (semi-positive)
symplectic manifold. Note that all contact manifolds contain overtwisted
submanifolds. In [14] it is shown that whenever $r<R<1$, there is a contact
isotopy of $(\mathbb{R}^{2n}\times S^{1},\ker(d\theta-\sum_{i}y_{i}dx_{i}))$
which squeezes $B^{2n}_{R}\times S^{1}$ inside $B^{2n}_{r}\times S^{1}$.
However, this is shown to be false when $r<1<R$. Though these results and
Theorem 1.2 all seem intuitively similar, no concrete connections between
these phenomena are presently understood.
## Appendix A Formal Legendrian Isotopy Classes in $\mathbb{R}^{2n+1}_{std}$
In order for the main result of this paper to useful in practice, we would
like to have an explicit way to tell when two knots are formally isotopic.
This is purely an issue about bundle theory and algebraic topology. The
calculations are not particularly deep, but they are somewhat involved. First
we define two invariants of formal Legendrian knots. Some of the details in
calculation are left to the reader, they can also be found in [9].
###### Definition A.1.
Let $L$ be a formal Legendrian knot in $(Y,\xi)$. $F_{1}$ is a bundle map
$TL\to\xi|_{L}$, so every fiber has Lagrangian image. The homotopy class of
this map in the space of Lagrangian bundle monomorphisms is called the
_rotation class_ of $L$. We denote this class $r(L)$.
Immersed Legendrian knots satisfy an h-principle [19], and the rotation class
classifies them up to regular Legendrian homotopy. If we have two Legendrian
knots which are smoothly homotopic, we can compare their rotation classes as
follows. A formal Legendrian defines an isomorphism $\xi|_{L}\cong
TL\otimes\mathbb{C}$, therefore two formal Legendrians together define a
difference element in $\operatorname{Aut}_{\mathbb{C}}(TL\otimes\mathbb{C})$,
also known as the gauge group of $\xi|_{L}$. Two Legendrians have the same
rotation class if and only if this difference element is in the component of
the identity. If $\xi|_{L}$ is trivial (which is always the case if $\xi|_{Y}$
is trivial) then
$\operatorname{Aut}_{\mathbb{C}}(\xi|_{L})\cong\operatorname{Map}(L,U_{n})$,
thus the difference class $r(L_{0})-r(L_{1})$ is an element of $K^{1}(L)$ in
this case.
###### Definition A.2.
Suppose $n$ is odd, and let $L$ be a formal Legendrian knot in $(Y,\xi)$.
Assume $L$ is orientable and nulhomologous. Extend $F_{s}$ to a path
$\tilde{F}_{s}$ in $\operatorname{Aut}_{\mathbb{R}}(TY|_{L})$. Let $R$ be a
vector field in $TY|_{L}$, positively transverse to $\xi$. Then
$\tilde{F}_{1}^{-1}(R)$ is nowhere tangent to $L$, and the linking number of
the knot with the vector field does not depend on the choice of lifting
$\tilde{F}_{s}$. This integer is called the _Thurston-Bennequin number_ of
$L$, denoted $tb(L)$.
###### Remark.
When $n$ is even, the definition makes sense but the invariant is
uninteresting. In the example $\mathbb{R}^{2n+1}_{std}$, we can equivalently
consider the signed count of self intersections in the Lagrangian projection
(regardless of dimension). If $n$ is odd, the intersection product is skew,
and the order of the inputs are given by height. For even $n$ the intersection
product is commutative, so all the data necessary to calculate $tb(L)$ is
contained in the Lagrangian projection. Together with the Lagrangian
neighborhood theorem, it follows that
$tb(L)=\tfrac{1}{2}(-1)^{{}^{n}/_{2}+1}\chi(L)$ in this case.
###### Proposition A.3.
Let $L$ be a Legendrian knot, and $M\subseteq D^{n}$. Then $r(s_{M}(L))=r(L)$
always. When $n$ is odd, $tb(s_{M}(L))=tb(L)-2\chi(M)$.
_Proof:_ $L$ and $s_{M}(L)$ are Legendrian regular homotopic by the homotopy
$\\{z=t\cdot h(x)\\}$ so the statement about rotation class is clear (see
Definition 2.4 for notation). We calculate $tb(L)$ by taking the signed count
of self intersections in the Lagrangian projection. In the course of the
homotopy $L$ will intersects itself once for each Morse critical point of $h$.
This corresponds to a a sign change for the associated intersection in the
Lagrangian projection, so $tb$ changes by $\pm 2$ for each Morse critical
point. By explicit calculation we see that the sign corresponds to the parity
of the Morse index, so the total change is $2\chi(M)$. $\hfill\square$
We now state a classification of formal Legendrian isotopy classes, assuming
our ambient manifold is $(\mathbb{R}^{2n+1},\xi_{std})$. This tells us that
all embeddings of $L$ are smoothly isotopic when $n>1$ [20]. Similar
calculations can be done for any $(Y,\xi)$, but there are smooth obstructions
and the bundle theory becomes more difficult.
###### Theorem A.4.
We describe formal Legendrian knots up to formal isotopy in
$\mathbb{R}^{2n+1}_{std}$.
(a) Suppose $n$ is odd. If two formal Legendrian knots have the same Thurston-
Bennequin number and rotation class, then they are formally Legendrian
isotopic.
(b) If two formal Legendrian surfaces in $\mathbb{R}^{5}_{std}$ have the same
rotation class, they are formally Legendrian isotopic.
(c) Suppose $n>2$ is even. Then for each rotation class there are at most two
formal Legendrian isotopy classes. If $L$ is simply connected, there are
exactly two.
###### Remarks.
Every set of invariants is realized by a formal Legendrian knot, with the
additional note in case (a) that the parity of $tb(L)$ is determined by
$r(L)$. However note that Proposition 7.3 is false if $n=1$: there is no
Legendrian realizing a formal Legendrian unknot with $tb=0$. For $n>3$ the
parity of $tb(L)$ is determined only by the topology of $L$, for example
$tb(S^{n})$ is odd for any Legendrian sphere in $\mathbb{R}^{2n+1}_{std}$. To
show this, first take the Lagrangian projection of $L$, which is an exact
Lagrangian immersion in $\mathbb{R}^{2n}_{std}$. Notice the parity of $tb(L)$
is equal to the $\bmod\,2$ count of self interesections of this Lagrangian
immersion, in fact this is an invariant of _smooth_ immersions in
$\mathbb{R}^{2n}$ up to regular homotopy. Both smooth and Lagrangian
immersions satisfy h-principles [19], thus the existence of Lagrangian
immersions of a given smooth regular homotopy class is governed by the
inclusion map $\pi_{n}U_{n}\to\pi_{n}V_{2n,n}$. For $n$ odd this is a map
$\mathbb{Z}\to\mathbb{Z}_{2}$, and (a stable shift of) Lemma A.6 implies this
is the zero map except when $n=1,3$.
It is unknown to the author if there exists a calculable invariant in
$\mathbb{Z}_{2}$ which distinguishes the formal isotopy classes in case (c).
Below it is defined as an invariant associated to a smooth isotopy between two
Legendrian knots, which is why the $\pi_{1}L=0$ assumption is needed. The
invariant in question should be a “Thurston-Bennequin-Kervaire
semicharacteristic”, see [2].
_Proof:_ We assume some basic facts about frame bundles, see [3], [23]. Given
two Legendrian knots construct a smooth isotopy $L_{t}$ between them, this
defines a path $\beta_{t}:L\to V_{2n+1,n}$ so that $\beta_{0}$ is a constant
map (here $V_{2n+1,n}$ is the Stiefel manifold of $n$-frames in
$\mathbb{R}^{2n+1}$). $L$ need not admit a global parallelization, since here
$\beta_{t}$ compares $dL_{t}$ to $dL_{0}$ at each point of $L$, and this
difference does not depend on a choice of framing at that point. Said
differently, maps $L\to Gr_{n}(\mathbb{R}^{2n+1}_{std})$ lifting the isotopy
$L_{t}$ can be identified with $\operatorname{Map}(L,V_{2n+1,n})$ by choosing
a connection on the tautological bundle over $Gr_{n}(\mathbb{R}^{2n+1})$.
Inside $V_{2n+1,n}$, identify $U_{n}$ as the subset of Legendrian frames.
(Though “which frames are Legendrian” depends on the point in
$\mathbb{R}^{2n+1}$, these inclusions are all homotopy equvialent to the
inclusion $U_{n}\subseteq O_{2n}\subseteq O_{2n+1}\to V_{2n+1,n}$.)
$\beta_{1}$ has image inside of $U_{n}$ since $L_{1}$ is Legendrian, and so
$\beta_{t}$ defines an element
$\beta\in\pi_{1}\left(\operatorname{Map}\left(L,V_{2n+1,n}\right),\operatorname{Map}\left(L,U_{n}\right)\right)$.
Notice that $\operatorname{Map}(L,V_{2n+1,n})$ is conected since $V_{2n+1,n}$
is $n$-connected.
Our smooth isotopy can be made into a formal Legendrian isotopy exactly when
$\beta=0$. Conversely, given any
$\beta\in\pi_{1}\left(\operatorname{Map}\left(L,V_{2n+1,n}\right),\operatorname{Map}\left(L,U_{n}\right)\right)$
and a Legendrian knot $L_{0}$, we can define a formal Legendrian knot
$(f,F_{s})=(L_{0},\beta_{s})$. If $L_{1}$ is a Legendrian realizing this
formal Legendrian (which exists by Proposition 7.3), then the obstruction
associated to the smooth isotopy between $L_{0}$ and $L_{1}$ is $\beta$.
In the long exact sequence for the pair, notice
$\partial_{*}\beta=r(L_{0})-r(L_{1})\in\pi_{0}\operatorname{Map}(L,U_{n})$.
Thus under the assumption $r(L_{0})=r(L_{1})$ we can lift $\beta$ to
$\tilde{\beta}\in\pi_{1}\operatorname{Map}(L,V_{2n+1,n})$. We pause to prove
some lemmas concerning the homotopy groups of frame bundles.
###### Lemma A.5.
Consider the fibration $O_{n+1}\to O_{2n+1}\to V_{2n+1,n}$. In the homotopy
long exact sequence, the map $\pi_{n+1}V_{2n+1,n}\to\pi_{n}O_{n+1}$ is
injective, except for $n=2,6$. For these two values, $\pi_{n}O_{n+1}$ is
trivial.
_Proof:_ First, consider the case where $n$ is odd. The kernel of our map is
the image of the group $\pi_{n+1}O_{2n+1}$. By Bott periodicity, this group is
finite. But $\pi_{n+1}V_{2n+1,n}\cong\mathbb{Z}$, so the image must be
trivial.
Next, consider the case where $n$ is even, and not equal to $2$ or $6$.
Consider the map $\pi_{n}O_{n+1}\to\pi_{n}O_{2n+1}$. The first group
classifies $(n+1)$-vector bundles on $S^{n+1}$, whereas the second group
classifies stable bundles. Since $TS^{n+1}$ is non-trivial, but stably trivial
[4], we know this map must have non-zero kernel. So
$\pi_{n+1}V_{2n+1,n}\to\pi_{n}O_{n+1}$ has non-zero image. Since
$\pi_{n+1}V_{2n+1,n}\cong\mathbb{Z}_{2}$, this implies the map is injective.
$\hfill\square$
###### Lemma A.6.
For all $n>2$, $\pi_{n+1}U_{n}\to\pi_{n+1}V_{2n+1,n}$ is the zero map. For
$n=2$, it is a surjection.
_Proof:_ Let $n\neq 2,6$. Notice that the inclusion $U_{n}\subseteq
V_{2n+1,n}$ factors through $U_{n}\subseteq O_{2n+1}\to V_{2n+1,n}$. By the
previous lemma, the second map is trivial on $\pi_{n+1}$.
Case $n=6$: Consider the map $\pi_{n+1}U_{n}\to\pi_{n+1}O_{2n+1}$. This is in
the stable range, so we can look at the exact sequence
$\pi_{n+1}U\to\pi_{n+1}O\to\pi_{n+1}(O/U)\to\pi_{n}U.$
By Bott periodicity, $\pi_{n}U\cong 0$, and
$\pi_{n+1}(O/U)\cong\pi_{n+1}(\Omega O)\cong\mathbb{Z}_{2}$. It follows that
the map $\pi_{n+1}U_{n}\to\pi_{n+1}O_{2n+1}$ is multiplication by $2$, as a
map $\mathbb{Z}\to\mathbb{Z}$. Therefore, the map
$\pi_{n+1}U_{n}\to\pi_{n+1}V_{2n+1,n}\cong\mathbb{Z}_{2}$ is zero.
Case $n=2$: Since $\pi_{n}O_{n+1}\cong 0$, we know $\pi_{n+1}O_{2n+1}$
surjects onto $\pi_{n+1}V_{2n+1,n}$. This, together with the fact that
$\pi_{n+1}U_{n}\to\pi_{n+1}O_{2n+1}$ is an isomorphism, implies the result.
$\hfill\square$
###### Lemma A.7.
Let $n$ be odd. From the fibrations $O_{n+1}\to O_{2n+1}\to V_{2n+1,n}$ and
$O_{n}\to O_{n+1}\to S^{n}$, form the composition map
$tb:\pi_{n+1}V_{2n+1,n}\to\pi_{n}O_{n+1}\to\pi_{n}S^{n}$. Then $tb$ is an
injection, in fact, it is the map $\mathbb{Z}\mapsto 2\mathbb{Z}$.
_Proof:_ We know from Lemma A.5 that the first map is an injection, so the
lemma is equivalent to “Is
$\operatorname{Im}(\pi_{n+1}V_{2n+1,n})\cap\ker(\to\pi_{n}S^{n})$ trivial in
$\pi_{n}O_{n+1}$?” By the exact sequences, this is equivalent to the
intersection
$\ker(\to\pi_{n}O_{2n+1})\cap\operatorname{Im}(\pi_{n}O_{n})\subseteq\pi_{n}O_{n+1}$.
This statement is then equivalent to “Suppose $\nu$ is an $(n+1)$-plane bundle
on $S^{n+1}$ which is both stably trivial and of zero euler class. Is $\nu$
trivial?”. But $\nu$ must be trivial; the tangent bundle of the sphere
generates the group of stably trivial vector bundles over $S^{n+1}$, and it
has nonzero euler class. The second statement follows since the euler class of
this generator is $2$. $\hfill\square$
Returning to the proof of the theorem, recall our isotopy is unobstructed if
$\tilde{\beta}\in\pi_{1}\operatorname{Map}(L,V_{2n+1,n})$ is in the image of
$\pi_{1}\operatorname{Map}(L,U_{n})$. Take any degree one map $L\to S^{n}$.
Since $V_{2n+1,n}$ is $n$-connected this map induces an isomorphism
$\pi_{1}\operatorname{Map}(L,V_{2n+1,n})\cong\pi_{n+1}V_{2n+1,n}$, identifying
the image of $\pi_{1}\operatorname{Map}(L,U_{n})$ with that of
$\pi_{n+1}U_{n}$.
For part (b), $n=2$: Lemma A.6 implies that that $\tilde{\beta}$ is in the
image of $\pi_{n+1}U_{n}$, thus $\beta=0$.
In part (a), $n$ is odd. We claim $tb(\tilde{\beta})=tb(L_{0})-tb(L_{1})$.
Since Lemma A.7 says $tb:\pi_{n+1}V_{2n+1,n}\to\pi_{n}S^{n}$ is an injection
and $tb(L_{0})=tb(L_{1})$ by hypothesis, this implies $\tilde{\beta}=0$.
Consider the geometric meaning of the maps in Lemma A.7. The first map to
$\pi_{n}O_{n+1}$ can be interpreted as a difference class of the Legendrian
framings of the normal bundle induced by the isotopy. The second map, induced
by $O_{n+1}\to S^{n}$, is simply “pick one vector in the frame”, here we think
of it as choosing the Reeb vector field. Thus $tb(\tilde{\beta})$ represents
the difference class of the Reeb framings, which equals $tb(L_{0})-tb(L_{1})$.
For part (c), $n>2$ is even.
$\tilde{\beta}\in\pi_{n+1}V_{2n+1,n}\cong\mathbb{Z}_{2}$, which implies there
are at most two formal Legendrian isotopy classes for the given rotation
class. However $\tilde{\beta}$ is an invariant of a smooth isotopy: one can
imagine a isotopy from a Legendrian to itself so that $\tilde{\beta}\neq 0$.
If such a case exists there will only be one formal isotopy class for the
given rotation class. Under the assumption $\pi_{1}L=0$, the space of smooth
embeddings $L\hookrightarrow\mathbb{R}^{2n+1}_{std}$ is simply connected [6]
and thus this cannot occur. $\hfill\square$
## References
* [1] M. Abouzaid, _Framed bordism and Lagrangian embeddings of exotic spheres_ , Ann. of Math. (to appear).
* [2] M. Atiyah, _Vector fields on manifolds_ , Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, Heft 200 (1970).
* [3] R. Bott, _The stable homotopy of the classical groups_ , Ann. of Math., (2) 70 (1959), 313-337.
* [4] R. Bott and J. Milnor, _On the parallelizability of spheres_ , Bull. Amer. Math. Soc. 64 (1958), 87-89.
* [5] K. Cieliebak and Y. Eliashberg, _From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds_ , Colloquium Publications, 59. AMS, 2012.
* [6] J.-P. Dax, _Etude homotopique des espaces de plongements_ , Ann. Scient. de l’Ècole Norm. Sup., 5 (1972), 303-377.
* [7] K. Dymara, _Legendrian knots in overtwisted contact structures_ , www.arxiv.org/abs/mathGT/0410122 (2004).
* [8] T. Ekholm, _Rational symplectic field theory over $\mathbb{Z}_{2}$ for exact Lagrangian cobordisms_, J. Eur. Math. Soc., 10 (2008), 641-704.
* [9] T. Ekholm, J. Etnyre, and M. Sullivan, _Non-isotopic Legendrian submanifolds in $\mathbb{R}^{2n+1}$_, J. Differential Geometry, 71 (2005), 85-128.
* [10] T. Ekholm, T. Kálmán _Isotopies of Legendrian 1-knots and Legendrian 2-tori_ , J. Symplectic Geom. 6, 4 (2008), 407-460.
* [11] Y. Eliashberg, _Classification of overtwisted contact structures_ , Invent. Math., 98 (1989), 623-637.
* [12] Y. Eliashberg, _Topological characterization of Stein manifolds of dimension $>2$_, Internat. J. Math., 1 (1990), 29-46.
* [13] Y. Eliashberg and M. Fraser, _Topologically trivial Legendrian knots_ , J. Symplectic Geom. 7, 2 (1998/2009), 77-127.
* [14] Y. Eliashberg, S. Kim, and L. Polterovich, _Geometry of contact transformations and domains: orderability versus squeezing_ , Geom. Topol. 10 (2006), 1635-1747.
* [15] Y. Eliashberg and N. Mishachev, _Introduction to the h-Principle_ , Graduate Studies in Mathematcs, 48. AMS, 2002.
* [16] Y. Eliashberg and N. Mishachev, _Wrinkled embeddings_ , Foliations, geometry, and topology; Comtemp. Math., 498 (2009), 207-232.
* [17] J. Etnyre and K. Honda, _On connected sums and Legendrian knots_ , Adv. Math., 179 (2003), 59-74.
* [18] D. Fuchs and S. Tabichnikov, _Invariants of Legendrian and transverse knots in the standard contact space_ , Topology, 5 (1997), 1025-1053.
* [19] M. Gromov, _Partial Differential Relations_ , Springer-Verlag, 1986.
* [20] A. Haefliger, _Plongements différentiables dans le domaine stable_ , Comment. Math. Helv., 37 (1962), 155-176.
* [21] K. Honda, _On the classification of tight contact structures I_ , Geom. Topol. 4 (2000), 309–368.
* [22] K. Niederkrüger and F. Presas, _Some remarks on the size of tubular neighborhoods in contact topology and fillability_ , Geom. Topol. 14, 2 (2010), 719-754.
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|
arxiv-papers
| 2012-01-11T05:39:39 |
2024-09-04T02:49:26.152637
|
{
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"authors": "Emmy Murphy",
"submitter": "Emmy Murphy",
"url": "https://arxiv.org/abs/1201.2245"
}
|
1201.2297
|
∎
11institutetext: Anh D. Phan and Lilia M. Woods 22institutetext: Department of
Physics, University of South Florida, Tampa, Florida 33620, USA
Tel.: +1-813-974-8489
Fax: +1-813-974-5813
22email: anhphan@mail.usf.edu 33institutetext: N. A. Viet 44institutetext:
Institute of Physics, 10 Daotan, Badinh, Hanoi, Vietnam
# Temperature phase transition model for the DNA-CNTs-based nanotweezers
Anh D. Phan Lilia M. Woods and N. A. Viet
(Received: date / Accepted: date)
###### Abstract
DNA and Carbon nanotubes (CNTs) have unique physical, mechanical and
electronic properties that make them revolutionary materials for advances in
technology. In state-of-the-art applications, these physical properties can be
exploited to design a type of bio-nanorobot. In this paper, we present the
behaviors of DNA-based nanotweezers and show the capabilities of controlling
the robotic device. The theoretical calculations are based on the Peyrard-
Bishop model for DNA. Furthermore, the influence of the van der Waals force
between two CNTs on the opening and closing of nanotweezers is studied in
comparison with the stretching forces of DNA.
###### Keywords:
van der Waals interaction Carbon nanotubes DNA model
## 1 Introduction
In the past several years, researchers have made much progress in synthesizing
new materials and developing fabrication techniques necessary for nanoscaled
device production. This progress has been particularly important for
applications utilizing physical systems intended for biological and medical
purposes.In this regard, biophysical devices at the nanoscale open up novel
possibilities for diagnostic and therapeutic applications.
DNA and carbon nanotubes (CNTs) are interesting and important systems in
nanoscience. They have been the subject of many investigations in the past two
decades1 ; 2 ; 3 ; 4 ; 23 . DNA is composed of two long polymer strands
organized in a double helical structure, where each strand consists of
repeating units (nucleotides)24 . CNTs are quasi-one dimensional cylindrically
wrapped graphene sheets with properties uniquely defined by theregistry
dependence of the wrapping given by a chirality index $(n,m)$ 25 . Various
applications of DNA/CNT complexes have been exploited with potential for
biosensors 6 , DNA transporters 7 , and field effect transistors 8 . The
DNA/CNT is a composite with complicated structure with temperature dependent
motion dynamics. Recently, using molecular dynamics simulations researchers
have proposed molecular tweezers combining DNA and CNTs 19 \- a device with
further technological and scientific potential.
A theoretical model of a geometrical soliton of DNA structure was constructed
for the first time by Englander 9 (E model). In this model, one of the
strands of the DNA is represented as a chain of pendula interacting with the
another fixed similar strand. The E model explains the existence of DNA open
state due to nonlinear excitations. In addition, the DNA structure and
dyanmics has been modeled in terms of the Peyrard-Bishop (PB) model 1 , which
has been succssessful in explaining DNA denaturation transitions, pre-melting
dynamics, and thermal transport. In the PB model, backbone of DNA is described
as chains of particles with nearest neighboring potentials. However, the
models ignore the helicoidal structure of the DNA molecule, the context of DNA
flexibility, and the properties associated with it.
CNTs are chemically inert and they interact with other materials via long-
ranged dispersive forces, such as van der Waals (vdW) forces. The vdW
interactions of graphitic nanostructures can be described via pairwise
interatomic Lennard-Jones (LJ) potentials 11 . This approach relies on
knowledge of the coupling Hamaker constants and it predicts the equilibrium
separation correctly. The LJ potential has been applied to model mutual
interaction between various CNTs as well as CNT based devices 26 ; 27 .
In this work we investigate the dynamics of hybrid DNA/CNT nanotweezers by
employing the PB and vdW-LJ models. This dynamics of stretching in terms of
its velocity and acceleartion due to environmental temperature changes is
investigated. The critical temperature where a melting transition of the
DNA/CNT takes place is presented. Comparisons between the strength of the
involved forces showing the temperature-dependent motion is dominated by the
stretching of the H bonds and bases, while the CNT vdW interaction is weaker.
The rest of the paper is organized as follows: In Sec. II, the theoretical
structure model, behavior and interactions of DNA-based nanotweezer are
introduced. In Sec. III, numerical results are presented. The conclusions are
given in Sec. IV.
## 2 Model and mathematical background
The proposed nanotweezer architecture is assembled by attaching the reactive
ends of two single wall CNTs to the DNA strands as shown in Fig. 1. The rest
of the end C bonds are saturated via H atoms. The size of this hybrid is quite
large, approximately thousands atoms, thus full quantum mechanical atomistic
treatment is not possible. The PB model is relatively simple 1 , which
describes the DNA two strands as a coupled pendulum system.
Figure 1: (Color online) Schematics of the DNA/CNTs-based nanotweezer.
### 2.1 Model of DNA dynamics
According to the PB model 1 , the DNA double strand is modeled by two parallel
chains of nucleotides via nearest-neighbor harmonic oscillator interactions.
The potential for the Hydrogen bonds is also included. The relevant
Hamiltonian is given as follows 1 ; 2
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\left[\frac{1}{2}m\left(\dot{u}_{n}^{2}+\dot{v}_{n}^{2}\right)+\frac{1}{2}k\left(u_{n}-u_{n-1}\right)^{2}+\right.$
(1)
$\displaystyle\left.\frac{1}{2}k\left(v_{n}-v_{n-1}\right)^{2}+V\left(u_{n}-v_{n}\right)\right],$
where $u_{n}$ and $v_{n}$ are the nucleotide displacements from equilibrium
along the direction of the hydrogen bonds for each strand. $m$ is the mass of
each nucleotide (taken to be the same for each unit), while $k$ is harmonic
oscillator coupling constant of the nearest-neighbor longitudinal interaction
along each strand in units of $eV/\AA^{2}$. The potential for the Hydrogen
bonds between the two strands is modeled via a Morse potential
$V(r)=D[e^{-\alpha r}-1]^{2}$. Here, $D$ is the dissociation energy and
$\alpha$ is a parameter. It is important to note that the Morse potential
represents the hydrogen bonds between complementary bases, the repulsive
interactions of the phosphate, and the influence of the solvent environment.
The dynamics of the system described by Eq.(1) is conveniently described using
a set of new variables $x_{n}=(u_{n}+v_{n})/\sqrt{2}$ and
$y_{n}=(u_{n}-v_{n})/\sqrt{2}$, representing the in-phase and out-of-phase
motion of the two strands, respectively. Using this separation of variables,
the Hamiltonian is decoupled. An important point is that $y_{n}$ represents
the relative displacements between two nucleoid at the site $n$ in different
strands. It reflects the stretching of DNA. Here we consider the out-of-phase
displacements stretch of the hydrogen bonds given by $H_{y}$
$\displaystyle
H_{y}=\sum_{n=1}^{N}\left[\frac{1}{2}m\dot{y}_{n}^{2}+\frac{1}{2}k\left(y_{n}-y_{n-1}\right)^{2}+V\left(2y_{n}\right)\right].$
(2)
In the case of large number of nucleotides $N\rightarrow\infty$ and $H$ is
independent on the particular site $n$. Perfroming statistical averaging in
the canonical ensemble, the Schrodinger equation of a single mode $y$ using
$H_{y}$ is given by 1 ; 15 ; 16
$\displaystyle\left(-\frac{1}{2\beta^{2}k}\frac{\partial^{2}}{\partial
y^{2}}+V(2y)\right)\varphi(y)=\varepsilon\varphi(y),$ (3)
where, $\beta=1/k_{B}T$, and $k_{B}$ is the Boltzmann constant. The exact
solution for eigenenergies is given 26
$\displaystyle\varepsilon_{n}=\frac{1}{2\beta}\ln\left(\frac{\beta
k}{2\pi}\right)+\frac{2\alpha}{\beta}\sqrt{\frac{D}{k}}\left(n+\frac{1}{2}\right)-\frac{\alpha^{2}}{\beta^{2}k}\left(n+\frac{1}{2}\right)^{2}.$
(4)
Eq.(4) has a discrete energy spectrum when $d=(\beta/\alpha)\sqrt{kD}>1/2$.
This allows one to obatin a critical temperature $T_{c}=2\sqrt{kD}/(\alpha
k_{B})$, which is considered as the melting temperature of DNA. The DNA states
are continuous for $T>T_{c}$ and discrete for $T<T_{c}$. For the parameters of
DNA, when we consider $T>200$ $K$, only the value of $n=0$ is taken into
account. There is no excitation state for DNA in our considerations.
From this, the ground state eigenfunction and eigenenergy in the
thermodynamics limit of a large system is obtained as 1 ; 2
$\displaystyle\varphi_{0}(y)=\sqrt{\sqrt{2}\alpha}\frac{(2d)^{d-1/2}}{\sqrt{\Gamma(2d-1)}}e^{-de^{-\sqrt{2}\alpha
y}}e^{-(d-1/2)\sqrt{2}\alpha y},$ (5)
$\displaystyle\varepsilon_{0}=\frac{1}{2\beta}\ln\left(\frac{\beta
k}{2\pi}\right)+\frac{\alpha}{\beta}\sqrt{\frac{D}{k}}-\frac{\alpha^{2}}{4\beta^{2}k}.$
(6)
In addition, the system described via Eq.(3) can be represented as a
quasiparticle with a tempereture dependent effective mass
$m^{*}=\hbar^{2}\beta^{2}k$. At room temperature, the value of the effective
mass is approximately $22.87$ $m_{0}$, here $m_{0}$ is the rest mass of
electron. The average stretching of the hydrogen bonds can also be calculated
via $\left\langle y\right\rangle=\int\varphi_{0}^{2}(y)ydy$ 1 ; 2 .
The stretching force is determined via the expression
$\displaystyle F_{s}=-\frac{\partial V(\left\langle
y\right\rangle)}{\partial\left\langle y\right\rangle}.$ (7)
To investigate thermal properties of DNA, we heated up and cooled down
temperature of the bio-systems flollowing an expression $T=1.14t+300$ (K) 19 .
Here $T$ (K) is the environment temperature, $t$ (ps) is time. Basing on the
average stretching $\left\langle y\right\rangle$ of the coupling constants
pointed out above, the velocity $v=d\left\langle y\right\rangle/dt$ and
acceleration $a=d^{2}\left\langle y\right\rangle/dt^{2}$ of the opening of the
nanotweezers obtained by taking the first and second derivative of the
stretching with respect to time, respectively, are presented in Fig. 2
Figure 2: (Color online) (Color online) The time-dependent velocity and
acceleration of the opening.
For $k=2.10^{-3}$ eV/$\AA^{2}$, the velocity of the opening increases and
reaches to the maximum with the value of $10.34$ m/s at around $t=20$ s. After
that, the velocity drops significantly to zero. It refers that the temperature
corresponding to the peak is $322.5$ K. On the other hand, initially, the
value of acceleration is positive and rises to the maximum value $0.187\times
10^{12}$ $m/s^{2}$ at $-5.2$ s or $294$ K before declining gradually to the
negative side, crossing the time axis at $19.4$ s or $322.5$ K, touching the
bottom $-0.164\times 10^{12}$ $m/s^{2}$ at around $45$ s and continuing to
approach to $0$. It can be easily explained due to the fact that below $322.5$
K, the stretching velocity climbs significantly, so the acceleration is
positive. Zero acceleration, of course, is at the relevant bending point of
the opening velocity. Above $322.5$ K, the unzipping velocity declines
notably, and is nearly unchanged. Therefore, the acceleration has the negative
values and goes to zero.
In the same way, for other values of $k=3.10^{-3}$ eV/$\AA^{2}$ and
$k=4.10^{-3}$ eV/$\AA^{2}$, the zero acceleration takes place at $88$ $s$ and
$138$ $s$, respectively. It means that the melting temperatures corresponds to
$401$ $K$ for $k=3.10^{-3}$ eV/$\AA^{2}$ and $456$ $K$ for $k=4.10^{-3}$
eV/$\AA^{2}$. As a result, there is a possibility to obtain the melting
temperature by observing the velocity of stretching.
### 2.2 CNT van der Waals interaction
The vdW interaction between the CNT parts of the DNA nanotweezers is described
via the Lennard-Jones (LJ) approximation. This approach is widely used in
calculating disperssive interactions between graphitic nanostructures because
of its relative simplicity and satisfactory results in determining their
equilibrium configurations 11 . The LJ potential is essentially a pairwise
apprximation, and for extended systems, one typically perfroms integration
over the volumes of the interacting objects. For CNTs, the integration is over
the surfaces of hollow cylinders with radii corresponding to the radii of the
nanotubes. The LJ-vdW potential per unit length for two parallel CNTs with
radii $R_{1}$ and $R_{2}$ is given by 10
Figure 3: (Color online) Sketch of van der Waals interaction between two CNTs.
$\displaystyle
V_{vdW}=\sigma^{2}\int\int\left(-\frac{A}{\rho^{6}}+\frac{B}{\rho^{12}}\right)dS_{1}dS_{2},$
(8)
where $A$ and $B$ are the Hamaker constants corresponding to the attractive
and repulsive contributions, respectively. For graphitic systems, one
typically takes the values for graphite $A=15.2$ eV$\AA^{6}$ and $B=24\times
10^{3}$ eV$\AA^{12}$ 11 . $\sigma=4/\sqrt{3}a^{2}$ is the mean surface density
of Carbon atoms with $a=2.49$ $\AA$ being the lattice constant. Also, the
distance between the CNT surfaces is $\rho$. Perfroming the integration over
the length of the two CNTs with radii $R_{1}$ and $R_{2}$, the LJ-vdW
interaction can be written as 10 :
$\displaystyle V_{vdW}$ $\displaystyle=$ $\displaystyle-\frac{3\pi
A\sigma^{2}R_{1}R_{2}}{8}\int_{0}^{2\pi}\int_{0}^{2\pi}\frac{1}{r^{5}}d\varphi_{1}d\varphi_{2}$
(9) $\displaystyle+\frac{63\pi
B\sigma^{2}R_{1}R_{2}}{256}\int_{0}^{2\pi}\int_{0}^{2\pi}\frac{1}{r^{11}}d\varphi_{1}d\varphi_{2},$
where the in-plane distance between two surface elements is defined as
$r^{2}=(R-R_{1}\cos\varphi_{1}+R_{2}\cos\varphi_{2})^{2}+(R_{1}\sin\varphi_{1}-R_{2}\sin\varphi_{2})^{2}$.
The definitions of $R_{1}$, $R_{2}$, $\varphi_{1}$, $\varphi_{2}$, and $r$ are
sketched in Fig. 3.
Then, applying the first derivative with respect to $R$, we obtain the van der
Waals interaction force per unit length
$\displaystyle F_{vdW}(R)=-\frac{\partial V_{vdW}}{\partial R}.$ (10)
## 3 Numerical results and discussions
As a prototype, we take that both CNTs are identical with the chiral vector
$(5,0)$ and $(6,0)$, and lengths $L_{1}=L_{2}=5$ $nm$. The total Hamiltonian
for the system is composed of two term, that account for the stretching and
van der Waals interaction - $H=H_{y}+V_{vdW}$. Because of the relatively weak
vdW force between the tubes, $V_{vdW}$ is treated as a perturbation compared
to $H_{y}$. The parameters of DNA are $D=0.33$ $eV$ and $\alpha=18$ $nm^{-1}$.
It is important to note that $\varphi_{0}(y)$ and $\varepsilon_{0}$ in the
previous section is the wave function and energy of the ground state of DNA
without the presence of CNTs.
In Fig. 4 and Fig. 5, we show results for the CNT vdW perturbative force
correction as a function of tempertaure and the stretching force. Fig. 5
indicates that $F_{s}$ decreases as $T$ increases. The stretching force goes
to zero at the critical temperature since the properties of DNA change when
$T$ reaches to $T_{c}$.
Figure 4: (Color online) The first-order energy is caused by the van der Waals
interactions between two CNTs.
Obviously, the wave function is temperature-dependent, so the energy and
energy shift are functions of temperature. The value of $\varepsilon_{0}$ for
three values of $k$ at this range of temperature varies from $220$ meV to
$280$ meV. It means that the influence of the van der Waals interaction on the
wave function and the energy in the ground state is minor. We can calculate
separately the interactions of DNA and CNTs. An additional point is that the
larger temperature is, the smaller the first-order pertubation of energy is. A
simple reason for this problem is that when temperature increases, two DNA
strands are opened 1 and it leads to a rapid growth of distance between two
CNTs.
It is remarkable that we have studied the van der Waals interaction and the
pertubation energy between two parallel CNTs. This configuration also is used
in order to calculate all of the van der Waals interactions below.
Nevertheless, in actual cases, we have two crossed CNTs. The dispersion
interaction in real biosystems is weaker than that in the parallel state.
Therefore, we can utilize the wave function $\varphi_{0}(y)$ in the following
calculations without addional terms due to the perturbation theory.
Figure 5: (Color online) The unzipping force as a function of temperature .
It is clearly seen in Fig. 5, at the critical temperature $T_{c}$, the
stretching force vanishes because two strands of DNA are broken for $T>T_{c}$.
The opening force of DNA is very large at low temperature. The smaller the
temperature is, the smaller distance between two strands is. This force
decreases when increasing temperature since the separation distance is larger
and larger.
These results have aggrement about the range of magnitude force with
experimental data and previous calculations 21 ; 22 . The increase of $k$
causes to the growth of stretching force due to the fact that the binding of
DNA rises.
Lets consider the interaction between two CNTs attached in the ends of DNA.
There are several types of DNA existing in nature such as B-DNA and Z-DNA.
Since the diameter of DNA is approximately $2.37$ nm for B-DNA and $1.84$ nm
for Z-DNA. We assume that the initial distance between two centers of CNTs is
$1.5$ nm. It is important to note that the van der Waals force is attractive
at this range of distance and the sign of this force should be minus. The
magnitude of van der Waals interaction between two CNTs is presented in Fig.
6.
Figure 6: (Color online) The van der Waals forces between two parallel CNTs
(5,0) and (6,0) as a function of the separation distance between two centers
of CNTs.
For $k=2.10^{-3}$ eV/$\AA^{2}$, if $T<277$ $K$, the stretching force is much
larger than the van der Waals force of CNTs (5,0) and (6,0) at the initial
state. Therefore, it is easy to control the opening and closing of DNA by
cooling down or heating up. At low temperature, the contribution of the
dispersion force in the movement of DNA strands is minor. However, it can rise
to significant role when $T>277$ $K$. We can do the same way with
$k=3.10^{-3}$ eV/$\AA^{2}$ and $k=4.10^{-3}$ eV/$\AA^{2}$.
Figure 7 shows the forces between CNT (5,0) and different CNTs at the certain
distances. In order to control the opening and closing of nanotweezers, the
van der Waals force is weaker than the stretching forces. It is difficult to
operate the movement of nanotweezers if two CNTs have large radii.
Figure 7: (Color online) The van der Waals forces between CNT (5,0) and
another CNT.
When we heat up the biosystem, two ends of DNA are separated by the stretching
force. At the larger temperature, the unzipping force is much larger than the
van der Waals interactions, the nanotweezers are opened. The obtained results
agree with the previous simulation study 19 . Therefore, in our nanorobots,
the movements of CNTs can be controlled by changing temperature. In addition,
the van der Waals interaction between two cylinders is proportional to the
length of tubes. If we want to have the smaller van der Waals interaction, it
is possible to choose the length 1 nm or 2 nm. Another point is that long CNTs
are bent because of the van der Waals interaction. As a consequence, the
length of tubes should not be large in designing the bio-nanorobots.
## 4 Conclusions
The use of intelligence, sensing and actuation nanodevices in surgery, medical
treatments and materials science is a reality which has become a hot topic in
the biomedical industry and research in recent years. Bio-nanorobots provide
further advance not only in the nanotechnology, but also efficient approaches
for disease treatment. Our studies showed the behavior and architecture of the
bio-nanotweezers. The temperature dependence of the opening displacements of
tweezers is presented and gives researchers some principles to understand the
operation of DNA-based molecular machines and devices. In addition, the
velocity and acceleration of the opening and closing tweezers as a function of
time are speculated. The theoretical calculations are easy to understand and
agree qualitatively with the previous works. Further research on these systems
can considerably extend interdisciplinary implications for the technology.
###### Acknowledgements.
We thank Professor M. Peyrard for helpful discussions and comments. We
gratefully acknowledge support through the Department of Energy under Contract
No. DE-FG02-06ER46297. The work was partly funded by the Nafosted Grant No.
103.06-2011.51.
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|
arxiv-papers
| 2012-01-11T12:51:11 |
2024-09-04T02:49:26.167638
|
{
"license": "Public Domain",
"authors": "Anh D. Phan, N. A. Viet",
"submitter": "Anh Phan Mr.",
"url": "https://arxiv.org/abs/1201.2297"
}
|
1201.2430
|
# A Well-typed Lightweight Situation Calculus††thanks: This work is also
offered to present at the 20th International Workshop on Functional and
(Constraint) Logic Programming (WFLP’11), Odense, Denmark, July 2011.
Li Tan Department of Computer Science and Engineering
University of California, Riverside
Riverside, CA, USA 92507 ltan003@cs.ucr.edu
###### Abstract
Situation calculus has been widely applied in Artificial Intelligence related
fields. This formalism is considered as a dialect of logic programming
language and mostly used in dynamic domain modeling. However, type systems are
hardly deployed in situation calculus in the literature. To achieve a correct
and sound typed program written in situation calculus, adding typing elements
into the current situation calculus will be quite helpful. In this paper, we
propose to add more typing mechanisms to the current version of situation
calculus, especially for three basic elements in situation calculus:
situations, actions and objects, and then perform rigid type checking for
existing situation calculus programs to find out the well-typed and ill-typed
ones. In this way, type correctness and soundness in situation calculus
programs can be guaranteed by type checking based on our type system. This
modified version of a lightweight situation calculus is proved to be a robust
and well-typed system.
## 1 Introduction
Introduced by John McCarthy in 1963 [11], situation calculus has been widely
applied in Artificial Intelligence related research areas and other fields.
This formalism is considered as a dialect of logic programming language and
mostly used in dynamic domain modeling. Based on First Order Logic (FOL) [2]
and Basic Action Theory [10], situation calculus can be used for reasoning
efficiently by virtue of dynamic elements, such as actions and fluents. Basic
concepts of situation calculus are fundamentals of First Order Logic and Set
Theory in Mathematical Logic, which greatly facilitate the process of action-
based reasoning in situation calculus.
In order to make programs sound and correct in semantics, people have proposed
type systems [13] to ensure such significant properties. A well-typed
programming language is determined by two semantic properties: preservation
and progress. The first property makes sure that types are invariant under the
evaluation and typing rules. And the progress property says a well-typed
program never gets stuck. Nevertheless, little attention has been put on
equipping formal languages good at dynamic modeling and reasoning, like
situation calculus, with strong typing mechanisms. Indeed, situation Calculus
is a typed second-order formal language, but from the viewpoint of type
checking, it is not enough to finish smoothly. For instance, in situation
calculus, only typed quantifiers are introduced for basic variables, while as
for other logical expressions consisting of variables and connectives, fluents
and predicates, current situation calculus emphasizes little on how to type
check whether they are well-typed, how to type them thoroughly. Therefore,
equipping other elements in current version of situation calculus with types
is greatly needed for a complete and robust programming language with its type
system, which is definitely feasible according to our investigation.
In this paper, in addition to the handy available typed variables, we propose
to add more typing mechanisms to three basic elements in situation calculus:
situations, actions and objects, consider a classical scenario for a piece of
program based on the modified lightweight situation calculus, and then perform
rigid type checking for the situation calculus program. If type errors are
found, we would provide corresponding recommendation on how to correct the
program into a well-typed one. Furthermore, to support our ideas in practice,
we implement a type checker to semi-automatically finish the type checking
work instead of working manually.
We organize our paper in the following way: section 2 introduces the related
work on typing situation calculus and its variants; Section 3 presents the
basic ideas on type systems and situation calculus in a straightforward way;
Section 4 illustrates the primary ideas on how to type a lightweight core of
the original situation calculus; Section 5 evaluates our typing mechanisms by
type checking an existing piece of program in situation calculus and section 6
concludes this paper.
## 2 Related Work
Due to its powerful action-based reasoning ability, situation calculus is
often chosen as the formalism to express other models and programming
languages which are either too complex to understand and use, like Artificial
Intelligence in games [6] and Planning Domain Definition Language (PDDL) [7],
or a little powerless to represent an entire complicated systems of different
types, like Action Description Language (ADL) [12]. In the literature
employing situation calculus as a formal method to express the semantics in
PDDL [4] and ADL [5], the authors have tried to introduce some typing
mechanisms, which is only limited to add type element in syntax, and only
applied to variables. Other significant terms, such as fluents and predicates,
are still typeless. Moreover, in semantics and reasoning, typing mechanisms
are hardly discussed in these papers, neither is type checking.
Yilan Gu et al. [8] proposed a modified version of the situation calculus
built using a two-variable fragment of the first-order logic extended with
counting quantifiers. By introducing several additional groups of axiom to
capture taxonomic reasoning and using similar regression operator in Raymond
Reiter’s work [14], the projection and executability problems are proved
decidable although an initial knowledge base is incomplete and open. While
their system concerns primarily on semantics of the new components proposed
but rarely talks about typing on them, our well-typed version of situation
calculus mentions typing mechanisms together with a modified situation
calculus version in an all around way.
There are also some attempts on modifying situation calculus only based on a
lightweight version of the original one. Gerhard Lakemeyer et al. [9] proposed
a new logic dialect of situation calculus with the situation terms suppressed,
namely, . That is, it is merely a similar formalism as a part of the current
situation calculus. Moreover, in this paper, the authors consider how to map
sentences between and situation calculus and try to prove is powerful enough
to handle most cases as the situation calculus does, but mention little about
how to type their new logic system as a fragment of situation calculus.
## 3 Background Knowledge
### 3.1 Type Systems
In the discipline of computer science, modern type systems are regarded as a
formal mechanism originated from Alonzo Church’s $\lambda$ calculus proposed
in 1940 [3]. One possible definition of a type system is “a tractable
syntactic framework for classifying phrases according to the kinds of values
they compute” [13]. By associating types with each computed value, a compiler
can detect meaningless or invalid code written in a given programming
language. For instance, the expression “mix = 29 + “Tan”” cannot get through
type checking since a string cannot be added to a number.
There are many branches in type systems, such as inferred typing and manifest
typing (implicit and explicit), and strong typing and weak typing. As for type
checking, people can utilize dynamic type checking and static type checking,
or a combination of both.
The primary and most obvious purpose of using type systems is to guarantee the
correctness of programs, i.e., detect potential errors, while a well-typed
system can further ensure the soundness (safety) of programs. The most
important characteristics of a well-typed system are properties of
preservation and progress. The former one makes sure a term can keep its type
passed into the term that it is evaluated to, and the latter keeps
reachability of a term: a typed term can either turn into a value or another
related term, which means a well-typed term will not get stuck. In this paper,
we plan to equip the current version of situation calculus with appropriate
type system besides several original ones for variables. Thus, a program
written in situation calculus can be easily type checked correct or not.
### 3.2 Situation Calculus
Situation Calculus [11] is a formal method based on First Order Logic and Set
Theory in Mathematical Logic, with a strong ability of action-based reasoning.
This formalism is considered as a dialect of logic programming language and
mostly used in dynamic domain modeling. In situation calculus, the world is
comprised of situations, actions and objects. The semantics of these three key
components in situation calculus is given informally below.
A situation represents a possible world history, simply a sequence of actions,
denoted by a first-order term. The constant $s_{0}$ is used to denote the
initial situation, namely, the empty sequence of actions.
An action represents any possible change to the world, denoted by a function,
for example, _drop(A)_ , _clean(B)_ and _check_in(ID)_.
An object represents an entity defined in the domain of a specific
application, denoted by a first-order term, for example, _x_ , _robot_A_ and
_table_.
Moreover, other than aforementioned three elements, there is another
significant symbol used frequently in situation calculus, namely, fluents. A
fluent represents a relation or a function whose truth values varies from one
situation to the next, called relational fluent or functional fluent
respectively.
Additionally, introduce two predefined binary symbols of fluents as follows:
Function symbol _do_ is defined as _do: Action $\times$ Situation
$\rightarrow$ Situation_, which maps an action _a_ and a situation _s_ to a
new situation called successor situation, which results from performing the
action _a_ in the situation _s_. This successor situation is denoted as _do(a,
s)_.
Predicate symbol _Poss_ is defined as _Poss: Action $\times$ Situation_.
Similarly as above, _Poss(a, s)_ means it is possible to execute the action
_a_ in the situation _s_. Note that in the original situation calculus, there
is no return value for _Poss_. For consistency, in our well-typed system, we
assign a _unit_ value for every _Poss_ predicate. In other words, _Poss_ is
defined as _Poss: Action $\times$ Situation $\rightarrow$ Unit_ (Capital ”U”
indicates it is a type but not a value.).
As mentioned before, fluents are used to represent a term whose value varies
according to the changing of situations. As a comparison, another symbol is
defined to denote a term whose value does varies with situations, namely,
predicate. For example, _hunger_status(person, time)_ and
_weather_condition(location, season)_ are relational fluents while
_drop(person, object)_ is a predicate, since in the first two fluents, the
second arguments are actually situations, namely in situation calculus, s, and
in the third term there is no specific situation, but only two objects, which
means the value of this term will not change when situation changes.
## 4 A Well-typed Mechanism in Situation Calculus
### 4.1 A Lightweight Situation Calculus
The situation calculus we study and try to extend here is a lightweight
version of its original form. Similarly as Featherweight Java (FJ), we only
grab some core features in situation calculus and skip derivable forms to keep
our ideas concise and efficient.
According to the language of situation calculus, we keep all the static domain
element: situations, actions and objects, and the majority of functional
elements like fluents _do_ and _poss_ , and all the predicates. The components
we ignore are those that either can derive from other elements or similarly be
expressed by others. For instance, the ordering predicate $\sqsubseteq$, which
defines an ordering relation on situations, can be expressed implicitly by the
return value of other fluents and predicates. Like the expression $s^{\prime}$
$\sqsubseteq$ $s$ which denotes that $s^{\prime}$ is a proper subsequence of
$s$, $s^{\prime}$ could be replaced with a fluent or predicate which leads $s$
to $s^{\prime}$, say, _do(findajob(person:Object, job:Object), s:Situation)_.
Likewise, we replace countably infinitely many predicate symbols with arity
_n_ , $(action\cup object)^{n}$ with $\overline{t}$, which is a shorthand of a
sequence of terms $t_{1},t_{2},\ldots,t_{n}~{}(n\geq 1)$.
### 4.2 Handy Typing Mechanism
In the original situation calculus, several elements such as quantifiers are
typed [14]. The handy typed elements are described formally as follows:
A typed notion _$\tau$(x)_ is used to denote _x_ associated with a finite set
of all possible types:
$\tau(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}x:T_{1}\vee
x:T_{2}\vee\ldots\vee x:T_{n}$, where $T_{1},T_{2},\ldots,T_{n}$ are types of
terms.
Moreover, typed quantifiers are given by virtue of:
$(\forall x:\tau)\phi(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(\forall
x).\tau(x)\supset\phi(x)$,
$(\exists x:\tau)\phi(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}(\exists
x).\tau(x)\wedge\phi(x)$.
Thus, expressions that contain such typed quantifiers could be rewritten as
sequences of conjunctions and disjunctions:
$(\forall
x:\tau)\phi(x)\equiv\phi(T_{1})\vee\phi(T_{2})\vee\ldots\vee\phi(T_{n})$,
$(\exists
x:\tau)\phi(x)\equiv\phi(T_{1})\wedge\phi(T_{2})\wedge\ldots\wedge\phi(T_{n})$.
### 4.3 A New Type System in the Lightweight Situation Calculus
Although the original version of situation calculus equips some components
with corresponding types and semantics, it is not enough to do type checking
based on these definitions. We proposed a new well-typed system to enable
potential task of type checking in a convenient way.
Syntactic Forms
$t::=\ldots\hfill\textbf{terms:}$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}x\hfill variable$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall x\hfill universal~{}quantified~{}variable$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}\exists x\hfill
existential~{}quantified~{}variable$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}\neg t\hfill negative~{}term$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}t_{1}\supset t_{2}\hfill
subset~{}logical~{}connection$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}t_{1}\wedge t_{2}\hfill
conjunction~{}logical~{}connection$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}t_{1}\vee t_{2}\hfill
disjunction~{}logical~{}connection$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}\overline{t}\hfill term~{}sequence$
$bt::=\ldots\hfill\textbf{behavioral terms:}$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}\neg bt\hfill negative~{}behavioral~{}term$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}r(\overline{t},s)\hfill relational~{}fluent$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}f(\overline{t})\hfill predicate$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}\textbf{\emph{do}}(bt,s)\hfill
functional~{}fluent$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}\textbf{\emph{poss}}(bt,s)\hfill
predicate~{}fluent$
$v::=\ldots\hfill\textbf{values:}$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}unit\hfill poss~{}predicate~{}value$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}true\hfill true~{}boolean~{}value$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}false\hfill false~{}boolean~{}value$
$T::=\ldots\hfill\textbf{types:}$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}Unit\hfill type~{}of~{}predicate~{}fluent$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}Bool\hfill type~{}of~{}booleans$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}Situation\hfill type~{}of~{}behavioral~{}terms$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}Action\hfill type~{}of~{}behavioral~{}terms$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}Object\hfill type~{}of~{}terms$
Semantics
Given a world _w_ comprised of situations, actions and objects and a set
_L(w)_ of all instances defined in _w_ , if a term _t_ holds in _w_ , we write
_w_ $\models$ _t_. Given a set of situations
$S=\\{s_{0},s_{1},\ldots,s_{n}\\}~{}(n\geq 0)$, we have:
$w\models x$ | $\Leftrightarrow~{}~{}~{}~{}x\in L(w)$
---|---
$w\models\forall x$ | $\Leftrightarrow~{}~{}~{}~{}\forall s_{i}\in S,w\models x$
$w\models\exists x$ | $\Leftrightarrow~{}~{}~{}~{}\exists s_{i}\in S,w\models x$
$w\models\neg x$ | $\Leftrightarrow~{}~{}~{}~{}w\not\models x$
$w\models t_{1}\supset t_{2}$ | $\Leftrightarrow~{}~{}~{}~{}w\models t_{1}\Rightarrow w\models t_{2}$
$w\models t_{1}\wedge t_{2}$ | $\Leftrightarrow~{}~{}~{}~{}w\models t_{1}~{}\mathrm{and}~{}w\models t_{2}$
$w\models t_{1}\vee t_{2}$ | $\Leftrightarrow~{}~{}~{}~{}w\models t_{1}~{}\mathrm{or}~{}w\models t_{2}$
$w\models\overline{t}$ | $\Leftrightarrow~{}~{}~{}~{}w\models t_{1},w\models t_{2},\ldots,w\models t_{n}$
$w\models\neg bt$ | $\Leftrightarrow~{}~{}~{}~{}w\not\models bt$
$w\models r(\overline{t},s)$ | $\Leftrightarrow~{}~{}~{}~{}w\models\overline{t}~{}\mathrm{and}~{}w\models s~{}in~{}r$
$w\models f(\overline{t})$ | $\Leftrightarrow~{}~{}~{}~{}w\models\overline{t}~{}in~{}f$
$w\models\textbf{\emph{do}}(bt,s)$ | $\Leftrightarrow~{}~{}~{}~{}\exists s_{i}\in S,bt~{}holds~{}in~{}s_{i}$
$w\models\textbf{\emph{poss}}(bt,s)$ | $\Leftrightarrow~{}~{}~{}~{}\exists s_{i}\in S,w\models\big{(}s_{i}\supset\textbf{\emph{do}}(bt,s_{i})\big{)}$
Evaluation Rules t $\rightarrow$ t’
$\frac{(x)bt~{}\rightarrow~{}(x^{\prime})bt}{(\forall
x)bt~{}\rightarrow~{}(\forall x^{\prime})bt}$ E-Unv
$\frac{(x)bt~{}\rightarrow~{}(x^{\prime})bt}{(\exists
x)bt~{}\rightarrow~{}(\exists x^{\prime})bt}$ E-Est
$\frac{t~{}\rightarrow~{}t^{\prime}}{\neg t~{}\rightarrow~{}\neg t^{\prime}}$,
$\frac{bt~{}\rightarrow~{}bt^{\prime}}{\neg bt~{}\rightarrow~{}\neg
bt^{\prime}}$ E-Neg
$\frac{t_{1}~{}\rightarrow~{}t_{1}^{\prime}}{t_{1}~{}\supset~{}t_{2}~{}\rightarrow~{}t_{1}^{\prime}~{}\supset~{}t_{2}}$
E-Spt
$\frac{t_{1}~{}\rightarrow~{}t_{1}^{\prime}}{t_{1}~{}\wedge~{}t_{2}~{}\rightarrow~{}t_{1}^{\prime}~{}\wedge~{}t_{2}}$
E-Conj
$\frac{t_{1}~{}\rightarrow~{}t_{1}^{\prime}}{t_{1}~{}\vee~{}t_{2}~{}\rightarrow~{}t_{1}^{\prime}~{}\vee~{}t_{2}}$
E-Disj
$\frac{t_{1}~{}\rightarrow~{}t_{1}^{\prime}}{t_{1},~{}t_{2},~{}\ldots,~{}t_{n}~{}\rightarrow~{}t_{1}^{\prime},~{}t_{2},~{}\ldots,~{}t_{n}}$
E-Seq
$\textbf{\emph{do}}(bt,s)\rightarrow[s\mapsto s^{\prime}]bt$ E-Do
$\textbf{\emph{poss}}(bt,s)\rightarrow s\supset[s\mapsto s^{\prime}]bt$ E-Poss
Typing Rules W $\vdash$ t : T
Here we continue to use _W_ (rather than the lower case _w_ used in semantics)
instead of conventional $\Gamma$ to denote a typing context. Formally, we
have:
$\emph{W}~{}\vdash~{}true:Bool$ T-True
$\emph{W}~{}\vdash~{}false:Bool$ T-False
$\frac{x~{}:~{}T~{}\in~{}\normalsize\emph{W}}{\normalsize\emph{W}~{}\vdash~{}x~{}:~{}T}$
T-Var
$\frac{\forall
r(x~{}:~{}T,~{}\overline{t}-x,~{}s)~{}\in~{}\normalsize\emph{W}}{\normalsize\emph{W}~{}\vdash~{}(\forall
x~{}:~{}T)~{}r(\overline{t},~{}s)}$ T-Unv1
$\frac{\exists
r(x~{}:~{}T,~{}\overline{t}-x,~{}s)~{}\in~{}\normalsize\emph{W}}{\normalsize\emph{W}~{}\vdash~{}(\exists
x~{}:~{}T)~{}r(\overline{t},~{}s)}$ T-Est1
$\frac{\forall
f(x~{}:~{}T,~{}\overline{t}-x)~{}\in~{}\normalsize\emph{W}}{\normalsize\emph{W}~{}\vdash~{}(\forall
x~{}:~{}T)~{}f(\overline{t})}$ T-Unv2
$\frac{\exists
f(x~{}:~{}T,~{}\overline{t}-x)~{}\in~{}\normalsize\emph{W}}{\normalsize\emph{W}~{}\vdash~{}(\exists
x~{}:~{}T)~{}f(\overline{t})}$ T-Est2
$\frac{\normalsize\emph{W}~{}\vdash~{}t~{}:~{}T}{\normalsize\emph{W}~{}\vdash~{}\neg
t~{}:~{}T}$,
$\frac{\normalsize\emph{W}~{}\vdash~{}bt~{}:~{}T}{\normalsize\emph{W}~{}\vdash~{}\neg
bt~{}:~{}T}$ T-Neg
$\frac{\normalsize\emph{W}~{}\vdash~{}(t_{1}~{}:~{}T_{1})~{}\supset~{}(t_{2}~{}:~{}T_{2})}{\normalsize\emph{W}~{}\vdash~{}(\forall
x~{}\in~{}t_{1})~{}x~{}:~{}T_{1}~{}\supset~{}(\forall
y~{}\in~{}t_{2})~{}y~{}:~{}T_{2}}$ T-Spt
$\frac{\normalsize\emph{W}~{}\vdash~{}(t_{1}~{}:~{}T_{1})~{}\wedge~{}(t_{2}~{}:~{}T_{2})}{\normalsize\emph{W}~{}\vdash~{}(\forall
x~{}\in~{}t_{1})~{}x~{}:~{}T_{1}~{}\wedge~{}(\forall
y~{}\in~{}t_{2})~{}y~{}:~{}T_{2}}$ T-Conj
$\frac{\normalsize\emph{W}~{}\vdash~{}(t_{1}~{}:~{}T_{1})~{}\vee~{}(t_{2}~{}:~{}T_{2})}{\normalsize\emph{W}~{}\vdash~{}(\forall
x~{}\in~{}t_{1})~{}x~{}:~{}T_{1}~{}\vee~{}(\forall
y~{}\in~{}t_{2})~{}y~{}:~{}T_{2}}$ T-Disj
$\frac{\normalsize\emph{W}~{}\vdash~{}(t_{1}~{}:~{}T_{1}),~{}(t_{2}~{}:~{}T_{2}),~{}\ldots,~{}(t_{n}~{}:~{}T_{n})}{\normalsize\emph{W}~{}\vdash~{}(\forall
x~{}\in~{}t_{1})~{}x~{}:~{}T_{1},~{}\ldots,~{}(\forall
z~{}\in~{}t_{n})~{}z~{}:~{}T_{n}}$ T-Seq
$\frac{\small\emph{W}~{}\vdash~{}\scriptsize r~{}:~{}Object\rightarrow
Situation\rightarrow
Situation,~{}\scriptsize~{}\overline{t}~{}:~{}Object,~{}\scriptsize~{}s~{}:~{}Situation}{\normalsize\emph{W}~{}\vdash~{}r(\overline{t},~{}s)~{}:~{}Situation}$
T-RelFlt
$\frac{\normalsize\emph{W}~{}\vdash~{}f~{}:~{}Object\rightarrow
Action~{}~{}\normalsize\emph{W}\vdash\overline{t}~{}:~{}Object}{\normalsize\emph{W}~{}\vdash~{}f(\overline{t})~{}:~{}Action}$
T-FunFlt
$\frac{\normalsize\emph{W},~{}bt~{}:~{}Action~{}\vdash~{}s~{}:~{}Situation}{\normalsize\emph{W}~{}\vdash~{}\textbf{\emph{do}}(bt,~{}s)~{}:~{}Situation}$
T-Do
$\frac{\normalsize\emph{W},~{}bt~{}:~{}Action~{}\vdash~{}s~{}:~{}Situation}{\normalsize\emph{W}~{}\vdash~{}\textbf{\emph{poss}}(bt,~{}s)~{}:~{}Unit}$
T-Poss
Item | Description
---|---
E-Unv | If one term $t^{\prime}$ occurred in a given behavioral term $bt$ derives from another term $t$ also in $bt$, then all such terms $t^{\prime}$ in $bt$ are also derivable from all such terms $t$ in $bt$.
E-Est | If one term $t^{\prime}$ occurred in a given behavioral term $bt$ derives from another term $t$ also in $bt$, then there exists such a term $t^{\prime}$ in $bt$ that is derivable from such a term $t$ in $bt$.
E-Neg | If one term/behavioral term $t^{\prime}$/$bt^{\prime}$ derives from another term $t$/$bt$, then not $t^{\prime}$/$bt^{\prime}$ also derives from not $t$/$bt$.
E-Spt | If one term $t^{\prime}$ derives from another term $t$, then this also holds in superset operation.
E-Conj | If one term $t^{\prime}$ derives from another term $t$, then this also holds in conjunction.
E-Disj | If one term $t^{\prime}$ derives from another term $t$, then this also holds in disjunction.
E-Seq | If one term $t^{\prime}$ derives from another term $t$, then this relationship holds if $t^{\prime}$ and $t$ are in a sequence of terms, respectively.
E-Do | In a specific situation $s$, behavioral term $bt$ gets executed means situation $s$ transits to its successor situation $s^{\prime}$ while doing $bt$.
E-Poss | In a specific situation $s$, behavioral term $bt$ is possible means current situation $s$ is a superset of its successor situation $s^{\prime}$.
T-True | As a Bool type value, true is within the typing map _W_.
T-False | As a Bool type value, false is within the typing map _W_.
T-Var | If a variable $x$ with type $T$ is within the typing map $W$, then $x:T$ derives from $W$.
T-Unv1 | If all relational fluents $r$ that have an argument $x$ with type $T$ hold, then all occurrence of $x$ in $r$ must have a type $T$.
T-Est1 | If there exists one relational fluent $r$ that has an argument $x$ with type $T$ hold, then there must be one occurrence of $x$ in $r$ with a type $T$.
T-Unv2 | If all functional fluents $f$ that have an argument $x$ with type $T$ hold, then all occurrence of $x$ in $r$ must have a type $T$.
T-Est2 | If there exists one functional fluent $f$ that has an argument $x$ with type $T$ hold, then there must be one occurrence of $x$ in $r$ with a type $T$.
T-Neg | If one term/behavioral term $t^{\prime}$/$bt^{\prime}$ with a type $T$ derives from the typing map $W$, then not $t^{\prime}$/$bt^{\prime}$ also derives from $W$ with the same type $T$.
T-Spt | If $t_{1}$ with a type $T_{1}$ as a superset of $t_{2}$ with a type $T_{2}$ derives from the typing map $W$, then in $W$, all subterms $x$ of $t_{1}$, $y$ of $t_{2}$ also have types $T_{1}$, $T_{2}$ respectively, and superset relationship still holds.
T-Conj | If the conjunction of $t_{1}$ with a type $T_{1}$ and $t_{2}$ with a type $T_{2}$ derives from the typing map $W$, then all subterms of $t_{1}$, $t_{2}$ also have types $T_{1}$, $T_{2}$, satisfying conjunction.
T-Disj | If the disjunction of $t_{1}$ with a type $T_{1}$ and $t_{2}$ with a type $T_{2}$ derives from the typing map $W$, then all subterms of $t_{1}$, $t_{2}$ also have types $T_{1}$, $T_{2}$, satisfying disjunction.
T-Seq | If a sequence of terms with its own types derives from the typing map $W$, then in $W$, all subterms of every term have the type their parent has.
T-RelFlt | Straightforward typing relationship of first-order logic.
T-FunFlt | Straightforward typing relationship of first-order logic.
T-Do | Straightforward typing relationship of first-order logic.
T-Poss | Straightforward typing relationship of first-order logic.
Table 1: A Directory of All Evaluation and Typing Rules in the Type System of
the Lightweight Situation Calculus
Note:
1\. $\overline{t}$ is a shorthand of a sequence of terms
$t_{1},t_{2},\ldots,t_{n}~{}(n\geq 1)$. Hence $\overline{t}$ cannot possibly
be empty.
2\. The type _Unit_ is defined as the type of value _unit_ , where
$unit\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\\{t|(\forall x\in
t)~{}x:Bool\vee Situation\vee Action\vee Object\\}$, which means all elements
in a _unit_ should have the same type.
3\. $[s\mapsto s^{\prime}]bt$ in the computation rules E-Do and E-Poss means
“the successor situation $s^{\prime}$ that results from executing the
behavioral term $bt$ in the situation $s$.” See the items for E-Do and E-Poss
in Table 1.
4\. The shadowed typing rules are adapted from the handy typing mechanism for
quantifiers in current version of situation calculus, which is discussed in
section 4.1.
5\. For simplicity, the detailed explanation is not given for typing rules
T-RelFlt, T-FunFlt, T-Do and T-Poss.
## 5 Evaluation
### 5.1 Case Description
Let us consider the following scenario: In face of an object _x_ on the floor,
say a vase, there is a robot _r_ who wants to pick up this vase and paints it
with some color, namely _c_. In situation calculus, we can describe this
scenario using three statements:
In a given situation _s_ that, say, there is a robot _r_ and a vase _x_ ready
for situations later on, if the robot _r_ then picked up the vase _x_ and
dropped it without holding it firmly, which made the vase became broken, then
the vase must be a fragile object:
$fragile(x,s)\supset broken(x,\textbf{\emph{do}}(drop(r,x),s))$ (1)
If the robot successfully picked up the vase _x_ and tried to paint it with
one color _c_ , holding it firmly, the vase would turn out to be in the color
_c_ :
$color(x,\textbf{\emph{do}}(paint(x,c),s))=c$ (2)
Finally, let us consider the conditions on which it is possible for the robot
_r_ to pick up the vase _x_ without any external help. The conditions should
be a combination of three: the robot _r_ is not holding any other object _z_ ,
it is next to _x_ , and _x_ is not heavy:
$\textbf{\emph{poss}}(pickup(r,x),s)\supset[(\forall z)\neg
holding(r,z,s)]\wedge\neg heavy(x)\wedge nextTo(r,x,s)$ (3)
### 5.2 Results and Analysis of Type Checking
Now let us do the type checking on the aforementioned three statements that
represent a scenario in which a robot _r_ wants to pick up a vase _x_ by
itself and paints it with some fancy color _c_. On the basis of the type
system defined in section 4.2, if all the typing goes through and does not get
stuck, the program written in situation calculus will be regarded as well-
typed.
First, we need to add predefined types for programs written in the original
situation calculus by virtue of our new type system. Hence we have:
$fragile(x:Object,s:Situation)\supset
broken(x:Object,\textbf{\emph{do}}(drop(r:Object,x:Object),s:Situation))$
$(1)^{\prime}$
$color(x:Object,\textbf{\emph{do}}(paint(x:Object,c:Object),s:Situation))=c:Object$
$(2)^{\prime}$
$\textbf{\emph{poss}}(pickup(r:Object,x:Object),s:Situation)\supset[(\forall
z:Object)\neg holding(r:Object,z:Object,s:Situation)]\wedge\neg
heavy(x:Object)\wedge nextTo(r:Object,x:Object,s:Situation)$ $(3)^{\prime}$
And then we know the world $w\equiv\\{x,s,r,c,z\\}$ and
$W\equiv\\{x:Object,s:Situation,r:Object,c:Object,z:Object\\}$.
Now, let us do typing derivation statement by statement. For $(1)^{\prime}$,
We notice that T-Spt cannot be applied since $(1)^{\prime}$ is of a superset
relationship between behavioral terms while T-Spt is for regular terms. Thus,
we turn to prove that the type of the left hand side of “$\supset$” is the
same as that of the right hand side.
For typesetting simplicity, we omit “ _W_ $\vdash$”, return types and the
final step of T-Var, and abbreviate “ _Object_ ”, “ _Situation_ ” and “
_Action_ ” to “ _Obj_ ”, “ _Stn_ ” and “ _Atn_ ”, respectively, in the
following typing derivation.
Left hand side of “$\supset$” in $(1)^{\prime}$:
$\frac{fragile:Obj\rightarrow Stn\rightarrow
Stn,~{}x:Obj,~{}s:Stn}{fragile(x,~{}s)}\textsc{T-RelFlt}$
Right hand side of “$\supset$” in $(1)^{\prime}$:
$\frac{drop:Obj\rightarrow
Atn,~{}r:Obj,~{}x:Obj,~{}s:Stn,~{}broken:Obj\rightarrow Stn\rightarrow
Stn}{drop(r:Obj,x:Obj),~{}s:Stn,~{}broken:Obj\rightarrow Stn\rightarrow
Stn}\textsc{T-FunFlt}$
$\frac{~{}}{~{}~{}~{}~{}~{}~{}~{}\textbf{\emph{do}}(drop(r:Obj,x:Obj),~{}s:Stn),~{}broken:Obj\rightarrow
Stn\rightarrow Stn~{}~{}~{}~{}~{}~{}~{}~{}}\textsc{T-Do}$
$\frac{~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}broken(x:Obj,~{}\textbf{\emph{do}}(drop(r:Obj,x:Obj),~{}s:Stn))~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}\textsc{T-RelFlt}$
According to this typing derivation, we know that both types of the left hand
side and right hand side are the same one: _Situation_. So this situation
calculus statement is proved to be well-typed.
For $(2)^{\prime}$, we have the similar form of typing derivation:
Left hand side of “=” in $(2)^{\prime}$:
$\frac{\hskip 2.27621ptpaint:Obj\rightarrow
Atn,~{}x:Obj,c:Obj,~{}s:Stn,~{}color:Obj\rightarrow Stn\rightarrow
Stn}{paint(x:Obj,c:Obj),~{}s:Stn,~{}color:Obj\rightarrow Stn\rightarrow
Stn}\textsc{T-FunFlt}$
$\frac{~{}}{~{}~{}~{}~{}~{}~{}~{}\textbf{\emph{do}}(paint(x:Obj,c:Obj),~{}s:Stn),~{}color:Obj\rightarrow
Stn\rightarrow Stn~{}~{}~{}~{}~{}~{}~{}}\textsc{T-Do}$
$\frac{~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}color(x:Obj,~{}\textbf{\emph{do}}(paint(x:Obj,c:Obj),~{}s:Stn))~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}\textsc{T-RelFlt}$
Right hand side of “=” in $(2)^{\prime}$:
_c: Object_
According to this typing derivation, we find that the type of the left hand
side is _Situation_ , while the right hand side has a type: _Object_ , which
obviously leads to a mismatch. So this situation calculus statement is proved
to be not well-typed. In fact, whatever type of _c_ will bring about stuck
terms or mismatches. Anyway, it can still be fixed. A possible correction is
to change the right hand side to _inColor(c: Object, s: Situation)_ , i.e., to
replace _c_ with a corresponding relational fluent to match the type of the
left hand side.
Let us check the final sample statement similarly as we did previously:
Left hand side of “$\supset$” in $(3)^{\prime}$:
$\frac{pickup:Obj\rightarrow Obj\rightarrow
Atn,~{}r:Obj,~{}x:Obj,~{}s:Stn}{pickup(r:Obj,~{}x:Obj),~{}s:Stn}\textsc{T-FunFlt}$
$\frac{~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\textbf{\emph{poss}}(pickup(r:Obj,~{}x:Obj),~{}s:Stn)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}\textsc{T-Poss}$
When coming to the right hand side of “$\supset$” in $(3)^{\prime}$, we notice
that T-Conj cannot be applied since the right hand side of $(3)^{\prime}$ is
of a conjunctive relationship between behavioral terms while T-Conj is for
regular terms. Thus, we turn to check whether types of each part of the
conjunction are the same. If so, the final type should be _Unit_ according to
its definition.
$\frac{holding:Obj\rightarrow Obj\rightarrow Stn\rightarrow
Stn,~{}r:Obj,~{}z:Obj,~{}s:Stn}{holding(r:Obj,~{}z:Obj,~{}s:Stn)}\textsc{T-RelFlt}$
$\frac{~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall\neg
holding(r:Obj,~{}z:Obj,~{}s:Stn)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}\textsc{T-Neg}$
$\frac{~{}}{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall z:Obj)\neg
holding(r:Obj,~{}z:Obj,~{}s:Stn)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}\textsc{T-Unv1}$
So we find that $(\forall z)\neg holding(r:Object,z:Object,s:Situation)$ has a
type _Situation_. Similarly, we can derive $\neg heavy(x:Object)$ to be of
type _Action_ by taking two typing derivation steps and
$nextTo(r:Object,x:Object,s:Situation)$ to be of type _Situation_ by taking
one step. By definition, the type of the right hand side of “$\supset$” in
$(3)^{\prime}$ is not _Unit_ since not all the subterms of the conjunction
have the same type. Therefore, we can claim that $(3)^{\prime}$ is not well-
typed as well. This time we can fix it intuitively by simply changing the
functional fluent $\neg heavy(x:Object)$ into a relational fluent $\neg
heavy(x:Object,s:Situation)$.
### 5.3 Implementation in OCaml
In the last section, types of every term and behavioral term written in a
modified lightweight situation calculus with our type system are checked for
consistency theoretically. Now we plan to implement a type checker in OCaml
which does the same job as we do manually, that is, all the type checking work
would be fulfilled by a type checker semi-automatically and efficiently, which
can give a great hand for those who are doing tedious type checking alone.
One piece of sample code in OCaml which typechecks situation calculus
statement $(1)^{\prime}$ as described in 5.2 is shown below:
`(* Types Definition *)`
`# type unit = Unit of unit;;`
`# type bool = Bool of bool;;`
`# type stn = Situation;;`
`# type atn = Action;;`
`# type obj = Object;;`
`(* T-RelFlt *)`
`# let r t s = `
` match t with`
` Object -> match s with`
` Situation -> Situation;;`
`(* T-FunFlt *)`
`# let f t1 t2 =`
` match t1 with`
` Object -> match t2 with`
` Object -> Action;;`
`(* T-Do *)`
`# let does bt s =`
` match bt with`
` Action -> match s with`
` Situation -> Situation;;`
`(* Left Hand Side *)`
`# let x = Object`
` and s = Situation`
` and fragile = r;;`
`val x : obj = Object`
`val s : stn = Situation`
`val fragile : obj -> stn -> stn = <fun>`
`# fragile (x:obj) (s:stn);;`
`- : stn = Situation`
`(* Right Hand Side *)`
`# let b = Object`
` and drop = f`
` and broken = r;;`
`val b : obj = Object`
`val drop : obj -> obj -> atn = <fun>`
`val broken : obj -> stn -> stn = <fun>`
`# broken (x:obj) (does (drop (b:obj) (x:obj)) (s:stn));;`
`- : stn = Situation`
`(* This statement is proved to be well-typed *)`
In the OCaml code above, we firstly defined the types in our type system, and
then implemented the T-RelFlt, T-FunFlt and T-Do typing rules. Finally some
necessary variables, two relational fluents _fragile_ and _broken_ , and a
funtional fluent _drop_ are declared. As a type checking process, these
fluents _fragile_ , _broken_ and _drop_ are invoked with inputs of pre-defined
variables to show the typing relationship among them, and the final types
calculated for the left hand side and right hand side indicate whether this
statement is well-typed.
In this way, all statements that we typechecked manually just now can be
dumped into this type checker for semi-automatical type checking. Due to some
limitation of typing rules in our system, we do need some additional manual
work occasionally. For instance, we need to check by ourselves that whether
the types of the left hand side and right hand side of a symbol ”$\supset$”
are the same. Anyway, the type checker indeed facilitate our process of
deciding whether a situation calculus program is well-typed or not.
## 6 Conclusions
Type systems have been proposed to guarantee the soundness of program types by
rigid typing mechanisms. As a popular formal language widely used in
Artificial Intelligence related fields, situation calculus itself has
insufficient methods to support a complete and robust type system, with a
rudimentary typing mechanism: only typed quantifiers for variables. It is
obviously not enough for type checking the current situation calculus
programs. By virtue of our newly-introduced type system for a lightweight
situation calculus which keep the core of the current one, we can easily do
basic type checking for existing situation calculus programs which are
referred a lot in various study of situation calculus. We also implemented the
theoretical type system in OCaml as a type checker to substantiate our ideas.
With the help of this type checker, precedent manual type checking work can be
greatly automated for a better performance. As for the programs checked out to
be ill-typed, we provide corresponding ways for correcting them into well-
typed forms.
Acknowledgements
The author would like to thank all anonymous reviewers for their generous and
constructive directives and comments on this paper.
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|
arxiv-papers
| 2012-01-11T21:44:18 |
2024-09-04T02:49:26.182399
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li Tan",
"submitter": "Li Tan",
"url": "https://arxiv.org/abs/1201.2430"
}
|
1201.2435
|
On the generalized method of cells and the prediction of
effective elastic properties of polymer bonded explosives
Biswajit Banerjee 111Corresponding author. E-mail: banerjee@eng.utah.edu
Phone: (801)-585-5239 Fax: (801)-585-9826 and Daniel O. Adams
Dept. of Mechanical Engineering, University of Utah,
50 S Central Campus Drive # 2202, Salt Lake City, UT 84112, USA.
## Abstract
The prediction of the effective elastic properties of polymer bonded
explosives using direct numerical simulations is computationally expensive
because of the high volume fraction of particles in these particulate
composites ($\sim$0.90) and the strong modulus contrast between the particles
and the binder ($\sim$20,000). The generalized method of cells (GMC) is an
alternative to direct numerical simulations for the determination of effective
elastic properties of composites. GMC has been shown to be more
computationally efficient than finite element analysis based approaches for a
range of composites. In this investigation, the applicability of GMC to the
determination of effective elastic properties of polymer bonded explosives is
explored. GMC is shown to generate excellent estimates of effective moduli for
composites containing square arrays of disks at volume fractions less than
0.60 and a modulus contrast of approximately 100. However, for high volume
fraction and strong modulus contrast polymer bonded explosives such as PBX
9501, the elastic properties predicted by GMC are found to be considerably
lower than finite element based estimates and experimental data. Simulations
of model microstructures are performed to show that normal stiffnesses are
underestimated by GMC when stress-bridging due to contact between particles is
dominant. Additionally, the computational efficiency of GMC decreases rapidly
with an increase in the number of subcells used to discretize a representative
volume element. The results presented in this work suggest that GMC may not be
suitable for calculating the effective elastic properties of high volume
fraction and strong modulus contrast particulate composites. Finally, a real-
space renormalization group approach called the recursive cell method (RCM) is
explored as an alternative to GMC and shown to provide improved estimates of
the effective properties of models of polymer bonded explosives.
Keywords : Effective Properties, Particulate Composites, High Volume Fraction,
Strong Modulus Contrast, Stress-Bridging, Method of Cells, Real-Space
Renormalization Group
## 1 Introduction
The generalized method of cells (GMC) [Aboudi96_1, Paley92] is a semi-
analytical method of determining the effective properties of composites. In
this method, a representative volume element (RVE) of the composite under
consideration is discretized into a regular grid of subcells. Equilibrium and
compatibility are satisfied on an average basis across subcells using
integrals over subcell boundaries. GMC generates a matrix of algebraic
expressions containing information about subcell material properties. The
effective stiffness of the composite can be obtained by inverting this matrix.
One advantage of GMC over other numerical techniques is that the full set of
effective elastic properties of a composite can be calculated in one step
instead of solving a number of boundary value problems with different boundary
conditions. GMC has also been found to be more computationally efficient that
finite element calculations for fiber reinforced composites [Aboudi96_1,
Wilt95], since far fewer GMC subcells than finite elements are necessary to
obtain the same degree of accuracy. The problem of discretization is also
minimized since a regular rectangular grid is used in GMC.
The generalized method of cells is discussed briefly in this work. Effective
stiffnesses predicted by this method are compared with accurate numerical
predictions for square arrays of disks [Greeng98]. The method is then applied
to two-dimensional models of a general polymer bonded explosive and to the
microstructure of PBX 9501 using a two-step procedure similar to that of Low
et al. [Low94]. GMC estimates of elastic properties are compared with
predictions from detailed finite element calculations. The performance of GMC
is explored for several microstructures with contacting particles and some
shortcomings of the method are identified. Finally, the recursive cell method
(RCM) is explored as an alternative to direct GMC calculations in the
prediction of effective properties of polymer bonded explosives.
## 2 The generalized method of cells
Figure 1 shows a schematic of the RVE, the subcells and the notation
[Aboudi91] used in GMC. In the figure, ($X_{1},X_{2},X_{3}$) is the global
coordinate system of the RVE and
($x^{(\alpha)}_{1}$,$x^{(\beta)}_{2}$,$x^{(\gamma)}_{3}$) is the coordinate
system local to a subcell denoted by $(\alpha\beta\gamma)$. It is assumed that
the displacement function $u^{(\alpha\beta\gamma)}_{i}$ varies linearly within
a subcell $(\alpha\beta\gamma)$ and can be written in the form
$u^{(\alpha\beta\gamma)}_{i}(x^{(\alpha)}_{1},x^{(\beta)}_{2},x^{(\gamma)}_{3})=w^{(\alpha\beta\gamma)}_{i}(X_{1},X_{2},X_{3})+\Phi^{(\alpha\beta\gamma)}_{i}x^{(\alpha)}_{1}+\Theta^{(\alpha\beta\gamma)}_{i}x^{(\beta)}_{2}+\Psi^{(\alpha\beta\gamma)}_{i}x^{(\gamma)}_{3}$
(1)
where $i$ represents the coordinate direction and takes the values 1, 2 or 3;
$w^{(\alpha\beta\gamma)}_{i}$ is the mean displacement at the center of the
subcell $(\alpha\beta\gamma)$; and $\Phi^{(\alpha\beta\gamma)}_{i}$,
$\Theta^{(\alpha\beta\gamma)}_{i}$, and $\Psi^{(\alpha\beta\gamma)}_{i}$ are
constants local to the subcell that represent gradients of displacement across
the subcell. The strain-displacement relations for the subcell are given by
$\epsilon^{(\alpha\beta\gamma)}_{ij}=\frac{1}{2}(\partial_{i}u^{(\alpha\beta\gamma)}_{j}+\partial_{j}u^{(\alpha\beta\gamma)}_{i})$
(2)
where $\partial_{1}=\partial/\partial x^{(\alpha)}_{1}$,
$\partial_{2}=\partial/\partial x^{(\beta)}_{2}$, and
$\partial_{3}=\partial/\partial x^{(\gamma)}_{3}$. Since polymer bonded
explosives are isotropic particulate composites, the following brief
description of GMC assumes that the RVE is cubic and all subcells are of equal
size. If each subcell $(\alpha\beta\gamma)$ has the same dimensions
$(2h,2h,2h)$ then the average strain in the subcell is defined as a volume
average of the strain field over the subcell as
$\left<\epsilon^{(\alpha\beta\gamma)}_{ij}\right>_{\text{S}}=\frac{1}{8h^{3}}\int_{-h}^{h}\int^{h}_{-h}\int_{-h}^{h}{\epsilon^{(\alpha\beta\gamma)}_{ij}}~{}dx^{(\alpha)}_{1}dx^{(\beta)}_{2}dx^{(\gamma)}_{3}$
(3)
The average strain in the subcell can be obtained in terms of the displacement
field variables. It is assumed that there is continuity of traction at the
interface of two subcells. The displacements and tractions are assumed to be
periodic at the boundaries the RVE. Applying the displacement continuity
equations on an average basis over the interfaces between subcells, the
average strain in the RVE can be expressed in terms of the subcell strains.
The average subcell stresses can be obtained from the subcell strains using
the traction continuity condition and the stress-strain relations of the
materials in the subcells. A relationship between the subcell stresses and the
average strains in the RVE is thus obtained.
For orthotropic, transversely isotropic or isotropic materials, the approach
discussed above leads to the decoupling of the normal and shear response of
the RVE. This decoupling leads to two systems of equations relating the
subcell stresses and the average strains in the RVE. For the normal components
of strain, the system of equations can be written as
$\begin{bmatrix}\mathbf{M1}_{1}&\mathbf{M1}_{2}&\mathbf{M1}_{3}\\\
\mathbf{M2}_{1}&\mathbf{M2}_{2}&\mathbf{M2}_{3}\\\
\mathbf{M3}_{1}&\mathbf{M3}_{2}&\mathbf{M3}_{3}\end{bmatrix}\begin{bmatrix}\mathbf{T}_{1}\\\
\mathbf{T}_{2}\\\ \mathbf{T}_{3}\end{bmatrix}=2N\begin{bmatrix}\mathbf{H}\\\
\mathbf{0}\\\
\mathbf{0}\end{bmatrix}\left<\epsilon_{11}\right>_{\text{V}}+2N\begin{bmatrix}\mathbf{0}\\\
\mathbf{H}\\\
\mathbf{0}\end{bmatrix}\left<\epsilon_{22}\right>_{\text{V}}+2N\begin{bmatrix}\mathbf{0}\\\
\mathbf{0}\\\ \mathbf{H}\end{bmatrix}\left<\epsilon_{33}\right>_{\text{V}}$
(4)
where $N$ is the number of subcells per side of the RVE. The corresponding
system of equations for the shear components is of the form
$\begin{bmatrix}\mathbf{M4}&\mathbf{0}&\mathbf{0}\\\
\mathbf{0}&\mathbf{M5}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&\mathbf{M6}\end{bmatrix}\begin{bmatrix}\mathbf{T}_{12}\\\
\mathbf{T}_{23}\\\
\mathbf{T}_{13}\end{bmatrix}=2N^{2}\begin{bmatrix}\mathbf{H}\\\ \mathbf{0}\\\
\mathbf{0}\end{bmatrix}\left<\epsilon_{12}\right>_{\text{V}}+2N^{2}\begin{bmatrix}\mathbf{0}\\\
\mathbf{H}\\\
\mathbf{0}\end{bmatrix}\left<\epsilon_{23}\right>_{\text{V}}+2N^{2}\begin{bmatrix}\mathbf{0}\\\
\mathbf{0}\\\ \mathbf{H}\end{bmatrix}\left<\epsilon_{13}\right>_{\text{V}}$
(5)
In equations (4) and (5) the $\mathbf{M}$ matrices contain material compliance
terms. The $\mathbf{T}$ matrices contain the average subcell stresses. The
vector $\mathbf{H}$ contains the dimensions of the subcells. Thus, a sparse
system of equations of size 3$N^{2}$ is produced that relates the subcell
stresses to the average strains in the RVE. After inverting these equations
and with some algebraic manipulation, explicit algebraic expressions for the
individual terms of the effective stiffness matrix can be obtained. These
stress-strain equations that relate the average RVE stresses to the average
RVE strains are of the form
$\begin{bmatrix}\left<\sigma_{11}\right>_{\text{V}}\\\
\left<\sigma_{22}\right>_{\text{V}}\\\ \left<\sigma_{33}\right>_{\text{V}}\\\
\left<\sigma_{23}\right>_{\text{V}}\\\ \left<\sigma_{31}\right>_{\text{V}}\\\
\left<\sigma_{12}\right>_{\text{V}}\end{bmatrix}=\begin{bmatrix}C^{\text{eff}}_{11}&C^{\text{eff}}_{12}&C^{\text{eff}}_{13}&0&0&0\\\
C^{\text{eff}}_{12}&C^{\text{eff}}_{22}&C^{\text{eff}}_{23}&0&0&0\\\
C^{\text{eff}}_{13}&C^{\text{eff}}_{23}&C^{\text{eff}}_{33}&0&0&0\\\
0&0&0&C^{\text{eff}}_{44}&0&0\\\ 0&0&0&0&C^{\text{eff}}_{55}&0\\\
0&0&0&0&0&C^{\text{eff}}_{66}\end{bmatrix}\begin{bmatrix}\left<\epsilon_{11}\right>_{\text{V}}\\\
\left<\epsilon_{22}\right>_{\text{V}}\\\
\left<\epsilon_{33}\right>_{\text{V}}\\\
2\left<\epsilon_{23}\right>_{\text{V}}\\\
2\left<\epsilon_{31}\right>_{\text{V}}\\\
2\left<\epsilon_{12}\right>_{\text{V}}\end{bmatrix}$ (6)
where $C^{\text{eff}}_{ij}$ are the terms of the effective stiffness matrix.
Details of the algebraic expressions for these terms have been published by
other researchers [Pindera97].
In GMC, the number of equations to be solved equals the number of subcells
raised to the $d$ th power, where $d$ is the number of dimensions in the
problem. As a result, the computational efficiency of GMC decreases as the
number of subcells increases. This issue has been partially resolved
[Orozco97] by identifying the sparsity characteristics of the system of
equations and by using the Harwell-Boeing suite of sparse solvers. The
computational efficiency of GMC has been further improved after a
reformulation [Pindera97, Bednar97] that takes advantage of the continuity of
tractions across subcells to obtain a system of $O(N^{2})$ equations in three
dimensions.
Due to decoupling of the normal and shear response of the RVE, the shear
components of the stiffness matrix obtained from GMC are the harmonic means of
the subcell shear stiffnesses and of the form
$1/C^{\text{eff}}_{66}=1/N^{3}\sum^{N}_{\alpha=1}\sum^{N}_{\beta=1}\sum^{N}_{\gamma=1}1/C^{(\alpha\beta\gamma)}_{66}.$
(7)
Bednarcyk and Arnold (2001) suggest that this lack of coupling makes for an
“ultra-efficient” micromechanics model. However, this lack of coupling can
lead to gross underestimation of shear moduli for high volume fraction and
high modulus contrast materials such as polymer bonded explosives. Recently,
researchers [Williams99, Gan00] have attempted to solve the problem by using
higher order expansions for the displacement and by explicitly satisfying both
subcell equilibrium and compatibility. However, these approaches decrease the
computational efficiency of GMC considerably and are not explored in this
work.
## 3 Validation - square arrays of disks
In this section, estimates of effective properties from GMC are compared with
accurate numerical results for square arrays of disks. Square RVEs containing
square arrays of disks exhibit square symmetry. The two-dimensional linear
elastic stress-strain relation for these RVEs can be written as
$\begin{bmatrix}\left<\sigma_{11}\right>_{\text{V}}\\\
\left<\sigma_{22}\right>_{\text{V}}\\\
\left<\sigma_{12}\right>_{\text{V}}\end{bmatrix}=\begin{bmatrix}K_{\text{eff}}+\mu_{\text{eff}}^{(1)}&K_{\text{eff}}-\mu_{\text{eff}}^{(1)}&0\\\
K_{\text{eff}}-\mu_{\text{eff}}^{(1)}&K_{\text{eff}}+\mu_{\text{eff}}^{(1)}&0\\\
0&0&\mu_{\text{eff}}^{(2)}\end{bmatrix}\begin{bmatrix}\left<\epsilon_{11}\right>_{\text{V}}\\\
\left<\epsilon_{22}\right>_{\text{V}}\\\
2\left<\epsilon_{12}\right>_{\text{V}}\end{bmatrix}$ (8)
where $K_{\text{eff}}$ is the two-dimensional effective bulk modulus,
$\mu_{\text{eff}}^{(1)}$ is the effective shear modulus when a shear stress is
applied along the diagonals of the RVE, and $\mu_{\text{eff}}^{(2)}$ is the
effective shear modulus when a shear stress is applied along the edges of the
RVE. These three effective moduli have been determined accurately, using an
integral equation approach, by Greengard and Helsing (1998) for square arrays
of disks containing a range of disk volume fractions.
To compare the effective moduli predicted by GMC with those from the integral
equation calculations [Greeng98], RVEs containing disk volume fractions from
0.10 to 0.70 were created. These RVEs were discretized into 64$\times$64 equal
sized subcells. The effective stiffness matrix of each RVE was calculated
using GMC. Finally, the two-dimensional effective moduli for each RVE were
calculated from the effective stiffness matrix using the relations
$K_{\text{eff}}=0.5(C^{\text{eff}}_{11}+C^{\text{eff}}_{12})~{},~{}\mu^{(1)}_{\text{eff}}=0.5(C^{\text{eff}}_{11}-C^{\text{eff}}_{12})~{},~{}\mu^{(2)}_{\text{eff}}=C^{\text{eff}}_{66}.$
(9)
Figure 2 shows the moduli predicted by GMC and those from the integral
equation method of Greengard and Helsing (G&H) for disk volume fractions from
0.10 to 0.70. The material properties of the disks and the binder used in the
calculations are shown in Table 1. The effective bulk moduli
($K_{\text{eff}}$) and diagonal shear moduli ($\mu_{\text{eff}}^{(1)}$)
obtained from the GMC calculations are within 4% of those obtained by the
integral equation method for all volume fractions up to 0.60. At a volume
fraction of 0.70, the GMC predictions for bulk modulus and diagonal shear
modulus are 4% and 11% less, respectively. For the shear modulus
$\mu_{\text{eff}}^{(2)}$, the GMC predictions are around 4% to 10% less than
the estimates of Greengard and Helsing for volume fractions from 0.10 to 0.60.
The difference is around 24% for a volume fraction of 0.70.
These results show that GMC estimates are quite accurate for composites
containing square arrays of disks with volume fractions up to 0.60, confirming
results reported elsewhere [Aboudi96_1]. In the next section, GMC is used to
determine the effective properties of models of polymer bonded explosives and
the results are compared to detailed finite element calculations and
experimental data.
## 4 Modeling polymer bonded explosives
Polymer bonded explosive (PBX) materials typically contain around 90% by
volume of particles surrounded by a binder. The particles consist of a mixture
of coarse and fine grains with the finer grains forming a filler between
coarser grains. Modeling the microstructure of these materials is difficult
due to the complex shapes of HMX particles and the large range of particle
sizes. Two-dimensional approximations of the microstructure of PBXs based on
digital images [Benson99] have been used to study some aspects of the
micromechanics of PBXs. However, such microstructures are difficult to
generate and require complex image processing techniques and excellent image
quality to accurately capture details of the material. A combination of Monte
Carlo and molecular dynamics techniques have also been used to generate three-
dimensional models of PBXs [Baer01]. Microstructures containing spheres and
oriented cubes have been generated using these techniques and appear to
represent PBX microstructures well. However, the generation of microstructures
using dynamics-based methods is extremely time consuming when tight particle
packing is required, as is the case for volume fractions above 0.70.
Comparisons of finite element predictions with exact relations for the
effective properties of composites [Banerjee02th] have shown that detailed
finite element estimates can be used as a benchmark to check the accuracy of
predictions from GMC. In this investigation, manually generated PBX
microstructures containing symmetrically distributed circular particles are
used initially to compare GMC and finite element predictions. The two-
dimensional microstructures contain 90% particles by volume and use two
particle length scales. Two-dimensional models containing randomly distributed
circular particles that reflect the actual particle size distribution of PBX
9501 are next modeled with GMC and the results compared to finite element
estimates.
The material properties used for the particles, the binder, and PBX9501 in
these calculations are shown in Table 2. These properties correspond to those
of HMX (the explosive particles), the binder, and PBX 9501 at 25o C and a
strain rate of 0.05/s [Wetzel99].
### 4.1 Simplified models of PBX materials
GMC and finite element calculations were performed for the six, manually
generated, simplified model microstructures of polymer bonded explosives shown
in Figure 3. These representative volume elements (RVEs) contain one or a few
relatively large particles surrounded by smaller particles to reflect common
particle size distributions of PBXs. The volume fraction of particles in each
of these models is around 0.90$\pm$0.005. The binder material surrounds all
particles in the six microstructures.
For the GMC calculations, a square grid was overlaid on the RVEs to generate
subcells. Two different approaches were used to assign materials to subcells
before the determination of effective properties of the RVE. In the first
approach, referred to as the “binary subcell approach”, a subcell was assigned
the material properties of particles if more than 50% of the subcell was
occupied by particles. Binder properties were assigned otherwise. Figure 4(a)
shows a schematic of the binary subcell approach. In the second approach,
called the “effective subcell approach”, a method of cells calculation
[Aboudi91] was used to determine the effective properties of a subcell based
on the cumulative volume fraction of particles in the subcell [Banerjee00].
Figure 4(b) shows a schematic of the effective subcell approach. After the
subcells were assigned material properties, the GMC technique was used to
compute the effective properties of the RVE.
Note that the particles are not resolved well when materials are assigned to
subcells in this manner if the number of subcells is small. However, the large
size of the matrix to be inverted in GMC limits the number of subcells that
can be used to discretize the RVE. If the binary subcell approach is used to
assign subcell materials, contacting particles are created where there are
none in the actual microstructure, leading to the prediction of higher than
actual stiffness values. The effective subcell approach improves upon the
binary subcell approach by “smearing”the material properties at the boundaries
of particles and thus reducing the particle contact artifacts caused by
discretization errors.
For validating the GMC results, detailed finite element (FEM) calculations
were performed using six-noded triangular elements to accurately model the
geometry of the particles. Around 65,000 nodes were used to discretize each of
the models. The volume average stress and strain in each RVE was determined
for applied normal and shear displacements. Periodicity was enforced through
displacement boundary conditions. Since these finite element calculations
serve to validate the GMC calculations, further mesh refinement was explored
and the results were found to converge those from the 65,000 node finite
element models.
Table 3 lists the effective stiffnesses of the six RVEs shown in Figure 3 from
GMC and FEM calculations. On average, the GMC calculations using the binary
subcell approach predict values of $C^{\text{eff}}_{11}$ that are around 2.5
times the FEM based values. The values of $C^{\text{eff}}_{11}$ from the
effective subcell approach based GMC calculations are closer to the FEM
estimates than the binary subcell based GMC estimates. The GMC and FEM
estimates of $C^{\text{eff}}_{12}$ are quite close. The values of
$C^{\text{eff}}_{66}$ from GMC are only 10% of the FEM values. The low values
of $C^{\text{eff}}_{66}$ are obtained because GMC predict effective shear
stiffnesses that are harmonic means of subcell shear stiffnesses. Models 3 and
4 (Figure 3) produce agreement in $C^{\text{eff}}_{11}$ and
$C^{\text{eff}}_{12}$ between the binary subcell approach based GMC
calculations and FEM (within 5%) whereas the other four models produce
considerable differences in predictions. These large differences are produced
by discretization errors introduced in the GMC approach that lead to
continuous stress-bridging paths across the RVEs and hence to increased
stiffness.
However, if the predicted effective stiffnesses shown in Table 3 are compared
with the experimental effective stiffness of PBX 9501 (shown in Table 2), it
can be observed that the models predict values of $C^{\text{eff}}_{11}$ and
$C^{\text{eff}}_{12}$ that are around 10% of the experimental values. Hence,
these simplified models are not appropriate for the modeling of PBX 9501. The
next section explores models based on the actual particle size distribution of
PBX 9501.
### 4.2 Models of PBX 9501
Coarse and fine particles of HMX are blended in a ratio of 1:3 by weight and
compacted in the process of manufacturing PBX 9501. Figure 5(a) shows four
RVEs of PBX 9501 based on the particle size distribution of the dry blend of
HMX [Wetzel99] prior to compaction. Figure 5(b) shows four RVEs based on the
particle size distribution of pressed PBX 9501 [Skid98]. The larger particles
are broken up in the pressing process leading to a larger proportion of
smaller particles in pressed PBX 9501. The models of the dry blend have been
labeled “DB” while those of pressed PBX 9501 have been labeled “PP”.
GMC calculations were performed on the PBX 9501 RVEs after discretizing each
RVE into 100$\times$100 subcells and assigning materials to subcells using the
effective subcell approach. In order to validate the GMC predictions, FEM
calculations were also performed on the RVEs after discretizing each RVE into
350$\times$350 four-noded square elements. The binary subcell approach was
applied to assign materials to elements for the FEM calculations.
The particles in each RVE were assigned properties of HMX from Table 2.
However, since particles occupy 92% of the total volume in actual PBX 9501
while the sample microstructures could be filled only up to $\sim$86%, an
intermediate homogenization step was required to determine the properties of
the binder. To produce the desired 92% volume of particles, a fine-particle
filled binder containing 36% particles by volume, or “dirty” binder, was
assumed. The effective elastic properties of the dirty binder were calculated
using the differential effective medium approximation [Markov00].
Table 4 shows the effective stiffness from FEM and GMC calculations for the
models of PBX 9501 shown in Figure 5. For all microstructures, the values of
$C^{\text{eff}}_{11}$ and $C^{\text{eff}}_{22}$ predicted by GMC are less than
5% of the FEM values and less than 10% of the experimental values for PBX 9501
(shown in Table 2). The FEM estimates increase with RVE size (varying from
150% to 450% of the experimental values), reflecting the dependence of
predicted stiffnesses on microstructure, discretization and particle size
distribution. The RVEs contain numerous particle to particle contacts, the
number of which increases with increase in RVE size. These contacts lead to
significant stress-bridging and hence relatively high values of stiffness as
is reflected in the FEM predictions. However, stress-bridging is not
incorporated accurately in the GMC approach leading to considerably lower
values of effective stiffness. The issue of stress-bridging is further
explored in the following section.
The values of $C^{\text{eff}}_{66}$ predicted by GMC are around 0.5% of the
FEM predictions. The reason for this large difference is that the effective
shear stiffness predicted by GMC is simply the harmonic mean of the subcell
shear moduli and only provides a lower bound on the shear stiffness.
## 5 Stress-Bridging
Comparisons of effective stiffness properties predicted by GMC with other
numerical estimates have shown that GMC performs quite well for low modulus
contrast materials with volume fractions below 60%. However, for high modulus
contrast materials with high particle volume fractions, GMC usually predicts
considerably lower effective stiffnesses than finite element calculations. In
this section, GMC is applied to selected microstructures containing stress-
bridging and the predicted properties are compared to finite element
estimates. The goal is to demonstrate that the effects of stress-bridging on
effective properties are inaccurately described by GMC.
### 5.1 Corner bridging : X-shaped microstructure
In the RVE shown in Figure 6, the particles are square, arranged in the form
of an ‘X’, and occupy a volume fraction of 25%. The particles transfer stress
through corner contacts. The effective properties of the X-shaped
microstructure shown in Figure 6 were calculated using the properties of HMX
and five different binders with Young’s moduli that range from 0.7 MPa to 7000
MPa, as shown in Table 5.
Figure 7 shows the variation in the effective stiffness properties
$C^{\text{eff}}_{11}$ and $C^{\text{eff}}_{66}$ of the X-shaped microstructure
with increasing Young’s modulus contrast between the particles and the binder
($E_{p}/E_{b}$). These effective stiffness properties have been calculated
using both finite elements (FEM) (256$\times$256 elements) and GMC
(64$\times$64 subcells). The FEM and GMC estimates are in good agreement for
Young’s modulus contrasts of 200 or less. For higher Young’s modulus
contrasts, the effective stiffness properties predicted by GMC are much lower
than those predicted by FEM. Note that the FEM estimates do not change
significantly with increased discretization, implying that the solution has
converged. The effect of corner singularities is also averaged out while
calculating the effective properties using FEM. If it is assumed that the FEM
estimates are close to the actual effective moduli of the RVE, the GMC
estimates for high modulus contrasts are orders of magnitude lower than the
actual effective moduli. Hence, GMC does not capture the stiffening effect of
corner contacts accurately.
Since the corner stress-bridging problem involves high stress concentrations
that are not resolved well by finite elements, it is possible that the FEM
calculations overestimate the effective properties of the X-shaped
microstructure. Such corner singularities are minimized in the microstructures
studied in the next section where the effect of stress-bridging along
particles edges is studied.
### 5.2 Edge bridging
Figure 8 shows five RVEs (A through E) in which the degree of stress-bridging
is increased progressively from corner bridging, to partial edge bridging, and
finally to continuous stress-bridging across the RVE. In Figure 8, the ‘1’
direction corresponds to the $x$-axis and the ‘2’ direction corresponds to the
$y$-axis.
Model A contains a square particle that occupies 25% of the volume, is
centered in the RVE, and does not have any stress-bridging. Model B contains
three particles that contact along a diagonal of the RVE. In model C, particle
contact is increased to produce a single line of stress-bridging in the
$x$-direction along the center of the RVE. Model D extends the line of contact
in the $x$-direction to an area of contact in the $x$-direction. In Model E,
particle bridging across the RVE is extended to both directions. The material
properties of the constituents of PBX 9501 at room temperature and low strain
rate were used for the calculations (Table 2). GMC simulations of the RVEs
were performed using 100$\times$100 subcells while the validating finite
element calculations were performed using approximately 10,000 eight-noded
quadrilateral elements. Table 6 shows the effective stiffnesses of the five
models obtained from GMC and finite element (FEM) calculations.
As expected in model A, GMC and FEM predict nearly the same values of
effective stiffness since there is no stress-bridging in the model (the
effective stiffness is determined primarily by the volume fraction occupied by
the square particle). However, FEM calculations for model B show that the
diagonal stress-bridge in the model produces a higher stiffness than would
occur if only the volume fraction occupied by the particles were considered in
the calculation of effective stiffness.
The GMC calculations for model B predict values of $C^{\text{eff}}_{11}$ and
$C^{\text{eff}}_{12}$ that are lower than the FEM estimates by a factor of 18.
This discrepancy implies that the diagonal stress-bridge in model B is not
detected by the GMC calculations. The value of $C^{\text{eff}}_{66}$ from FEM
is around 1,400 times that from GMC. This difference shows that, in the
presence of stress-bridging, the shear stiffness can be considerably
underestimated by GMC, even for low volume fraction composites.
Model C has a continuous path through particles along the $x$-axis (the ‘1’
direction) and another continuous particle path along one diagonal.
Intuitively, the stress-bridge path along the ‘1’ direction is expected to
primarily affect the normal components of stiffness ($C^{\text{eff}}_{11}$,
$C^{\text{eff}}_{12}$, $C^{\text{eff}}_{22}$) while particle contact along the
diagonal is expected to affect the shear stiffness ($C^{\text{eff}}_{66}$).
These paths are shown by dashed lines (for normal stress-bridging) and by
dotted lines (for shear stress-bridging) in Figure 9. Results for model C in
Table 6 show that FEM predicts a considerable stiffening in the ‘1’ direction
while GMC does not appear to account for these stress-bridges. Since the shear
stiffness from GMC is simply a harmonic means of the subcell stiffnesses,
$C^{\text{eff}}_{66}$ is not affected at all by geometry and only increases in
proportion with the volume fraction of particles in the RVE.
The estimates of $C^{\text{eff}}_{11}$ for model D and of
$C^{\text{eff}}_{11}$, $C^{\text{eff}}_{12}$, $C^{\text{eff}}_{22}$ for model
E show that GMC can capture the effect of stress-bridging, provided there are
continuous rows of particles with edge-to-edge contacts extending completely
across the RVE.
These studies of stress-bridging explain why GMC underestimates the effective
modulus of the PBX 9501 models shown in Figure 5. In all these models, if
100$\times$100 subcells are used to discretize the RVE, there are no rows or
columns of subcells extending across the RVE that contain no binder. Though
corner contacts and other continuous stress paths exist in the PBX 9501
models, the effects of these stress-bridging paths are not incorporated into
the GMC estimates of effective stiffness. The strain-compatible or shear-
coupled method of cells [Williams99, Gan00] approaches may be able to overcome
some of these deficiencies of GMC. However, the computational efficiency of
GMC is greatly reduced when these modifications are incorporated into GMC and
hence the attractiveness of this micromechanics approach as an alternative to
finite element analysis is also reduced.
## 6 The recursive cell method
The recursive cell method (RCM) [Banerjee02th] is a real-space renormalization
group [Wilson71, Wilson79] approach for calculating the effective elastic
properties of composites that has been developed to address the shortcomings
of GMC while retaining high computational efficiency. A schematic of RCM is
shown in Figure 10. In RCM, as in GMC, the RVE is first discretized into a
regular grid of subcells. For the first iteration of the recursive process,
the subcells are assigned material properties based on the particle
distribution in the RVE using the binary subcell approach discussed earlier.
The subcells in the original grid are then grouped into blocks of n$\times$n
subcells. The effective elastic stiffness matrix of each of the blocks is
calculated using a suitable homogenization approach such as GMC or FEM.
Effective stiffnesses are assigned to each block, resulting in a new, coarser
grid. This procedure is repeated until only one homogeneous block remains. The
properties of this homogeneous block are the effective properties of the RVE.
Studies on the recursive cell method [Banerjee02th] have shown that the method
leads to an upper bound on the effective elastic properties if a FEM approach
is used to homogenize blocks of subcells. As the number of elements used to
discretized a block is increased, the value of the upper bound decreases and a
more accurate estimate of the effective properties is obtained. GMC is an
attractive alternative to the FEM approach for homogenization since less
discretization is required to arrive at the same level of accuracy.
In the previous section, GMC has been shown to not properly account for
stress-bridging in the absence of continuous stress-bridge paths across a RVE.
However, error due to improper stress-bridging is reduced when GMC is used as
the homogenizer in RCM because the probability of the existence of continuous
stress-bridging paths across blocks of subcells is greater than that for the
whole RVE. In addition, homogenization errors due to the overestimation or
underestimation of stress-bridging in sections of the RVE are averaged out if
the particle distribution is sufficiently random.
A second source of error in GMC is the underestimation of the shear stiffness
term $C^{\text{eff}}_{66}$. However, this error can be avoided while using RCM
to determine the effective elastic properties of PBXs because, for
macroscopically isotropic materials such as PBXs, relatively accurate
estimates of the effective shear stiffness can be obtained from the effective
normal stiffness terms [Banerjee02th] and therefore direct estimates of
$C^{\text{eff}}_{66}$ are not required. On the other hand, if the composite is
not macroscopically isotropic, a FEM homogenizer [Banerjee02th] can be used to
determine the value of $C^{\text{eff}}_{66}$ instead of GMC.
The RCM technique has been applied to the four microstructures of the dry
blend and pressed PBX 9501 shown in Figure 5(a) and 5(b), respectively. Each
RVE was discretized into blocks of 256$\times$256 square subcells of equal
size. At each stage of recursion, blocks of 2$\times$2 subcells were
homogenized using GMC.
The values of $C^{\text{eff}}_{11}$ for the four dry blend microstructures
obtained from finite element (FEM) calculations, GMC calculations and RCM
calculations are compared in Figure 11(a). The RCM estimates of
$C^{\text{eff}}_{11}$ for the four microstructures vary from 90% to 150% of
the FEM estimates. These RCM estimates are a considerable improvement over the
GMC estimates shown as black bars in Figure 11(a). Comparisons of
$C^{\text{eff}}_{11}$ for the four pressed PBX 9501 microstructures (shown in
Figure 5(b)) are shown in Figure 11(b). For pressed PBX 9501, the RCM
estimates vary between 84% and 180% of the finite element estimates. These RCM
estimates are also a considerable improvement over the GMC estimates of
effective properties. RCM estimates of $C^{\text{eff}}_{22}$ and
$C^{\text{eff}}_{12}$ for the dry blend and pressed PBX 9501 have also been
found to be in much better agreement with FEM results than the GMC estimates.
As was expected, the estimated value of $C^{\text{eff}}_{66}$ from RCM is
quite low compared to both finite element estimates and experimental data. An
improved estimate of $C^{\text{eff}}_{66}$ can be obtained if the shear
stiffness of each RCM block is calculated using finite elements
[Banerjee02th]. The normal stiffnesses $C^{\text{eff}}_{11}$,
$C^{\text{eff}}_{12}$, and $C^{\text{eff}}_{22}$ can be still be calculated
using GMC, taking advantage of the absence of shear coupling.
These results show that the RCM approach, in conjunction with a GMC
homogenizer, can be used to arrive at reasonably accurate estimates of the
effective properties of PBX materials. The RCM approach can therefore be used
as an alternative to direct GMC calculations for high volume fraction, strong
modulus contrast materials such as polymer bonded explosives.
## 7 Summary and conclusions
The generalized method of cells (GMC) has been found to accurately predict the
effective elastic properties of composites containing square arrays of disks
for volume fractions up to 0.60. However, for two-dimensional models of the
polymer bonded explosive PBX 9501, estimates of effective elastic properties
from GMC have been found to be considerably lower than both experimental
values and estimates based on finite element (FEM) calculations.
The lower values of normal stiffness predicted by GMC for PBX 9501 are due to
inadequate incorporation of particle stress bridging into the approach. Model
representative volume elements (RVEs) with corner and edge stress bridging
show that corner bridging is ignored by GMC while edge stress bridging is
incorporated only if continuous stress bridges exist along entire rows or
columns of subcells that traverse the length of the RVE. Low values of
effective shear stiffness predicted by GMC can be attributed to the use of a
harmonic mean of subcell shear stiffnesses to determine the effective shear
stiffness of a RVE. The harmonic mean is a lower bound on the effective shear
stiffness and is not applicable for microstructures where there is significant
interaction between particles.
Improvements suggested to GMC that incorporate normal-shear coupling and
strain compatibility across subcells have the potential to overcome some of
these weaknesses of GMC. However, these improvements lead to much larger
systems of equations and a considerable increase in the computational cost of
the method. The requirement of inverting a large matrix to obtain the
effective properties makes the generalized method of cells very inefficient as
the number of subcells increases. When materials such as PBX 9501 are modeled,
the number of subcells needed to represent a random distribution of particles
necessarily becomes large. In such situations, the generalized method of cells
becomes inefficient and it may be preferable to perform finite element
analyses to determine the effective properties. Thus, GMC does not appear to
be an improvement over finite element analyses for high volume fraction, high
modulus contrast particulate composites such as polymer bonded explosives.
A computationally efficient alternative to both direct GMC and finite elements
is the recursive cell method (RCM) with GMC being used to homogenize blocks of
subcells. RCM estimates of normal stiffness terms for models of PBX 9501 show
considerable improvement compared to GMC estimates. The RCM estimates of shear
stiffness can be improved if FEM is used, rather than GMC, to determine the
effective shear stiffness of blocks of subcells. RCM, with a combination of
GMC and FEM being used to homogenize blocks of subcells, has the potential of
providing fast and accurate estimates of the effective properties of polymer
bonded explosives.
## Acknlowledgements
This research was supported by the University of Utah Center for the
Simulation of Accidental Fires and Explosions (C-SAFE), funded by Department
of Energy grant DE-FG03-02ER45914. The authors would also like to thank Prof.
Graeme Milton and Dr. Brett Bednarcyk for their suggestions.
## References
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* [6] Bednarcyk Pindera1997Bednar97 Bednarcyk, B. A. Pindera, M.-J. 1997, Micromechanical modeling of woven metal matrix composites, Technical Report NASA-CR 204153, National Aeronautics and Space Administration, Lewis Research Center, USA.
* [7] Benson Conley1999Benson99 Benson, D. J. Conley, P. 1999, ‘Eulerian finite-element simulations of experimentally acquired HMX microstructures’, Modelling Simul. Mater. Sci. Eng. 7(3), 333–354.
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* [9] Greengard Helsing1998Greeng98 Greengard, L. Helsing, J. 1998, ‘On the numerical evaluation of elastostatic fields in locally isotropic two-phase composites’, J. Mech. Phys. Solids 46, 1441–1462.
* [10] [Low et al.]Low, Gardner Pittman1994Low94 Low, B. Y., Gardner, S. D. Pittman, C. U. 1994, ‘A micromechanical characterization of graphite-fiber/epoxy composites containing a heterogeneous interphase region’, Composites Sci. Tech. 52, 589–606.
* [11] Markov2000Markov00 Markov, K. Z. 2000, Elementary micromechanics of heterogeneous media, in K. Z. Markov L. Preziosi, eds, ‘Heterogeneous Media : Micromechanics Modeling Methods and Simulations’, Birkhauser, Boston, pp. 1–162.
* [12] Orozco1997Orozco97 Orozco, C. E. 1997, ‘Computational aspects of modeling complex microstructure composites using GMC’, Composites Part B 28B, 167–175.
* [13] Paley Aboudi1992Paley92 Paley, M. Aboudi, J. 1992, ‘Micromechanical analysis of composites by the generalized cells model’, Mechanics of Materials 14, 127–139.
* [14] Pindera Bednarcyk1997Pindera97 Pindera, M.-J. Bednarcyk, B. A. 1997, An efficient implementation of the gmc micromechanics model for multi-phased materials with complex microstructures, Technical Report NASA-CR 202350, National Aeronautics and Space Administration, Lewis Research Center, USA.
* [15] [Skidmore et al.]Skidmore, Phillips, Howe, Mang Romero1998Skid98 Skidmore, C. B., Phillips, D. S., Howe, P. M., Mang, J. T. Romero, J. A. 1998, The evolution of microstructural changes in pressed HMX explosives, in ‘Proc., 11th International Detonation Symposium’, Snowmass, Colorado, pp. 556–564.
* [16] Wetzel1999Wetzel99 Wetzel, B. J. 1999, Investigation of the creep behavior of nonlinear viscoelastic simulant materials for high explosives, Master’s thesis, California Institute of Technology, Pasadena, California.
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* [18] Wilson1971Wilson71 Wilson, K. G. 1971, ‘Renormalization group and critical phenomena. I. renormalization group and Kadanoff scaling picture’, Physical Review B 4(9), 3174–3183.
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* [21]
Figure 1: RVE, subcells and notation used in GMC. Figure 2: Comparison of
effective moduli of square arrays of disks from Greengard and Helsing (1998)
(G&H) and GMC calculations. Figure 3: Manually generated microstructures
containing $\sim$ 90% circular particles by volume. Figure 4: Schematics of
the application of the binary subcell approach and the effective subcell
approach in GMC calculations. Figure 5: Microstructures containing circular
particles based on the particle size distribution of the dry blend (DB) of PBX
9501 and of pressed (PP) PBX 9501. Figure 6: RVE used for corner stress-
bridging model. Figure 7: Variation of effective stiffness with modulus
contrast for ‘X’-shaped microstructure. The Young’s modulus contrast is the
ratio of the Young’s modulus of the particles to that of the binder. Figure 8:
Progressive stress-bridging models A through E. Figure 9: Stress-bridging
paths for Model C. Figure 10: Schematic of the recursive cell method. Figure
11: Comparisons of estimates of $C^{\text{eff}}_{11}$ for (a) models of the
dry blend of PBX 9501 (b) models of pressed PBX 9501.
Table 1: Component properties used by Greengard and Helsing (1998). | Young’s | Poisson’s | Two-Dimensional | Shear
---|---|---|---|---
| Modulus | Ratio | Bulk Modulus | Modulus
| (MPa) | | (MPa) | (MPa)
Disks | 324 | 0.20 | 225 | 135
Binder | 2.7 | 0.35 | 3.3 | 1
Table 2: Experimentally determined elastic moduli and stiffness of PBX 9501 and its constituents (Wetzel 1999). $C_{ij}$ are components of the stiffness matrix. Material | Young’s | Poisson’s | $C_{11}$ = $C_{22}$ | $C_{12}$ | $C_{66}$
---|---|---|---|---|---
| Modulus | Ratio | | |
| (MPa) | | (MPa) | (MPa) | (MPa)
Particles | 15300 | 0.32 | 21894 | 10303 | 5795
Binder | 0.7 | 0.49 | 11.97 | 11.51 | 0.235
PBX 9501 | 1013 | 0.35 | 1626 | 875 | 375
Table 3: Effective stiffnesses of the six model microstructures from GMC and FEM calculations. | $C^{\text{eff}}_{11}$ (MPa) | $C^{\text{eff}}_{12}$ (MPa) | $C^{\text{eff}}_{66}$ (MPa)
---|---|---|---
| FEM | GMC | FEM | GMC | FEM | GMC
| | Binary | Effective | | Binary | Effective | | Binary | Effective
| | Subcell | Subcell | | Subcell | Subcell | | Subcell | Subcell
Model 1 | 177 | 814 | 479 | 90 | 119 | 103 | 11 | 2.4 | 2.3
Model 2 | 181 | 807 | 477 | 86 | 112 | 103 | 12 | 2.3 | 2.3
Model 3 | 186 | 815 | 193 | 88 | 108 | 89 | 15 | 2.3 | 2.2
Model 4 | 143 | 116 | 142 | 114 | 112 | 124 | 33 | 2.4 | 2.6
Model 5 | 237 | 132 | 323 | 94 | 100 | 104 | 38 | 2.3 | 2.5
Model 6 | 229 | 132 | 334 | 76 | 93 | 100 | 9 | 2.1 | 2.5
Mean | 192 | 471 | 325 | 91 | 107 | 104 | 20 | 2.3 | 2.4
Table 4: Effective stiffness of the model PBX 9501 microstructures from GMC and FEM calculations. | $C^{\text{eff}}_{11}$ (MPa) | $C^{\text{eff}}_{22}$ (MPa) | $C^{\text{eff}}_{12}$ (MPa) | $C^{\text{eff}}_{66}$ (MPa)
---|---|---|---|---
| FEM | GMC | FEM | GMC | FEM | GMC | FEM | GMC
Model DB1 | 2385 | 152 | 2094 | 148 | 633 | 122 | 792 | 4.9
Model DB2 | 3618 | 146 | 1643 | 144 | 656 | 122 | 750 | 4.9
Model DB3 | 3546 | 149 | 3385 | 148 | 1142 | 125 | 1317 | 5.0
Model DB4 | 5274 | 144 | 5124 | 146 | 1712 | 120 | 1703 | 4.8
Model PP1 | 3180 | 180 | 3570 | 188 | 989 | 131 | 1262 | 4.8
Model PP2 | 3886 | 170 | 3683 | 190 | 1032 | 132 | 1278 | 5.7
Model PP3 | 6302 | 181 | 6221 | 181 | 2043 | 133 | 2077 | 6.6
Model PP4 | 7347 | 182 | 7587 | 186 | 2547 | 129 | 2542 | 6.9
Table 5: The elastic properties of the components of the X shaped microstructure. $C_{ij}$ are components of the stiffness matrix. | Young’s | Poisson’s | $C_{11}$ | $C_{12}$ | $C_{66}$
---|---|---|---|---|---
| Modulus | Ratio | | |
| (MPa) | | (MPa) | (MPa) | (MPa)
Particles | 15300 | 0.32 | 21894 | 10303 | 5795
Binder a | 0.7 | 0.49 | 12 | 11.5 | 0.2
Binder b | 7 | 0.49 | 120 | 115 | 2.4
Binder c | 70 | 0.49 | 1198 | 1151 | 23.5
Binder d | 700 | 0.49 | 11980 | 11510 | 235
Binder e | 7000 | 0.49 | 119799 | 115101 | 2349
Table 6: Effective properties of edge bridging models. | $C^{\text{eff}}_{11}$ (MPa) | $C^{\text{eff}}_{22}$ (MPa) | $C^{\text{eff}}_{12}$ (MPa) | $C^{\text{eff}}_{66}$ (MPa)
---|---|---|---|---
| FEM | GMC | FEM | GMC | FEM | GMC | FEM | GMC
Model A | 16 | 16 | 16 | 16 | 15 | 15 | 0.4 | 0.3
Model B | 336 | 19 | 343 | 19 | 337 | 18 | 537 | 0.4
Model C | 4095 | 25 | 889 | 24 | 1470 | 23 | 1093 | 0.5
Model D | 8992 | 8540 | 1361 | 32 | 523 | 23 | 1182 | 0.6
Model E | 10017 | 9042 | 10052 | 9042 | 2892 | 2143 | 1799 | 0.9
Figure 1
Figure 2
Figure 3
(a) Binary subcell approach.
(b) Effective subcell approach.
Figure 4
0.65$\times$0.65 mm2 0.94$\times$0.94 mm2 1.13$\times$1.13 mm2
1.33$\times$1.33 mm2
DB1 DB2 DB3 DB4
(a) Dry Blend of PBX 9501
0.36$\times$0.36 mm2 0.42$\times$0.42 mm2 0.54$\times$0.54 mm2
0.68$\times$0.68 mm2
PP1 PP2 PP3 PP4
(b) Pressed PBX 9501
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
(a) Dry Blend (b) Pressed PBX
Figure 11
|
arxiv-papers
| 2012-01-11T22:14:40 |
2024-09-04T02:49:26.190070
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Biswajit Banerjee and Daniel O. Adams",
"submitter": "Biswajit Banerjee",
"url": "https://arxiv.org/abs/1201.2435"
}
|
1201.2444
|
# On exact relations for the calculation of effective properties of composites
Biswajit Banerjee (b.banerjee.nz@gmail.com) and Daniel O. Adams
Dept. of Mechanical Engineering, University of Utah, Salt Lake City, USA
(June 2002)
## Abstract
Numerous exact relations exist that relate the effective elastic properties of
composites to the elastic properties of their components. These relations can
not only be used to determine the properties of certain composites, but also
provide checks on the accuracy on numerical techniques for the calculation of
effective properties. In this work, some exact relations are discussed and
estimates from finite element calculations, the generalized method of cells
and the recursive cell method are compared with estimates from the exact
relations. Comparisons with effective properties predicted using exact
relations show that the best estimates are obtained from the finite element
calculations while the moduli are overestimated by the recursive cell method
and underestimated by the generalized method of cells. However, not all exact
relations can be used to make such a distinction.
## 1 Introduction
Exact relations for the effective elastic properties of two-component
composites can be classified into three types. The first type consists of
relations that have been determined from the similarity of the two-dimensional
stress and strain fields for certain types of materials. These exact relations
are called duality relations [1]. The second type of exact relations, called
translation-based relations, state that if a constant quantity is added to the
elastic moduli of the component materials then the effective elastic moduli
are also “translated” by the same amount. Microstructure independent exact
relations, valid for special combinations of the elastic properties of the
components, form the third category [2]. The known exact relations are
directly applicable only to a limited range of properties of the components.
Therefore the utility of these relations lies not only in determining the
effective elastic properties of a small range of composites but also in
evaluating the accuracy of numerical and analytical methods of computing
effective properties. In this work, predictions from exact relations are
compared with estimates from finite element calculations, the generalized
method of cells (GMC) [3], and the recursive cell method (RCM) [4]. The goal
is to assess the effectiveness of these relations in evaluating the accuracy
of the three numerical methods, especially with regard to high modulus
contrast materials such as polymer bonded explosives.
Five exact relations are explored in this work. The first is a duality-based
identity for the effective shear modulus that is valid for phase-
interchangeable materials [5]. The second is a set of duality relations that
are valid for materials that are rigid with respect to shear [1]. Two
translation-based relations are explored next - the CLM theorem [6] and a
relation for symmetric composites with equal bulk modulus [5]. The
microstructure independent Hill’s relation [7] is explored last.
## 2 Phase interchange identity
A symmetric composite is one that is invariant with respect to interchange of
the components. A checkerboard, as shown in Figure 1, is an example of a
symmetric composite.
Figure 1: Representative volume element for a checkerboard composite.
The phase interchange identity [5] for the effective shear modulus of a
symmetric two-dimensional two-component isotropic composite is a duality-based
exact relation that states that
$G_{\text{eff}}=\sqrt{G_{1}G_{2}}$ (1)
where $G_{1}$, $G_{2}$ are the shear moduli of the two components and
$G_{\text{eff}}$ is the effective shear modulus.
The phase interchange identity is valid only for isotropic composites. In a
finite-sized representative volume element (RVE) for a checkerboard composite
the shear modulus is not the same all directions and hence isotropy is not
achieved. The two-dimensional stress-strain relation for such a RVE with
“square symmetry” can be written as
$\left[\begin{array}[]{c}\sigma_{11}\\\ \sigma_{22}\\\
\sigma_{12}\end{array}\right]=\left[\begin{array}[]{lll}K+\mu^{1}&K-\mu^{1}&0\\\
K-\mu^{1}&K+\mu^{1}&0\\\
0&0&\mu^{2}\end{array}\right]\left[\begin{array}[]{c}\epsilon_{11}\\\
\epsilon_{22}\\\ 2\left<\epsilon_{12}\right>_{\text{V}}\end{array}\right]$ (2)
where $\sigma_{11}$, $\sigma_{22}$, $\sigma_{12}$ are the stresses;
$\epsilon_{11}$, $\epsilon_{22}$, $\epsilon_{12}$ are the strains; $K$ is the
two-dimensional bulk modulus, $\mu^{1}$ is the shear modulus when shear is
applied along the diagonals of the RVE, and $\mu^{2}$ is the shear modulus for
shear along the edges of the RVE.
The numerical verification of the phase interchange identity, therefore,
requires that the components of the composite be chosen so that the difference
between $\mu^{1}$ and $\mu^{2}$ for the composite is minimal. This implies
that the components should have a weak modulus contrast.
Numerical estimates of the effective elastic properties of the checkerboard
composite shown in Figure 1 were obtained using finite elements (FEM), the
recursive cell method (RCM) and the generalized method of cells (GMC).
Following the requirement of low modulus contrast, both components were
assigned a Young’s modulus of 15,300 MPa. The Poisson’s ratio of the first
component was fixed at 0.32 while that of the second component was varied from
0.1 to 0.49. The FEM calculations were performed using a mesh of
256$\times$256 four-noded square elements. The RCM calculations used a grid of
64$\times$64 subcells with blocks of 2$\times$2 subcells and each subcell was
modeled using one nine-noded element. The GMC calculations used 64$\times$64
square subcells to discretize the RVE.
Figure 2 shows a comparison of the exact effective shear modulus for the
checkerboard composite with estimates of $\mu^{1}$ and $\mu^{2}$ from the
three numerical approaches.
Figure 2: Validation of FEM, RCM and GMC using the phase interchange identity
for a checkerboard composite.
The results show that all the three methods perform well (the maximum error is
0.1%) in predicting the effective shear modulus when the modulus contrast is
small, i.e., when the composite is nearly isotropic. It can also be observed
that the values of $\mu^{1}$ and $\mu^{2}$ are within 1% of each other for the
chosen component moduli.
### 2.1 Range of applicability
The question that arises at this point is whether the three numerical
approaches can predict the phase interchange identity for larger modulus
contrasts. Numerical calculations have been performed on the checkerboard
microstructure to explore this issue. The first component of the checkerboard
was assigned a Young’s modulus of 15,300 MPa and a Poisson’s ratio of 0.32.
For the second component, the Poisson’s ratio was fixed at 0.49 and the
Young’s modulus was varied from 0.7 MPa to 7000 MPa.
Figure 3 shows plots of the effective $\mu^{1}$ and $\mu^{2}$ versus shear
modulus contrast for a checkerboard RVE.
Figure 3: Variation of effective shear moduli with modulus contrast for a
checkerboard composite.
The plots confirm that when the modulus contrast between the components of the
checkerboard exceeds 2, the material can no longer be considered isotropic
since the values of $\mu^{1}$ and $\mu^{2}$ are considerably different from
each other. However, the values of $\mu^{1}$ predicted by FEM are quite close
to the effective shear modulus $G_{\text{eff}}$ predicted by the phase
interchange identity. This result suggests that the simulation of a diagonal
shear may not be necessary to predict the effective shear modulus of an
isotropic composite when the finite element approach is used. It also implies
that the phase interchange identity can be used for a much larger range of
modulus contrasts. The effective shear moduli predicted by GMC are
considerably lower than that from the exact relation while the values from RCM
are consistently higher. The RCM estimates worsen with increasing modulus
contrast. If only the value of $\mu^{1}$ is examined, the phase interchange
identity indicates that the FEM approach is much more accurate than the GMC
and RCM approaches. However, it is difficult to choose between GMC and RCM for
high modulus contrasts composites. While the exact value of $\mu^{1}$ is 10
times the value predicted by GMC, the corresponding RCM estimate is 10 times
the exact value. These results confirm the findings of detailed numerical
studies on high modulus contrast, high volume fraction polymer bonded
explosives [8, 9, 4].
### 2.2 Convergence of FEM calculations
The checkerboard material provides an extreme case to test the convergence of
the FEM solution because the corner singularities lead to high stresses that
can only be resolved with refined meshes. Figure 4 shows the convergence of
the effective $\mu^{1}$ and $\mu^{2}$ with increasing mesh refinement for a
checkerboard with a shear modulus contrast of about 25,000.
Figure 4: Convergence of effective moduli predicted by FEM with increase in
mesh refinement for a checkerboard composite with a shear modulus contrast of
25,000.
The effective $\mu^{1}$ converges to a steady value when 128$\times$128
elements are used to discretize the RVE. The shear modulus $\mu^{2}$ reaches a
steady value when 256$\times$256 elements are used. This is why the finite
element calculations in this work were performed using 256$\times$256 elements
or more.
RCM uses a finite element approach to homogenize blocks of subcells. When
blocks of 2$\times$2 subcells are used, some of these blocks can resemble
checkerboards - especially at the first level of recursion for a two-component
composite. The finite element convergence result suggests that RCM may
overestimate the effective shear moduli by a factor of two if a block of four
subcells is simulated using only four finite elements.
## 3 Materials rigid in shear
The stress-strain response of two-dimensional materials that are rigid with
respect to shear can be represented by
$\left[\begin{array}[]{c}\epsilon_{11}\\\ \epsilon_{22}\\\
\epsilon_{12}\end{array}\right]=\left[\begin{array}[]{ccc}S_{11}&S_{12}&0\\\
S_{12}&S_{22}&0\\\
0&0&0\end{array}\right]\left[\begin{array}[]{c}\sigma_{11}\\\ \sigma_{22}\\\
\sigma_{12}\end{array}\right]$ (3)
where $\sigma_{11}$, $\sigma_{22}$, $\sigma_{12}$ are the stresses;
$\epsilon_{11}$, $\epsilon_{22}$ and $\epsilon_{12}$ are the strains, and
$S_{ij}$ are the components of the compliance matrix.
Two duality-based exact relations that are valid for two-component composites
composed of such materials are [1]:
Relation RS1
If $S_{11}S_{22}-(S_{12})^{2}=\Delta$ for each phase (where $\Delta$ is a
constant), then the effective compliance tensor also satisfies the same
relationship, i.e.,
$S^{\text{eff}}_{11}S^{\text{eff}}_{22}-(S^{\text{eff}}_{12})^{2}=\Delta_{\text{eff}}$.
This relation is true for all microstructures.
Relation RS2
If the compliance tensors of the two phases are of the form
$\mathbf{S}_{1}=\alpha_{1}\mathbf{A}$ and
$\mathbf{S}_{2}=\alpha_{2}\mathbf{A}$ where $\mathbf{A}$ is a constant matrix,
then the effective compliance tensor of a checkerboard of the two phases
satisfies the relation
$\det{\mathbf{S}_{\text{eff}}}=S^{\text{eff}}_{11}S^{\text{eff}}_{22}-(S^{\text{eff}}_{12})^{2}=\alpha_{1}\alpha_{2}(A_{11}A_{22}-(A_{12})^{2})$.
### 3.1 Relation RS1
Figure 5 shows a square array of disks occupying an area fraction of 0.7.
Figure 5: RVE for a square array of disks.
Numerical experiments have been performed on this array of disks to check if
Relation RS1 can be reproduced by finite element analyses, GMC and RCM. The
$\mathbf{S}$ matrices that have been used for the disks (superscript $1$) and
the matrix (superscript $2$), and the corresponding values of $\Delta$ are
shown below. These matrices have been chosen so that the value of $\Delta$ is
constant.
$\mathbf{S}_{1}=\left[\begin{array}[]{ccc}1000&-300&0\\\ -300&1000&0\\\
0&0&0.001\end{array}\right],~{}\Delta=9.1\times 10^{5},$
and
$\mathbf{S}_{2}=\left[\begin{array}[]{ccc}1094.3&-536.21&0\\\
-536.21&1094.3&0\\\ 0&0&0.001\end{array}\right],~{}\Delta=9.1\times 10^{5}.$
The shear modulus for both materials is $1000$ (arbitrary units) - around
$10^{6}$ times the Young’s modulus. Higher values of shear modulus have been
tested and found not to affect the effective stiffness matrix significantly.
Table 1 shows the values of $S^{\text{eff}}_{11}$, $S^{\text{eff}}_{12}$ and
$\Delta_{\text{eff}}$ calculated using finite elements (350$\times$350
elements), GMC (64$\times$64 subcells) and RCM (256$\times$256 subcells).
Table 1: Two-dimensional effective compliance matrix for a square array of disks. | $S^{\text{eff}}_{11}$ | $S^{\text{eff}}_{12}$ | $\Delta_{\text{eff}}$ $(\times 10^{5})$ | $\Delta_{\text{eff}}$/$\Delta$
---|---|---|---|---
FEM | 850.35 | -536.32 | 4.35 | 0.48
RCM | 847.25 | -538.17 | 4.28 | 0.47
GMC | 871.75 | -517.14 | 4.93 | 0.54
The ratio of the calculated $\Delta_{\text{eff}}$ to the original $\Delta$ are
also shown in the table. The modulus contrast between the two components of
the composite is small, so the calculated effective properties are expected to
be accurate (based on the results on the phase interchange identity for shear
moduli). However, the results in Table 1 show that all the three numerical
methods predict values of $\Delta_{\text{eff}}$ that are around half the
original $\Delta$. These results imply that all three methods (FEM, GMC and
RCM) overestimate the effective normal stiffness of the array of disks.
Relation RS1 for materials rigid in shear may therefore be a very sensitive
test of the accuracy of numerical methods even though the modulus contrast
that can be used is small.
### 3.2 Relation RS2
The second duality relation for materials that are rigid in shear requires
(Relation RS2) is valid for the checkerboard geometry shown in Figure 1. The
following values of the elastic properties have been used to test the accuracy
of FEM, RCM and GMC in predicting this relation.
$\displaystyle\mathbf{S}_{1}$
$\displaystyle=100\left[\begin{array}[]{cc}10&-3\\\
-3&10\end{array}\right]~{};~{}\mathbf{S}_{2}=1000\left[\begin{array}[]{cc}10&-3\\\
-3&10\end{array}\right]$ $\displaystyle\alpha_{1}$
$\displaystyle=100~{};~{}\alpha_{2}=1000$ $\displaystyle\mathbf{A}$
$\displaystyle=\left[\begin{array}[]{cc}10&-3\\\ -3&10\end{array}\right]$
The duality relation requires that the effective compliance matrix of the
checkerboard composite should be such that
$\det(\mathbf{S}_{\text{eff}})=S^{\text{eff}}_{11}S^{\text{eff}}_{22}-(S^{\text{eff}}_{12})^{2}=9.10\times
10^{6}~{}.$
The FEM calculations were performed using 350$\times$350 four-noded elements,
the RCM calculations used 64$\times$64 subcells (blocks of 2$\times$2
subcells) and the GMC calculations used 64$\times$64 subcells too. The results
from these three methods are tabulated in Table 2.
Table 2: Effective compliance matrix for a checkerboard composite with components rigid in shear. | $S^{\text{eff}}_{11}$ | $S^{\text{eff}}_{12}$ | $\det(\mathbf{S}_{\text{eff}})(\times 10^{6})$ | $\det(\mathbf{S}^{\text{eff}})/\alpha_{1}\alpha_{2}\det{\mathbf{A}}$
---|---|---|---|---
FEM | 3282 | -2004 | 6.75 | 0.74
RCM | 1655 | -7090 | 2.23 | 0.24
GMC | 5007 | -2146 | 2.05 | 2.25
The finite element calculations lead to quite an accurate effective compliance
matrix and the deviation from the exact result is only around 25%. The GMC
calculations overestimate the compliance matrix and the determinant of the
compliance matrix is around 2.3 times higher than the exact result. On the
other hand, the RCM calculations predict a compliance matrix that has a
determinant that is only around 20
## 4 The CLM theorem
The Cherkaev, Lurie and Milton (CLM) theorem is a well known “translation”
based exact relation for two-component planar composites [6]. For a two-
dimensional two-component isotropic composite, this theorem can be stated as
follows.
Let the isotropic bulk moduli of the components be $K_{1}$ and $K_{2}$. Let
the shear moduli of the two components be $G_{1}$ and $G_{2}$. The effective
bulk and shear modulus of a two-dimensional composite made of these two
components are $K_{\text{eff}}$ and $G_{\text{eff}}$ respectively. Let us now
create two new materials that are “translated” from the original component
materials by a constant amount $\lambda$. That is, let the bulk and shear
moduli of the translated component materials be given by
$\displaystyle
1/K^{T}_{1}=1/K_{1}-\lambda~{};~{}1/K^{T}_{2}=1/K_{2}-\lambda~{};$
$\displaystyle
1/G^{T}_{1}=1/G_{1}+\lambda~{};~{}1/G^{T}_{2}=1/G_{2}+\lambda~{}.$
The CLM theorem states that the effective bulk and shear moduli of a two-
dimensional composite of the two translated materials, having the same
microstructure as the original composite, are given by
$1/K^{T}_{\text{eff}}=1/K_{\text{eff}}-\lambda~{};~{}1/G^{T}_{\text{eff}}=1/G_{\text{eff}}+\lambda.$
(6)
The requirement of isotropy can be satisfied approximately by choosing
component material properties that are close to each other. Since our goal is
to determine how well GMC and RCM perform for high modulus contrast, choosing
materials with small modulus contrast is not adequate. Another alternative is
to choose a RVE that represents a hexagonal packing of disks. However, such an
RVE is necessarily rectangular and cannot be modeled using RCM in its current
form. It should be noted that RCM can easily be modified to deal with elements
that are not square and hence to model rectangular regions.
Another problem in the application of the CLM theorem is that the value of
$\lambda$ has to be small if the difference between the original and the
translated moduli is large and vice versa. If the value of $\lambda$ is small,
floating point errors can accumulate and exceed the value of $\lambda$. On the
other hand, if $\lambda$ is large, the original and the translated moduli are
very close to each other and the difference between the two can be lost
because of errors in precision. Hence, the numbers have to be chosen carefully
keeping in mind the limits on the value of the Poisson’s ratio.
The translation relation has been tested on the square array of disks
occupying a volume fraction of 0.70 from Figure 5. This RVE exhibits square
symmetry, i.e., the shear moduli $\mu^{1}$ and $\mu^{2}$ shown in equation (2)
are not equal. A unique value of the effective shear modulus cannot be
calculated for this RVE. Instead, he value of the effective translated shear
modulus is calculated from equation (6) by first setting $G_{\text{eff}}$
equal to $\mu^{1}$ and then to $\mu^{2}$. These “exact” values are compared
with the $\mu^{1}$ and $\mu^{2}$ values predicted using finite element
analyses, GMC and RCM.
The original set of elastic moduli for the RVE is chosen to reflect the
elastic moduli of the constituents of polymer bonded explosives. These moduli
are then translated by a constant $\lambda=0.001$. The original and the
translated constituent two-dimensional moduli are shown in Table 3 (phase ’p’
represents the particles and phase ’b’ represents the binder). It can be
observed that the translation process creates quite a large change in the bulk
modulus of the particles.
Table 3: Original and translated two-dimensional constituent moduli for checking the CLM condition. | $K_{p}$ | $G_{p}$ | $K_{b}$ | $G_{b}$ | $K_{p}/K_{b}$ | $G_{p}/G_{b}$
---|---|---|---|---|---|---
| $(\times 10^{2})$ | $(\times 10^{2})$ | | | $(\times 10^{2})$ | $(\times 10^{2})$
Original | 9.60 | 4.80 | 10.07 | 0.20 | 0.95 | 23.8
Translated | 240.0 | 3.24 | 10.17 | 0.20 | 23.5 | 16.1
Table 4 shows the effective bulk and shear moduli of the original and the
translated material calculated using finite elements (350$\times$350
elements), GMC (64$\times$64 subcells) and RCM (256$\times$256 subcells). The
values of $\lambda_{\text{err}}$ shown in the table have been calculated using
the equation
$\displaystyle\lambda_{\text{err}}$ $\displaystyle=(\lambda/0.001-1)\times
100,$
$\displaystyle\lambda=1/K_{\text{eff}}-1/K^{T}_{\text{eff}}=1/\mu^{i(T)}_{\text{eff}}-1/\mu^{i}_{\text{eff}}.$
Table 4: Comparison of effective moduli for the original and the translated composites. | $K_{\text{eff}}$ | $\mu^{1}_{\text{eff}}$ | $\mu^{2}_{\text{eff}}$
---|---|---|---
| Orig. | Trans. | $\lambda_{\text{err}}$(%) | Orig. | Trans. | $\lambda_{\text{err}}$(%) | Orig. | Trans. | $\lambda_{\text{err}}$(%)
FEM | 36.4 | 37.8 | -0.8 | 10.1 | 10 | -3.1 | 0.9 | 0.9 | 22
RCM | 42.5 | 44.5 | 6.1 | 29.8 | 29 | -6.9 | 1.3 | 1.3 | -292
GMC | 34.0 | 35.1 | -1.3 | 3.8 | 3.8 | 5.3 | 0.7 | 0.7 | 30
Even though the modulus contrast between the two components of the composite
is high, the effective properties predicted by FEM, GMC and RCM are close to
each other in magnitude. The effective moduli of the translated composite are
also quite close to that of the original composite as predicted by the CLM
condition. The interesting fact is that all the three methods satisfy the CLM
condition and the error is small (as seen by the values of
$\lambda_{\text{err}}$. Of the three methods, FEM and GMC produce the least
error while RCM produces the most error.
## 5 Composites with equal bulk modulus
The translation procedure can also be used to generate an exact solution for
the effective shear modulus of two-dimensional symmetric two-component
composites with both components having the same bulk modulus [5]. This
relation is
$\displaystyle K_{\text{eff}}$ $\displaystyle=~{}K~{}=~{}K_{1}~{}=~{}K_{2}$
$\displaystyle G_{\text{eff}}$
$\displaystyle=\frac{K}{-1+\sqrt{\left(1+\frac{K}{G_{1}}\right).\left(1+\frac{K}{G_{2}}\right)}}$
(7)
This relation has been tested on the checkerboard model shown in Figure 1
using the component material properties given in Table 5. The exact effective
properties for the composite, calculated using equation (7), are also given in
the table. The values of the effective moduli calculated using finite elements
(FEM), GMC and RCM are also shown in Table 5.
Table 5: Component properties, exact effective properties and numerically
computed effective properties for two-component symmetric composite with equal
component bulk moduli.
| $E$ | $\nu$ | $K$ | $G$
---|---|---|---|---
| $(\times 10^{2})$ | | $(\times 10^{3})$ | $(\times 10^{2})$
Component 1 | 25.00 | 0.25 | 2.0 | 10.0
Component 2 | 1.19 | 0.49 | 2.0 | 0.4
Composite | 5.12 | 0.46 | 2.0 | 1.76
| $K_{\text{eff}}$ | $\mu^{1}_{\text{eff}}$ | Diff. | $\mu^{2}_{\text{eff}}$ | Diff.
---|---|---|---|---|---
| $(\times 10^{2})$ | $(\times 10^{2})$ | % | $(\times 10^{2})$ | %
FEM | 20 | 1.29 | -26.8 | 2.54 | 44.4
GMC | 20 | 0.77 | -56.3 | 0.77 | -56.3
RCM | 20 | 2.96 | 68.0 | 4.41 | 150.9
These results show that the effective two-dimensional bulk modulus is
calculated correctly by all the three methods. However, the shear moduli
calculated for the checkerboard microstructure are quite different from the
exact result. This exact result also shows that the FEM calculations are the
most accurate, followed by GMC and then RCM. The values of
$\mu^{1}_{\text{eff}}$ are also found to most closely approximate the value of
$G_{\text{eff}}$.
## 6 Hill’s equation
Hill’s equation [7] is an exact relation that is independent of
microstructure. This equation is valid for composites composed of isotropic
components that have the same shear modulus. For a two-dimensional two-
component composite, this equation can be written as
$\frac{1}{K_{\text{eff}}+G}=\frac{f_{p}}{K_{p}+G}+\frac{f_{b}}{K_{b}+G}$ (8)
where $f$ represents a volume fraction, $K$ represents a bulk modulus, and $G$
represents a shear modulus. The subscript ’$p$’ represents a particle
property, ’$b$’ represents a binder property, and ’eff’ represents the
effective property of the composite.
This relationship is verified using the RVE containing an array of disks
occupying 70% of the volume that is shown in Figure 5. Table 6 shows the
properties of the two components used to compare the predictions of finite
elements, GMC and RCM with the exact value of bulk modulus predicted by Hill’s
equation. It should be noted that the materials chosen are not quite
representative of polymer bonded explosive materials.
Table 6: Phase properties used for testing Hill’s equation and the exact effective moduli of the composite. | Vol. | $E$ | $\nu$ | $G$ | $K$
---|---|---|---|---|---
| Frac. | $(\times 10^{3})$ | | $(\times 10^{3})$ | $(\times 10^{3})$
Disks | 0.7 | 3.00 | 0.25 | 1.20 | 2.40
Binder | 0.3 | 3.58 | 0.49 | 1.20 | 60.00
Composite | 1.0 | 3.22 | 0.34 | 1.20 | 3.82
Since the modulus contrast is small, the square array of disks is expected to
exhibit nearly isotropic behavior. Therefore, the predictions of finite
elements, GMC and RCM are expected to be close to the exact values of the
effective properties of the composite. The numerically calculated values of
the effective two-dimensional bulk and shear moduli of the composite are shown
in Table 7. The percentage difference of the effective bulk modulus from the
exact value is also shown in the table.
Table 7: Numerically computed effective properties for a square array of disks with equal component shear moduli. | $K_{\text{eff}}$ | % Diff. | $\mu^{1}_{\text{eff}}$ | $\mu^{2}_{\text{eff}}$
---|---|---|---|---
| ($\times 10^{3}$) | | ($\times 10^{3}$) | ($\times 10^{3}$)
FEM | 3.98 | 4.4 | 1.20 | 1.20
RCM | 3.92 | 2.7 | 1.20 | 1.20
GMC | 3.66 | -4.2 | 1.20 | 1.20
The effective shear moduli predicted by all the three methods are exact. In
case of the effective bulk moduli, the RCM predictions are the most accurate
followed by GMC and the finite element based calculations. The finite element
based calculations overestimate the effective two-dimensional bulk modulus by
around 4.4% while GMC underestimates the bulk modulus by around 4.2%. Since
the error of estimation of all the three methods is small, it is suggested
that all three methods are accurate for low contrasts in the shear modulus.
However, Hill’s equation does not appear to be suitable for determining the
best numerical method of the three.
## 7 Summary and conclusions
Predictions from the phase interchange identity for the shear modulus are
closely approximated by the finite element approach (FEM), the recursive cell
method (RCM) and the generalized method of cells (GMC) for checkerboard
composites with low modulus contrast. However, for higher modulus contrasts
the FEM approximations of shear moduli are the most accurate. The RCM
predictions overestimate the shear modulus while GMC underestimates the shear
modulus. The exact relations for materials that are rigid in shear show that
all three numerical techniques are inaccurate. The exact relation for this
class of materials that is applicable to checkerboard materials shows that the
FEM calculations are the most accurate while both RCM and GMC perform poorly
in comparison. Though the predictions of the CLM theorem are quite accurately
predicted by all three numerical methods for high modulus contrast composites,
the FEM results show the least error between the original and the translated
effective properties while the RCM results show the largest error. The exact
relation for isotropic composites with components that have the same bulk
moduli also shows that the FEM predictions are the most accurate though they
are somewhat higher than the exact values. However, no such distinction
between the three methods can be made using Hill’s equation. These results
reflect previous studies for high modulus contrast, high volume fraction
polymer bonded explosives using FEM, RCM and RCM and show that exact relations
can be used to determine the accuracy of numerical methods without performing
detailed numerical studies.
## Acknowledgements
This research was supported by the University of Utah Center for the
Simulation of Accidental Fires and Explosions (C-SAFE), funded by the
Department of Energy, Lawrence Livermore National Laboratory, under
subcontract B341493.
## Appendix
The components of the two-dimensional stiffness matrix can be computed from
two-dimensional plane strain finite element analyses. However, the components
of the two-dimensional compliance matrix cannot be directly determined from
two-dimensional plane strain finite element analyses. The reasons for these
are discussed in this appendix. The approach taken to approximate the two-
dimensional compliance matrix is also discussed.
### A.1 Two-Dimensional Stiffness Matrix
The stress-strain relation for an anisotropic linear elastic material is given
by
$\left[\begin{array}[]{c}\sigma_{11}\\\ \sigma_{22}\\\ \sigma_{33}\\\
\sigma_{23}\\\ \sigma_{31}\\\
\sigma_{12}\end{array}\right]=\left[\begin{array}[]{cccccc}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\\
C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\\
C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\\
C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\\
C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\\
C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{array}\right]\left[\begin{array}[]{c}\epsilon_{11}\\\
\epsilon_{22}\\\ \epsilon_{33}\\\ \epsilon_{23}\\\ \epsilon_{31}\\\
\epsilon_{12}\end{array}\right].$ (9)
For the plane strain assumption, we have,
$\epsilon_{33}~{}=~{}\epsilon_{23}~{}=~{}\epsilon_{31}~{}=~{}0.$ (10)
Therefore, the stress-strain relation can be reduced to
$\left[\begin{array}[]{c}\sigma_{11}\\\ \sigma_{22}\\\
\sigma_{12}\end{array}\right]=\left[\begin{array}[]{ccc}C_{11}&C_{12}&C_{16}\\\
C_{12}&C_{22}&C_{26}\\\
C_{16}&C_{26}&C_{66}\end{array}\right]\left[\begin{array}[]{c}\epsilon_{11}\\\
\epsilon_{22}\\\ \epsilon_{12}\end{array}\right].$ (11)
The six terms in the apparent two-dimensional stiffness matrix reduce to four
is the material is orthotropic, i.e.,
$\left[\begin{array}[]{c}\sigma_{11}\\\ \sigma_{22}\\\
\sigma_{12}\end{array}\right]=\left[\begin{array}[]{ccc}C_{11}&C_{12}&0\\\
C_{12}&C_{22}&0\\\
0&0&C_{66}\end{array}\right]\left[\begin{array}[]{c}\epsilon_{11}\\\
\epsilon_{22}\\\ \epsilon_{12}\end{array}\right].$ (12)
The three constants $C_{11}$, $C_{12}$ and $C_{22}$ can be determined by the
application of normal displacements in the ’1’ and ’2’ directions
respectively. The constant $C_{66}$ can be determined using shear displacement
boundary conditions in a finite element analysis. Hence, it can be seen that
the stiffness matrix can be calculated directly from two-dimensional plane
strain based finite element analyses. This is not true for the compliance
matrix.
### A.2 Two-Dimensional Compliance Matrix
The strain-stress relation for an anisotropic linear elastic material can be
written as
$\left[\begin{array}[]{c}\epsilon_{11}\\\ \epsilon_{22}\\\ \epsilon_{33}\\\
\epsilon_{23}\\\ \epsilon_{31}\\\
\epsilon_{12}\end{array}\right]=\left[\begin{array}[]{cccccc}S_{11}&S_{12}&S_{13}&S_{14}&S_{15}&S_{16}\\\
S_{12}&S_{22}&S_{23}&S_{24}&S_{25}&S_{26}\\\
S_{13}&S_{23}&S_{33}&S_{34}&S_{35}&S_{36}\\\
S_{14}&S_{24}&S_{34}&S_{44}&S_{45}&S_{46}\\\
S_{15}&S_{25}&S_{35}&S_{45}&S_{55}&S_{56}\\\
S_{16}&S_{26}&S_{36}&S_{46}&S_{56}&S_{66}\end{array}\right]\left[\begin{array}[]{c}\sigma_{11}\\\
\sigma_{22}\\\ \sigma_{33}\\\ \sigma_{23}\\\ \sigma_{31}\\\
\sigma_{12}\end{array}\right].$ (13)
The relationship between the stiffness matrix and the compliance matrix is
$\left[\begin{array}[]{cccccc}S_{11}&S_{12}&S_{13}&S_{14}&S_{15}&S_{16}\\\
S_{12}&S_{22}&S_{23}&S_{24}&S_{25}&S_{26}\\\
S_{13}&S_{23}&S_{33}&S_{34}&S_{35}&S_{36}\\\
S_{14}&S_{24}&S_{34}&S_{44}&S_{45}&S_{46}\\\
S_{15}&S_{25}&S_{35}&S_{45}&S_{55}&S_{56}\\\
S_{16}&S_{26}&S_{36}&S_{46}&S_{56}&S_{66}\end{array}\right]=\left[\begin{array}[]{cccccc}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\\
C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\\
C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\\
C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\\
C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\\
C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{array}\right]^{-1}$ (14)
or,
$\mathbf{S}=\mathbf{C}^{-1}.$ (15)
It is obvious from the above equation that the apparent two-dimensional
compliance matrix is not equal to the inverse of the apparent two-dimensional
stiffness matrix, i.e.,
$\left[\begin{array}[]{ccc}S_{11}&S_{12}&S_{16}\\\ S_{12}&S_{22}&S_{26}\\\
S_{16}&S_{26}&S_{66}\end{array}\right]\neq\left[\begin{array}[]{ccc}C_{11}&C_{12}&C_{16}\\\
C_{12}&C_{22}&C_{26}\\\ C_{16}&C_{16}&C_{66}\end{array}\right]^{-1}.$ (16)
Hence, we cannot determine the two-dimensional compliance matrix if we only
know the two-dimensional stiffness matrix.
Let us again examine the effect of the plane-strain assumption on the stress-
strain relation. We then have
$\left[\begin{array}[]{c}\epsilon_{11}\\\ \epsilon_{22}\\\ 0\\\
\epsilon_{12}\end{array}\right]=\left[\begin{array}[]{cccc}S_{11}&S_{12}&S_{13}&S_{16}\\\
S_{12}&S_{22}&S_{23}&S_{26}\\\ S_{13}&S_{23}&S_{33}&S_{36}\\\
S_{16}&S_{26}&S_{36}&S_{66}\end{array}\right]\left[\begin{array}[]{c}\sigma_{11}\\\
\sigma_{22}\\\ \sigma_{33}\\\ \sigma_{12}\end{array}\right].$ (17)
For orthotropic materials, this relation simplifies to
$\left[\begin{array}[]{c}\epsilon_{11}\\\ \epsilon_{22}\\\ 0\\\
\epsilon_{12}\end{array}\right]=\left[\begin{array}[]{cccc}S_{11}&S_{12}&S_{13}&0\\\
S_{12}&S_{22}&S_{23}&0\\\ S_{13}&S_{23}&S_{33}&0\\\
0&0&0&S_{66}\end{array}\right]\left[\begin{array}[]{c}\sigma_{11}\\\
\sigma_{22}\\\ \sigma_{33}\\\ \sigma_{12}\end{array}\right].$ (18)
This equation shows that we need to know the stress $\sigma_{33}$ to determine
the terms of the compliance matrix and hence three-dimensional analyses are
necessary. If we assume plane stress, we can determine the terms of the matrix
$\mathbf{S}$ directly. However, the apparent two-dimensional compliance matrix
for plane stress is not equal to that for plane strain and hence we cannot
apply this method for our purposes. This is why the plane strain compliance
matrix cannot be determined using two-dimensional finite element analyses
only.
### A.3 Approximation of Compliance Matrix
The two-dimensional compliance matrix can be determined approximately for
materials with square symmetry by assuming that $S_{13}$, $S_{23}$ and
$S_{33}$ are known. Let,
$S_{13}=S_{23}=-\frac{\nu_{3}}{E_{3}}\\\ S_{33}=\frac{1}{E_{3}}$ (19)
where, $\nu_{3}$ is the Poisson’s ratio in the out-of-plane direction and
$E_{3}$ is the Young’s ratio in that direction. Then, for a material with
square symmetry,
$\left[\begin{array}[]{c}\epsilon_{11}\\\ \epsilon_{22}\\\ 0\\\
\epsilon_{12}\end{array}\right]=\left[\begin{array}[]{cccc}S_{11}&S_{12}&-\frac{\nu_{3}}{E_{3}}&0\\\
S_{12}&S_{11}&-\frac{\nu_{3}}{E_{3}}&0\\\
-\frac{\nu_{3}}{E_{3}}&-\frac{\nu_{3}}{E_{3}}&\frac{1}{E_{3}}&0\\\
0&0&0&S_{66}\end{array}\right]\left[\begin{array}[]{c}\sigma_{11}\\\
\sigma_{22}\\\ \sigma_{33}\\\ \sigma_{12}\end{array}\right].$ (20)
Inverting the relation, we have,
$\left[\begin{array}[]{c}\sigma_{11}\\\ \sigma_{22}\\\ \sigma_{33}\\\
\sigma_{12}\end{array}\right]=\left[\begin{array}[]{cccc}C_{11}&C_{12}&C_{13}&0\\\
C_{12}&C_{11}&C_{23}&0\\\ C_{13}&C_{23}&C_{33}&0\\\
0&0&0&C_{66}\end{array}\right]\left[\begin{array}[]{c}\epsilon_{11}\\\
\epsilon_{22}\\\ 0\\\ \epsilon_{12}\end{array}\right].$ (21) where,
$\displaystyle
C_{11}=\frac{E_{3}S_{11}-\nu_{3}^{2}}{E_{3}S_{11}^{2}-2\nu_{3}^{2}S_{11}-E_{3}S_{12}^{2}+2\nu^{2}S_{12}}~{},$
$\displaystyle
C_{12}=\frac{-E_{3}S_{12}+\nu_{3}^{2}}{E_{3}S_{11}^{2}-2\nu_{3}^{2}S_{11}-E_{3}S_{12}^{2}+2\nu^{2}S_{12}}.$
Note that it is not necessary to know $C_{13}$, $C_{23}$ and $C_{33}$ to
determine $S_{11}$ and $S_{12}$.
We can write the above relations between $C_{11},C_{12}$ and $S_{11},S_{12}$
in the form
$\displaystyle
E_{3}S_{11}^{2}-\left(\frac{E_{3}}{C_{11}}+2\nu_{3}^{2}\right)S_{11}-\left(E_{3}S_{12}^{2}-2\nu_{3}^{2}S_{12}-\frac{\nu_{3}^{2}}{C_{11}}\right)$
$\displaystyle=0,$ (22) $\displaystyle
E_{3}S_{12}^{2}-\left(\frac{E_{3}}{C_{12}}+2\nu_{3}^{2}\right)S_{12}-\left(E_{3}S_{11}^{2}-2\nu_{3}^{2}S_{11}+\frac{\nu_{3}^{2}}{C_{12}}\right)$
$\displaystyle=0.$ (23)
In simplified form,
$\displaystyle A_{1}S_{11}^{2}+B_{1}S_{11}+C_{1}$ $\displaystyle=0,$ (24)
$\displaystyle A_{2}S_{12}^{2}+B_{2}S_{12}+C_{2}$ $\displaystyle=0.$ (25)
We can solve these quadratic equations to get expressions for $S_{11}$ and
$S_{12}$ as
$\displaystyle S_{11}$ $\displaystyle=\frac{-B+\sqrt{B^{2}-4AC}}{2A},$ (26)
$\displaystyle S_{12}$ $\displaystyle=\frac{-B-\sqrt{B^{2}-4AC}}{2A}.$ (27)
Knowing $C_{11}$, $C_{12}$, $E_{3}$ and $\nu_{3}$ these two equations can be
solved iteratively to determine $S_{11}$ and $S_{12}$. The values of $C_{11}$
and $C_{12}$ can be determined using the procedure outlined at the beginning
of this section. It remains to be discussed how $E_{3}$ and $\nu_{3}$ are to
be determined.
### A.4 Determination of $E_{3}$ and $\nu_{3}$
Two methods can be used to determine the values of $E_{3}$ and $\nu_{3}$ for
our calculations. The first method is to assume that the rule of mixtures is
accurate enough to determine the effective properties in the ’3’ direction.
Thus, if the volume fraction of the first component is $f_{1}$ and that of the
second component is $f_{2}$, we have,
$\displaystyle E_{3}$ $\displaystyle=f_{1}E_{1}+f_{2}E_{2},$ (28)
$\displaystyle\nu_{3}$ $\displaystyle=f_{1}\nu_{1}+f_{2}\nu_{2}.$ (29)
where $E_{i}$ and $\nu_{i}$ are the Young’s modulus and the Poisson’s ratio of
the $i$th component.
The other option is to use the values of $S_{13}$, $S_{23}$ and $S_{33}$
obtained from GMC since these are also quite accurate for the out of plane
direction. Thus, we have,
$\displaystyle E_{3}$ $\displaystyle=\frac{1}{S_{33}^{\text{GMC}}},$ (30)
$\displaystyle\nu_{3}$ $\displaystyle=-S_{13}^{\text{GMC}}E_{3}.$ (31)
This is the procedure we have use to determine the effective compliance
matrices discussed in this paper.
## References
* [1] J. Helsing, G. W. Milton, and A. B. Movchan. Duality relations, correspondences, and numerical results for planar elastic composites. J. Mech. Phys. Solids, 45(4):565–590, 1997.
* [2] G. W. Milton. Composites : a myriad of microstructure independent relations. In T. Tatsumi, E. Watanabe, and T. Kambe, editors, Theoretical and Applied Mechanics (Proc. XIX International Congress of Theoretical and Applied Mechanics, Kyoto, 1996), pages 443–459. Elsevier, Amsterdam, 1997.
* [3] J. Aboudi. Micromechanical analysis of composites by the method of cells - update. Appl. Mech. Rev, 49(10):S83–S91, 1996.
* [4] B. Banerjee and D. O. Adams. On predicting the effective elastic properties of polymer bonded explosives using the recursive cell method. 2002\.
* [5] G. W. Milton. Theory of Composites. Cambridge University Press, New York, 2002.
* [6] A. V. Cherkaev, K. A. Lurie, and G. W. Milton. Invariant properties of the stress in plane elasticity and equivalence classes of composites. Proc. R. Soc. Lond. A, 438(1904):519–529, 1992.
* [7] R. Hill. Theory of mechanical properties of fibre-strengthened materials: I. elastic behaviour. J. Mech. Phys. Solids, 12:199–212, 1964.
* [8] B. Banerjee and D. O. Adams. Effective elastic moduli of polymer bonded explosives from finite element simulations. 2002\.
* [9] B. Banerjee and D. O. Adams. Application of the generalized method of cells to polymer bonded explosives. 2002\.
|
arxiv-papers
| 2012-01-11T23:58:04 |
2024-09-04T02:49:26.199187
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Biswajit Banerjee and Daniel O. Adams",
"submitter": "Biswajit Banerjee",
"url": "https://arxiv.org/abs/1201.2444"
}
|
1201.2460
|
# Limits on Large Extra Dimensions Based on Observations of Neutron Stars with
the _Fermi_ -LAT
M. Ajello L. Baldini G. Barbiellini D. Bastieri K. Bechtol R. Bellazzini
B. Berenji E.D. Bloom E. Bonamente A.W. Borgland J. Bregeon M. Brigida
P. Bruel R. Buehler S. Buson G.A. Caliandro R.A. Cameron P.A. Caraveo
J.M. Casandjian C. Cecchi E. Charles A. Chekhtman J. Chiang S. Ciprini
R. Claus J. Cohen-Tanugi J. Conrad S. Cutini A. de Angelis F. de Palma
C.D. Dermer E. do Couto e Silva P.S. Drell A. Drlica-Wagner T. Enoto C.
Favuzzi S.J. Fegan E.C. Ferrara Y. Fukazawa P. Fusco F. Gargano D.
Gasparrini S. Germani N. Giglietto F. Giordano M. Giroletti T. Glanzman
G. Godfrey P. Graham I.A. Grenier S. Guiriec M. Gustafsson D. Hadasch M.
Hayashida R.E. Hughes A.S. Johnson T. Kamae H. Katagiri J. Kataoka J.
Knödlseder M. Kuss J. Lande L. Latronico A.M. Lionetto F. Longo F.
Loparco M.N. Lovellette P. Lubrano M.N. Mazziotta P.F. Michelson W.
Mitthumsiri T. Mizuno C. Monte M.E. Monzani A. Morselli I.V. Moskalenko
S. Murgia J.P. Norris E. Nuss T. Ohsugi A. Okumura E. Orlando J.F. Ormes
M. Ozaki D. Paneque M. Pesce-Rollins M. Pierbattista F. Piron G. Pivato
S. Rainò M. Razzano S. Ritz M. Roth P.M. Saz Parkinson J.D. Scargle T.L.
Schalk C. Sgrò E.J. Siskind G. Spandre P. Spinelli D.J. Suson H. Tajima
H. Takahashi T. Tanaka J.G. Thayer J.B. Thayer L. Tibaldo M. Tinivella
D.F. Torres E. Troja Y. Uchiyama T.L. Usher J. Vandenbroucke V. Vasileiou
G. Vianello V. Vitale A.P. Waite B.L. Winer K.S. Wood M. Wood Z. Yang
S. Zimmer
###### Abstract
We present limits for the compactification scale in the theory of Large Extra
Dimensions (LED) proposed by Arkani-Hamed, Dimopoulos, and Dvali. We use 11
months of data from the Fermi Large Area Telescope (Fermi-LAT) to set gamma
ray flux limits for 6 gamma-ray faint neutron stars (NS). To set limits on LED
we use the model of Hannestad and Raffelt (HR) that calculates the Kaluza-
Klein (KK) graviton production in supernova cores and the large fraction
subsequently gravitationally bound around the resulting NS. The predicted
decay of the bound KK gravitons to $\gamma\gamma$ should contribute to the
flux from NSs. Considering 2 to 7 extra dimensions of the same size in the
context of the HR model, we use Monte Carlo techniques to calculate the
expected differential flux of gamma-rays arising from these KK gravitons,
including the effects of the age of the NS, graviton orbit, and absorption of
gamma-rays in the magnetosphere of the NS. We compare our Monte Carlo-based
differential flux to the experimental differential flux using maximum
likelihood techniques to obtain our limits on LED. Our limits are more
restrictive than past EGRET-based optimistic limits that do not include these
important corrections. Additionally, our limits are more stringent than LHC
based limits for 3 or fewer LED, and comparable for 4 LED. We conclude that if
the effective Planck scale is around a TeV, then for 2 or 3 LED the
compactification topology must be more complicated than a torus.
## 1 Introduction
In the Standard Model of particle physics, gravity is not unified with the
other 3 fundamental forces .This is manifested by the _hierarchy problem_ ,
the fact that the electroweak mass scale $M_{\operatorname{EW}}\sim
1\operatorname{TeV}$ is many orders of magnitude smaller than the Planck mass
scale $M_{\operatorname{P}}\approx 1.22\times 10^{16}\operatorname{TeV}$ [1].
Arkani-Hamed, Dimopoulos, and Dvali (ADD) propose a model of Large Extra
Dimensions (LED) as a solution for the hierarchy problem. The ADD scenario may
be embedded into string theory, which allows for the existence of compactified
extra dimensions. Due to the presence of $n$ extra dimensions, at length
scales smaller than the size of the extra dimensions, the gravitational
potential between test masses has a $1/r^{n+1}$ dependence; however, on scales
larger than the size of the extra dimensions the gravitational potential
reverts to the ordinary $1/r$ dependence. For a given $n$, if all the extra
dimensions are toroidally compactified, _i.e._ , have the same size $R$, the
effective Planck mass in the $(n+4)-$dimensional space, $M_{D}$, is related to
the reduced Planck mass $\bar{M}_{P}=M_{P}/\sqrt{8\pi}$ by the relation:
$\bar{M}^{2}_{P}=R^{n}M^{n+2}_{D}.$ (1.1)
In the ADD model, the hierarchy problem is solved because the presence of LED
brings the effective Planck mass, $M_{D}$, to the TeV scale, the truly
fundamental scale of gravity. As a consequence, the associated Kaluza-Klein
gravitons, denoted by $G_{KK}$, are massless in the bulk, but they have mass
on the 3-brane related to their momentum in the bulk (unlike the gravitons of
General Relativity).
According to the ADD model, it is possible to place constraints on extra
dimensions by $G_{KK}$ emission from nucleon-nucleon gravibremsstrahlung in
type II supernova cores, $NN\rightarrow NNG_{KK}$. ADD obtain limits from
Supernova (SN) 1987A, assuming pion-exchange mediated by the strong force as
the dominant process. Hanhart, Reddy and Savage (2001) assume a different
process[2]. They use nucleon-nucleon gravibremsstrahlung mediated by nucleon-
nucleon scattering to obtain the emission rate for KK gravitons. Furthermore,
they indicate that the actual details of the scattering process are not
important in the soft-radiation limit, where the energy of the outgoing
gravitons is much less than the energy other incoming nucleons[2]. Then they
proceed to obtain limits for $n=2$ and $n=3$ extra dimensions from SN1987A.
This is based on the argument that the observed neutrino luminosity sets an
upper bound of $10^{19}$ ergs g-1s-1 on the energy loss rate into particles
other than neutrinos such as $G_{KK}$[3].
Hannestad and Raffelt (henceforth HR [4]) extend this idea to neutron stars,
proposing that if the KK gravitons are bound in the gravitational potential of
a proto-neutron star as it evolves into a neutron star, then the flux of
photons from KK graviton decays, $G_{KK}\rightarrow 2\gamma$, could be used to
set a limit on extra dimensions. They use EGRET results to set limits on LED.
However, they do not place direct flux limits on the neutron stars not
detected by EGRET. Rather, they quote their flux upper limit as the 1 yr
point-source sensitivity of EGRET for a high latitude point source with a
$E^{-2}$ spectrum (see section 2.1), and derive limits on LED more restrictive
than from arguments based on KK graviton emissivities from SN 1987A. To obtain
upper limits, we follow similar theoretical arguments as HR, but we perform a
very different analysis, including spectral corrections and upper limit
spectral analysis with Fermi-LAT data, on 6 gamma-ray faint NS.
## 2 Data Analysis
### 2.1 Experimental Methods
The Fermi-LAT is a gamma-ray imaging pair-conversion telescope, consisting of
an anti-coincidence detector, tracker, calorimeter, and electronics modules.
The details of the Fermi-LAT are discussed by Atwood _et al._[5]. The Fermi-
LAT features improved performance compared to its predecessor $\gamma$-ray
observatory, EGRET. Some of these specifications, relevant to this study, are
compared in Table 1.
specification | Fermi-LAT | EGRET
---|---|---
68% containment PSF (∘) at 200 MeV | 2.8 | 3.3
Effective Area (cm2) at 200 MeV | 3000 | 1000
Energy Resolution (%) at 200 MeV | 13 | 9.3
flux sensitivity (cm-2s-1) | $6\times 10^{-9}$ | $1.3\times 10^{-7}$
Table 1: Comparison of performance specifications of Fermi-LAT and EGRET, as
relevant for the gamma-ray energies considered in this paper[5, 6]. Flux
sensitivity is evaluated for a high-latitude point source with a $E^{-2}$
spectrum, with 1 year of data, for $E>100$ MeV. EGRET effective area is quoted
for Class A events.
Setting flux limits on sources requires knowledge of background point sources
and diffuse emission. We make use of the publicly available diffuse models
developed by the Fermi-LAT collaboration: the Galactic diffuse emission model,
_gll_iem_v02.fit_[7, 8]; and the isotropic model, _isotropic_iem_v02.txt_[9].
The Galactic diffuse model is allowed to vary in a region of interest (ROI)
around each source by multiplying by a power-law spectral function, as
described in [10], effectively making the spectrum harder or softer. The scale
for the power-law is 100 MeV, and the index is allowed to vary between -0.1
and 0.1 (a value of 0 implies no correction to the model). The background
point sources are modeled by fixing the spectral parameters from the first
year Fermi-LAT (1FGL) catalog [10].
The data sample consists of a selection of 11 months of all-sky data obtained
with the Fermi-LAT instrument, using a time interval beginning with the start
of survey mode, August 4, 2008, until July 4, 2009. This time interval is
chosen to be consistent with the 1FGL catalog, so that nearby point sources
detected with high significance may be modeled appropriately as power-law
sources [10]. The instrument response function (IRF) chosen is
_P6_V3_DIFFUSE_[11], as is the case for the 1FGL catalog. This IRF specifies a
parametrization of effective area, energy resolution, and point spread
function. We select data from the 1FGL catalog dataset for regions of interest
(ROIs) corresponding to each source described further in this paper. This
dataset excludes events for which the rocking angle is larger than 43∘,
because of contamination from the Earth’s limb due to interactions of cosmic
rays with Earth’s upper atmosphere. For the same reason, for each ROI, events
for which the zenith angle is larger than 105∘ are excluded. There is also a
good time interval (GTI) cut applied, as described in Ref. [10].
### 2.2 Selection of Neutron Stars
A query is made on the Australia Telescope National Facility (ATNF) radio
pulsar catalog to select NS [12]. In order to obtain the best limits, we
reject candidate sources that have associations with $\gamma$-ray sources
detected in the 1FGL catalog. To minimize attenuation of the putative KK
graviton decay signal, as discussed in section IV, we choose NS that satisfy
the following criteria: distance $d<0.40$ kpc; surface magnetic field
$B_{\operatorname{surf}}<5\times 10^{13}$ G; and characteristic age
$t_{\operatorname{age}}<2\times 10^{8}$ yr. We take the NS ages as the spin-
down ages of the pulsars, for consistency over all sources; the corrected ages
may differ, as discussed in Ref. [13]. However, consideration of the corrected
ages hardly affects the limits presented here. In addition, the $\gamma$-ray
sky as viewed by Fermi-LAT is filled with sources near the Galactic plane, and
diffuse components are also dominant and have large systematic uncertainties
at low latitudes; we require for Galactic latitude ($b$), that
$|b|>15^{\circ}$ for candidate neutron stars. Applying all the above selection
criteria to the ATNF catalog, 6 sources remain for analysis, with parameters
as shown in Table 2.
source name | RA | Dec. | $\ell$ | $b$ | $P$ | $d$ | Age | $B_{\mbox{surf}}$
---|---|---|---|---|---|---|---|---
| (∘) | (∘) | (∘) | (∘) | (s) | (kpc) | (Myr) | (G)
RX J1856$-$3754 | $284.15$ | $-37.90$ | $358.61$ | $-17.21$ | $7.05$ | 0.16 | 3.76 | 1.47$\times 10^{13}$
J0108$-$1431 | $17.04$ | $-14.35$ | $140.93$ | $-76.82$ | $0.808$ | 0.24 | 166 | 2.52$\times 10^{11}$
J0953$+$0755 | $148.29$ | $7.93$ | $228.91$ | $43.7$ | $0.25$ | 0.26 | 17.5 | 2.44$\times 10^{11}$
J0630$-$2834 | $97.71$ | $-28.58$ | $236.95$ | $-16.76$ | $1.24$ | 0.33 | 2.77 | 3.01$\times 10^{12}$
J1136$+$1551 | $174.01$ | $15.85$ | $241.90$ | $69.20$ | $1.19$ | 0.36 | 5.04 | 2.13$\times 10^{12}$
J0826$+$2637 | $126.71$ | $26.62$ | $196.96$ | $31.74$ | $0.53$ | 0.36 | 4.92 | 9.64$\times 10^{11}$
Table 2: Astrophysical properties of neutron star sources analyzed in this
work, with sources in increasing order of distance. Coordinates, periods.
distances, ages, and surface magnetic field strengths are obtained from the
ATNF Catalog [12].
### 2.3 Gamma-ray Spectral Limits
Fermi-LAT gamma-ray events are selected in a 12∘ radius ROI centered on each
NS source listed in Table 2. Given the limitations of the dataset we use and
the expected spectral energy distribution from gamma rays from trapped KK
graviton decay, only gamma-ray events with energies in the range 100 MeV to
400 MeV are considered. Although desirable, going below 100 MeV is not
feasible for this analysis using _P6_V3_DIFFUSE_. Before obtaining upper
limits for each source, a model for the corresponding ROI is developed,
inclusive of 1FGL sources and the 2 components of diffuse emission (no
putative neutron star source is included in this step). 1FGL catalog sources
within a 14∘ radius are parametrized as point sources with a power-law
spectral energy distribution, with fluxes and spectral indices _fixed_ at
catalog values; those sources farther than 14∘ away are not considered. These
parameters are fixed due to the small gamma-ray energy range (100 MeV-400 MeV)
that we use, which leaves little spectral range to perform accurate fitting.
An initial unbinned likelihood fit is done in order to determine _only_ the
diffuse parameters. For the isotropic diffuse component, the parameter to be
determined is normalization, while for the Galactic diffuse component, we
consider the normalization and the spectral index. The analysis, including the
diffuse fitting and upper limit determination, is performed with the Fermi
_ScienceTools_ program pyLikelihood, featuring maximum likelihood-based
fitting. Version _09-17-00_ of the Fermi-LAT _ScienceTools_ is used[14]. For
the neutron star RX J1856$-$3754, a counts map of 100-400 MeV photons in a
$10^{\circ}\times 10^{\circ}$ region, convolved with a Gaussian approximation
to the Fermi-LAT PSF, is shown in the left panel of Figure 1. The residual
counts map (determined from comparison of the counts map to the model-based
map), for source RX J1856$-$3754, is displayed in the right panel of Figure 1.
Figure 2 shows a residual counts plot versus energy, obtained by integrating
over the counts map spatial dependence, and subtracting the data counts from
model counts and dividing by the model counts.
Figure 1: Left: Counts map from data, for source RX J1856$-$3754, convolved
with a Gaussian approximation to the Fermi-LAT PSF, in order to reduce
statistical fluctuations without dramatically reducing the angular resolution.
The colorbar shows counts per pixel. The white dashed circle shows the 200 MeV
PSF. Right: Residual map, (counts-model)/model, for the same source, based on
the 1FGL model with the fitted diffuse model. The pixel size is $0.4^{\circ}$
for both. Green crosses show 1FGL point sources, and the putative $\gamma$-ray
source is at the center. Figure 2: Residual plot of counts over the 10∘
square region around source RX J1856$-$3754\. Black horizontal bands indicate
the energy range for each point, and black vertical bands represent
statistical uncertainties, while red vertical bands represent systematic
uncertainties of about 10% at 100 MeV, and decreasing to 5% at 562 MeV[15].
A spectral model, which determines the differential flux $d\Phi/dE$, for each
source and number of extra dimensions $n$, is developed in the next section. A
significant difference in the data analysis technique from Hannestad and
Raffelt lies in considering the differential flux rather than the integral
flux, in determining limits on $R$; this is a more accurate and optimal method
in setting limits in a Fermi-LAT analysis, when comparing the data to a pre-
defined theoretical distribution. Complete details of the theoretical
development of the differential flux, as well as analysis methods, can be
found in [13].
## 3 Calculating the Spectral Model for KK Graviton Decay $\gamma-$Rays from
NS
In the following subsections, we explain how we calculate the gamma-ray
spectrum. Important departures from the analysis of HR in forming the
theoretical differential flux, $d\Phi/dE$, to be compared to Fermi-LAT
observations include: attenuation of the signal due to the age of the neutron
star, orbital position and velocity of the $G_{KK}$, decay of $G_{KK}\to
2\gamma$, and attenuation of the signal due to magnetic field (which is
position and velocity dependent). These features are included via a Monte
Carlo simulation of about $10^{7}\ G_{KK}$ in orbit for each NS source and for
$n=2,3,\ldots,7$ extra dimensions.
### 3.1 Theoretical Model
Following HR, we start with the differential distribution of $G_{KK}$ created
during the proto-neutron star core collapse with total energy $\omega$ and
mass $m$:
$\frac{{\rm d}^{2}N_{\rm KK,n}}{{\rm d}\omega{\rm d}\mu}=\frac{{\rm
d}^{2}Q_{n}}{\omega{\rm d}\omega{\rm d}\mu}\Delta t_{NS}V_{NS}\ .$ (3.1)
We will make use of notations defined in HR, that we rewrite here for
completeness: $Q_{n}$ is the total energy loss rate per unit volume into KK
gravitons which depends on $n$; $\mu=m/\omega$ is the inverse of the initial
Lorentz factor of the $G_{KK}$; $\Delta t_{NS}\simeq 7.5$ s is the time-scale
for emission of $G_{KK}$ during the core collapse; and $V_{NS}=\frac{4}{3}\pi
R_{NS}^{3}$ is the volume of the proto-neutron star (and current neutron star)
of radius $R_{NS}\simeq 13$ km. According to HR, we have:
$\frac{{\rm d}^{2}Q_{n}}{{\rm d}\omega{\rm
d}\mu}=Q_{0}(RT)^{n}\Omega_{n}G_{n-1}(\mu)F_{n}\left(\frac{\omega}{T}\right)\
,$ (3.2)
where $R$ is the extra dimension size, as in eq. (1), $T\gtrsim 30$ MeV is the
supernova core temperature (see Section 5), and
$\Omega_{n}=2\pi^{n/2}/\Gamma(n/2)$ is the surface of the $n$-dimensional unit
hypersphere, with $\Gamma(...)$ as the Gamma function. We also have,
$\displaystyle{Q_{0}}$
$\displaystyle=\frac{{512}}{{5{\pi^{3/2}}}}\frac{G_{N}\sigma
n_{B}^{2}T^{7/2}}{M^{1/2}}$ (3.3)
$\displaystyle=1.100\times{10^{22}}{\rm{MeV}}\
{{\rm{cm}}^{-3}}{\rm{}}{{\rm{s}}^{-1}}\left(T/30\ {\rm
MeV}\right)^{7/2}\left(\rho/3\times 10^{14}\ {\rm g\
cm}^{-3}\right)^{2}(f_{KK}/0.0075)\ ,$ (3.4)
where Newton’s constant is $G_{N}=6.708\times 10^{-33}\hbar c\,({\rm
MeV}/c^{2})^{-2}$, $\sigma$ is the nucleon-nucleon scattering cross section of
25 mb, $n_{B}$ is the number density of baryons of 0.16 fm-3, and $M$ is the
isospin-averaged nucleon mass of 938 MeV/$c^{2}$. $f_{KK}\simeq 0.01$ is the
estimated fraction of core-collapse energy radiated away as $G_{KK}$[16, 17].
In eq. (3.2), the following functions are defined, where $q$ is an integer:
$\displaystyle G_{q}(\mu)$
$\displaystyle=\mu^{q}\sqrt{1-\mu^{2}}\left(\frac{19}{18}+\frac{11}{9}\mu^{2}+\frac{2}{9}\mu^{4}\right)$
(3.5) $\displaystyle F_{q}(\omega/T)$
$\displaystyle=\frac{(\omega/T)^{q}}{1+\exp(\omega/T)}.$ (3.6)
In the previous equation, in writing $F(\omega/T)$, we are assuming that the
structure function of the nuclear medium, in the notation of HR,
$s(\omega/T)$, of the nuclear medium is unity, which is accurate to first
order[4]. An expansion to the next order, $(\omega/T)^{2}$, would likely shift
the expected energy distribution of the differential flux to higher energies.
Therefore, our assumption of $s(\omega/T)\simeq 1$ is in the direction of
making the associated limits more conservative. Finally, the integral for the
case of trapped KK gravitons, which make up the initial cloud bound to the NS,
is given by:
$N_{KK,n}(t=0)=3\int_{0}^{\infty}d\omega\ \int_{0}^{1}r^{2}\
dr\int_{1+U(r)}^{1}d\mu\ \frac{{\rm d}^{2}N_{KK,n}}{{\rm d}\omega{\rm d}\mu}.$
(3.7)
In eq. (3.7), HR assume that the graviton creation is isotropic at the
dimensionless radial distance from the neutron star center, $r$, scaled to the
neutron star radius, $R_{NS}$. The integration over $r$ is performed from the
proto-neutron star’s center to its surface, where $r=1$, and the condition
$\mu>1+U(r)$ selects the $G_{KK}$ that are gravitationally trapped. As in HR,
we model the neutron star’s potential as Newtonian:
$U(r)=-\frac{G_{\operatorname{N}}M_{\operatorname{NS}}}{R_{\operatorname{NS}}c^{2}}\times\left\\{\begin{array}[]{ll}\left(\frac{3}{2}-\frac{1}{2}r^{2}\right)\
,&r<1\\\ \frac{1}{r}\ ,&r\geq 1\end{array}\right.$ (3.8)
with $U_{NS}=-G_{N}M_{NS}/(R_{NS}c^{2})=-0.159(M_{NS}/1.4M_{\odot})(13{\rm
km}/R_{NS})$.
The $G_{KK}$ lifetime is [18]111This takes into account competing decays to
$e^{+}e^{-}$ and $\nu\bar{\nu}$.
$\tau(m)=1\times
10^{9}\mbox{yr}\left(\frac{100\mbox{MeV}}{m}\right)^{3}\equiv\kappa^{-1}m^{-3},$
(3.9)
where $\kappa=3.17\times 10^{-23}\ \mbox{MeV}^{-3}\mbox{s}^{-1}$. Assuming an
exponential decay of the KK gravitons, the number of KK gravitons remaining at
time $t$ after the core collapse is given by:
$N_{KK,n}(t)=N_{KK,n}(t=0)\exp\left(-\frac{\mu t}{\tau(m)}\right)$ (3.10)
Then, the time derivative (absolute value), of eq. (3.1), is given by:
$\begin{split}\left|\frac{{\rm d}^{2}\dot{N}_{KK,n}}{{\rm d}\mu{\rm
d}\omega}\right|&=\kappa\frac{m^{4}}{\omega}N_{KK,n}(t)\\\
&=Q_{0}(RT)^{n}\Omega_{n}\Delta t_{NS}V_{NS}\kappa
T^{2}G_{n+3}(\mu)F_{n+2}(\omega/T)\exp\left(-\frac{\mu
t}{\tau(m)}\right).\end{split}$ (3.11)
### 3.2 Determining the Differential Flux by Monte Carlo Simulation
We determine the differential flux according to a Monte Carlo simulation that
uses eq. (3.11). We calculate the mass distributions and the Lorentz
parameters of the decaying gravitons. We then determine the momentum and
energy distributions of the $G_{KK}$, considering the geometry of the decays.
At the same time, the age of the NS determines the remaining number of
gravitons. We also consider whether a given gamma ray can escape the NS
magnetosphere. Finally, this determines the differential flux of gamma rays
from the NS.
We carry out the Monte Carlo simulation of the differential flux in the
following steps:
1. 1
) Sample $\omega$ from $F_{n+2}(\omega/T)$, as in eq. (3.11), for
$0<\omega/T<20$ ($T=30$ MeV). For $\omega/T>20$, there is negligible
contribution from the integral of $F_{n+2}(\omega/T)$.
2. 2
) Sample $\mu$ from $G_{n+3}(\mu)$, as in eq. (3.11), between $\mu_{\min}$ and
$\mu_{\max}$. (Note that the sampling steps 1 & 2 are independent of each
other, see HR.) To simplify the orbit calculation with only a small error, we
assume that all of the created $G_{KK}$ start their orbit at the center of the
NS ($r=0$)222We have calculated that this approximation is in the direction of
making our limits more conservative.. Thus we have $\mu_{\min}=0.807$,
corresponding to the $G_{KK}$ escape velocity, and $\mu_{\max}=0.926$,
corresponding to the minimum velocity to reach the neutron star surface, from
$r=0$. Having determined a value of $\mu$, we determine the initial $G_{KK}$
Lorentz factor $\gamma=1/\mu$ and initial velocity $\beta=\sqrt{1-\mu^{2}}$.
Given the geometry of the SN explosion, we assume, as do HR, that the $G_{KK}$
orbits are radial. Using $\omega$ and $\mu$, from steps 1 and 2, we determine
a value for the mass, $m=\mu\omega$. Representative distributions of $m$, for
different values of $n$, are shown in Figure 3.
Figure 3: Unit-normalized distributions of KK graviton masses for $n=2,5,7$,
as determined according to Monte Carlo simulation.
Since we know the mass at this point, and given the age of the neutron star,
$t_{\operatorname{age}}$, we calculate the exponential decay fraction,
$F_{\operatorname{decay}}=\exp\left(-\frac{\mu
t_{\operatorname{age}}}{\tau(m)}\right)$. We sample a real number $u$
uniformly in the interval [0,1]: if $u>F_{\operatorname{decay}}$, then the
event is rejected.
3. 3
) Sample the decay vertex $r_{0}$ for a given $\mu$. The probability density
function $P(r_{0};\mu)$, which is shown in Figure 4 for two values of $\mu$,
is obtained as described in Appendix A. In Figure 5, unit-normalized radial
profile of decay vertices as a function of radial coordinate, for n = 2, is
plotted.
(a) $\mu=0.83$
(b) $\mu=0.90$
Figure 4: Radial probability density functions, $P(r;\mu)$, for 2 different
values of $\mu$. There are 20 linearly-spaced bins over the interval
$[0,r_{\max}\times R_{NS}]$, and the $y-$axis is the fraction of events per
bin. Figure 5: Unit-normalized radial distribution of decay vertices as a
function of radial coordinate, $r_{\operatorname{km}}$, for $n=2$. On the
y-axis is plotted $\left<P(r;\mu)\right>_{\mu}$, averaged over all $\mu$
between $\mu_{\min}$ and $\mu_{\max}$, for $1<r<5.5$.
4. 4
) Sample the orbit direction isotropically ($-1<\cos\theta<1$,
$-\pi<\phi<\pi$). This selects an orbit direction:
$\hat{r}_{0}=\sin\theta\cos\phi\ \hat{x}+\sin\theta\sin\phi\
\hat{y}+\cos\theta\ \hat{z}$ (3.12)
where $\hat{x},\hat{y},\hat{z}$ are the unit coordinate directions in the NS
frame. The $\hat{z}$ direction is chosen to align with the magnetic dipole
axis of the NS. At the sampled decay vertex, $r_{0}$, we then obtain a
velocity,
$\beta^{\prime}=R_{NS}\dot{r}/c,$ (3.13)
where $\dot{r}=\left.dr/dt\right|_{r=r_{0}}$ is obtained numerically, and the
Lorentz factor,
$\gamma^{\prime}=1/\sqrt{1-\beta^{\prime\ 2}}.$ (3.14)
Using the determined values of $m$ and $\beta^{\prime}$, thus yields
$p_{KK}^{\nu}$, the $G_{KK}$ 4-momentum in the NS frame at the decay vertex.
5. 5
) Determine the energy and momentum distribution of one of the two decay
photons at decay point on the orbit with direction $\hat{r}_{0}$. We treat the
other decay photon by multiplying the final flux by two. Full details of this
procedure, as implemented in the Monte Carlo simulation, are given in Appendix
B.
6. 6
) Determine whether the photon pair-produces in the neutron star
magnetosphere: this probability is given by
$P_{\operatorname{pp}}(E_{\gamma},\vec{r},\vec{p}_{\gamma})$. The probability
for photon survival from pair production in the Monte Carlo simulation is then
taken as
$P_{\operatorname{pp}}(E_{\gamma},\vec{r},\vec{p}_{\gamma})=\exp(-\tau_{pp})$.
In Appendix C, we describe the computation of $\tau_{pp}$. We sample a real
number $v$ uniformly on the interval [0,1]: if $v>P_{\operatorname{pp}}$, then
the event is rejected.
### 3.3 The Final Flux Result from the Monte Carlo Simulation
The distribution defined by the Monte Carlo simulation, $dN_{n}/dE_{\gamma}$,
is related to the differential flux by:
$\frac{{d\Phi}}{{dE_{\gamma}}}={k_{n}}{R^{n}}(d/{\rm
kpc})^{-2}\frac{dN_{n}}{dE_{\gamma}},$ (3.15)
where the $n-$dependent constant $k_{n}$ is given as:
${k_{n}}=\frac{1}{{4\pi{{(3.086\times{{10}^{21}})}}}^{2}}T^{2}\kappa\left(\frac{T}{\hbar
c}\right)^{n}\frac{2}{3}N_{0,n}\
{\rm{c}}{{\rm{m}}^{-2}}{{\rm{s}}^{-1}}{{\rm{m}}^{-n}}.$ (3.16)
and
$N_{0,n}=N_{KK,n}(t=0)/(RT)^{n},$ (3.17)
where the factor $\kappa$ is related to the decay rate as in eq. (3.9), and
the factor $T/(\hbar c)$ is a conversion constant, which is numerically
$1.52033\times 10^{14}\ \mbox{m}^{-1}$ at $T=30$ MeV. Values of $N_{0,n}$ and
$k_{n}$ are tabulated in Table 3.
In the computation of $dN_{n}/dE_{\gamma}$, steps (3) and (7) of Section 3.2
reject events based on the decay from the lifetime and the pair production
optical depth, respectively. In the case of a zero-age, zero-magnetic field
neutron star source, the spectrum $dN_{n}/dE_{\gamma}$ is normalized to 1.
Formally, the distribution $dN_{n}/dE_{\gamma}$ is defined by:
$\frac{dN_{n}}{dE_{\gamma}}=\frac{1}{N_{ev}}\frac{dN_{n}^{\prime}}{dE_{\gamma}}$
(3.18)
where $N_{ev}$ is the number of events in the Monte Carlo simulation.
$N_{rem}$, the number of events remaining after the effects of decay and pair
production are taken into account, is given by the integral:
$N_{rem}=\int_{0}^{600\operatorname{MeV}}\frac{dN_{n}^{\prime}}{dE_{\gamma}}dE_{\gamma}.$
(3.19)
The upper limit of 600 MeV is determined by the condition $\omega/T\leq
20$333The range of gamma-ray energies used to generate $dN_{n}^{\prime}/dE$ in
the Monte Carlo is $0<E_{\gamma}<600$ MeV.. The parameter $\eta$, defined as,
$\eta\equiv{\int_{100\ {\rm MeV}}^{400\ {\rm
MeV}}\frac{dN_{n}}{dE_{\gamma}}{\rm
d}E_{\gamma}}\mathord{\left/{\vphantom{{\int_{100\ {\rm MeV}}^{400\ {\rm
MeV}}\frac{dN_{n}}{dE_{\gamma}}{\rm d}E_{\gamma}}{\int_{100\ {\rm MeV}}^{400\
{\rm MeV}}\left.\frac{dN_{n}}{dE_{\gamma}}\right|_{\rm non-atten}{\rm
d}E_{\gamma}}}}\right.\kern-1.2pt}{\int_{100\ {\rm MeV}}^{400\ {\rm
MeV}}\left.\frac{dN_{n}}{dE_{\gamma}}\right|_{\rm non-atten}{\rm
d}E_{\gamma}}.$ (3.20)
parameterizes the efficiency with which photons contribute to the spectrum,
after signal attenuation effects of lifetime and pair production have been
taken into account. Values for each source and $n$ are shown in Table 4.
$n$ | $N_{0,n}$ | $k_{n}({\rm{c}}{{\rm{m}}^{-2}}{{\rm{s}}^{-1}}{{\rm{m}}^{-n}})$
---|---|---
2 | 6.47$\times 10^{40}$ | 7.126$\times 10^{6}$
3 | 3.46$\times 10^{41}$ | 5.799$\times 10^{21}$
4 | 1.94$\times 10^{42}$ | 4.963$\times 10^{36}$
5 | 7.40 $\times 10^{43}$ | 4.511$\times 10^{51}$
6 | 7.05$\times 10^{43}$ | 4.355$\times 10^{66}$
7 | 4.97$\times 10^{44}$ | 4.452$\times 10^{81}$
Table 3: $n-$dependent constants, as defined in equations (3.17) and (3.16). n | RX J1856$-$3754 | J0108$-$1431 | J0953$+$0755 | J0630$-$2834 | J1136$+$1551 | J0826$+$2637
---|---|---|---|---|---|---
2 | 0.335 | 0.031 | 0.221 | 0.359 | 0.309 | 0.332
3 | 0.350 | 0.037 | 0.249 | 0.382 | 0.332 | 0.360
4 | 0.361 | 0.041 | 0.276 | 0.402 | 0.351 | 0.385
5 | 0.368 | 0.043 | 0.302 | 0.416 | 0.365 | 0.406
6 | 0.370 | 0.042 | 0.325 | 0.424 | 0.374 | 0.419
7 | 0.365 | 0.037 | 0.334 | 0.424 | 0.373 | 0.423
Table 4: Table of values of the attenuation parameter $\eta$, defined by eq.
(3.20), for the different sources analyzed. These attenuation effects can be
quite large. HR used only source RX J1856$-$3754 and J0953+0755. These values
are calculated for 100 MeV$\leq E_{\gamma}\leq$ 400 MeV.
(a) $n=2$ (b) $n=5$
Figure 6: For $n=2$ and $n=5$, representative distributions of
$dN_{n}/dE_{\gamma}$, according to eq. (3.18), for the non-attenuated spectrum
and all neutron star sources considered, corrected for magnetic pair-
production and age attenuation effects.
## 4 Limits on LED Results
### 4.1 Individual Limits
With all parameters for the ROI fixed, namely catalog sources and diffuse
components, as determined in Section 2.3, upper limits on $R^{n}$ are
determined from spectral fitting based on the method of maximum likelihood.
Fit values are determined by the MINUIT optimizer[19], and one-sided 95%
confidence level upper limits are determined by performing a scan of the log-
likelihood function in each ROI[19]. Statistical parameters of the fit are
consistent with non-detection of the KK graviton decay signal for all NS
considered.As a check of this method, we compare upper limits obtained in this
manner against upper limits computed using profile likelihood implemented by a
different method in the Fermi-LAT _ScienceTools_ , and find agreement within
10%. Additional systematic checks, due to uncertainties in the parameters of
the background sources in the ROI, are also performed to verify the accuracy
of the limits, for source RX J1856$-$3754 (the source with the best limits):
the agreement in the flux upper limits is found to be 15% or better. The flux
upper limits for each source and $n$ are displayed in Table 5, and the
corresponding limits on the LED size $R$ are shown in Table 6.
n | J1856$-$3754 | J0108$-$1431 | J0953+0755 | J1136+1551 | J0630$-$2834 | J0826+2637
---|---|---|---|---|---|---
2 | 3.8 | 4.3 | 5.3 | 4.1 | 5.8 | 6.6
3 | 4.0 | 4.4 | 5.4 | 4.2 | 6.8 | 8.4
4 | 3.7 | 4.2 | 6.2 | 4.4 | 9.4 | 9.9
5 | 4.0 | 4.1 | 6.3 | 4.4 | 11 | 13
6 | 4.1 | 4.0 | 6.7 | 4.2 | 14 | 15
7 | 4.2 | 4.0 | 7.8 | 3.5 | 19 | 17
Table 5: Table of 95% C.L. flux upper limits ($10^{-9}$ cm-2s-1) for the sources analyzed. n | J1856$-$3754 | J0108-14 | J0953+0755 | J1136+1551 | J0630$-$2834 | J0826+2637
---|---|---|---|---|---|---
2 | 9.5 | 49 | 22 | 23 | 24 | 29
3 | 3.9$\times 10^{-2}$ | 0.11 | 6.7$\times 10^{-2}$ | 6.9$\times 10^{-2}$ | 7.4$\times 10^{-2}$ | 8.4$\times 10^{-2}$
4 | 2.5$\times 10^{-3}$ | 5.4$\times 10^{-3}$ | 3.8$\times 10^{-3}$ | 3.9$\times 10^{-3}$ | 4.3$\times 10^{-3}$ | 4.8$\times 10^{-3}$
5 | 5.0$\times 10^{-4}$ | 9.1$\times 10^{-4}$ | 7.0$\times 10^{-4}$ | 7.1$\times 10^{-4}$ | 8.1$\times 10^{-4}$ | 8.6$\times 10^{-4}$
6 | 1.7$\times 10^{-4}$ | 2.8$\times 10^{-4}$ | 2.3$\times 10^{-4}$ | 2.3$\times 10^{-4}$ | 2.7$\times 10^{-4}$ | 2.8$\times 10^{-4}$
7 | 8.2$\times 10^{-5}$ | 1.3$\times 10^{-4}$ | 1.0$\times 10^{-4}$ | 1.0$\times 10^{-4}$ | 1.2$\times 10^{-4}$ | 1.3$\times 10^{-4}$
Table 6: Table of limits on extra dimensions size $R$ (nm) for the sources
analyzed.
### 4.2 Combined Limits
We use the following method to combine limits from multiple neutron star
sources. A scan over the log-likelihood function in each ROI is done with
respect to the parameter $R^{n}$, as shown in Figure 7. A curve of the change
in log-likelihood, $|2\Delta\log\mathcal{L}|$, versus parameter value $R^{n}$,
is generated for each source. Then the sum of these curves is taken for all
the sources, and the parameter value corresponding to intersection of that
curve with a value of 2.71, corresponding to a one-sided 95% confidence level,
is quoted as the combined limit value. The results of combining limits on $R$
from this method, as well as results from HR, are presented in Table 7.
Figure 7: Plot of $|2\Delta\log\mathcal{L}|$ versus parameter value of $R^{n}$ for $n=2$. The 95% confidence level upper limit corresponds to a $y-$axis value of 2.71, shown by a dashed line. The sum of the curves, solid black, is used to obtain the posterior combined limit at the intersection with 2.71. | $R$ (nm) | $R$ (nm)
---|---|---
n | Combined | HR
2 | 8.7 | 51
3 | 0.037 | 0.11
4 | 2.5$\times 10^{-3}$ | 5.5$\times 10^{-3}$
5 | 5.0$\times 10^{-4}$ | 9.1$\times 10^{-4}$
6 | 1.7$\times 10^{-4}$ | 2.8$\times 10^{-4}$
7 | 8.2$\times 10^{-5}$ | 1.2$\times 10^{-4}$
Table 7: 95% CL upper limits on $R$ (nm) for each $n$, compared to HR2003
EGRET-based results[4].
### 4.3 Dependence of LED Limits on Model Parameters
n | $T=30\ \mbox{MeV}$ | $T=45\ \mbox{MeV}$
---|---|---
2 | $9.5$ | $1.2$
3 | $3.9\times 10^{-2}$ | $9.3\times 10^{-3}$
4 | $2.5\times 10^{-3}$ | $8.5\times 10^{-4}$
5 | $5.0\times 10^{-4}$ | $2.1\times 10^{-4}$
6 | $1.7\times 10^{-4}$ | $8.1\times 10^{-5}$
7 | $8.2\times 10^{-5}$ | $4.1\times 10^{-5}$
Table 8: A comparison of upper limits on $R$ (nm), evaluated for different
values of $T$, for source RX J1856$-$3754.
Dependence of the limits on model parameters, namely $T$, $f_{KK},$ and
$\Delta t_{NS}$, have been evaluated. We have determined that the bounds on
extra dimensions are quite sensitive to changes in $T$. Limits evaluated for
source RX J1856$-$3754 for $T=30$ MeV and a higher value $T=45$ MeV, are
compared in Table 8. The limits on LED are a strong function of temperature.
The dependence enters through two effects: changing the constant $k_{n}$ and
changing the distribution of gamma-ray energies. The limits are affected,
since $k_{n}\sim T^{-n-5.5}$; in other words, by modifying $k_{n}$, the bounds
on LED size improve as:
$R\sim\left(\frac{T}{30\ \mbox{MeV}}\right)^{-1-5.5/n}.$ (4.1)
In addition, for higher temperatures, the distribution of energies is shifted
to higher gamma-ray energies. Quantitatively, this increases the integral of
the distribution function above 100 MeV, $\int_{100\mbox{MeV}}^{\infty}dN/dE\
dE$, which tends to improve the bounds. Limits placed on $R$ from source RX
J1856$-$3754 may vary by an order of magnitude, as shown in Table 8. We do not
consider lower values of $T$, since according to [3], $T=30$ MeV is a
conservative lower limit on the SN core temperature.
By varying the timescale of core collapse, $\Delta t_{NS}$, the limits on $R$
vary as $\left(\Delta t_{NS}\right)^{-1/n}$. Estimates for this parameter vary
from 5 s to 20 s[20], while we use the value of 7.5 s from HR. Thus, we see
that the limits depend only weakly on variations of $\Delta t_{NS}$ and
$f_{KK}$.
$n$ | $f_{KK}/10^{-3}$ | $R$ (nm)
---|---|---
2 | 6.3 | 9.5
3 | 8.7 | 0.035
4 | 7.4 | 2.5 $\times 10^{-3}$
5 | 5.1 | 5.3 $\times 10^{-4}$
6 | 9.1 | 1.7 $\times 10^{-4}$
7 | 9.0 | 8.0 $\times 10^{-5}$
Table 9: Table of $f_{KK}$ values and combined upper limits on $R$ (nm) for
each value of $n$, assuming a Gaussian prior on $f_{KK}$ (with mean 0.0075 and
sigma 0.00144), as discussed in Sec. 4.4.
### 4.4 Effect of Uncertainties on $f_{KK}$ on the Limits
Varying the fraction of energy lost into the graviton channel, $f_{KK}$, the
limits on $R$ vary as $f_{KK}^{-1/n}$. HR assumed $f_{KK}\simeq 0.01$, as
consistent with diffuse gamma-ray measurements according to EGRET[16, 17].
However, a more accurate treatment from EGRET low energy diffuse measurements
constrains $f_{KK}$ such that $0.005<f_{KK}<0.01$. To take this range of
values for $f_{KK}$ into account when computing limits, we perform an analysis
allowing for a Gaussian prior on the $f_{KK}$ parameter, with a mean of 0.0075
and a sigma of 0.00144 (as obtained from the variance for a uniform PDF for
$f_{KK}$ between 0.005 and 0.01). We constrain this parameter to be the same
across the 6 ROIs, for each value of $n$. This is possible within the
framework of the Fermi-LAT _ScienceTools_ ; a similar technique was used to
constrain dark matter signals from a combined analysis of Milky Way satellites
with the Fermi-LAT[21]. Limits obtained in this manner are shown in Table 9.
## 5 Discussion and Conclusions
If $M_{D}$ is at a TeV, then for $n<4$, the results presented here imply that
the compactification topology is more complicated than a torus, i.e., all LED
having the same size. For flat LED of the same size, the lower limits on
$M_{D}$ results are consistent with $n\geq 4$. The constraints on LED based on
neutron star gamma ray emission yield improvements over previously reported
neutron star limits, based on gamma-ray measurements and combination of
individual sources, as shown in Table 10. In addition, the results for the
$n$-dimensional Planck mass are much better than collider limits from LEP and
Tevatron for $n<4$, and are comparable or slightly better for $n=4$.
n | Combined | CDF | DØ | LEP | ATLAS | CMS
---|---|---|---|---|---|---
2 | 230 | 2.09 | 1.40 | 1.60 | 1.5 | 3.2
3 | 16 | 1.94 | 1.15 | 1.20 | 1.1 | 3.3
4 | 2.5 | 1.62 | 1.04 | 0.94 | 1.8 | 3.4
5 | 0.67 | 1.46 | 0.98 | 0.77 | 2.0 | 3.4
6 | 0.25 | 1.36 | 0.94 | 0.66 | 2.0 | 3.4
7 | 0.11 | 1.29 | - | - | - | -
Table 10: Comparison of 95% CL lower limits on $M_{D}$ (TeV) with previous
astrophysical limits and collider limits. _Combined_ limits are obtained in
this paper. Collider limits are taken from references [22, 23, 24, 25]. ATLAS
and CMS results are quoted where $\Lambda/M_{D}=1$. ATLAS results are quoted
with 3.1 pb-1 of data; CMS results are quoted with 36 pb-1 of data.
These limits may prove useful, especially for $n=4$ case (where the limits are
comparable to collider results), in the context of constraining phase space in
searches for extra dimensions underway at the LHC. These results are also more
stringent than those reported by short distance gravity experiments probing
for deviations from the inverse square law. The most sensitive such experiment
to KK graviton emission presented a result of 37 $\mu$m for $n=2$ at 95%
C.L.[26]; this is several orders of magnitude larger than the combined result
reported here, of 8.7 nm.
Cassé _et al._ obtain upper limits significantly better than ours [27] (a
factor $\sim 20$ for $n=2$, though decreasing approximately as $1/n^{2}$ with
increasing $n$), summing the contribution of all the expected NS in the
Galactic bulge and comparing to the EGRET data. But it is should be noted that
they do not account for age nor magnetic field attenuation, while the present
analysis shows that both impact the photon distribution in a significant way.
As these effects are not taken into account in HR, which the Cassé _et al._
paper is based on, their upper limits are necessarily underestimated.
Furthermore, it should be emphasized that the 6 neutron stars analyzed here
are chosen at high latitude to avoid the large systematic uncertainties
involved in modeling the diffuse Galactic background. These are even larger in
the low energy range that we are interested in here. An analysis of the bulge
with the current Fermi-LAT instrument response functions and the current
Fermi-LAT diffuse models could nominally improve the flux upper limits, but at
the cost of a much less robust analysis, the systematics of which are
difficult to evaluate. Within the Fermi-LAT collaboration, a better inner
galaxy model is in process, which is necessary before approaching a Galactic
Bulge analysis. A Large Extra Dimensions analysis of the Galactic Bulge will
be the subject of future studies.
## Appendix A Appendix: Sampling Decay Vertices from Graviton Trajectories
In our model, due to the $G_{KK}$ emission radially outward during the SN core
collapse, the $G_{KK}$ are not given any initial angular momentum; thus we
assume that the $G_{KK}$ oscillate on radial paths (completely eccentric
orbits) through the center of the neutron star. In spherical coordinates
$(r,\theta,\phi)$, this is equivalent to the following constraints:
$\dot{\theta}=\dot{\phi}=0$. The orbital radial distribution, $P(r;\mu)$, is
defined outside the neutron star by the radial Kepler equation, in which time
is given as a function of the radial coordinate $r$[28]444The radial Kepler
equation is not manifestly periodic. However, the full orbit cycle includes an
interval over $0<r<r_{\max}$, or a quarter of a cycle. Due to the symmetry of
the orbit, our treatment of sampling the decay vertex from $t=0$ to
$t(r_{\max})$ is sufficient to obtain the full distribution of $r_{0}$.:
$t(r)=t_{k}\left(\arcsin\left(\sqrt{kr}\right)-\sqrt{kr(1-kr)}+c_{1}\right),$
(A.1)
where:
$k=r_{\max}^{-1}=\frac{1+1.5|U_{NS}|-\gamma}{|U_{NS}|}\ ,$ (A.2)
and:
$t_{k}=\frac{R_{NS}}{\beta
ck^{2}}\sqrt{\frac{k(1-k)}{1-\left|U_{NS}\right|/\beta^{2}}}.$ (A.3)
The solution inside the NS ($r<1$) is defined as:
$r(t)=\frac{\beta}{\sqrt{\left|U_{NS}\right|}}\sin(\Omega t)\ ,$ (A.4)
where the parameter $\Omega=\sqrt{|U_{NS}|}c/R_{NS}=9.13\times 10^{3}$ s-1.
Given that $t=0$ is the time when the $G_{KK}$ is created at $r=0$, the radial
distributions are determined by sampling time uniformly between the $t=0$ and
$t_{\max}$. $t_{\max}$ is given by the time to achieve the maximum distance,
$r_{\max}=1/k$, as in eq. (A.2). $c_{1}$ is determined from boundary
conditions of position and velocity at the surface of the neutron star by
solving the full equation of motion inside and outside the neutron star. The
trajectories for a couple of values of $\beta$ are plotted in Figure 8, while
the radial distributions for representative values of $\mu$, $P(r;\mu)$, are
shown in Figure 4.
Figure 8: Trajectories for $\beta=0.4$ and $\beta=0.5$. Notice that for larger
$\beta$, the KK graviton may achieve a farther maximum distance,
$r_{\max}\times R_{NS}$. This figure shows one quarter of the orbit cycle.
## Appendix B Appendix: Relativistic Decay Kinematics of KK Gravitons
The energy is given by:
$E_{\gamma}=\frac{1}{2}\gamma^{\prime}m\left(1+\beta^{\prime}\cos\theta^{*}\right),$
(B.1)
while the components of the photon momentum,
$\vec{p}_{\gamma}^{\prime}=p_{x^{\prime},\gamma}\hat{x}^{\prime}\ +\
p_{y^{\prime},\gamma}\hat{y}^{\prime}\ +\
p_{z^{\prime},\gamma}\hat{z}^{\prime},$ (B.2)
in the neutron star frame relative to the direction of the $G_{KK}$ are given
by:
$\displaystyle p_{x^{\prime},\gamma}$
$\displaystyle=\frac{1}{2}m\sin\theta^{*}\cos\phi^{*}$ (B.3) $\displaystyle
p_{y^{\prime},\gamma}$ $\displaystyle=\frac{1}{2}m\sin\theta^{*}\sin\phi^{*}$
(B.4) $\displaystyle p_{z^{\prime},\gamma}$
$\displaystyle=\frac{1}{2}\gamma^{\prime}m(\beta^{\prime}+\cos\theta^{*}).$
(B.5)
The $z^{\prime}$ axis is defined by: $\hat{z}^{\prime}=\hat{r}_{0}$.
$\theta^{*}$ is the polar angle between the direction of the $G_{KK}$ in the
lab frame ($z^{\prime}$) and the decay photon in the rest frame of the
$G_{KK}$, and $\phi^{*}$ is the angle in the ($x^{\prime}-y^{\prime}$)plane.
In this frame, $\hat{x}^{\prime}$ is taken as perpendicular to
$\hat{z}^{\prime}$ in the $z-z^{\prime}$ plane and in the direction of
increasing $\theta$, and
$\hat{y}^{\prime}=\hat{z}^{\prime}\times\hat{x}^{\prime}$. The coordinate
systems used are depicted in Fig. 9. We sample a $\cos\theta^{*}$ value
uniformly over the interval [-1,1], and $\phi^{*}$ uniformly over the interval
[$-\pi,\pi$], given isotropic emission of photons in the rest frame of the
gravitons. Subsequently, we obtain momentum components of the gamma ray in the
frame of the neutron star, $\vec{p}_{\gamma}$, by rotating back into the frame
of the neutron star, using the direction of the momentum vector,
$\vec{p}_{KK}$, as defined by step (5). This is needed for the next step.
Figure 9: The coordinate system, as described in Appendix B.
## Appendix C Appendix: Determining Photon Pair Production Optical Depths
Approximations for the pair-production attenuation are according to the
treatment in Refs. [29, 30]. The attenuation coefficient depends on the
parameter:
$\chi(E_{\gamma},\vec{p}_{\gamma},\vec{r})=\frac{E_{\gamma}}{m_{e}c^{2}}\frac{B_{\perp}(\vec{r},\vec{p}_{\gamma})}{B_{\operatorname{cr}}},$
(C.1)
where the critical field is given by
$B_{\operatorname{cr}}=\frac{m^{2}_{e}c^{3}}{e\hbar}=4.414\times 10^{13}\
\mbox{G},$ (C.2)
$B_{\perp}$ is the magnetic field component of the neutron star perpendicular
to the photon’s momentum vector $\vec{p}_{\gamma}$, and $E_{\gamma}$ is the
photon energy. For the magnetic field of the neutron star, we assume a static
dipole field,
$\vec{B}(\vec{r})=\frac{3(\vec{m}\cdot\hat{r})\hat{r}-\vec{m}}{r^{3}}$ (C.3)
with dipole moment $\vec{m}=\frac{1}{2}B_{surf}R_{NS}^{3}\hat{z}$. The
attenuation coefficient, $\alpha$, is given by:
$\alpha(\chi(E_{\gamma},\vec{p_{\gamma}},\vec{r}))=\frac{\alpha_{\operatorname{fs}}}{\lambdabar_{e}}\frac{B_{\perp}(\vec{r},\vec{p}_{\gamma})}{B_{\operatorname{cr}}}\alpha_{1}(\chi),$
(C.4)
where the reduced attenuation coefficient, $\alpha_{1}(\chi)$, is expressible
as a function of $\chi$ in terms of a modified Bessel function of the second
kind, with asymptotic limiting expressions for small and large values of
$\chi$ (as plotted in Figure 10):
$\alpha_{1}(\chi)=0.16\frac{1}{\chi}K_{1/3}^{2}\left(\frac{2}{3\chi}\right)=\begin{array}[]{c}\end{array}\left\\{\begin{array}[]{l}0.377e^{-\frac{4}{3\chi}},\chi\leq
0.1\\\ \\\ 0.597\chi^{-1/3},\chi\geq 100\end{array}\right.$ (C.5)
In eq. (C.5), $\lambdabar_{e}=3.861\times 10^{-11}$ cm is the reduced electron
Compton wavelength and $\alpha_{\operatorname{fs}}$ is the fine structure
constant. The asymptotic expressions are used in the Monte Carlo simulation,
where appropriate, in order to save computer time.
Figure 10: Plot of the reduced attenuation coefficient $\alpha_{1}(\chi)$ and
limiting asymptotic expressions corresponding to eq. (C.5).
The optical depth $\tau_{pp}$ is calculated by path integrating the
attenuation coefficient along the direction of the photon, from the point of
decay, $\vec{r}_{0}\equiv<x_{0},y_{0},z_{0}>$, out to where $r_{km}=7R_{NS}$
(where the field has attenuated to 0.3% of the surface field strength),
according to
$\begin{split}\tau_{pp}&=\int_{\operatorname{path}}\alpha\ {\rm d}s\\\
&=\int_{0}^{s_{\max}}\alpha\left(\chi\left(E_{\gamma},\vec{p}_{\gamma},\vec{r}_{0}+\hat{p}_{\gamma}s\right)\right){\rm
d}s\\\ \end{split}$ (C.6)
In the preceding equation, $s_{\max}$, given by:
$s_{\max}=-x_{0}p_{1}-y_{0}p_{2}-z_{0}p_{3}+\sqrt{(x_{0}p_{1}+y_{0}p_{2}+z_{0}p_{3})^{2}+(7R_{NS})^{2}-x_{0}^{2}-y_{0}^{2}-z_{0}^{2}},$
(C.7)
refers to the path length where the photon with direction unit vector *
$\hat{p}_{\gamma}=\frac{\vec{p}_{\gamma}}{\left|\vec{p}_{\gamma}\right|}=p_{1}\hat{x}+p_{2}\hat{y}+p_{3}\hat{z}$
(C.8)
is considered to have escaped the magnetosphere.
## Acknowledgments
The Fermi-LAT Collaboration acknowledges generous ongoing support from a
number of agencies and institutes that have supported both the development and
the operation of the LAT as well as scientific data analysis. These include
the National Aeronautics and Space Administration and the Department of Energy
in the United States, the Commissariat à l’Energie Atomique and the Centre
National de la Recherche Scientifique / Institut National de Physique
Nucléaire et de Physique des Particules in France, the Agenzia Spaziale
Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry
of Education, Culture, Sports, Science and Technology (MEXT), High Energy
Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency
(JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research
Council and the Swedish National Space Board in Sweden.
Additional support for science analysis during the operations phase is
gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy
and the Centre National d’Études Spatiales in France.
## References
* [1] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, The hierarchy problem and new dimensions at a millimeter, Phys. Lett. B429 (1998) 263, [hep-ph/9803315].
* [2] C. Hanhart, D. R. Phillips, S. Reddy, and M. J. Savage, Extra dimensions, sn1987a, and nucleon-nucleon scattering data, nucl-th/0007016.
* [3] G. G. Raffelt, Stars as Laboratories for Fundamental Physics. The University of Chicago Press, 1996.
* [4] S. Hannestad and G. G. Raffelt, Supernova and neutron-star limits on large extra dimensions reexamined, Phys. Rev. D 67 (Jun, 2003) 125008, [hep-ph/0304029v2].
* [5] _Fermi_ -LAT Collaboration, W. Atwood et. al., The Large Area Telescope on the Fermi Gamma-Ray Space Telescope Mission, ApJ 697 (June, 2009) 1071–1102, [arXiv:0902.1089].
* [6] D. J. Thompson et. al. Ap. J. Supplements 86 (June, 1993) 629–656.
* [7] A. Strong, I. Moskalenko, and O. Reimer Ap. J. 613 (2004) 962–976.
* [8] A. Strong Astron. and Astrophys. Supplement Series 309 (2007) 35.
* [9] http://fermi.gsfc.nasa.gov/ssc/data/analysis/software/aux/isotropic_iem_v02.txt.
* [10] _Fermi_ -LAT Collaboration, A. A. Abdo et. al., Fermi large area telescope first source catalog, Astrophysical Journal Supplement Series 188 (2010), no. 2 405, [arXiv:1002.2280].
* [11] _Fermi_ -LAT Collaboration, R. Rando et. al. Proceedings of the 31st ICRC, July, 2009. astro-ph/0907.0626.
* [12] R. Manchester, G. Hobbs, A. Teoh, and M. Hobbs, The ATNF Pulsar Catalogue, astro-ph/0412641.
* [13] B. Berenji, Search for Large Extra Dimensions Based on Observations of Neutron Stars with Fermi-LAT. PhD thesis, Stanford University, 2011. http://purl.stanford.edu/sj534tb9150.
* [14] http://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/.
* [15] A. A. Abdo et. al., Fermi lat measurements of the diffuse gamma-ray emission at intermediate galactic latitudes, Phys. Rev. Lett. 103 (2009) 251101, [astro-ph/0912.0973].
* [16] P. Sreekumar et. al. Ap. J. 494 (1998).
* [17] S. Hannestad and G. Raffelt, New supernova limit on large extra dimensions, Phys. Rev. Lett. 87 (2001) 051301, [hep-ph/0103201v2].
* [18] T. Han, J. D. Lykken, and R.-J. Zhang, On Kaluza-Klein states from large extra dimensions, Phys. Rev. D59 (1999) 105006, [hep-ph/9811350].
* [19] F. James and M. Winkler, “Minuit user’s guide.” http://seal.web.cern.ch/seal/snapshot/work-packages/mathlibs/minuit/, June, 2004\.
* [20] S. Rosswog and M. Brüggen, Introduction to High Energy Astrophysics. Cambridge University Press, 2007.
* [21] _Fermi_ -LAT Collaboration, M. Ackermann et. al., Constraining dark matter models from a combined analysis of Milky Way satellites with the _Fermi_ -Large Area Telescope, Phys. Rev. Lett. 107 (2011) [arXiv:1111.0320].
* [22] R. Franceschini, P. P. Giardino, G. F. Giudice, P. Lodone, and A. Strumia, LHC Bounds on Large Extra Dimensions, JHEP 05 (2011) 092, [hep-ph/1101.4919].
* [23] http://lepexotica.web.cern.ch/LEPEXOTICA, 2004.
* [24] V. Abazov et. al., Search for large extra spatial dimensions in the dielectron and diphoton channels in $\bm{p\bar{p}}$ collisions at $\sqrt{}$s =1.96 tev, Phys. Rev. Lett. 102 (2009) 051601, [hep-ex/0809.2813v1].
* [25] 16th International Workshop on Deep Inelastic Scattering and Related Subjects, Searches for Large Extra Dimensions at the Tevatron, July, 2008.
* [26] D. Kapner et. al., Tests of the gravitational inverse-square law below the dark-energy length scale, Phys. Rev. Lett. 98 (January, 2007) 021101.
* [27] M. Cassé, J. Paul, G. Bertone, and G. Sigl, Gamma rays from the galactic bulge and large extra dimensions, Phys. Rev. Lett. 92 (Mar, 2004) 111102, [hep-ph/0309173].
* [28] H. Goldstein, C. Poole, and J. Safko, Classical Mechanics. Addison Wesley, 3 ed., 2002.
* [29] T. Erber, High-energy electromagnetic conversion processes in intense magnetic fields, Rev. Mod. Phys. 38 (1966) 626.
* [30] V. Berestetskii, E. Lifshitz, and L. Pitaeveskii, Quantum Electrodynamics , vol. 4 of Landau and Lifshitz Course of Theoretical Physics. Pergamon Press, 2nd ed., 1982.
|
arxiv-papers
| 2012-01-12T01:42:01 |
2024-09-04T02:49:26.206984
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bijan Berenji, Elliott Bloom, Johann Cohen-Tanugi (for the Fermi-LAT\n Collaboration)",
"submitter": "Bijan Berenji",
"url": "https://arxiv.org/abs/1201.2460"
}
|
1201.2476
|
# Report No. C-SAFE-CD-IR-04-004
MPM VALIDATION: A MYRIAD OF TAYLOR IMPACT TESTS
B. Banerjee (b.banerjee.nz@gmail.com)
Department of Mechanical Engineering, University of Utah,
Salt Lake City, UT 84112, USA
August 21, 2004
###### Abstract
Taylor impacts tests were originally devised to determine the dynamic yield
strength of materials at moderate strain rates. More recently, such tests have
been used extensively to validate numerical codes for the simulation of
plastic deformation. In this work, we use the material point method to
simulate a number of Taylor impact tests to compare different Johnson-Cook,
Mechanical Threshold Stress, and Steinberg-Guinan-Cochran plasticity models
and the vob Mises and Gurson-Tvergaard-Needleman yield conditions. In addition
to room temperature Taylor tests, high temperature tests have been performed
and compared with experimental data.
## 1 INTRODUCTION
The Taylor impact test (Taylor [1]) was originally devised as a means of
determining the dynamic yield strength of solids. The test involves the impact
of a flat-nosed cylindrical projectile on a hard target at normal incidence.
Taylor provided an analytical solutions for the dynamic yield strength of the
material of the projectile based on the length of the elastic region and the
radius of the region of permanent set. As described by Whiffin [2], that use
of the test was limited to relatively small deformations obtained from low
velocity impacts. Though the Taylor impact test continues to be used to
determine yield strengths of materials at high strain rates, the test is
limited to peak strains of around 0.6 at the center of the specimen (Johnson
and Holmquist [3]). For higher strains and strain rates, the Taylor test is
currently used more as a means of validating plasticity models in numerical
codes for the simulation of high rate phenomena such as impact and explosive
deformation as suggested by Zerilli and Armstrong [4].
In this paper, we describe our experience in validating the plasticity models
in a parallel, multiphysics code that uses the material point method (Sulsky
et al. [5, 6]) using Taylor impact tests for various strain rates and
temperatures. A number of metrics are used to compare simulations and
experiments and suggesstions are made regarding the use of Taylor impacts
tests for the validation of the plasticity portion of such codes.
The organization of this paper is as follows. Section 2 provides the
background for the current study and describes the mutiphysics code Uintah,
the material point method, and the stress update algorithm, and various
plasticity models and yield conditions. A few validation metrics are
identified and their significance is discussed in Section 3. Comparisons
between experimental data and simulations of Taylor impact tests using the
validation metrics are described in Section 4. Finally, conclusions and
suggestions are presented in Section 5.
## 2 BACKGROUND
The goal of this work is to present some results and insights we have obatined
during the process of validation of plasticity models used in the simulation
of the deformation and failure of a steel container that expands under the
effect of gases produced by an explosively reacting high energy material (PBX
9501) contained inside. The entire process is simulated using the massively
parallel, Common Component Architecture [7] based, Uintah Computational
Framework (UCF) [8].
The high energy material reacts at temperatures of 450 K and above, This
elevated temperature is achieved through external heating of the steel
container. Experiments conducted at the University of Utah have shown that
failure of the container can be due to ductile fracture associated with void
coalescence and adiabatic shear bands. If shear bands dominate the steel
container fragments, otherwise a few large cracks propagate along the cylinder
and pop it open. Figure 1 shows the result of a simulation of a coupled fire-
container-explosion using the Uintah.
Figure 1: Simulation of exploding cylinder.
The dynamics of the solid materials - steel and PBX 9501 - is modeled using
the Lagrangian Material Point Method (MPM) [5]. Gases are generated from solid
PBX 9501 using a burn model [9]. Gas-solid interaction is accomplished using
an Implicit Continuous Eulerian (ICE) multi-material hydrodynamic code [10]. A
single computational grid is used for all the materials.
The constitutive response of PBX 9501 is modeled using ViscoSCRAM [11], which
is a five element generalized Maxwell model for the viscoelastic response
coupled with statistical crack mechanics. Solid PBX 9501 is progressively
converted into a gas with an appropriate equation of state. The temperature
and pressure in the gas increase rapidly as the reaction continues. As a
result, the steel container is pressurized, undergoes plastic deformation, and
finally fragments.
The main issues regarding the constitutive modeling of the steel container are
the selection of appropriate models for nonlinear elasticity, plasticity,
damage, loss of material stability, and failure. The numerical simulation of
the steel container involves the choice of appropriate algorithms for the
integration of balance laws and constitutive equations, as well as the
methodology for fracture simulation. Models and simulation methods for the
steel container are required to be temperature sensitive and valid for large
distortions, large rotations, and a range of strain rates (quasistatic at the
beginning of the simulation to approximately $10^{6}$ s-1 at fracture).
The approach chosen for the present work is to use hypoelastic-plastic
constitutive models that assume an additive decomposition of the rate of
deformation tensor into elastic and plastic parts. Hypoelastic materials are
known not to conserve energy in a loading-unloading cycle unless a very small
time step is used. However, the choice of this model is justified under the
assumption that elastic strains are expected to be small for the problem under
consideration and unlikely to affect the computation significantly.
Two plasticity models for flow stress are considered along with a two
different yield conditions. Explicit fracture simulation is computationally
expensive and prohibitive in the large simulations under consideration. The
choice, therefore, has been to use damage models and stability criteria for
the prediction of failure (at material points) and particle erosion for the
simulation of fracture propagation.
### 2.1 The Material Point Method
The Material Point Method (MPM) [5] is a particle method for structural
mechanics simulations. In this method, the state variables of the material are
described on Lagrangian particles or “material points”. In addition, a
regular, structured Eulerian grid is used as a computational scratch pad to
compute spatial gradients and to solve the governing conservation equations.
An explicit time-stepping version of the Material Point Method has been used
in the simulations presented in this paper. The MPM algorithm is summarized
below [6].
It is assumed that an particle state at the beginning of a time step is known.
The mass ($m$), external force ($\mathbf{f}^{\text{ext}}$), and velocity
($\mathbf{v}$) of the particles are interpolated to the grid using the
relations
$m_{g}=\sum_{p}S_{gp}~{}m_{p}~{},~{}~{}~{}~{}\mathbf{v}_{g}=(1/m_{g})\sum_{p}S_{gp}~{}m_{p}~{}\mathbf{v}_{p}~{},~{}~{}~{}~{}\mathbf{f}^{\text{ext}}_{g}=\sum_{p}S_{gp}~{}\mathbf{f}^{\text{ext}}_{p}$
(1)
where the subscript ($g$) indicates a quantity at a grid node and a subscript
($p$) indicates a quantity on a particle. The symbol $\sum_{p}$ indicates a
summation over all particles. The quantity ($S_{gp}$) is the interpolation
function of node ($g$) evaluated at the position of particle ($p$). Details of
the interpolants used can be found elsewhere [12].
Next, the velocity gradient at each particle is computed using the grid
velocities using the relation
$\boldsymbol{\nabla}\mathbf{v}_{p}=\sum_{g}\mathbf{G}_{gp}\mathbf{v}_{g}$ (2)
where $\mathbf{G}_{gp}$ is the gradient of the shape function of node ($g$)
evaluated at the position of particle ($p$). The velocity gradient at each
particle is used to determine the Cauchy stress ($\boldsymbol{\sigma}_{p}$) at
the particle using a stress update algorithm.
The internal force at the grid nodes ($\mathbf{f}^{\text{int}}_{g}$) is
calculated from the divergence of the stress using
$\mathbf{f}^{\text{int}}_{g}=\sum_{p}\mathbf{G}_{gp}~{}\boldsymbol{\sigma}_{p}~{}V_{p}$
(3)
where $V_{p}$ is the particle volume.
The equation for the conservation of linear momentum is next solved on the
grid. This equation can be cast in the form
$\mathbf{m}_{g}~{}a_{g}=\mathbf{f}^{\text{ext}}_{g}-\mathbf{f}^{\text{int}}_{g}$
(4)
where $\mathbf{a}_{g}$ is the acceleration vector at grid node ($g$).
The velocity vector at node ($g$) is updated using an explicit (forward Euler)
time integration, and the particle velocity and position are then updated
using grid quantities. The relevant equations are
$\displaystyle\mathbf{v}_{g}(t+\Delta t)$
$\displaystyle=\mathbf{v}_{g}(t)+\mathbf{a}_{g}~{}\Delta t$ (5)
$\displaystyle\mathbf{v}_{p}(t+\Delta t)$
$\displaystyle=\mathbf{v}_{p}(t)+\sum_{g}S_{gp}~{}\mathbf{a}_{g}~{}\Delta
t~{};~{}~{}~{}~{}\mathbf{x}_{p}(t+\Delta
t)=\mathbf{x}_{p}(t)+\sum_{g}S_{gp}~{}\mathbf{v}_{g}~{}\Delta t$ (6)
The above sequence of steps is repeated for each time step. The above
algorithm leads to particularly simple mechanisms for handling contact.
Details of these contact algorithms can be found elsewhere [13].
### 2.2 Plasticity and Failure Simulation
A hypoelastic-plastic, semi-implicit approach [14] has been used for the
stress update in the simulations presented in this paper. An additive
decomposition of the rate of deformation tensor into elastic and plastic parts
has been assumed. One advantage of this approach is that it can be used for
both low and high strain rates. Another advantage is that many strain-rate and
temperature-dependent plasticity and damage models are based on the assumption
of additive decomposition of strain rates, making their implementation
straightforward.
The stress update is performed in a co-rotational frame which is equivalent to
using the Green-Naghdi objective stress rate. An incremental update of the
rotation tensor is used instead of a direct polar decomposition of the
deformation gradient. The accuracy of model is good if elastic strains are
small compared to plastic strains and the material is not unloaded. It is also
assumed that the stress tensor can be divided into a volumetric and a
deviatoric component. The plasticity model is used to update only the
deviatoric component of stress assuming isochoric behavior. The hydrostatic
component of stress is updated using a solid equation of state.
Since the material in the container may unload locally after fracture, the
hypoelastic-plastic stress update may not work accurately under certain
circumstances. An improvement would be to use a hyperelastic-plastic stress
update algorithm. Also, the plasticity models are temperature dependent. Hence
there is the issue of severe mesh dependence due to change of the governing
equations from hyperbolic to elliptic in the softening regime [15, 16, 17].
Viscoplastic stress update models or nonlocal/gradient plasticity models [18,
19] can be used to eliminate some of these effects and are currently under
investigation.
A particle is tagged as “failed” when its temperature is greater than the
melting point of the material at the applied pressure. An additional condition
for failure is when the porosity of a particle increases beyond a critical
limit. A final condition for failure is when a bifurcation condition such as
the Drucker stability postulate is satisfied. Upon failure, a particle is
either removed from the computation by setting the stress to zero or is
converted into a material with a different velocity field which interacts with
the remaining particles via contact. Either approach leads to the simulation
of a newly created surface.
In the parallel implementation of the stress update algorithm, sockets have
been added to allow for the incorporation of a variety of plasticity, damage,
yield, and bifurcation models without requiring any change in the stress
update code. The algorithm is shown in Algorithm 1. The equation of state,
plasticity model, yield condition, damage model, and the stability criterion
are all polymorphic objects created using a factory idiom in C++ [20].
Data: Persistent:Initial moduli, temperature, porosity, scalar damage,
equation of state, plasticity model, yield condition, stability criterion,
damage model
Temporary:Particle state at time $t$
Result: Particle state at time $t+\Delta t$
for _all the patches in the domain_ do
Read the particle data and initialize updated data storage;
for _all the particles in the patch_ do
Compute the velocity gradient, the rate of deformation tensor and the spin
tensor;
Compute the updated left stretch tensor, rotation tensor, and deformation
gradient;
Rotate the input Cauchy stress and the rate of deformation tensor to the
material configuration;
Compute the current shear modulus and melting temperature;
Compute the pressure using the equation of state, update the hydrostatic
stress, and compute the trial deviatoric stress;
Compute the flow stress using the plasticity model;
Evaluate the yield function;
if _particle is elastic_ then
Rotate the stress back to laboratory coordinates;
Update the particle state;
else
Find derivatives of the yield function;
Do radial return adjustment of deviatoric stress;
Compute updated porosity, scalar damage, and temperature increase due to
plastic work;
Compute elastic-plastic tangent modulus and evaluate stability condition;
Rotate the stress back to laboratory coordinates;
Update the particle state;
if _Temperature $>$ Melt Temperature or Porosity $>$ Critical Porosity or
Unstable_ then
Tag particle as failed;
end if
end if
end for
end for
Convert failed particles into a material with a different velocity field;
Algorithm 1 Stress Update Algorithm
### 2.3 Models
The stress in the solid is partitioned into a volumetric part and a deviatoric
part. Only the deviatoric part of stress is used in the plasticity
calculations assuming isoschoric plastic behavior.
The hydrostatic pressure ($p$) is calculated either using the bulk modulus
($K$) and shear modulus ($\mu$) or from a temperature-corrected Mie-Gruneisen
equation of state of the form [14]
$p=\frac{\rho_{0}C_{0}^{2}\zeta\left[1+\left(1-\frac{\Gamma_{0}}{2}\right)\zeta\right]}{\left[1-(S_{\alpha}-1)\zeta\right]^{2}+\Gamma_{0}C_{p}T}~{},~{}~{}~{}~{}\zeta=(\rho/\rho_{0}-1)$
(7)
where $C_{0}$ is the bulk speed of sound, $\rho_{0}$ is the initial density,
$\rho$ is the current density, $C_{p}$ is the specific heat at constant
volume, $T$ is the temperature, $\Gamma_{0}$ is the Gruneisen’s gamma at
reference state, and $S_{\alpha}$ is the linear Hugoniot slope coefficient.
Depending on the plasticity model being used, the pressure and temperature-
dependent shear modulus ($\mu$) and the pressure-dependent melt temperature
($T_{m}$) are calculated using the relations [21]
$\displaystyle\mu$
$\displaystyle=\mu_{0}\left[1+A\frac{p}{\eta^{1/3}}-B(T-300)\right]$ (8)
$\displaystyle T_{m}$
$\displaystyle=T_{m0}\exp\left[2a\left(1-\frac{1}{\eta}\right)\right]\eta^{2(\Gamma_{0}-a-1/3)}$
(9)
where, $\mu_{0}$ is the shear modulus at the reference state($T$ = 300 K, $p$
= 0, $\epsilon_{p}$ = 0), $\epsilon_{p}$ is the plastic strain.
$\eta=\rho/\rho_{0}$ is the compression, $A=(1/\mu_{0})(d\mu/dp)$,
$B=(1/\mu_{0})(d\mu/dT)$, $T_{m0}$ is the melt temperature at $\rho=\rho_{0}$,
and $a$ is the coefficient of the first order volume correction to Gruneisen’s
gamma.
We have explored two temperature and strain rate dependent plasticity models -
the Johnson-Cook plasticity model [22] and the Mechanical Threshold Stress
(MTS) plasticity model [23, 24]. The flow stress ($\sigma_{f}$) from the
Johnson-Cook model is given by
$\sigma_{f}=[A+B(\epsilon_{p})^{n}][1+C\ln(\dot{\epsilon_{p}^{*}})][1-(T^{*})^{m}]~{};~{}~{}\dot{\epsilon_{p}^{*}}=\cfrac{\dot{\epsilon_{p}}}{\dot{\epsilon_{p0}}}~{};~{}~{}T^{*}=\cfrac{(T-T_{r})}{(T_{m}-T_{r})}$
(10)
where $\dot{\epsilon_{p0}}$ is a user defined plastic strain rate, A, B, C, n,
m are material constants, $T_{r}$ is the room temperature, and $T_{m}$ is the
melt temperature.
The flow stress for the MTS model is given by
$\sigma_{f}=\sigma_{a}+\frac{\mu}{\mu_{0}}S_{i}\hat{\sigma}_{i}+\frac{\mu}{\mu_{0}}S_{e}\hat{\sigma}_{e}$
(11)
where
$\displaystyle\mu$
$\displaystyle=\mu_{0}-\frac{D}{\exp\left(\frac{T_{0}}{T}\right)-1}$
$\displaystyle S_{i}$ $\displaystyle=\left[1-\left(\frac{kT}{g_{0i}\mu
b^{3}}\ln\frac{\dot{\epsilon}_{0i}}{\dot{\epsilon}}\right)^{1/qi}\right]^{1/pi}~{};~{}~{}S_{e}=\left[1-\left(\frac{kT}{g_{0e}\mu
b^{3}}\ln\frac{\dot{\epsilon}_{0e}}{\dot{\epsilon}}\right)^{1/qe}\right]^{1/pe}$
$\displaystyle\theta$
$\displaystyle=\theta_{0}[1-F(X)]+\theta_{IV}F(X)~{};~{}~{}\theta_{0}=a_{0}+a_{1}\ln\dot{\epsilon}+a_{2}\sqrt{\dot{\epsilon}}-a_{3}T$
$\displaystyle X$
$\displaystyle=\cfrac{\hat{\sigma}_{e}}{\hat{\sigma}_{es}}~{};~{}~{}F(X)=\tanh(\alpha
X)~{};~{}~{}\ln(\hat{\sigma}_{es}/\hat{\sigma}_{es0})=\left(\frac{kT}{\mu
b^{3}g_{0es}}\right)\ln\left(\cfrac{\dot{\epsilon}}{\dot{\epsilon}_{es0}}\right)$
$\displaystyle\hat{\sigma}_{e}^{(n+1)}$
$\displaystyle=\hat{\sigma}_{e}^{(n)}+\theta\Delta\epsilon$
and $\sigma_{a}$ is the athermal component of mechanical threshold stress,
$\mu_{0}$ is the shear modulus at 0 K, $D,T_{0}$ are empirical constants,
$\hat{\sigma}_{i}$ represents the stress due to intrinsic barriers to
thermally activated dislocation motion and dislocation-dislocation
interactions, $\hat{\sigma}_{e}$ represents the stress due to microstructural
evolution with increasing deformation, $k$ is the Boltzmann constant, $b$ is
the length of the Burger’s vector, $g_{0[i,e]}$ are the normalized activation
energies, $\dot{\epsilon}_{0[i,e]}$ are constant strain rates,
$q_{[i,e]},p_{[i,e]}$ are constants, $\theta_{0}$ is the hardening due to
dislocation accumulation, $a_{0},a_{1},a_{2},a_{3},\theta_{IV},\alpha$ are
constants, $\hat{\sigma}_{es}$ is the stress at zero strain hardening rate,
$\hat{\sigma}_{es0}$ is the saturation threshold stress for deformation at 0
K, $g_{0es}$ is a constant, and $\dot{\epsilon}_{es0}$ is the maximum strain
rate.
We have decided to focus on ductile failure of the steel container.
Accordingly, two yield criteria have been explored - the von Mises condition
and the Gurson-Tvergaard-Needleman (GTN) yield condition [25, 26] which
depends on porosity. An associated flow rule is used to determine the plastic
rate parameter in either case. The von Mises yield condition is given by
$\Phi=\left(\frac{\sigma_{eq}}{\sigma_{f}}\right)^{2}-1=0~{};~{}~{}~{}\sigma_{eq}=\sqrt{\frac{3}{2}\sigma^{d}:\sigma^{d}}$
(12)
where $\sigma_{eq}$ is the von Mises equivalent stress, $\sigma^{d}$ is the
deviatoric part of the Cauchy stress, and $\sigma^{f}$ is the flow stress. The
GTN yield condition can be written as
$\Phi=\left(\frac{\sigma_{eq}}{\sigma_{f}}\right)^{2}+2q_{1}f_{*}\cosh\left(q_{2}\frac{Tr(\sigma)}{2\sigma_{f}}\right)-(1+q_{3}f_{*}^{2})=0$
(13)
where $q_{1},q_{2},q_{3}$ are material constants and $f_{*}$ is the porosity
(damage) function given by
$f*=\begin{cases}f&\text{for}~{}~{}f\leq f_{c},\\\
f_{c}+k(f-f_{c})&\text{for}~{}~{}f>f_{c}\end{cases}$ (14)
where $k$ is a constant and $f$ is the porosity (void volume fraction). The
flow stress in the matrix material is computed using either of the two
plasticity models discussed earlier. Note that the flow stress in the matrix
material also remains on the undamaged matrix yield surface and uses an
associated flow rule.
The evolution of porosity is calculated as the sum of the rate of growth and
the rate of nucleation [27]. The rate of growth of porosity and the void
nucleation rate are given by the following equations [28]
$\displaystyle\dot{f}$
$\displaystyle=\dot{f}_{\text{nucl}}+\dot{f}_{\text{grow}}$ (15)
$\displaystyle\dot{f}_{\text{grow}}$
$\displaystyle=(1-f)\text{Tr}(\mathbf{D}_{p})$ (16)
$\displaystyle\dot{f}_{\text{nucl}}$
$\displaystyle=\cfrac{f_{n}}{(s_{n}\sqrt{2\pi})}\exp\left[-\frac{1}{2}\cfrac{(\epsilon_{p}-\epsilon_{n})^{2}}{s_{n}^{2}}\right]\dot{\epsilon}_{p}$
(17)
where $\mathbf{D}_{p}$ is the rate of plastic deformation tensor, $f_{n}$ is
the volume fraction of void nucleating particles , $\epsilon_{n}$ is the mean
of the distribution of nucleation strains, and $s_{n}$ is the standard
deviation of the distribution.
Part of the plastic work done is converted into heat and used to update the
temperature of a particle. The increase in temperature ($\Delta T$) due to an
increment in plastic strain ($\Delta\epsilon_{p}$) is given by the equation
[29]
$\Delta T=\cfrac{\chi\sigma_{f}}{\rho C_{p}}\Delta\epsilon_{p}$ (18)
where $\chi$ is the Taylor-Quinney coefficient, and $C_{p}$ is the specific
heat. A special equation for the dependence of $C_{p}$ upon temperature is
also used for steel [30].
$C_{p}=10^{3}(0.09278+7.454\times 10^{-4}T+12404.0/T^{2})$ (19)
Under normal conditions, the heat generated at a material point is conducted
away at the end of a time step using the heat equation. If special adiabatic
conditions apply (such as in impact problems), the heat is accumulated at a
material point and is not conducted to the surrounding particles. This
localized heating can be used to simulate adiabatic shear band formation.
After the stress state has been determined on the basis of the yield condition
and the associated flow rule, a scalar damage state in each material point can
be calculated using either of two damage models - the Johnson-Cook model [31]
or the Hancock-MacKenzie model [32]. While the Johnson-Cook model has an
explicit dependence on temperature, the Hancock-McKenzie model depends on the
temperature implicitly, via the stress state. Both models depend on the strain
rate to determine the value of the scalar damage parameter.
The damage evolution rule for the Johnson-Cook damage model can be written as
$\dot{D}=\cfrac{\dot{\epsilon_{p}}}{\epsilon_{p}^{f}}~{};~{}~{}\epsilon_{p}^{f}=\left[D_{1}+D_{2}\exp\left(\cfrac{D_{3}}{3}\sigma^{*}\right)\right]\left[1+D_{4}\ln(\dot{\epsilon_{p}}^{*})\right]\left[1+D_{5}T^{*}\right]~{};~{}~{}\sigma^{*}=\cfrac{\text{Tr}(\boldsymbol{\sigma})}{\sigma_{eq}}~{};~{}~{}$
(20)
where $D$ is the damage variable which has a value of 0 for virgin material
and a value of 1 at fracture, $\epsilon_{p}^{f}$ is the fracture strain,
$D_{1},D_{2},D_{3},D_{4},D_{5}$ are constants, $\boldsymbol{\sigma}$ is the
Cauchy stress, and $T^{*}$ is the scaled temperature as in the Johnson-Cook
plasticity model.
The Hancock-MacKenzie damage evolution rule can be written as
$\dot{D}=\cfrac{\dot{\epsilon_{p}}}{\epsilon_{p}^{f}}~{};~{}~{}\epsilon_{p}^{f}=\frac{1.65}{\exp(1.5\sigma^{*})}$
(21)
The determination of whether a particle has failed can be made on the basis of
either or all of the following conditions:
* •
The particle temperature exceeds the melting temperature.
* •
The TEPLA-F fracture condition [33] is satisfied. This condition can be
written as
$(f/f_{c})^{2}+(\epsilon_{p}/\epsilon_{p}^{f})^{2}=1$ (22)
where $f$ is the current porosity, $f_{c}$ is the maximum allowable porosity,
$\epsilon_{p}$ is the current plastic strain, and $\epsilon_{p}^{f}$ is the
plastic strain at fracture.
* •
An alternative to ad-hoc damage criteria is to use the concept of bifurcation
to determine whether a particle has failed or not. Two stability criteria have
been explored in this paper - the Drucker stability postulate [34] and the
loss of hyperbolicity criterion (using the determinant of the acoustic tensor)
[35, 36].
The simplest criterion that can be used is the Drucker stability postulate
[34] which states that time rate of change of the rate of work done by a
material cannot be negative. Therefore, the material is assumed to become
unstable (and a particle fails) when
$\dot{}\boldsymbol{\sigma}:\mathbf{D}^{p}\leq 0$ (23)
Another stability criterion that is less restrictive is the acoustic tensor
criterion which states that the material loses stability if the determinant of
the acoustic tensor changes sign [35, 36]. Determination of the acoustic
tensor requires a search for a normal vector around the material point and is
therefore computationally expensive. A simplification of this criterion is a
check which assumes that the direction of instability lies in the plane of the
maximum and minimum principal stress [37]. In this approach, we assume that
the strain is localized in a band with normal $\mathbf{n}$, and the magnitude
of the velocity difference across the band is $\mathbf{g}$. Then the
bifurcation condition leads to the relation
$R_{ij}g_{j}=0~{};~{}~{}~{}R_{ij}=M_{ikjl}n_{k}n_{l}+M_{ilkj}n_{k}n_{l}-\sigma_{ik}n_{j}n_{k}$
(24)
where $M_{ijkl}$ are the components of the co-rotational tangent modulus
tensor and $\sigma_{ij}$ are the components of the co-rotational stress
tensor. If $\det(R_{ij})\leq 0$, then $g_{j}$ can be arbitrary and there is a
possibility of strain localization. If this condition for loss of
hyperbolicity is met, then a particle deforms in an unstable manner and
failure can be assumed to have occurred at that particle.
## 3 VALIDATION METRICS
The attractiveness of the Taylor impact test arises because of the simplicity
and inexpensiveness of the test. A flat-ended cylinder is fired on a target at
a large enough velocity and the final deformed shape is measured. The drawback
of this test is that intermediate states of the cylinder are difficult to
measure and hence are generally not. The validation metrics that we consider
in this paper are based on the final shape of the cylinder though other
metrics may be considered if measurements of these are made during the course
of an impact test. We note that the Taylor test could also be used to validate
simulations of dynamic fracture though we do not address that issue in this
paper.
There is a large literature on the systematic verification and validation of
computational codes (see Oberkampf et al. [38], Babuska and Oden [39] and
references therein). It has been suggested that validation metrics be
developed that can be used to compare experimental data and simulation
results. The metrics discussed in this paper are intended to be a step in that
direction but they are not intended to be complete or comprehensive.
The most common metric used in the literature is the “calibrated eyeball”
approach or “view-graph norm” (Oberkampf et al. [38]) where a plot of the
simulated deformed configuration is superimposed on the experimental data and
a subjective judgement of accuracy is made. We believe that there is value to
this approach and present all our data in this form. However, we also believe
that more quantitative descriptions of the difference between experiment and
simulations can be obtained and present comparisons using other metrics.
Metrics, sensitivity studies, and determination of experimental variability
are essential. Some quantities of interest are:
1. 1.
Metrics
1. (a)
Regression between profiles
2. (b)
Length change
3. (c)
Diameter change
4. (d)
Volume change
5. (e)
Middle bulge difference
6. (f)
Length of elastic zone
7. (g)
plastic strain
8. (h)
temperature
9. (i)
time of impact
10. (j)
energy conversion at impact
2. 2.
Sensitivity studies
1. (a)
mesh size (quantify discretization errors)
2. (b)
plasticity model parameters
3. (c)
plasticity model
4. (d)
impact velocity
5. (e)
temperature
6. (f)
length and diameter
3. 3.
Variability in experimental data
1. (a)
material
2. (b)
geometry
3. (c)
velocity
4. (d)
temperature
5. (e)
measurement error
## 4 Taylor impact simulations
In this section, we compare the final deformed shapes from simulations of
Taylor impact tests with experimentally obtained data. In cases where images
or profiles of the deformed shapes of the cylinders were available, these were
digitized using a scanner and then imported into XFig (Sutanthavibul et al.
[40]). The scanned images were overlaid with manually digitized lines that
were drawn as accurately as possible after expanding the images to a resultion
of 1024$\times$1260\. The digitized curves were then rotated so that the axes
were aligned with the grid. The XFig coordinates were then scaled to length
units using cues from the digitized images and their axes (if any were
provided). Some small errors (1%-2%) are expected in this procedure. However,
the overall profiles of the cylinders are captured accurately in most cases.
The simulations were run for 150 $\mu$s - 200 $\mu$s depending on the problem.
The simulation times were chosen such that the cylinders bounced off the anvil
and moved away for at least 20 $\mu$s. It was observed at beyond this time,
the deformed shape of the cylinder reamined constant and all elastic strains
and rotations had been recovered.
In the paper by Carrington and Gayler [41] a highly deformed mild steel
specimen has been shown (plate 1, figure 3). To determine if MPM could be used
to simulate such large deformations, we ran a Taylor impact test on the
problem geometry using the Johnson-Cook plasticity model for 4340 steel. The
final deformed shape from Carrington’s paper is compared with our predicted
shape in Figure 2. The initial velocity is $V_{0}=2140ft/s=652.3m/s$, the
initial diameter of the cylinder is $D_{0}=0.5in=12.7mm$, and the initial
length of the cylinder is $L_{0}=0.999in=25.37mm$.
(a) Actual profile (Carrington and Gayler [41]).
(b) Computed vs. actual profile.
Figure 2: Comparison of experimental vs. computed shapes. $L_{0}$ = 25.37 mm ,
$D_{0}$ = 12.7 mm, $V_{0}$ = 652.3 m/s.
In our simulation, the 4340 steel cylinder was impacted against a stiff anvil
using frictional contact. The 4340 steel flows much more readily than the mild
steel used by Carrington and Gayler [41]. The experiment also shows that the
tips of the “mushroom” have broken off. We did not simulate any fracture and
hence we do not see that effect. However, the overall shape of the deformed
specimen suggests that our simulations can provide good qualitative
descriptions of large deformations. To quantify how well our simulations fit
experimental data, we ran a series of Taylor impact tests on various materials
and compared them against experimental data. Some of those results are
presented in this report.
### 4.1 Taylor impact tests on copper
In this section we present the results from Taylor tests on copper specimens
for different initial temperatures and impact velocities. Table 1 shows the
initial dimensions, velocity, and temperature of the specimens (along with the
type of copper used and the source of the data) that we have simulated and
compared with experimental data.
Table 1: Initial data for copper simulations. OFHC = oxygen free high conductivity. ETP = electrolytic tough pitch. Case | Material | Initial | Initial | Initial | Initial | Source
---|---|---|---|---|---|---
| | Length | Diameter | Velocity | Temperature |
| | ($L_{0}$ mm) | ($D_{0}$ mm) | ($V_{0}$ m/s) | ($T_{0}$ K) |
Cu-A | OFHC Cu | 23.47 | 7.62 | 210 | 298 | Wilkins and Guinan [42]
Cu-B | OFHC Cu | 25.4 | 7.62 | 130 | 298 | Johnson and Cook [22]
Cu-C | OFHC Cu | 25.4 | 7.62 | 146 | 298 | Johnson and Cook [22]
Cu-D | OFHC Cu | 25.4 | 7.62 | 190 | 298 | Johnson and Cook [22]
Cu-E | ETP Cu | 30 | 6.00 | 277 | 295 | Gust [43]
Cu-F | ETP Cu | 30 | 6.00 | 188 | 718 | Gust [43]
Cu-G | ETP Cu | 30 | 6.00 | 211 | 727 | Gust [43]
Cu-H | ETP Cu | 30 | 6.00 | 178 | 1235 | Gust [43]
Cu-I | Annealed Cu. | | | | | Zocher et al. [14]
Cu-J | With porosity | | | | | Addessio et al. [44]
#### 4.1.1 Room temperature impact of copper
Comparisons between the computed and experimental profiles of annealed copper
specimen Cu-I are shown in Figure 3. The MTS model predicts the final length
quite accurately (at this is true for other room temperature simulations of
copper). The profile shape is also computed accurately. The Johnson-Cook model
overestimate the final length. However, the difference is small and may be
attributed to material variability.
(a) Johnson-Cook.
(b) Mechanical Threshold Stress.
Figure 3: Comparison of experimental and computed shapes of annealed copper
cylinder Cu-I using the Johnson-Cook and Mechanical Threshold Stress
plasticity models. The axes are shown in cm units.
Simulations of impact case Cu-I with increasing mesh refinement are shown in
Figure 4. The number of MPM particles is doubled with each refinement. We
observe that the solution does not change much as we refine the mesh. However,
this is true only at low temperatures and moderate impact velocities.
Significant mesh dependence is observed at high temperatures where softening
becomes dominant as wee will see in our calculations with 6061-T6 aluminum.
(a) With friction.
(b) Without friction.
Figure 4: Comparison of experimental and computed shapes of 6061T6 aluminum
cylinders using the Johnson-Cook (JC) with increasing mesh refinement. The
axes are in cm.
#### 4.1.2 High temperature impact of copper
At higher temperatures, the response of the three plasticity models is quite
different. Comparisons between the computed and experimental profiles of ETP
copper specimen Cu-F are shown in Figure 5(a), (b), and (c). Those for
specimen Cu-G are shown in Figure 5(d), (e), and (f). If frictional contact at
the impact surface is simulated, the final shapes of the specimens Cu-F and
Cu-G are as shown in Figure 5(g), (h), (i), (j), (k), and (l).
Notice that though both specimens are nominally at the same temperature and
has almost identical impact velocities, the final profile is quite different
even though the final lengths are nearly identical. It is likely that most of
the difference is due the initial conditions with a small contribution from
material variability. This conjecture is partially supported by the fact that
the profiles predicted by the Johnson-Cook model match the experiments quite
well.
We observe that the Johnson-Cook and Steinberg-Guinan models perform well for
specimen Cu-F when friction is not included in the calculation. In the
presence of frictional contact, the predicted profiles deviate significantly
from the experimental profiles in the mushroom region. This indicates that
there is a possibility of inaccurate contact force calculation when friction
is included.
From specimen Cu-G, the slightly higher impact velocity leads to an
underestimation of the final length by the Johnson-Cook model, even though the
mushroom region is predicted accurately. The MTS model overestimates the
length and underestimates the mushroom diameter while the Steinberg-Guinan
model predicts the final length best but fails to predict the mushroom shape.
Once again, frictional contact appears to reduce the accuracy of the
prediction.
(a) JC (Cu-F).
(g) JC (Cu-F) with friction.
(b) MTS (Cu-F).
(h) MTS (Cu-F) with friction.
(c) SCG (Cu-F).
(i) SCG (Cu-F) with friction.
(d) JC (Cu-G).
(j) JC (Cu-G) with friction.
(e) MTS (Cu-G).
(k) MTS (Cu-G) with friction.
(f) SCG (Cu-G).
(l) SCG (Cu-G) with friction.
Figure 5: Comparison of experimental and computed shapes of ETP copper
cylinders using the Johnson-Cook (JC), Mechanical Threshold Stress (MTS), and
Steinberg-Cochran-Guinan (SCG) plasticity models. Specimen C-F has an initial
temperature of 718 K and Cu-G is initially at 727 K. The initial velocities
are 188 m/s and 211 m/s, respectively. The axes are shown in cm units.
#### 4.1.3 Comparisons with FEM
To determine how our MPM simulations compare with FEM simulations we have run
two high temperature ETP copper impact tests using LS-DYNA (with the coupled
structural-thermal option). Figure 6 shows the final deformed shapes for the
two cases from the MPM and FEM simulations using Johnson-Cook plasticity. In
this case frictional contact has been used.
The FEM simulations consistently overestimate the final length of the specimen
though the mushroom diameter is more accurately predicted by FEM. For the case
where no contact friction is applied, MPM predictions are consistently better
than FEM predictions.
(a) MPM (Cu-F)
(b) FEM (Cu-F)
(c) MPM (Cu-G)
(d) FEM (Cu-G)
Figure 6: Comparison of experimental and computed shapes of ETP copper
cylinders using MPM and FEM. The axes are in cm.
### 4.2 Taylor impact tests on 6061-T6 aluminum alloy
In this section we present the results from Taylor tests on 6061-T6 aluminum
specimens for different initial temperatures and impact velocities. We have
chosen to study this material as it is a well characterized face centered
cubic material that has been utilized by Chhabildas et al. [45] for the
validation of high velocity impacts that formed the basis of the second stage
of our validation simulations. Table 2 shows the initial dimensions, velocity,
and temperature of the specimens (along with the type of copper used and the
source of the data) that we have simulated and compared with experimental
data.
Table 2: Initial data for 6061-T6 aluminum simulations. Case | Material | Initial | Initial | Initial | Initial | Source
---|---|---|---|---|---|---
| | Length | Diameter | Velocity | Temperature |
| | ($L_{0}$ mm) | ($D_{0}$ mm) | ($V_{0}$ m/s) | ($T_{0}$ K) |
Al-A | 6061-T6 Al | 23.47 | 7.62 | 373 | 298 | Wilkins and Guinan [42]
Al-B | 6061-T6 Al | 23.47 | 7.62 | 603 | 298 | Wilkins and Guinan [42]
Al-C | 6061-T6 Al | 46.94 | 7.62 | 275 | 298 | Wilkins and Guinan [42]
Al-D | 6061-T6 Al | 46.94 | 7.62 | 484 | 298 | Wilkins and Guinan [42]
Al-E | 6061-T6 Al | 30 | 6.00 | 200 | 295 | Gust [43]
Al-F | 6061-T6 Al | 30 | 6.00 | 358 | 295 | Gust [43]
Al-G | 6061-T6 Al | 30 | 6.00 | 194 | 635 | Gust [43]
Al-H | 6061-T6 Al | 30 | 6.00 | 354 | 655 | Gust [43]
Al-I | 6061-T6 Al | | | | | Addessio et al. [44]
#### 4.2.1 Room temperature impact: 6061-T6 Al
Comparisons between the computed and experimental profiles of 6061T6 aluminum
alloy specimen Al-A are shown in Figure 7(a), (b), and (c). Those for specimen
Al-C are shown in Figure 7(d), (e), and (f). If frictional contact at the
impact surface is simulated, the final shapes of the specimens Al-A and Al-C
are as shown in Figure 7 (g), (h), (i), (j), (k), and (l).
We note that all three models predict essentially identical profiles. The
higher velocity impact of the shorter specimen Al-A is best predicted by the
MTS model as far as final length is concerned. The mushroom width is predicted
better when some friction is included at the anvil-specimen interface. There
is a noticeable amount of curvature under frictional contact. We believe that
this partly due to the contact algorithm that has been used.
The longer specimens have lower impact velocities. However, all three models
predict a final length that is shorter than that observed in experiment. We
believe that is discrepancy is due to material variability. Note the accuracy
with which the profiles are predicted and the noticeably lower curvature of
the mushroom under frictional contact compared to specimen Al-A.
(a) JC (Al-A).
(g) JC (Al-A) Friction.
(b) MTS (Al-A).
(h) MTS (Al-A) Friction.
(c) SCG (Al-A).
(i) SCG (Al-A) Friction.
(d) JC (Al-C).
(j) JC (Al-C) Friction.
(e) MTS (Al-C).
(k) MTS (Al-C) Friction.
(f) SCG (Al-C).
(l) SCG (Al-C) Friction.
Figure 7: Comparison of experimental and computed shapes of 6061T6 aluminum
cylinders using the Johnson-Cook (JC), Mechanical Threshold Stress (MTS), and
Steinberg-Cochran-Guinan (SCG) plasticity models. The figure in the top row
are from simulations without friction while those in the bottom row are with
friction. The axes are shown in cm units.
#### 4.2.2 High temperature impact: 6061-T6 Al
At higher temperatures, the response of the three plasticity models is quite
different. Comparisons between the computed and experimental profiles of
6061T6 aluminum alloy specimens have been performed under conditions of
frictional contact. The final shapes of the specimens Al-G and Al-H are as
shown in Figure 8. If failure simulation is included, the profiles are as
shown in Figures 8(g), (h), (i), (j), (k), and (l).
For the lower impact velocity of specimen Al-G, the Johnson-Cook model
performs the best at predicting both the final length and the mushroom
diameter. Both the MTS and SGC models overestimate the final length and
underestimate the mushroom diameter. The MTS model fares slightly worse than
the SCG model. However, the differences are small enough that they can be
attributed to material variability. Including erosion effects in the
simulation does not affect the result significantly.
At the higher impact velocity represented by specimen Al-H, all models fail to
predict the final length accurately. The Johnson-Cook model comes closest but
overestimates the length and has an excessively deformed mushroom region. The
MTS and SCG models have more reasonably shaped mushroom regions but fail to
predict the final length by almost 100%. The SCG model is slightly better than
the MTS model.
(a) JC (Al-G).
(g) JC (Al-G) with erosion.
(b) MTS (Al-G).
(h) MTS (Al-G) with erosion.
(c) SCG (Al-G).
(i) SCG (Al-G) with erosion.
(d) JC (Al-H).
(j) JC (Al-H) with erosion.
(e) MTS (Al-H).
(k) MTS (Al-H) with erosion.
(f) SCG (Al-H).
(l) SCG (Al-H) with erosion.
Figure 8: Comparison of experimental and computed shapes of 6061T6 aluminum
cylinders using the Johnson-Cook (JC), Mechanical Threshold Stress (MTS), and
Steinberg-Cochran-Guinan (SCG) plasticity models. Specimens Al-G and Al-H are
both initially at 635 K. Al-G has an impact velocity of 194 m/s while Al-H
impacts at 354 m/s. The axes are shown in cm units.
It is possible that the discrepancy that we observe for specimen Al-H is due
to inadequate discretization. Simulations of impact specimen Al-H with
increasing mesh refinement are shown in Figure 9. The number of grid cells in
the plane of the specimen profile has been doubled with each refinement.
If we examine the profiles shown in Figure 9(a), we observe that the cylinder
does appear to shorten with increasing refinement. However, there is
unphysical curling of the end of the specimen. On the other hand, if we
eliminate friction from the calculation, the mushroom appears to increase with
increased refinement while the length decreases. This indicates that there is
some amount of mesh dependence of the solution that is probably due to the
softening behavior of the material.
(a) Al-H with friction.
(b) Al-H without friction.
Figure 9: Comparison of experimental and computed shapes of 6061T6 aluminum
cylinders (Al-H) using the Johnson-Cook (JC) with increasing mesh refinement.
The axes are in cm.
#### 4.2.3 Comparisons with FEM
To determine how our MPM simulations compare with FEM simulations we have run
two high temperature aluminum impact tests using LS-DYNA (with the coupled
structural-thermal option). Figure 10 shows the final deformed shapes for the
two cases from the MPM and FEM simulations using Johnson-Cook plasticity. The
FEM simulations consistently overestimate the final length and underestimate
the mushroom diameter at high temperatures.
(a) MPM (Al-G)
(b) FEM (Al-G)
(c) MPM (Al-H)
(d) FEM (Al-H)
Figure 10: Comparison of experimental and computed shapes of 6061T6 aluminum
cylinders using MPM and FEM. The axes are in cm.
### 4.3 Taylor impact tests on 4340 steel
In this section we present the results from Taylor tests on 4340 steel
specimens for different initial temperatures and impact velocities. Table 3
shows the initial dimensions, velocity, and temperature of the specimens
(along with the type of copper used and the source of the data) that we have
simulated and compared with experimental data. Note that only a few
representative results are shown in this report.
Table 3: Initial data for 4340 steel simulations. Case | Hardness | Initial | Initial | Initial | Initial | Source
---|---|---|---|---|---|---
| | Length | Diameter | Velocity | Temperature |
| | ($L_{0}$ mm) | ($D_{0}$ mm) | ($V_{0}$ m/s) | ($T_{0}$ K) |
St-A | $R_{c}=40$ | 30 | 6.00 | 158 | 295 | Gust [43]
St-B | $R_{c}=40$ | 30 | 6.00 | 232 | 295 | Gust [43]
St-C | $R_{c}=40$ | 30 | 6.00 | 183 | 715 | Gust [43]
St-D | $R_{c}=40$ | 30 | 6.00 | 312 | 725 | Gust [43]
St-E | $R_{c}=40$ | 30 | 6.00 | 136 | 1285 | Gust [43]
St-F | $R_{c}=40$ | 30 | 6.00 | 160 | 1285 | Gust [43]
St-G | $R_{c}=30$ | 25.4 | 7.62 | 208 | 298 | Johnson and Cook [22]
St-H | $R_{c}=30$ | 12.7 | 7.62 | 282 | 298 | Johnson and Cook [22]
St-I | $R_{c}=30$ | 8.1 | 7.62 | 343 | 298 | Johnson and Cook [22]
St-J | | | | | | Addessio et al. [44]
#### 4.3.1 Room temperature impact: steel
Figure 11 shows the simulated profile of case St-G without friction. The
Johnson-Cook model performs quite well in predicting the deformed profile of
the specimen. An almost identical profile is obtained if we incorporate
friction at the impact surface. Similar results are obtained for the other
room temperature specimens.
Figure 11: Comparison of experimental and computed shape of 4340 steel
cylinder (St-G) without friction. The axes are in cm.
#### 4.3.2 High temperature impact: steel
For high temperature impacts tests, the effect of friction is more obvious in
that there is a curling of the edges. Figures 12(a),(b),(c),(d) show the
simulated profiles of cases St-D and St-F with friction. Specimen D is at a
lower temperature than specimen F but the impact velocity of the form is
almost double that of the latter. The Johnson-Cook model predicts the final
length of the St-D accurately but underestimates the final length of St-G.
This indicates that the high temperature behavior of the model is not quite
correct even though the rate dependence is captured well. On the other hand,
the Steinberg-Guinan model fails miserably at predicting the high velocity
response but does well for the low velocity/high temperature response.
Figures 12(e), (f), (g), and (h) show Taylor impact simulations for cases St-D
and St-F with particle erosion. No significant difference can be seen in the
computed profiles when we compare these to the plots in Figure 12, except for
the SCG model for the St-D sample. Erosion and fracture of the mushroom end
does not appear to have a first-order effect on the final length of the impact
specimen.
(a) JC (St-D).
(e) JC (St-D) with erosion.
(b) SCG (St-D).
(f) SCG (St-D) with erosion.
(c) JC (St-F).
(g) JC (St-F) with erosion.
(d) SCG (St-F).
(h) SCG (St-F) with erosion.
Figure 12: Comparison of experimental and computed shapes of 4340 steel
cylinder with friction. St-D is at 725 K and 312 m/s. St-F is at 1285 K and
160 m/s. The axes are in cm.
## 5 CONCLUSION
Lower temperature simulations lead to predicted profiles that are close to
those observed in experiment. This is true over a range of impact velocities.
However, high temperature impacts do not fare so well.
For the copper specimens, the Johnson-Cook and Steinberg-Guinan models perform
better than the MTS model both at low and high temperatures. In future work we
show that this is partially due to the stress integration algorithm used in
these calculations. Also, FEM simulations consistently overestimate the final
length of specimens at high temperatures.
For the aluminum specimens, the three models, Johnson-Cook, MTS and Steinberg-
Guinan, predict accurate final lengths and mushroom diameters at room
temperature. However, at high temperatures all three models deviate from
experiment, especially when the strain rate is increased. This indicates a
coupling of strain rate and temperature that is either not captured by these
models or requires further calibration. Mesh dependence due to softening
appears to be an issue in the MPM simulations. FEM simulations with LS-DYNA
predict profiles (at high temperatures) that are less deformed than those
predicted by MPM suggesting that the constitutive model evaluation is not as
accurate in FEM calculations.
For the steel specimens, both Johnson-Cook and Steinberg-Guinan perform well
at room temperature. However, the Steinberg-Guinan model fails under a
combination of high impact velocities and high temperatures. The Johnson-Cook
model is not very accurate at high temperatures but captures rate-dependent
effects quite well. Failure of the material at the mushroom end does not
appear to affect the final length of the specimen significantly.
## Acknowledgments
This work was supported by the the U.S. Department of Energy through the
Center for the Simulation of Accidental Fires and Explosions, under grant
W-7405-ENG-48.
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|
arxiv-papers
| 2012-01-12T04:53:41 |
2024-09-04T02:49:26.219165
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Biswajit Banerjee",
"submitter": "Biswajit Banerjee",
"url": "https://arxiv.org/abs/1201.2476"
}
|
1201.2501
|
# Nearly Optimal Sparse Fourier Transform
Haitham Hassanieh
MIT Piotr Indyk
MIT Dina Katabi
MIT Eric Price
MIT
({haithamh,indyk,dk,ecprice}@mit.edu)
###### Abstract
We consider the problem of computing the $k$-sparse approximation to the
discrete Fourier transform of an $n$-dimensional signal. We show:
* •
An $O(k\log n)$-time randomized algorithm for the case where the input signal
has at most $k$ non-zero Fourier coefficients, and
* •
An $O(k\log n\log(n/k))$-time randomized algorithm for general input signals.
Both algorithms achieve $o(n\log n)$ time, and thus improve over the Fast
Fourier Transform, for any $k=o(n)$. They are the first known algorithms that
satisfy this property. Also, if one assumes that the Fast Fourier Transform is
optimal, the algorithm for the exactly $k$-sparse case is optimal for any
$k=n^{\Omega(1)}$.
We complement our algorithmic results by showing that any algorithm for
computing the sparse Fourier transform of a general signal must use at least
$\Omega(k\log(n/k)/\log\log n)$ signal samples, even if it is allowed to
perform _adaptive_ sampling.
## 1 Introduction
The discrete Fourier transform (DFT) is one of the most important and widely
used computational tasks. Its applications are broad and include signal
processing, communications, and audio/image/video compression. Hence, fast
algorithms for DFT are highly valuable. Currently, the fastest such algorithm
is the Fast Fourier Transform (FFT), which computes the DFT of an
$n$-dimensional signal in $O(n\log n)$ time. The existence of DFT algorithms
faster than FFT is one of the central questions in the theory of algorithms.
A general algorithm for computing the exact DFT must take time at least
proportional to its output size, i.e., $\Omega(n)$. In many applications,
however, most of the Fourier coefficients of a signal are small or equal to
zero, i.e., the output of the DFT is (approximately) sparse. This is the case
for video signals, where a typical 8x8 block in a video frame has on average 7
non-negligible frequency coefficients (i.e., 89% of the coefficients are
negligible) [CGX96]. Images and audio data are equally sparse. This sparsity
provides the rationale underlying compression schemes such as MPEG and JPEG.
Other sparse signals appear in computational learning theory [KM91, LMN93],
analysis of Boolean functions [KKL88, O’D08], compressed sensing [Don06,
CRT06], multi-scale analysis [DRZ07], similarity search in databases [AFS93],
spectrum sensing for wideband channels [LVS11], and datacenter monitoring
[MNL10].
For sparse signals, the $\Omega(n)$ lower bound for the complexity of DFT no
longer applies. If a signal has a small number $k$ of non-zero Fourier
coefficients – the exactly $k$-sparse case – the output of the Fourier
transform can be represented succinctly using only $k$ coefficients. Hence,
for such signals, one may hope for a DFT algorithm whose runtime is sublinear
in the signal size, $n$. Even for a general $n$-dimensional signal $x$ – the
general case – one can find an algorithm that computes the best k-sparse
approximation of its Fourier transform, $\widehat{x}$, in sublinear time. The
goal of such an algorithm is to compute an approximation vector
$\widehat{x}^{\prime}$ that satisfies the following $\ell_{2}/\ell_{2}$
guarantee:
$\|\widehat{x}-\widehat{x}^{\prime}\|_{2}\leq C\min_{k\text{-sparse
}y}\|\widehat{x}-y\|_{2},$ (1)
where $C$ is some approximation factor and the minimization is over $k$-sparse
signals. We allow the algorithm to be _randomized_ , and only succeed with
constant (say, 2/3) probability.
The past two decades have witnessed significant advances in sublinear sparse
Fourier algorithms. The first such algorithm (for the Hadamard transform)
appeared in [KM91] (building on [GL89]). Since then, several sublinear sparse
Fourier algorithms for complex inputs have been discovered [Man92, GGI+02,
AGS03, GMS05, Iwe10, Aka10, HIKP12b]. These algorithms provide111The algorithm
of [Man92], as stated in the paper, addresses only the exactly $k$-sparse
case. However, it can be extended to the general case using relatively
standard techniques. the guarantee in Equation (1).222All of the above
algorithms, as well as the algorithms in this paper, need to make some
assumption about the precision of the input; otherwise, the right-hand-side of
the expression in Equation (1) contains an additional additive term. See
Preliminaries for more details.
The main value of these algorithms is that they outperform FFT’s runtime for
sparse signals. For very sparse signals, the fastest algorithm is due to
[GMS05] and has $O(k\log^{c}(n)\log(n/k))$ runtime, for some333The paper does
not estimate the exact value of $c$. We estimate that $c\approx 3$. $c>2$.
This algorithm outperforms FFT for any $k$ smaller than $\Theta(n/\log^{a}n)$
for some $a>1$. For less sparse signals, the fastest algorithm is due to
[HIKP12b], and has $O(\sqrt{nk}\log^{3/2}n)$ runtime. This algorithm
outperforms FFT for any $k$ smaller than $\Theta(n/\log n)$.
Despite impressive progress on sparse DFT, the state of the art suffers from
two main limitations:
1. 1.
None of the existing algorithms improves over FFT’s runtime for the whole
range of sparse signals, i.e., $k=o(n)$.
2. 2.
Most of the aforementioned algorithms are quite complex, and suffer from large
“big-Oh” constants (the algorithm of [HIKP12b] is an exception, but has a
running time that is polynomial in $n$).
#### Results.
In this paper, we address these limitations by presenting two new algorithms
for the sparse Fourier transform. We require that the length $n$ of the input
signal is a power of 2. We show:
* •
An $O(k\log n)$-time algorithm for the exactly $k$-sparse case, and
* •
An $O(k\log n\log(n/k))$-time algorithm for the general case.
The key property of both algorithms is their ability to achieve $o(n\log n)$
time, and thus improve over the FFT, for any $k=o(n)$. These algorithms are
the first known algorithms that satisfy this property. Moreover, if one assume
that FFT is optimal and hence the DFT cannot be computed in less than $O(n\log
n)$ time, the algorithm for the exactly $k$-sparse case is optimal444One also
needs to assume that $k$ divides $n$. See Section 5 for more details. as long
as $k=n^{\Omega(1)}$. Under the same assumption, the result for the general
case is at most one $\log\log n$ factor away from the optimal runtime for the
case of “large” sparsity $k=n/\log^{O(1)}n$.
Furthermore, our algorithm for the exactly sparse case (depicted as Algorithm
1 on page 5) is quite simple and has low big-Oh constants. In particular, our
preliminary implementation of a variant of this algorithm is faster than FFTW,
a highly efficient implementation of the FFT, for $n=2^{22}$ and $k\leq
2^{17}$ [HIKP12a]. In contrast, for the same signal size, prior algorithms
were faster than FFTW only for $k\leq 2000$ [HIKP12b].555Note that both
numbers ($k\leq 2^{17}$ and $k\leq 2000$) are for the exactly k-sparse case.
The algorithm in [HIKP12b] can deal with the general case, but the empirical
runtimes are higher.
We complement our algorithmic results by showing that any algorithm that works
for the general case must use at least $\Omega(k\log(n/k)/\log\log n)$ samples
from $x$. The lower bound uses techniques from [PW11], which shows a lower
bound of $\Omega(k\log(n/k))$ for the number of arbitrary linear measurements
needed to compute the $k$-sparse approximation of an $n$-dimensional vector
$\widehat{x}$. In comparison to [PW11], our bound is slightly worse but it
holds even for adaptive sampling, where the algorithm selects the samples
based on the values of the previously sampled coordinates.666Note that if we
allow arbitrary adaptive linear measurements of a vector $\widehat{x}$, then
its $k$-sparse approximation can be computed using only $O(k\log\log(n/k))$
samples [IPW11]. Therefore, our lower bound holds only where the measurements,
although adaptive, are limited to those induced by the Fourier matrix. This is
the case when we want to compute a sparse approximation to $\widehat{x}$ from
samples of $x$. Note that our algorithms are non-adaptive, and thus limited by
the more stringent lower bound of [PW11].
#### Techniques – overview.
We start with an overview of the techniques used in prior works. At a high
level, sparse Fourier algorithms work by binning the Fourier coefficients into
a small number of bins. Since the signal is sparse in the frequency domain,
each bin is likely777One can randomize the positions of the frequencies by
sampling the signal in time domain appropriately [GGI+02, GMS05]. See
Preliminaries for the description. to have only one large coefficient, which
can then be located (to find its position) and estimated (to find its value).
The binning has to be done in sublinear time, and thus these algorithms bin
the Fourier coefficients using an $n$-dimensional filter vector $G$ that is
concentrated both in time and frequency. That is, $G$ is zero except at a
small number of time coordinates, and its Fourier transform $\hat{G}$ is
negligible except at a small fraction (about $1/k$) of the frequency
coordinates, representing the filter’s “pass” region. Each bin essentially
receives only the frequencies in a narrow range corresponding to the pass
region of the (shifted) filter $\hat{G}$, and the pass regions corresponding
to different bins are disjoint. In this paper, we use filters introduced in
[HIKP12b]. Those filters (defined in more detail in Preliminaries) have the
property that the value of $\hat{G}$ is “large” over a constant fraction of
the pass region, referred to as the “super-pass” region. We say that a
coefficient is “isolated” if it falls into a filter’s super-pass region and no
other coefficient falls into filter’s pass region. Since the super-pass region
of our filters is a constant fraction of the pass region, the probability of
isolating a coefficient is constant.
To achieve the stated running times, we need a fast method for locating and
estimating isolated coefficients. Further, our algorithm is iterative, so we
also need a fast method for updating the signal so that identified
coefficients are not considered in future iterations. Below, we describe these
methods in more detail.
#### New techniques – location and estimation.
Our location and estimation methods depends on whether we handle the exactly
sparse case or the general case. In the exactly sparse case, we show how to
estimate the position of an isolated Fourier coefficient using only two
samples of the filtered signal. Specifically, we show that the phase
difference between the two samples is linear in the index of the coefficient,
and hence we can recover the index by estimating the phases. This approach is
inspired by the frequency offset estimation in orthogonal frequency division
multiplexing (OFDM), which is the modulation method used in modern wireless
technologies (see [HT01], Chapter 2).
In order to design an algorithm888We note that although the two-sample
approach employed in our algorithm works in theory only for the exactly
$k$-sparse case, our preliminary experiments show that using a few more
samples to estimate the phase works surprisingly well even for general
signals. for the general case, we employ a different approach. Specifically,
we can use two samples to estimate (with constant probability) individual bits
of the index of an isolated coefficient. Similar approaches have been employed
in prior work. However, in those papers, the index was recovered bit by bit,
and one needed $\Omega(\log\log n)$ samples per bit to recover all bits
correctly with constant probability. In contrast, in this paper we recover the
index one block of bits at a time, where each block consists of $O(\log\log
n)$ bits. This approach is inspired by the fast sparse recovery algorithm of
[GLPS10]. Applying this idea in our context, however, requires new techniques.
The reason is that, unlike in [GLPS10], we do not have the freedom of using
arbitrary “linear measurements” of the vector $\hat{x}$, and we can only use
the measurements induced by the Fourier transform.999In particular, the method
of [GLPS10] uses measurements corresponding to a random error correcting code.
As a result, the extension from “bit recovery” to “block recovery” is the most
technically involved part of the algorithm. Section 4.1 contains further
intuition on this part.
#### New techniques – updating the signal.
The aforementioned techniques recover the position and the value of any
isolated coefficient. However, during each filtering step, each coefficient
becomes isolated only with constant probability. Therefore, the filtering
process needs to be repeated to ensure that each coefficient is correctly
identified. In [HIKP12b], the algorithm simply performs the filtering $O(\log
n)$ times and uses the median estimator to identify each coefficient with high
probability. This, however, would lead to a running time of $O(k\log^{2}n)$ in
the $k$-sparse case, since each filtering step takes $k\log n$ time.
One could reduce the filtering time by subtracting the identified coefficients
from the signal. In this way, the number of non-zero coefficients would be
reduced by a constant factor after each iteration, so the cost of the first
iteration would dominate the total running time. Unfortunately, subtracting
the recovered coefficients from the signal is a computationally costly
operation, corresponding to a so-called non-uniform DFT (see [GST08] for
details). Its cost would override any potential savings.
In this paper, we introduce a different approach: instead of subtracting the
identified coefficients from the signal, we subtract them directly from the
bins obtained by filtering the signal. The latter operation can be done in
time linear in the number of subtracted coefficients, since each of them
“falls” into only one bin. Hence, the computational costs of each iteration
can be decomposed into two terms, corresponding to filtering the original
signal and subtracting the coefficients. For the exactly sparse case these
terms are as follows:
* •
The cost of filtering the original signal is $O(B\log n)$, where $B$ is the
number of bins. $B$ is set to $O(k^{\prime})$, where $k^{\prime}$ is the the
number of yet-unidentified coefficients. Thus, initially $B$ is equal to
$O(k)$, but its value decreases by a constant factor after each iteration.
* •
The cost of subtracting the identified coefficients from the bins is $O(k)$.
Since the number of iterations is $O(\log k)$, and the cost of filtering is
dominated by the first iteration, the total running time is $O(k\log n)$ for
the exactly sparse case.
For the general case, we need to set $k^{\prime}$ and $B$ more carefully to
obtain the desired running time. The cost of each iterative step is multiplied
by the number of filtering steps needed to compute the location of the
coefficients, which is $\Theta(\log(n/B))$. If we set $B=\Theta(k^{\prime})$,
this would be $\Theta(\log n)$ in most iterations, giving a
$\Theta(k\log^{2}n)$ running time. This is too slow when $k$ is close to $n$.
We avoid this by decreasing $B$ more slowly and $k^{\prime}$ more quickly. In
the $r$-th iteration, we set $B=k/\text{poly}(r)$. This allows the total
number of bins to remain $O(k)$ while keeping $\log(n/B)$ small—at most
$O(\log\log k)$ more than $\log(n/k)$. Then, by having $k^{\prime}$ decrease
according to $k^{\prime}=k/r^{\Theta(r)}$ rather than $k/2^{\Theta(r)}$, we
decrease the number of rounds to $O(\log k/\log\log k)$. Some careful analysis
shows that this counteracts the $\log\log k$ loss in the $\log(n/B)$ term,
achieving the desired $O(k\log n\log(n/k))$ running time.
#### Organization of the paper.
In Section 2, we give notation and definitions used throughout the paper.
Sections 3 and 4 give our algorithm in the exactly $k$-sparse and the general
case, respectively. Section 5 gives the reduction to the exactly $k$-sparse
case from a $k$-dimensional DFT. Section 6 gives the sample complexity lower
bound for the general case. Section 7 describes how to efficiently implement
our filters. Finally, Section 8 discusses open problems arising from this
work.
## 2 Preliminaries
This section introduces the notation, assumptions, and definitions used in the
rest of this paper.
#### Notation.
We use $[n]$ to denote the set $\\{1,\dotsc,n\\}$, and define
$\omega=e^{-2\pi\mathbf{i}/n}$ to be an $n$th root of unity. For any complex
number $a$, we use $\phi(a)\in[0,2\pi]$ to denote the phase of $a$. For a
complex number $a$ and a real positive number $b$, the expression $a\pm b$
denotes a complex number $a^{\prime}$ such that $\left|a-a^{\prime}\right|\leq
b$. For a vector $x\in{\mathbb{C}}^{n}$, its support is denoted by
$\operatorname{supp}(x)\subset[n]$. We use $\left\lVert x\right\rVert_{0}$ to
denote $\left|\operatorname{supp}(x)\right|$, the number of non-zero
coordinates of $x$. Its Fourier spectrum is denoted by $\widehat{x}$, with
$\widehat{x}_{i}=\frac{1}{\sqrt{n}}\sum_{j\in[n]}\omega^{ij}x_{j}.$
For a vector of length $n$, indices should be interpreted modulo $n$, so
$x_{-i}=x_{n-i}$. This allows us to define _convolution_
$(x*y)_{i}=\sum_{j\in[n]}x_{j}y_{i-j}$
and the _coordinate-wise product_ $(x\cdot y)_{i}=x_{i}y_{i}$, so
$\widehat{x\cdot y}=\widehat{x}*\widehat{y}$.
When $i\in{\mathbb{Z}}$ is an index into an $n$-dimensional vector, sometimes
we use $\left|i\right|$ to denote $\min_{j\equiv i\pmod{n}}\left|j\right|$.
#### Definitions.
The paper uses two tools introduced in previous papers: (pseudorandom)
spectrum permutation [GGI+02, GMS05, GST08] and flat filtering windows
[HIKP12b].
###### Definition 2.1.
Suppose $\sigma^{-1}$ exists mod $n$. We define the _permutation_
$P_{\sigma,a,b}$ by
$(P_{\sigma,a,b}x)_{i}=x_{\sigma(i-a)}\omega^{\sigma bi}.$
We also define $\pi_{\sigma,b}(i)=\sigma(i-b)\bmod n$.
###### Claim 2.2.
$\widehat{P_{\sigma,a,b}x}_{\pi_{\sigma,b}(i)}=\widehat{x}_{i}\omega^{a\sigma
i}$.
###### Proof.
$\displaystyle\widehat{P_{\sigma,a,b}x}_{\sigma(i-b)}$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{j\in[n]}\omega^{\sigma(i-b)j}(P_{\sigma,a,b}x)_{j}$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{j\in[n]}\omega^{\sigma(i-b)j}x_{\sigma(j-a)}\omega^{\sigma
bj}$ $\displaystyle=\omega^{a\sigma
i}\frac{1}{\sqrt{n}}\sum_{j\in[n]}\omega^{i\sigma(j-a)}x_{\sigma(j-a)}$
$\displaystyle=\widehat{x}_{i}\omega^{a\sigma i}.$
∎
###### Definition 2.3.
We say that
$(G,\widehat{G^{\prime}})=(G_{B,\delta,\alpha},\widehat{G^{\prime}}_{B,\delta,\alpha})\in\mathbb{R}^{n}\times\mathbb{R}^{n}$
is a _flat window function_ with parameters $B\geq 1$, $\delta>0$, and
$\alpha>0$ if
$\left|\operatorname{supp}(G)\right|=O(\frac{B}{\alpha}\log(n/\delta))$ and
$\widehat{G^{\prime}}$ satisfies
* •
$\widehat{G^{\prime}}_{i}=1$ for $\left|i\right|\leq(1-\alpha)n/(2B)$
* •
$\widehat{G^{\prime}}_{i}=0$ for $\left|i\right|\geq n/(2B)$
* •
$\widehat{G^{\prime}}_{i}\in[0,1]$ for all $i$
* •
$\left\lVert\widehat{G^{\prime}}-\widehat{G}\right\rVert_{\infty}<\delta$.
The above notion corresponds to the
$(1/(2B),(1-\alpha)/(2B),\delta,O(B/\alpha\log(n/\delta))$-flat window
function in [HIKP12b]. In Section 7 we give efficient constructions of such
window functions, where $G$ can be computed in
$O(\frac{B}{\alpha}\log(n/\delta))$ time and for each $i$,
$\widehat{G^{\prime}}_{i}$ can be computed in $O(\log(n/\delta))$ time. Of
course, for $i\notin[(1-\alpha)n/(2B),n/(2B)]$,
$\widehat{G^{\prime}}_{i}\in\\{0,1\\}$ can be computed in $O(1)$ time.
The fact that $\widehat{G^{\prime}}_{i}$ takes $\omega(1)$ time to compute for
$i\in[(1-\alpha)n/(2B),n/(2B)]$ will add some complexity to our algorithm and
analysis. We will need to ensure that we rarely need to compute such values. A
practical implementation might find it more convenient to precompute the
window functions in a preprocessing stage, rather than compute them on the
fly.
We use the following lemma from [HIKP12b]:
###### Lemma 2.4 (Lemma 3.6 of [HIKP12b]).
If $j\neq 0$, $n$ is a power of two, and $\sigma$ is a uniformly random odd
number in $[n]$, then $\Pr[\sigma j\in[-C,C]\pmod{n}]\leq 4C/n$.
#### Assumptions.
Through the paper, we require that $n$, the dimension of all vectors, is an
integer power of $2$. We also make the following assumptions about the
precision of the vectors $\widehat{x}$:
* •
For the exactly $k$-sparse case, we assume that
$\widehat{x}_{i}\in\\{-L,\ldots,L\\}$ for some precision parameter $L$. To
simplify the bounds, we assume that $L=n^{O(1)}$; otherwise the $\log n$ term
in the running time bound is replaced by $\log L$.
* •
For the general case, we only achieve Equation (1) if
$\left\lVert\widehat{x}\right\rVert_{2}\leq n^{O(1)}\cdot\min_{k\text{-sparse
}y}\left\lVert\widehat{x}-y\right\rVert_{2}$. In general, for any parameter
$\delta>0$ we can add $\delta\left\lVert\widehat{x}\right\rVert_{2}$ to the
right hand side of Equation (1) and run in time $O(k\log(n/k)\log(n/\delta))$.
## 3 Algorithm for the exactly sparse case
In this section we assume $\widehat{x}_{i}\in\\{-L,\dotsc,L\\}$, where $L\leq
n^{c}$ for some constant $c>0$, and $\widehat{x}$ is $k$-sparse. We choose
$\delta=1/(4n^{2}L)$. The algorithm (NoiselessSparseFFT) is described as
Algorithm 1. The algorithm has three functions:
* •
HashToBins. This permutes the spectrum of $\widehat{x-z}$ with
$P_{\sigma,a,b}$, then “hashes” to $B$ bins. The guarantee will be described
in Lemma 3.3.
* •
NoiselessSparseFFTInner. Given time-domain access to $x$ and a sparse vector
$\widehat{z}$ such that $\widehat{x-z}$ is $k^{\prime}$-sparse, this function
finds “most” of $\widehat{x-z}$.
* •
NoiselessSparseFFT. This iterates NoiselessSparseFFTInner until it finds
$\widehat{x}$ exactly.
Algorithm 1 Exact $k$-sparse recovery
procedure HashToBins($x$, $\widehat{z}$, $P_{\sigma,a,b}$, $B$, $\delta$,
$\alpha$)
Compute $\widehat{y}_{jn/B}$ for $j\in[B]$, where
$y=G_{B,\alpha,\delta}\cdot(P_{\sigma,a,b}x)$
Compute
$\widehat{y^{\prime}}_{jn/B}=\widehat{y}_{jn/B}-(\widehat{G^{\prime}_{B,\alpha,\delta}}*\widehat{P_{\sigma,a,b}z})_{jn/B}$
for $j\in[B]$
return $\widehat{u}$ given by $\widehat{u}_{j}=\widehat{y^{\prime}}_{jn/B}$.
end procedure
procedure NoiselessSparseFFTInner($x$, $k^{\prime}$, $\widehat{z}$, $\alpha$)
Let $B$ = $k^{\prime}/\beta$, for sufficiently small constant $\beta$.
Let $\delta=1/(4n^{2}L)$.
Choose $\sigma$ uniformly at random from the set of odd numbers in $[n]$.
Choose $b$ uniformly at random from $[n]$.
$\widehat{u}\leftarrow\textsc{HashToBins}(x,\widehat{z},P_{\sigma,0,b},B,\delta,\alpha)$.
$\widehat{u}^{\prime}\leftarrow\textsc{HashToBins}(x,\widehat{z},P_{\sigma,1,b},B,\delta,\alpha)$.
$\widehat{w}\leftarrow 0$.
Compute $J=\\{j:|\widehat{u}_{j}|>1/2\\}$.
for $j\in J$ do
$a\leftarrow\widehat{u}_{j}/\widehat{u}^{\prime}_{j}$.
$i\leftarrow\sigma^{-1}(\text{round}(\phi(a)\frac{n}{2\pi}))\bmod n$.
$\triangleright$ $\phi(a)$ denotes the phase of $a$.
$v\leftarrow\text{round}(\widehat{u}_{j})$.
$\widehat{w}_{i}\leftarrow v$.
end for
return $\widehat{w}$
end procedure
procedure NoiselessSparseFFT($x$, $k$)
$\widehat{z}\leftarrow 0$
for $t\in 0,1,\dotsc,\log k$ do
$k_{t}\leftarrow k/2^{t}$, $\alpha_{t}\leftarrow\Theta(2^{-t})$.
$\widehat{z}\leftarrow\widehat{z}+\textsc{NoiselessSparseFFTInner}(x,k_{t},\widehat{z},\alpha_{t})$.
end for
return $\widehat{z}$
end procedure
We analyze the algorithm “bottom-up”, starting from the lower-level
procedures.
#### Analysis of NoiselessSparseFFTInner and HashToBins.
For any execution of NoiselessSparseFFTInner, define the support
$S=\operatorname{supp}(\widehat{x}-\widehat{z})$. Recall that
$\pi_{\sigma,b}(i)=\sigma(i-b)\bmod n$. Define
$h_{\sigma,b}(i)=\text{round}(\pi_{\sigma,b}(i)B/n)$ and
$o_{\sigma,b}(i)=\pi_{\sigma,b}(i)-h_{\sigma,b}(i)n/B$. Note that therefore
$\left|o_{\sigma,b}(i)\right|\leq n/(2B)$. We will refer to $h_{\sigma,b}(i)$
as the “bin” that the frequency $i$ is mapped into, and $o_{\sigma,b}(i)$ as
the “offset”. For any $i\in S$ define two types of events associated with $i$
and $S$ and defined over the probability space induced by $\sigma$ and $b$:
* •
“Collision” event $E_{coll}(i)$: holds iff $h_{\sigma,b}(i)\in
h_{\sigma,b}(S\setminus\\{i\\})$, and
* •
“Large offset” event $E_{off}(i)$: holds iff
$|o_{\sigma,b}(i)|\geq(1-\alpha)n/(2B)$.
###### Claim 3.1.
For any $i\in S$, the event $E_{coll}(i)$ holds with probability at most
$4|S|/B$.
###### Proof.
Consider distinct $i,j\in S$. By Lemma 2.4,
$\displaystyle\Pr[h_{\sigma,b}(i)=h_{\sigma,b}(j)]$
$\displaystyle\leq\Pr[\pi_{\sigma,b}(i)-\pi_{\sigma,b}(j)\bmod
n\in[-n/B,n/B]]$ $\displaystyle=\Pr[\sigma(i-j)\bmod n\in[-n/B,n/B]]$
$\displaystyle\leq 4/B.$
By a union bound over $j\in S$, $\Pr[E_{coll}(i)]\leq 4\left|S\right|/B$. ∎
###### Claim 3.2.
For any $i\in S$, the event $E_{off}(i)$ holds with probability at most
$\alpha$.
###### Proof.
Note that $o_{\sigma,b}(i)\equiv\pi_{\sigma,b}(i)\equiv\sigma(i-b)\pmod{n/B}$.
For any odd $\sigma$ and any $l\in[n/B]$, we have that
$\Pr_{b}[\sigma(i-b)\equiv l\pmod{n/B}]=B/n$. Since only $\alpha n/B$ offsets
$o_{\sigma,b}(i)$ cause $E_{off}(i)$, the claim follows. ∎
###### Lemma 3.3.
Suppose $B$ divides $n$. The output $\widehat{u}$ of HashToBins satisfies
$\widehat{u}_{j}=\sum_{h_{\sigma,b}(i)=j}\widehat{(x-z)}_{i}\widehat{(G^{\prime}_{B,\delta,\alpha})}_{-o_{\sigma,b}(i)}\omega^{a\sigma
i}\pm\delta\left\lVert\widehat{x}\right\rVert_{1}.$
Let $\zeta=\left|\\{i\in\operatorname{supp}(\widehat{z})\mid
E_{off}(i)\\}\right|$. The running time of HashToBins is
$O(\frac{B}{\alpha}\log(n/\delta)+\left\lVert\widehat{z}\right\rVert_{0}+\zeta\log(n/\delta))$.
###### Proof.
Define the flat window functions $G=G_{B,\delta,\alpha}$ and
$\widehat{G^{\prime}}=\widehat{G^{\prime}}_{B,\delta,\alpha}$. We have
$\displaystyle\widehat{y}$ $\displaystyle=\widehat{G\cdot
P_{\sigma,a,b}}x=\widehat{G}*\widehat{P_{\sigma,a,b}x}$
$\displaystyle\widehat{y^{\prime}}$
$\displaystyle=\widehat{G^{\prime}}*\widehat{P_{\sigma,a,b}(x-z)}+(\widehat{G}-\widehat{G^{\prime}})*\widehat{P_{\sigma,a,b}x}$
By Claim 2.2, the coordinates of $\widehat{P_{\sigma,a,b}x}$ and $\widehat{x}$
have the same magnitudes, just different ordering and phase. Therefore
$\left\lVert(\widehat{G}-\widehat{G^{\prime}})*\widehat{P_{\sigma,a,b}x}\right\rVert_{\infty}\leq\left\lVert\widehat{G}-\widehat{G^{\prime}}\right\rVert_{\infty}\left\lVert\widehat{P_{\sigma,a,b}x}\right\rVert_{1}\leq\delta\left\lVert\widehat{x}\right\rVert_{1}$
and hence
$\displaystyle\widehat{u}_{j}=\widehat{y^{\prime}}_{jn/B}$
$\displaystyle=\sum_{\left|l\right|<n/(2B)}\widehat{G^{\prime}}_{-l}\widehat{(P_{\sigma,a,b}(x-z))}_{jn/B+l}\pm\delta\left\lVert\widehat{x}\right\rVert_{1}$
$\displaystyle=\sum_{\left|\pi_{\sigma,b}(i)-jn/B\right|<n/(2B)}\widehat{G^{\prime}}_{jn/B-\pi_{\sigma,b}(i)}\widehat{(P_{\sigma,a,b}(x-z))}_{\pi_{\sigma,b}(i)}\pm\delta\left\lVert\widehat{x}\right\rVert_{1}$
$\displaystyle=\sum_{h_{\sigma,b}(i)=j}\widehat{G^{\prime}}_{-o_{\sigma,b}(i)}\widehat{(x-z)}_{i}\omega^{a\sigma
i}\pm\delta\left\lVert\widehat{x}\right\rVert_{1}$
as desired.
We can compute HashToBins via the following method:
1. 1.
Compute $y$ with $\left\lVert
y\right\rVert_{0}=O(\frac{B}{\alpha}\log(n/\delta))$ in
$O(\frac{B}{\alpha}\log(n/\delta))$ time.
2. 2.
Compute $v\in{\mathbb{C}}^{B}$ given by $v_{i}=\sum_{j}y_{i+jB}$.
3. 3.
Because $B$ divides $n$, by the definition of the Fourier transform (see also
Claim 3.7 of [HIKP12b]) we have $\widehat{y}_{jn/B}=\widehat{v}_{j}$ for all
$j$. Hence we can compute it with a $B$-dimensional FFT in $O(B\log B)$ time.
4. 4.
For each coordinate $i\in\operatorname{supp}(\widehat{z})$, decrease
$\widehat{y}_{\frac{n}{B}h_{\sigma,b}(i)}$ by
$\widehat{G^{\prime}}_{-o_{\sigma,b}(i)}\widehat{z}_{i}\omega^{a\sigma i}$.
This takes $O(\left\lVert\widehat{z}\right\rVert_{0}+\zeta\log(n/\delta))$
time, since computing $\widehat{G^{\prime}}_{-o_{\sigma,b}(i)}$ takes
$O(\log(n/\delta))$ time if $E_{off}(i)$ holds and $O(1)$ otherwise.
∎
###### Lemma 3.4.
Consider any $i\in S$ such that neither $E_{coll}(i)$ nor $E_{off}(i)$ holds.
Let $j=h_{\sigma,b}(i)$. Then
$\text{round}(\phi(\widehat{u}_{j}/\widehat{u}^{\prime}_{j}))\frac{n}{2\pi})=\sigma
i\pmod{n},$ $\text{round}(\widehat{u}_{j})=\widehat{x}_{i}-\widehat{z}_{i},$
and $j\in J$.
###### Proof.
We know that $\left\lVert\widehat{x}\right\rVert_{1}\leq
k\left\lVert\widehat{x}\right\rVert_{\infty}\leq kL<nL$. Then by Lemma 3.3 and
$E_{coll}(i)$ not holding,
$\widehat{u}_{j}=\widehat{(x-z)}_{i}\widehat{G^{\prime}}_{-o_{\sigma,b}(i)}\pm\delta
nL.$
Because $E_{off}(i)$ does not hold,
$\widehat{G^{\prime}}_{-o_{\sigma,b}(i)}=1$, so
$\displaystyle\widehat{u}_{j}=\widehat{(x-z)}_{i}\pm\delta nL.$ (2)
Similarly,
$\widehat{u}_{j}^{\prime}=\widehat{(x-z)}_{i}\omega^{\sigma i}\pm\delta nL$
Then because $\delta nL<1\leq\left|\widehat{(x-z)}_{i}\right|$, the phase is
$\phi(\widehat{u}_{j})=0\pm\sin^{-1}(\delta nL)=0\pm 2\delta nL$
and $\phi(\widehat{u}_{j}^{\prime})=-\sigma i\frac{2\pi}{n}\pm 2\delta nL$.
Thus $\phi(\widehat{u}_{j}/\widehat{u}^{\prime}_{j})=\sigma i\frac{2\pi}{n}\pm
4\delta nL=\sigma i\frac{2\pi}{n}\pm 1/n$ by the choice of $\delta$. Therefore
$\text{round}(\phi(\widehat{u}_{j}/\widehat{u}^{\prime}_{j})\frac{n}{2\pi})=\sigma
i\pmod{n}.$
Also, by Equation (2),
$\text{round}(\widehat{u}_{j})=\widehat{x}_{i}-\widehat{z}_{i}$. Finally,
$\left|\text{round}(\widehat{u}_{j})\right|=\left|\widehat{x}_{i}-\widehat{z}_{i}\right|\geq
1$, so $|\widehat{u}_{j}|\geq 1/2$. Thus $j\in J$. ∎
For each invocation of NoiselessSparseFFTInner, let $P$ be the the set of all
pairs $(i,v)$ for which the command $\widehat{w}_{i}\leftarrow v$ was
executed. Claims 3.1 and 3.2 and Lemma 3.4 together guarantee that for each
$i\in S$ the probability that $P$ does not contain the pair
$(i,(\widehat{x}-\widehat{z})_{i})$ is at most $4|S|/B+\alpha$. We complement
this observation with the following claim.
###### Claim 3.5.
For any $j\in J$ we have $j\in h_{\sigma,b}(S)$. Therefore, $|J|=|P|\leq|S|$.
###### Proof.
Consider any $j\notin h_{\sigma,b}(S)$. From Equation (2) in the proof of
Lemma 3.4 it follows that $|\widehat{u}_{j}|\leq\delta nL<1/2$. ∎
###### Lemma 3.6.
Consider an execution of NoiselessSparseFFTInner, and let
$S=\operatorname{supp}(\widehat{x}-\widehat{z})$. If $|S|\leq k^{\prime}$,
then
$E[\|\widehat{x}-\widehat{z}-\widehat{w}\|_{0}]\leq 8(\beta+\alpha)|S|.$
###### Proof.
Let $e$ denote the number of coordinates $i\in S$ for which either
$E_{coll}(i)$ or $E_{off}(i)$ holds. Each such coordinate might not appear in
$P$ with the correct value, leading to an incorrect value of
$\widehat{w}_{i}$. In fact, it might result in an arbitrary pair
$(i^{\prime},v^{\prime})$ being added to $P$, which in turn could lead to an
incorrect value of $\widehat{w}_{i^{\prime}}$. By Claim 3.5 these are the only
ways that $\widehat{w}$ can be assigned an incorrect value. Thus we have
$\|\widehat{x}-\widehat{z}-\widehat{w}\|_{0}\leq 2e.$
Since $E[e]\leq(4|S|/B+\alpha)|S|\leq(4\beta+\alpha)|S|$, the lemma follows. ∎
#### Analysis of NoiselessSparseFFT.
Consider the $t$th iteration of the procedure, and define
$S_{t}=\operatorname{supp}(\widehat{x}-\widehat{z})$ where $\widehat{z}$
denotes the value of the variable at the beginning of loop. Note that
$|S_{0}|=|\operatorname{supp}(\widehat{x})|\leq k$.
We also define an indicator variable $I_{t}$ which is equal to $0$ iff
$|S_{t}|/|S_{t-1}|\leq 1/8$. If $I_{t}=1$ we say the the $t$th iteration was
not successful. Let $\gamma=8\cdot 8(\beta+\alpha)$. From Lemma 3.6 it follows
that $\Pr[I_{t}=1\mid|S_{t-1}|\leq k/2^{t-1}]\leq\gamma$. From Claim 3.5 it
follows that even if the $t$th iteration is not successful, then
$|S_{t}|/|S_{t-1}|\leq 2$.
For any $t\geq 1$, define an event $E(t)$ that occurs iff
$\sum_{i=1}^{t}I_{i}\geq t/2$. Observe that if none of the events $E(1)\ldots
E(t)$ holds then $|S_{t}|\leq k/2^{t}$.
###### Lemma 3.7.
Let $E=E(1)\cup\ldots\cup E(\lambda)$ for $\lambda=1+\log k$. Assume that
$(4\gamma)^{1/2}<1/4$. Then $\Pr[E]\leq 1/3$.
###### Proof.
Let $t^{\prime}=\lceil t/2\rceil$. We have
$\Pr[E(t)]\leq\binom{t}{t^{\prime}}\gamma^{t^{\prime}}\leq
2^{t}\gamma^{t^{\prime}}\leq(4\gamma)^{t/2}$
Therefore
$\Pr[E]\leq\sum_{t}\Pr[E(t)]\leq\frac{(4\gamma)^{1/2}}{1-(4\gamma)^{1/2}}\leq
1/4\cdot 4/3=1/3.$
∎
###### Theorem 3.8.
Suppose $\widehat{x}$ is $k$-sparse with entries from $\\{-L,\dotsc,L\\}$ for
some known $L=n^{O(1)}$. Then the algorithm NoiselessSparseFFT runs in
expected $O(k\log n)$ time and returns the correct vector $\widehat{x}$ with
probability at least $2/3$.
###### Proof.
The correctness follows from Lemma 3.7. The running time is dominated by
$O(\log k)$ executions of HashToBins.
Assuming a correct run, in every round $t$ we have
$\left\lVert\widehat{z}\right\rVert_{0}\leq\left\lVert\widehat{x}\right\rVert_{0}+\left|S_{t}\right|\leq
k+k/2^{t}\leq 2k.$
Therefore
$\operatorname{\mathbb{E}}[\left|\\{i\in\operatorname{supp}(z)\mid
E_{off}(i)\\}\right|]\leq\alpha\left\lVert\widehat{z}\right\rVert_{0}\leq
2\alpha k,$
so the expected running time of each execution of HashToBins is
$O(\frac{B}{\alpha}\log(n/\delta)+k+\alpha
k\log(n/\delta))=O(\frac{B}{\alpha}\log n+k+\alpha k\log n)$. Setting
$\alpha=\Theta(2^{-t/2})$ and $\beta=\Theta(1)$, the expected running time in
round $t$ is $O(2^{-t/2}k\log n+k+2^{-t/2}k\log n)$. Therefore the total
expected running time is $O(k\log n)$. ∎
## 4 Algorithm for the general case
This section shows how to achieve Equation (1) for $C=1+\epsilon$. Pseudocode
is in Algorithm 1 and 2.
### 4.1 Intuition
Let $S$ denote the “heavy” $O(k/\epsilon)$ coordinates of $\widehat{x}$. The
overarching algorithm SparseFFT works by first “locating” a set $L$ containing
most of $S$, then “estimating” $\widehat{x}_{L}$ to get $\widehat{z}$. It then
repeats on $\widehat{x-z}$. We will show that each heavy coordinate has a
large constant probability of both being in $L$ and being estimated well. As a
result, $\widehat{x-z}$ is probably nearly $k/4$-sparse, so we can run the
next iteration with $k\to k/4$. The later iterations then run faster and
achieve a higher success probability, so the total running time is dominated
by the time in the first iteration and the total error probability is bounded
by a constant.
In the rest of this intuition, we will discuss the first iteration of
SparseFFT with simplified constants. In this iteration, hashes are to
$B=O(k/\epsilon)$ bins and, with $3/4$ probability, we get $\widehat{z}$ so
$\widehat{x-z}$ is nearly $k/4$-sparse. The actual algorithm will involve a
parameter $\alpha$ in each iteration, roughly guaranteeing that with
$1-\sqrt{\alpha}$ probability, we get $\widehat{z}$ so $\widehat{x-z}$ is
nearly $\sqrt{\alpha}k$-sparse; the formal guarantee will be given by Lemma
4.8. For this intuition we only consider the first iteration where $\alpha$ is
a constant.
#### Location.
As in the noiseless case, to locate the “heavy” coordinates we consider the
“bins” computed by HashToBins with $P_{\sigma,a,b}$. This roughly corresponds
to first permuting the coordinates according to the (almost) pairwise
independent permutation $P_{\sigma,a,b}$, partitioning the coordinates into
$B=O(k/\epsilon)$ “bins” of $n/B$ consecutive indices, and observing the sum
of values in each bin. We get that each heavy coordinate $i$ has a large
constant probability that the following two events occur: no other heavy
coordinate lies in the same bin, and only a small (i.e., $O(\epsilon/k)$)
fraction of the mass from non-heavy coordinates lies in the same bin. For such
a “well-hashed” coordinate $i$, we would like to find its location
$\tau=\pi_{\sigma,b}(i)=\sigma(i-b)$ among the $\epsilon n/k<n/k$ consecutive
values that hash to the same bin. Let
$\displaystyle\theta^{*}_{j}=\frac{2\pi}{n}(j+\sigma b)\pmod{2\pi}.$ (3)
so $\theta^{*}_{\tau}=\frac{2\pi}{n}\sigma i$. In the noiseless case, we
showed that the difference in phase in the bin using $P_{\sigma,0,b}$ and
using $P_{\sigma,1,b}$ is $\theta^{*}_{\tau}$ plus a negligible $O(\delta)$
term. With noise this may not be true; however, we can say for any
$\beta\in[n]$ that the difference in phase between using $P_{\sigma,a,b}$ and
$P_{\sigma,a+\beta,b}$, as a distribution over uniformly random $a\in[n]$, is
$\beta\theta^{*}_{\tau}+\nu$ with (for example)
$\operatorname{\mathbb{E}}[\nu^{2}]=1/100$ (all operations on phases modulo
$2\pi$). We can only hope to get a constant number of bits from such a
“measurement”. So our task is to find $\tau$ within a region $Q$ of size $n/k$
using $O(\log(n/k))$ “measurements” of this form.
One method for doing so would be to simply do measurements with random
$\beta\in[n]$. Then each measurement lies within $\pi/4$ of
$\beta\theta^{*}_{\tau}$ with at least
$1-\frac{\operatorname{\mathbb{E}}[\nu^{2}]}{\pi^{2}/16}>3/4$ probability. On
the other hand, for $j\neq\tau$ and as a distribution over $\beta$,
$\beta(\theta^{*}_{\tau}-\theta^{*}_{j})$ is roughly uniformly distributed
around the circle. As a result, each measurement is probably more than $\pi/4$
away from $\beta\theta^{*}_{j}$. Hence $O(\log(n/k))$ repetitions suffice to
distinguish among the $n/k$ possibilities for $\tau$. However, while the
number of measurements is small, it is not clear how to decode in polylog
rather than $\Omega(n/k)$ time.
To solve this, we instead do a $t$-ary search on the location for
$t=\Theta(\log n)$. At each of $O(\log_{t}(n/k))$ levels, we split our current
candidate region $Q$ into $t$ consecutive subregions $Q_{1},\dotsc,Q_{t}$,
each of size $w$. Now, rather than choosing $\beta\in[n]$, we choose
$\beta\in[\frac{n}{16w},\frac{n}{8w}]$. By the upper bound on $\beta$, for
each $q\in[t]$ the values $\\{\beta\theta^{*}_{j}\mid j\in Q_{q}\\}$ all lie
within $\beta w\frac{2\pi}{n}\leq\pi/4$ of each other on the circle. On the
other hand, if $\left|j-\tau\right|>16w$, then
$\beta(\theta^{*}_{\tau}-\theta^{*}_{j})$ will still be roughly uniformly
distributed about the circle. As a result, we can check a single candidate
element $e_{q}$ from each subregion: if $e_{q}$ is in the same subregion as
$\tau$, each measurement usually agrees in phase; but if $e_{q}$ is more than
$16$ subregions away, each measurement usually disagrees in phase. Hence with
$O(\log t)$ measurements, we can locate $\tau$ to within $O(1)$ consecutive
subregions with failure probability $1/t^{\Theta(1)}$. The decoding time is
$O(t\log t)$.
This primitive LocateInner lets us narrow down the candidate region for $\tau$
to a subregion that is a $t^{\prime}=\Omega(t)$ factor smaller. By repeating
LocateInner $\log_{t^{\prime}}(n/k)$ times, LocateSignal can find $\tau$
precisely. The number of measurements is then $O(\log
t\log_{t}(n/k))=O(\log(n/k))$ and the decoding time is $O(t\log
t\log_{t}(n/k))=O(\log(n/k)\log n)$. Furthermore, the “measurements” (which
are actually calls to HashToBins) are non-adaptive, so we can perform them in
parallel for all $O(k/\epsilon)$ bins, with $O(\log(n/\delta))$ average time
per measurement. This gives $O(k\log(n/k)\log(n/\delta))$ total time for
LocateSignal.
This lets us locate every heavy and “well-hashed” coordinate with
$1/t^{\Theta(1)}=o(1)$ failure probability, so every heavy coordinate is
located with arbitrarily high constant probability.
#### Estimation.
By contrast, estimation is fairly simple. As in Algorithm 1, we can estimate
$\widehat{(x-z)}_{i}$ as $\widehat{u}_{h_{\sigma,b}(i)}$, where $\widehat{u}$
is the output of HashToBins. Unlike in Algorithm 1, we now have noise that may
cause a single such estimate to be poor even if $i$ is “well-hashed”. However,
we can show that for a random permutation $P_{\sigma,a,b}$ the estimate is
“good” with constant probability. EstimateValues takes the median of
$R_{est}=O(\log\frac{1}{\epsilon})$ such samples, getting a good estimate with
$1-\epsilon/64$ probability. Given a candidate set $L$ of size $k/\epsilon$,
with $7/8$ probability at most $k/8$ of the coordinates are badly estimated.
On the other hand, with $7/8$ probability, at least $7k/8$ of the heavy
coordinates are both located and well estimated. This suffices to show that,
with $3/4$ probability, the largest $k$ elements $J$ of our estimate
$\widehat{w}$ contains good estimates of $3k/4$ large coordinates, so
$\widehat{x-z-w_{J}}$ is close to $k/4$-sparse.
procedure SparseFFT($x$, $k$, $\epsilon$, $\delta$)
$R\leftarrow O(\log k/\log\log k)$ as in Theorem 4.9.
$\widehat{z}^{(1)}\leftarrow 0$
for $r\in[R]$ do
Choose $B_{r},k_{r},\alpha_{r}$ as in Theorem 4.9.
$R_{est}\leftarrow O(\log(\frac{B_{r}}{\alpha_{r}k_{r}}))$ as in Lemma 4.8.
$L_{r}\leftarrow\textsc{LocateSignal}(x,\widehat{z}^{(r)},B_{r},\alpha_{r},\delta)$
$\widehat{z}^{(r+1)}\leftarrow\widehat{z}^{(r)}+\textsc{EstimateValues}(x,\widehat{z}^{(r)},3k_{r},L_{r},B_{r},\delta,R_{est})$.
end for
return $\widehat{z}^{(R+1)}$
end procedure
procedure EstimateValues($x$, $\widehat{z}$, $k^{\prime}$, $L$, $B$, $\delta$,
$R_{est}$)
for $r\in[R_{est}]$ do
Choose $a_{r},b_{r}\in[n]$ uniformly at random.
Choose $\sigma_{r}$ uniformly at random from the set of odd numbers in $[n]$.
$\widehat{u}^{(r)}\leftarrow\textsc{HashToBins}(x,\widehat{z},P_{\sigma,a_{r},b},B,\delta)$.
end for
$\widehat{w}\leftarrow 0$
for $i\in L$ do
$\widehat{w}_{i}\leftarrow\operatorname*{median}_{r}\widehat{u}_{h_{\sigma,b}(i)}^{(r)}\omega^{-a_{r}\sigma
i}$.$\triangleright$ Separate median in real and imaginary axes.
end for
$J\leftarrow\operatorname*{arg\,max}_{\left|J\right|=k^{\prime}}\left\lVert\widehat{w}_{J}\right\rVert_{2}$.
return $\widehat{w}_{J}$
end procedure
Algorithm 1 $k$-sparse recovery for general signals, part 1/2.
procedure LocateSignal($x$, $\widehat{z}$, $B$, $\alpha$, $\delta$)
Choose uniformly at random $\sigma,b\in[n]$ with $\sigma$ odd.
Initialize $l^{(1)}_{i}=(i-1)n/B$ for $i\in[B]$.
Let $w_{0}=n/B,t=\log n,t^{\prime}=t/4,D_{max}=\log_{t^{\prime}}(w_{0}+1)$.
Let $R_{loc}=\Theta(\log_{1/\alpha}(t/\alpha))$ per Lemma 4.5.
for $D\in[D_{max}]$ do
$l^{(D+1)}\leftarrow\textsc{LocateInner}(x,\widehat{z},B,\delta,\alpha,\sigma,\beta,l^{(D)},w_{0}/(t^{\prime})^{D-1},t,R_{loc})$
end for
$L\leftarrow\\{\pi_{\sigma,b}^{-1}(l^{(D_{max}+1)}_{j})\mid j\in[B]\\}$
return $L$
end procedure
$\triangleright$ $\delta,\alpha$ parameters for $G$, $G^{\prime}$
$\triangleright$ $(l_{1},l_{1}+w),\dotsc,(l_{B},l_{B}+w)$ the plausible
regions.
$\triangleright$ $B\approx k/\epsilon$ the number of bins
$\triangleright$ $t\approx\log n$ the number of regions to split into.
$\triangleright$ $R_{loc}\approx\log t=\log\log n$ the number of rounds to run
procedure LocateInner($x$, $\widehat{z}$, $B$, $\delta$, $\alpha$, $\sigma$,
$b$, $l$, $w$, $t$, $R_{loc}$)
Let $s=\Theta(\alpha^{1/3})$.
Let $v_{j,q}=0$ for $(j,q)\in[B]\times[t]$.
for $r\in[R_{loc}]$ do
Choose $a\in[n]$ uniformly at random.
Choose $\beta\in\\{\frac{snt}{4w},\dotsc,\frac{snt}{2w}\\}$ uniformly at
random.
$\widehat{u}\leftarrow\textsc{HashToBins}(x,\widehat{z},P_{\sigma,a,b},B,\delta,\alpha)$.
$\widehat{u}^{\prime}\leftarrow\textsc{HashToBins}(x,\widehat{z},P_{\sigma,a+\beta,b},B,\delta,\alpha)$.
for $j\in[B]$ do
$c_{j}\leftarrow\phi(\widehat{u}_{j}/\widehat{u}^{\prime}_{j})$
for $q\in[t]$ do
$m_{j,q}\leftarrow l_{j}+\frac{q-1/2}{t}w$
$\theta_{j,q}\leftarrow\frac{2\pi(m_{j,q}+\sigma b)}{n}\bmod 2\pi$
if
$\min(\left|\beta\theta_{j,q}-c_{j}\right|,2\pi-\left|\beta\theta_{j,q}-c_{j}\right|)<s\pi$
then
$v_{j,q}\leftarrow v_{j,q}+1$
end if
end for
end for
end for
for $j\in[B]$ do
$Q^{*}\leftarrow\\{q\in[t]\mid v_{j,q}>R_{loc}/2\\}$
if $Q^{*}\neq\emptyset$ then
$l_{j}^{\prime}\leftarrow\min_{q\in Q^{*}}l_{j}+\frac{q-1}{t}w$
else
$l_{j}^{\prime}\leftarrow\perp$
end if
end for
return $l^{\prime}$
end procedure
Algorithm 2 $k$-sparse recovery for general signals, part 2/2.
### 4.2 Formal definitions
As in the noiseless case, we define $\pi_{\sigma,b}(i)=\sigma(i-b)\bmod n$,
$h_{\sigma,b}(i)=\text{round}(\pi_{\sigma,b}(i)B/n)$ and
$o_{\sigma,b}(i)=\pi_{\sigma,b}(i)-h_{\sigma,b}(i)n/B$. We say
$h_{\sigma,b}(i)$ is the “bin” that frequency $i$ is mapped into, and
$o_{\sigma,b}(i)$ is the “offset”. We define
$h_{\sigma,b}^{-1}(j)=\\{i\in[n]\mid h_{\sigma,b}(i)=j\\}$.
Define
$\operatorname{Err}(x,k)=\min_{k\text{-sparse\ }y}\left\lVert
x-y\right\rVert_{2}.$
In each iteration of SparseFFT, define
$\widehat{x}^{\prime}=\widehat{x}-\widehat{z}$, and let
$\displaystyle\rho^{2}$
$\displaystyle=\operatorname{Err}^{2}(\widehat{x^{\prime}},k)+\delta^{2}n\left\lVert\widehat{x}\right\rVert_{1}^{2}$
$\displaystyle\mu^{2}$ $\displaystyle=\epsilon\rho^{2}/k$ $\displaystyle S$
$\displaystyle=\\{i\in[n]\mid|\widehat{x^{\prime}}_{i}|^{2}\geq\mu^{2}\\}$
Then $\left|S\right|\leq(1+1/\epsilon)k=O(k/\epsilon)$ and
$\left\lVert\widehat{x^{\prime}}-\widehat{x^{\prime}}_{S}\right\rVert_{2}^{2}\leq(1+\epsilon)\rho^{2}$.
We will show that each $i\in S$ is found by LocateSignal with probability
$1-O(\alpha)$, when $B=\Omega(\frac{k}{\alpha\epsilon})$.
For any $i\in S$ define three types of events associated with $i$ and $S$ and
defined over the probability space induced by $\sigma$ and $b$:
* •
“Collision” event $E_{coll}(i)$: holds iff $h_{\sigma,b}(i)\in
h_{\sigma,b}(S\setminus\\{i\\})$;
* •
“Large offset” event $E_{off}(i)$: holds iff
$|o_{\sigma,b}(i)|\geq(1-\alpha)n/(2B)$; and
* •
“Large noise” event $E_{noise}(i)$: holds iff
$\left\lVert\widehat{x^{\prime}}_{h_{\sigma,b}^{-1}(h_{\sigma,b}(i))\setminus
S}\right\rVert_{2}^{2}\geq\operatorname{Err}^{2}(\widehat{x^{\prime}},k)/(\alpha
B)$.
By Claims 3.1 and 3.2, $\Pr[E_{coll}(i)]\leq 4\left|S\right|/B=O(\alpha)$ and
$\Pr[E_{off}(i)]\leq 2\alpha$ for any $i\in S$.
###### Claim 4.1.
For any $i\in S$, $\Pr[E_{noise}(i)]\leq 4\alpha$.
###### Proof.
For each $j\neq i$, $\Pr[h_{\sigma,b}(j)=h_{\sigma,b}(i)]\leq\Pr[\left|\sigma
j-\sigma i\right|<n/B]\leq 4/B$ by Lemma 2.4. Then
$\operatorname{\mathbb{E}}[\left\lVert\widehat{x^{\prime}}_{h_{\sigma,b}^{-1}(h_{\sigma,b}(i))\setminus
S}\right\rVert_{2}^{2}]\leq 4\left\lVert\widehat{x^{\prime}}_{[n]\setminus
S}\right\rVert_{2}^{2}/B$
The result follows by Markov’s inequality. ∎
We will show for $i\in S$ that if none of $E_{coll}(i),E_{off}(i)$, and
$E_{noise}(i)$ hold then SparseFFTInner recovers $\widehat{x}^{\prime}_{i}$
with $1-O(\alpha)$ probability.
###### Lemma 4.2.
Let $a\in[n]$ uniformly at random, $B$ divide $n$, and the other parameters be
arbitrary in
$\widehat{u}=\textsc{HashToBins}(x,\widehat{z},P_{\sigma,a,b},B,\delta,\alpha).$
Then for any $i\in[n]$ with $j=h_{\sigma,b}(i)$ and none of $E_{coll}(i)$,
$E_{off}(i)$, or $E_{noise}(i)$ holding,
$\operatorname{\mathbb{E}}[\left|\widehat{u}_{j}-\widehat{x^{\prime}}_{i}\omega^{a\sigma
i}\right|^{2}]\leq 2\frac{\rho^{2}}{\alpha B}$
###### Proof.
Let $\widehat{G^{\prime}}=\widehat{G^{\prime}}_{B,\delta,\alpha}$. Let
$T=h_{\sigma,b}^{-1}(j)\setminus\\{i\\}$. We have that $T\cap S=\\{\\}$ and
$\widehat{G^{\prime}}_{-o_{\sigma,b}(i)}=1$. By Lemma 3.3,
$\displaystyle\widehat{u}_{j}-\widehat{x^{\prime}}_{i}\omega^{a\sigma i}$
$\displaystyle=\sum_{i^{\prime}\in
T}\widehat{G^{\prime}}_{-o_{\sigma}(i^{\prime})}\widehat{x^{\prime}}_{i^{\prime}}\omega^{a\sigma
i^{\prime}}\pm\delta\left\lVert\widehat{x}\right\rVert_{1}.$
Because the $\sigma i^{\prime}$ are distinct for $i^{\prime}\in T$, we have by
Parseval’s theorem
$\displaystyle\operatorname{\mathbb{E}}_{a}\left|\sum_{i^{\prime}\in
T}\widehat{G^{\prime}}_{-o_{\sigma}(i^{\prime})}\widehat{x^{\prime}}_{i^{\prime}}\omega^{a\sigma
i^{\prime}}\right|^{2}$ $\displaystyle=\sum_{i^{\prime}\in
T}(\widehat{G^{\prime}}_{-o_{\sigma}(i^{\prime})}\widehat{x^{\prime}}_{i^{\prime}})^{2}\leq\left\lVert\widehat{x^{\prime}_{T}}\right\rVert_{2}^{2}$
Since $\left|X+Y\right|^{2}\leq 2\left|X\right|^{2}+2\left|Y\right|^{2}$ for
any $X,Y$, we get
$\displaystyle\operatorname{\mathbb{E}}_{a}[\left|\widehat{u}_{j}-\widehat{x^{\prime}}_{i}\omega^{a\sigma
i}\right|^{2}]$ $\displaystyle\leq 2\left\lVert
x^{\prime}_{T}\right\rVert_{2}^{2}+2\delta^{2}\left\lVert\widehat{x}\right\rVert_{1}^{2}$
$\displaystyle\leq 2\operatorname{Err}^{2}(\widehat{x^{\prime}},k)/(\alpha
B)+2\delta^{2}\left\lVert\widehat{x}\right\rVert_{1}^{2}$ $\displaystyle\leq
2\rho^{2}/(\alpha B).$
∎
### 4.3 Properties of LocateSignal
In our intuition, we made a claim that if $\beta\in[n/(16w),n/(8w)]$ uniformly
at random, and $i>16w$, then $\frac{2\pi}{n}\beta i$ is “roughly uniformly
distributed about the circle” and hence not concentrated in any small region.
This is clear if $\beta$ is chosen as a random real number; it is less clear
in our setting where $\beta$ is a random integer in this range. We now prove a
lemma that formalizes this claim.
###### Lemma 4.3.
Let $T\subset[m]$ consist of $t$ consecutive integers, and suppose $\beta\in
T$ uniformly at random. Then for any $i\in[n]$ and set $S\subset[n]$ of $l$
consecutive integers,
$\Pr[\beta i\bmod n\in S]\leq\left\lceil im/n\right\rceil(1+\left\lfloor
l/i\right\rfloor)/t\leq\frac{1}{t}+\frac{im}{nt}+\frac{lm}{nt}+\frac{l}{it}$
###### Proof.
Note that any interval of length $l$ can cover at most $1+\left\lfloor
l/i\right\rfloor$ elements of any arithmetic sequence of common difference
$i$. Then $\\{\beta i\mid\beta\in T\\}\subset[im]$ is such a sequence, and
there are at most $\left\lceil im/n\right\rceil$ intervals $an+S$ overlapping
this sequence. Hence at most $\left\lceil im/n\right\rceil(1+\left\lfloor
l/i\right\rfloor)$ of the $\beta\in[m]$ have $\beta i\bmod n\in S$. Hence
$\Pr[\beta i\bmod n\in S]\leq\left\lceil im/n\right\rceil(1+\left\lfloor
l/i\right\rfloor)/t.$
∎
###### Lemma 4.4.
Let $i\in S$. Suppose none of $E_{coll}(i),E_{off}(i)$, and $E_{noise}(i)$
hold, and let $j=h_{\sigma,b}(i)$. Consider any run of LocateInner with
$\pi_{\sigma,b}(i)\in[l_{j},l_{j}+w]$ . Let $f>0$ be a parameter such that
$B=\frac{Ck}{\alpha f\epsilon}.$
for $C$ larger than some fixed constant. Then
$\pi_{\sigma,b}(i)\in[l^{\prime}_{j},l^{\prime}_{j}+4w/t]$ with probability at
least $1-tf^{\Omega(R_{loc})}$,
###### Proof.
Let $\tau=\pi_{\sigma,b}(i)\equiv\sigma(i-b)\pmod{n}$, and for any $j\in[n]$
define
$\theta^{*}_{j}=\frac{2\pi}{n}(j+\sigma b)\pmod{2\pi}$
so $\theta^{*}_{\tau}=\frac{2\pi}{n}\sigma i$. Let $g=\Theta(f^{1/3})$, and
$C^{\prime}=\frac{B\alpha\epsilon}{k}=\Theta(1/g^{3})$.
To get the result, we divide $[l_{j},l_{j}+w]$ into $t$ “regions”,
$Q_{q}=[l_{j}+\frac{q-1}{t}w,l_{j}+\frac{q}{t}w]$ for $q\in[t]$. We will first
show that in each round $r$, $c_{j}$ is close to $\beta\theta^{*}_{\tau}$ with
$1-g$ probability. This will imply that $Q_{q}$ gets a “vote,” meaning
$v_{j,q}$ increases, with $1-g$ probability for the $q^{\prime}$ with $\tau\in
Q_{q^{\prime}}$. It will also imply that $v_{j,q}$ increases with only $g$
probability when $\left|q-q^{\prime}\right|>3$. Then $R_{loc}$ rounds will
suffice to separate the two with $1-f^{-\Omega(R_{loc})}$ probability. We get
that with $1-tf^{-\Omega(R_{loc})}$ probability, the recovered $Q^{*}$ has
$\left|q-q^{\prime}\right|\leq 3$ for all $q\in Q^{*}$. If we take the minimum
$q\in Q^{*}$ and the next three subregions, we find $\tau$ to within $4$
regions, or $4w/t$ locations, as desired.
In any round $r$, define $\widehat{u}=\widehat{u}^{(r)}$ and $a=a_{r}$. We
have by Lemma 4.2 and that $i\in S$ that
$\displaystyle\operatorname{\mathbb{E}}[\left|\widehat{u}_{j}-\omega^{a\sigma
i}\widehat{x^{\prime}}_{i}\right|^{2}]$ $\displaystyle\leq
2\frac{\rho^{2}}{\alpha B}=\frac{2k}{B\alpha\epsilon}\mu^{2}$
$\displaystyle=\frac{2}{C^{\prime}}\mu^{2}\leq\frac{2}{C^{\prime}}|\widehat{x^{\prime}}_{i}|^{2}.$
Note that $\phi(\omega^{a\sigma i})=-a\theta^{*}_{\tau}$. Thus for any $p>0$,
with probability $1-p$ we have
$\displaystyle\left|\widehat{u}_{j}-\omega^{a\sigma
i}\widehat{x^{\prime}}_{i}\right|$
$\displaystyle\leq\sqrt{\frac{2}{C^{\prime}p}}\left|\widehat{x^{\prime}}_{i}\right|$
$\displaystyle\left\lVert\phi(\widehat{u}_{j})-(\phi(\widehat{x^{\prime}}_{i})-a\theta^{*}_{\tau})\right\rVert_{\bigcirc}$
$\displaystyle\leq\sin^{-1}(\sqrt{\frac{2}{C^{\prime}p}})$
where $\left\lVert
x-y\right\rVert_{\bigcirc}=\min_{\gamma\in{\mathbb{Z}}}\left|x-y+2\pi\gamma\right|$
denotes the “circular distance” between $x$ and $y$. The analogous fact holds
for $\phi(\widehat{u^{\prime}}_{j})$ relative to
$\phi(\widehat{x^{\prime}}_{i})-(a+\beta)\theta^{*}_{\tau}$. Therefore with at
least $1-2p$ probability,
$\displaystyle\left\lVert c_{j}-\beta\theta^{*}_{\tau}\right\rVert_{\bigcirc}$
$\displaystyle=\left\lVert\phi(\widehat{u}_{j})-\phi(\widehat{u^{\prime}}_{j})-\beta\theta^{*}_{\tau}\right\rVert_{\bigcirc}$
$\displaystyle=\bigg{\|}\left(\phi(\widehat{u}_{j})-(\phi(\widehat{x^{\prime}}_{i})-a\theta^{*}_{\tau})\right)-\left(\phi(\widehat{u^{\prime}}_{j})-(\phi(\widehat{x^{\prime}}_{i})-(a+\beta)\theta^{*}_{\tau})\right)\bigg{\|}_{\bigcirc}$
$\displaystyle\leq\left\lVert\phi(\widehat{u}_{j})-(\phi(\widehat{x^{\prime}}_{i})-a\theta^{*}_{\tau})\right\rVert_{\bigcirc}+\left\lVert\phi(\widehat{u^{\prime}}_{j})-(\phi(\widehat{x^{\prime}}_{i})-(a+\beta)\theta^{*}_{\tau})\right\rVert_{\bigcirc}$
$\displaystyle\leq 2\sin^{-1}(\sqrt{\frac{2}{C^{\prime}p}})$
by the triangle inequality. Thus for any $s=\Theta(g)$ and $p=\Theta(g)$, we
can set $C^{\prime}=\frac{2}{p\sin^{2}(s\pi/4)}=\Theta(1/g^{3})$ so that
$\displaystyle\left\lVert
c_{j}-\beta\theta^{*}_{\tau}\right\rVert_{\bigcirc}<s\pi/2$ (4)
with probability at least $1-2p$.
Equation (4) shows that $c_{j}$ is a good estimate for $i$ with good
probability. We will now show that this means the approprate “region”
$Q_{q^{\prime}}$ gets a “vote” with “large” probability.
For the $q^{\prime}$ with
$\tau\in[l_{j}+\frac{q^{\prime}-1}{t}w,l_{j}+\frac{q^{\prime}}{t}w]$, we have
that $m_{j,q^{\prime}}=l_{j}+\frac{q^{\prime}-1/2}{t}w$ satisfies
$\left|\tau-m_{j,q^{\prime}}\right|\leq\frac{w}{2t}$
so
$\left|\theta^{*}_{\tau}-\theta_{j,q^{\prime}}\right|\leq\frac{2\pi}{n}\frac{w}{2t}.$
Hence by Equation (4), the triangle inequality, and the choice of
$B\leq\frac{snt}{2w}$,
$\displaystyle\left\lVert
c_{j}-\beta\theta_{j,q^{\prime}}\right\rVert_{\bigcirc}$
$\displaystyle\leq\left\lVert
c_{j}-\beta\theta^{*}_{\tau}\right\rVert_{\bigcirc}+\left\lVert\beta\theta^{*}_{\tau}-\beta\theta_{j,q^{\prime}}\right\rVert_{\bigcirc}$
$\displaystyle<\frac{s\pi}{2}+\frac{\beta\pi w}{nt}$
$\displaystyle\leq\frac{s\pi}{2}+\frac{s\pi}{2}$ $\displaystyle=s\pi.$
Thus, $v_{j,q^{\prime}}$ will increase in each round with probability at least
$1-2p$.
Now, consider $q$ with $\left|q-q^{\prime}\right|>3$. Then $\left|\tau-
m_{j,q}\right|\geq\frac{7w}{2t}$, and (from the definition of
$\beta>\frac{snt}{4w}$) we have
$\displaystyle\beta\left|\tau-m_{j,q}\right|\geq\frac{7sn}{8}>\frac{3sn}{4}.$
(5)
We now consider two cases. First, suppose that $\left|\tau-
m_{j,q}\right|\leq\frac{w}{st}$. In this case, from the definition of $\beta$
it follows that
$\beta\left|\tau-m_{j,q}\right|\leq n/2.$
Together with Equation (5) this implies
$\Pr[\beta(\tau-m_{j,q})\bmod n\in[-3sn/4,3sn/4]]=0.$
On the other hand, suppose that $\left|\tau-m_{j,q}\right|>\frac{w}{st}$. In
this case, we use Lemma 4.3 with parameters $l=3sn/2$, $m=\frac{snt}{2w}$,
$t=\frac{snt}{4w}$, $i=(\tau-m_{j,q})$ and $n=n$, to conclude that
$\displaystyle\Pr[\beta(\tau-m_{j,q})\bmod n\in[-3sn/4,3sn/4]]$
$\displaystyle\leq\frac{4w}{snt}+2\frac{\left|\tau-
m_{j,q}\right|}{n}+3s+\frac{3sn}{2}\frac{st}{w}\frac{4w}{snt}$
$\displaystyle\leq\frac{4w}{snt}+\frac{2w}{n}+9s$
$\displaystyle<\frac{6}{sB}+9s$ $\displaystyle<10s$
where we used that $\left|i\right|\leq w\leq n/B$, the assumption
$\frac{w}{st}<|i|$, $t\geq 1$, $s<1$, and that $s^{2}>6/B$ (because
$s=\Theta(g)$ and $B=\omega(1/g^{3})$).
Thus in either case, with probability at least $1-10s$ we have
$\displaystyle\left\lVert\beta\theta_{j,q}-\beta\theta^{*}_{\tau}\right\rVert_{\bigcirc}=\left\lVert\frac{2\pi\beta(m_{j,q}-\tau)}{n}\right\rVert_{\bigcirc}>\frac{2\pi}{n}\frac{3sn}{4}=\frac{3}{2}s\pi$
for any $q$ with $\left|q-q^{\prime}\right|>3$. Therefore we have
$\left\lVert
c_{j}-\beta\theta_{j,q}\right\rVert_{\bigcirc}\geq\left\lVert\beta\theta_{j,q}-\beta\theta^{*}_{\tau}\right\rVert_{\bigcirc}-\left\lVert
c_{j}-\beta\theta^{*}_{\tau}\right\rVert_{\bigcirc}>s\pi$
with probability at least $1-10s-2p$, and $v_{j,q}$ is not incremented.
To summarize: in each round, $v_{j,q^{\prime}}$ is incremented with
probability at least $1-2p$ and $v_{j,q}$ is incremented with probability at
most $10s+2p$ for $\left|q-q^{\prime}\right|>3$. The probabilities
corresponding to different rounds are independent.
Set $s=g/20$ and $p=g/4$. Then $v_{j,q^{\prime}}$ is incremented with
probability at least $1-g$ and $v_{j,q}$ is incremented with probability less
than $g$. Then after $R_{loc}$ rounds, if $\left|q-q^{\prime}\right|>3$,
$\Pr[v_{j,q}>R_{loc}/2]\leq\binom{R_{loc}}{R_{loc}/2}g^{R_{loc}/2}\leq(4g)^{R_{loc}/2}=f^{\Omega(R_{loc})}$
for $g=f^{1/3}/4$. Similarly,
$\Pr[v_{j,q^{\prime}}<R_{loc}/2]\leq f^{\Omega(R_{loc})}.$
Hence with probability at least $1-tf^{\Omega(R_{loc})}$ we have
$q^{\prime}\in Q^{*}$ and $\left|q-q^{\prime}\right|\leq 3$ for all $q\in
Q^{*}$. But then $\tau-l^{\prime}_{j}\in[0,4w/t]$ as desired.
Because
$\operatorname{\mathbb{E}}[\left|\\{i\in\operatorname{supp}(\widehat{z})\mid
E_{off}(i)\\}\right|]=\alpha\left\lVert\widehat{z}\right\rVert_{0}$, the
expected running time is
$O(R_{loc}Bt+R_{loc}\frac{B}{\alpha}\log(n/\delta)+R_{loc}\left\lVert\widehat{z}\right\rVert_{0}(1+\alpha\log(n/\delta)))$.
∎
###### Lemma 4.5.
Suppose $B=\frac{Ck}{\alpha^{2}\epsilon}$ for $C$ larger than some fixed
constant. The procedure LocateSignal returns a set $L$ of size
$\left|L\right|\leq B$ such that for any $i\in S$, $\Pr[i\in L]\geq
1-O(\alpha)$. Moreover the procedure runs in expected time
$O((\frac{B}{\alpha}\log(n/\delta)+\left\lVert\widehat{z}\right\rVert_{0}(1+\alpha\log(n/\delta)))\log(n/B)).$
###### Proof.
Consider any $i\in S$ such that none of $E_{coll}(i),E_{off}(i)$, and
$E_{noise}(i)$ hold, as happens with probability $1-O(\alpha)$.
Set $t=\log n,t^{\prime}=t/4$ and $R_{loc}=O(\log_{1/\alpha}(t/\alpha))$. Let
$w_{0}=n/B$ and $w_{D}=w_{0}/(t^{\prime})^{D-1}$, so $w_{D_{max}+1}<1$ for
$D_{max}=\log_{t^{\prime}}(w_{0}+1)<t$. In each round $D$, Lemma 4.4 implies
that if $\pi_{\sigma,b}(i)\in[l^{(D)}_{j},l^{(D)}_{j}+w_{D}]$ then
$\pi_{\sigma,b}(i)\in[l^{(D+1)}_{j},l^{(D+1)}_{j}+w_{D+1}]$ with probability
at least $1-\alpha^{\Omega(R_{loc})}=1-\alpha/t$. By a union bound, with
probability at least $1-\alpha$ we have
$\pi_{\sigma,b}(i)\in[l^{(D_{max}+1)}_{j},l^{(D_{max}+1)}_{j}+w_{D_{max}+1}]=\\{l^{(D_{max}+1)}_{j}\\}$.
Thus $i=\pi_{\sigma,b}^{-1}(l^{(D_{max}+1)}_{j})\in L$.
Since $R_{loc}D_{max}=O(\log_{1/\alpha}(t/\alpha)\log_{t}(n/B))=O(\log(n/B))$,
the running time is
$\displaystyle
O(D_{max}(R_{loc}\frac{B}{\alpha}\log(n/\delta)+R_{loc}\left\lVert\widehat{z}\right\rVert_{0}(1+\alpha\log(n/\delta))))$
$\displaystyle={}$ $\displaystyle
O((\frac{B}{\alpha}\log(n/\delta)+\left\lVert\widehat{z}\right\rVert_{0}(1+\alpha\log(n/\delta)))\log(n/B)).$
∎
### 4.4 Properties of EstimateValues
###### Lemma 4.6.
For any $i\in L$,
$\Pr[\left|\widehat{w}_{i}-\widehat{x^{\prime}}_{i}\right|^{2}>\mu^{2}]<e^{-\Omega(R_{est})}$
if $B>\frac{Ck}{\alpha\epsilon}$ for some constant $C$.
###### Proof.
Define $e_{r}=\widehat{u}_{j}^{(r)}\omega^{-a_{r}\sigma
i}-\widehat{x^{\prime}}_{i}$ in each round $r$. Suppose none of
$E_{coll}^{(r)}(i),E_{off}^{(r)}(i)$, and $E_{noise}^{(r)}(i)$ hold, as
happens with probability $1-O(\alpha)$. Then by Lemma 4.2,
$\displaystyle\operatorname{\mathbb{E}}_{a_{r}}[\left|e_{r}\right|^{2}]$
$\displaystyle\leq 2\frac{\rho^{2}}{\alpha B}=\frac{2k}{\alpha\epsilon
B}\mu^{2}<\frac{2}{C}\mu^{2}$
Hence with $3/4-O(\alpha)>5/8$ probability in total,
$\left|e_{r}\right|^{2}<\frac{8}{C}\mu^{2}<\mu^{2}/2$
for sufficiently large $C$. Then with probability at least
$1-e^{-\Omega(R_{est})}$, both of the following occur:
$\displaystyle\left|\operatorname*{median}_{r}\text{real}(e_{r})\right|^{2}$
$\displaystyle<\mu^{2}/2$
$\displaystyle\left|\operatorname*{median}_{r}\text{imag}(e_{r})\right|^{2}$
$\displaystyle<\mu^{2}/2.$
If this is the case, then
$\left|\operatorname*{median}_{r}e_{r}\right|^{2}<\mu^{2}$. Since
$\widehat{w}_{i}=\widehat{x^{\prime}}_{i}+\operatorname*{median}e_{r}$, the
result follows. ∎
###### Lemma 4.7.
Let $R_{est}\geq C\log\frac{B}{\gamma fk}$ for some constant $C$ and
parameters $\gamma,f>0$. Then if EstimateValues is run with input
$k^{\prime}=3k$, it returns $\widehat{w_{J}}$ for $\left|J\right|=3k$
satisfying
$\operatorname{Err}^{2}(\widehat{x^{\prime}_{L}}-\widehat{w_{J}},fk)\leq\operatorname{Err}^{2}(\widehat{x^{\prime}_{L}},k)+O(k\mu^{2})$
with probability at least $1-\gamma$.
###### Proof.
By Lemma 4.6, each index $i\in L$ has
$\Pr[\left|\widehat{w}_{i}-\widehat{x^{\prime}}_{i}\right|^{2}>\mu^{2}]<\frac{\gamma
fk}{B}.$
Let $U=\\{i\in
L\mid\left|\widehat{w}_{i}-\widehat{x^{\prime}}_{i}\right|^{2}>\mu^{2}\\}$.
With probability $1-\gamma$, $\left|U\right|\leq fk$; assume this happens.
Then
$\displaystyle\left\lVert(\widehat{x^{\prime}}-\widehat{w})_{L\setminus
U}\right\rVert_{\infty}^{2}\leq\mu^{2}.$ (6)
Let $T$ contain the top $2k$ coordinates of $\widehat{w}_{L\setminus U}$. By
the analysis of Count-Sketch (most specifically, Theorem 3.1 of [PW11]), the
$\ell_{\infty}$ guarantee (6) means that
$\displaystyle\left\lVert\widehat{x^{\prime}}_{L\setminus
U}-\widehat{w}_{T}\right\rVert_{2}^{2}\leq\operatorname{Err}^{2}(\widehat{x^{\prime}}_{L\setminus
U},k)+3k\mu^{2}.$ (7)
Because $J$ is the top $3k>(2+f)k$ coordinates of $\widehat{w_{L}}$, $T\subset
J$. Let $J^{\prime}=J\setminus(T\cup U)$, so $\left|J^{\prime}\right|\leq k$.
Then
$\displaystyle\operatorname{Err}^{2}(\widehat{x^{\prime}_{L}}-\widehat{w_{J}},fk)$
$\displaystyle\leq\left\lVert\widehat{x^{\prime}_{L\setminus
U}}-\widehat{w_{J\setminus U}}\right\rVert_{2}^{2}$
$\displaystyle=\left\lVert\widehat{x^{\prime}}_{L\setminus(U\cup
J^{\prime})}-\widehat{w_{T}}\right\rVert_{2}^{2}+\left\lVert(\widehat{x^{\prime}}-\widehat{w})_{J^{\prime}}\right\rVert_{2}^{2}$
$\displaystyle\leq\left\lVert\widehat{x^{\prime}}_{L\setminus
U}-\widehat{w_{T}}\right\rVert_{2}^{2}+\left|J^{\prime}\right|\left\lVert(\widehat{x^{\prime}}-\widehat{w})_{J^{\prime}}\right\rVert_{\infty}^{2}$
$\displaystyle\leq\operatorname{Err}^{2}(\widehat{x^{\prime}}_{L\setminus
U},k)+3k\mu^{2}+k\mu^{2}$
$\displaystyle=\operatorname{Err}^{2}(\widehat{x^{\prime}}_{L\setminus
U},k)+O(k\mu^{2})$
where we used Equations (6) and (7). ∎
### 4.5 Properties of SparseFFT
We will show that $\widehat{x}-\widehat{z}^{(r)}$ gets sparser as $r$
increases, with only a mild increase in the error.
###### Lemma 4.8.
Define $\widehat{x}^{(r)}=\widehat{x}-\widehat{z}^{(r)}$. Consider any one
loop $r$ of SparseFFT, running with parameters
$(B,k,\alpha)=(B_{r},k_{r},\alpha_{r})$ such that
$B\geq\frac{Ck}{\alpha^{2}\epsilon}$ for some $C$ larger than some fixed
constant. Then for any $f>0$,
$\operatorname{Err}^{2}(\widehat{x}^{(r+1)},fk)\leq(1+\epsilon)\operatorname{Err}^{2}(\widehat{x}^{(r)},k)+O(\epsilon\delta^{2}n\left\lVert\widehat{x}\right\rVert_{1}^{2})$
with probability $1-O(\alpha/f)$, and the running time is
$O((\lVert\widehat{z}^{(r)}\rVert_{0}(1+\alpha\log(n/\delta))+\frac{B}{\alpha}\log(n/\delta))(\log\frac{1}{\alpha\epsilon}+\log(n/B))).$
###### Proof.
We use $R_{est}=O(\log\frac{B}{\alpha k})=O(\log\frac{1}{\alpha\epsilon})$
rounds inside EstimateValues.
The running time for LocateSignal is
$O((\frac{B}{\alpha}\log(n/\delta)+\lVert\widehat{z}^{(r)}\rVert_{0}(1+\alpha\log(n/\delta)))\log(n/B)),$
and for EstimateValues is
$O((\frac{B}{\alpha}\log(n/\delta)+\lVert\widehat{z}^{(r)}\rVert_{0}(1+\alpha\log(n/\delta)))\log\frac{1}{\alpha\epsilon})$
for a total running time as given.
Recall that in round $r$,
$\mu^{2}=\frac{\epsilon}{k}(\operatorname{Err}^{2}(\widehat{x}^{(r)},k)+\delta^{2}n\left\lVert\widehat{x}\right\rVert_{1}^{2})$
and $S=\\{i\in[n]\mid\left|\widehat{x}^{(r)}_{i}\right|^{2}>\mu^{2}\\}$. By
Lemma 4.5, each $i\in S$ lies in $L_{r}$ with probability at least
$1-O(\alpha)$. Hence $\left|S\setminus L\right|<fk$ with probability at least
$1-O(\alpha/f)$. Then
$\displaystyle\operatorname{Err}^{2}(\widehat{x}^{(r)}_{[n]\setminus L},fk)$
$\displaystyle\leq\left\lVert\widehat{x}^{(r)}_{[n]\setminus(L\cup
S)}\right\rVert_{2}^{2}$
$\displaystyle\leq\operatorname{Err}^{2}(\widehat{x}^{(r)}_{[n]\setminus(L\cup
S)},k)+k\left\lVert\widehat{x}^{(r)}_{[n]\setminus(L\cup
S)}\right\rVert_{\infty}^{2}$
$\displaystyle\leq\operatorname{Err}^{2}(\widehat{x}^{(r)}_{[n]\setminus
L},k)+k\mu^{2}.$ (8)
Let
$\widehat{w}=\widehat{z}^{(r+1)}-\widehat{z}^{(r)}=\widehat{x}^{(r)}-\widehat{x}^{(r+1)}$
by the vector recovered by EstimateValues. Then
$\operatorname{supp}(\widehat{w})\subset L$, so
$\displaystyle\operatorname{Err}^{2}(\widehat{x}^{(r+1)},2fk)$
$\displaystyle=\operatorname{Err}^{2}(\widehat{x}^{(r)}-\widehat{w},2fk)$
$\displaystyle\leq\operatorname{Err}^{2}(\widehat{x}^{(r)}_{[n]\setminus
L},fk)+\operatorname{Err}^{2}(\widehat{x}^{(r)}_{L}-\widehat{w},fk)$
$\displaystyle\leq\operatorname{Err}^{2}(\widehat{x}^{(r)}_{[n]\setminus
L},fk)+\operatorname{Err}^{2}(\widehat{x}^{(r)}_{L},k)+O(k\mu^{2})$
by Lemma 4.7. But by Equation (4.5), this gives
$\displaystyle\operatorname{Err}^{2}(\widehat{x}^{(r+1)},2fk)$
$\displaystyle\leq\operatorname{Err}^{2}(\widehat{x}^{(r)}_{[n]\setminus
L},k)+\operatorname{Err}^{2}(\widehat{x}^{(r)}_{L},k)+O(k\mu^{2})$
$\displaystyle\leq\operatorname{Err}^{2}(\widehat{x}^{(r)},k)+O(k\mu^{2})$
$\displaystyle=(1+O(\epsilon))\operatorname{Err}^{2}(\widehat{x}^{(r)},k)+O(\epsilon\delta^{2}n\left\lVert\widehat{x}\right\rVert_{1}^{2}).$
The result follows from rescaling $f$ and $\epsilon$ by constant factors. ∎
Given the above, this next proof follows a similar argument to [IPW11],
Theorem 3.7.
###### Theorem 4.9.
With $2/3$ probability, SparseFFT recovers $\widehat{z}^{(R+1)}$ such that
$\left\lVert\widehat{x}-\widehat{z}^{(R+1)}\right\rVert_{2}\leq(1+\epsilon)\operatorname{Err}(\widehat{x},k)+\delta\left\lVert\widehat{x}\right\rVert_{2}$
in $O(\frac{k}{\epsilon}\log(n/k)\log(n/\delta))$ time.
###### Proof.
Define $f_{r}=O(1/r^{2})$ so $\sum f_{r}<1/4$. Choose $R$ so $\prod_{r\leq
R}f_{r}<1/k\leq\prod_{r<R}f_{r}$. Then $R=O(\log k/\log\log k)$, since
$\prod_{r\leq R}f_{r}<(f_{R/2})^{R/2}=(2/R)^{R}$.
Set $\epsilon_{r}=f_{r}\epsilon$, $\alpha_{r}=\Theta(f_{r}^{2})$,
$k_{r}=k\prod_{i<r}f_{i}$, $B_{r}=O(\frac{k}{\epsilon}\alpha_{r}f_{r})$. Then
$B_{r}=\omega(\frac{k_{r}}{\alpha_{r}^{2}\epsilon_{r}})$, so for sufficiently
large constant the constraint of Lemma 4.8 is satisfied. For appropriate
constants, Lemma 4.8 says that in each round $r$,
$\displaystyle\operatorname{Err}^{2}(\widehat{x}^{(r+1)},k_{r+1})$
$\displaystyle=\operatorname{Err}^{2}(\widehat{x}^{(r+1)},f_{r}k_{r})\leq(1+f_{r}\epsilon)\operatorname{Err}^{2}(\widehat{x}^{(r)},k_{r})+O(f_{r}\epsilon\delta^{2}n\left\lVert\widehat{x}\right\rVert_{1}^{2})$
(9)
with probability at least $1-f_{r}$. The error accumulates, so in round $r$ we
have
$\displaystyle\operatorname{Err}^{2}(\widehat{x}^{(r)},k_{r})$
$\displaystyle\leq\operatorname{Err}^{2}(\widehat{x},k)\prod_{i<r}(1+f_{i}\epsilon)+\sum_{i<r}O(f_{r}\epsilon\delta^{2}n\left\lVert\widehat{x}\right\rVert_{1}^{2})\prod_{i<j<r}(1+f_{j}\epsilon)$
with probability at least $1-\sum_{i<r}f_{i}>3/4$. Hence in the end, since
$k_{R+1}=k\prod_{i\leq R}f_{i}<1$,
$\displaystyle\left\lVert\widehat{x}^{(R+1)}\right\rVert_{2}^{2}$
$\displaystyle=\operatorname{Err}^{2}(\widehat{x}^{(R+1)},k_{R+1})\leq\operatorname{Err}^{2}(\widehat{x},k)\prod_{i\leq
R}(1+f_{i}\epsilon)+O(R\epsilon\delta^{2}n\left\lVert\widehat{x}\right\rVert_{1}^{2})\prod_{i\leq
R}(1+f_{i}\epsilon)$
with probability at least $3/4$. We also have
$\prod_{i}(1+f_{i}\epsilon)\leq e^{\epsilon\sum_{i}f_{i}}\leq e$
making
$\prod_{i}(1+f_{i}\epsilon)\leq 1+e\sum_{i}f_{i}\epsilon<1+2\epsilon.$
Thus we get the approximation factor
$\left\lVert\widehat{x}-\widehat{z}^{(R+1)}\right\rVert_{2}^{2}\leq(1+2\epsilon)\operatorname{Err}^{2}(\widehat{x},k)+O((\log
k)\epsilon\delta^{2}n\left\lVert\widehat{x}\right\rVert_{1}^{2})$
with at least $3/4$ probability. Rescaling $\delta$ by $\text{poly}(n)$, using
$\left\lVert\widehat{x}\right\rVert_{1}^{2}\leq
n\left\lVert\widehat{x}\right\rVert_{2}$, and taking the square root gives the
desired
$\left\lVert\widehat{x}-\widehat{z}^{(R+1)}\right\rVert_{2}\leq(1+\epsilon)\operatorname{Err}(\widehat{x},k)+\delta\left\lVert\widehat{x}\right\rVert_{2}.$
Now we analyze the running time. The update
$\widehat{z}^{(r+1)}-\widehat{z}^{(r)}$ in round $r$ has support size
$3k_{r}$, so in round $r$
$\lVert\widehat{z}^{(r)}\rVert_{0}\leq\sum_{i<r}3k_{r}=O(k).$
Thus the expected running time in round $r$ is
$\displaystyle
O((k(1+\alpha_{r}\log(n/\delta))+\frac{B_{r}}{\alpha_{r}}\log(n/\delta))(\log\frac{1}{\alpha_{r}\epsilon_{r}}+\log(n/B_{r})))$
$\displaystyle={}$ $\displaystyle
O((k+\frac{k}{r^{4}}\log(n/\delta)+\frac{k}{\epsilon
r^{2}}\log(n/\delta))(\log\frac{r^{2}}{\epsilon}+\log(n\epsilon/k)+\log r))$
$\displaystyle={}$ $\displaystyle O((k+\frac{k}{\epsilon
r^{2}}\log(n/\delta))(\log r+\log(n/k)))$
We split the terms multiplying $k$ and $\frac{k}{\epsilon
r^{2}}\log(n/\delta)$, and sum over $r$. First,
$\displaystyle\sum_{r=1}^{R}(\log r+\log(n/k))$ $\displaystyle\leq R\log
R+R\log(n/k)$ $\displaystyle\leq O(\log k+\log k\log(n/k))$
$\displaystyle=O(\log k\log(n/k)).$
Next,
$\displaystyle\sum_{r=1}^{R}\frac{1}{r^{2}}(\log r+\log(n/k))=O(\log(n/k))$
Thus the total running time is
$\displaystyle O(k\log
k\log(n/k)+\frac{k}{\epsilon}\log(n/\delta)\log(n/k))=O(\frac{k}{\epsilon}\log(n/\delta)\log(n/k)).$
∎
## 5 Reducing the full $k$-dimensional DFT to the exact $k$-sparse case in
$n$ dimensions
In this section we show the following lemma. Assume that $k$ divides $n$.
###### Lemma 5.1.
Suppose that there is an algorithm $A$ that, given an $n$-dimensional vector
$y$ such that $\hat{y}$ is $k$-sparse, computes $\hat{y}$ in time $T(k)$. Then
there is an algorithm $A^{\prime}$ that given a $k$-dimensional vector $x$
computes $\hat{x}$ in time $O(T(k)))$.
###### Proof.
Given a $k$-dimensional vector $x$, we define $y_{i}=x_{i\bmod k}$, for
$i=0\ldots n-1$. Whenever $A$ requests a sample $y_{i}$, we compute it from
$x$ in constant time. Moreover, we have that $\hat{y}_{i}=\hat{x}_{i/(n/k)}$
if $i$ is a multiple of $(n/k)$, and $\hat{y}_{i}=0$ otherwise. Thus $\hat{y}$
is $k$-sparse. Since $\hat{x}$ can be immediately recovered from $\hat{y}$,
the lemma follows. ∎
###### Corollary 5.2.
Assume that the $n$-dimensional DFT cannot be computed in $o(n\log n)$ time.
Then any algorithm for the $k$-sparse DFT (for vectors of arbitrary dimension)
must run in $\Omega(k\log k)$ time.
## 6 Lower Bound
In this section, we show any algorithm satisfying Equation (1) must access
$\Omega(k\log(n/k)/\log\log n)$ samples of $x$.
We translate this problem into the language of compressive sensing:
###### Theorem 6.1.
Let $F\in{\mathbb{C}}^{n\times n}$ be orthonormal and satisfy
$\left|F_{i,j}\right|=1/\sqrt{n}$ for all $i,j$. Suppose an algorithm takes
$m$ adaptive samples of $Fx$ and computes $x^{\prime}$ with
$\left\lVert x-x^{\prime}\right\rVert_{2}\leq 2\min_{k\text{-sparse
}x^{*}}\left\lVert x-x^{*}\right\rVert_{2}$
for any $x$, with probability at least $3/4$. Then it must have
$m=\Omega(k\log(n/k)/\log\log n)$.
###### Corollary 6.2.
Any algorithm computing the approximate Fourier transform must access
$\Omega(k\log(n/k)/\log\log n)$ samples from the time domain.
If the samples were chosen non-adaptively, we would immediately have
$m=\Omega(k\log(n/k))$ by [PW11]. However, an algorithm could choose samples
based on the values of previous samples. In the sparse recovery framework
allowing general linear measurements, this adaptivity can decrease the number
of measurements to $O(k\log\log(n/k))$ [IPW11]; in this section, we show that
adaptivity is much less effective in our setting where adaptivity only allows
the choice of Fourier coefficients.
We follow the framework of Section 4 of [PW11]. In this section we use
standard notation from information theory, including $I(x;y)$ for mutual
information, $H(x)$ for discrete entropy, and $h(x)$ for continuous entropy.
Consult a reference such as [CT91] for details.
Let $\mathcal{F}\subset\\{S\subset[n]:\left|S\right|=k\\}$ be a family of
$k$-sparse supports such that:
* •
$\left|S\oplus S^{\prime}\right|\geq k$ for $S\neq S^{\prime}\in\mathcal{F}$,
where $\oplus$ denotes the exclusive difference between two sets, and
* •
$\log\left|\mathcal{F}\right|=\Omega(k\log(n/k))$.
This is possible; for example, a random code on $[n/k]^{k}$ with relative
distance $1/2$ has these properties.
For each $S\in\mathcal{F}$, let $X^{S}=\\{x\in\\{0,\pm
1\\}^{n}\mid\operatorname{supp}(x^{S})=S\\}$. Let $x\in X^{S}$ uniformly at
random. The variables $x_{i}$, $i\in S$, are i.i.d. subgaussian random
variables with parameter $\sigma^{2}=1$, so for any row $F_{j}$ of $F$,
$F_{j}x$ is subgaussian with parameter $\sigma^{2}=k/n$. Therefore
$\Pr_{x\in X^{S}}[\left|F_{j}x\right|>t\sqrt{k/n}]<2e^{-t^{2}/2}$
hence for each $S$, we can choose an $x^{S}\in X^{S}$ with
$\displaystyle\left\lVert Fx^{S}\right\rVert_{\infty}<O(\sqrt{\frac{k\log
n}{n}}).$ (10)
Let $X=\\{x^{S}\mid S\in\mathcal{F}\\}$ be the set of such $x^{S}$.
Let $w\sim N(0,\alpha\frac{k}{n}I_{n})$ be i.i.d. normal with variance $\alpha
k/n$ in each coordinate.
Consider the following process:
#### Procedure.
First, Alice chooses $S\in\mathcal{F}$ uniformly at random, then selects the
$x\in X$ with $\operatorname{supp}(x)=S$. Alice independently chooses $w\sim
N(0,\alpha\frac{k}{n}I_{n})$ for a parameter $\alpha=\Theta(1)$ sufficiently
small. For $j\in[m]$, Bob chooses $i_{j}\in[n]$ and observes
$y_{j}=F_{i_{j}}(x+w)$. He then computes the result $x^{\prime}\approx x$ of
sparse recovery, rounds to $X$ by $\hat{x}=\operatorname*{arg\,min}_{x^{*}\in
X}\left\lVert x^{*}-x^{\prime}\right\rVert_{2}$, and sets
$S^{\prime}=\operatorname{supp}(\hat{x})$. This gives a Markov chain $S\to
x\to y\to x^{\prime}\to\hat{x}\to S^{\prime}$.
We will show that deterministic sparse recovery algorithms require large $m$
to succeed on this input distribution $x+w$ with $3/4$ probability. By Yao’s
minimax principle, this means randomized sparse recovery algorithms also
require large $m$ to succeed with $3/4$ probability.
Our strategy is to give upper and lower bounds on $I(S;S^{\prime})$, the
mutual information between $S$ and $S^{\prime}$.
###### Lemma 6.3 (Analog of Lemma 4.3 of [PW11] for $\epsilon=O(1)$).
There exists a constant $\alpha^{\prime}>0$ such that if
$\alpha<\alpha^{\prime}$, then $I(S;S^{\prime})=\Omega(k\log(n/k))$ .
###### Proof.
Assuming the sparse recovery succeeds (as happens with 3/4 probability), we
have $\left\lVert x^{\prime}-(x+w)\right\rVert_{2}\leq 2\left\lVert
w\right\rVert_{2}$, which implies $\left\lVert
x^{\prime}-x\right\rVert_{2}\leq 3\left\lVert w\right\rVert_{2}$. Therefore
$\displaystyle\left\lVert\hat{x}-x\right\rVert_{2}$
$\displaystyle\leq\left\lVert\hat{x}-x^{\prime}\right\rVert_{2}+\left\lVert
x^{\prime}-x\right\rVert_{2}$ $\displaystyle\leq 2\left\lVert
x^{\prime}-x\right\rVert_{2}$ $\displaystyle\leq 6\left\lVert
w\right\rVert_{2}.$
We also know $\left\lVert
x^{\prime}-x^{\prime\prime}\right\rVert_{2}\geq\sqrt{k}$ for all distinct
$x^{\prime},x^{\prime\prime}\in X$ by construction. Because
$\operatorname{\mathbb{E}}[\left\lVert w\right\rVert_{2}^{2}]=\alpha k$, with
probability at least $3/4$ we have $\left\lVert
w\right\rVert_{2}\leq\sqrt{4\alpha k}<\sqrt{k}/6$ for sufficiently small
$\alpha$. But then $\left\lVert\hat{x}-x\right\rVert_{2}<\sqrt{k}$, so
$\hat{x}=x$ and $S=S^{\prime}$. Thus $\Pr[S\neq S^{\prime}]\leq 1/2$.
Fano’s inequality states $H(S\mid S^{\prime})\leq 1+\Pr[S\neq
S^{\prime}]\log\left|\mathcal{F}\right|$. Thus
$I(S;S^{\prime})=H(S)-H(S\mid
S^{\prime})\geq-1+\frac{1}{2}\log\left|\mathcal{F}\right|=\Omega(k\log(n/k))$
as desired. ∎
We next show an analog of their upper bound (Lemma 4.1 of [PW11]) on
$I(S;S^{\prime})$ for adaptive measurements of bounded $\ell_{\infty}$ norm.
The proof follows the lines of [PW11], but is more careful about dependencies
and needs the $\ell_{\infty}$ bound on $Fx$.
###### Lemma 6.4.
$I(S;S^{\prime})\leq O(m\log(1+\frac{1}{\alpha}\log n)).$
###### Proof.
Let $A_{j}=F_{i_{j}}$ for $j\in[m]$, and let $w^{\prime}_{j}=A_{j}w$. The
$w^{\prime}_{j}$ are independent normal variables with variance
$\alpha\frac{k}{n}$. Because the $A_{j}$ are orthonormal and $w$ is drawn from
a rotationally invariant distribution, the $w^{\prime}$ are also independent
of $x$.
Let $y_{j}=A_{j}x+w^{\prime}_{j}$. We know $I(S;S^{\prime})\leq I(x;y)$
because $S\to x\to y\to S^{\prime}$ is a Markov chain. Because the variables
$A_{j}$ are deterministic given $y_{1},\dotsc,y_{j-1}$,
$\displaystyle I(x;y_{j}\mid y_{1},\dotsc,y_{j-1})$
$\displaystyle=I(x;A_{j}x+w^{\prime}_{j}\mid y_{1},\dotsc,y_{j-1})$
$\displaystyle=h(A_{j}x+w^{\prime}_{j}\mid
y_{1},\dotsc,y_{j-1})-h(A_{j}x+w^{\prime}_{j}\mid x,y_{1},\dotsc,y_{j-1})$
$\displaystyle=h(A_{j}x+w^{\prime}_{j}\mid
y_{1},\dotsc,y_{j-1})-h(w^{\prime}_{j}).$
By the chain rule for information,
$\displaystyle I(S;S^{\prime})$ $\displaystyle\leq I(x;y)$
$\displaystyle=\sum_{j=1}^{m}I(x;y_{j}\mid y_{1},\dotsc,y_{j-1})$
$\displaystyle=\sum_{j=1}^{m}h(A_{j}x+w^{\prime}_{j}\mid
y_{1},\dotsc,y_{j-1})-h(w^{\prime}_{j})$
$\displaystyle\leq\sum_{j=1}^{m}h(A_{j}x+w^{\prime}_{j})-h(w^{\prime}_{j}).$
Thus it suffices to show
$h(A_{j}x+w^{\prime}_{j})-h(w^{\prime}_{j})=O(\log(1+\frac{1}{\alpha}\log n))$
for all $j$.
Note that $A_{j}$ depends only on $y_{1},\dotsc,y_{j-1}$, so it is independent
of $w^{\prime}_{j}$. Thus
$\displaystyle\operatorname{\mathbb{E}}[(A_{j}x+w^{\prime}_{j})^{2}]=\operatorname{\mathbb{E}}[(A_{j}x)^{2}]+\operatorname{\mathbb{E}}[(w^{\prime}_{j})^{2}]\leq
O(\frac{k\log n}{n})+\alpha\frac{k}{n}$
by Equation (10). Because the maximum entropy distribution under an $\ell_{2}$
constraint is a Gaussian, we have
$\displaystyle h(A_{j}x+w^{\prime}_{j})-h(w^{\prime}_{j})$ $\displaystyle\leq
h(N(0,O(\frac{k\log n}{n})+\alpha\frac{k}{n}))-h(N(0,\alpha\frac{k}{n}))$
$\displaystyle=\frac{1}{2}\log(1+\frac{O(\log n)}{\alpha})$
$\displaystyle=O(\log(1+\frac{1}{\alpha}\log n)).$
as desired. ∎
Theorem 6.1 follows from Lemma 6.3 and Lemma 6.4, with $\alpha=\Theta(1)$.
## 7 Efficient Constructions of Window Functions
###### Claim 7.1.
Let $\operatorname{cdf}$ denote the standard Gaussian cumulative distribution
function. Then:
1. 1.
$\operatorname{cdf}(t)=1-\operatorname{cdf}(-t)$.
2. 2.
$\operatorname{cdf}(t)\leq e^{-t^{2}/2}$ for $t<0$.
3. 3.
$\operatorname{cdf}(t)<\delta$ for $t<-\sqrt{2\log(1/\delta)}$.
4. 4.
$\int_{x=-\infty}^{t}\operatorname{cdf}(x)dx<\delta$ for
$t<-\sqrt{2\log(3/\delta)}$.
5. 5.
For any $\delta$, there exists a function
$\widetilde{\operatorname{cdf}}_{\delta}(t)$ computable in $O(\log(1/\delta))$
time such that
$\left\lVert\operatorname{cdf}-\widetilde{\operatorname{cdf}}_{\delta}\right\rVert_{\infty}<\delta$.
###### Proof.
1. 1.
Follows from the symmetry of Gaussian distribution.
2. 2.
Follows from a standard moment generating function bound on Gaussian random
variables.
3. 3.
Follows from (2).
4. 4.
Property (2) implies that $\operatorname{cdf}(t)$ is at most $\sqrt{2\pi}<3$
times larger than the Gaussian pdf. Then apply (3).
5. 5.
By (1) and (3), $\operatorname{cdf}(t)$ can be computed as $\pm\delta$ or
$1\pm\delta$ unless $\left|t\right|<\sqrt{2(\log(1/\delta))}$. But then an
efficient expansion around $0$ only requires $O(\log(1/\delta))$ terms to
achieve precision $\pm\delta$.
For example, we can truncate the representation [Mar04]
$\operatorname{cdf}(t)=\frac{1}{2}+\frac{e^{-t^{2}/2}}{\sqrt{2\pi}}\left(t+\frac{t^{3}}{3}+\frac{t^{5}}{3\cdot
5}+\frac{t^{7}}{3\cdot 5\cdot 7}+\dotsb\right)$
at $O(\log(1/\delta))$ terms.
∎
###### Claim 7.2.
Define the continuous Fourier transform of $f(t)$ by
$\widehat{f}(s)=\int_{-\infty}^{\infty}e^{-2\pi\mathbf{i}st}f(t)dt.$
For $t\in[n]$, define
$g_{t}=\sqrt{n}\sum_{j=-\infty}^{\infty}f(t+nj)$
and
$g^{\prime}_{t}=\sum_{j=-\infty}^{\infty}\widehat{f}(t/n+j).$
Then $\widehat{g}=g^{\prime}$, where $\widehat{g}$ is the $n$-dimensional DFT
of $g$.
###### Proof.
Let $\Delta_{1}(t)$ denote the Dirac comb of period $1$: $\Delta_{1}(t)$ is a
Dirac delta function when $t$ is an integer and zero elsewhere. Then
$\widehat{\Delta_{1}}=\Delta_{1}$. For any $t\in[n]$, we have
$\displaystyle\widehat{g}_{t}$
$\displaystyle=\sum_{s=1}^{n}\sum_{j=-\infty}^{\infty}f(s+nj)e^{-2\pi\mathbf{i}ts/n}$
$\displaystyle=\sum_{s=1}^{n}\sum_{j=-\infty}^{\infty}f(s+nj)e^{-2\pi\mathbf{i}t(s+nj)/n}$
$\displaystyle=\sum_{s=-\infty}^{\infty}f(s)e^{-2\pi\mathbf{i}ts/n}$
$\displaystyle=\int_{-\infty}^{\infty}f(s)\Delta_{1}(s)e^{-2\pi\mathbf{i}ts/n}ds$
$\displaystyle=\widehat{(f\cdot\Delta_{1})}(t/n)$
$\displaystyle=(\widehat{f}*\Delta_{1})(t/n)$
$\displaystyle=\sum_{j=-\infty}^{\infty}\widehat{f}(t/n+j)$
$\displaystyle=g^{\prime}_{t}.$
∎
###### Lemma 7.3.
For any parameters $B\geq 1,\delta>0,$ and $\alpha>0$, there exist flat window
functions $G$ and $\widehat{G^{\prime}}$ such that $G$ can be computed in
$O(\frac{B}{\alpha}\log(n/\delta))$ time, and for each $i$
$\widehat{G^{\prime}}_{i}$ can be evaluated in $O(\log(n/\delta))$ time.
###### Proof.
We will show this for a function $\widehat{G^{\prime}}$ that is a Gaussian
convolved with a box-car filter. First we construct analogous window functions
for the continuous Fourier transform. We then show that discretizing these
functions gives the desired result.
Let $D$ be the pdf of a Gaussian with standard deviation $\sigma>1$ to be
determined later, so $\widehat{D}$ is the pdf of a Gaussian with standard
deviation $1/\sigma$. Let $\widehat{F}$ be a box-car filter of length $2C$ for
some parameter $C<1$; that is, let $\widehat{F}(t)=1$ for $\left|t\right|<C$
and $F(t)=0$ otherwise, so $F(t)=2C\text{sinc}(t/(2C))$. Let $G^{*}=D\cdot F$,
so $\widehat{G^{*}}=\widehat{D}*\widehat{F}$.
Then $\left|G^{*}(t)\right|\leq 2C\left|D(t)\right|<2C\delta$ for
$\left|t\right|>\sigma\sqrt{2\log(1/\delta)}$. Furthermore, $G^{*}$ is
computable in $O(1)$ time.
Its Fourier transform is
$\widehat{G^{*}}(t)=\operatorname{cdf}(\sigma(t+C))-\operatorname{cdf}(\sigma(t-C))$.
By Claim 7.1 we have for $\left|t\right|>C+\sqrt{2\log(1/\delta)}/\sigma$ that
$\widehat{G^{*}}(t)=\pm\delta$. We also have, for
$\left|t\right|<C-\sqrt{2\log(1/\delta)}/\sigma$, that
$\widehat{G^{*}}(t)=1\pm 2\delta$.
Now, for $i\in[n]$ let $H_{i}=\sqrt{n}\sum_{j=\infty}^{\infty}G^{*}(i+nj)$. By
Claim 7.2 it has DFT
$\widehat{H}_{i}=\sum_{j=\infty}^{\infty}\widehat{G^{*}}(i/n+j)$. Furthermore,
$\displaystyle\sum_{\left|i\right|>\sigma\sqrt{2\log(1/\delta)}}\left|G^{*}(i)\right|$
$\displaystyle\leq 4C\sum_{i<-\sigma\sqrt{2\log(1/\delta)}}\left|D(i)\right|$
$\displaystyle\leq
4C\left(\int_{-\infty}^{-\sigma\sqrt{2\log(1/\delta)}}\left|D(x)\right|dx+D(-\sigma\sqrt{2\log(1/\delta)})\right)$
$\displaystyle\leq
4C(\operatorname{cdf}(-\sqrt{2\log(1/\delta)})+D(-\sigma\sqrt{2\log(1/\delta)}))$
$\displaystyle\leq 8C\delta\leq 8\delta.$
Thus if we let
$G_{i}=\sqrt{n}\sum_{\begin{subarray}{c}\left|j\right|<\sigma\sqrt{2\log(1/\delta)}\\\
j\equiv i\pmod{n}\end{subarray}}G^{*}(j)$
for $\left|i\right|<\sigma\sqrt{2\log(1/\delta)}$ and $G_{i}=0$ otherwise,
then $\left\lVert G-H\right\rVert_{1}\leq 8\delta\sqrt{n}$.
Now, note that for integer $i$ with $\left|i\right|\leq n/2$,
$\displaystyle\widehat{H}_{i}-\widehat{G^{*}}(i/n)$
$\displaystyle=\sum_{\begin{subarray}{c}j\in{\mathbb{Z}}\\\ j\neq
0\end{subarray}}\widehat{G^{*}}(i/n+j)$
$\displaystyle\left|\widehat{H}_{i}-\widehat{G^{*}}(i/n)\right|$
$\displaystyle\leq 2\sum_{j=0}^{\infty}\widehat{G^{*}}(-1/2-j)$
$\displaystyle\leq 2\sum_{j=0}^{\infty}\operatorname{cdf}(\sigma(-1/2-j+C))$
$\displaystyle\leq
2\int_{-\infty}^{-1/2}\operatorname{cdf}(\sigma(x+C))dx+2\operatorname{cdf}(\sigma(-1/2+C))$
$\displaystyle\leq 2\delta/\sigma+2\delta\leq 4\delta$
by Claim 7.1, as long as
$\displaystyle\sigma(1/2-C)>\sqrt{2\log(3/\delta)}.$ (11)
Let
$\widehat{G^{\prime}}_{i}=\left\\{\begin{array}[]{cl}1&\left|i\right|\leq
n(C-\sqrt{2\log(1/\delta)}/\sigma)\\\ 0&\left|i\right|\geq
n(C+\sqrt{2\log(1/\delta)}/\sigma)\\\
\widetilde{\operatorname{cdf}}_{\delta}(\sigma(i+C)/n)-\widetilde{\operatorname{cdf}}_{\delta}(\sigma(i-C)/n)&\text{otherwise}\end{array}\right.$
where $\widetilde{\operatorname{cdf}}_{\delta}(t)$ computes
$\operatorname{cdf}(t)$ to precision $\pm\delta$ in $O(\log(1/\delta))$ time,
as per Claim 7.1. Then $\widehat{G^{\prime}}_{i}=\widehat{G^{*}}(i/n)\pm
2\delta=\widehat{H}_{i}\pm 6\delta$. Hence
$\displaystyle\left\lVert\widehat{G}-\widehat{G^{\prime}}\right\rVert_{\infty}$
$\displaystyle\leq\left\lVert\widehat{G^{\prime}}-\widehat{H}\right\rVert_{\infty}+\left\lVert\widehat{G}-\widehat{H}\right\rVert_{\infty}$
$\displaystyle\leq\left\lVert\widehat{G^{\prime}}-\widehat{H}\right\rVert_{\infty}+\left\lVert\widehat{G}-\widehat{H}\right\rVert_{2}$
$\displaystyle=\left\lVert\widehat{G^{\prime}}-\widehat{H}\right\rVert_{\infty}+\left\lVert
G-H\right\rVert_{2}$
$\displaystyle\leq\left\lVert\widehat{G^{\prime}}-\widehat{H}\right\rVert_{\infty}+\left\lVert
G-H\right\rVert_{1}$ $\displaystyle\leq(8\sqrt{n}+6)\delta.$
Replacing $\delta$ by $\delta/n$ and plugging in
$\sigma=\frac{4B}{\alpha}\sqrt{2\log(n/\delta)}>1$ and
$C=(1-\alpha/2)/(2B)<1$, we have the required properties of flat window
functions:
* •
$\left|G_{i}\right|=0$ for
$\left|i\right|\geq\Omega(\frac{B}{\alpha}\log(n/\delta))$
* •
$\widehat{G^{\prime}}_{i}=1$ for $\left|i\right|\leq(1-\alpha)n/(2B)$
* •
$\widehat{G^{\prime}}_{i}=0$ for $\left|i\right|\geq n/(2B)$
* •
$\widehat{G^{\prime}}_{i}\in[0,1]$ for all $i$.
* •
$\left\lVert\widehat{G^{\prime}}-\widehat{G}\right\rVert_{\infty}<\delta$.
* •
We can compute $G$ over its entire support in
$O(\frac{B}{\alpha}\log(n/\delta))$ total time.
* •
For any $i$, $\widehat{G^{\prime}}_{i}$ can be computed in $O(\log(n/\delta))$
time for $\left|i\right|\in[(1-\alpha)n/(2B),n/(2B)]$ and $O(1)$ time
otherwise.
The only requirement was Equation (11), which is that
$\frac{4B}{\alpha}\sqrt{2\log(n/\delta)}(1/2-\frac{1-\alpha/2}{2B})>\sqrt{2\log(3n/\delta)}.$
This holds if $B\geq 2$. The $B=1$ case is trivial using the constant function
$\widehat{G^{\prime}}_{i}=1$. ∎
## 8 Open questions
* •
Design an $O(k\log n)$-time algorithm for general signals. Alternatively,
prove that no such algorithm exists, under “reasonable” assumptions.101010The
$\Omega(k\log(n/k)/\log\log n)$ lower bound for the sample complexity shows
that the running time of our algorithm, $O(k\log n\log(n/k))$, is equal to the
sample complexity of the problem times (roughly) $\log n$. One could speculate
that this logarithmic discrepancy is due to the need for using FFT to process
the samples. Although we do not have any evidence for the optimality of our
general algorithm, the “sample complexity times $\log n$” bound appears to be
a natural barrier to further improvements.
* •
Reduce the sample complexity of the algorithms. Currently, the number of
samples used by each algorithm is only bounded by their running times.
* •
Extend the results to other (related) tasks, such as computing the sparse
Walsh-Hadamard Transform.
* •
Extend the algorithm to the case when $n$ is not a power of $2$. Note that
some of the earlier algorithms, e.g., [GMS05], work for any $n$.
* •
Improve the failure probability of the algorithms. Currently, the algorithms
only succeed with constant probability. Straightforward amplification would
take a $\log(1/p)$ factor slowdown to succeed with $1-p$ probability. One
would hope to avoid this slowdown.
## Acknowledgements
The authors would like to thank Martin Strauss and Ludwig Schmidt for many
helpful comments about the writing of the paper. This work is supported by the
Space and Naval Warfare Systems Center Pacific under Contract No.
N66001-11-C-4092, David and Lucille Packard Fellowship, and NSF grants
CCF-1012042 and CNS-0831664. E. Price is supported in part by an NSF Graduate
Research Fellowship.
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|
arxiv-papers
| 2012-01-12T08:34:46 |
2024-09-04T02:49:26.230298
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price",
"submitter": "Eric Price",
"url": "https://arxiv.org/abs/1201.2501"
}
|
1201.2531
|
# DREAM: DiffeRentially privatE smArt Metering
Technical report
Gergely Acs and Claude Castelluccia
INRIA Rhone Alpes, Montbonnot, France
gergely.acs, claude.castelluccia@inrialpes.fr
###### Abstract
This paper presents a new privacy-preserving smart metering system. Our scheme
is private under the differential privacy model and therefore provides strong
and provable guarantees. With our scheme, an (electricity) supplier can
periodically collect data from smart meters and derive aggregated statistics
while learning only limited information about the activities of individual
households. For example, a supplier cannot tell from a user’s trace when he
watched TV or turned on heating. Our scheme is simple, efficient and
practical. Processing cost is very limited: smart meters only have to add
noise to their data and encrypt the results with an efficient stream cipher.
###### Contents
1. 1 Introduction
2. 2 Related Work
3. 3 The model
1. 3.1 Network model
2. 3.2 Adversary model
3. 3.3 Privacy model
4. 3.4 Output perturbation: achieving differential privacy
5. 3.5 Utility definition
4. 4 Objectives
5. 5 Overview of approaches
1. 5.1 Fully decentralized approach (without aggregator)
2. 5.2 Aggregation with a trusted aggregator
3. 5.3 Our approach: aggregation without trusted entity
1. 5.3.1 Distributed noise generation: a new approach
2. 5.3.2 Encryption
6. 6 Protocol description
1. 6.1 System setup
2. 6.2 Smart meter processing
3. 6.3 Supplier processing
7. 7 Adding robustness
1. 7.1 Sanitization phase extension
2. 7.2 Encryption phase extension
1. 7.2.1 A simple approach
2. 7.2.2 Our proposal
3. 7.3 Utility evaluation
8. 8 Security Analysis
1. 8.1 Deploying malicious nodes
2. 8.2 Lying supplier
9. 9 Simulation results
1. 9.1 A high-resolution electricity trace simulator
2. 9.2 Error according to the cluster size
1. 9.2.1 Random clustering
2. 9.2.2 Consumption based clustering
3. 9.3 Privacy over multiple slots
1. 9.3.1 Privacy of appliances
10. 10 Conclusion
11. A Proof of Theorem 2 (Utility)
12. B Privacy of some ordinary appliances
## 1 Introduction
Several countries throughout the world are planning to deploy smart meters in
households in the very near future. The main motivation, for governments and
electricity suppliers, is to be able to match consumption with generation.
Traditional electrical meters only measure total consumption on a given period
of time (i.e., one month or one year). As such, they do not provide accurate
information of when the energy was consumed. Smart meters, instead, monitor
and report consumption in intervals of few minutes. They allow the utility
provider to monitor, almost in real-time, consumption and possibly adjust
generation and prices according to the demand. Billing customers by how much
is consumed and at what time of day will probably change consumption habits to
help matching consumption with generation. In the longer term, with the advent
of smart appliances, it is expected that the smart grid will remotely control
selected appliances to reduce demand.
##### Problem Statement:
Although smart metering might help improving energy management, it creates
many new privacy problems [1]. Smart meters provide very accurate consumption
data to electricity providers. As the interval of data collected by smart
meters decreases, the ability to disaggregate low-resolution data increases.
Analyzing high-resolution consumption data, Nonintrusive Appliance Load
Monitoring (NALM) [12] can be used to identify a remarkable number of electric
appliances (e.g., water heaters, well pumps, furnace blowers, refrigerators,
and air conditioners) employing exhaustive appliance signature libraries.
Researchers are now focusing on the myriad of small electric devices around
the home such as personal computers, laser printers, and light bulbs [16].
Moreover, it has also been shown that even simple off-the-shelf statistical
tools can be used to extract complex usage patterns from high-resolution
consumption data [17]. This extracted information can be used to profile and
monitor users for various purposes, creating serious privacy risks and
concerns. As data recorded by smart meters is lowering in resolution, and
inductive algorithms are quickly improving, it is urgent to develop privacy-
preserving smart metering systems that provide strong and provable guarantees.
##### Contributions:
We propose a privacy-preserving smart metering scheme that guarantees users’
privacy while still preserving the benefits and promises of smart metering.
Our contributions are many-fold and summarized as follows:
* •
We provide the first provably private and distributed solution for smart
metering that optimizes utility without relying on a third trusted party
(i.e., an aggregator). We were able to avoid the use of a third trusted party
by proposing a new distributed Laplacian Perturbation Algorithm (DLPA).
In our scheme, smart meters are grouped into clusters, where a cluster is a
group of hundreds or thousands of smart meters corresponding, for example, to
a quarter of a city. Each smart meter sends, at each sampling period, their
measures to the supplier. These measures are noised and encrypted such that
the supplier can compute the noised aggregated electricity consumption of the
cluster, at each sampling period, without getting access to individual values.
The aggregate is noised just enough to provide differential privacy to each
participating user, while still providing high utility (i.e., low error). Our
scheme is secure under the differential privacy model and therefore provides
strong and provable privacy guarantees. In particular, we guarantee that the
supplier can retrieve information about any user consumption only up to a
predefined threshold. Our scheme is simple, efficient and practical. It
requires either one or two rounds of message exchanges between a meter and the
supplier. Furthermore, processing cost is very limited: smart meters only have
to add noise to their data and encrypt the results with an efficient stream
cipher. Finally, our scheme is robust against smart meter failures and
malicious nodes. More specifically, it is secure even if an $\alpha$ fraction
of all nodes of a cluster collude with the supplier, where $\alpha$ is a
security parameter.
* •
We provide a detailed analysis of the security and performance of our
proposal. The security analysis is performed analytically. The performance,
which is evaluated using the utility metric, is performed using simulation. We
implemented a new electricity trace generation tool based on [21] which
generates one-minute resolution synthetic consumption data of different
households.
## 2 Related Work
Several papers addressed the privacy problems of smart metering in the recent
past [8, 17, 1, 18, 2, 3, 20, 10]. However, only a few of them have proposed
technical solutions to protect users’ privacy. In [1, 2], the authors discuss
the different security aspects of smart metering and the conflicting interests
among stakeholders. The privacy of billing is considered in [20, 17]. These
techniques uses zero-knowledge proofs to ensure that the fee calculated by the
user is correct without disclosing any consumption data.
Seemingly, the privacy of monitoring the sum consumption of multiple users may
be solved by simply anonymizing individual measurements like in [8] or using
some mixnet. However, these “ad-hoc” techniques are dangerous and do not
provide any real assurances of privacy. Several prominent examples in the
history have shown that ad-hoc methods do not work [14]. Moreover, these
techniques require an existing trusted third party who performs anonymization.
The authors in [3] perturb the released aggregate with random noise and use a
different model from ours to analyze the privacy of their scheme. However,
they do not encrypt individual measurements which means that the added noise
must be large enough to guarantee reasonable privacy. As individual noise
shares sum up at the aggregation, the final noise makes the aggregate useless.
In contrast to this, [10] uses homomorphic encryption to guarantee privacy for
individual measurements. However, the aggregate is not perturbed which means
that it is not differential private.
The notion of differential privacy was first proposed in [7]. The main
advantage of differential privacy over other privacy models is that it does
not specify the prior knowledge of the adversary and provides rigorous privacy
guarantee if each users’ data is statistically independent [13]. Initial works
on differential privacy focused on the problem how a trusted curator
(aggregator), who collects all data from users, can differential privately
release statistics. By contrast, our scheme ensures differential privacy even
if the curator is untrusted. Although [6] describes protocols for generating
shares of random noise which is secure against malicious participants, it
requires communication between users and it uses expensive secret sharing
techniques resulting in high overhead in case of large number of users.
Similarly, traditional Secure Multiparty Computation (SMC) techniques [11] [5]
also require interactions between users. All these solutions are impractical
for resource constrained smart meters where all the computation is done by the
aggregator and users are not supposed to communicate with each other.
Two closely related works to ours are [19] and [22]. In [19], the authors
propose a scheme to differential privately aggregate sums over multiple slots
when the aggregator is untrusted. However, they use the threshold Paillier
cryptosystem [9] for homomorphic encryption which is much more expensive
compared to [4] that we use. They also use different noise distribution
technique which requires several rounds of message exchanges between the users
and the aggregator. By contrast, our solution is much more efficient and
simple: it requires only a single message exchange if there are no node
failures, otherwise, we only need one extra round. In addition, our solution
does not rely on expensive public key cryptography during aggregation.
A recent paper [22] proposes another technique to privately aggregate time
series data. This work differs from ours as follows: (1) they use a Diffie-
Hellman-based encryption scheme, whereas our construction is based on a more
efficient construction that only use modular additions. This approach is
better adapted to resource constrained devices like smart meters. (2) Although
[22] does not require the establishment (and storage) of pairwise keys between
nodes as opposed to our approach, it is unclear how [22] can be extended to
tolerate node and communication failures. By contrast, our scheme is more
robust, as the encryption key of non-responding nodes is known to other nodes
in the network that can help to recover the aggregate. (3) Finally, [22] uses
a different noise generation method from ours, but this technique only
satisfies the relaxed $(\varepsilon,\delta)$-differential privacy definition.
Indeed, in their scheme, each node adds noise probabilistically which means
that none of the nodes add noise with some positive probability $\delta$.
Although $\delta$ can be arbitrarily small, this also decreases the utility.
By contrast, in our scheme, $\delta=0$ while ensuring nearly optimal utility.
## 3 The model
### 3.1 Network model
The network is composed of four major parts: the _supplier/aggregator_ , the
_electricty distribution network_ , the _communication network_ , and the
_users_ (customers). Every user is equipped with an electricity smart meter,
which measures the electricity consumption of the user in every $T_{p}$ long
period, and, using the communication network, sends the measurement to the
aggregator at the end of every slot (in practice, $T_{p}$ is around 1-30
minutes). Note that the communication and distribution network can be the same
(e.g., when PLC technology is used to transfer data). The measurement of user
$i$ in slot $t$ is denoted by $X_{t}^{i}$. The consumption profile of user $i$
is described by the vector $(X_{1}^{i},X_{2}^{i},\ldots)$, where the
measurements of different users are statistically independent. Privacy
directly correlates with $T_{p}$; finer-grained samples means more accurate
profile, but also entails weaker privacy. The supplier is interested in the
sum of all measurements in every slot (i.e.,
$\sum_{i=1}^{N}X_{t}^{i}\stackrel{{\scriptstyle\mathsf{def}}}{{=}}\mathbf{X}_{t}$).
As in [3], we also assume that smart meters are trusted devices (i.e., tamper-
resistant) which can store key materials and perform crypto computations. This
realistic assumption has also been confirmed in [2]. We assume that each node
is configured with a private key and gets the corresponding certificate from a
trusted third party. For example, each country might have a third party that
generates these certificate and can additionally generate the “supplier”
certificates to supplier companies [2]. As in [2], we also assume that public
key operations are employed only for initial key establishment, probably when
a meter is taken over by a new supplier. Messages exchanged between the
supplier and the meters are authenticated using pairwise MACs 111Please refer
to [18] for a more detailed discussion about key management issues in smart
metering systems.. Smart meters are assumed to have bidirectional
communication channel (using some wireless or PLC technology) with the
aggregator, but the meters cannot communicate with each other. We suppose that
nodes may (randomly) fail, and in these cases, cannot send their measurements
to the aggregator. However, nodes are supposed to use some reliable transport
protocol to overcome the transient communication failures of the channel.
Finally, we note that smart meters also allow the supplier to perform fine-
grained billing based on time-dependant variable tariffs. Here, we are not
concerned with the privacy and security problems of this service. Interested
readers are referred to [20, 17].
### 3.2 Adversary model
In general, the objective of the adversary is to infer detailed information
about household activity (e.g, how many people are in home and what they are
doing at a given time). In order to do that, it needs to extract complex usage
patterns of appliances which include the level of power consumption,
periodicity, and duration. It has been shown in [17] that different data
mining techniques can be easily applied to a raw consumption profile to obtain
this information.
In terms of its capability, we distinguish three types of adversary. The first
is the a _honest-but-curious (HC) adversary_ , who attempts to obtain private
information about a user, but it follows the protocol faithfully and do not
provide false information [17]. It only uses the (non-manipulated) collected
data.
The _dishonest-but-non-intrusive (DN) adversary_ may not follow the protocol
correctly and is allowed to provide false information to manipulate the
collected data. Some users can also be malicious and collude even with the
supplier to collect information about honest users. However, the DN adversary
is not allowed to access and modify the distribution network to mount attacks.
In particular, he is not allowed to install wiretapping devices to eavesdrop
on the victim’s consumption.
Likewise the DN adversary, the strongest _dishonest-and-intrusive (DI)
adversary_ may not follow all protocols either, but that can, in addition,
invade the distribution network to gather more information about clients. In
other words, the DI adversary can monitor the electricity consumption of the
clients by installing meters on the power line that is outside of the client’s
control (like outside from his household).
We suppose that all types of adversary can have any kind of extra knowledge
about honest users, beyond the collected measurements, which might help to
infer private information about them. For instance, it can observe their daily
activities222Similarly to monitoring neighbors. Indeed, neighbors can also be
malicious users, which is included in our model., or obtain extra information
by doing personal interviews, surveys, etc.
### 3.3 Privacy model
We use differential privacy [7] that models the adversary described above. In
particular, differential privacy guarantees that a user’s privacy should not
be threatened substantially more if he provides his measurement to the
supplier.
###### Definition 1 ($\varepsilon$-differential privacy).
An algorithm $\mathcal{A}$ is $\varepsilon$-differential private, if for all
data sets $D_{1}$ and $D_{2}$, where $D_{1}$ and $D_{2}$ differ in at most a
single user, and for all subsets of possible answers
$S\subseteq\mathit{Range}(\mathcal{A})$,
$P(\mathcal{A}(D_{1})\in S)\leq e^{\varepsilon}\cdot P(\mathcal{A}(D_{2})\in
S)$
Differential private algorithms produce indistinguishable outputs for similar
inputs (more precisely, differing by a single entry), and thus, the
modification of any single user’s data in the dataset changes the probability
of any output only up to a multiplicative factor $e^{\varepsilon}$. The
parameter $\varepsilon$ allows us to control the level of privacy. Lower
values of $\varepsilon$ implies stronger privacy, as they restrict further the
influence of a user’s data on the output. Note that this model, if users’ data
are independent, guarantees privacy for a user even if all other users’ data
is known to the adversary (e.g., it knows all measurements comprising the
aggregate except the target user’s), like when $N-1$ out of $N$ users are
malicious and cooperate with the supplier.
###### Example 1 (Illustration of $\varepsilon$-differential privacy).
There is a dataset $D$ containing a list of patients’ entries. Each entry has
an attribute that indicates whether the corresponding patient has cancer or
not. Suppose an $\varepsilon$-differential private query $\mathcal{A}$ that
returns the sanitized number of patients in $D$ that have cancer. We assume
that the adversary knows the exact number of cancer patients, $x$, before
adding Alice to $D$, and wants to learn from the random output $O$ of
$\mathcal{A}(D\cup\\{\text{Alice}\\})$ whether Alice has cancer or not. The
adversary has no prior knowledge about Alice (i.e., the probability that Alice
has cancer is 0.5 before accessing $O$). The adversary either infers Alice as
a cancer or a non-cancer patient. The success probability of this inference
has a maximum of $\frac{1}{1+e^{-\varepsilon}}$ (and $\geq 0.5$)333Let $A$
denote the event that Alice has cancer. Using a bayesian reasoning,
$P(A|O)=\frac{P(O|A)}{P(O|A)+P(O|\overline{A})}=\frac{P(\mathcal{A}(x+1)=O)}{P(\mathcal{A}(x+1)=O)+P(\mathcal{A}(x)=O)}\leq\frac{1}{1+e^{-\varepsilon}}$,
where we used that $P(A)=P(\overline{A})$ and
$e^{-\varepsilon}\leq\frac{P(\mathcal{A}(x)=O)}{P(\mathcal{A}(x+1)=O)}\leq
e^{\varepsilon}$. Moreover, the optimal inference strategy is the maximum
likelihood decision: the adversary infers Alice as a cancer patient if
$P(\mathcal{A}(x+1)=O)>P(\mathcal{A}(x)=O)$ or with probability 0.5 if
$P(\mathcal{A}(x+1)=O)=P(\mathcal{A}(x)=O)$, otherwise as a non-cancer
patient.. For example, the values 2, 1, 0.5, 0.1 of $\varepsilon$ yield
correct inferences with a maximum probability of 0.88, 0.73, 0.62, 0.52, resp.
The definition of differential privacy also maintains a _composability
property_ : the composition of differential private algorithms remains
differential private and their $\varepsilon$ parameters are accumulated. In
particular, a protocol having $t$ rounds, where each round is individually
$\varepsilon$ differential private, is itself $t\cdot\varepsilon$ differential
private.
### 3.4 Output perturbation: achieving differential privacy
Let’s say that we want to publish in a differentially private way the output
of a function $f$. The following theorem says that this goal can be achieved
by perturbing the output of $f$; simply adding a random noise to the value of
$f$, where the noise distribution is carefully calibrated to the global
sensitivity of $f$, results in $\varepsilon$-differential privacy. The global
sensitivity of a function is the maximum ”change” in the value of the function
when its input differs in a single entry. For instance, if $f$ is the sum of
all its inputs, the sensitivity is the maximum value that an input can take.
###### Theorem 1 (Laplacian Perturbation Algorithm (LPA) [7]).
For all $f:\mathbb{D}\rightarrow\mathbb{R}^{r}$, the following mechanism
$\mathcal{A}$ is $\varepsilon$-differential private:
$\mathcal{A}(D)=f(D)+\mathcal{L}(S(f)/\varepsilon)$, where
$\mathcal{L}(S(f)/\varepsilon)$ is an independently generated random variable
following the Laplace distribution and $S(f)$ denotes the global sensitivity
of $f$444Formally, let $f:\mathbb{D}\rightarrow\mathbb{R}^{r}$, then the
global sensitivity of $f$ is $S(f)=\max||f(D_{1})-f(D_{2})||_{1}$, where
$D_{1}$ and $D_{2}$ differ in a single entry and $||\cdot||_{1}$ denotes the
$L_{1}$ distance..
###### Example 2.
To illustrate these definitions, consider a mini smart metering application,
where users $U_{1}$, $U_{2}$, and $U_{3}$ need to send the sum of their
measurements in two consecutive slots. The measurements of $U_{1}$, $U_{2}$
and $U_{3}$ are $(X_{1}^{1}=300,X_{2}^{1}=300)$,
$(X_{1}^{2}=100,X_{2}^{2}=400)$, and $(X_{1}^{3}=50,X_{2}^{3}=150)$, resp. The
nodes want differential privacy for the released sums with at least a
$\varepsilon=0.5$. Based on Theorem 1, they need to add
$\mathcal{L}(\lambda=\max_{i}\sum_{t}X_{t}^{i}/0.5=1200)$ noise to the
released sum in each slot. This noise ensures
$\varepsilon=\sum_{t}X_{t}^{1}/\lambda=0.5$ individual indistinguishability
for $U_{1}$, $\varepsilon=0.42$ for $U_{2}$, and $\varepsilon=0.17$ for
$U_{3}$. Hence, the global $\varepsilon=0.5$ bound is guaranteed to all.
Another interpretation is that $U_{1}$ has
$\varepsilon_{1}=X_{1}^{1}/\lambda=0.25$,
$\varepsilon_{2}=X_{2}^{1}/\lambda=0.25$ privacy in each individual slot, and
$\varepsilon=\varepsilon_{1}+\varepsilon_{2}=0.5$ considering all two slots
following from the composition property of differential privacy.
### 3.5 Utility definition
Let $f:\mathbb{D}\rightarrow\mathbb{R}$. In order to measure the utility, we
quantify the difference between $f(D)$ and its perturbed value (i.e.,
$\hat{f}(D)=f(D)+\mathcal{L}(\lambda)$) which is the error introduced by LPA.
A common scale-dependant error measure is the Mean Absolute Error (MAE), which
is $\mathbb{E}|f(D)-\hat{f}(D)|$ in our case. However, the error should be
dependent on the non-perturbed value of $f(D)$; if $f(D)$ is greater, the
added noise becomes small compared to $f(D)$ which intuitively results in
better utility. Hence, we rather use a slightly modified version of a scale-
independent metric called Mean Absolute Percentage Error (MAPE), which shows
the proportion of the error to the data, as follows.
###### Definition 2 (Error function).
Let $D_{t}\in\mathbb{D}$ denote a dataset in time-slot $t$. Furthermore, let
$\delta_{t}=\frac{|f(D_{t})-\hat{f}(D_{t})|}{f(D_{t})+1}$ (i.e., the value of
the error in slot $t$). The error function is defined as
$\mu(t)=\mathbb{E}(\delta_{t})$. The expectation is taken on the randomness of
$\hat{f}(D_{t})$. The standard deviation of the error is
$\sigma(t)=\sqrt{\mathit{Var}(\delta_{t})}$ in time $t$.
In the rest of this paper, the terms ”utility” and ”error” are used
interchangeably.
## 4 Objectives
Our goal is to develop a practical scheme that should not introduce more
privacy risks for users than traditional metering systems while retaining the
benefits of smart meters. More specifically, the scheme should be
* •
_differentially private_ : Considering DN adversary, the scheme differential
privately releases sanitized aggregates $\hat{\mathbf{X}}_{t}$ where the
leaked information about users is measured by $\varepsilon$.
* •
_robust and easily configurable_ : It tolerates (random) node failures.
* •
_efficient_ : It has low overhead which includes low computation load on smart
meters, and low communication overhead between the supplier and individual
meters. It should use pubic key operations only for initial key establishment.
Afterwards, all communication is protected using more efficient symmetric
crypto-based techniques.
* •
_distributed_ : Besides a certificate authority, the protocol does not require
any trusted third party such as a trusted aggregator as in [3]. The smart
meters communicate directly with the supplier.
* •
_useful for the supplier_ : The sanitized and the original (non-sanitized)
aggregate should be “similar” (i.e., the error should be as small as
possible). For instance, the supplier should be able to perform efficient
management of the resource using the sanitized data: to monitor the
consumption at the granularity of a maximum few hundred households, and to
detect consumption peaks or abnormal consumption.
## 5 Overview of approaches
Our task is to enable the supplier to calculate the sum of maximum $N$
measurements (i.e., $\sum_{i=1}^{N}X_{t}^{i}=\mathbf{X}_{t}$ in all $t$)
coming from $N$ different users while ensuring $\varepsilon$-differential
privacy for each user. This is guaranteed if the supplier can only access
$\mathbf{X}_{t}+\mathcal{L}(\lambda(t))$, where $\mathcal{L}(\lambda(t))$
555We will use the notation $\lambda$ instead of $\lambda(t)$ if the
dependency on time is obvious in the context. is the Laplace noise calibrated
to $\varepsilon$ as it has been described in Section 3.4. There are (at least)
3 possible approaches to do this which are detailed as follows.
Node 1Node 2$\ldots$Node
$N$AggregatorSupplier$\mathit{Dec}(\sum_{i}\mathit{Enc}(X_{t}^{i}+\mathcal{L}(\lambda)))=\mathbf{X}_{t}+\mathcal{L}(\lambda)$$\sum_{i}\mathit{Enc}(X_{t}^{i})+\mathit{Enc}(\mathcal{L}(\lambda))$$\mathit{Enc}(X_{t}^{1})$$\mathit{Enc}(X_{t}^{2})$$\mathit{Enc}(X_{t}^{N})$
(a) Centralized approach: aggregation with trusted aggregator.
Node 1Node 2$\ldots$Node
$N$Supplier/Aggregator$\mathit{Dec}(\sum_{i}\mathit{Enc}(X_{t}^{i}+\sigma_{i}))=\mathbf{X}_{t}+\mathcal{L}(\lambda)$$\mathit{Enc}(X_{t}^{1}+\sigma_{1})$$\mathit{Enc}(X_{t}^{2}+\sigma_{2})$$\mathit{Enc}(X_{t}^{N}+\sigma_{N})$
(b) Our approach: aggregation without trusted entity. If
$\sigma_{i}=\mathcal{G}_{1}(N,\lambda)+\mathcal{G}_{2}(N,\lambda)$, where
$\mathcal{G}_{1}$, $\mathcal{G}_{2}$ are i.i.d gamma noise, then
$\sum_{i=1}^{N}\sigma_{i}=\mathcal{L}(\lambda)$.
Figure 1: Aggregating measurements while guaranteeing differential privacy.
### 5.1 Fully decentralized approach (without aggregator)
Our first attempt is that each user adds some noise to its own measurement,
where the noise is drawn from a Laplace distribution. In particular, each node
$i$ sends the value of $X_{t}^{i}+\mathcal{L}(\lambda)$ directly to the
supplier in time $t$. It is easy to see that $\varepsilon$ is guaranteed to
all users, but in fact the final noise added to the aggregate (i.e.,
$\sum_{i=1}^{N}\mathcal{L}(\lambda)$) is $N$ times larger than
$\mathcal{L}(\lambda)$, and hence, the error is
$\mu(t)=\frac{1}{\mathbf{X}_{t}+1}\mathbb{E}|\sum_{i=1}^{N}\mathcal{L}(\lambda)|=\frac{N\cdot\lambda}{\mathbf{X}_{t}+1}$.
### 5.2 Aggregation with a trusted aggregator
Our second attempt can be to aggregate the measurements of some users, and
send the perturbed aggregate to the supplier. In particular, nodes are grouped
into $N$ sized clusters and each node of a cluster sends its measurement
$X_{t}^{i}$ to the (trusted) cluster aggregator, that is a trusted entity
different from the supplier. The aggregator computes
$\mathbf{X}_{t}=\sum_{i=1}^{N}X_{t}^{i}$ and obtains
$\hat{\mathbf{X}}_{t}=\mathbf{X}_{t}+\mathcal{L}(\lambda)$ by adding noise to
the aggregate. This perturbed aggregate is then sent to the supplier as it is
illustrated in Figure 1(a).
The utility of this approach is better than in the previous case, as the noise
is only added to the sum and not to each measurement $X_{t}^{i}$. Formally,
$\mu(t)=\frac{1}{\mathbf{X}_{t}+1}\mathbb{E}|\mathcal{L}(\lambda)|=\frac{\lambda}{\mathbf{X}_{t}+1}$.
Similary,
$\delta(t)=\frac{1}{\mathbf{X}_{t}+1}\cdot\sqrt{\mathbb{E}|\mathcal{L}(\lambda)|^{2}-(\mathbb{E}|\mathcal{L}(\lambda)|)^{2}}=\frac{\lambda}{\mathbf{X}_{t}+1}$.
However, the main drawback of this approach is that the aggregator must be
fully trusted since it receives each individual measurement from the users.
This can make this scheme impractical if there is no such trusted entity.
### 5.3 Our approach: aggregation without trusted entity
Although the previous scheme is differential private, it works only if the
aggregator is trustworthy and faithfully adds the noise to the measurement. In
particular, the scheme will not be secure if the aggregator omits to add the
noise.
Our scheme, instead, does not rely on any centralized aggregator. The noise is
added by each smart meter on their individual data and encrypted in such a way
that the aggregator can only compute the (noisy) aggregate. Note that with our
approach the aggregator and the supplier do need to be separate entities. The
supplier can even play the role of the aggregator, as the encryption prevents
it to access individual measurements, and the distributed generation of the
noise ensures that it cannot manipulate the noise.
Our proposal is composed of 2 main steps: distributed generation of the
Laplacian noise and encryption of individual measurements. These 2 steps are
described in the remainder of this section.
#### 5.3.1 Distributed noise generation: a new approach
In our proposal, the Laplacian noise is generated in a fully distributed way
as is illustrated in Figure 1(b). We use the following lemma that states that
the Laplace distribution is divisible and be constructed as the sum of i.i.d.
gamma distributions. As this divisibility is infinite, it works for arbitrary
number of users.
###### Lemma 1 (Divisibility of Laplace distribution [15]).
Let $\mathcal{L}(\lambda)$ denote a random variable which has a Laplace
distribution with PDF
$f(x,\lambda)=\frac{1}{2\lambda}e^{\frac{|x|}{\lambda}}$. Then the
distribution of $\mathcal{L}(\lambda)$ is infinitely divisible. Furthermore,
for every integer $n\geq 1$,
$\mathcal{L}(\lambda)=\sum_{i=1}^{n}[\mathcal{G}_{1}(n,\lambda)-\mathcal{G}_{2}(n,\lambda)]$,
where $\mathcal{G}_{1}(n,\lambda)$ and $\mathcal{G}_{2}(n,\lambda)$ are i.i.d.
random variables having gamma distribution with PDF
$g(x,n,\lambda)=\frac{(1/\lambda)^{1/n}}{\Gamma(1/n)}x^{\frac{1}{n}-1}e^{-x/\lambda}$
where $x\geq 0$.
The lemma comes from the fact that $\mathcal{L}(\lambda)$ can be represented
as the difference of two i.i.d exponential random variables with rate
parameter $1/\lambda$. Moreover,
$\sum_{i=1}^{n}\mathcal{G}_{1}(n,\lambda)-\sum_{i=1}^{n}\mathcal{G}_{2}(n,\lambda)=\mathcal{G}_{1}(1/\sum_{i=1}^{n}\frac{1}{n},\lambda)-\mathcal{G}_{2}(1/\sum_{i=1}^{n}\frac{1}{n},\lambda)=\mathcal{G}_{1}(1,\lambda)-\mathcal{G}_{2}(1,\lambda)$
due to the summation property of the gamma distribution666The sum of i.i.d.
gamma random variables follows gamma distribution (i.e.,
$\sum_{i=1}^{n}\mathcal{G}(k_{i},\lambda)=\mathcal{G}(1/\sum_{i=1}^{n}\frac{1}{k_{i}},\lambda)$)..
Here, $\mathcal{G}_{1}(1,\lambda)$ and $\mathcal{G}_{2}(1,\lambda)$ are i.i.d
exponential random variable with rate parameter $1/\lambda$ which completes
the argument.
Our distributed sanitization algorithm is simple; user $i$ calculates value
$\hat{X}_{t}^{i}=X_{t}^{i}+\mathcal{G}_{1}(N,\lambda)-\mathcal{G}_{2}(N,\lambda)$
in slot $t$ and sends it to the aggregator, where $\mathcal{G}_{1}(N,\lambda)$
and $\mathcal{G}_{2}(N,\lambda)$ denote two random values independently drawn
from the same gamma distribution. Now, if the aggregator sums up all values
received from the $N$ users of a cluster, then
$\sum_{i=1}^{N}\hat{X}_{t}^{i}=\sum_{i=1}^{N}X_{t}^{i}+\sum_{i=1}^{N}[\mathcal{G}_{1}(N,\lambda)-\mathcal{G}_{2}(N,\lambda)]=\mathbf{X}_{t}+\mathcal{L}(\lambda)$
based on Lemma 1.
The utility of our distributed scheme is defined as
$\mu(t)=\frac{1}{\mathbf{X}_{t}+1}\mathbb{E}|\mathbf{X}_{t}-\mathbf{X}_{t}+\sum_{i=1}^{n}[\mathcal{G}_{1}(N,\lambda)-\mathcal{G}_{2}(N,\lambda)]|=\frac{\mathbb{E}|\mathcal{L}(\lambda)|}{\mathbf{X}_{t}+1}=\frac{\lambda}{\mathbf{X}_{t}+1}$,
and $\delta(t)=\frac{\lambda}{\mathbf{X}_{t}+1}$.
#### 5.3.2 Encryption
The previous step is not enough to guarantee privacy as only the sum of the
measurements (i.e., $\hat{\mathbf{X}}_{t}$) is differential private but not
the individual measurements. In particular, the aggregator has access to
$\hat{X}_{t}^{i}$, and even if $\hat{X}_{t}^{i}$ is noisy,
$\mathcal{G}_{1}(N,\lambda)-\mathcal{G}_{2}(N,\lambda)$ is usually
insufficient to provide reasonable privacy for individual users if $N\gg 1$.
This is illustrated in Figure 2, where an individual’s noisy and original
measurements slightly differ.
(a) $X_{t}^{i}$ (b)
$X_{t}^{i}+\mathcal{G}_{1}(N,\lambda)-\mathcal{G}_{2}(N,\lambda)$
Figure 2: The original and noisy measurements of user $i$, where the added
noise is $\mathcal{G}_{1}(N,\lambda)-\mathcal{G}_{2}(N,\lambda)$ ($N=100$,
$T_{p}$ is 10 min).
To address this problem, each contribution is encrypted using a modulo
addition-based encryption scheme, inspired by [4], such that the aggregator
can only decrypt the sum of the individual values, and cannot access any of
them. In particular, let $k_{i}$ denote a random key generated by user $i$
inside a cluster such that $\sum_{i=1}^{N}k_{i}=0$, and $k_{i}$ is not known
to the aggregator. Furthermore, $Enc()$ denotes a probabilistic encryption
scheme such that $Enc(p,k,m)=p+k\mod m$, where $p$ is the plaintext, $k$ is
the encryption key, and $m$ is a large integer. The adversary cannot decrypt
any $\mathit{Enc}(\hat{X}_{t}^{i},k_{i},m)$, since it does not know $k_{i}$,
but it can easily retrieve the noisy sum by adding the encrypted noisy
measurements of all users;
$\sum_{i=1}^{N}Enc(\hat{X}_{t}^{i},k_{i},m)=\sum_{i=1}^{N}\hat{X}_{t}^{i}+\sum_{i=1}^{N}k_{i}=\sum_{i=1}^{N}\hat{X}_{t}^{i}\mod
m$. If $z=\max_{i,t}(\hat{X}_{t}^{i})$ then $m$ should be selected as
$m=2^{\lceil\log_{2}(z\cdot N)\rceil}$ [4]. The generation of $k_{i}$ is
described in Section 6.2.
## 6 Protocol description
### 6.1 System setup
In our scheme, nodes are grouped into clusters of size $N$, where $N$ is a
parameter. The protocol requires the establishment of pairwise keys between
each pair of nodes inside a cluster that can be done by using traditional
Diffie-Hellman key exchange as follows. When a node $v_{i}$ is installed, it
provides a self-signed DH component and its certificate to the supplier. Once
all the nodes of a cluster are installed, or a new node is deployed, the
supplier broadcasts the certificates and public DH components of all nodes.
Finally, each node $v_{i}$ of the cluster can compute a pairwise key $K_{i,j}$
shared with any other node $v_{j}$ in the networks. Note that no communication
is required between $v_{i}$ and $v_{j}$.
### 6.2 Smart meter processing
Each node $v_{i}$ sends at time $t$ its periodic measurement, $X^{i}_{t}$, to
the supplier as follows:
Phase 1 (Data sanitization):
Node $v_{i}$ calculates value
$\hat{X}_{t}^{i}=X_{t}^{i}+\mathcal{G}_{1}(N,\lambda)-\mathcal{G}_{2}(N,\lambda)$,
where $\mathcal{G}_{1}(N,\lambda)$ and $\mathcal{G}_{2}(N,\lambda)$ denote two
random values independently drawn from the same gamma distribution and $N$ is
the cluster size.
Phase 2 (Data encryption):
Each noisy data $\hat{X}_{t}^{i}$ is then encrypted into
$\mathit{Enc}(\hat{X}_{t}^{i})$ using the modulo addition-based encryption
scheme detailed in Section 5.3.2. The following extension is then applied to
generate the encryption keys: Each node, $v_{i}$, selects $\ell$ other nodes
randomly, such that if $v_{i}$ selects $v_{j}$, then $v_{j}$ also selects
$v_{i}$. Afterwards, both nodes generate a common dummy key $k$ from their
pairwise key $K_{i,j}$; $v_{i}$ adds $k$ to $\mathit{Enc}(\hat{X}_{t}^{i})$
and $v_{j}$ adds $-k$ to $\mathit{Enc}(\hat{X}_{t}^{j})$. As a result, the
aggregator cannot decrypt the individual ciphertexts (it does not know the
dummy key $k$). However, it adds all the ciphertexts of a given cluster, the
dummy keys cancel out and it retrieves the encrypted sum of the (noisy)
contributions. The more formal description is as follows:
1. 1.
node $v_{i}$ selects some nodes of the cluster randomly (we call them
participating nodes) using a secure pseudo random function (PRF) such that if
$v_{i}$ selects $v_{j}$, then $v_{j}$ also selects $v_{i}$. In particular,
$v_{i}$ selects $v_{j}$ if mapping $\mathit{PRF}(K_{i,j},r_{1})$ to a value
between 0 and 1 is less or equal than $\frac{w}{N-1}$, where $r_{1}$ is a
public value changing in each slot. We denote by $\ell$ the number of selected
participating nodes, and $\mathsf{ind}_{i}[j]$ (for $j=1,\ldots,\ell$) denotes
the index of the $\ell$ nodes selected by node $v_{i}$. Note that, for the
supplier, the probability that $v_{i}$ selects $v_{j}$ is $\frac{w}{N-1}$ as
it does not know $K_{i,j}$. The expected value of $\ell$ is $w$.
2. 2.
$v_{i}$ computes for each of its $\ell$ participating nodes a dummy key. A
dummy key between $v_{i}$ and $v_{j}$ is defined as
$\mathsf{dkey}_{i,j}=(i-j)/|i-j|\cdot\mathit{PRF}(K_{i,j},r_{2})$, where
$K_{i,j}$ is the key shared by $v_{i}$ and $v_{j}$, and $r_{2}\neq r_{1}$ is
public value changing in each slot. Note that
$\mathsf{dkey}_{i,j}=-\mathsf{dkey}_{j,i}$.
3. 3.
$v_{i}$ then computes
$\mathit{Enc}(\hat{X}_{t}^{i})=\hat{X}_{t}^{i}+K^{\prime}_{i}+\sum^{\ell}_{j=1}\mathsf{dkey}_{i,\mathsf{ind}_{i}[j]}\pmod{m}$,
where $K^{\prime}_{i}$ is the keystream shared by $v_{i}$ and the aggregator
which can be established using the DH protocol as above, and $m$ is a large
integer (see [4]). Note that $m$ must be larger than the sum of all
contributions (i.e., final aggregate) plus the Laplacian noise.777Note that
the noise is a random value from an infinite domain and this sum might be
larger than $m$. However, choosing sufficiently large $m$, the probability
that the sum exceeds $m$ can be made arbitrary small due to the exponential
tail of the Laplace distribution.
Note that $\hat{X}_{t}^{i}$ is encrypted multiple times: it is first encrypted
with the keystream $K^{\prime}_{i}$ and then with several dummy keys.
$K^{\prime}_{i}$ is needed to ensure confidentiality between a user and the
aggregator. The dummy keys are needed to prevent the aggregator (supplier)
from retrieving $\hat{X}_{t}^{i}$.
4. 4.
$\mathit{Enc}(\hat{X}_{t}^{i})$ is sent to the aggregator (supplier).
### 6.3 Supplier processing
Phase 1 (Data aggregation):
At each slot, the supplier aggregates the $N$ measurements received from the
cluster smart meters by summing them, and obtains
$\sum^{N}_{i=1}\mathit{Enc}(X_{t}^{i})$. In particular,
$\mathit{Enc}(\hat{\mathbf{X}}_{t})=\sum^{N}_{i=1}(\hat{X}_{t}^{i}+K^{\prime}_{i})+\sum^{N}_{i=1}\sum^{\ell}_{j=1}\mathsf{dkey}_{i,\mathsf{ind}_{i}[j]}\pmod{m}$
where $\sum^{N}_{i=1}\sum^{\ell}_{j=1}\mathsf{dkey}_{i,\mathsf{ind}_{i}[j]}=0$
because $\mathsf{dkey}_{i,j}=-\mathsf{dkey}_{j,i}$. Hence,
$\mathit{Enc}(\hat{\mathbf{X}}_{t})=\sum^{N}_{i=1}(\hat{X}_{t}^{i}+K^{\prime}_{i})=\sum^{N}_{i=1}\mathit{Enc}(\hat{X}_{t}^{i})$
Phase 2 (Data decryption):
The aggregator then decrypts the aggregated value by subtracting the sum of
the node’s keystream, and retrieves the sum of the noisy measures:
$\sum^{N}_{i=1}\mathit{Enc}(\hat{X}_{t}^{i})-\sum^{N}_{i=1}K^{\prime}_{i}=\sum^{N}_{i=1}\hat{X}_{t}^{i}\pmod{m}$
where
$\sum^{N}_{i=1}\hat{X}_{t}^{i}=\sum^{N}_{i=1}X_{t}^{i}+\sum^{N}_{i=1}\mathcal{G}_{1}(N,\lambda)-\sum^{N}_{i=1}\mathcal{G}_{2}(N,\lambda)=\sum^{N}_{i=1}X_{t}^{i}+\mathcal{L}(\lambda)$
based on Lemma 1.
The main idea of the scheme is that the aggregator is not able to decrypt the
individual encrypted values because it does not know the dummy keys. However,
by adding the different encrypted contributions, dummy keys cancel each other
and the aggregator can retrieve the sum of the plaintext. The resulting
plaintext is then the perturbed sums of the measurements, where the noise
ensures the differential privacy of each user.
##### Complexity:
Let $b$ denote the size of the pairwise keys (i.e., $K_{i,j}$). Our scheme has
$O(N\cdot b)$ storage complexity, as each node needs to store $\ell\leq N$
pairwise keys. The computational overhead is dominated by the encryption and
the key generation complexity. The encryption is composed of $\ell\leq N$
modular addition of $\log_{2}m$ bits long integers, while the key generation
needs the same number of PRF executions. This results in a complexity of
$O(N\cdot(\log_{2}m+c(b)))$, where $c(b)$ is the complexity of the applied PRF
function. 888For instance, if $\log_{2}m=32$ bits (which should be sufficient
in our application), $b=128$, and $N=1000$, a node needs to store 16 Kb of key
data and perform maximum 1000 additions along with 1000 subtractions (for
modular reduction) on 32 bits long integers, and maximum 1000 PRF executions.
This overhead should be negligible even on constrained embedded devices.
## 7 Adding robustness
We have assumed so far that all the $N$ nodes of a cluster participated in the
protocol. However, it might happen that, for several different reasons (e.g.,
node or communication failures) some nodes are not able to participate in each
epoch. This would have two effects: first, security will be reduced since the
sum of the noise added by each node will not be equivalent to
$\mathcal{L}(\lambda)$. Hence, differential privacy may not be guaranteed.
Second, the aggregator will not be able to decrypt the aggregated value since
the sum of the dummy keys will not cancel out.
In this section, we extend our scheme to resist node failures. We propose a
scheme which resists the failure of up to $M$ out of $N$ nodes, where $M$ is a
configuration parameter. We will study later the impact of the value $M$ on
the scheme performance.
### 7.1 Sanitization phase extension
In order to resist the failure of $M$ nodes, each node should add the
following noise to their individual measurement:
$\mathcal{G}_{1}(N-M,\lambda)-\mathcal{G}_{2}(N-M,\lambda)$. Note that
$\sum_{i=1}^{N-M}[\mathcal{G}_{1}(N-M,\lambda)-\mathcal{G}_{2}(N-M,\lambda)]=\mathcal{L}(\lambda)$.
Therefore, this sanitization algorithm remains differential private, if at
least $N-M$ nodes participate in the protocol. Note that in that case each
node adds extra noise to the aggregate in order to ensure differential privacy
even if fewer than $M$ nodes fail to send their noise share to the aggregator.
### 7.2 Encryption phase extension
#### 7.2.1 A simple approach
As described previously, all the dummy keys cancel out at the aggregator.
However, this is not the case if not all the nodes participate in the
protocol. In order to resist the failure of nodes, one can extend the
encryption scheme with an additional round where the aggregator asks the
participating nodes of non-responding nodes for the missing dummy keys:
1. 1.
Once the aggregator received all contributions, it broadcasts the ids of the
non-responding nodes.
2. 2.
Upon the reception of this message, each node $v_{i}$ verifies whether any of
the ids in the broadcast message are in its participating node list (i.e., it
can be found in $\mathsf{ind}_{i}$). For each of such id, the node sends the
corresponding dummy key to the aggregator.
3. 3.
The aggregator then subtracts all received dummy keys from
$\mathit{Enc}(\hat{\mathbf{X}}_{t})$ and retrieves
$\sum^{N}_{i=1}(\hat{X}_{t}^{i}+K^{\prime}_{i})$ which can be decrypted.
This approach has a severe problem: if the aggregator is untrusted, it can
easily retrieve the measurement of a $v_{i}$: broadcasting its id in Step 2,
the participating nodes of $v_{i}$ reply with the dummy keys of $v_{i}$ which
can be removed from $\mathit{Enc}(\hat{X}_{t}^{i})$.
#### 7.2.2 Our proposal
In this approach, each node adds a secret random value to its encrypted value
before releasing it in the first round. This is needed to prevent the
adversary to recover the noisy measurement through combining different
messages of the nodes. Then, in the second round when the aggregator asks for
the missing dummy keys, every node reveals its random keys along with the
missing dummy keys that it knows:
1. 1.
Each node $v_{i}$ sends
$\mathit{Enc}(\hat{X}_{t}^{i})=\hat{X}_{t}^{i}+K^{\prime}_{i}+\sum^{\ell}_{j=1}\mathsf{dkey}_{i,\mathsf{ind}_{i}[j]}+C_{i}\pmod{m}$
where $C_{i}$ is the secret random key of $v_{i}$ generated randomly in each
round.
2. 2.
After receiving all measurements, the aggregator asks all nodes for their
random keys and the missing dummy keys through broadcasting the id of the non-
responding nodes.
3. 3.
Each node $v_{i}$ verifies whether any ids in this broadcast message are in
its participating node list, where the set of the corresponding participating
nodes is denoted by $S$. Then, $v_{i}$ replies with $\sum_{j\in
S}\mathsf{dkey}_{i,\mathsf{ind}_{i}[j]}+C_{i}\pmod{m}$.
4. 4.
The aggregator subtracts all received values from
$\sum_{i=1}^{N}\mathit{Enc}(\hat{X}_{t}^{i})$ which results in
$\sum^{N}_{i=1}(\hat{X}_{t}^{i}+K^{\prime}_{i})$, as the random keys as well
as the dummy keys cancel out.
Note that as the supplier does not know the random keys, it cannot remove them
from any messages but only from the final aggregate; adding each node’s
response to the aggregate all the dummy keys and secret random keys cancel out
and the supplier obtains $\hat{\mathbf{X}}_{t}$. Although the supplier can
still recover $\hat{X}_{t}^{i}$ if it knows $v_{i}$’s participating nodes (the
supplier simply asks for all the dummy keys of $v_{i}$ in Step 2 and subtracts
$v_{i}$’s response in Step 4 from $\mathit{Enc}(\hat{X}_{t}^{i})$), we will
show later that this probability can be made practically small by adjusting
$w$ and $N$ correctly.
Note that the protocol fails if, for some reasons, a node does not send its
random key to the aggregator (as only the node itself knows its random key, it
cannot be reconstructed by other parties). However, it is very unlikely that a
node between the two rounds fails, and an underlying reliable transport
protocol helps to overcome communication errors.
Finally, also note that this random key approach always requires two rounds of
communication (even if the aggregator receives all encrypted values correctly
in the first round), as the random keys are needed to be removed from
$\mathit{Enc}(\hat{\mathbf{X}}_{t})$ in the second round.
### 7.3 Utility evaluation
If all $N$ nodes participate in the protocol, the added noise will be larger
than $\mathcal{L}(\lambda)$ which is needed to ensure differential privacy. In
particular,
$\sum_{i=1}^{N}[\mathcal{G}_{1}(N-M,\lambda)-\mathcal{G}_{2}(N-M,\lambda)]=\mathcal{L}(\lambda)+\sum_{i=1}^{M}[\mathcal{G}_{1}(N-M,\lambda)-\mathcal{G}_{2}(N-M,\lambda)]$,
where the last summand is the extra noise needed to tolerate the failure of
maximum $M$ nodes. Clearly, this extra noise increases the error if all $N$
nodes operate correctly and add their noise shares faithfully. In what
follows, we calculate the error and its standard deviation if we add this
extra noise to the aggregate.
###### Theorem 2 (Utility).
Let $\alpha=M/N$ and $\alpha<1$. Then,
$\mu(t)\leq\frac{2}{B(1/2,\frac{1}{1-\alpha})}\cdot\frac{\lambda(t)}{\mathbf{X}_{t}+1}$
and
$\sigma(t)\leq\sqrt{\left(\frac{2}{1-\alpha}-\frac{4}{B(1/2,\frac{1}{1-\alpha})^{2}}\right)}\cdot\frac{\lambda(t)}{\mathbf{X}_{t}+1}$
where $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ is the beta function.
The derivation can be found in Appendix A. Based on Theorem 2,
$\sigma(t)=\mu(t)\cdot\left(\frac{2}{B(1/2,\frac{1}{1-\alpha})}\right)^{-1}\cdot\sqrt{\left(\frac{2}{1-\alpha}-\frac{4}{B(1/2,\frac{1}{1-\alpha})^{2}}\right)}$.
It is easy to check that $\sigma(t)$ is always less or equal than $\mu(t)$. In
particular, if $\alpha=0$ (there are no malicious nodes and node failures),
then $\sigma(t)=\mu(t)$. If $\alpha>0$ then $\sigma(t)<\mu(t)$ but
$\sigma(t)\approx\mu(t)$.
## 8 Security Analysis
### 8.1 Deploying malicious nodes
In the proposed scheme, each measurement is perturbed and encrypted.
Therefore, a honest-but-curious attacker cannot gain any information (up to
$\varepsilon$) about individual measurements in any slot. This is guaranteed
by the encryption scheme and the added noise.
However, a DN adversary (see Section 3.2), which deploys $T$ malicious nodes,
may be able to:
* •
reduce the noise level by limiting (or omitting) the gamma noise added by
malicious nodes. As a result, the sum of the noise shares will not equal to
the Laplacian noise which can decrease the privacy of users. However, recall
that, due to the robustness property of our scheme detailed in Section 7, we
add extra noise to tolerate $M$ node failures. Adding extra noise calibrated
to $M+T$ is sufficient to tolerate this type of attack.
* •
decrypt $\mathit{Enc}(\hat{X}_{t}^{i})$ of a node $v_{i}$ and retrieve the
perturbed data. As individual data is only weakly noised, the attacker might
infer some information from them, and therefore, compromise privacy. However,
the encryption scheme that we used is provably secure [4], and nodes are
assumed to be tamper-resistant. Thus, the only way to break privacy is to
retrieve the dummy keys of $v_{i}$. Because the participating nodes are
selected randomly for each message, this can only be achieved if all
participating nodes of $v_{i}$ are malicious and the supplier is also
malicious (i.e., the adversary knows $K^{\prime}_{i}$). This happens if
$v_{i}$ does not select any honest participating node that has a probability
of $(1-\frac{w}{N-1})^{N-T-1}$. For instance, it is easy to check that if
$N=100$ and 50% of the nodes are malicious (which anyway should be a quite
strong assumption), then setting $w$ to 30 results in a success probability of
$1.8\cdot 10^{-8}$. This means that if an epoch is 5 min long, then the
adversary will compromise 1 measurement during 458 years in average.
Finally, also note that this is the success probability of the adversary in a
single slot. This means that a supplier that succeeds the previous attack only
gets a single (noisy) measurement of the customer (corresponding to a single
epoch). As a node selects different participating nodes in each slot, the
probability that the adversary gets $k$ different measurements of the node is
$(1-\frac{w}{N-1})^{k(N-T-1)}$, which is even smaller.
### 8.2 Lying supplier
#### Lying about non-responding nodes
In addition to deploying malicious (fake) nodes, a malicious supplier can lie
about the non-responding nodes. In order to recover $\hat{X}_{t}^{i}$, the
supplier needs $\sum^{\ell}_{j=1}\mathsf{dkey}_{i,\mathsf{ind}_{i}[j]}+C_{i}$.
The supplier has two options to retrieve this sum. First, it might pretend
that a node $v_{i}$ did not respond in the first round, and asks for $v_{i}$’s
dummy keys to its participating nodes. At the same time, the supplier claims
to $v_{i}$ that its participating nodes are responding. Hence, as described in
Section 7.2.2, the participating nodes of $v_{i}$ will disclose $v_{i}$’s
dummy keys and $v_{i}$ will disclose $C_{i}$. However, the random keys of
$v_{i}$’s participating nodes prevent the supplier to retrieve $v_{i}$’s dummy
keys from their messages.
Second, the supplier can pretend that $v_{i}$’s participating nodes do not
respond in the first round, and asks $v_{i}$ for their dummy keys in the
second round. In particular, there are three types of dummy keys: the first is
shared with a malicious node, and hence, known to the supplier. The second is
asked to $v_{i}$ by the supplier in the second round (the supplier pretends
that these nodes are non-responding), and $v_{i}$ replies with the sum of
$C_{i}$ and the requested keys. Finally, the rest is shared with honest
participating nodes and they are not asked to $v_{i}$ in the second round.
Apparently, if $v_{i}$ has at least one dummy key from the last group, its
measurement cannot be recovered. This is because if $v_{j}$ is a participating
honest node of $v_{i}$ and $\mathsf{dkey}_{i,\mathsf{ind}_{i}[j]}$ is not
asked to $v_{i}$ in the second round, it could only be recovered from
$v_{j}$’s messages. However, $v_{j}$ sends
$C_{j}+\mathsf{dkey}_{i,\mathsf{ind}_{i}[j]}$, where $C_{j}$ is only known to
$v_{j}$.
Nevertheless, it might happen that $v_{i}$ does not have any third-type dummy
key (i.e., the supplier asks $v_{i}$ for all the dummy keys shared with honest
nodes). Then, the supplier can easily recover $v_{i}$’s measurement, since it
knows $\sum^{\ell}_{j=1}\mathsf{dkey}_{i,\mathsf{ind}_{i}[j]}+C_{i}$ (they are
malicious keys or provided by $v_{i}$). However, the supplier can only guess
$v_{i}$’s participating nodes and target them randomly since $v_{i}$ also
selects them randomly999Note that _all_ nodes send responses in the second
round, and the randomness of $C_{i}$ ensures that the supplier cannot gain any
knowledge about the participating nodes of any nodes.. Assuming that the
supplier can ask $v_{i}$ for maximum $M$ dummy keys in the second round, the
probability that all participating nodes of $v_{i}$ are either malicious or
specified as non-responding nodes by the supplier is less than
$(1-\frac{w}{N-1})^{N-(T+M)-1}$. Using $\alpha=(T+M)/N$ and $\beta=w/N$, then
$(1-\frac{w}{N-1})^{N-(T+M)-1}=(1-\frac{\beta}{1-N^{-1}})^{N(1-\alpha)-1}$.
This probability is depicted in Figure 3 depending on $\alpha,\beta$ and $N$.
(a) $N$=100 (b) $N$=300
Figure 3: Success probability of guessing participating nodes depending on
$\beta$ and different values of $\alpha$ and $N$.
#### Lying about cluster size
Another strategy for the supplier to compromise the privacy of users is to lie
about the cluster size. If the supplier pretends that the cluster size
$N^{\prime}$ is larger than it really is (i.e., $N^{\prime}>N$), the noise
added by each node will be underestimated. In fact, each node will calibrate
its gamma noise using $N^{\prime}$ instead of $N$. As a result, the aggregated
noise at the supplier will be smaller than necessary to guarantee sufficient
differential privacy.
In order to prevent this attack, a solution would be to set the cluster size
to a fixed value. For example, all clusters should have a size of 100.
Although simple and efficient, this solution is not flexible and might not be
applicable to all scenarios. Another option is that the supplier publishes,
together with the list of cluster nodes, a self-signed certificate of each
node of the cluster (containing a timestamp, the cluster id and the node
information). That way, each node could verify the cluster size and get
information about other member nodes.
## 9 Simulation results
### 9.1 A high-resolution electricity trace simulator
Due to the lack of high-resolution real world data, we implemented a domestic
electricity demand model [21] that can generate one-minute resolution
synthetic consumption data of different households101010Available at
http://www.crysys.hu/~acs/misc/. It is an extended version of the simulator
developed in [21]. The simulator includes 33 different appliances and
implements a separate lighting model which takes into account the level of
natural daylight depending on the month of the year. The number of residents
in each household is randomly selected between 1 and 5. A trace is associated
to a household and generated as follows: (1) A number of active persons is
selected according to some distribution derived from real statistics. This
number may vary as some members can enter or leave the house. (2) A set of
appliances is then selected and activated at different time of the day
according to another distribution, which was also derived from real
statistics.
The input of the simulator is the number of households, the month of the day,
and the type of the day (either a working or weekend day). The output is the
power demand model (1-min profile) of all appliances in each household on the
given day. Using this simulator, we generated 3000 electricity traces
corresponding to different households on a working day in November, where the
number of residents in each household was randomly selected between 1 and 5.
Each trace was then sanitized according to our scheme. The noise added in each
slot (i.e., $\lambda(t)$) is set to the maximum consumption in the slot (i.e.,
$\lambda(t)=\max_{1\leq i\leq N}X_{t}^{i}$ where the maximum is taken on all
users in the cluster). This amount of noise ensures $\varepsilon=1$
indistinguishability for individual measurements in all slots. Although one
can increase $\lambda(t)$ to get better privacy, the error will also increase.
Note that the error $\mu_{\varepsilon^{\prime}}(t)$ for other
$\varepsilon^{\prime}\neq\varepsilon$ values if $\mu_{\varepsilon}(t)$ is
given is
$\mu_{\varepsilon^{\prime}}(t)=\frac{\varepsilon}{\varepsilon^{\prime}}\cdot\mu_{\varepsilon}(t)$.
We assume that $\lambda(t)=\max_{i}X_{t}^{i}$ is known a priori.
### 9.2 Error according to the cluster size
The error introduced by our scheme depends on the cluster size $N$. In this
section, we present how the error varies according to $N$.
#### 9.2.1 Random clustering
The most straightforward scheme to build $N$-sized clusters is to select $N$
users uniformly at random. The advantage of this approach is that users only
need to send the noisy aggregate to the supplier. Figure 4(a) and 4(b) show
the average error value and its standard deviation, resp., depending on the
size of the cluster. The average error of a given cluster size $N$ is the
average of $\mathsf{mean}_{t}(\mu(t))$ of all $N$-sized clusters111111In fact,
the average error is approximated in Figure 4(a): we picked up 200 different
clusters for each $N$, and plotted the average of their
$\mathsf{mean}_{t}(\mu(t))$. 200 is chosen according to experimental analysis.
Above 200, the average error do not change significantly.. Obviously, higher
$N$ causes smaller error. Furthermore, a high $\alpha$ results in larger noise
added by each meters, as described in Section 7.3, which also implies larger
error. Interestingly, increasing the sampling period (i.e., $T_{p}$) results
in slight error decrease121212This increase is less than 0.01 even if $N$ is
small when the sampling period is changed from 5 min to 15 min., hence, we
only considered 10 min sampling period. Otherwise noted explicitly, we assume
10 min sampling period in the sequel.
(a) Average error
(b) Standard deviation of the average error
(c) Maximum error
Figure 4: The error depending on $N$ using random clustering. $T_{p}$ is 10
min.
#### 9.2.2 Consumption based clustering
As $\lambda(t)$ is set to the maximum consumption at $t$ inside a cluster, we
could get lower error if the maximum consumption is close to the mean of the
measurements within a cluster in every $t$. Hence, instead of randomly
clustering users, a more clever approach is to cluster them based on the
“similarity” of their consumption profiles. Intuitively, the measurements in
similar profiles are close, and thus, the difference between the maximum
consumption and the average should also be smaller than in a random cluster.
We measure profile similarity by the average daily consumption: the $N$-sized
clusters are created by calculating daily consumption levels
$\ell_{1},\ell_{2},\ldots,\ell_{n}$ (where $\ell_{i}\leq\ell_{i+1}$ for all
$1\leq i\leq n-1$) such that the number of users whose daily average is
between $\ell_{i}$ and $\ell_{i+1}$ for all $i$ is exactly $N$. Then, all
users being in the same level form a cluster. In contrast to random
clustering, users need to provide the supplier with their daily averages which
may leak some private information. However, this can also be derived from the
(monthly) aggregate consumption of each user, which is generally revealed for
the purpose of billing.
Figure 5(a) and 5(b) show the average error and its deviation, resp.,
calculated identically to random clustering. Comparing Figure 5 and 4,
consumption based clustering has lower error than the random one. The
improvement varies up to 5% depending on $N$. For instance, while random
clustering provides an average error of 0.13 with $N=100$ users in a cluster,
consumption based clustering has 0.07. The difference decreases as $N$
increases. There are more significant differences between the standard
deviations and the worst cases: at lower values of $N$, the standard deviation
of the average error in random clustering is almost twice as large as in
consumption based clustering (Figure 5(b) and 4(b)). To compute the worst case
error, at a given $N$, the maximum error is computed in all slots, which is
the highest cluster error that can occur in a slot with cluster size $N$.
Then, the average of these maximum errors (the average is taken on all slots)
are plotted in Figure 4(c) and 5(c). Apparently, the worst case error in
random clustering is much higher than in consumption based clustering, as
random clustering may put high and low consuming users into the same cluster.
(a) Average error
(b) Standard deviation of the average error
(c) Maximum error
Figure 5: The error depending on $N$ using consumption based clustering.
$T_{p}$ is 10 min.
### 9.3 Privacy over multiple slots
So far, we have considered the privacy of individual slots, i.e. added noise
to guarantee $\varepsilon=1$ privacy in each slot of size 10 minutes. However,
a trace is composed of several slots. For instance, if a user watches TV
during multiple slots, we have guaranteed that an adversary cannot tell if the
TV is watched in any particular slot (up to $\varepsilon=1$). However, by
analysing $s$ consecutive slots corresponding to a given period, it may be
able to tell whether the TV was watched during that period (the privacy bound
of this is $\varepsilon_{s}=\varepsilon\cdot s$ due to the composition
property of differential privacy). Based on Theorem 1, we need to add noise
$\lambda(t)=\sum_{i=1}^{s}\max_{i}X_{t}^{i}$ to _each_ aggregate to guarantee
$\varepsilon_{s}=1$ bound in consecutive $s$ slots, which, of course, results
in higher error than in the case of $s=1$ that we have assumed so far.
Obviously, using the LPA technique, we cannot guarantee reasonably low error
if $s$ increases, as the necessary noise
$\lambda(t)=\sum_{i=1}^{s}\max_{i}X_{t}^{i}$ can be large. In order to keep
the error $\lambda(t)/\sum_{i=1}^{N}X_{t}^{i}$ low while ensuring better
privacy than $\varepsilon_{s}=s\cdot\varepsilon$, one can increase the number
of users inside each cluster (i.e., $N$).
Figure 6(a) shows what average privacy of a user has, in our dataset, as a
function of the cluster size and value $s$. As the cluster size increases, the
privacy bound decreases (i.e. privacy increases). The reason is that when the
cluster size increases, the maximum consumption also increases with high
probability. Since the noise is calibrated according to the maximum
consumption within the cluster, it will be larger. This results in better
privacy.
(a) All appliances
(b) Active appliances
Figure 6: Privacy of appliances in $s$ long time windows (where $s$ is 10 min,
15 min, 30 min, 1 h, 4 h, 8 h, 1 day).
#### 9.3.1 Privacy of appliances
In the previous section, we analysed how a user’s privacy varies over time. In
this section, we consider the privacy of the different appliances. For
example, we aim at answering the following question: what was the user’s
privacy when he was watching TV last evening between 18:00 and 20:00? More
specifically, we consider two privacy threats:
* •
_Presence of appliances_ : Can the adversary tell that the user watched TV
yesterday? In order to compute the corresponding privacy (i.e.
$\varepsilon_{s}$), we compute $\sum_{t=108}^{120}\varepsilon(t)$, where
$\varepsilon(t)=\\{\text{TV's consumption in $t$}\\}/\lambda(t)$.
* •
_Activation time of appliances_ : If the adversary knows that the user watched
TV, can he tell what time he did it? We use statistical inference to detect
the position of an appliance signature in the noisy trace.
##### Presence of appliance:
We summarized some of the appliance privacy in Table 1 in Appendix B. Each
value is computed by averaging the privacy provided in our 3000 traces. The
appliances can be divided into two major groups: the usage of active
appliances indicate that the user is at home and uses the appliance (their
consumption significantly changes during their active usage such as iron,
vacuum, kettle, etc.), whereas passive appliances (like fridge, freezers,
storage heater, etc.) have more or less identical consumption regardless the
user is at home or not. In general, appliances having lower consumption
threats privacy less than devices with higher energy demands. Obviously,
$\varepsilon_{s}$ increases when $s$ increases since an appliance is used more
frequently within longer periods.
Finally, we want to measure the privacy of active appliances. This is
equivalent to answer the question if the user was at home in any $s$ long
period. The average privacy are depicted in Figure 6(b). Observe that there is
no considerable differences between Figures 6(b) and 6(a), as a profile is
primarily shaped by active appliances (because they typically consume much
more than passive appliances).
##### Activation time of appliances:
Consider the consumption profile $\mathbf{V}=(V_{1},V_{2},\ldots,V_{n})$ of a
given appliance of a user (on a single day), where the appliance is switched
on at $t_{s}$ first and switched off at $t_{s}+d$ last (i.e., $V_{i}=0$ for
$1\leq i<t_{s}$ and $t_{s}+d<i\leq n$). The signature of the appliance
$\mathsf{Sig}(\mathbf{V})=(V_{t_{s}},V_{t_{s}+1},\ldots,V_{t_{s}+d})$ is the
consumption profile of the appliance between $t_{s}$ and $t_{s}+d$. The
adversary is provided with the noisy consumption profile of the appliance
(i.e., $\hat{\mathbf{V}}$) and, in addition, knows the signature of the
appliance, but it does not know $t_{s}$ (i.e., it knows that the appliance was
used with the given signature but does not know when).
The goal of the adversary is to infer the starting slot $t_{s}$ in
$\mathbf{V}$ using $\hat{\mathbf{V}}$. If the adversary’s guess is
$t^{\prime}$, the inference accuracy is measured by $|t^{\prime}-t_{s}|$. We
consider the following adversaries:
* •
$\mathsf{RG\textendash Adv}$: This is the simple random guesser and serves as
a baseline. If there are $n-d$ possible values of $t_{s}$, then the guess
$t^{\prime}$ is selected out of them uniformly at random.
* •
$\mathsf{ST\textendash Adv}$: This adversary knows the relative frequency of
each slot occuring as a starting slot (denoted by $f_{i}$ at slot $i$), and
guesses the most likely starting slot: $t^{\prime}=\max_{i}f_{i}$ ($1\leq
i\leq n-d$). This information is publicly available from several surveys [21].
* •
$\mathsf{Bayesian\textendash Adv}$: This adversary performs bayesian inference
on $t_{s}$. In particular, let $\mathbf{V}^{t}$ denote a profile where the
signature starts at slot $t$ (i.e., $\mathbf{V}^{t}$ is obtained by shifting
$\mathbf{V}$ with $|t-t_{s}|$ positions to left/right if $t-t_{s}$ is
negative/positive.131313More formally, $\mathbf{V}^{t}=0$ for all $1\leq i<t$
and $t+d<i\leq n$, and $V_{i}=\mathsf{Sig}(\mathbf{V})_{i}$ for $1\leq i\leq
t+d$). Assuming that the adversary has no prior knowledge about the
distribution of starting slots (i.e., they are distributed uniformly at
random), the posterior distribution is computed as
$P(\mathcal{T}=i)=\frac{\prod_{k=1}^{n}P(V^{i}_{k}+\mathcal{L}(\lambda_{k})=\hat{V}_{k})}{\sum_{j=1}^{n-d}\prod_{k=1}^{n}P(V^{j}_{k}+\mathcal{L}(\lambda_{k})=\hat{V}_{k})}$
where $\mathcal{T}$ describes the posterior distribution of starting slots. As
the bayes risk is ”linear” in our case (i.e., $|t^{\prime}-t_{s}|$), the
bayes’ estimate (i.e., $t^{\prime}$) is the posterior median (i.e.,
$t^{\prime}$ satisfies $P(\mathcal{T}\leq t^{\prime})\geq 0.5$ and
$P(\mathcal{T}\geq t^{\prime})\leq 0.5$).
* •
$\mathsf{Bayesian\textendash ST\textendash Adv}$: We expect better results if
the bayesian adversary uses the relative frequencies as a prior knowledge. In
particular, the adversary knows the probability distribution of starting slots
a priori, denoted by $\theta=\\{f_{1},f_{2},\ldots,f_{n-d}\\}$, which is
described by the relative frequencies:
$P(\mathcal{T}=i|\theta)=\frac{\prod_{k=1}^{n}f_{i}\cdot
P(V^{i}_{k}+\mathcal{L}(\lambda_{k})=\hat{V}_{k})}{\sum_{j=1}^{n-d}\prod_{k=1}^{n}P(V^{j}_{k}+\mathcal{L}(\lambda_{k})=\hat{V}_{k})\cdot
f_{j}}$
As before, the bayes’ estimate is the posterior median.
The inference accuracy of each adversary is shown in Table 2 in Appendix B.
The inference is performed on our dataset within a single day.
$\mathsf{Bayesian\textendash ST\textendash Adv}$ outperforms all adversaries
especially for active devices, however, its accuracy never falls below 1.7
hour. Regarding the passive appliances, $\mathsf{ST\textendash Adv}$ overcomes
$\mathsf{Bayesian\textendash ST\textendash Adv}$ in general. This is explained
by the fact that passive appliances usually follow a regular operation cycle
with less user intervention in all households, and the accuracy of
$\mathsf{ST\textendash Adv}$’s is always within the length of one operation
cycle independently of the added noise141414The same type of appliance is used
in all households..
## 10 Conclusion
Our measurements show two different, and conflicting, results. Figure 6(a)
shows that it may actually be difficult to hide the presence of activities in
a household. In fact, computed $\varepsilon$ values are quite high, even for
large clusters. However, results presented in Tables 1 and 2 are more
encouraging. They show that, although, it might be difficult to hide a user’s
presence, it is still possible to hide his actual activity. In fact,
appliances privacy bounds ($\varepsilon$ values) are quite small, which
indicates that an adversary will have difficulty telling whether the user is,
for example, using his computer or watching TV during a given period of time.
Furthermore, in Table 2, results show that it is even more difficult for an
adversary to tell when a given activity actually started. Finally, we recall
that in order to keep the error $\lambda(t)/\sum_{i=1}^{N}X_{t}^{i}$ low while
ensuring better privacy one can always increase the number of users inside
each cluster. For instance, doubling $N$ from 100 to 200 allows to double the
noise while keeping approximately the same error value (0.118 in Figure 5(a)
if $\alpha=0$). This results in much better privacy, since, on average,
doubling the noise halves the privacy parameter $\varepsilon_{s}$.
Although more work and research is needed, we believe this is a encouraging
result for privacy.
## Acknowledgements
The work presented in this paper was supported in part by the European
Commission within the STREP WSAN4CIP project. The views and conclusions
contained herein are those of the authors and should not be interpreted as
representing the official policies or endorsement of the WSAN4CIP project or
the European Commission.
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## Appendix A Proof of Theorem 2 (Utility)
###### Lemma 2 (Integral property of the Bessel function [15]).
Let
$K_{\vartheta}(x)=\frac{1}{2}\left(\frac{x}{2}\right)^{\vartheta}\int_{0}^{\infty}t^{-\vartheta-1}\exp\left(-t-\frac{x^{2}}{4t}\right)dt,\qquad
x>0$
define the modified Bessel function of the third kind with index
$\vartheta\in\mathbb{R}$. For any $\gamma>0$ and $\gamma,\nu$ such that
$\gamma+1\pm\nu>0$
$\int_{0}^{\infty}x^{\gamma}K_{\nu}(ax)dx=\frac{2^{\gamma-1}}{a^{\gamma+1}}\Gamma\left(\frac{1+\gamma+\nu}{2}\right)\Gamma\left(\frac{1+\gamma-\nu}{2}\right)$
###### Lemma 3.
Let $\mathcal{G}_{1},\mathcal{G}_{2}$ be i.i.d gamma random variables with
parameters $(n,\lambda)$. Then,
$\displaystyle\mathbb{E}|\mathcal{G}_{1}(n,\lambda)-\mathcal{G}_{2}(n,\lambda)|=\frac{2\lambda}{B\left(\frac{1}{2},\frac{1}{n}\right)}$
(1)
and
$\displaystyle\mathit{Var}|\mathcal{G}_{1}(n,\lambda)-\mathcal{G}_{2}(n,\lambda)|=\left(\frac{2}{n}-\frac{4}{B\left(\frac{1}{2},\frac{1}{n}\right)^{2}}\right)\lambda^{2}$
(2)
where $B(x,y)$ is the beta function defined as
$B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$.
###### Proof (of Lemma 3)
Consider $\mathcal{Y}=\mathcal{G}_{1}-\mathcal{G}_{2}$. The characteristic
function of $\mathcal{Y}$ is
$\phi_{\mathcal{Y}}(t)=\left(\frac{1}{1+i\lambda
t}\right)^{\frac{1}{n}}\cdot\left(\frac{1}{1-i\lambda
t}\right)^{\frac{1}{n}}=\left(\frac{1}{1+\lambda^{2}t^{2}}\right)^{\frac{1}{n}}$
which is a special case of the characteristic function of the Generalized
Asymetric Laplace distribution (GAL) with parameters
$(\theta,\kappa,\omega,\tau)$:
$\phi_{\mathit{GAL}}(t)=e^{i\theta
t}\left(\frac{1}{1+i\frac{\sqrt{2}}{2}\omega\kappa
t}\right)^{\tau}\cdot\left(\frac{1}{1-i\frac{\sqrt{2}}{2\kappa}\omega
t}\right)^{\tau}$
where $\theta=0,\kappa=1,\omega=\sqrt{2}\lambda$, and $\tau=1/n$. The density
function of $GAL(\theta,\kappa,\omega,\tau)$ when $\theta=0$ and $\kappa=1$ is
$f_{\mathit{GAL}}(x)=\frac{\sqrt{2}}{\omega^{\tau+1/2}\Gamma(\tau)\sqrt{\pi}}\left(\frac{|x|}{\sqrt{2}}\right)^{\tau-1/2}K_{\tau-1/2}(\sqrt{2}|x|/\omega)$
where $K_{\tau-1/2}(\frac{\sqrt{2}}{\omega}|x|)$ is the Bessel function
defined in Lemma 2. In addition,
$\mathbb{E}|\mathcal{Y}|=\int_{-\infty}^{\infty}|x|f_{\mathit{GAL}}(x)dx=2\cdot\int_{0}^{\infty}x\frac{\sqrt{2}}{\omega^{\tau+1/2}\Gamma(\tau)\sqrt{\pi}}\left(\frac{x}{\sqrt{2}}\right)^{\tau-1/2}K_{\tau-1/2}(\sqrt{2}x/\omega)dx$
which follows from the symmetry property of $f_{\mathit{GAL}}(x)$
($\phi_{\mathcal{Y}}(t)$ is is real valued). After reformulation, we have
$\mathbb{E}|\mathcal{Y}|=\frac{2\sqrt{2}}{\sqrt{2}^{\tau-1/2}\omega^{\tau+1/2}\Gamma(\tau)\sqrt{\pi}}\int_{0}^{\infty}x^{\tau+1/2}K_{\tau-1/2}(\sqrt{2}x/\omega)dx$
Now, we can apply Lemma 2 for the integral and we obtain
$\mathbb{E}|\mathcal{Y}|=\sqrt{2}\cdot
w\cdot\frac{\Gamma(\tau+\frac{1}{2})}{\Gamma(\frac{1}{2})\sqrt{\pi}}$
after simple derivation. Using that $\sqrt{\pi}=\Gamma(1/2)$ and
$B(x,y)=\frac{\Gamma(x)(y)}{\Gamma(x+y)}$, we have
$\mathbb{E}|\mathcal{Y}|=\frac{\sqrt{2}}{B(1/2,\tau)}\cdot w$
Applying $\omega=\sqrt{2}\lambda$ and $\tau=1/n$, we arrive at Equation (1).
To prove Equation (2), consider that
$\mathit{Var}(|\mathcal{Y}|)=\mathbb{E}|\mathcal{Y}|^{2}-[\mathbb{E}|\mathcal{Y}|]^{2}$
where
$\mathbb{E}|\mathcal{Y}|^{2}=\mathbb{E}(\mathcal{Y}^{2})=\mathbb{E}(\mathcal{G}_{1}^{2})+\mathbb{E}(\mathcal{G}_{2}^{2})-2\cdot\mathbb{E}(\mathcal{G}_{1})\cdot\mathbb{E}(\mathcal{G}_{2})$
Using that
$\mathbb{E}(\mathcal{G}_{1}^{2})=\mathbb{E}(\mathcal{G}_{2}^{2})=(1/n^{2}+1/n)\lambda^{2}$,
we obtain Equation (2).
Now, we can easily prove Theorem 2.
###### Proof (of Theorem 2)
$\mathbb{E}|\sum_{i=1}^{N}(X_{t}^{i}-\hat{X}_{t}^{i})|=$
$=\mathbb{E}|\sum_{i=1}^{N}\mathcal{G}_{1}(N-M,\lambda)-\sum_{i=1}^{N}\mathcal{G}_{2}(N-M,\lambda)|=$
(using that
$\sum_{i=1}^{n}\mathcal{G}(k_{i},\lambda)=\mathcal{G}(1/\sum_{i=1}^{n}\frac{1}{k_{i}},\lambda)$)
$=\mathbb{E}|\mathcal{G}_{1}(1-M/N,\lambda)-\mathcal{G}_{2}(1-M/N,\lambda)|=$
(using $\alpha=M/N$ and applying Lemma 3)
$=\frac{2}{B(1/2,\frac{1}{1-\alpha})}\lambda$
The standard deviation
$\sqrt{\mathit{Var}|\sum_{i=1}^{N}(X_{t}^{i}-\hat{X}_{t}^{i})|}$ can be
derived identically.
## Appendix B Privacy of some ordinary appliances
| _Appliance_ | $s=30\,\text{min}$ | $s=1\,\text{h}$ | $s=4\,\text{h}$ | $s=8\,\text{h}$ | $s=24\,\text{h}$
---|---|---|---|---|---|---
| _mean_ | _dev_ | _max_ | _mean_ | _dev_ | _max_ | _mean_ | _dev_ | _max_ | _mean_ | _dev_ | _max_ | _mean_ | _dev_ | _max_
Active appliances | Lighting | 0.91 | 1.28 | 17.87 | 1.29 | 1.37 | 18.84 | 2.68 | 1.82 | 19.38 | 3.63 | 2.29 | 21.49 | 4.89 | 2.97 | 25.37
Cassette / CD Player | 0.02 | 0.04 | 0.79 | 0.04 | 0.04 | 0.81 | 0.05 | 0.05 | 0.82 | 0.07 | 0.05 | 0.88 | 0.09 | 0.07 | 0.96
Hi-Fi | 0.10 | 0.17 | 4.43 | 0.16 | 0.19 | 4.59 | 0.17 | 0.20 | 4.62 | 0.18 | 0.21 | 4.62 | 0.19 | 0.21 | 4.62
Iron | 0.75 | 1.81 | 42.91 | 0.82 | 1.82 | 42.99 | 0.92 | 1.83 | 42.99 | 1.00 | 1.86 | 42.99 | 1.02 | 1.89 | 42.99
Vacuum | 1.67 | 7.59 | 134.54 | 1.70 | 7.59 | 134.54 | 1.82 | 7.58 | 134.54 | 1.90 | 7.60 | 134.54 | 1.94 | 7.63 | 134.54
Fax | 0.04 | 0.10 | 1.55 | 0.04 | 0.10 | 1.55 | 0.04 | 0.10 | 1.55 | 0.05 | 0.10 | 1.56 | 0.05 | 0.10 | 1.56
Personal computer | 0.21 | 0.32 | 7.48 | 0.34 | 0.36 | 7.48 | 0.83 | 0.49 | 7.48 | 1.09 | 0.58 | 7.53 | 1.42 | 0.83 | 8.37
Printer | 0.07 | 0.30 | 7.78 | 0.08 | 0.31 | 7.78 | 0.09 | 0.31 | 7.78 | 0.10 | 0.31 | 7.78 | 0.11 | 0.31 | 7.83
TV | 0.15 | 0.47 | 7.41 | 0.22 | 0.48 | 7.45 | 0.37 | 0.52 | 7.45 | 0.45 | 0.58 | 8.37 | 0.50 | 0.63 | 8.37
VCR / DVD | 0.05 | 0.16 | 2.81 | 0.07 | 0.17 | 2.84 | 0.10 | 0.17 | 2.89 | 0.13 | 0.18 | 2.95 | 0.14 | 0.19 | 3.01
TV Receiver box | 0.03 | 0.11 | 2.12 | 0.05 | 0.11 | 2.21 | 0.08 | 0.12 | 2.32 | 0.10 | 0.13 | 2.40 | 0.11 | 0.14 | 2.42
Hob | 1.90 | 9.58 | 132.86 | 1.96 | 9.58 | 132.86 | 2.15 | 9.57 | 132.86 | 2.28 | 9.59 | 132.86 | 2.34 | 9.67 | 132.86
Oven | 1.50 | 3.91 | 96.19 | 1.58 | 3.92 | 96.19 | 1.74 | 3.94 | 96.19 | 1.85 | 3.97 | 96.19 | 1.91 | 4.07 | 98.51
Microwave | 1.13 | 4.23 | 82.73 | 1.20 | 4.24 | 82.73 | 1.26 | 4.24 | 82.73 | 1.29 | 4.27 | 83.17 | 1.31 | 4.29 | 83.57
Kettle | 0.55 | 2.71 | 63.59 | 0.59 | 2.71 | 63.59 | 0.72 | 2.73 | 63.87 | 0.83 | 2.76 | 64.22 | 1.02 | 2.79 | 64.22
Small cooking (group) | 0.25 | 1.61 | 26.16 | 0.25 | 1.61 | 26.16 | 0.26 | 1.61 | 26.16 | 0.27 | 1.61 | 26.16 | 0.27 | 1.62 | 26.16
Dish washer | 0.93 | 2.67 | 55.64 | 1.49 | 2.67 | 55.64 | 1.78 | 2.71 | 55.64 | 1.97 | 2.95 | 60.15 | 2.03 | 2.97 | 60.15
Tumble dryer | 2.57 | 8.05 | 152.33 | 3.93 | 8.16 | 154.99 | 5.24 | 8.20 | 155.08 | 6.30 | 8.33 | 155.08 | 7.01 | 8.68 | 155.08
Washing machine | 1.23 | 1.43 | 31.57 | 1.30 | 1.45 | 31.72 | 1.96 | 1.63 | 33.24 | 2.55 | 1.76 | 33.24 | 3.07 | 2.07 | 34.62
Washer dryer | 1.82 | 1.08 | 19.22 | 3.17 | 1.33 | 19.27 | 4.70 | 1.99 | 25.82 | 6.39 | 2.38 | 25.82 | 7.92 | 3.49 | 33.66
E-INST | 1.47 | 1.12 | 6.54 | 1.93 | 1.15 | 6.54 | 3.47 | 1.16 | 7.58 | 4.70 | 1.49 | 9.00 | 7.06 | 2.13 | 10.99
Electric shower | 2.13 | 14.78 | 249.24 | 2.16 | 14.78 | 249.24 | 2.28 | 14.78 | 249.24 | 2.34 | 14.78 | 249.24 | 2.38 | 14.80 | 249.24
Passive app. | DESWH | 3.34 | 14.01 | 249.29 | 4.04 | 14.04 | 251.01 | 6.13 | 14.06 | 253.21 | 7.83 | 14.23 | 255.20 | 10.85 | 14.57 | 257.76
Storage heaters | 3.22 | 0.32 | 3.96 | 5.64 | 0.56 | 6.95 | 20.20 | 1.99 | 24.87 | 30.45 | 4.23 | 41.48 | 30.45 | 4.23 | 41.48
Elec. space heating | 1.64 | 0.85 | 6.14 | 2.86 | 1.07 | 7.54 | 7.49 | 2.15 | 13.03 | 8.50 | 2.49 | 14.57 | 10.06 | 4.08 | 26.25
Chest freezer | 0.61 | 0.74 | 15.94 | 0.61 | 0.74 | 15.95 | 1.39 | 0.92 | 17.20 | 1.85 | 1.07 | 18.10 | 2.55 | 1.24 | 18.96
Fridge freezer | 0.91 | 0.39 | 7.56 | 0.91 | 0.40 | 7.61 | 2.19 | 0.95 | 8.67 | 2.94 | 1.25 | 10.58 | 4.07 | 1.61 | 11.69
Refrigerator | 0.44 | 0.22 | 3.83 | 0.45 | 0.23 | 4.00 | 1.06 | 0.49 | 4.77 | 1.40 | 0.64 | 5.68 | 1.92 | 0.80 | 6.50
| Upright freezer | 0.67 | 0.39 | 8.37 | 0.67 | 0.39 | 8.42 | 1.63 | 0.80 | 9.09 | 2.16 | 1.03 | 10.99 | 2.98 | 1.31 | 11.98
Table 1: $\varepsilon_{s}$ of different appliances in case of different $s$. $N=100$ and the sampling period is 10 min. | _Appliance_ | # of users | $\mathsf{RG\textendash Adv}$ | $\mathsf{ST\textendash Adv}$ | $\mathsf{Bayesian\textendash Adv}$ | $\mathsf{Bayesian\textendash ST\textendash Adv}$
---|---|---|---|---|---|---
| _mean_ | _dev_ | _mean_ | _dev_ | _mean_ | _dev_ | _mean_ | _dev_
Active appliances | Lighting | 2998 | 2.53 | 2.54 | 5.16 | 3.66 | 1.87 | 1.73 | 1.71 | 2.34
Cassette / CD Player | 2650 | 4.70 | 4.09 | 3.34 | 3.70 | 3.96 | 2.50 | 3.19 | 3.46
Hi-Fi | 744 | 7.49 | 5.49 | 4.10 | 3.29 | 5.58 | 3.29 | 4.04 | 2.65
Iron | 1247 | 6.53 | 4.40 | 3.89 | 3.19 | 3.62 | 2.94 | 2.95 | 2.28
Vacuum | 1192 | 6.61 | 4.47 | 4.00 | 3.22 | 3.54 | 3.02 | 2.92 | 2.45
Fax | 241 | 6.85 | 5.66 | 7.78 | 4.81 | 5.76 | 3.26 | 4.19 | 2.85
Personal computer | 1970 | 5.35 | 4.50 | 5.32 | 4.55 | 4.79 | 3.20 | 4.03 | 3.49
Printer | 1608 | 6.21 | 5.06 | 5.73 | 4.69 | 4.71 | 3.01 | 4.07 | 3.04
TV | 2519 | 5.41 | 4.07 | 4.22 | 3.41 | 3.73 | 2.44 | 2.50 | 2.50
VCR / DVD | 2299 | 5.55 | 4.09 | 4.29 | 3.44 | 3.72 | 2.42 | 2.53 | 2.57
TV Receiver box | 2413 | 5.58 | 4.09 | 4.27 | 3.42 | 3.71 | 2.37 | 2.53 | 2.58
Hob | 857 | 6.53 | 4.49 | 3.64 | 3.19 | 3.55 | 2.89 | 2.95 | 2.48
Oven | 760 | 6.31 | 4.50 | 3.78 | 3.13 | 3.35 | 2.99 | 2.74 | 2.41
Microwave | 505 | 6.41 | 4.24 | 3.96 | 3.17 | 3.39 | 2.97 | 2.90 | 2.44
Kettle | 2808 | 4.81 | 4.13 | 3.62 | 3.84 | 3.83 | 2.67 | 3.29 | 3.48
Small cooking (group) | 1441 | 6.55 | 4.41 | 3.92 | 3.18 | 3.51 | 2.65 | 3.00 | 2.40
Dish washer | 434 | 6.32 | 4.46 | 4.57 | 3.39 | 3.28 | 3.00 | 2.71 | 2.19
Tumble dryer | 1018 | 5.79 | 4.15 | 4.32 | 3.37 | 2.23 | 2.56 | 2.03 | 2.57
Washing machine | 2228 | 5.28 | 4.02 | 3.58 | 3.31 | 2.85 | 2.67 | 2.36 | 2.80
Washer dryer | 417 | 5.05 | 3.77 | 3.26 | 3.07 | 1.94 | 2.21 | 1.79 | 2.66
E-INST | 29 | 3.05 | 2.42 | 1.87 | 3.09 | 1.84 | 2.35 | 1.71 | 2.88
Electric shower | 1039 | 6.10 | 4.31 | 3.76 | 3.12 | 3.47 | 2.96 | 2.89 | 2.36
Passive app. | DESWH | 510 | 3.70 | 3.41 | 2.54 | 3.14 | 1.22 | 1.73 | 1.54 | 2.46
Storage heaters | 84 | 8.50 | 5.84 | 0.00 | 0.00 | 0.27 | 0.25 | 0.00 | 0.00
Elec. space heating | 73 | 6.52 | 5.05 | 6.75 | 5.02 | 2.85 | 3.42 | 2.14 | 3.14
Chest freezer | 466 | 0.56 | 0.47 | 0.51 | 0.44 | 0.42 | 0.32 | 0.40 | 0.34
Fridge freezer | 1954 | 0.49 | 0.42 | 0.28 | 0.30 | 0.34 | 0.29 | 0.34 | 0.33
Refrigerator | 1301 | 0.56 | 0.48 | 0.35 | 0.39 | 0.40 | 0.33 | 0.41 | 0.42
| Upright freezer | 866 | 0.55 | 0.46 | 0.35 | 0.37 | 0.38 | 0.30 | 0.37 | 0.36
Table 2: Inference accuracy of starting slots. $N=100$, $T_{p}=10$ min, and ”#
of users” means the number of users who have the given appliance in our
dataset. The accuracy ($|t^{\prime}-t_{s}|$) is given in hours.
|
arxiv-papers
| 2012-01-12T11:15:02 |
2024-09-04T02:49:26.245116
|
{
"license": "Public Domain",
"authors": "Gergely Acs and Claude Castelluccia",
"submitter": "Gergely Acs",
"url": "https://arxiv.org/abs/1201.2531"
}
|
1201.2660
|
Fermi National Accelerator Laboratory (Fermilab), USA
E-mail: ajung@fnal.gov
FERMILAB-CONF-11-677-PPD 13,14 13,14
# Top differential cross section measurements (Tevatron)
Andreas W. Jungins:x (for the collaboration) ins:xins:x
###### Abstract
Differential cross sections in the top quark sector measured at the Fermilab
Tevatron collider are presented. CDF used $2.7~{}\mathrm{fb^{-1}}$ of data and
measured the differential cross section as a function of the invariant mass of
the $t\bar{t}$ system. The measurement shows good agreement with the standard
model and furthermore is used to derive limits on the ratio $\kappa/M_{Pl}$
for gravitons which decay to top quarks in the Randall-Sundrum model. used
$1.0~{}\mathrm{fb^{-1}}$ of data to measure the differential cross section as
a function of the transverse momentum of the top quark. The measurement shows
a good agreement to the higher order perturbative QCD prediction and various
predictions based on various Monte-Carlo generators.
## 1 Introduction
The $top$ quark is the heaviest known elementary particle and was discovered
at the Tevatron $p\bar{p}$ collider in 1995 by the CDF and collaboration [1,
2] at a mass of around $175~{}\mathrm{GeV}$. The production is dominated by
the $q\bar{q}$ annihilation process with 85% as opposed to gluon-gluon fusion
which contributes only 15%. Both measurements presented here are performed
using the $l+$jets channel, where one of the $W$ bosons (stemming from the
decay of the $top$ quarks) decays leptonically. The other $W$ boson decays
hadronically. The $l+$jets channel is a good compromise between signal and
background contribution whilst having high event statistics. The branching
fraction for top quarks decaying into $Wb$ is almost 100%. Jets containing a
beauty quark are identified by means of a neural network (NN) build by the
combination of variables describing the properties of secondary vertices and
of tracks with large impact parameters relative to the primary vertex.
## 2 Measurement of the transverse momentum distribution of the top quarks
The measurement of the transverse momentum distribution of the top quarks [3]
selects events with an isolated lepton with a transverse momentum $p_{T}$ of
at least $20~{}\mathrm{GeV}$ and a pseudo-rapidity of $|\eta|<1.1$ ($e+$jets)
or $|\eta|<2.0$ ($\mu+$jets). A cut on the missing transverse energy
($\not\\!\\!E_{T}$) of $20~{}\mathrm{GeV}$ is applied. Furthermore at least
four jets are required with $p_{T}>20~{}\mathrm{GeV}$ and $|\eta|<2.5$, an
additional cut of $p_{T}>40~{}\mathrm{GeV}$ is applied for the leading jet.
Finally at least one jet needs to be identified as a $b$-jet. For the
reconstruction of the event kinematics additional constraints are used: the
masses of the two $W$ bosons are constrained to $80.4~{}\mathrm{GeV}$.
Furthermore the masses of the two reconstructed top quarks are assumed to be
equal. All possible permutations of objects are considered where the final
solution is the one with the smallest $\chi^{2}$.
Figure 1: a) compares the background-subtracted reconstructed top-quark
$p_{T}$ distribution [3] with the one corrected for the effects of finite
experimental resolution (two entries per event). Inner error bars represent
the statistical uncertainty, whereas the outer one is statistical and
systematic added in quadrature. b) shows the unfolded invariant mass
distribution of the $t\bar{t}$ system [4] compared to signal $t\bar{t}$ MC.
Figure 1a) shows the background-subtracted reconstructed top-quark $p_{T}$
distribution compared to the one corrected for finite experimental resolution.
The latter is derived by using regularized matrix unfolding. Figure 1b)
compares the unfolded invariant mass distribution of the $t\bar{t}$ system to
the expectation using $t\bar{t}$ signal MC. The correction for finite detector
resolution is again done using regularized unfolding.
Figure 2a) shows the differential cross section as a function of top-quark
$p_{T}$, where the leptonic and hadronic decay of the $W$ boson to the top-
quark cross section are combined. All predictions use the proton parton
density function (PDF) CTEQ61 with the scale set to $\mu_{r}=\mu_{f}=m_{t}$
($m_{t}=170~{}\mathrm{GeV}$) except for the approximate NNLO perturbative QCD
(pQCD) prediction which uses the MSTW08 PDF. The normalization is nicely
described by pQCD in (N)NLO, however there is an offset for PYTHIA and ALPGEN
in normalization. Figure 2b) shows that the shape is reasonable described by
all predictions. The inclusive total cross section for $t\bar{t}$ production
is measured to $\sigma=8.31\pm 1.28(\mathrm{stat.})~{}\mathrm{pb}$ and in good
agreement with the latest theoretical predictions of
$\sigma=6.41\pm^{0.51}_{0.42}~{}\mathrm{pb}$ [5] and
$\sigma=7.46\pm^{0.48}_{0.67}~{}\mathrm{pb}$ [6].
Figure 2: a) Differential cross section data (points) as a function of top-
quark $p_{T}$ (two entries per event) [3] compared with expectations from NLO
pQCD (solid lines), from an approximate NNLO pQCD calculation, and for several
event generators (dashed and dotdashed lines). The gray band reflects
uncertainties on the pQCD scale and parton distribution functions. Inner error
bars represent the statistical uncertainty, whereas the outer one is
statistical and systematic added in quadrature. b) shows the ratio of
$(1/\sigma)d\sigma/dp_{T}$ relative to NLO pQCD for an approximate NNLO pQCD
calculation and of predictions for several event generators.
## 3 Measurement of the invariant mass distribution of the $t\bar{t}$ system
This measurement of the invariant mass distribution of the $t\bar{t}$ system
$M_{t\bar{t}}$ [4] selects events with an isolated lepton with a $p_{T}$ of at
least $20~{}\mathrm{GeV}$ and a pseudo-rapidity of $|\eta|<1.1$. A cut on the
missing transverse energy of $20~{}\mathrm{GeV}$ is applied. Furthermore at
least four jets are required with $p_{T}>20~{}\mathrm{GeV}$ and $|\eta|<2.0$.
Finally at least one jet needs to be identified as a $b$-jet. The hadronic $W$
decay is used to constrain the Jet Energy Scale (JES). $M_{t\bar{t}}$ is
reconstructed by using the four-vectors of the b-tagged jet and the three
remaining leading jets in the event, the lepton and the transverse components
of the neutrino momentum, given by $\not\\!\\!E_{T}$.
Figure 3a) shows the differential $t\bar{t}$ cross section as a function of
$M_{t\bar{t}}$ compared to the standard model expectation using the proton PDF
CTEQ5L PDF with a top mass of $175~{}\mathrm{GeV}$. The SM uncertainty
reflects all systematic uncertainties, except for the luminosity uncertainty
in each bin. Especially the tail of $M_{t\bar{t}}$ is sensitive to broad
enhancements as well as to narrow resonances, which is why the agreement
between data and SM expectation has been evaluated. There is no indication of
beyond standard model contributions to the differential cross section. The
analysis also measured the inclusive total cross section for $t\bar{t}$
production to: $\sigma=6.9\pm 1.0(\mathrm{stat.+JES})~{}\mathrm{pb}$, which is
in good agreement with latest theoretical predictions [5, 6] as well as with
the result. Furthermore the distribution has been used to derive a limit on
gravitons which decay to top quarks in the Randall-Sundrum model. The mass of
the first resonance is fixed to 600 GeV and gravitons are modeled using
MadEvent plus Pythia. Figure 3b) shows the derived limits, values of
$\kappa/M_{Pl}>0.16$ are excluded at the 95% confidence level.
Figure 3: a) shows the differential $t\bar{t}$ cross section (circles) as a
function of $M_{t\bar{t}}$ [4] compared to the standard model expectation
(line). The SM uncertainty (green band) reflects all systematic uncertainties,
except for the luminosity uncertainty in each bin. b) shows limits on the
ratio $\kappa/M_{Pl}$ for gravitons which decay to top quarks in the Randall-
Sundrum model, where the mass of the first resonance is fixed at 600 GeV.
Values of $\kappa/M_{Pl}>0.16$ are excluded at the 95% confidence level.
## 4 Conclusion
Two differential cross section measurements have been presented. The cross
section as a function of the transverse momentum of the top quark by [3] and
as a function of the invariant mass of the $t\bar{t}$ system by CDF [4]. Both
presented results are consistent with the standard model cross section
predictions. The final Tevatron data sample has $5-10$ times the presented
statistics allowing for more precise measurements in the near future being one
of the legacy measurements of the Tevatron.
## References
* [1] F. Abe et al. (CDF Collaboration), Phys. Rev. Lett. 74, 2626 (1995) [arXiv:hep-ex/9503002].
* [2] S. Abachi et al. ( Collaboration), Phys. Rev. Lett. 74, 2632 (1995) [arXiv:hep-ex/9503003].
* [3] V. Abazov et al ( Collaboration), Phys. Lett. B 693, 515 (2010), [arXiv.org:1001.1900].
* [4] T. Aaltonen et al. (CDF Collaboration), PRL 102 222003, [arxiv.org:0903.2850].
* [5] V. Ahrens, A. Ferroglia, M. Neubert, B. D. Pecjak, and L. L. Yang, J. High Energy Phys. 09, 097 (2010); V. Ahrens, A. Ferroglia, M. Neubert, B. D. Pecjak, and L. L. Yang, Nucl. Phys. Proc. Suppl. 205-206, 48 (2010).
* [6] S. Moch and P. Uwer, Phys. Rev. D 78, 034003 (2008); U. Langenfeld, S. Moch, and P. Uwer, Phys. Rev. D 80, 054009 (2009);M. Aliev et al., Comput. Phys. Commun. 182, 1034 (2011).
|
arxiv-papers
| 2012-01-12T20:23:40 |
2024-09-04T02:49:26.268088
|
{
"license": "Public Domain",
"authors": "Andreas W. Jung",
"submitter": "Andreas Werner Jung",
"url": "https://arxiv.org/abs/1201.2660"
}
|
1201.2661
|
# Dimensional Reduction without Continuous Extra Dimensions
Ali H. Chamseddine American University of Beirut, Physics Department, Beirut,
Lebanon and I.H.E.S. F-91440 Bures-sur-Yvette, France J. Fröhlich, B.
Schubnel ETHZ, Mathematics and Physics Departments, Zürich, Switzerland D.
Wyler Inst. of Theoretical Physics, University of Zürich, Switzerland
###### Abstract
We describe a novel approach to dimensional reduction in classical field
theory. Inspired by ideas from noncommutative geometry, we introduce extended
algebras of differential forms over space-time, generalized exterior
derivatives and generalized connections associated with the "geometry" of
space-times with discrete extra dimensions. We apply our formalism to theories
of gauge- and gravitational fields and find natural geometrical origins for an
axion- and a dilaton field, as well as a Higgs field.
## I Introduction
Introducing extra dimensions in order to unify physical laws and identify
natural geometrical origins of various gauge- and scalar fields has quite a
long history, beginning in the 1920’s with attempts by Kaluza and Klein (see
[KA, ], [KL, ]) to unify Maxwell’s theory with general relativity in a five-
dimensional space-time, continuing with Pauli’s construction of non-abelian
SU(2)-gauge fields in a six dimensional space-time and culminating with
string- and M-theory; (see, e.g., [OR, ]). All these attempts are plagued with
the appearance of infinite towers of modes of ever larger mass. In theories
where all modes are coupled to the gravitational field such towers may seem to
be a problem.
Within the general framework of noncommutative geometry, Connes has proposed
to consider generalized notions of differential geometry to describe extra
dimensions and to construct classical field theories where certain scalar
fields, such as the Higgs field of the standard model, appear for geometrical
reasons, but towers of very massive modes do not arise; see [CO1, ], [CO2, ].
Connes’ attempts are based on generalizations of spin geometry. The
fundamental geometrical data are encoded in so-called "spectral triples",
$(\mathcal{A},D,\mathcal{H})$, where $\mathcal{A}$ is a (possibly non
commutative) ∗algebra of operators represented on a separable Hilbert space
$\mathcal{H}$, and $D$ is an elliptic operator acting on $\mathcal{H}$
generalizing the Dirac operator.
In this note, we present an alternative approach to "dimensional reduction",
based on certain extensions of the graded differential algebra, $\Omega(M)$,
of differential forms over space-time M, that does not involve introducing
continuous extra dimensions, but involves generalized notions of "exterior
derivative", "connection" and "metric". Our approach is inspired by Connes’
ideas ([CO1, ],[CO2, ]), but we attempt to generalize general Riemannian -
rather than spin-geometry; (see [FRO1, ]). Thus, besides a *algebra of
operators, it involves two anti-commutaing Kähler-Dirac operators,
$\mathcal{D}$ and $\bar{\mathcal{D}}$, acting on a Hilbert space of
generalized differential forms (rather than a single Dirac operator acting on
a Hilbert space of generalized spinors). Classical fields are identified with
elements of a (sub-)space of "zero modes"on which
$\mathcal{D}^{2}=\bar{\mathcal{D}}^{2}$. The linear combinations
$d:=\mathcal{D}-i\bar{\mathcal{D}}$ and
$d^{*}:=\mathcal{D}+i\bar{\mathcal{D}}$ can then be interpreted as
generalizations of the exterior derivative and its adjoint; (see [FRO1, ]).
The purpose of our note is to provide natural geometrical interpretations of
various scalar fields, such as an axion-, a dilaton and a Higgs field, using
ideas and results from [FRO1, ]. As in Connes’ approach, "space-time" will
have the structure of two copies of the usual four-dimensional space-time
carrying ( a priori massless) left-handed and right-handed spinors,
respectively. This is reminiscent of a five-dimensional generalization of the
quantum Hall effect discussed in [FRO2, ], the extra fifth dimension being
treated as a discrete two-point set.
The axion will turn out to be the "fifth" component of the electromagnetic
vector potential, the dilaton to be a gravitational degree of freedom
associated with the discrete fifth dimension, and the Higgs field will appear
as a component of the electroweak gauge field that induces tunneling processes
between the two sheets of "space-time"and provides masses to the fermions and
to the W- and Z gauge bosons, as sketched in figure 1.
Our paper is organized as follows. In section II, we summarize, in a sketchy
way, some elements of noncommutative geometry that are needed in subsequent
sections. For further details, the reader is referred to [CO1, ],[CO2, ] and
[FRO1, ]. In section III, we first recover an axion field (section III.1) by
identifying it with the fifth component of the electromagnetic vector
potential. This represents the simplest application of our formalism. In
section III.2, we proceed to generalize the Einstein-Hilbert gravitational
action to our two-sheeted space-time and find that this leads to the
appearance of a dilaton field. Finally, in section III.3, we show how the
Higgs field of the electroweak theory finds a natural geometrical
interpretation within our formalism. Some additional remarks and conclusions
are sketched in section IV.
$\psi_{R}$$\psi_{L}$$\phi$ Figure 1: A schematic view of the Yukawa coupling
between left- and right-handed fermions, interpreted in a five dimensional
space time. The left- and right-handed fermions live on separate four-
dimensional sheets. The Higgs field couples left- to right-handed spinors via
quantum tunnelling.
## II Generalized differential geometry
Gauge theories are intimately related to differential geometry. The reader may
remember an undergraduate course on electromagnetism where the Maxwell
equations were entirely rewritten in terms of differential forms. Classical
fields in a gauge theory with gauge group $G$ are sections of some vector
bundles over space-time $M$ associated to a principal $G$-bundle over $M$.
Gauge potentials (such as the U(1)- or SU(2)- gauge potentials) are
$\mathcal{G}$-valued one-forms appearing in the definition of covariant
derivatives in a local basis of sections of associated vector bundles, and
$\mathcal{G}$ is the Lie algebra of $G$. A well-known theorem of Serre and
Swan ([SW, ]) tells us that all finite-dimensional vector bundles over a
smooth compact manifold $M$ correspond to finitely generated projective
$\mathcal{C}^{\infty}(M,\mathbb{C})$-modules. This result motivates our
present approach. To generalize classical gauge theories, we will introduce
∗algebras $\mathcal{A}$ (in particular non-commutative algebras) generalizing
the commutative algebra $\mathcal{C}^{\infty}(M,\mathbb{C})$, and then
consider finitely generated projective $\mathcal{A}$-modules and define a
generalization of the $\mathbb{Z}$-graded algebra of differential forms over
$M$. This furnishes the right kind of geometrical data enabling us to
generalize the notion of gauge theories.
### II.1 Basic definitions
Let $\mathcal{A}$ be a unital ∗algebra over the field $K=\mathbb{R}$ or
$\mathbb{C}$. We denote by
$\Omega(\mathcal{A})=\bigoplus_{p}\Omega^{p}(\mathcal{A})$ any
$\mathbb{Z}$-graded differential algebra with
$\mathcal{A}=\Omega^{0}(\mathcal{A})$. The graded product over the "algebra of
generalized differential forms" $\Omega(\mathcal{A})$ is denoted by
$\omega\omega^{\prime}$, where $\omega$, $\omega^{\prime}$ are elements of
$\Omega(\mathcal{A})$. The degree of a homogeneous element
$\omega\in\Omega(\mathcal{A})$ is denoted by $\text{deg}(\omega)$.
###### Definition 1.
Vector bundles over $\mathcal{A}$
Inspired by the theorem of Serre and Swan, one defines a noncommutative vector
bundle, $\mathcal{M}(\mathcal{A})$, over $\mathcal{A}$ as a finitely generated
projective (left) $\mathcal{A}-$module (see [CO1, ]). Every such module admits
a generating family, i.e., there exist
$s_{1},...,s_{n}\in\text{Hom}(\mathcal{M}(\mathcal{A}),\mathcal{A})$,
$e_{1},...,e_{n}\in\mathcal{M}(\mathcal{A})$ such that, for all
$x\in\mathcal{M}(\mathcal{A})$,
$x=\sum_{i=1}^{n}s_{i}(x)e_{i}.$
The set $\\{e_{i}\in\mathcal{M}(\mathcal{A}),\text{
}i=1,...,n\in\mathbb{N}\\}$, is called a generating family of sections of the
vector bundle $\mathcal{M}(\mathcal{A})$.
Next, we assume that there exists a $\mathbb{Z}_{2}$-graded nilpotent operator
$d_{\mathcal{A}}$ ($d_{\mathcal{A}}^{2}=0$) acting on a $\mathbb{Z}$-graded
differential algebra $\Omega(\mathcal{A})$. Since $\Omega(\mathcal{A})$ is a
left $\Omega(\mathcal{A})$-module, we may define the differential
$\delta_{\mathcal{A}}:=\left[d_{\mathcal{A}},\cdot\right]_{g}$ (1)
on the algebra $\Omega(\mathcal{A})$, where the commutator
$\left[\cdot,\cdot\right]_{g}$ respects the $\mathbb{Z}_{2}$-grading of
$\Omega(\mathcal{A})$, i.e.,
$\left[d_{\mathcal{A}},\omega_{p}\right]_{g}=d_{\mathcal{A}}\omega_{p}+(-1)^{p+1}\omega_{p}d_{\mathcal{A}}$
(2)
for any $\omega_{p}$ of degree $p$. For all homogeneous
$\omega\in\Omega(\mathcal{A})$, we may assume that $d_{\mathcal{A}}\omega$ is
homogeneous. Note that $\delta_{\mathcal{A}}$ is nilpotent:
$\displaystyle\delta_{\mathcal{A}}^{2}\omega$ $\displaystyle=$ $\displaystyle
d_{\mathcal{A}}^{2}\omega+(-1)^{\text{deg}(\omega)+1}d_{\mathcal{A}}\omega
d_{\mathcal{A}}+(-1)^{\text{deg}(d_{\mathcal{A}}\omega)+1}d_{\mathcal{A}}\omega
d_{\mathcal{A}}+\omega d_{\mathcal{A}}^{2}$ $\displaystyle=$ $\displaystyle
d_{\mathcal{A}}^{2}\omega+\omega d_{\mathcal{A}}^{2}=0.$
Furthermore, $\delta_{A}$ obeys the Leibniz’s rule
$\delta_{\mathcal{A}}(\omega\omega^{\prime})=\delta_{\mathcal{A}}(\omega)\omega^{\prime}+(-1)^{\text{deg}(\omega)}\omega\delta_{\mathcal{A}}\omega^{\prime}$
(3)
and $\delta_{\mathcal{A}}(1_{\mathcal{A}})=0$.
###### Definition 2.
Connections
Let $\mathcal{M}(\mathcal{A})$ be a projective, finitely generated (left)
$\mathcal{A}$-module, and let $\delta_{\mathcal{A}}$ be defined as in (1). A
connection, $\nabla$, on $\mathcal{M}(\mathcal{A})$ associated to
$\delta_{\mathcal{A}}$ is a $\mathbb{C}$-linear map
$\displaystyle\nabla:\mathcal{M}(\mathcal{A})$ $\displaystyle\longrightarrow$
$\displaystyle\Omega^{\text{odd}}(\mathcal{A})\otimes_{\mathcal{A}}\mathcal{M}(\mathcal{A})$
such that, for all $a\in\mathcal{A}$, $s\in\mathcal{M}(\mathcal{A})$,
$\nabla(as)=\delta_{\mathcal{A}}a\otimes s+a\nabla s.$ (4)
$\delta_{\mathcal{A}}a$ in (4) is understood as
$(\delta_{\mathcal{A}}a)1_{\mathcal{A}}=d_{\mathcal{A}}a-ad_{\mathcal{A}}1_{\mathcal{A}}$.
Every projective finitely generated module having a generating family
$\\{e_{i}\\}_{i=1}^{n}$ of sections, connections are entirely determined by
their action on the $e_{i}$’s
$\nabla(e_{i})=-\Omega^{j}_{i}\otimes e_{j},$
where $\Omega^{j}_{i}\in\Omega^{\text{odd}}(\mathcal{A})$. The forms
$\Omega^{j}_{i}$ correspond to the gauge potential in classical gauge
theories. If the module is free and the generating family is a basis, one can
choose arbitrary forms $\Omega^{j}_{i}$. If the module is not free one has to
impose some restrictions on the coefficients $\Omega^{j}_{i}$ ([CO1, ],[CH2,
]).
We require that
$\nabla(\omega\otimes s)=\delta_{\mathcal{A}}\omega\otimes
s+(-1)^{\text{deg}(\omega)}\omega\nabla s$ (5)
for all homogeneous $\omega\in\Omega(\mathcal{A})$,
$s\in\mathcal{M}(\mathcal{A})$, where the product is between forms, i.e.,
$\omega(\omega_{1}\otimes s)=(\omega\omega_{1})\otimes s$. As in (4),
$\delta_{\mathcal{A}}\omega$ in (5) is understood as
$(\delta_{\mathcal{A}}\omega)1_{\mathcal{A}}$. Using (5), we can extend the
definition of a connection to
$\Omega(\mathcal{A})\otimes_{\mathcal{A}}\mathcal{M}(\mathcal{A})$ in a unique
way and define curvature as follows.
###### Definition 3.
Curvature
The curvature of a connection $\nabla$ is the left $\mathcal{A}$-linear map:
$-\nabla^{2}:\mathcal{M}(\mathcal{A})\longrightarrow\Omega^{\text{even}}(\mathcal{A})\otimes_{\mathcal{A}}\mathcal{M}(\mathcal{A}).$
(6)
### II.2 Generalization of the algebra of differential forms
Let $\mathcal{A}$ and $\mathcal{B}$ be unital algebras over the field
$K=\mathbb{R}$ or $\mathbb{C}$. We consider $\mathbb{Z}$-graded differential
algebras $\Omega(\mathcal{A})$ and $\Omega(\mathcal{B})$, with
$\mathcal{A}=\Omega^{0}(\mathcal{A})$, $\mathcal{B}=\Omega^{0}(\mathcal{B})$.
We write $\mathcal{C}:=\mathcal{A}\otimes_{K}\mathcal{B}$. Then
$\Omega(\mathcal{A})\otimes_{K}\Omega(\mathcal{B})$ is a left
$\mathcal{C}$-module and can be equipped with a graded product. Henceforth we
usually omit the "$K$" in $\otimes_{K}$.
###### Definition 4.
Graded product over $\Omega(\mathcal{A})\otimes\Omega(\mathcal{B})$
The graded product, $\wedge$, over the algebra
$\Omega(\mathcal{A})\otimes\Omega(\mathcal{B})$ is defined as follows: For all
homogeneous elements $\omega,\omega^{\prime}\in\Omega(\mathcal{A})$ and
$\sigma,\sigma^{\prime}\in\Omega(\mathcal{B})$,
$(\omega\otimes\sigma)\wedge(\omega^{\prime}\otimes\sigma^{\prime})=(-1)^{\text{deg}(\sigma)\text{deg}(\omega^{\prime})}\omega\omega^{\prime}\otimes\sigma\sigma^{\prime}.$
(7)
With this product, $\Omega(\mathcal{A})\otimes\Omega(\mathcal{B})$ is a
$\mathbb{Z}$-graded algebra, and we have that
$(\Omega(\mathcal{A})\otimes\Omega(\mathcal{B}))^{n}=\underset{p+q=n}{\bigoplus}\Omega(\mathcal{A})^{p}\otimes\Omega(\mathcal{B})^{q}$
where $\Omega^{p}(.)$ is the subspace of $\Omega(.)$ of degree $p$.
We assume that there exist $\mathbb{Z}_{2}$-graded nilpotent operators
$d_{\mathcal{A}}$ on $\Omega(\mathcal{A})$ and $d_{\mathcal{B}}$ on
$\Omega(\mathcal{B})$.
###### Definition 5.
Extension of $(d_{\mathcal{A}},d_{\mathcal{B}})$
An extension of $(d_{\mathcal{A}},d_{\mathcal{B}})$ is a
$\mathbb{Z}_{2}$-graded, linear nilpotent operator $\tilde{d}$ acting on the
left $\mathcal{C}$-module
$\Omega(\mathcal{C}):=\Omega(\mathcal{A})\otimes\Omega(\mathcal{B})$ that can
be written in the form
$\tilde{d}=\alpha d_{\mathcal{A}}\otimes
1_{\mathcal{B}}+\beta\Gamma_{\mathcal{A}}\otimes d_{\mathcal{B}}+\sigma$ (8)
where $\sigma$ is an odd element of $\Omega(\mathcal{C})$, $\alpha,\beta\in
K$, and $\Gamma_{\mathcal{A}}$ is the involution on $\Omega(\mathcal{A})$
defined by
$\Gamma_{\mathcal{A}}(\omega)=(-1)^{\text{deg}(\omega)}\omega,$
for a homogeneous $\omega\in\Omega(\mathcal{A})$.
As in (1), we define a differential
$\tilde{\delta}:=\left[\tilde{d},\cdot\right]_{g}$ on the graded algebra
$\Omega(\mathcal{C})$, as well as connections and curvature on any
(noncommutative) vector bundle $\mathcal{M}(\mathcal{C})$. When $\sigma=0$ in
(8), it is easy to check that $\tilde{d}^{2}=0$. Let
$\kappa:=\omega\otimes\omega^{\prime}\in\Omega(\mathcal{C})$, with $\omega$
homogeneous. One then has that
$\displaystyle\tilde{d}^{2}\kappa$ $\displaystyle=$
$\displaystyle\tilde{d}(\alpha
d_{\mathcal{A}}\omega\otimes\omega^{\prime}+\beta(-1)^{\text{deg}(\omega)}\omega\otimes
d_{\mathcal{B}}\omega^{\prime})$ $\displaystyle=$
$\displaystyle\alpha^{2}d_{\mathcal{A}}^{2}\omega\otimes\omega^{\prime}+\alpha\beta(-1)^{\text{deg}(d_{\mathcal{A}}\omega)}d_{A}\omega\otimes
d_{\mathcal{B}}\omega^{\prime}+\alpha\beta(-1)^{\text{deg}(\omega)}d_{\mathcal{A}}\omega\otimes
d_{B}\omega^{\prime}+\beta^{2}\omega\otimes
d_{\mathcal{B}}^{2}\omega^{\prime}$ $\displaystyle=$ $\displaystyle 0.$
If $\sigma\neq 0$ one must add the conditions that $\left[\alpha
d_{\mathcal{A}}\otimes 1_{\mathcal{B}}+\beta\Gamma_{\mathcal{A}}\otimes
d_{\mathcal{B}},\sigma\right]_{g}=0$ and $\sigma^{2}=0$.
Below, we will choose for $\Omega(\mathcal{B})$ the exterior algebra of a
finite-dimensional vector space $V$ over $K$, which we denote by
$\mathcal{G}(V)$; ( $\mathcal{G}$ stands for "Grassmann Algebra"). This is a
graded commutative algebra over the field $K$. The algebra $\mathcal{B}$ is
the field $K$. We denote by $\times$ the exterior product on $\mathcal{G}(V)$,
and, with $\mathcal{C}=\mathcal{A}\otimes_{K}K\approx\mathcal{A}$,
$\Omega(\mathcal{A})_{V}:=\Omega(\mathcal{C})=\Omega(\mathcal{A})\otimes\mathcal{G}(V)$.
Let $\xi_{p}\in\mathcal{G}(V)$ be a homogeneous element of odd degree $p$. The
operator $d_{\mathcal{B}}:=\xi_{p}\times(.)$ acting on $\mathcal{G}(V)$ is
linear, $\mathbb{Z}_{2}$-graded and nilpotent. For any $\alpha,\beta\in K$,
$\tilde{d}:=\alpha d_{\mathcal{A}}\otimes
1+\beta\Gamma_{\mathcal{A}}\otimes(\xi_{p}\times.)=\alpha
d_{\mathcal{A}}\otimes 1+\beta(1_{\mathcal{A}}\otimes\xi_{p})\wedge\cdot$ (9)
is linear, $\mathbb{Z}_{2}$-graded and nilpotent. More generally, we have the
following proposition.
###### Proposition 1.
Let $\xi_{p}\in\mathcal{G}(V)$ be a homogeneous element of odd degree, and,
let $\omega\in\Omega(\mathcal{A})$ be an even differential form such that
$\delta_{\mathcal{A}}(\omega)=0$. Then, for all $\alpha\in K$,
$\tilde{d}=\alpha d_{\mathcal{A}}\otimes 1+(\omega\otimes\xi_{p})\wedge\cdot$
(10)
is a linear nilpotent $\mathbb{Z}_{2}$-graded operator on
$\Omega(\mathcal{A})_{V}$.
If $\omega\in Z(\Omega(\mathcal{A}))$ (the center of $\Omega(\mathcal{A})$)
and if the vector space $V$ is one dimensional, then
$\tilde{\delta}=\left[d_{\mathcal{A}}\otimes
1+(\omega\otimes\xi_{1})\wedge,.\right]_{g}$ maps $\Omega(\mathcal{A})\otimes
1$ to itself,
$\tilde{\delta}(\omega^{\prime}\otimes
1)=\delta_{\mathcal{A}}\omega^{\prime}\otimes 1$
for any $\omega^{\prime}\in\Omega(\mathcal{A})$. The action of
$\tilde{\delta}$ on $\Omega(\mathcal{A})\otimes\xi_{1}$ is also of the form
$\tilde{\delta}(\omega^{\prime}\otimes\xi_{1})=\delta_{\mathcal{A}}\omega^{\prime}\otimes\xi_{1}$.
In other words, $\tilde{\delta}=\delta_{\mathcal{A}}\otimes 1$.
###### Corollary 1.
Let $\omega_{i}$, $i=1,...,n$, be commuting differential forms of homogeneous
even degree in $\Omega(\mathcal{A})$ such that $\delta(\omega_{i})=0$, and let
$\xi_{p_{i}}\in\mathcal{G}(V)$ be homogeneous elements of odd degree. Then,
for all $\alpha\in K$,
$\tilde{d}=\alpha d_{\mathcal{A}}\otimes
1+\sum_{i}(\omega_{i}\otimes\xi_{p_{i}})\wedge\cdot$ (11)
is a linear nilpotent $\mathbb{Z}_{2}$-graded operator on
$\Omega(\mathcal{A})_{V}$.
### II.3 Hermitian structure on $\Omega(\mathcal{A})_{V}$
Until now, the algebras $\mathcal{A}$ and $\Omega(\mathcal{A})$ have been
quite general. From now on, we focus on the case where
$\mathcal{A}=\mathcal{C}^{\infty}(M,M_{n}(K))$ and
$\Omega(\mathcal{A})=\Omega(M,M_{n}(K))$ is the $\mathbb{Z}$-graded algebra of
$M_{n}(K)$\- ($n\times n$ matrices) valued forms, where $M$ is a compact,
orientable, smooth manifold of dimension $m$. In this section, we suppose that
$M$ is Riemannian. The exterior derivative on $\Omega(M,M_{n}(K))$ is denoted
by $d$. We construct a hermitian structure on $\Omega^{p}(\mathcal{A})_{V}$
using a generalization of the Hodge operator on $\Omega(\mathcal{M})$. Let
$\text{dim}(V)=k$. We choose a basis $(\xi^{m+1},...,\xi^{m+k})$ of $V$. This
basis has the same properties as the fermionic superspace coordinates used in
the theory of supermanifolds. We introduce the notion of Berezin integration
on $\mathcal{G}(V)$ well known from fermionic functional integrals.
###### Definition 6.
Berezin integration
Let $\int_{b}$ denote Berezin integration on $\mathcal{G}(V)$, i.e.,
$\int_{b}d\xi^{i}\xi^{i}=1$, $\int_{b}d\xi^{i}=0$, and
$\int_{b}d\xi^{m+k}...\text{ }d\xi^{m+1}\xi^{m+1}...\text{ }\xi^{m+k}=1.$ (12)
Take $(dx^{1},...,dx^{m})$ to be a coordinate basis of $1$-forms on $M$. To
define the extended Hodge operator, we write: $\xi^{i}:=dx^{i}\otimes 1\equiv
dx^{i}$. The metric on the manifold M is denoted by $g$. To raise the indices
of the totally antisymmetric tensor $\epsilon_{\mu\nu...}$, we extend $g$ by
imposing $g^{(m+i)j}=g^{j(m+i)}=\delta^{(m+i)j}$, for all
$i\in\\{{1,...,k\\}}$ and $j\in\\{{1,...,m+k\\}}$. This choice is consistent
because it is not affected by any change of coordinates on $M$.
###### Definition 7.
Extended Hodge operator
The extended Hodge $*$-operator is the map
$*.:\Omega^{p}(\mathcal{A})_{V}\longrightarrow\Omega^{m+k-p}(\mathcal{A})_{V}$
defined by:
$*(\xi^{\mu_{1}}\wedge...\wedge\xi^{\mu_{p}})=\frac{\sqrt{\mid
g\mid}}{(m+k-p)!}\epsilon^{\mu_{1}...\mu_{p}}_{\text{ }\text{
}\nu_{p+1}...\nu_{m+k}}\xi^{\nu_{p+1}}\wedge...\wedge\xi^{\nu_{m+k}}$ (13)
and if
$\omega=\frac{1}{p!}\omega_{\mu_{1}...\mu_{p}}\xi^{\mu_{1}}\wedge...\wedge\xi^{\mu_{p}}\in\Omega^{p}(\mathcal{A})_{V}$,
by
$*(\omega)=\frac{\sqrt{\mid
g\mid}}{(m+k-p)!p!}(\omega_{\mu_{1}...\mu_{p}})^{\dagger}\text{
}\epsilon^{\mu_{1}...\mu_{p}}_{\text{ }\text{
}\nu_{p+1}...\nu_{m+k}}\xi^{\nu_{p+1}}\wedge...\wedge\xi^{\nu_{m+k}},$ (14)
where † is the adjoint on $M_{n}(K)$.
Next, we construct a hermitian structure $(\cdot,\cdot)$ on the
$\mathcal{A}$-module $\Omega^{p}(\mathcal{A})_{V}$, for any $p\in\mathbb{N}$.
A hermitian structure is a sesquilinear form
$(\cdot,\cdot):\Omega^{p}(\mathcal{A})_{V}\times\Omega^{p}(\mathcal{A})_{V}\rightarrow\mathcal{A}$,
such that
$\begin{array}[]{ll}i)&(as,bs^{\prime})=a(s,s^{\prime})b^{\dagger}\mbox{, for
all }a,b\in\mathcal{A},\text{ }s,s^{\prime}\in\Omega^{p}(\mathcal{A})_{V},\\\
ii)&(s,s)\geq 0,\mbox{ for all }s\in\Omega^{p}(\mathcal{A})_{V}\mbox{, and
}(s,s)=0\Rightarrow s=0.\\\ \end{array}$ (15)
For arbitrary $\omega,\omega^{\prime}\in\Omega^{p}(\mathcal{A})_{V}$, we
define $(\cdot,\cdot)$ by
$\omega\wedge(*\omega^{\prime})=:(\omega,\omega^{\prime})d\mathcal{V}$ (16)
where $d\mathcal{V}=\sqrt{\mid g\mid}\xi^{1}...\text{ }\xi^{m+k}$ is the
invariant extended volume form. The fact that $(\cdot,\cdot)$ defined in (16)
satisfies properties i) and ii) of (15) is obvious from the definitions.
The space $\Omega^{p}(\mathcal{A})_{V}$ of $p-$forms also carries a scalar
product
$\displaystyle\langle\cdot,\cdot\rangle:\Omega^{p}(\mathcal{A})_{V}\times\Omega^{p}(\mathcal{A})_{V}$
$\displaystyle\longrightarrow$ $\displaystyle\mathbb{C}\cup\\{\pm\infty\\}$
$\displaystyle(\omega,\omega^{\prime})$ $\displaystyle\longmapsto$
$\displaystyle\langle\omega,\omega^{\prime}\rangle:=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int
tr(\omega\wedge*(\omega^{\prime})),$
where we have set
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\omega:=(-1)^{mk}\int_{M}\int_{b}d\xi^{m+k}...\text{
}d\xi^{m+1}\omega.$ (17)
The factor $(-1)^{mk}$ ensures positivity of the scalar product and comes from
the anticommutation relations $\\{\xi^{i},\xi^{j}\\}=0$, for
$i,j\in\\{1,...,m+k\\}$. On the right-hand side, the Berezin integration is
defined in the following way: For all
$\omega=\frac{1}{p!}\omega_{\mu_{1}...\mu_{p}}\xi^{\mu_{1}}...\text{
}\xi^{\mu_{p}}\in\Omega^{p}(\mathcal{A})_{V}$,
$\int_{b}d\xi^{m+k}...\text{
}d\xi^{m+1}\omega:=\frac{1}{p!}\omega_{\mu_{1}...\mu_{p}}\int_{b}d\xi^{m+k}...\text{
}d\xi^{m+1}\xi^{\mu_{1}}...\text{ }\xi^{\mu_{p}}$ (18)
and the Berezin integration is carried out by putting all the Berezin
variables $\xi^{\mu_{i}}$ on the left after passing them through the
coordinate 1-forms. For instance,
$\int_{b}d\xi^{1}dxdy\xi^{1}=\left(\int_{b}d\xi^{1}\xi^{1}\right)dxdy=dxdy.$
## III Dimensional Reduction
In this section, we apply "generalized differential geometry" to some examples
from classical field theory in order to show that various classical fields,
such as the axion, acquire a natural geometrical interpretation. We begin with
the axion field that has appeared in [FRO2, ] by dimensional reduction of
Maxwell theory, starting from a five-dimensional bulk space-time.
### III.1 Axion field
To recover the axion field, only a little change of the differential geometric
formulation of electromagnetism is necessary. Let $M$ be a compact four-
dimensional Lorentzian manifold without boundary. We consider the algebra
$\mathcal{A}=\mathcal{C}^{\infty}(M,\mathbb{C})$ ($K=\mathbb{C}$). The new
ingredient that makes the axion field appear is the modification of the graded
algebra of differential forms over $M$. We choose
$V=\\{\lambda\xi_{1},\lambda\in\mathbb{C}\\}$ the one-dimensional vector space
spanned by $\xi_{1}$, and its exterior algebra $\mathcal{G}(V)$. On
$\Omega(\mathcal{A})_{V}=\Omega(M,\mathbb{C})\otimes\mathcal{G}(V)$, we define
a natural generalization of the exterior derivative satisfying the hypotheses
of Proposition 1.
$\tilde{d}=d\otimes 1+\alpha(1\otimes\xi_{1})\wedge\cdot$ (19)
with $\alpha\in\mathbb{C}$. A connection $\nabla$ on
$\mathcal{M}(\mathcal{A}):=\mathcal{A}=\mathcal{C}^{\infty}(M,\mathbb{C})$ is
a $\mathbb{C}$-linear map
$\nabla:\mathcal{C}^{\infty}(M,\mathbb{C})\longrightarrow\Omega^{1}(\mathcal{A})_{V}\otimes_{\mathcal{C}^{\infty}(M)}\mathcal{C}^{\infty}(M,\mathbb{C})\cong\Omega^{1}(\mathcal{A})_{V}.$
###### Proposition 2.
Let $\nabla$ be any connection on $\mathcal{C}^{\infty}(M,\mathbb{C})$ and
$f\in\mathcal{C}^{\infty}(M,\mathbb{C})$. Then
$\nabla f=-\Omega\otimes f$ (20)
where $\Omega=\omega+\phi\xi_{1}$, with $\omega\in\Omega^{1}(M,\mathbb{C})$,
$\phi\in\mathcal{C}^{\infty}(M,\mathbb{C})$.
This proposition follows directly from the definition of $\nabla$. The module
being free, we require that $\phi\in\mathcal{C}^{\infty}(M,\mathbb{R})$, so
that the field $\phi$ has zero charge. $\phi$ will turn out to be the axion
field. The curvature (see (6)) associated to a connection is
$\displaystyle-\nabla^{2}f$ $\displaystyle=$
$\displaystyle-\nabla(-\Omega\otimes f)=(\tilde{\delta}\Omega)\otimes
f-\Omega\wedge\nabla f$ $\displaystyle=$
$\displaystyle(\tilde{\delta}\Omega)\otimes f=:F_{\nabla^{2}}\otimes f$
with
$F_{\nabla^{2}}=\tilde{\delta}\Omega=\left[\tilde{d},\omega+\phi\xi_{1}\right]_{+}1=d\omega+d\phi\xi_{1}.$
(21)
In [FRO2, ], the integral of the Chern-Simons five-form led to an axion term
in the action. The corresponding extended integral of the extended Chern-
Simons five-form $\Omega\wedge F_{\nabla^{2}}\wedge F_{\nabla^{2}}$ is given
by
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\Omega\wedge
F_{\nabla^{2}}\wedge F_{\nabla^{2}}$ $\displaystyle=$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int(\omega+\phi\xi_{1})\wedge\left(d\omega+d\phi\xi_{1}\right)\wedge\left(d\omega+d\phi\xi_{1}\right)$
$\displaystyle=$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int(\omega+\phi\xi_{1})\left((d\omega)^{2}+2d\omega
d\phi\xi_{1}\right)$ $\displaystyle=$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\left(\omega
d\omega d\omega+\phi\xi_{1}(d\omega)^{2}+2\omega d\omega d\phi\xi_{1}\right).$
The Berezin integration $\int_{b}d\xi_{1}=0$ implies that
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\omega
d\omega d\omega=0$.
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\Omega\wedge
F_{\nabla^{2}}\wedge F_{\nabla^{2}}$ $\displaystyle=$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\left(\phi(d\omega)^{2}\xi_{1}+2\omega
d\omega d\phi\xi_{1}\right)$ $\displaystyle=$
$\displaystyle\int_{M}\left(\phi(d\omega)^{2}+2\omega d\omega d\phi\right).$
The manifold $M$ has no boundary, and therefore
$0=\int_{M}d(\omega d\omega\phi)=\int_{M}d\omega d\omega\phi-\int_{M}\omega
d\omega d\phi,$
which finally yields
$\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\Omega\wedge
F_{\nabla^{2}}\wedge F_{\nabla^{2}}=3\int_{M}\phi(d\omega)^{2}$ (22)
with $d\omega$ in (22) the electromagnetic field strength in four-dimensional
space-time. We see that $\phi$ can be interpreted as an axion field that
couples to the electromagnetic field. We find the same result as in [FRO2, ].
However, we have not added any extra continuous dimension. We recover the
kinetic term for the axion by dimensional reduction of the Maxwell action
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int
F_{\nabla^{2}}\wedge(*F_{\nabla^{2}})$ $\displaystyle=$
$\displaystyle\int_{M}d\omega\wedge*(d\omega)_{4}+\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int(d\phi\xi_{1})\wedge*(d\phi\xi_{1})$
$\displaystyle=$
$\displaystyle\int_{M}d\omega\wedge*(d\omega)_{4}+\int_{M}\partial^{\mu}\phi\text{
}\partial_{\mu}\phi\sqrt{\mid g\mid}\text{ }d^{4}x$
where $*(.)_{4}$ is the Hodge operator on $\Omega(M)$.
### III.2 Gravity with dilaton
We derive an Einstein-Hilbert action with dilaton using our formalism. We
consider a four-dimensional compact Lorentzian manifold $M$ without boundary
and choose $K=\mathbb{R}$, $\mathcal{A}=\mathcal{C}^{\infty}(M,\mathbb{R})$
and $V=\\{\lambda\xi_{1},\lambda\in\mathbb{R}\\}$. On
$\Omega(\mathcal{A})_{V}:=\Omega(M,\mathbb{R})\otimes\mathcal{G}(V)$, we take
$\tilde{d}=d\otimes 1+\alpha(1\otimes\xi_{1})\wedge\cdot.$ (23)
We consider the vector bundle
$\mathcal{M}(\mathcal{A})=\Omega^{1}(\mathcal{A})_{V}$. It generalizes the
cotangent bundle of the manifold $M$. Connections, $\nabla$, on
$\mathcal{M}(\mathcal{A})$ are linear maps:
$\nabla:\Omega^{1}(\mathcal{A})_{V}\longrightarrow\Omega^{1}(\mathcal{A})_{V}\otimes_{\mathcal{A}}\Omega^{1}(\mathcal{A})_{V}.$
To keep our notation simple in the following calculations, we identify
$\xi_{1}\equiv dx^{4}$, as if $\xi_{1}$ were the coordinate one-form
corresponding to an extra dimension. We introduce an extension of the Cartan
basis
$E^{A}=e^{A}_{C}dx^{C},$ (24)
where $A,C=0,...,4$ and $(E^{A},E^{B})=\eta^{AB}$; $(\cdot,\cdot)$ is the
hermitian structure on $\Omega^{1}(\mathcal{A})_{V}$ defined in (16), and
$\eta^{AB}$ is the Minkowski metric tensor in five dimensions with signature
$(-,+,+,+,+)$.
###### Proposition 3.
Let $\nabla$ be a connection on $\Omega^{1}(\mathcal{A})_{V}$. With respect to
the Cartan basis,
$\nabla E^{A}=-\Omega_{B}^{A}\otimes E^{B}$ (25)
where $\Omega_{B}^{A}\in\Omega^{1}(M,\mathbb{R})$, for
$A,B\in\\{{0,...,4\\}}$, i.e.,
$\Omega_{B}^{A}=\omega_{B}^{A}+\phi_{B}^{A}dx^{4}$ (26)
with $\omega_{B}^{A}\in\Omega^{1}(\mathcal{A})_{V}$,
$\phi_{B}^{A}\in\mathcal{C}^{\infty}(M,\mathbb{R})$.
The curvature two form associated to $\nabla$ takes the form:
$\displaystyle-\nabla^{2}(\alpha_{A}E^{A})$ $\displaystyle=$
$\displaystyle-\nabla((\tilde{\delta}\alpha_{A})\otimes
E^{A}-\alpha_{A}\Omega_{B}^{A}\otimes E^{B})$ $\displaystyle=$
$\displaystyle-\left[(\tilde{\delta}\alpha_{A}\wedge\Omega_{B}^{A})\otimes
E^{B}-\tilde{\delta}(\alpha_{A}\Omega_{B}^{A})\otimes
E^{B}-\alpha_{A}\Omega_{B}^{A}\wedge\Omega_{C}^{B}\otimes E^{C}\right]$
$\displaystyle=$
$\displaystyle\alpha_{A}(\tilde{\delta}\Omega_{C}^{A}+\Omega_{B}^{A}\wedge\Omega_{C}^{B})\otimes
E^{C}=\alpha_{A}\mathcal{R}_{C}^{A}\otimes E^{C}$
where
$\displaystyle\mathcal{R}_{C}^{A}$ $\displaystyle=$
$\displaystyle\tilde{\delta}\Omega_{C}^{A}+\Omega_{B}^{A}\wedge\Omega_{C}^{B}.$
(27)
We can compute the scalar curvature using (27). In the following calculations,
we denote by capital letters $A,B,...$ indices that take values in
$\\{0,1,2,3,4\\}$ and by $a,b,...$ indices in the range $0$ to $3$. To
simplify matters, we suppose that the Cartan basis is of the form
$E^{A}=\delta^{A}_{a}e^{a}_{\mu}dx^{\mu}+\delta^{A}_{4}e^{\sigma}dx^{4}$ (28)
where $\sigma\in\mathcal{C}^{\infty}(M,\mathbb{R})$. With this ansatz, we
tacitly assume that the added dimension does not "warp" when one moves along
$M$. The hermitian structure $(\cdot,\cdot)$ defined in (16) satisfies
$(dx^{\mu},dx^{4})=0$. We make the following hypotheses:
* •
The connection is torsion free, i.e., $T(\nabla)=0$; (for the definition of
torsion see [FRO1, ])
* •
The connection is unitary with respect to the metric on the extended tangent
space, i.e.,
$\tilde{\delta}(\omega_{1},\omega_{2})=(\nabla\omega_{1},\omega_{2})+(\omega_{1},\nabla\omega_{2})$
(29)
for arbitrary $\omega_{1},\omega_{2}\in\Omega^{1}(\mathcal{A})_{V}$.
These constraints characterize the Levi-Civita connection.
#### III.2.1 Torsion-free condition
One has that
$\displaystyle T(\nabla)E^{A}$ $\displaystyle=$
$\displaystyle\tilde{\delta}E^{A}+\Omega^{A}_{B}\wedge E^{B}=0.$
By writing $e^{a}=e^{a}_{\mu}dx^{\mu}$, it is easy to show that this condition
leads to the following identities.
* •
$A=a$:
$\left\\{\begin{array}[]{ll}de^{a}+\omega_{b}^{a}e^{b}&=0\\\ \omega_{\nu
4}^{a}e^{\sigma}-\phi_{b}^{a}e^{b}_{\nu}&=0\end{array}\right.$ (30)
* •
$A=4$:
$\left\\{\begin{array}[]{ll}\omega^{4}_{\mu b}e^{b}_{\nu}-\omega^{4}_{\nu
b}e^{b}_{\mu}&=0\\\ (\partial_{\nu}\sigma)e^{\sigma}+\omega_{\nu
4}^{4}e^{\sigma}-\phi_{b}^{4}e^{b}_{\nu}&=0\end{array}\right.$ (31)
#### III.2.2 Unitarity condition
Next, we use the unitarity condition (29)
$\displaystyle\tilde{\delta}(E^{A},E^{B})$ $\displaystyle=$
$\displaystyle(\nabla E^{A},E^{B})+(E^{A},\nabla E^{B})$ $\displaystyle=$
$\displaystyle-(\Omega^{A}_{C}E^{C},E^{B})-(E^{A},\Omega^{B}_{D}E^{D})$
$\displaystyle=$
$\displaystyle-\Omega^{A}_{C}\eta^{CB}-\Omega^{B}_{D}\eta^{AD}.$
By definition,
$\tilde{\delta}(E^{A},E^{B})=\tilde{\delta}(\eta^{AB})=0.$
Consequently, we are led to
$\left\\{\begin{array}[]{ll}\omega^{A}_{C}\eta^{CB}+\omega^{B}_{D}\eta^{AD}&=0\\\
\phi^{A}_{C}\eta^{CB}+\eta^{AD}\phi^{B}_{D}&=0\end{array}\right.$ (32)
Listing all the possibilities for the components $A$ and $B$ of $\omega$ and
$\phi$, and using equations (30) and (31), we see that the components of the
connection satisfy the identities
$\left\\{\begin{array}[]{ll}\phi^{4}_{b}&=e^{\nu}_{b}(\partial_{\nu}\sigma)e^{\sigma}\\\
\phi^{b}_{4}&=-e^{\nu}_{c}(\partial_{\nu}\sigma)e^{\sigma}\eta^{cb}\end{array}\right.$
(33)
with all other components of $\phi^{A}_{B}$ vanishing. For $\omega^{A}_{B}$,
only the forms $\omega^{a}_{b}$ may be non zero.
#### III.2.3 Components of the curvature tensor
We have to find an expression for the components of the curvature tensor in
terms of the components of the connection calculated in (33). According to
(27),
$\displaystyle\mathcal{R}^{A}_{B}$ $\displaystyle=$
$\displaystyle\tilde{\delta}\Omega^{A}_{B}+\Omega^{A}_{C}\Omega^{C}_{B}$
$\displaystyle=$ $\displaystyle\frac{1}{2}R^{A}_{BCD}\text{ }E^{C}\wedge
E^{D}.$
An easy identification leads to
$\displaystyle R^{A}_{Bcd}$ $\displaystyle=$ $\displaystyle
e^{\mu}_{c}e^{\nu}_{d}\left(\partial_{\mu}\omega_{\nu
B}^{A}-\partial_{\nu}\omega_{\mu B}^{A}+\omega_{\mu E}^{A}\omega_{\nu
B}^{E}-\omega_{\nu E}^{A}\omega_{\mu B}^{E}\right)$ (34) $\displaystyle
R^{A}_{B4d}$ $\displaystyle=$ $\displaystyle
e^{-\sigma}e^{\nu}_{d}\left(-\partial_{\nu}\phi_{B}^{A}+\phi_{E}^{A}\omega_{\nu
B}^{E}-\omega_{\nu E}^{A}\phi_{B}^{E}\right).$ (35)
As our main goal is to compute the scalar curvature, we have to find the
components of the Ricci tensor using that
$R_{BD}=R^{A}_{BAD}=R^{a}_{BaD}+R^{4}_{B4D}.$ (36)
Because the scalar curvature is given by
$R=\eta^{BD}R_{BD},$ (37)
we only have to determine $R_{bd}$ and $R_{44}$. For instance,
$\displaystyle R_{bd}$ $\displaystyle=$ $\displaystyle
R^{a}_{bad}+R^{4}_{b4d}$ $\displaystyle=$
$\displaystyle\underbrace{e^{\mu}_{a}e^{\nu}_{d}\left(\partial_{\mu}\omega_{\nu
b}^{a}-\partial_{\nu}\omega_{\mu b}^{a}+\omega_{\mu c}^{a}\omega_{\nu
b}^{c}-\omega_{\nu c}^{a}\omega_{\mu b}^{c}\right)}_{=R^{(4)}_{bd}}+$
$\displaystyle+$
$\displaystyle\underbrace{e^{-\sigma}e^{\nu}_{d}\left(-\partial_{\nu}\phi_{b}^{4}+\phi_{c}^{4}\omega_{\nu
b}^{c}-\cancel{\omega_{\nu c}^{4}\phi_{b}^{c}}+\cancel{\phi_{4}^{4}\omega_{\nu
b}^{4}}-\cancel{\omega_{\nu 4}^{4}\phi_{b}^{4}}\right)}_{(I)}.$
It is possible to evaluate (I) using properties of the Cartan basis. One finds
that
$\displaystyle(I)$ $\displaystyle=$ $\displaystyle
e^{\nu}_{d}\left(-\underbrace{\left[\partial_{\nu}e^{\mu}_{b}-\omega_{\nu
b}^{c}e^{\mu}_{c}\right]}_{(II)}\partial_{\mu}\sigma-e^{\mu}_{b}(\partial_{\nu}\partial_{\mu}\sigma)-e^{\mu}_{b}(\partial_{\nu}\sigma)(\partial_{\mu}\sigma)\right).$
The term (II) underlined above reduces to
$(II)=-e_{b}^{\alpha}\Gamma^{\mu}_{\nu\alpha}$ where
$\Gamma^{\mu}_{\nu\alpha}$ are the Christoffel symbols, defined, in any
coordinate basis, by
$\nabla^{(4)}(dx^{\mu})=-\Gamma^{\mu}_{\nu\alpha}dx^{\nu}\otimes dx^{\alpha}$
and $\nabla^{(4)}$ is the Levi-Civita connection on
$\Omega^{1}(M,\mathbb{R})$, given by
$\nabla^{(4)}(E^{a})=-\omega^{a}_{b}\otimes E^{b}.$
Indeed,
$\displaystyle\nabla^{(4)}(E^{a})$ $\displaystyle=$
$\displaystyle\nabla^{(4)}(e^{a}_{\kappa}dx^{\kappa})=\partial_{\nu}e^{a}_{\kappa}dx^{\nu}\otimes
dx^{\kappa}-e^{a}_{\delta}\Gamma^{\delta}_{\nu\kappa}dx^{\nu}\otimes
dx^{\kappa}$ $\displaystyle=$ $\displaystyle-\omega^{a}_{\nu
c}e^{c}_{\kappa}dx^{\nu}\otimes dx^{\kappa}$
which yields
$\partial_{\nu}e^{a}_{\kappa}-e^{a}_{\delta}\Gamma^{\delta}_{\nu\kappa}=-\omega^{a}_{\nu
c}e^{c}_{\kappa}.$ (38)
Moreover, as $e^{a}_{\mu}e^{\mu}_{b}=\delta^{a}_{b}$,
$\partial_{\nu}e^{\mu}_{b}=-e^{\kappa}_{b}e^{\mu}_{a}\partial_{\nu}e^{a}_{\kappa}.$
(39)
Plugging (38) and (39) into (II),
$\displaystyle(II)$ $\displaystyle=$
$\displaystyle-e^{\kappa}_{b}e^{\mu}_{a}(-\omega^{a}_{\nu
c}e^{c}_{\kappa}+e^{a}_{\delta}\Gamma^{\delta}_{\nu\kappa})-\omega_{\nu
b}^{c}e^{\mu}_{c}$ $\displaystyle=$
$\displaystyle-e^{\alpha}_{b}\Gamma^{\mu}_{\nu\alpha}.$
Thus,
$\displaystyle(I)$ $\displaystyle=$
$\displaystyle-e^{\nu}_{d}e_{b}^{\mu}\left(\underbrace{\left[-\Gamma^{\alpha}_{\nu\mu}\partial_{\alpha}\sigma+\partial_{\mu}\partial_{\nu}\sigma\right]}_{=\nabla^{(4)}_{\mu}\partial_{\nu}\sigma}+(\partial_{\nu}\sigma)(\partial_{\mu}\sigma)\right)$
where we have identified the components of the covariant derivative of
$\partial_{\nu}\sigma$. Then,
$R_{bd}=R_{bd}^{(4)}-e^{\nu}_{d}e_{b}^{\mu}\left(\nabla^{(4)}_{\mu}\partial_{\nu}\sigma+(\partial_{\nu}\sigma)(\partial_{\mu}\sigma)\right).$
(40)
In the same way, one finds for $R_{44}$
$\displaystyle R_{44}$ $\displaystyle=$
$\displaystyle-e^{\nu}_{a}e^{\mu}_{c}\eta^{ca}\left[\nabla^{(4)}_{\nu}\partial_{\mu}\sigma+(\partial_{\nu}\sigma)(\partial_{\mu}\sigma)\right].$
(41)
Using (37), the extended scalar curvature is given by
$R^{(5)}=R^{(4)}-2g^{\mu\nu}\left[\nabla^{(4)}_{\nu}\partial_{\mu}\sigma+(\partial_{\nu}\sigma)(\partial_{\mu}\sigma)\right].$
(42)
#### III.2.4 Einstein-Hilbert action and Dilaton
The generalized Einstein-Hilbert action reads
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\sqrt{\mid
g\mid}\text{ }R^{(5)}e^{\sigma}d^{4}x\xi_{1}$ $\displaystyle=$
$\displaystyle\int_{M}d^{4}x\sqrt{\mid
g\mid}R^{(5)}e^{\sigma}=\int_{M}d^{4}x\sqrt{\mid
g\mid}e^{\sigma}\left(R^{(4)}-2g^{\mu\nu}\left[\nabla_{\nu}\partial_{\mu}\sigma+(\partial_{\nu}\sigma)(\partial_{\mu}\sigma)\right]\right)$
where we have replaced $\nabla^{(4)}_{\nu}$ by $\nabla_{\nu}$, as there is no
risk of confusion, anymore. One can use a conformal transformation to change
the form of the integrand. Suppose that we rescale the metric,
$\tilde{g}_{\mu\nu}=e^{2\Phi}g_{\mu\nu}.$
For a manifold $M$ of dimension $d$ (cf. [CHO, ]), this rescaling changes the
scalar curvature by
$e^{2\Phi}\tilde{R}-R=-2(d-1)\nabla^{\nu}\partial_{\nu}\Phi-(d-2)(d-1)(\partial^{\nu}\Phi)(\partial_{\nu}\Phi).$
$M$ is four-dimensional and if we choose $\Phi=\frac{1}{2}\sigma$, we find
that
$e^{\sigma}\tilde{R}-R=-3\nabla^{\nu}\partial_{\nu}\sigma-\frac{3}{2}(\partial^{\nu}\sigma)(\partial_{\nu}\sigma).$
Here $R=R^{(4)}$. Consequently, the generalized Hilbert-Einstein action is
given by
$\displaystyle\int_{M}d^{4}x\sqrt{\mid g\mid}e^{\sigma}\text{ }R^{(5)}$
$\displaystyle=$ $\displaystyle\int_{M}d^{4}x\sqrt{\mid\tilde{g}\mid}\text{
}\left(\tilde{R}-\frac{1}{2}\tilde{g}^{\mu\nu}(\partial_{\nu}\sigma)(\partial_{\mu}\sigma)+\tilde{\nabla}^{\nu}\partial_{\nu}\sigma\right).$
(43)
$\tilde{\nabla}^{\nu}\partial_{\nu}\sigma$ can be rewritten as
$\frac{1}{\sqrt{\mid\tilde{g}\mid}}\partial_{\nu}(\sqrt{\mid\tilde{g}\mid}\partial^{\nu}\sigma)$.
As $M$ is without boundary,
$\int_{M}d^{4}x\sqrt{\mid\tilde{g}\mid}\text{
}\tilde{\nabla}^{\nu}\partial_{\nu}\sigma=0$
and only the kinetic term for the dilaton remains (cf. for instance [DA, ]):
$S=\int_{M}d^{4}x\sqrt{\mid\tilde{g}\mid}\text{
}\left(\tilde{R}-\frac{1}{2}\tilde{g}^{\mu\nu}(\partial_{\nu}\sigma)(\partial_{\mu}\sigma)\right).$
(44)
### III.3 Electroweak theory with a Higgs field
Let M be a four-dimensional compact Lorentzian manifold without boundary and
$\mathcal{A}=\mathcal{C}^{\infty}(M,\mathbb{C})$. We consider the
$\mathcal{A}$-bimodule
$\tilde{\mathcal{M}}(\mathcal{A})=S^{2}(M)\otimes_{\mathcal{A}}\mathcal{M}(\mathcal{A})$
where
$\mathcal{M}(\mathcal{A})=\mathcal{C}^{\infty}(M,\mathbb{C}^{2}\oplus\mathbb{C})$
and $S^{2}(M)$ is the Hilbert space of square integrable spinors on $M$.
$\tilde{\mathcal{M}}(\mathcal{A})$ is projective and finitely generated. We
consider the one-dimensional vector space
$V=\\{\lambda\xi_{1},\lambda\in\mathbb{C}\\}$ and introduce the exterior
derivative
$\tilde{d}=d\otimes 1$ (45)
on $\Omega(\mathcal{A})_{V}$. Connections on $\mathcal{M}(\mathcal{A})$ are
linear maps
$\displaystyle\nabla:\mathcal{M}(\mathcal{A})$ $\displaystyle\longrightarrow$
$\displaystyle\Omega^{\text{odd}}(\mathcal{A})_{V}\otimes_{\mathcal{A}}\mathcal{M}(\mathcal{A}).$
Once we have constructed a connection on $\mathcal{M}(\mathcal{A})$, we can
construct a connection on $\tilde{\mathcal{M}}(\mathcal{A})$ in the following
way. Let $\nabla_{S^{2}}$ be the canonical spin connection on $S^{2}(M)$. We
define
$\displaystyle\tilde{\nabla}:\tilde{\mathcal{M}}(\mathcal{A})$
$\displaystyle\longrightarrow$
$\displaystyle\Omega^{\text{odd}}(\mathcal{A})_{V}\otimes_{\mathcal{A}}\tilde{\mathcal{M}}(\mathcal{A})$
by
$\tilde{\nabla}(\psi\otimes f)=\nabla_{S^{2}}\psi\otimes
f+\pi(\psi\otimes\nabla f)$
where $\pi(\psi\otimes\omega\otimes f)=\omega\otimes\psi\otimes f$, for all
$\psi\in S^{2}(M)$, $\omega\in\Omega(\mathcal{A})_{V}$ and
$f\in\mathcal{M}(\mathcal{A})$.
We construct a connection $\nabla$ on the free $\mathcal{A}$-module
$\mathcal{M}(\mathcal{A})$. Let $(s_{1},s_{2},s_{3})$ be a basis of
$\mathcal{M}(\mathcal{A})$.
$\nabla
s_{i}=-\underbrace{\Omega^{j}_{i}}_{\in\Omega^{1}(\mathcal{A})_{V}}\otimes
s_{j}$
Similarly to (27), the components of the curvature tensor are given by
$(F_{\Omega})^{i}_{j}=\tilde{\delta}\Omega^{i}_{j}+\Omega^{i}_{k}\Omega^{k}_{j}.$
(46)
The general form of $\Omega:=(\Omega^{j}_{i})$ reads, in matrix notation,
$\Omega=A\otimes 1+B\otimes\xi_{1}$ (47)
where $A\in\Omega(M,M_{3}(\mathbb{C})),B\in M_{3}(\mathbb{C})$. The module
being free, we can take an arbitrary consistent choice for $A,B$ in (47). We
first introduce a Hermitian structure on $\mathcal{M}(\mathcal{A})$ in which
the basis is orthonormal, i.e., we choose $(\cdot,\cdot)$ such that
$(s_{i},s_{j})=\delta_{ij}$ and
$(f,f^{\prime})=\sum_{i=1}^{3}f_{i}\bar{f}^{\prime}_{i}$, $\bar{(\cdot)}$
denoting complex conjugation. We require $\Omega$ to be unitary with respect
to this metric, i.e., $\Omega$ must be skew-hermitian. We would like $A$ to be
chosen as in the Standard Model of particle physics (see, e.g. [WE, ]); i.e.,
$A=\left(\begin{array}[]{cc}\omega_{\tiny 2\times 2}&0_{2\times 1}\\\ 0_{\tiny
1\times 2}&\alpha_{1\times 1}\\\ \end{array}\right)$
where $\omega=\omega_{\mu}dx^{\mu}$ and $\alpha_{1\times
1}=\alpha_{\mu}dx^{\mu}$ are the $U(2)$ and $U(1)$ gauge potentials,
respectively. The form $\Omega$ being skew-hermitian, $\omega_{\mu}$ must be
skew-hermitian and $\alpha_{\mu}\in i\mathbb{R}$. We would like $B\notin
M_{2}(\mathbb{C})\oplus\mathbb{C}$ to exchange left- and right-handed spinors,
describing tunneling processes between the two sheets of space-time as
explained in section I.
$\displaystyle B=\left(\begin{array}[]{cc}0_{\tiny 2\times 2}&H\\\
-H^{\dagger}&0_{1\times 1}\\\ \end{array}\right)$
where $H\in\mathcal{C}^{\infty}(M,M_{2\times 1}(\mathbb{C}))$. We can add to
$B$ an axion field $\phi\in\mathcal{C}^{\infty}(M,\mathbb{R})$, as considered
in section III.1. Then the final form for $\Omega$ is given by
$\Omega=\left(\begin{array}[]{cc}\omega_{2\times 2}&0_{2\times 1}\\\
0_{1\times 2}&\alpha_{1\times 1}\\\ \end{array}\right)\otimes
1+\left(\begin{array}[]{cc}i\phi 1_{2\times 2}&H\\\ -H^{\dagger}&i\phi\\\
\end{array}\right)\otimes\xi_{1}.$ (49)
Next, we determine the components of the curvature two-form. Before doing so,
we propose to investigate how the components of $\Omega$ transform under a
gauge transformation.
#### III.3.1 Gauge transformations
Consider two bases of sections, $\\{s^{\prime}_{i}\\}$, $\\{s_{j}\\}$ such
that
$s^{\prime}_{i}=g^{j}_{i}s_{j}.$
One has that
$(\Omega^{\prime})_{i}^{l}=-(\tilde{d}g^{k}_{i})(g^{-1})^{l}_{k}+g^{j}_{i}\Omega^{k}_{j}(g^{-1})^{l}_{k}.$
(50)
The matrix-valued function $g$ maps a basis of sections to another basis of
sections. Since eft- and right-handed spinors should not be mixed by gauge
transformations, the most general form for $g$ is
$g=\left(\begin{array}[]{cc}A_{2\times 2}&0\\\ 0&e^{i\theta}\\\
\end{array}\right),$
where $A\in U(2)$, $\theta\in\mathbb{R}$. One then finds that
$\displaystyle\omega$ $\displaystyle\rightarrow$ $\displaystyle A\omega
A^{\dagger}-dAA^{\dagger}$ $\displaystyle\alpha$ $\displaystyle\rightarrow$
$\displaystyle\alpha-id\theta$ $\displaystyle H$ $\displaystyle\rightarrow$
$\displaystyle AHe^{-i\theta}$ $\displaystyle\phi$ $\displaystyle\rightarrow$
$\displaystyle\phi.$
$H$ transforms as the standard Higgs field under a gauge transformation. We
will need these formulas to check that the gauge field strength transforms
correctly under gauge transformations, i.e.,
$F_{\Omega}\rightarrow gF_{\Omega}g^{-1}.$
#### III.3.2 Curvature 2-form
We use the notations
$\displaystyle DH$ $\displaystyle:=$ $\displaystyle dH+\omega H-\alpha H$
$\displaystyle F_{\omega}$ $\displaystyle:=$ $\displaystyle
d\omega+\omega^{2}$ $\displaystyle F_{\alpha}$ $\displaystyle:=$
$\displaystyle d\alpha.$
Then
$F_{\Omega}=\left(\begin{array}[]{cc}F_{\omega}+i(d\phi)\xi_{1}1_{2\times
2}&(DH)\xi_{1}\\\
-(DH)^{\dagger}\xi_{1}&id\phi\xi_{1}+F_{\alpha}\end{array}\right).$ (51)
Under a gauge transformation $g$, $F_{\Omega}$ given in (51) satisfies the
transformation law
$F_{\Omega}\rightarrow\left(\begin{array}[]{cc}A_{2\times 2}&0\\\
0&e^{i\theta}\\\
\end{array}\right)F_{\Omega}\left(\begin{array}[]{cc}A^{\dagger}_{2\times
2}&0\\\ 0&e^{-i\theta}\\\ \end{array}\right).$ (52)
Indeed,
$\displaystyle DH$ $\displaystyle\rightarrow$ $\displaystyle
d(AHe^{-i\theta})+(A\omega A^{\dagger}-dAA^{\dagger})(AHe^{-i\theta})-(\alpha-
id\theta)(AHe^{-i\theta})$ $\displaystyle=$
$\displaystyle\cancel{dAHe^{-i\theta}}+e^{-i\theta}AdH-\bcancel{iAe^{-i\theta}d\theta}+A\omega
He^{-i\theta}-\cancel{dAHe^{-i\theta}}-\alpha
AHe^{-i\theta}+\bcancel{i(d\theta)AHe^{-i\theta}}$ $\displaystyle=$
$\displaystyle ADHe^{-i\theta}.$
All the other components are easily determined. It follows from (52) that the
action functional
$S=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int
tr\left[F_{\Omega}\wedge(*F_{\Omega})\right]$ (53)
is gauge-invariant.
#### III.3.3 Yukawa coupling and kinetic energy term for the Higgs field
It is not difficult to compute the Hodge dual of $F_{\Omega}$ in the basis
($dx^{0},dx^{1},dx^{2},dx^{3},dx^{4}=\xi_{1})$. If
$\omega=\frac{1}{p!}\omega_{\mu_{1}...\mu_{p}}dx^{\mu_{1}}\wedge...\wedge
dx^{\mu_{p}}\in\Omega^{p}(M,M_{3}(\mathbb{C}))_{V}$,
$*(w)=\frac{\sqrt{\mid
g\mid}}{(m+1-p)!p!}(\omega_{\mu_{1}...\mu_{p}})^{\dagger}\text{
}\epsilon^{\mu_{1}...\mu_{p}}_{\text{ }\text{
}\nu_{p+1}...\nu_{m+k}}dx^{\nu_{p+1}}\wedge...\wedge dx^{\nu_{m+k}}.$
As in II.3, to raise the lower indices of $\epsilon_{AB...}$, we extend the
metric tensor $g$ of the manifold M by defining $g^{4A}=\delta_{4}^{A}$, for
$A\in\\{0,1,2,3,4\\}$. In what follows, $A,B,...$ are indices that range from
0 to 4, whereas $\mu,\nu,...$ take values in $\\{0,1,2,3\\}$. We work in the
signature $(-,+,+,+)$ for the Minkowski metric $\eta_{\mu\nu}$.
If the two-form $\omega=\frac{1}{2}\omega_{\mu\nu}dx^{\mu}dx^{\nu}$ does not
contain any term with $\xi_{1}$ then
$\displaystyle*(\omega)$ $\displaystyle=$ $\displaystyle\frac{\sqrt{\mid
g\mid}}{3!2!}(\omega_{\mu\nu})^{\dagger}\text{ }\epsilon^{\mu\nu}_{\text{
}\text{ }A...C}dx^{A}\wedge...\wedge dx^{C}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\mid g\mid}}{2!2!}(\omega_{\mu\nu})^{\dagger}\text{
}\epsilon^{\mu\nu}_{\text{ }\text{ }\delta\gamma}dx^{\delta}\wedge
dx^{\gamma}\wedge\xi_{1}$ $\displaystyle=$
$\displaystyle*(\omega)_{4}\wedge\xi_{1}$
where $*(\omega)_{4}$ denotes the Hodge dual on $\Omega(M,M_{3}(\mathbb{C}))$.
We then find that
$*(F_{\Omega})=\left(\begin{array}[]{cc}*(F_{\omega})-i(\partial_{\mu}\phi)*(dx^{\mu}\xi_{1})1_{2\times
2}&-(DH)_{\mu}*(dx^{\mu}\xi_{1})\\\
(DH)_{\mu}^{\dagger}*(dx^{\mu}\xi_{1})&-i(\partial_{\mu}\phi)*(dx^{\mu}\xi_{1})+*(F_{\alpha})\end{array}\right).$
(54)
The usual Yang-Mills type action (53) involves only four terms
$S=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\left[(I)+(II)+(III)+(IV)\right]$
where:
$\displaystyle(I)$ $\displaystyle=$ $\displaystyle
tr(F_{\omega}\wedge*(F_{\omega}))+2\partial^{\mu}\phi\partial_{\mu}\phi\sqrt{\mid
g\mid}d^{5}x$ $\displaystyle(II)$ $\displaystyle:=$ $\displaystyle
tr\left((DH)_{\mu}(DH)^{\mu\dagger}\right)\sqrt{\mid g\mid}d^{5}x$
$\displaystyle(III)$ $\displaystyle:=$
$\displaystyle(DH)_{\mu}^{\dagger}(DH)^{\mu}\sqrt{\mid g\mid}d^{5}x$
$\displaystyle(IV)$ $\displaystyle:=$ $\displaystyle
F_{\alpha}\wedge*(F_{\alpha})+\partial^{\mu}\phi\partial_{\mu}\phi\sqrt{\mid
g\mid}d^{5}x.$
and $d^{5}x=d^{4}x\xi_{1}$. Using the fact that
$tr(AB^{\dagger})=B^{\dagger}A$ for any $A,B\in M_{2\times 1}(\mathbb{C})$, we
finally find that
$\displaystyle S$ $\displaystyle=$
$\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int\left[tr(F_{\omega}\wedge*(F_{\omega}))+F_{\alpha}\wedge*(F_{\alpha})+2(DH)_{\mu}^{\dagger}(DH)^{\mu}\sqrt{\mid
g\mid}d^{5}x+3\partial^{\mu}\phi\partial_{\mu}\phi\sqrt{\mid
g\mid}d^{5}x\right],$
i.e., after dimensional reduction,
$S=\int_{M}\left[tr(F_{\omega}\wedge*(F_{\omega})_{4})+F_{\alpha}\wedge*(F_{\alpha})_{4}+2(DH)_{\mu}^{\dagger}(DH)^{\mu}\sqrt{\mid
g\mid}dx^{4}+3\partial^{\mu}\phi\partial_{\mu}\phi\sqrt{\mid
g\mid}d^{4}x\right].$ (55)
The first two terms in (55) are the Yang-Mills actions of the $U(2)$ and
$U(1)$ gauge fields. To recover the classical $SU(2)$ and $U(1)$ gauge field
strengths, we further impose the constraint that
$tr(\omega)=\alpha.$
There is no mass term and no quartic potential for the Higgs field, but such
terms are gauge-invariant and are generated under renormalization. We will
elucidate why such terms are absent in section IV.
To determine the Yukawa couplings, we recall the definition of
$\tilde{\mathcal{M}}(\mathcal{A})$ and note that $\Omega^{1}(\mathcal{A})_{V}$
has 5 generators. The Clifford action
$c:\Omega^{1}(\mathcal{A})_{V}\rightarrow
End(\tilde{\mathcal{M}}(\mathcal{A}))$ is then given by
$\displaystyle c(dx^{\mu}):=i\tilde{\gamma}^{\mu}\otimes 1$ $\displaystyle
c(\xi_{1}):=\gamma^{5}\otimes 1$
where $\tilde{\gamma}^{\mu}$’s are the Dirac matrices in curved spacetime,
i.e., $\tilde{\gamma}^{\mu}=e^{\mu}_{a}\gamma^{a}$ with
$\\{\gamma^{a},\gamma^{b}\\}=-2\eta^{ab}$ (we work with the signature
$(-,+,+,+)$), $\gamma^{a\dagger}=-\gamma^{a}$ for $a=1,2,3$ and
$\gamma^{0\dagger}=\gamma^{0}$; $\gamma^{5}$ is given by the product
$\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$. One checks that
$\gamma^{5\dagger}=\gamma^{5}$, $(\gamma^{5})^{2}=1$. The Dirac Operator is
given by
$D_{c}:=c\circ\tilde{\nabla}:\tilde{\mathcal{M}}(\mathcal{A})\rightarrow\tilde{\mathcal{M}}(\mathcal{A}).$
This yields
$\displaystyle D_{c}$ $\displaystyle=$ $\displaystyle
i\gamma^{a}e_{a}^{\mu}(\partial_{\mu}-\frac{1}{2}i\omega^{bc}_{\mu}\Sigma_{bc}+\Omega_{\mu})+\gamma^{5}\Omega_{4}$
(56)
where $\Sigma_{ab}=\frac{i}{4}\left[\gamma_{a},\gamma_{b}\right]$ and
$\omega^{bc}_{\mu}$ are the components of the spin connection. We use the
notations
$\displaystyle\Psi=\left(\begin{array}[]{c}\psi_{1L}\\\ \psi_{2L}\\\
\psi_{3R}\end{array}\right),\qquad$ $\displaystyle
H=\left(\begin{array}[]{c}H_{1}\\\ H_{2}\end{array}\right),\qquad$
$\displaystyle\bar{\psi}_{i}=\psi_{i}^{\dagger}\gamma^{0}.$
The fermionic action, defined by
$S=\int d^{4}x\sqrt{\mid g\mid}\bar{\Psi}D_{c}\Psi,$ (58)
for arbitrary $\Psi\in\tilde{\mathcal{M}}(\mathcal{A})$, with
$\bar{\Psi}=(\bar{\psi}_{1L}\bar{\psi}_{2L}\bar{\psi}_{3R})$, gives rise to
the Yukawa and axion-fermion couplings through the term
$\bar{\Psi}\gamma^{5}\Omega_{4}\Psi$:
$\displaystyle\bar{\Psi}\gamma^{5}\Omega_{4}\Psi$ $\displaystyle=$
$\displaystyle\left(\bar{\psi}_{1L}H_{1}\gamma^{5}\psi_{3R}+\bar{\psi}_{2L}H_{2}\gamma^{5}\psi_{3R}-\bar{\psi}_{3R}\bar{H}_{1}\gamma^{5}{\psi}_{1L}-\bar{\psi}_{3R}\bar{H}_{2}\gamma^{5}{\psi}_{2L}\right)+i\sum_{i=1}^{3}\bar{\psi}_{i}\phi\gamma^{5}\psi_{i}$
$\displaystyle=$
$\displaystyle-\left(\bar{\psi}_{1L}H_{1}\psi_{3R}+\bar{\psi}_{2L}H_{2}\psi_{3R}+\bar{\psi}_{3R}\bar{H}_{1}{\psi}_{1L}+\bar{\psi}_{3R}\bar{H}_{2}{\psi}_{2L}\right)+i\sum_{i=1}^{3}\bar{\psi}_{i}\phi\gamma^{5}\psi_{i}.$
Coupling constants can be introduced by rescaling the fields.
## IV Remarks and Conclusions
We should explain why we do not find any quartic and quadratic terms for the
Higgs field. For this purpose, we outline a parallel between our point of view
and Connes’ point of view of noncommutative geometry. To do so, we introduce
the same toy model as Connes did in [CO1, ], (p.563-567). We consider the
discrete space $X=\\{{a,b\\}}$, formed by two separate points. Suppose that
there is a complex vector space $W_{a}$ of dimension $n_{a}$ attached to $a$,
and a complex vector space $W_{b}$ of dimension $n_{b}$ attached to $b$.
$W=W_{a}\oplus W_{b}$ is a projective finitely generated left
$\mathbb{C}\oplus\mathbb{C}$-module, and one could, exactly as in [CO1, ],
consider the algebra $\mathcal{A}=\mathbb{C}\oplus\mathbb{C}$ and introduce a
connection. In Connes’ formalism, the space of "noncommutative" one forms
$\Omega^{1}(\mathcal{A})$ is 2-dimensional. This is the reason why his Yang-
Mills action (p.567 of [CO1, ]) exhibits quartic and quadratic terms that
mimic the Higgs potential.
In our approach, we consider the vector bundle $W$ as a free
$\mathbb{C}$-module. We can choose a basis of sections
$a_{1},...,a_{n_{a}},b_{1},...,b_{n_{b}}$ with $a_{i}\in W_{a}$ and $b_{j}\in
W_{b}$ for all $i,j$. A connection is a linear map that determines the
variation of this basis when one moves along space-time. Here it quantifies
the variation due to jumping from $a$ to $b$. To quantify this jump, we
introduce the one-dimensional vector space
$V=\\{\lambda\xi_{1},\lambda\in\mathbb{C}\\}$ spanned by $\xi_{1}$, where
$\xi_{1}$ plays the role of $dx$ in the direction of the jump. The variation
$\Delta a_{i}=a^{\prime}_{i}-a_{i}$ can be written $\Delta
a_{i}=\phi_{i1}b_{1}+...+\phi_{in_{b}}b_{n_{b}}$, with
$\phi_{ij}\in\mathbb{C}$. In the same way, $\Delta
b_{i}=\phi^{\prime}_{i1}a_{1}+...+\phi^{\prime}_{in_{a}}a_{n_{a}}$, with the
$\phi^{\prime}_{ij}\in\mathbb{C}$. The connection has the form
$\nabla\left(\begin{array}[]{c}a_{1}\\\ ..\\\ a_{n_{a}}\\\ b_{1}\\\ ..\\\
b_{n_{b}}\end{array}\right)=\left(\begin{array}[]{cccccc}0&...&0&\phi_{11}&...&\phi_{1n_{b}}\\\
...&...&...&...&...&...\\\ 0&...&0&\phi_{n_{a}1}&...&\phi_{n_{a}n_{b}}\\\
\phi^{\prime}_{11}&...&\phi^{\prime}_{1n_{a}}&0&..&0\\\
...&...&...&...&...&...\\\
\phi^{\prime}_{n_{a}1}&...&\phi^{\prime}_{n_{b}n_{a}}&0&..&0\end{array}\right)\xi_{1}\otimes\left(\begin{array}[]{c}a_{1}\\\
..\\\ a_{n_{a}}\\\ b_{1}\\\ ..\\\ b_{n_{b}}\end{array}\right).$
Our approach is thus different from Connes’ approach and leads to different
results. For instance, quartic and quadratic terms in the $\phi_{ij}$’s vanish
because $\Omega(\mathcal{B})_{V}$ is one-dimensional. To get such terms one
must enlarge $V$, e.g. take a two-dimensional vector space. We have carried
out such generalizations for the Higgs field but they lead to fermion
doubling. Indeed, if $V$ has two generators $\xi_{1}$ and $\xi_{2}$, one can
write the connection $\Omega$ of the last section in the form (neglecting the
axion field)
$\Omega=\left(\begin{array}[]{cc}\omega_{2\times 2}&0_{2\times 1}\\\
0_{1\times 2}&\alpha_{1\times 1}\\\ \end{array}\right)\otimes
1+\left(\begin{array}[]{cc}0_{2\times 2}&H\\\ -H^{\dagger}&0\\\
\end{array}\right)\otimes\xi_{1}+\left(\begin{array}[]{cc}0_{2\times
2}&H^{\prime}\\\ -H^{\prime\dagger}&0\\\ \end{array}\right)\otimes\xi_{2}$
(59)
where $H^{\prime}=iH$ if one wants to recover a quartic term for the Higgs
field in the action. Then the Clifford action
$c:\Omega^{1}(\mathcal{A})_{V}\rightarrow
End(\tilde{\mathcal{M}}(\mathcal{A}))$ is given by
$\displaystyle c(dx^{\mu}):=\Gamma^{\mu}\otimes 1$ $\displaystyle
c(\xi_{1}):=\Gamma^{5}\otimes 1$ $\displaystyle c(\xi_{2}):=\Gamma^{6}\otimes
1$
where $\Gamma^{A}$, $A\in\\{0,1,2,3,5,6\\}$, are $8\times 8$ complex matrices.
The number of spinors has to be multiplied by a factor of two to make sense of
$\Gamma^{A}\psi$, and we end up with fermion doubling. As there is, a priori,
no obstruction against adding gauge-invariant terms to the action, we prefer a
five dimensional model.
The introduction of right-handed neutrinos is possible within our formalism.
The see-saw mechanism ( see [BO, ], [KI, ] for reviews) furnishes a potential
explanation of the origin of the mass of the left-handed neutrinos of the
Standard Model. It is based on the presence, in the action, of a Majorana mass
term for the right-handed neutrinos, of the form
$M_{rr}\bar{\nu}_{r}\nu_{r}^{c}$, and a small Dirac mass
$m_{lr}\bar{\nu}_{l}\nu_{r}+h.c.$, with $m_{lr}<<M_{rr}$, coming from Yukawa
couplings. The mass matrix can be written in the form
$(\bar{\nu}_{l}\bar{\nu}_{r}^{c})\left(\begin{array}[]{cc}0&m_{lr}\\\
m_{lr}^{\dagger}&M_{rr}\end{array}\right)\left(\begin{array}[]{c}\nu_{l}^{c}\\\
\nu_{r}\end{array}\right).$
The diagonalization of this mass matrix leads to a small mass for the left-
handed neutrinos, of the order of $m_{lr}M_{rr}^{-1}m_{lr}^{\dagger}$, whereas
the Majorana masses for the right handed neutrinos are left essentially
unchanged. We can introduce a Dirac mass in our model. Consider a toy model,
where we only add one right-handed neutrino, described by a Majorana spinor
$\nu_{R}^{c}=\nu_{R}$. On the free $\mathcal{C}^{\infty}(M,\mathbb{C})$-module
$\mathcal{C}^{\infty}(M,\mathbb{C}^{2}\oplus\mathbb{C}^{2})$, we can choose
the connection
$\Omega=\left(\begin{array}[]{cc}\omega_{2\times 2}&0_{2\times 2}\\\
0_{2\times 2}&\left(\begin{array}[]{cc}\alpha&0\\\ 0&0\end{array}\right)\\\
\end{array}\right)\otimes 1+\left(\begin{array}[]{cc}0_{2\times 2}&-m_{lr}\\\
m_{lr}^{\dagger}&0\\\ \end{array}\right)\otimes\xi_{1}.$ (60)
This connection leads to a Dirac mass term in the action.
###### Acknowledgements.
A. H. C is supported in part by the National Science Foundation under Grant
No. Phys-0854779.
## References
* [1] A. Boyarsky, O. Ruchayskiy, and M. Shaposhnikov. The role of sterile neutrinos in cosmology and astrophysics. Annu. Rev. Nucl. Part. S., 59(1):191–214, 2009.
* [2] A.H. Chamseddine, A. Connes, and M. Marcolli. Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys., 11:991–1089, 2007.
* [3] A.H. Chamseddine, G. Felder, and J. Fröhlich. Unified gauge theories in noncommutative geometry. Phys. Lett. B, 296(1):109–116, 1992.
* [4] A.H. Chamseddine, J. Fröhlich, and O. Grandjean. The gravitational sector in the connes–lott formulation of the standard model. J. Math. Phys., 36:6255–6275, 1995.
* [5] Y. Choquet-Bruhat. General Relativity and the Einstein Equations. Oxford Univ. Press, Oxford, 2009.
* [6] A. Connes. Noncommutative Geometry. Academic Press, San Diego, CA, 1994.
* [7] A. Connes and M. Marcolli. Noncommutative Geometry, Quantum Fields and Motives, volume 55. Amer. Math. Soc. Coll. Publ., 2008.
* [8] T. Damour, G.W. Gibbons, and C. Gundlach. Dark matter, time-varying g, and a dilaton field. Phys. Rev. Lett., 64(2):123–126, 1990.
* [9] J. Fröhlich, O. Grandjean, and A. Recknagel. Supersymmetric quantum theory and differential geometry. Commun. Math. Phys., 193(3):527–594, 1998.
* [10] J. Fröhlich and B. Pedrini. New applications of the chiral anomaly. Mathematical physics 2000, pages 9–47, 2000.
* [11] T. Kaluza. Zum Unitätsproblem der Physik. Sitz. Preuss. Akad. Wiss. Phys. Math. K, 1:966–972, 1921.
* [12] S.F. King. Neutrino mass models. Rep. Prog. Phys., 67:107–157, 2004.
* [13] O. Klein. Quantentheorie und fünfdimensionale Relativitätstheorie. Z. Phys. A-Hadron Nucl., 37(12):895–906, 1926.
* [14] M.L. Michelson and H.B. Lawson. Spin Geometry. Princeton Univ. Press, Princeton, 1990.
* [15] L. O’Raifeartaigh and N. Straumann. Gauge theory: Historical origins and some modern developments. Rev. Mod. Phys., 72(1):1–23, 2000.
* [16] R.G. Swan. Vector bundles and projective modules. T. Am. Math. Soc., 105(2):264–277, 1962.
* [17] S. Weinberg and O.W. Greenberg. The Quantum Theory of Fields, Vol. II: Modern Applications. Cambridge Univ. Press, Cambridge, 1996.
|
arxiv-papers
| 2012-01-12T20:26:10 |
2024-09-04T02:49:26.273879
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ali Chamseddine, Juerg Froehlich, Baptiste Schubnel, Daniel Wyler",
"submitter": "Baptiste Schubnel",
"url": "https://arxiv.org/abs/1201.2661"
}
|
1201.2714
|
# How I Learned to Stop Worrying and Love QFT
LMU, Summer 2011
Robert C. Helling
(helling@atdotde.de)
Notes by Constantin Sluka and Mario Flory
(August 4, 2011)
###### Abstract
Lecture notes of a block course explaining why quantum field theory might be
in a better mathematical state than one gets the impression from the typical
introduction to the topic. It is explained how to make sense of a perturbative
expansion that fails to converge and how to express Feynman loop integrals and
their renormalization using the language of distribtions rather than
divergent, ill-defined integrals.
## 1 Introduction
Physicists are often lax when it comes to mathematical rigor and use objects
that do not exist according to strict mathematical standards or happily
exchange limits without justification. This different culture of “everything
is allowed as long as it is not proven to be wrong and even then it sometimes
ok because we do not actually mean what we are writing” is preferred by many
as it allows to “focus on the content rather than the formal aspects” and to
progress at a much faster pace.
This attitude can be seen when physicists talk about quantum mechanics and
treat operators as if they were matrices and plane waves as if they are
elements of the relevant Hilbert space. This is generally accepted since one
has the feeling that these arguments can easily be repaired at the expense of
clarity by talking about wave packets instead of plane waves and (like it is
discussed at length in our “Mathematical Quantum Mechanics” course) by talking
about quadratic forms instead of the operators directly.
The situation appears to be very different in the case of quantum field
theory: There, most of the time, one deals with perturbative series expansions
in the coupling constant without thinking about convergence (or if one spends
some thought on this one easily sees that the radius of convergence has to be
zero) and the individual terms in the series turn out to be divergent and one
obtains reasonable, finite expressions after some very doubtful formal
manipulations (often presented as subtracting infinity from infinity in the
“right way”). The typical QFT course, unlike quantum mechanics above, does not
indicate any way to “repair” these mathematical shortcomings. Often, one is
left with the impression that there is some blind faith required on the side
of the physicists or at least that some black magic is helping to obtain
numerical values that fit so impressively what is measured in experiments from
very doubtful expressions.
In these notes we will indicate some ways in which these treatments can be
made more exact mathematically thus providing some cure to the mathematical
uneasiness related to quantum field theory. In particular, we will argue that
QFT is not “obviously wrong” as claimed by some mistakenly confusing
mathematical rigor with correctness.
Concretely, we want to explain how two (mostly independent) crucial steps in
QFT can be understood more mathematically:
In a simplified example, we will explore what conclusions can be drawn from
the perturbative expansion even though the series does not converge for any
finite value of the coupling constant. In particular we will discuss the role
of non-perturbative contributions like instantons in the full interacting
theory. We will find that up to a certain level of accuracy (depending on the
strength of coupling), the first terms of the perturbative expansion do
represent the full answer even though summing up all terms leads to infinite,
meaningless expressions. Furthermore, at least in principle, using the
technique of “Borel resummation” one can express the true expression for all
values of the coupling constant in terms of just the perturbative expansion.
As a second step, at each order in perturbation theory, we will see how by
correctly using the language of distributions one can set up the calculation
of Feynman diagrams without diverging momentum integrals. We will find that
these divergences can be understood to arise from trying to multiply
distributions. We will set this up as the problem to extend distributions from
a subset of all test functions at the expense of a finite number of
undetermined quantities that we will identify as the “renormalized coupling
constants”. Finally we will understand how these vary when we change
regulating functions that were introduced in the procedure which leads to an
understanding of the renormalization group in this formalism of “causal
perturbation theory”.
The aim is to argue how the techniques of physicists could be embedded in a
more mathematical language without actually doing this. At many places we just
claim results without proof or argue by analogy (for example we will discuss a
one dimensional integral instead of an infinite dimensional path-integral). To
really discuss the topic at a mathematical level of rigor requires a lot more
work and to large extend still needs to be done for theories of relevance to
particle physics.
All this material is not new but well known to experts in the field. Still, we
hope that these notes will be a useful complement to standard introductions to
quantum field theory for (beginning) practitioners.
## 2 Perturbative expansion — making sense of divergent series
Before we take a look at divergent series, we will first give a brief review
of how perturbative expansion is used in quantum field theory.
### 2.1 Brief overview on path integrals
A quantum field theory in Minkowski spacetime is described by a Lagrangian
density $\mathcal{L}(\phi,\partial\phi)$ and a generating functional of
correlation functions111This subsection displays some standard expressions to
set the context. For many more details see for example [1].
$\mathcal{Z}[J]=\int\mathcal{D}\phi
e^{i\int\mathrm{d}^{4}x(\mathcal{L}+J\phi)}.$ (1)
The correlation functions can be obtained by functional derivatives of (1)
with respect to $J$.
$\langle\phi(x_{1})\phi(x_{2})\dots\phi(x_{n})\rangle=\frac{1}{\mathcal{Z}[0]}\left(-i\frac{\delta}{\delta
J(x_{1})}\right)\left(-i\frac{\delta}{\delta
J(x_{2})}\right)\dots\left(-i\frac{\delta}{\delta
J(x_{n})}\right)\mathcal{Z}[J]\Bigg{|}_{J=0}$ (2)
In this lecture we will use Euclidean signature for the metric instead of
Minkowski. The change between the metrics can be performed as rotation of the
time axis in the complex plane $t\rightarrow-i\tau$ if all expressions are
analytic. In Euclidean metric, the exponent in the generating functional is
real and falls of at large field values. This gives the path integral a chance
to have a mathematical definition in terms of Wiener measures but that will
not concern us in these notes.
$\mathcal{Z}[J]=\int\mathcal{D}\phi
e^{\int\mathrm{d}^{4}x(\mathcal{L}+J\phi)}$ (3)
In general, the integral (3) cannot be computed exactly. For a scalar quantum
field theory in Euclidean space the Lagrangian has the form
$\mathcal{L}=\frac{1}{2}\phi(\Box-m^{2})\phi-V(\phi)$ (4)
with $\Box\equiv(\partial_{\tau})^{2}+(\nabla)^{2}$.
If the potential $V(\phi)$ vanishes, equation (3) can be formally computed as
it becomes an integral of Gaussian type. One therefore arbitrarily splits the
Lagrangian into its “kinetic part” $\frac{1}{2}\phi(\Box-m^{2})\phi$ and its
“interaction part” $-V(\phi)$.
$\displaystyle\mathcal{Z}[J]$ $\displaystyle=\int\mathcal{D}\phi
e^{\frac{1}{2}\int\mathrm{d}^{4}x\phi(\Box-m^{2})\phi}e^{-\int\mathrm{d}^{4}xV(\phi)}e^{-\int\mathrm{d}^{4}xJ\phi}$
$\displaystyle=e^{-\int\mathrm{d}^{4}xV(\frac{\delta}{\delta
J})}\int\mathcal{D}\phi
e^{\int\mathrm{d}^{4}x\frac{1}{2}\phi(\Box-m^{2})\phi-J\phi}$ (5)
To obtain the Gaussian integral one has to complete the square in the
exponent. This is achieved by shifting the field $\phi$:
$\phi^{\prime}=\phi+(\Box-m^{2})^{-1}J$ (6)
The inverse of $\Box-m^{2}$, called “Green’s function” $G(x-y)$, is a
distribution defined by
$(\Box-m^{2})G(x-y)=\delta(x-y).$ (7)
Changing variables in the functional integral (5) leads to
$\mathcal{Z}[J]=e^{-\int\mathrm{d}^{4}xV(\frac{\delta}{\delta
J})}\int\mathcal{D}\phi^{\prime}e^{\int\mathrm{d}^{4}x\frac{1}{2}\phi^{\prime}(\Box-m^{2})\phi^{\prime}}e^{-\int\mathrm{d}^{4}x\int\mathrm{d}^{4}y\frac{1}{2}J(x)G(x-y)J(y)}.$
(8)
The complicated expression in the middle of equation (8) does not depend on
$J$ and will in fact cancel out in equation (2) for the correlation function,
so we will just denote it $C$ and forget about it:
$\mathcal{Z}[J]=Ce^{-\int\mathrm{d}^{4}xV(\frac{\delta}{\delta
J})}e^{-\int\mathrm{d}^{4}x\int\mathrm{d}^{4}y\frac{1}{2}J(x)G(x-y)J(y)}$
Now let us take a look on a specific example for a quantum field theory by
choosing a potential for the scalar field. We will consider our favorite
$\phi^{4}$ theory given by the potential
$V(\phi)=\lambda\phi^{4}.$ (9)
The next step is to insert this potential in equation (8) and write the
exponential as a power series in the coupling strength $\lambda$.
$\mathcal{Z}[J]=C\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{k!}\int\mathrm{d}^{4}x_{1}\frac{\delta^{4}}{\delta
J(x_{1})^{4}}\dots\int\mathrm{d}^{4}x_{k}\frac{\delta^{4}}{\delta
J(x_{k})^{4}}e^{-\int\mathrm{d}^{4}x\int\mathrm{d}^{4}y\frac{1}{2}J(x)G(x-y)J(y)}$
(10)
We now found an expression for any general correlation function in terms of an
power series expansion in the coupling strength.
$\displaystyle\langle\phi(y_{1})\dots\phi(y_{n})\rangle=$
$\displaystyle\frac{\delta}{\delta J(y_{1})}\cdots\frac{\delta}{\delta
J(y_{1})}\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{k!}\int\mathrm{d}^{4}x_{1}\frac{\delta^{4}}{\delta
J(x_{1})^{4}}\cdots\int\mathrm{d}^{4}x_{k}\frac{\delta^{4}}{\delta
J(x_{k})^{4}}\times$ $\displaystyle
e^{-\int\mathrm{d}^{4}x\int\mathrm{d}^{4}y\frac{1}{2}J(x)G(x-y)J(y)}\Bigg{|}_{J=0}$
(11)
The combinatorics of the occurring expressions in terms of integrals over
interaction points $x_{i}$, Green’s functions and external fields can be
summarized in terms of Feynman diagrams each standing for a single term in the
power series in the coupling constant $\lambda$222The careful reader wishing
to avoid ill-defined expressions using path-integrals, can use this formula as
the definition of the terms in the perturbative series.. In the following, we
want to study the convergence behavior of this power series.
### 2.2 Radius of convergence of correlation functions
Let us briefly review the definition of the radius of convergence for a power
series from introductory analyis. It is useful to think of a power series to
be defined in the complex plane:
$\sum_{k}^{\infty}\lambda^{k}(\dots)\quad\lambda\in{\mathbbm{C}}$ (12)
(If one does not like the idea of a complex coupling strength in a quantum
field theory, just restrict to the special $\lambda\in{\mathbbm{C}}$ that
happen to be real.). Every power series has a radius of convergence
$R\in[0,\infty]$ such that
$\sum_{k}^{\infty}\lambda^{k}(\dots)\begin{cases}\text{converges}&\forall\left|\lambda\right|<R\\\
\text{diverges}&\forall\left|\lambda\right|>R.\end{cases}$ (13)
Now we want to find out the radius of convergence for the correlation
functions (11) in a quantum field theory. A physicist’s argument was given by
Freeman Dyson in 1952[2]. Let us take a look on the potential, for example in
our $\phi^{4}$ theory as shown in figure 1. For positive coupling strength
$\lambda$ the potential is bounded from below and large values of $\phi$ are
strongly disfavored. This behavior, however, gets radically different in case
of a negative $\lambda$. The potential becomes unbounded from below and the
field $\phi$ will want to run off to $\phi=\pm\infty$. Obviously, such a
behavior is highly unphysical, since ever increasing values of $\phi$ would
lead to an infinite energy gain. It is thus clear that such a theory cannot
lead to healthy correlation functions, in other words for any negative
$\lambda$ the power series (12) will diverge333We expect at least a phase
transition when $\lambda$ is changed from positive to negative values.. From
this we can conclude the radius of convergence being $R=0$!
$\sum_{k}^{\infty}\lambda^{k}(\dots)\qquad\text{diverges }\forall\lambda>0$
(14)
$V(\phi)$$\phi$ (a) $\lambda>0$
$V(\phi)$$\phi$ (b) $\lambda<0$
Figure 1: Potentials of $\phi^{4}$ theory
For readers not satisfied by this argument using physics of unstable
potentials for determining the radius of convergence, let us mention an
alternative line of argument. Again, consider equation (11), this time,
however, we will focus on the Feynman diagrams. At any order $k$ in the
perturbation expansion there is a sum of different Feynman diagrams expressing
the integrals in (11), where $k$ counts the number of vertices. The
combinatorics of all Feynman diagrams shows that the number of Feynman
diagrams grows like $k!$. The power series, therefore, will behave like
$\sum_{k}^{\infty}\lambda^{k}k!(\ldots).$ (15)
Assuming that $(\ldots)$ is not surprisingly suppressed for large $k$, the
coefficients of $\lambda^{k}$ grow faster than any power, we again find the
radius of convergence $R=0$.
In the following, we want to give an example, how one can nevertheless make
sense of (some) divergent power series.
### 2.3 Non-perturbative corrections
In order to get a feeling for the problem of divergent power series, we will
consider a one dimensional toy problem (rather than the infinite dimensional
problem of a path integral):
$\mathcal{Z}(\lambda)=\int_{-\infty}^{\infty}\mathrm{d}x\,e^{-x^{2}-\lambda
x^{4}}$ (16)
We take $\lambda\geq 0$, so this integral yields some finite, positive number.
For $\lambda=0$ the solution is well known
$\mathcal{Z}(0)=\sqrt{\pi}.$ (17)
In general, equation (16) can be expressed in terms of special functions, e.g.
Mathematica gives the solution
$\mathcal{Z}(\lambda)=\frac{e^{\frac{1}{8\lambda}}K_{1/4}({1/8\lambda})}{2\sqrt{\lambda}}$
(18)
with $K_{n}(x)$ being the modified Bessel function of the second kind. We call
solution (18) the “full, non-perturbative answer”. Now we will do the same as
in quantum field theory and split the integral into a “kinetic” and an
“interaction” part, respectively.
#### 2.3.1 Treating the toy model perturbatively
Following the same procedure, we will again expand the “interaction part”
$-\lambda x^{4}$ in a power series:
$\mathcal{Z}(\lambda)=\int_{-\infty}^{\infty}\mathrm{d}xe^{-x^{2}-\lambda
x^{4}}=\int_{-\infty}^{\infty}\mathrm{d}xe^{-x^{2}}\sum_{k=0}^{\infty}\frac{(-\lambda
x^{4})^{k}}{k!}$ (19)
Now comes the crucial step and “root of all evil”. Following precisely the
same steps leading towards equation (11) for correlation functions in quantum
field theory, we will change the order of integration and summation, leading
to the interpretation of a power series of Feynman diagrams:
$\mathcal{Z}(\lambda)\text{``}=\text{''}\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{k!}\int_{-\infty}^{\infty}\mathrm{d}x\
x^{4k}e^{-x^{2}}$ (20)
From this step, as we will see later, the problems arise. Although this step
is forbidden (as roughly speaking, we are changing the behavior of the
integrand at $x=\pm\infty$), we are interested in to what extent a
“perturbative solution” obtained from equation (20) will agree with the full,
non-perturbative solution (18). Carrying on, we observe that the integral in
(20) is now of the type “polynomial times Gaussian” and can be computed with
standard methods. We smuggle an addtional factor $a$ into the exponent
allowing us to write the integrand as derivatives of $e^{-ax^{2}}$ with
respect to $a$ at the point $a=1$.
$\mathcal{Z}(\lambda)=\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{k!}\int_{-\infty}^{\infty}\mathrm{d}x\frac{\partial^{2k}}{\partial
a^{2k}}e^{-ax^{2}}\Big{|}_{a=1}=\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{k!}\frac{\partial^{2k}}{\partial
a^{2k}}\sqrt{\frac{\pi}{a}}\Big{|}_{a=1}$ (21)
Of course we can easily evaluate the derivatives:
$\frac{\partial^{2k}}{\partial
a^{2k}}a^{-\frac{1}{2}}\Big{|}_{a=1}=\underbrace{\frac{1}{2}\frac{3}{2}\cdot\frac{5}{2}\frac{7}{2}\cdot\frac{9}{2}\frac{11}{2}\cdot\,\cdots}_{\text{total
of $2k$ factors}}$ (22)
In order to find an explicit expression for (22) one can insert factors of $1$
between all factors, such that the nominator becomes $(4k)!$:
$\displaystyle\frac{\partial^{2k}}{\partial
a^{2k}}a^{-\frac{1}{2}}\Big{|}_{a=1}$
$\displaystyle=\underbrace{\frac{1}{2}\frac{2}{2}\frac{3}{2}\frac{4}{4}\frac{5}{2}\frac{6}{6}\frac{7}{2}\frac{8}{8}\frac{9}{2}\frac{10}{10}\frac{11}{2}\frac{12}{12}\dots}_{\text{total
of $4k$
factors}}=\frac{(4k)!}{2^{2k}}\underbrace{\frac{1}{2}\frac{1}{4}\frac{1}{6}\frac{1}{8}\frac{1}{10}\frac{1}{12}\dots}_{\text{total
of $2k$ factors}}$
$\displaystyle=\frac{(4k)!}{2^{2k}}\frac{1}{2^{2k}(2k)!}=\frac{(4k)!}{2^{4k}(2k)!}$
(23)
Thus we obtain the “perturbative solution” of problem (16)
$\mathcal{Z}(\lambda)=\sum_{k=0}^{\infty}\sqrt{\pi}\frac{(-\lambda)^{k}(4k)!}{2^{4k}(2k)!k!}.$
(24)
Let us take a closer look at this expression. By observing that the
denominator of the summand eventually contains smaller factors than the
nominator for all $k$ larger than a critical integer, we can realize that the
series is divergent. More carefully we can apply Stirling’s formula
$n!\approx\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}$ for large values of $k$:
$\frac{(4k)!}{2^{4k}(2k)!k!}\approx\frac{4^{k}}{\sqrt{\pi
k}}\left(\frac{k}{e}\right)^{k}\approx\frac{1}{\sqrt{2}\pi}4^{k}k!$ (25)
We already know that the sum
$\sum_{k=0}^{\infty}(-4\lambda)^{k}k!$ (26)
will diverge. This shows that the power series (24) is divergent and in
particular it is not the finite number that we are looking for as an
expression for (16).
#### 2.3.2 The perturbative and the full solution compared
Even though the perturbative series will diverge, we want to study its
numerical usefulness at finite order. After all, one usually computes only a
finite number of Feynman diagrams to obtain only the first few summands of the
perturbative expansion. Is there a way to approximate the full, non-
perturbative solution (18) from (24)? Let us choose one value for $\lambda$,
e.g. $\frac{1}{50}$, and evaluate (18) numerically:
$\mathcal{Z}\left(\frac{1}{50}\right)=1.7478812\dots$ (27)
For the same value of $\lambda$ the evaluation of the first few terms of the
infinite sum (24)
$\mathcal{Z}_{N}(\lambda)=\sum_{k=0}^{N}\sqrt{\pi}\frac{(-\lambda)^{k}(4k)!}{2^{4k}(2k)!k!}.$
(28)
gives
$\displaystyle\mathcal{Z}_{5}$ $\displaystyle=1.7478728\dots$ (29a)
$\displaystyle\mathcal{Z}_{10}$ $\displaystyle=1.7478818\dots$ (29b)
The first terms of the perturbative solution agree up to six digits! We can
use conveniently a figure for plotting higher orders of the perturbative
series. Figure 2 shows that the perturbative solution gets in a certain regime
very close to the result of the full solution, before the series starts to
diverge.
Figure 2: Values of the perturbative series (24) evaluated to order $N$ Figure
3: $\mathcal{Z}(\lambda)$ obtained from the full solution (thick) and first
approximations from the perturbative series
We can use a figure as well to compare the solutions for variable $\lambda$.
Figure 3 shows nicely the non-perturbative solution and compares it to the
perturbative solution for orders of one to twelve. We can see that at some
point all approximations given by the perturbative solution will disagree
strongly from the full solution!
The question that arises is, how long does the perturbative solution become
better before it starts to diverge? Obviously, the fact that it approximates
the non-perturbative solution to high precision leads to the great success of
quantum field theory, even if for higher orders the series diverges! As we
will see now, the perturbative solution (24) is a good approximation as long
as we only consider terms up to order $N=O(\frac{1}{\lambda})$. Remembering
the dimensionless coupling strength of Quantum Electrodynamics being the
Sommerfeld finestructure constant $\alpha\approx\frac{1}{137}$ we can be
ensured that perturbation theory will lead to great precision given that the
most elaborate QED calculations for $(g-2)$ are to order $N=7$!
#### 2.3.3 The method of steepest descend
But what is the origin of the eventual divergence and complete loss of
numerical accuracy? It turns out that there are “non-perturbative” terms that
do not show up in a Taylor expansion but that become dominant when the
perturbative expansion breaks down. To see this, let us substitute
$x^{2}\equiv\frac{u^{2}}{\lambda}$ in equation (16):
$\mathcal{Z}(\lambda)=\frac{1}{\sqrt{\lambda}}\int\mathrm{d}ue^{-\frac{u^{2}+u^{4}}{\lambda}}$
(30)
The exponent is strictly negative and its absolute value becomes very large in
the limit of small $\lambda$. This allows to perform the method of steepest
descent: The main contribution to the integral, as $\lambda\to 0$ comes from
the extrema of the integrand
$u^{2}+u^{4}.$ (31)
In general the method works as follow: For $\Lambda\to\infty$ we want to solve
an integral of the general form
$\int\mathrm{d}xA(x)e^{(i)\phi(x)\Lambda}.$ (32)
One expands now around its extrema444Notice that in field theory
$\phi^{\prime}(x)=0$ is the equation of motion $\phi^{\prime}(x_{0})=0$ and
obtains again an integral of “Gaussian times polynomial” type555Corrections
from $(x-x_{0})A^{\prime}(x_{0})$ can be obtained by doing again the trick of
smuggling an $a$ into the exponent and write the term as derivative with
respect of $a$ evaluated at $a=1$.
$\displaystyle=\sum_{x_{0}:\phi^{\prime}(x_{0})=0}\int\mathrm{d}x(A(x_{0})+(x-x_{0})A^{\prime}(x_{0})\dots)e^{\Lambda(\phi(x_{0})+(x-x_{0})^{2}\phi^{\prime\prime}(x_{0})+\dots)}$
$\displaystyle=\sum_{x_{0}:\phi^{\prime}(x_{0})=0}A(x_{0})e^{\Lambda\phi(x_{0})}\sqrt{\frac{2\pi}{\phi^{\prime\prime}(x_{0})\Lambda}}\left(1+O\left(\frac{1}{\Lambda}\right)\right).$
(33)
In our case, the extrema of (31) are $u=0$ and $u=\pm i/\sqrt{2}$. Expansion
around the first yields the perturbative expansion of above. The other two
yield contributions like $e^{\frac{1}{4\lambda}}$ that are invisible to a
Taylor expansion around $\lambda=0$, as all derivative vanish here. We have
found an example of a “non-perturbative contribution”.
The perturbative solution, however, gives meaningful results, as long as its
terms are bigger than to the non-perturbative contributions. This allows an
estimate, to what order in the perturbative series the expansion around
$\lambda=0$ dominates. This happens also to be the order at which the
divergence from the exat solution starts as we are missing the non-
perturbative terms:
$\displaystyle e^{-\frac{1}{4\lambda}}$ $\displaystyle\approx\lambda^{k}$
$\displaystyle-\frac{1}{4\lambda}$ $\displaystyle\approx k\cdot\ln(\lambda)$
$\displaystyle k$ $\displaystyle\approx\frac{1}{\lambda}$ (34)
We have seen that the perturbative analysis of (30) requires as well
expansions around the other extrema, besides $\lambda=0$! Combining all power
series together, the resulting perturbative solution has a chance to converge.
Before we continue to the mathematical discussion of the problem how finite
results can be obtained from divergent series, we will take a look on examples
of non-perturbative contributions in physics.
#### 2.3.4 Instantons
“Field configurations” contributing $e^{-\frac{1}{\sim\lambda}}$ are called
“instantons”. Usually these contributions are hard to calculate, in some
situations, however, one can find the result. Consider for example a gauge
theory666Hodge $\ast$ operator: $\ast
F^{\mu\nu}\equiv\epsilon^{\mu\nu\sigma\tau}F_{\sigma\tau}$. More details can
be found in chapters 1.10 and 10.5 of [3].
$S=\int\mathcal{L}=\int\frac{1}{g^{2}}\operatorname{tr}(F\wedge\ast F)$ (35)
The stationary point we use for the expansion is given by the equations of
motion
$\displaystyle\mathrm{d}F$ $\displaystyle=0$ (36a)
$\displaystyle\mathrm{d}\ast F$ $\displaystyle=0$ (36b)
The first equation (36a) is automatically fulfilled once we express the field-
strength in terms of a vector potential $F=\mathrm{d}A$. The second (36b) is
automatically solved if it happens that
$F=\ast F.$ (37)
One calls solutions to (37) instantons. In terms of the vector potential $A$
(37) is a first order partial differential equation as compared to (36b) which
is second order. One can easily see that there exist no solution in Lorentzian
metric as the Hodge star squares to $-1$ on 2-forms.
$F=\ast F=\ast\ast F=-F.$ (38)
In Euclidean metric, however, such solutions exist because of $\ast\ast F=F$.
As it turns out (as one can for example argue using the Atiya-Singer index
theorem), for a compact manifold $M$, the action in the instanton case yields
an integer (up to a pre-factor):
$\int_{M}\operatorname{tr}(F\wedge F)\in 8\pi^{2}{\mathbbm{Z}}$ (39)
This leads to
$e^{\frac{1}{g^{2}}\int\operatorname{tr}(F\wedge\ast
F)}=e^{\frac{1}{g^{2}}\int\operatorname{tr}(F\wedge
F)}=e^{-\frac{1}{g^{2}}8\pi^{2}N}$ (40)
#### 2.3.5 Dual theories
Sometimes a quantum field theory with coupling constant $\lambda$ can be
rewritten in terms of another (or possibly the same) quantum field theory with
coupling $\tilde{\lambda}=1/{\lambda}$. One calls such a relation between two
theories a “duality”. In many examples, such theories arise from string theory
constructions, where the coupling $\lambda$ can be given a geometric meaning.
Imagine for example a problem of a quantum field theory on a torus. A torus
can be viewed as ${\mathbbm{C}}/({\mathbbm{Z}}+\tau{\mathbbm{Z}})$ with
$\tau\in{\mathbbm{C}}\backslash{\mathbbm{R}}$. The torus has a basis of two
non-contractible circles, one that goes along the real axis from 0 to 1 and
one that goes from 0 to $\tau$. This choice of basis, however, is not unique:
For example, swapping these cycles corresponds to a substitution
$\tau\rightarrow-1/{\tau}$. If the torus parameter $\tau$ is identified with
the coupling strength a duality has been found since both $\tau$ and $-1/\tau$
describe geometricaly the same torus! To make contact with our discussion
above, we should identify $\lambda$ with the imaginary part of $\tau$. The
duality allows for a Taylor expansion of the non-perturbative contributions
via
$e^{-\frac{1}{4\lambda}}=e^{-\frac{\tilde{\lambda}}{4}}=\sum_{k}^{\infty}\frac{\left(-\tilde{\lambda}\right)^{k}}{k!4^{k}}$
(41)
Troublesome terms in one theory are therefore perfectly defined in the dual
theory. The caveat however is the difficulty of actually proving that
$\lambda\rightarrow 1/{\lambda}$ is a symmetry of the quantum field theory at
hand.
### 2.4 Asymptotic series and Borel summation
In the following, we take a look on the mathematical situation of asymptotic
series. This discussion is based on chapter $XII$ of [4].
###### Definition 1
Let $f\colon{\mathbbm{R}}_{\geq 0}\to{\mathbbm{C}}$. The series
$\sum_{n}^{\infty}a_{n}z^{n}$ is called asymptotic to $f$ as $z\searrow 0$ iff
$\forall N\in{\mathbbm{N}}:\lim_{z\searrow
0}\frac{f(z)-\sum_{n}^{N}a_{n}z^{n}}{z^{n}}=0$ (42)
For $z\in{\mathbbm{C}}$ a analog definition is possible.
Obviously, every function can have at most one asymptotic expansion. This can
be seen by assuming two asymptotic expansions $a_{n}$ and $\tilde{a}_{n}$.
(42) requires that $a_{n}=\tilde{a}_{n}$. Otherwise, let $n$ be the smallest
index for which $a_{n}\neq\tilde{a}_{n}$ and
$\lim_{z\searrow
0}\frac{\sum_{k}(a_{k}-\tilde{a}_{k})z^{k}}{z^{n}}=a_{n}-\tilde{a}_{n}\stackrel{{\scriptstyle!}}{{=}}0.$
(43)
The other way around is not true, as can be seen by $f(z)=e^{-\frac{1}{z}}$
and $\tilde{f}(z)=0$ having both the asymptotic series
$\sum_{k}^{\infty}0\cdot z^{k}$. This means that knowing the asymptotic series
of a function tells us nothing about $f(z)$ for a non vanishing $z$, we only
know how $f(z)$ approaches $f(0)$ as $z\searrow 0$.
We try to find a stronger definition of an asymptotic series, allowing us to
uniquely recover one function. The following theorem helps us to find the
necessary conditon:
###### Theorem 2
(Carleman’s theorem) Let $g$ be an analytic function in the interior of
$S=\\{z\in{\mathbbm{C}}|\left|z\right|\leq B,\left|\mathrm{arg}\
z\right|\leq\frac{\pi}{2}\\}$ and continous on $S$. If for all
$n\in{\mathbbm{N}}$ and $z\in S$ we have
$\left|g(z)\right|~{}\leq~{}b_{n}\left|z\right|^{n}$ and
$\sum_{n}^{\infty}b_{n}^{-\frac{1}{n}}=\infty$, then $g$ is identically zero.
A simpler special case of the theorem is found by considering $g$ an analytic
function in the interior of
$S_{\epsilon}=\\{z\in{\mathbbm{C}}|\left|z\right|\leq R,\left|\mathrm{arg}\
z\right|\leq\frac{\pi}{2}+\epsilon\\}$ for some $\epsilon>0$ and continous on
$S_{\epsilon}$. If there exist $C$ and $B$ so that
$\left|g(z)\right|<CB^{n}n!\left|z\right|^{n}\ \forall z\in S\ and\ \forall
n$, then $g$ is identically zero.
In order to find a unique function for an asymptotic series, we use Carleman’s
theorem to define “strong asymptotic series”.
###### Definition 3
Let $f$ be an analytic function on the interior of
$S_{\epsilon}=\\{z\in{\mathbbm{C}}|\left|z\right|\leq R,\left|\mathrm{arg}\
z\right|\leq\frac{\pi}{2}+\epsilon\\}\rightarrow{\mathbbm{R}}$. The series
$\sum_{n}^{\infty}a_{n}z^{n}$ is a strong asymptotic series if there exist
$C,\sigma$ so that $\forall N\in{\mathbbm{N}},z\in S_{\epsilon}$ the strong
asymptotic condition
$\left|f(z)-\sum_{n}^{N}a_{n}z^{n}\right|\leq
C\sigma^{N+1}(N+1)!\left|z\right|^{N+1}$ (44)
is fulfilled.
This means, if we are given a strong asymptotic series, we can recover by
theorem 2 the function! Assume for example $\sum_{n}^{\infty}a_{n}z^{n}$ is a
strong asymptotic series for two functions $f$ and $g$, respectively. Then
$\left|f(z)-g(z)\right|\leq
2C\sigma^{N+1}(N+1)!\left|z\right|^{N+1}\Rightarrow f=g$ (45)
The strong asymptotic condition (44) implies $\left|a_{n}\right|\leq
C\sigma^{n}n!$. This is precisely the growth behavior of (24) we found in our
toy example, where $C=\frac{1}{\sqrt{2\pi}}$ and $\sigma=4$. The necessary
conditions, therefore, are fulfilled in our toy model (assuming analyticity
away from 0 of course).
By now, we learned that a strong asymptotic series (in particular the type we
obtain in quantum field theory) although not converging has the chance to be a
unique approximation to one function. The final question is, how one can
obtain this function $f$ from its strong asymptotic series. In the last
theorem we introduce the method of “Borel summation” to obtain a final result.
We can define a convergent series by taking out a factor of $n!$ from the
coefficients:
###### Theorem 4
(Watson’s theorem) If $f:S_{\epsilon}\to{\mathbbm{R}}$ has a strong asymptotic
series $\sum_{n}^{\infty}a_{n}z^{n}$, we define the Borel transform
$g(z)=\sum_{n}^{\infty}\frac{a_{n}}{n!}z^{n}.$ (46)
The Borel transform converges for
$\left|z\right|<\frac{1}{\left|\sigma\right|}$. We obtained a convergent power
series with finite radius of convergence, which, as it turns out, can be
analytically continued to all complex $z\in{\mathbbm{C}}$ with
$\left|\mathrm{arg}\ z\right|<\epsilon$. Then the function $f$ is given by the
Laplace transform
$f(z)=\int_{0}^{\infty}\mathrm{d}b\ g(bz)e^{-b}.$ (47)
This Laplace transform is called “inverse Borel transform” and the method
outlined here is known as “Borel summability method”. It describes how to
obtain a finite answer from divergent series, that is formally a sum for the
series.
Let us make a sanity check. Using $\int_{0}^{\infty}\mathrm{d}x\
x^{k}e^{-x}=k!$ we can plug the definition of the Borel transform into (47),
formally interchange the sum and the integration and obtain
$f(z)=\int_{0}^{\infty}\mathrm{d}b\
g(bz)e^{-b}=\int_{0}^{\infty}\mathrm{d}b\sum_{n}\frac{a_{n}}{n!}b^{n}z^{n}e^{-b}\
\hbox{``=''}\sum_{n}a_{n}z^{n}.$ (48)
So at least for analytic functions we do recover the original function.
### 2.5 Summary
We have learned why for $N<\mathcal{O}(\frac{1}{\lambda})$ the sum of the
first $N$ terms of the perturbation expansion is numerically good, even when
the original series $\sum_{n}a_{n}z^{n}=\infty$ diverges. This way we
approximate the true function up to instantonic terms of the order of
$e^{1/\lambda}$ which a Taylor expansion cannot resolve. Given that the
coefficients $a_{n}$ obey the strong asymptotic condition
$\left|a_{n}\right|\leq C\sigma^{n}n!$, which is usually the case when using
Feynman diagrams, the Borel transform exists and one can compute the Borel
summation. Unfortunately this is more a theoretical assurance that
perturbation theory can be given a mathematical meaning even though it does
not converge, since in order to really compute the integral (47) one has to
know the analytic continuation of $g$ which requires knowledge of all
coefficients $a_{n}$ and not just the first $N$.
## 3 Regularization and renormalization as extensions of distributions
In the previous section, we learned how to make sense of (some) divergent
series of the form $\sum^{\infty}_{k=0}a_{k}=\infty$, but in QFT the factors
$a_{k}$ are typically complicated mathematical expressions described by
Feynman diagramms, and generically, these expressions diverge themselves,
creating a need for renormalization techniques.
A typical example of a divergent diagramm (in 4 dimensions) is shown in Figure
4.
Figure 4: Divergent 1-loop diagramm
The term described by this diagramm reads (without unimportant factors):
$\displaystyle\int{d^{4}k\frac{1}{k^{2}+m^{2}}\frac{1}{(p_{1}+p_{2}+k)^{2}+m^{2}}}\stackrel{{\scriptstyle
k\gg p,m}}{{\longrightarrow}}\int\frac{k^{3}dk}{k^{4}}=\infty$
where $m$ denotes the mass of the scalar particles we are scattering. This
integral obviously diverges logarithmically for $k\rightarrow\infty$ as shown
above. The most straightforward approach to this problem is to introduce a
cut-off energy-scale $\Lambda$, such that the divergence at the upper boundary
becomes
$\displaystyle\int^{\Lambda}{\frac{k^{3}dk}{k^{4}}}\sim\log(\Lambda),$
Usually, such blunt cut-off regularization is incompatible with the symmetries
of the theory at hand and is thus only useful to estimate “how divergent” a
diagram is (a notion we will below formalize as the “singular degree”) and has
to be replaced by more sophisticated methods like dimensional or Pauli-Villars
regularization in more practical applications.
In these notes, instead of momentum representation, we will work in position
space where instead of loop momenta one integrates over the position of the
interaction vertices.
What was $1/(k^{2}+m^{2})$, is now the propagator $G$ defined by the equation
$\displaystyle(\Box+m^{2})G(x)=\delta(x),$ (49)
Figure 5: Divergent 1-loop diagramm in position-space language
we can compute the same diagramm in position space language, which then reads
(see Figure 5):
$\displaystyle\int d^{4}x\int
d^{4}y\;\phi^{2}_{0}(x)G^{2}(x-y)\phi^{2}_{0}(y)=\int d^{4}x\int
d^{4}u\;\phi^{2}_{0}(x)G^{2}(u)\phi^{2}_{0}(x-u)$ (50)
The approach of “causal perturbation theory” or “Epstein-Glaser
regularization” is to take seriously the fact that the propagator is really a
distribution and in the above expression, we are trying to multiply
distributions which in general is undefined. This approach is advocated in the
book by Scharf [5]. Here, we will follow (a simplified, flat space version) of
[6] and in particular [7].
Specifically, in the defining equation (49) $\delta$ is not a function but a
distribution (physicists writing $\delta(x)$ are trying to imply that this is
the kernel of the distribution $\delta$, i.e. that $\delta$ arises by
muliplying the testfunction by a function $\delta(x)$ and then integrating
over $x$, which of course does not exist). Thus, we should interpret $G(x)$ as
a distribution as well (a priori it is only a weak solution of the
differential equation (49)). But as it is in general not possible to multiply
distributions, as we will see later, we do not have a naive way to obtain
“$G^{2}$” as a distribution. In this chapter, our goal will be to understand
renormalization techniques in terms of distributions. Our route will be led by
the question how to define the product of two distributions that are almost
everywhere functions (which can be multiplied). We will first, therefore,
recapitulate what distributions actually are. Then, we will see in which cases
it is possible to multiply distributions and in which it is not. This will
lead us to the renormalization techniques we are searching for.
### 3.1 Recapitulation of distributions
Distributions are generalized functions. Like in many other cases of
generalizations this is done via dualization: Starting from an ordinatry
function $f$ (in our case locally integrable, that is
$f\colon{\mathbbm{R}}^{n}\to{\mathbbm{C}}$ with $\int_{K}|f|<\infty$ for each
compact $K\subset{\mathbbm{R}}^{n}$, so that “divergence of the integral
$\int|f|$ at infinity” is tolerated) one can view it as a linear functional
$T_{f}$ (called a “regular distribution”) on the functions of compact support
via
$T_{f}\colon\phi\mapsto\int f\phi.$ (51)
As the map $f\mapsto T_{f}$ is injective we can use the $T_{f}$’s to
distinguish the different $f$’s and view $T_{f}$ in place of $f$. This
suggests to generalize the construction to all linear functionals
$T:\phi\mapsto T(\phi)$ called distributions of which the regular ones arising
from functions $f$ as above are a subset.
Specifically, distributions are defined to be linear and continuous
functionals on the space of test functions
$D(\mathbb{R}^{n})=\mathcal{C}^{\infty}_{0}(\mathbb{R}^{n})$ (the subscript
$0$ meaning compact support) equipped with an appropriate topology that will
not concern us here. So formally, we can denote the distributions to be
elements of a space:
$\displaystyle
D^{\prime}(\mathbb{R}^{n})=\left\\{T:D(\mathbb{R}^{n})\rightarrow\mathbb{C}\mid\text{$T$
is linear and continuous}\right\\}$
Besides the regular distibutions $T_{f}$ encountered above (of which the
function $f$ is called the kernel) the typical example of a singular
distribution is the $\delta$-distribution: if we take a given test function
$\phi(x)\in D$, then the $\delta$-distribution is defined to be the functional
$\delta[\phi]=\phi(0)$. This distribution is not regular even though
physicists pretend it to be with a kernel $\delta(x)$ that is so singular at
$x=0$ that $\int\delta(x)=1$ even though it vanishes for all $x\neq 0$.
Later, we will make use of the fact that distributions can be differentiated.
Using integration by parts in the integral representation of a regular
distribution, we easily obtain $T_{f^{\prime}}[\phi]=-T_{f}[\phi^{\prime}]$
which enables us to define the derivative of a distribution to be
$T^{\prime}[\phi]\equiv-T[\phi^{\prime}]$. Thus we can take the derivative of
a regular distribution $T_{f}$ even if the kernel $f$ is not differentiable.
The only operation defined on functions that does not directly carry over to
distributions is (pointwise) multiplication $(f\cdot g)(x)=f(x)g(x)$. Already
$L^{1}_{loc}$ is not closed under multiplication (recall that in order for a
function to be in $L^{1}_{loc}$ it must not have singularities that go like
$1/x^{\alpha}$ with $\alpha\geq 1$, a property not stable under
multiplication) and in general the product of distributions is not defined. Of
course, as long as with $f$ and $g$ also $f\cdot g\in L^{1}_{loc}$ we still
have the regular distribution $T_{f\cdot g}$ and, from a technical
perspective, in this sections, we will deal with the problem to extend a
distribution that can be written as $T_{f\cdot g}$ for a subset of test
functions (those that vanish where $f\cdot g$ is too singular to be in
$L^{1}_{loc}$) to all test functions.
To this end, for any distribution $T\in D^{\prime}$ we define the singular
support of T ($\operatorname{singsupp}(T)$) as the smallest closed set in
$\mathbb{R}^{n}$ such that there exists a function $f\in L^{1}_{\text{loc}}$
with $T[\phi]=T_{f}[\phi]$ for all $\phi\in D$ with
supp$\phi\cap\operatorname{singsupp}(T)=\emptyset$. For example
$\operatorname{singsupp}(\delta)=\\{0\\}$, and the corresponding function
$f\in L^{1}_{\text{loc}}$ is simply $f(x)=0$. So the idea behind this
definition is that every distribution can be written as regular distributions
as long as it is only applied to test functions which vanish in a
neighbourhood of the distribution’s singular support, which enables us to
multiply distributions if we manage to take care of the singular support.
### 3.2 Definition of $G^{2}$ in
$D^{\prime}(\mathbb{R}^{4}\text{\textbackslash}\\{0\\})$
Coming back to our concrete field-theoretic problem for a moment, we now have
to answer one important question: what is the singular support of $G$? Here,
we can utilize the fact that $\Box+m^{2}$ is an elliptic operator777Explaining
it without going into details, a differential operator which is defined as
polynomial of $\vec{\partial}$ (with possible coordinate dependent
coefficients) is elliptic if it is non-zero if we replace $\vec{\partial}$
with any non-zero vector $\vec{y}$. In our euclidian examples,
$\Box=\sum^{4}_{i=1}\partial^{2}_{i}\rightarrow|\vec{y}|^{2}>0$ foy any non-
zero $\vec{y}$ and it can be shown that if two distributions $T$ and $S$ are
related by $\sigma T=S$ with an elliptic operator $\sigma$, then
$\operatorname{singsupp}(T)\subset\operatorname{singsupp}(S)$, so we
immediately see $\operatorname{singsupp}(G)\subset\\{0\\}$.
It is clear that the singularity of $G(x)$ at $x=0$ corresponds to the
divergence of the momentum-integral at high energies
$\Lambda\rightarrow\infty$, because in order to probe small distances, short
wavelengths which correspond to high momenta are needed, therefore we speak of
UV-divergencies. In this regime, we can set $m^{2}\approx 0$, and so (49)
simplifies to $\Box G(x)=\delta(x)\Rightarrow G(x)\sim\frac{1}{x^{2}}$ for
small $x$.888The relation
$\delta(\vec{x})=-\frac{1}{4\pi}\Box\frac{1}{\left|\vec{x}\right|}$ is well
known to hold in 3 dimensions. In general, $\Box|x|^{2-n}\propto\delta$ in $n$
dimensions
Using this, we see that the position-space integral
$\int_{|x|>\frac{1}{\Lambda}}d^{4}x\,G^{2}(x)$ again diverges as
log$(\Lambda)$. Because of this divergence $G^{2}(x)\notin
L^{1}_{\text{loc}}(\mathbb{R}^{4})$, so $G^{2}$ is still not defined as
distribution in $D^{\prime}(\mathbf{R}^{4})$. Nevertheless, we can use
$G^{2}(x)$ as kernel of a distribution in
$D^{\prime}(\mathbf{R}^{4}\text{\textbackslash}\\{0\\})=\left\\{T:D(\mathbb{R}^{4}\text{\textbackslash}\\{0\\})\rightarrow\mathbb{C}\mid\text{linear
and continuous}\right\\}$ where
$D(\mathbb{R}^{4}\text{\textbackslash}\\{0\\})$ is the set of test functions
with $\\{0\\}\notin\text{supp}\phi$.
We now managed to define a distribution $G^{2}$, but we still have to extend
it from $D(\mathbb{R}^{4}\text{\textbackslash}\\{0\\})$ to
$D(\mathbb{R}^{4})$. Formally, as a linear map, we have to say what values the
extension takes on $D({\mathbbm{R}}^{4})/D({\mathbbm{R}}^{4}\setminus\\{0\\})$
which is still an infinite dimensional vector space. To control this infinity,
we will use the scaling degree.
### 3.3 The scaling degree and extensions of distributions
Consider a scaling-map $\Lambda$ acting on test functions:
$\displaystyle\mathbb{R}_{>0}\times D(\mathbb{R}^{n})$
$\displaystyle\rightarrow D(\mathbb{R}^{n})$
$\displaystyle\left(\lambda\text{,}\phi\right)$
$\displaystyle\mapsto\phi_{\lambda}(x)\equiv\lambda^{-n}\phi(\lambda^{-1}x)$
The pullback of this map to the space distributions reads
$\displaystyle\left(\Lambda^{\ast}T\right)[\phi]=T[\phi_{\lambda}]\equiv
T_{\lambda}[\phi],$
which for regular distributions gives
$\displaystyle
T_{f,\lambda}[\phi]=\int{\frac{d^{n}x}{\lambda^{n}}f(x)\phi\left(\frac{x}{\lambda}\right)}=\int{d^{n}xf(\lambda
x)\phi(x)},$
so power of $\lambda$ in the scaling map acting on test functions is chosen
such that the kernel $f$ transforms in a simple manner without prefactor. We
now define the scaling degree ($sd$) of $T\in
D^{\prime}(M\subset\mathbb{R}^{n})$:
$\displaystyle
sd(T)=\inf\Big{\\{}\omega\in\mathbb{R}\Big{|}\lim_{\lambda\searrow
0}\lambda^{\omega}T_{\lambda}=0\Big{\\}}$
To understand this definition, we have to note several properties:
* •
$sd(T)\in[-\infty,\infty[$
* •
For regular distributions $sd(T_{f})\leq 0$
* •
$sd(\delta)=n$
* •
$sd(\partial^{\alpha}T)\leq sd(T)+|\alpha|$ with some multi-index $\alpha$
* •
$sd(x^{\alpha}T)\leq sd(T)-|\alpha|$ with some multi-index $\alpha$999Remember
that distributions do not depend on coordinates, only their kernels. Here we
used the definition $(x^{\alpha}T)[\phi]\equiv T[x^{\alpha}\phi]$
* •
$sd(T_{1}+T_{2})=\max\\{sd(T_{1}),sd(T_{2})\\}$
This leads us to the following important theorem:
###### Theorem 5
If $T_{0}\in D^{\prime}(\mathbb{R}^{n}\text{\textbackslash}\\{0\\})$ is a
distribution with $sd(T_{0})<n$, then there is a unique distribution $T\in
D^{\prime}(\mathbb{R}^{n})$ with $sd(T)=sd(T_{0})$ extending $T_{0}$.
The proof of uniqueness is quite easy: We do it by assuming the existence of
two solutions $T$ and $\tilde{T}$ extending $T_{0}$, and showing a
contradiction. Obviously supp$(T-\tilde{T})=\\{0\\}$ and from this it follows
that $T-\tilde{T}=P(\partial)\delta$ with some polynomial $P$. As can be seen
from the above notes, $sd(P(\partial)\delta)\geq n$ and this would be a
contradiction to $sd(T)=sd(\tilde{T})=sd(T_{0})<n$. Existence is shown
constructively using a smooth cut-off function $c_{\epsilon}(x)$ that is 1
outside a ball of radius $2\epsilon$ and and vanishes in a ball of radius
$\epsilon$. Then we can define
$T[\phi]=\lim_{\epsilon\searrow 0}T_{0}[c_{\epsilon}\phi],$ (52)
where one still has to show that the above limit exists in the sense of
distributions.
The theorem above now enables us to uniquely extend distributions of low
scaling degree to the full space $D^{\prime}(\mathbb{R}^{n})$, but what about
distributions with scaling degree $\geq n$? We will solve this problem in the
next section, and afterwards we will be able to return to our field-theoretic
problem of understanding the nature of $G^{2}$.
But first we have to determine what the scaling degree of the massive
propagator $G$, defined by $\delta=(\Box+m^{2})G$. We know that
$sd(\delta)=n$, and therefore $sd((\Box+m^{2})G)=n$ too. If we denote $sd(G)$
by $w$, from the above items it follows that $sd(\Box G)=w+2$, $sd(m^{2}G)=w$
and therefore $sd((\Box+m^{2})G)=w+2$. From this it follows that $w=n-2$ even
for the massive propagator.
### 3.4 Case of distributions with high scaling degree
Considering now a distribution $T_{0}\in
D^{\prime}(\mathbb{R}^{n}\text{\textbackslash}\\{0\\})$ with $sd(T_{0})\geq
n$, uniqueness as in the above theorem does not hold anymore. But if we take a
test function $\phi\in D(\mathbb{R})$ which vanishes of order $\omega\equiv
sd(T_{0})-n$ (“singular order”) at $x=0$, i.e. which can be written as
$\phi(x)=\sum_{|\alpha|=\left\lfloor\omega\right\rfloor+1}x^{\alpha}\phi_{\alpha}(x)$
where $\phi_{\alpha}(0)$ is finite and $\lfloor\omega\rfloor$ denotes the
largest integer not bigger than $\omega$, we can define
$T[\phi]\equiv\sum_{|\alpha|=\left\lfloor\omega\right\rfloor+1}(x^{\alpha}T_{0})[\phi_{\alpha}]$.
Then the distribution $x^{\alpha}T_{0}$ has scaling degree less than $n$ and
can thus be uniquely extended.
A general test function can of course be written as a sum of a function
vanishing of order $\omega$ and a polynomial of degree at most $\omega$ by
subtracting and adding the order $\omega$ Taylor polynomial at $x=0$:
$\displaystyle\phi_{s}(x)\equiv\phi(x)-\sum_{|\alpha|\leq\omega}\frac{x^{\alpha}}{|\alpha|!}\partial^{\alpha}\phi(0)$
(53)
This procedure of subtracting the terms leading to divergencies is the
regularization in this framework. Since the extended distribution $T$ beeing
applied to $\phi_{s}$is unique, by linearity, we still have to define $T$ only
on the monomials in $x$ of maximal degree $\omega$. There is no further
restriction on doing this and this ambiguity in the extension $T$ is what one
would have expected: Changing the value of $T$ on a monomial $x^{\alpha}$
correspond to adding a multiple of $\partial^{\alpha}\delta$ to $T$.
Note well that the arbitratry values of $T[x^{\alpha}]$ are exactly those
where $T_{0}[x^{\alpha}]$ was undefined (divergent in physicists’ parlance)
and selecting a certain value corresponds to picking a counter term. procedure
known as renormalization, as formally infinite values are replaced by finte
ones (that have to be fixed by further physical input like the measurement of
the “physical mass” or the physical “coupling constant”). In the following
small sections, we will try out this method in a few easy, concrete examples.
#### 3.4.1 Example in $n=1$
In order to let our steps so far become clearer, we are going to apply them to
a simple example in $n=1$. In fact, this example shows already the full
regularization and renormalization procedure.
As can be easily checked, the function $f(x)=\frac{1}{|x|}$ is not an element
of $L^{1}_{loc}(\mathbb{R})$ because of its pole at $x=0$ is not integrable
(it is of course log-divergent), so we can not a priori use it as kernel of a
distribution $T_{f}\in D^{\prime}({\mathbb{R}})$ as we have seen in section
3.1. But $f(x)\in L^{1}_{loc}({\mathbbm{R}}\text{\textbackslash}\\{0\\})$ and
$sd(T_{f})=1=n$, therefore $\omega=0$. This means that for a test function
$\phi(x)$ with $\phi(0)=0$ we can define
$T_{f}[\phi]\equiv\int{dx\frac{\phi(x)}{|x|}}$ which gives a finite result:
Using l’Hôpital’s rule, we see $\lim_{x\to\pm
0}\frac{\phi(x)}{|x|}=\lim_{x\to\pm
0}\frac{\phi^{\prime}(x)}{\text{sign}(x)}=\text{finite}$ and thus the
integrand is finite everywhere. This is similar to what we have done in
sections 3.2 and 3.3. For other test functions, we can again (as in this
section above) define $\phi_{s}(x)\equiv\phi(x)-\phi(0)$. Afterwards, we write
the general extension for a distribution acting on $\phi$ as
$T_{f}[\phi]\equiv T_{f}[\phi_{s}]+c\phi(0)$ with one arbitrary constant $c$
of our choice.
The careful reader will have realised that there is still a problem as
$\phi_{s}$ fails to have compact support when $\phi(0)\neq 0$ and thus the
integration now diverges at the boundary $x\to\pm\infty$. We will deal with
this problem below but the important observation is that the divergence in the
ultraviolet, that is at small $x$ is cured.
#### 3.4.2 Example in $n=4$
In our field theoretic problem (50) from above, we have
$G^{2}\sim\frac{1}{x^{4}}$ in $\mathbb{R}^{4}$ which is quite similar to the
previous example, as it is the kernel of a distribution
$T_{G^{2}}=T_{\frac{1}{x^{4}}}\in
D^{\prime}({\mathbbm{R}}^{4}\setminus\\{0\\})$. Again, we are looking for an
extension. Once more, we have $sd(G^{2})=4=n$. Regularization and
renormalization are as in the example above and yield
$\displaystyle
T_{\frac{1}{x^{4}}}^{r}[\phi]=\int{d^{4}x\frac{\phi(x)-\phi(0)}{x^{4}}}+c\delta[\phi]$
(54)
with $T_{\frac{1}{x^{4}}}^{r}\in D^{\prime}(\mathbb{R}^{4})$ and arbitrary
$c$. Again, we successfully got rid of the problems at $x=0$ (at the cost of
introducing one constant $c$).
This concludes our calculation of the fish diagram Fig. 4 that computes a
contribution of the form $\phi(x)^{2}\phi(y)^{2}$ to the effective action of
the theory. Since the ambiguous term we found is $c\delta(x-y)$, the ambiguity
in the effective action is indeed $\phi(x)^{4}\delta(x-y)$. We see, that it
corresponds to the counter term Fig. 6 and renormalizes the coupling constant
(the coefficient of the $\phi^{4}$-term in the action).
Figure 6: Counter-term diagramm.
#### 3.4.3 Example with $sd(T)>n$ and preservation of symmetry
In the two examples above, we had both times distributions $T$ with $sd(T)=n$
which led us to the introduction of one arbitrary constant $c$. This amount of
ambiguity increases with $sd(T)$, but not all possible polynomials
$P(\partial)\delta$ allowed by the counting can arise physically. In
particular, we require that our theory is still Lorentz invariant after
renormalization and if it has a local gauge symmetry before that needs to be
maintained as well (otherwise one has an anomaly that renders the theory ill-
defined at the quantum level since the number of degrees of freedom changes
upon renormalization).
Let us consider one example where SO(4) invariance (the euclidian version of
the Lorentz group SO(3,1)) selects a subset of the possible counter terms.
In a theory with potential $\propto\phi^{4}$ (quartic interaction), there
cannot only be diagramms like Figure 4, but also such ones like Figure 7,
known as the setting sun diagram.
Figure 7: Setting sun diagramm in quartic interaction
The term encoded by this diagramm obviously contains
$G^{3}\sim\frac{1}{x^{6}}$, which has $sd(G^{3})=6>n=4$. By performing the
same steps as above, in this case we a priori get an ambiguity
$c_{1}\delta+c_{2}^{i}\partial_{i}\delta+c^{ij}_{3}\partial_{i}\partial_{j}\delta$
with in total $1+4+\frac{4(4+1)}{2}=15$ arbitrary constants, but upon imposing
SO(4)-invariance this reduces to $c_{1}\delta+c_{3}\Delta\delta$ with only 2
arbitrary constants.
In the effective action, as above, they contribute to the quadratic terms (as
the diagram Fig. 7 has two external lines)
$\phi(x)(c_{1}\delta(x-y)+c_{3}\Delta\delta(x-y))\phi(y)=\phi(x)(c_{3}\Delta+c_{1})\phi(x)\delta(x-y)$.
We recognize that $c_{3}$ is a wave function renormalization while $c_{1}$
renormalizes the mass-term $m^{2}\phi^{2}$.
The fact that $\phi^{4}$-theory is renormalizable in $n=4$ means that these
two counter terms and the one in the previous subsection are the only
ambiguities that arise when any Feynman diagram of the theory is renormalized,
a proof of being well beyond the scope of these notes.
### 3.5 Regaining compact support and RG flow
In the above calculations, we ignored an important problem:
$\phi_{s}(x)=\phi(x)-\phi(0)$ is not necessarily a test function, as it
obviously has $\lim_{x\rightarrow\infty}=-\phi(0)$, therefore for example the
integral $\int{dx\frac{\phi(x)-\phi(0)}{|x|}}$ that we encountered in section
3.4.1 may diverge at infinity. We can solve this by introducing a function
$w(x)\in D(\mathbb{R}^{n})$ with (without loss of generality) $w(0)=1$. We
then change the regularized part (i.e. the part without arbitrary constants)
of the integral in (54) to
$\displaystyle
T_{\frac{1}{|x|}}[\phi]\equiv\int{dx\frac{\phi(x)-w(x)\frac{\phi(0)}{w(0)}}{|x|}}.$
(55)
This is a special case of the general formula
$\displaystyle\phi_{s}(x)\equiv\phi(x)-\sum_{|\alpha|\leq\omega}\frac{x^{\alpha}w(x)}{|\alpha|!}\left(\partial^{\alpha}\frac{\phi(x)}{w(x)}\right)\bigg{\lvert}_{x=0},$
which replaces equation (53). Starting from (55) we can now write
$\displaystyle
T_{\frac{1}{|x|}}[\phi]=\int{dx\frac{w(x)(\phi(x)-\phi(0))}{|x|}}+\underbrace{\int{dx\frac{(1-w(x))\phi(x)}{|x|}}}_{=T_{\frac{1-w(x)}{|x|}}[\phi]}.$
The second term already is a perfectly fine distribution, the first term can
be manipulated in the following way:
$\displaystyle\int{dx\frac{w(x)(\phi(x)-\phi(0))}{|x|}}=\int{dx\frac{w(x)}{|x|}\int^{x}_{0}du\phi^{\prime}(u)}$
Now, in the inner integral we can substitute $u=tx$ and afterwards interchange
the integrals:
$\displaystyle\int{dx\frac{w(x)}{|x|}\int^{1}_{0}dtx\phi^{\prime}(tx)}=\int^{1}_{0}{dt\int
dx\;\text{sign}(x)w(x)\phi^{\prime}(tx)}$
After this, we can substitute $y=tx$ in the inner integral, giving us
$\displaystyle\int^{1}_{0}dt\int\frac{dy}{t}w\left(\frac{y}{t}\right)\text{sign}\left(\frac{y}{t}\right)\phi^{\prime}(y)$
$\displaystyle=\int{dy\underbrace{\int^{1}_{0}\frac{dt}{t}w\left(\frac{y}{t}\right)\text{sign}(y)}_{\equiv
f(y)}\phi^{\prime}(y)}$ $\displaystyle=T_{f}[\partial\phi]=-(\partial
T_{f})[\phi].$
So also the first term is a good distribution. The function $f(y)$ defined
above as function of $y$ is well behaved, as $w(x)$ which enters its
definition is a test function, and therefore extremely well behaved, in
particular vanishes for large arguments, so taking $t\to 0$ does not introduce
problems.
As an example, let us now set $w(x)=\theta(1-M|x|)$ (or actually a smoothed
out version of this non-continuous function)
$\displaystyle f(y)$
$\displaystyle=\int^{1}_{0}\frac{dt}{t}\theta\left(1-M\frac{|y|}{t}\right)\text{sign}(y)$
$\displaystyle=\int^{1}_{M|y|}\frac{dt}{t}\theta\left(1-M|y|\right)\text{sign}(y)$
$\displaystyle=-\ln(M|y|)\theta\left(1-M|y|\right)\text{sign}(y)$
Morally, we regularized our distribution with non-integrable kernel
$\propto\frac{1}{|x|}$ by substituting the derivative of a distribution with
kernel $\propto\log(|y|)$, which is integrable.
In the above calculations, we introduced a mass/energy-scale $M$. It is now an
important question to ask how the distribution changes under transformations
of this scale, i.e. renormalization group (RG) transformations generated by
$M\frac{\partial}{\partial M}$, so called RG flows. We will now show that it
is only the part $const\cdot\delta(x)$, i.e. the part that is fixed by
arbitrary renormalization constants that will change.
First of all, using $\partial_{x}\text{sign}(x)=2\delta(x)$, we see:
$\displaystyle
f^{\prime}(x)=\frac{-1}{x}\theta\left(1-M|x|\right)\text{sign}(x)-\log(M|x|)\theta\left(1-M|x|\right)2\delta(x)$
Then, we start with:
$\displaystyle M\frac{\partial}{\partial
M}T_{\frac{1}{|x|}}[\phi]=M\frac{\partial}{\partial
M}\left(\int{dx\frac{(1-w(x))\phi(x)}{|x|}}+T_{-\partial f}[\phi]\right)$
Because of $\frac{1-w(x)}{|x|}=\frac{\theta\left(M|x|-1\right)}{|x|}$ in our
example, the first term becomes a distribution with kernel
$\displaystyle M\frac{\partial}{\partial
M}\frac{\theta\left(M|x|-1\right)}{|x|}=M\delta(M|x|-1).$ (56)
The second term in contrast becomes a distribution with kernel
$\displaystyle M\frac{\partial}{\partial
M}(-f^{\prime}(x))=-M\delta(M|x|-1)+M\frac{\partial}{\partial
M}\left[2\log(M|x|)\theta\left(1-M|x|\right)\delta(x)\right].$
The first term of this expression obviously cancels with the contribution from
(56), so $M\frac{\partial}{\partial M}T_{\frac{1}{|x|}}$ turns out to be a
distribution with kernel:
$\displaystyle M\frac{\partial}{\partial
M}\left[2\log(M|x|)\theta\left(1-M|x|\right)\delta(x)\right]$
$\displaystyle=2\delta(x)\left[\theta\left(1-M|x|\right)-\log(M|x|)M\delta\left(1-M|x|\right)|x|\right]$
$\displaystyle=2\delta(x)$
In the last step, we used the presence of the factor $\delta(x)$ (under an
integral!) to set $\log(M|x|)|x|=0$ and $\theta\left(1-M|x|\right)=1$. So,
under a renormalization group transformation, the distribution changes by
$\delta T\propto const\cdot\delta(x)$, that means that a change of energy-
scale corresponds to a change of the (at the beginning) arbitrarily selected
renormalization coefficients.
### 3.6 What we have achieved in this section
We have seen a way to recast what looks like divergent Feynman diagrams as to
what looks like distributions for non-integrable functions. We could then turn
these into proper distributions by first restricting the space of test-
functions and then extend them to a full distribution, possibly at the price
of a finite number of undetermined numerical constants. Those have to be
determined by a finite number of measurements.
In order for the number of introduced parameters for all Feynman diagrams of
the theory to be finite, the scaling degrees of all appearing distributions in
all diagramms have to be below some maximum, otherwise the theory is not
renormalizable.
## 4 Summary
The material in these notes will not be useful for any concrete calculation in
quantum field theory that a physicist might be interested in. But they might
give him or her some confidence that the calculation envisaged has a chance to
be meaningful.
We tried to present material that is in no sense original but still is
probably not covered in most introductions to quantum field theory. Hopefully,
it helps to refute some of the prejudices against (perturbative) quantum field
theory that mathematically minded people may have and helps others to better
understand how far the hand waving arguments that we use in our daily work can
carry.
In particular, we put our emphasis on two points: Even if the perturbative
expansion is divergent as a power series it can serve two purposes: The first
terms do provide a numerically good approximation to the true, non-
perturbative result and all terms taken together can indeed recover the full
result but only in terms of Borel resummation rather than as a power series.
Second, unphysical infinite momentum integrals in the computation of Feynman
diagrams can be avoided when properly expressed in terms of distributions. The
renormalization of coupling constants is then expressed as the problem to
extend a distribution from a subspace to all test functions. The language of
distribution theory allows one to avoid mathematically ill-defined divergent
expressions altogether.
## Acknowledgements
A lot of the material presented we learned from Klaus Fredenhagen, Dirk Prange
and Marcel Vonk. We would like to thank the Elitemasterprogramme “Theoretical
and Mathematical Physics” and Elitenetzwerk Bayern.
## References
* [1] H. Osborn, “Advanced quantum field theory lecture notes.” available at http://www.damtp.cam.ac.uk/user/ho/Notes.pdf, April, 2007.
* [2] F. J. Dyson, Divergence of perturbation theory in quantum electodynamics, Phys.Rev. 85 (1952) 631–632.
* [3] M. Nakahara, Geometry, Topology and Physics. Taylor & Francis, 2003.
* [4] M. Reed and B. Simon, Analysis of operators, vol. 4 of Methods of modern mathematical physics. Academic Press, 1978.
* [5] G. Scharf, Finite Quantum Electrodynamics: The Causal Approach. Springer, New York, 1995.
* [6] R. Brunetti and K. Fredenhagen, Quantum field theory on curved backgrounds, 0901.2063.
* [7] D. Prange, Epstein-glaser renormalization and differential renormalization, J.Phys.A A32 (1999) 2225–2238, [hep-th/9710225].
|
arxiv-papers
| 2012-01-13T00:23:43 |
2024-09-04T02:49:26.286145
|
{
"license": "Public Domain",
"authors": "Mario Flory and Robert C. Helling and Constantin Sluka",
"submitter": "Robert C. Helling",
"url": "https://arxiv.org/abs/1201.2714"
}
|
1201.2748
|
# Mott-Hubbard localization in model of electronic subsystem of doped
fullerides
Yu. Dovhopyaty, L. Didukh, O. Kramar, Yu. Skorenkyy, Yu. Drohobitskyy
Ternopil National Technical University, 56, Ruska Str., Ternopil, 46001,
Ukraine
###### Abstract
Microscopical model of a doped fulleride electronic subsystem taking into
account the triple orbital degeneracy of energy states is considered within
the configurational-operator approach. Using the Green function method the
energy spectrum of the model at integer band filling $n=1$ is calculated,
which case corresponds to $AC_{60}$ compounds. Possible correlation-driven
metal-insulator transition in the model is discussed.
###### pacs:
71.27.+a;72.80.Rj
## I Introduction
Electrical, optical and mechanical properties of fullerenes elec95 ; mani06
in condensed state demonstrate considerable physical content of phenomena
which take place in fullerenes and show that the use of such materials in
electronics has significant perspectives. Fullerene crystals and films are
semiconductors with an energy gap of $1.2-1.9eV$ sait91 ; achi91 and have
photoconductivity under visible light irradiation. Fullerene crystals have
comparatively small binding energy and at room temperature the phase
transition connected with orientational disordering of fullerene molecules
take place in such crystals heyn91 .
Addition of radicals containing platinum group metals hawk92 to fullerenes
C60 allows to obtain ferromagnetic material based on fullerene. In polycrystal
C60 doped by alkali metal superconductivity at temperature lower then $33K$ is
observed flem91 ; holc91 . Large binding energy is typical for
metallocarbohedrenes M8C12, where $M=Ti,V,Hg,Zr$. For example, in Ti8C12
molecule binding energy per atom is $6.1eV$ redd92 (for C60 molecule this
energy is $7.4-7.6eV$ sait91 ).
Fullerenes in solid state (fullerites) are the molecular crystals, where
interaction between atoms in C60 molecule is much larger then interaction
between nearest molecules. In tightly packed structure each fullerene molecule
has $12$ nearest neighbors. Depending on peculiarities of molecular
interaction, face-centered cubic lattice or hexagonal lattice is realized
beth90 . Phase transition in C60 crystal occurs at the temperature of $257K$
and this is the first order transition. At high temperatures molecules can
freely rotate whereas at low temperatures rotation is stopped and anisotropy
of neighbor molecule C60 interaction becomes important. This leads to small
sharp change of distance between the nearest molecules. According to results
of X-ray structure analysis regu91 lattice constant changes from $1.4154\pm
0.0003nm$ to $1.4111\pm 0.0003nm$ (that is by $0.43\pm 0.06$ percent).
At low temperature, when C60 -molecules are oriented in space, crystal lattice
symmetry does not coincide with the symmetry of single molecule C60
(icosahedral symmetry $Y$). In a unit cell of fullerite crystal lattice there
are four C60-molecules. These molecules form tetrahedron in which orientations
of all molecules are the same. Tetrahedra, in their turn, form simple cubic
lattice.
Fullerites are semiconductors with energy gap of $1.5-1.95eV$ sait91 .
Electrical resistivity of polycrystals C60 regu91 monotonically changes with
changing temperature and energy gap has monotonic dependence on the pressure
value: an increase of energy gap under the pressure, higher than $2\times
10^{5}$ atm indicates the absence of metal-insulator transition at $p\simeq
10^{6}$atm. In the temperature region $150-400K$ the relaxation time is
temperature-independent what indicates that the carriers are localized and
hopping mechanism of recombination, which includes tunneling of electrons
between localized states, is realized.
It has been shown in 1991 flem91 that doping of solid fullerenes C60 by small
quantity of alkaline metal leads to formation of material with metallic type
of conductivity and this material becomes superconducting at low temperatures
($T_{c}$ from $2.5K$ for Na2KC60 to $33K$ for RbCs2C60). At changes of
temperature, concentration of alkaline metal, parameters and structure of
lattice various phases of these compounds have been realized. In particular,
at various filling $n$ ($n$ may change from 0 to 6) of lowest unoccupied
molecular orbital (LUMO) the metallic, insulating or superconducting phases
have been realized. Superconductivity in doped fullerenes KxC60 has been
studied theoretically in paper zai93 and strong electron correlations have
been shown to play a crucial role in superconducting state stabilization.
Recently, strong electron correlation were also proven zai11 to be
responsible for superconductivity of planar carbon systems of graphene type.
Let us consider the electronic structure of C60 in detail. In single-particle
approximation, neglecting electron correlations, the following spectrum has
been calculated mani06 : 50 of 60 $p_{z}$ electrons of a neutral molecule fill
all orbitals up to $L=4$. The lowest $L=0,1,2$ orbitals correspond to
icosahedral states $a_{g},t_{1u},h_{g}$. All states with greater $L$ values
undergo the icosahedral-field splitting. There are 10 electrons in partially
filled $L=5$ state. Icosahedral splitting ($L=5\rightarrow
h_{u}+t_{1u}+t_{2u}$) of this 11-fold degenerate orbital leads to the
electronic configuration shown below. Microscopic calculations and
experimental data show that the completely filled highest occupied molecular
orbital is of $h_{u}$ symmetry, and LUMO (3-fold degenerate) has $t_{1u}$
symmetry. At such conditions HOMO-LUMO gap appears due to icosahedral
perturbation in the shell with $L=5$; energy gap found experimentally is about
$1eV$ for molecules in vacuum. A $t_{2g}$ (LUMO+1)-state, originated from
$L=6$ shell, is found approximately $1eV$ above the $t_{1u}$ LUMO.
Electron-electron correlations in C60 are described by two main parameters:
intra-molecular Coulomb repulsion $U$ and Hund’s coupling $J_{H}$. In
fullerenes the competition between intra-site Coulomb interaction (Hubbard
$U$) and delocalization processes, connected with translational motion of
electrons (which determines the bandwidth), causes the realization of
insulator or metallic state gunn97 . Majority of the experimental data and
theoretical calculations indicate that all materials with ions C${}^{-n}_{60}$
at integer $n$ are Mott-Hubbard insulators as $U$ is quite large for all doped
compounds AxC60. Fullerides AxC60 doped with alkali metals A attract much
attention of researchers due to unusual metal-insulator transition in these
compounds. Only A3C60 is metallic and other phases AC60, A2C60 and A4C60 are
insulator poir93 . This experimental fact contradicts to the results of band
structure calculations (see sath92 for example) which predict purely metallic
behavior. It has been noted in paper lu94 , that for explanation of metallic
behavior of Mott-Hubbard system ($x=3$ corresponds to the half-filling of the
conduction band) one has to take into account a degeneracy of energy band. On
the base of Gutzwiller variational approach the metal-insulator transition has
been proven lu94 to exist for all integer band fillings. It is shown that the
critical value of Coulomb interaction parameter depends essentially on the
band filling and degeneracy (in case of half filling $\frac{U_{c}}{2w}\simeq
2,8$ for double degeneracy, $\frac{U_{c}}{2w}\simeq 3,9$ for triple
degeneracy). The present study is devoted to investigation of Mott-Hubbard
localization in electronic subsystem of fullerides with strong electron
correlations within the model taking into account the orbital degeneracy of
energy levels, strong Coulomb interaction and correlated hopping of electrons.
## II The Hamiltonian of doped fulleride electronic subsystem
Within the second quantization formalism the Hamiltonian of interacting
electron systems can be written fett71 as
$\displaystyle
H=-\mu\sum_{i\lambda\sigma}a_{i\lambda\sigma}^{+}a_{i\lambda\sigma}+{\sum_{ij\lambda\sigma}}^{\prime}t_{ij}a_{i\lambda\sigma}^{+}a_{j\lambda\sigma}+\frac{1}{2}{\sum_{ijkl}}{\sum_{\alpha\beta\gamma\delta}}{\sum_{\sigma\sigma^{\prime}}}J^{\alpha\beta\gamma\delta}_{ijkl}a_{i\alpha\sigma}^{+}a_{j\beta\sigma^{\prime}}^{+}a_{l\delta\sigma^{\prime}}a_{k\gamma\sigma},$
(1)
where the first sum with matrix element
$\displaystyle t_{ij}=\int{d^{3}}r{\phi}_{\lambda i}^{*}({\bf r}-{\bf
R}_{i})\times\left[-\frac{\hbar^{2}}{2m}\Delta+V^{ion}({\bf
r})\right]\phi_{\lambda i}({\bf r}-{\bf R}_{j})$ (2)
describes translational motion (hopping) of electrons in the crystal field
$V^{ion}(\bf{r})$ and the second sum is the general expression for pair
electron interactions described by matrix elements
$\displaystyle
J^{\alpha\beta\gamma\delta}_{ijkl}=\int{\int{{\phi}_{\alpha}^{*}({\bf r}-{\bf
R}_{i}){\phi}_{\beta}({\bf r}-{\bf
R}_{j})}}\times\frac{e^{2}}{|r-r^{\prime}|}{\phi}_{\delta}^{*}({\bf r}-{\bf
R}_{l}){\phi}_{\gamma}({\bf r}-{\bf R}_{k})drdr^{\prime}.$ (3)
In the above formulae $a_{i\lambda\sigma}^{+}$, $a_{i\lambda\sigma}$ are
operators of spin-$\sigma$ electron creation and annihilation in orbital state
$\lambda$ on lattice site $i$, respectively, indices $\alpha$, $\beta$,
$\gamma$, $\delta$, $\lambda$ denote orbital states, ${\phi}_{\lambda i}$ is
wave-function in Wannier (site) representation other notation are standard.
Hamiltonian (1) is essentially non-diagonal and hard to treat mathematically.
The problem can be greatly simplified by neglecting the matrix elements of
interaction of the third and further orders of magnitude and restrict oneself
to consideration of a single orbital per site. In this way, Hamiltonian of
Hubbard model and many other backbone models of strongly correlated electrons
theory were derived. However, it has been shown that these models lack the
possibility of description of electron-hole asymmetry, observed in real
correlated electron systems. To maintain such possibility we are to consider
the energy levels structure and estimate interaction parameters prior to make
simplifications. Following papers d_act00 ; dsdh_prb we derive the
Hamiltonial which takes into account the correlated hopping of electrons (the
site-occupation dependence of hopping parameters results from taking into
account the interactions with second order of magnitude matrix elements) and
variety of intra-cite interactions caused by triple orbital degeneracy of LUMO
in doped fullerites. Interaction integral of zeroth-order magnitude is on-site
Coulomb correlation (characterized by Hubbard parameter $U$):
$\displaystyle U=\int{\int{|{\phi}_{\lambda}^{*}({\bf r}-{\bf
R}_{i})|^{2}\frac{e^{2}}{|r-r^{\prime}|}|\phi_{\lambda}({\bf r^{\prime}}-{\bf
R}_{i})|^{2}drdr^{\prime}}},$ (4)
In orbitally degenerate system, the on-site (Hund’s rule) exchange integral
$\displaystyle J_{H}=\int{\int{\phi}_{\lambda}^{*}}({\bf r}-{\bf
R}_{i})\phi_{\lambda^{{}^{\prime}}}({\bf r}-{\bf
R}_{i}){e^{2}\over|{r}-{r}^{{}^{\prime}}|}\times\phi^{*}{{}_{\lambda^{{}^{\prime}}}}({\bf
r}^{{}^{\prime}}-{\bf R}_{i})\phi_{\lambda}({\bf r}^{{}^{\prime}}-{\bf
R}_{i})d{\bf r}d{\bf r}^{{}^{\prime}},$ (5)
is of principal importance, too. Parameter $U$ value for fullerenes have been
estimated within different methods. Use of local density approximation (LDA)
gives 3.0 eV ped92 ; antr92 . Experimental estimation of electron repulsion
energy hett91 gives $U\simeq 2.7$ eV.
It’s worth to note, that in solid state molecules are placed close enough to
provide substantial screening of interaction. Calculation with screening
effect took into account give $U$ 2.7 ped92 ; antr92 . Combining Auger
spectroscopy and photoemission spectroscopy lead to value 1.4-1.6 eV lof92 ;
bruh93 for $U$. We also note that energy cost of electron configurations with
spins aligned in parallel is considerably less than for anti-parallel
alignment. Orbitally degenerate levels are filled according to Hund’s rule.
Experimental methods lof92 for singlet-triplet splitting give 0.2 eV $\pm$
0.1 eV; and in work mart93 has the values close to 0.05 eV. The relevant
inter-site parameters are electron hopping integral and inter-site exchange
coupling $J(i\lambda j{\lambda}^{\prime}j\lambda i{\lambda}^{{}^{\prime}})$.
The resulting Hamiltonian of doped fulleride electronic subsystem reads as
$\displaystyle H$
$\displaystyle=-\mu\sum_{i\lambda\sigma}a_{i\lambda\sigma}^{+}a_{i\lambda\sigma}+U\sum_{i\lambda}n_{i\lambda\uparrow}n_{i\lambda\downarrow}+\frac{U^{\prime}}{2}\sum_{i\lambda\sigma}n_{i\lambda\sigma}n_{i\lambda^{\prime}\bar{\sigma}}+\frac{U^{\prime}-J_{H}}{2}\sum_{i\lambda\lambda^{\prime}\sigma}n_{i\lambda\sigma}n_{i\lambda^{\prime}\sigma}+$
(6) $\displaystyle+$
$\displaystyle{\sum_{ij\lambda\sigma}}^{\prime}t_{ij}(n)a_{i\lambda\sigma}^{+}a_{j\lambda\sigma}+{\sum_{ij\lambda\sigma}}^{\prime}t^{{}^{\prime}}_{ij}\left(a_{i\lambda\sigma}^{+}a_{j\lambda\sigma}n_{i\bar{\lambda}}+h.c.\right)+{\sum_{ij\lambda\sigma}}^{\prime}t^{{}^{\prime\prime}}_{ij}\left(a_{i\lambda\sigma}^{+}a_{j\lambda\sigma}n_{i\lambda\bar{\sigma}}+h.c.\right),$
where $n_{i\lambda\sigma}=a_{i\lambda\sigma}^{+}a_{i\lambda\sigma}$,
$U^{\prime}=U-2J_{H}$ and hopping integrals $t_{ij}(n)$,
$t^{{}^{\prime}}_{ij},t^{{}^{\prime\prime}}_{ij}$ taking into account three
types of correlated hopping of electrons did97 are introduced.
Figure 1: Possible site configurations in threefold degenerate model. The
first symbol in the state notation correspond to $\alpha$ orbital, the second
and the third - to $\beta$ and $\gamma$ orbitals, correspondingly.
In a model of triply degenerate band, every site can be in one of 64
configurations (see fig. 1). To pass from electron operator to Hubbard
operators $X^{pl}$ of site transition from state $|l\rangle$ to state
$|p\rangle$ we use relations of type
$\displaystyle\hat{a}_{\alpha\uparrow}^{+}$ $\displaystyle=$ $\displaystyle
X^{\uparrow 00,000}+X^{200,\downarrow 00}+X^{\uparrow\uparrow 0,0\uparrow
0}+X^{\uparrow\downarrow 0,0\downarrow 0}+X^{\uparrow
0\uparrow,00\uparrow}+X^{\uparrow 0\downarrow,00\downarrow}+X^{\uparrow
20,020}+X^{\uparrow 02,002}$ $\displaystyle+$ $\displaystyle X^{2\downarrow
0,\downarrow\downarrow 0}+X^{20\downarrow,\downarrow
0\downarrow}+X^{\uparrow\uparrow\uparrow,0\uparrow\uparrow}+X^{\uparrow\downarrow\downarrow,0\downarrow\downarrow}+X^{2\uparrow
0,\downarrow\uparrow 0}+X^{20\uparrow,\downarrow
0\uparrow}+X^{\uparrow\uparrow\downarrow,0\uparrow\downarrow}+X^{\uparrow\downarrow\uparrow,0\downarrow\uparrow}$
$\displaystyle+$ $\displaystyle
X^{2\downarrow\downarrow,\downarrow\downarrow\downarrow}+X^{2\downarrow\uparrow,\downarrow\downarrow\uparrow}+X^{2\uparrow\downarrow,\downarrow\uparrow\downarrow}+X^{2\uparrow\uparrow,\downarrow\uparrow\uparrow}+X^{220,\downarrow
20}+X^{\uparrow 2\uparrow,02\uparrow}+X^{\uparrow
2\downarrow,02\downarrow}+X^{202,\downarrow 02}$ $\displaystyle+$
$\displaystyle X^{\uparrow\uparrow 2,0\uparrow 2}+X^{\uparrow\downarrow
2,0\downarrow 2}+X^{\uparrow 22,022}+X^{22\downarrow,\downarrow
2\downarrow}+X^{2\downarrow 2,\downarrow\downarrow 2}+X^{22\uparrow,\downarrow
2\uparrow}+X^{2\uparrow 2,\downarrow\uparrow 2}+X^{222,\downarrow 22},$
$\displaystyle\hat{a}_{\alpha\downarrow}^{+}$ $\displaystyle=$ $\displaystyle
X^{\downarrow 00,000}-X^{200,\uparrow 00}+X^{\downarrow\uparrow 0,0\uparrow
0}+X^{\downarrow\downarrow 0,0\downarrow 0}+X^{\downarrow
0\uparrow,00\uparrow}+X^{\downarrow 0\downarrow,00\downarrow}+X^{\downarrow
20,020}+X^{\downarrow 02,002}$ (7) $\displaystyle-$ $\displaystyle
X^{2\uparrow 0,\uparrow\uparrow 0}-X^{20\uparrow,\uparrow
0\uparrow}+X^{\downarrow\uparrow\uparrow,0\uparrow\uparrow}+X^{\downarrow\downarrow\downarrow,0\downarrow\downarrow}-X^{2\downarrow
0,\uparrow\downarrow 0}-X^{20\downarrow,\uparrow
0\downarrow}+X^{\downarrow\uparrow\downarrow,0\uparrow\downarrow}+X^{\downarrow\downarrow\uparrow,0\downarrow\uparrow}$
$\displaystyle-$ $\displaystyle
X^{2\uparrow\uparrow,\uparrow\uparrow\uparrow}-X^{2\uparrow\downarrow,\uparrow\uparrow\downarrow}-X^{2\downarrow\uparrow,\uparrow\downarrow\uparrow}-X^{2\downarrow\downarrow,\uparrow\downarrow\downarrow}-X^{220,\uparrow
20}+X^{\downarrow 2\uparrow,02\uparrow}+X^{\downarrow
2\downarrow,02\downarrow}-X^{202,\uparrow 02}$ $\displaystyle+$ $\displaystyle
X^{\downarrow\uparrow 2,0\uparrow 2}+X^{\downarrow\downarrow 2,0\downarrow
2}+X^{\downarrow 22,022}-X^{22\uparrow,\uparrow 2\uparrow}-X^{2\uparrow
2,\uparrow\uparrow 2}-X^{22\downarrow,\uparrow 2\downarrow}-X^{2\downarrow
2,\uparrow\downarrow 2}-X^{222,\uparrow 22},$
which ensure the fulfilment of anticommutation relations
$\\{X_{i}^{pl};X_{j}^{kt}\\}=\delta_{ij}(\delta_{lk}X_{i}^{pt}+\delta_{pt}X_{i}^{kl})$,
and normalizing condition $\sum\limits_{i}X_{i}^{p}=1$, for number operators
$X_{i}^{p}=X_{i}^{pl}X_{i}^{lp}$ of $|p>$-state on site $i$. Such type of
electronic operators representation is typical for models of strongly-
correlated electron systems as superconducting cuprates ovch94 , manganites
gavr11 , cobaltites ovch11 , optical lattices stas09 ; stas10 . Using the root
vector notations introduced in paper zai76 allows to obtain much more compact
form of Hamiltonian in configurational representation. However, in our case
number of subbands is relatively small and we use bulky but simple notations
which make the projection procedure used below more transparent.
In the configurational representation the model Hamiltonian takes the form
$H=H_{0}+T$. Here $H_{0}$ sums the ”atomic limit” terms and the translational
part may decomposed as $T=\sum\limits_{n,m}T_{nm}$, where $n,m$ serve for
numbering ”atomic” states. Terms $T_{nn}$ of the Hamiltonian form the energy
subbands and terms of $T_{nm}$ describe the hybridization of these subbands.
Different hopping integrals correspond to transitions in (or between) the
different subbands. The subbands of higher-energy processes appear to be
narrower due to the correlated hopping of electrons. The relative positions
and overlapping of the subbands depends on the relations between the energy
parameters. At integer values of electron concentration ($n=1,2,3,4,5$) in the
system the metal-insulator transition is possible.
In the partial case of band filling $n=1$, strong Coulomb correlation and
strong Hund’s coupling (parameter $U-3J_{H}$ is much greater than the
bandwidth, see estimations in papers ped92 ; antr92 ) the states with three
and more electrons on the same site are excluded. Then the influence of
correlated hopping can be described by three different hopping integrals. The
bare band hopping integral $t_{ij}$ is renormalized to take into account the
band narrowing caused by concentration dependent correlated hopping as
$t_{ij}(n)=t_{ij}(1-\tau_{1}n)$. This hopping integral characterizes lower
Hubbard subband. Parameter $\tau_{1}$ is usually neglected, but it is of
principle important for a consistent description of correlation effects in
narrow band systems (see d_act00 ; dsdh_prb for a detailed discussion). The
hopping integral for upper Hubbard subband is
$\tilde{t}_{ij}(n)=t_{ij}(n)+2t^{\prime}_{ij}$ and
$\bar{t}_{ij}(n)=t_{ij}(n)+t^{\prime}_{ij}$ describes a hybridization of lower
and upper Hubbard subbands. In the following only the case $n=1$ is considered
so we omit the explicit notation of concentration dependence. Then the
Hamiltonian in $X-$operator representation hubb65 has the form
$\displaystyle H$
$\displaystyle=H_{0}+\sum_{\lambda=\alpha,\beta,\gamma}\left(H_{b}^{(\lambda)}+H_{h}^{(\lambda)}\right),$
$\displaystyle H_{0}$ $\displaystyle=-\mu\sum_{i\sigma}(X_{i}^{\sigma
00}+X_{i}^{0\sigma 0}+X_{i}^{00\sigma}+2\left(X_{i}^{\sigma\sigma
0}+X_{i}^{\sigma
0\sigma}+X_{i}^{0\sigma\sigma}\right))+(U-3J_{H})\sum_{i\sigma}\left(X_{i}^{\sigma\sigma
0}+X_{i}^{\sigma 0\sigma}+X_{i}^{0\sigma\sigma}\right),$ $\displaystyle
H_{b}^{(\alpha)}$ $\displaystyle={\sum\limits_{ij\sigma}}(t_{ij}X_{i}^{\sigma
00,000}X_{j}^{000,\sigma 00}+\tilde{t}_{ij}X_{i}^{\sigma\sigma 0,0\sigma
0}X_{j}^{0\sigma 0,\sigma\sigma 0}+\tilde{t}_{ij}X_{i}^{\sigma
0\sigma,00\sigma}X_{j}^{00\sigma,\sigma 0\sigma}+$ $\displaystyle+$
$\displaystyle\tilde{t}_{ij}X_{i}^{\sigma\sigma 0,0\sigma
0}X_{j}^{00\sigma,\sigma 0\sigma}+\tilde{t}_{ij}X_{i}^{\sigma
0\sigma,00\sigma}X_{j}^{0\sigma 0,\sigma\sigma 0}),$ $\displaystyle
H_{h}^{(\alpha)}$
$\displaystyle={\sum\limits_{ij\sigma}}\bar{t}_{ij}(X_{i}^{\sigma
00,000}X_{j}^{0\sigma 0,\sigma\sigma 0}+X_{i}^{\sigma\sigma 0,0\sigma
0}X_{j}^{000,\sigma 00}+$ $\displaystyle+$ $\displaystyle X_{i}^{\sigma
00,000}X_{j}^{00\sigma,\sigma 0\sigma}+X_{i}^{\sigma
0\sigma,00\sigma}X_{j}^{000,\sigma 00}),$ $\displaystyle H_{b}^{(\beta)}$
$\displaystyle={\sum\limits_{ij\sigma}}(t_{ij}X_{i}^{0\sigma
0,000}X_{j}^{000,0\sigma 0}+\tilde{t}_{ij}X_{i}^{\sigma\sigma 0,\sigma
00}X_{j}^{\sigma 00,\sigma\sigma
0}+\tilde{t}_{ij}X_{i}^{0\sigma\sigma,00\sigma}X_{j}^{00\sigma,0\sigma\sigma}-$
$\displaystyle-$ $\displaystyle\tilde{t}_{ij}X_{i}^{\sigma\sigma 0,\sigma
00}X_{j}^{00\sigma,0\sigma\sigma}-\tilde{t}_{ij}X_{i}^{0\sigma\sigma,00\sigma}X_{j}^{\sigma
00,\sigma\sigma 0}),$ $\displaystyle H_{h}^{(\beta)}$
$\displaystyle={\sum\limits_{ij\sigma}}\bar{t}_{ij}(X_{i}^{0\sigma
0,000}X_{j}^{00\sigma,0\sigma\sigma}+X_{i}^{0\sigma\sigma,00\sigma}X_{j}^{000,0\sigma
0}-X_{i}^{0\sigma 0,000}X_{j}^{\sigma 00,\sigma\sigma 0}-X_{i}^{\sigma\sigma
0,\sigma 00}X_{j}^{000,0\sigma 0}),$ $\displaystyle H_{b}^{(\gamma)}$
$\displaystyle={\sum\limits_{ij\sigma}}(t_{ij}X_{i}^{00\sigma,000}X_{j}^{000,00\sigma}+\tilde{t}_{ij}X_{i}^{\sigma
0\sigma,\sigma 00}X_{j}^{\sigma 00,\sigma
0\sigma}-\tilde{t}_{ij}X_{i}^{0\sigma\sigma,0\sigma 0}X_{j}^{0\sigma
0,0\sigma\sigma}+$ $\displaystyle+$ $\displaystyle\tilde{t}_{ij}X_{i}^{\sigma
0\sigma,\sigma 00}X_{j}^{0\sigma
0,0\sigma\sigma}+\tilde{t}_{ij}X_{i}^{0\sigma\sigma,0\sigma 0}X_{j}^{\sigma
00,\sigma 0\sigma}),$ $\displaystyle H_{h}^{(\gamma)}$
$\displaystyle=-{\sum\limits_{ij\sigma}}\bar{t}_{ij}(X_{i}^{00\sigma,000}X_{j}^{\sigma
00,\sigma 0\sigma}+X_{i}^{\sigma 0\sigma,\sigma
00}X_{j}^{000,00\sigma}+X_{i}^{00\sigma,000}X_{j}^{0\sigma
0,0\sigma\sigma}+X_{i}^{0\sigma\sigma,0\sigma 0}X_{j}^{000,00\sigma}).$
Green functions technique allows us to calculate the energy spectrum of the
model which corresponds to the electronic subsystem of AxC60 in the case of
electron concentration $n=1$. One can rewrite the single-particle Green
function $\langle\langle
a_{i\lambda\sigma}|a_{j\lambda\sigma}^{+}\rangle\rangle$ on the basis of
relation between electronic operators and Hubbard’s X-operators:
$\displaystyle a_{p\alpha\uparrow}$ $\displaystyle=$ $\displaystyle
X_{p}^{000,\uparrow 00}+X_{p}^{0\uparrow 0,\uparrow\uparrow
0}+X_{p}^{00\uparrow,\uparrow 0\uparrow}\equiv X_{p}^{000,\uparrow 00}+Y_{p},$
(9)
where the operator $Y_{p}$ describes the transition processes between doubly
occupied Hund’s state and single occupied state. The processes involving other
type of doubly occupied states, empty states, states with three or more
electrons is improbable due to energy scaling.
In this way we obtain the following expression for the single electron Green
function
$\displaystyle\langle\langle
a_{p\alpha\uparrow}|a_{p^{\prime}\alpha\uparrow}^{+}\rangle\rangle=\langle\langle
X_{p}^{000,\uparrow 00}|X_{p^{\prime}}^{\uparrow
00,000}\rangle\rangle++\langle\langle X_{p}^{000,\uparrow
00}|Y_{p^{\prime}}^{+}\rangle\rangle+\langle\langle
Y_{p}|X_{p^{\prime}}^{000,\uparrow 00}\rangle\rangle+\langle\langle
Y_{p}|Y_{p^{\prime}}^{+}\rangle\rangle.$ (10)
Equation of motion for Green function $\langle\langle X_{p}^{000,\uparrow
00}|X_{p^{\prime}}^{\uparrow 00,000}\rangle\rangle$ has the form
$\displaystyle(E+\mu)\langle\langle X_{p}^{000,\uparrow
00}|X_{p^{\prime}}^{\uparrow 00,000}\rangle\rangle$
$\displaystyle=\delta_{pp^{\prime}}\frac{X_{p}^{000}+X_{p}^{\uparrow
00}}{2\pi}+\langle\langle[X_{p}^{000,\uparrow
00};\sum_{\lambda}{H_{b}^{(\lambda)}}]|X_{p^{\prime}}^{\uparrow
00,000}\rangle\rangle$ (11) $\displaystyle+\langle\langle[X_{p}^{000,\uparrow
00};\sum_{\lambda}{H_{h}^{(\lambda)}}]|X_{p^{\prime}}^{\uparrow
00,000}\rangle\rangle$
and equation of motion for Green function $\langle\langle
Y_{p}|X_{p^{\prime}}^{000,\uparrow 00}\rangle\rangle$ -
$\displaystyle(E+\mu-U+3J_{H})\langle\langle
Y_{p}|X_{p^{\prime}}^{000,\uparrow
00}\rangle\rangle=\langle\langle[Y_{p};\sum_{\lambda}{H_{b}^{(\lambda)}}]|X_{p^{\prime}}^{\uparrow
00,000}\rangle\rangle+\langle\langle
Y_{p};\sum_{\lambda}{H_{h}^{(\lambda)}}]|X_{p^{\prime}}^{\uparrow
00,000}\rangle\rangle.$
To obtain closed system of equations for Green functions $\langle\langle
X_{p}^{000,\uparrow 00}|X_{p^{\prime}}^{\uparrow 00,000}\rangle\rangle$ and
$\langle\langle Y_{p}|X_{p^{\prime}}^{\uparrow 00,000}\rangle\rangle$ we use
the projection procedure similar to the work did97 :
$\displaystyle[X_{p}^{000,\uparrow 00};\sum_{\lambda}{H_{b}^{(\lambda)}}]$
$\displaystyle=$ $\displaystyle\sum_{i}\varepsilon_{pi}^{b}X_{i}^{000,\uparrow
00};$ (12) $\displaystyle[X_{p}^{000,\uparrow
00};\sum_{\lambda}{H_{h}^{(\lambda)}}]$ $\displaystyle=$
$\displaystyle\sum_{i}\varepsilon_{pi}^{h}Y_{i};$
$\displaystyle[Y_{p};\sum_{\lambda}{H_{b}^{(\lambda)}}]$ $\displaystyle=$
$\displaystyle\sum_{i}\tilde{\varepsilon}_{pi}^{b}Y_{i};$
$\displaystyle[Y_{p};\sum_{\lambda}{H_{h}^{(\lambda)}}]$ $\displaystyle=$
$\displaystyle\sum_{i}\tilde{\varepsilon}_{pi}^{h}X_{i}^{000,\uparrow 00}.$
As a result after Fourier transformation we obtain the Green function in the
form:
$\displaystyle\langle\langle X_{i}^{000,\uparrow 00}|X_{j}^{\uparrow
00,000}\rangle\rangle_{\bf k}=\frac{X^{000}+X^{\uparrow
00}}{2\pi}\times\frac{E+\mu-U+3J_{H}-\tilde{\varepsilon}^{b}({\bf
k})}{(E-E_{1}({\bf k}))(E-E_{2}({\bf k}))},$ (13)
where the quasi-particle energy spectrum
$\displaystyle E_{1,2}({\bf
k})=-\mu+\frac{U-3J_{H}}{2}+\frac{\varepsilon^{b}({\bf
k})+\tilde{\varepsilon}^{b}({\bf
k})}{2}\mp\frac{1}{2}\sqrt{(U-3J_{H}-\varepsilon^{b}({\bf
k})+\tilde{\varepsilon}^{b}({\bf k}))^{2}+4\varepsilon^{h}({\bf
k})\tilde{\varepsilon}^{h}({\bf k})}.$ (14)
In the absence of orbital order the energy spectrum for $\beta$ and $\gamma$
electrons is the same as for $\alpha$ electrons.
The non-operator coefficients $\varepsilon^{b}({\bf
k}),\tilde{\varepsilon}^{b}({\bf k}),\varepsilon^{h}({\bf
k}),\tilde{\varepsilon}^{h}({\bf k})$ one can obtain by the anticommutation of
Eq.(12) with basis operators $X_{i}^{000,\uparrow 00}$ and $Y_{i}^{+}$ and
following replacement of operators by $c$-numbers (see in this connection
d_act00 ).
$\displaystyle\varepsilon_{\bf k}^{b}=\frac{1}{C_{1}}[t_{\bf k}(\langle
X_{p}^{000}(X_{p^{\prime}}^{000}+X_{p^{\prime}}^{\uparrow 00})\rangle++\langle
X_{p}^{\uparrow 00}(X_{p^{\prime}}^{000}+X_{p^{\prime}}^{\uparrow
00})\rangle+\langle X_{p}^{\downarrow 00,\uparrow 00}X_{p^{\prime}}^{\uparrow
00,\downarrow 00}\rangle+\langle X_{p}^{0\uparrow 0,\uparrow
00}X_{p^{\prime}}^{\uparrow 00,0\uparrow 0}\rangle+$ $\displaystyle+\langle
X_{p}^{0\downarrow 0,\uparrow 00}X_{p^{\prime}}^{\uparrow 00,0\downarrow
0}\rangle++\langle X_{p}^{00\uparrow,\uparrow 00}X_{p^{\prime}}^{\uparrow
00,00\uparrow}\rangle+\langle X_{p}^{00\downarrow,\uparrow
00}X_{p^{\prime}}^{\uparrow 00,00\downarrow}\rangle)--\tilde{t}_{\bf
k}(\langle X_{p}^{\uparrow\uparrow 0,000}X_{p^{\prime}}^{000,\uparrow\uparrow
0}\rangle+\langle X_{p}^{\uparrow 0\uparrow,000}X_{p^{\prime}}^{000,\uparrow
0\uparrow}\rangle)],$ $\displaystyle\varepsilon_{\bf
k}^{h}=\frac{1}{C_{2}}\bar{t}_{\bf k}[\langle(X_{p}^{000}+X_{p}^{\uparrow
00})\times(X_{p^{\prime}}^{0\uparrow
0}+X_{p^{\prime}}^{00\uparrow}+X_{p^{\prime}}^{\uparrow\uparrow
0}+X_{p^{\prime}}^{\uparrow 0\uparrow})\rangle+\langle X_{p}^{0\uparrow
0,\uparrow 00}X_{p^{\prime}}^{\uparrow
0\uparrow,0\uparrow\uparrow}\rangle+\langle X_{p^{\prime}}^{\uparrow\uparrow
0,000}X_{p}^{000,\uparrow\uparrow 0}\rangle-$ $\displaystyle-\langle
X_{p^{\prime}}^{0\uparrow 0,\uparrow 00}X_{p}^{\uparrow 00,0\uparrow
0}\rangle-\langle X_{p}^{00\uparrow,\uparrow 00}X_{p^{\prime}}^{\uparrow
00,00\uparrow}\rangle+\langle X_{p^{\prime}}^{\uparrow
0\uparrow,000}X_{p}^{000,\uparrow 0\uparrow}\rangle-\langle
X_{p}^{00\uparrow,\uparrow 00}X_{p^{\prime}}^{\uparrow\uparrow
0,0\uparrow\uparrow}\rangle],$ $\displaystyle\tilde{\varepsilon}_{\bf
k}^{b}=-\frac{t_{\bf k}}{C_{2}}[\langle X_{p^{\prime}}^{\uparrow\uparrow
0,000}X_{p}^{000,\uparrow\uparrow 0}\rangle+\langle X_{p^{\prime}}^{\uparrow
0\uparrow,000}X_{p}^{000,\uparrow 0\uparrow}\rangle]+$
$\displaystyle+\frac{\tilde{t}_{\bf k}}{C_{2}}[\langle X_{p}^{0\uparrow
0}+X_{p}^{\uparrow\uparrow 0}+X_{p}^{00\uparrow}+X_{p}^{\uparrow
0\uparrow}\rangle\times\langle X_{p^{\prime}}^{0\uparrow
0}+X_{p^{\prime}}^{\uparrow\uparrow
0}+X_{p^{\prime}}^{00\uparrow}+X_{p^{\prime}}^{\uparrow
0\uparrow}\rangle+\langle X_{p}^{0\uparrow 0,\uparrow
00}X_{p^{\prime}}^{\uparrow 00,0\uparrow 0}\rangle+\langle
X_{p}^{0\uparrow\uparrow,\uparrow 0\uparrow}X_{p}^{\uparrow
0\uparrow,0\uparrow\uparrow}\rangle-$ $\displaystyle-\langle X_{p}^{0\uparrow
0,\uparrow 00}X_{p^{\prime}}^{\uparrow
0\uparrow,0\uparrow\uparrow}\rangle-\langle X_{p}^{0\uparrow\uparrow,\uparrow
0\uparrow}X_{p^{\prime}}^{\uparrow 00,0\uparrow 0}\rangle+\langle
X_{p}^{00\uparrow,\uparrow 00}X_{p^{\prime}}^{\uparrow
00,00\uparrow}\rangle+\langle X_{p}^{0\uparrow\uparrow,\uparrow\uparrow
0}X_{p^{\prime}}^{\uparrow\uparrow 0,0\uparrow\uparrow}\rangle+$
$\displaystyle+\langle X_{p}^{00\uparrow,\uparrow
00}X_{p^{\prime}}^{\uparrow\uparrow 0,0\uparrow\uparrow}\rangle+\langle
X_{p}^{0\uparrow\uparrow,\uparrow\uparrow 0}X_{p^{\prime}}^{\uparrow
00,00\uparrow}\rangle],$ $\displaystyle\tilde{\varepsilon}_{\bf
k}^{h}=-\frac{\bar{t}_{\bf
k}}{C_{1}}[\langle(X_{p^{\prime}}^{000}+X_{p^{\prime}}^{\uparrow
00})\times(X_{p}^{\uparrow\uparrow 0}+X_{p}^{\uparrow
0\uparrow}+X_{p}^{0\uparrow 0}+X_{p}^{00\uparrow})\rangle+\langle
X_{p}^{0\uparrow\uparrow,\uparrow 0\uparrow}X_{p^{\prime}}^{\uparrow
00,0\uparrow 0}\rangle-\langle X_{p}^{0\uparrow 0,\uparrow
00}X_{p^{\prime}}^{\uparrow 00,0\uparrow 0}\rangle+$ $\displaystyle+\langle
X_{p^{\prime}}^{\uparrow\uparrow 0,000}X_{p}^{000,\uparrow\uparrow
0}\rangle++\langle X_{p}^{00\uparrow,\uparrow 00}X_{p^{\prime}}^{\uparrow
00,00\uparrow}\rangle-\langle X_{p^{\prime}}^{\uparrow
0\uparrow,000}X_{p}^{000,\uparrow 0\uparrow}\rangle\langle
X_{p}^{0\uparrow\uparrow,\uparrow\uparrow 0}X_{p^{\prime}}^{\uparrow
00,00\uparrow}\rangle],$
where $C_{1}=\langle X_{p}^{000}+X_{p}^{\uparrow 00}\rangle$, $C_{2}=\langle
X_{p}^{0\uparrow 0}+X_{p}^{00\uparrow}+X_{p}^{\uparrow\uparrow
0}+X_{p}^{\uparrow 0\uparrow}\rangle$. It is worth to note that in the partial
case of band filling $n=1$ and strong Coulomb correlation we work with reduced
Hilbert space of electronic states, so $C_{1}+C_{2}=1$.
Let us denote the concentration of empty lattice sites by $e$, concentration
of singly occupied sites with spin $\sigma$ electron in orbital state
$\lambda$ by $s_{\lambda\sigma}$, Hund’s doublons concentration by
$d_{\sigma}$, Hubbard doublons by $d_{2}$ and non-Hund doublons by
$\tilde{d}$. In a paramagnetic state $s_{\lambda\sigma}=s$, $d_{\sigma}=d$.
For the case of strong Hund’s coupling the high energy doublon configurations
are excluded, $d_{2}=\tilde{d}=0$. We can utilize the completeness condition
for the $X$-operator set to have constraint $e+6s+6d=1$, which, at condition
$e=6d$, leads to the equation
$\displaystyle s=\frac{1-12d}{6}.$ (15)
Finally in the paramagnetic case at $n=1$ we obtain
$\displaystyle\varepsilon^{b}=\frac{216d^{2}-12d+1}{24d+1}t_{\bf
k}+\frac{72d^{2}}{24d+1}\tilde{t}_{\bf k};$ (16)
$\displaystyle\varepsilon^{h}=\bar{t}_{\bf k}\frac{7d-12d^{2}}{1-6d},$ (17)
$\displaystyle\tilde{\varepsilon}^{b}=t_{\bf
k}\frac{36d^{2}}{1-6d}+\frac{\tilde{t}_{\bf k}}{2(1-6d)},$ (18)
$\displaystyle\tilde{\varepsilon}^{h}=t_{\bf
k}\frac{24d+1-216d^{2}}{3(24d+1)},$ (19)
In this way, the energy spectrum depends on the concentration of doublons $d$
(through the dependence of non-operator coefficients). The doublon
concentration is determined by the condition
$\displaystyle 6d={1\over 2N}\sum_{\bf k}{\left(\frac{A_{e}(\bf
k)}{\exp(\frac{E_{1}(\bf k)}{kT}+1)}+\frac{B_{e}(\bf k)}{\exp(\frac{E_{2}(\bf
k)}{kT}+1)}\right)},$ (20)
where
$\displaystyle A_{e}({\bf k})={1\over
2}\left(1+\frac{U-3J_{H}+\tilde{\varepsilon}^{b}-\varepsilon^{b}}{\sqrt{(U-3J_{H}-\varepsilon^{b}+\tilde{\varepsilon}^{b})^{2}+4\varepsilon^{h}\tilde{\varepsilon}^{h}}}\right),$
$\displaystyle B_{e}({\bf k})=1-A_{e}({\bf k}).$ (21)
Using the model rectangular density of states at zero temperature one obtains
$\displaystyle 6d={1\over
4w}\int^{w}_{-w}{\frac{A_{e}(\varepsilon)\Theta(-E_{1}(\varepsilon))}{E-E_{1}(\varepsilon)}d\varepsilon}{{1\over
4w}\int^{w}_{-w}\frac{B_{e}(\varepsilon)\Theta(-E_{2}(\varepsilon))}{E-E_{2}(\varepsilon)}d\varepsilon},$
(22)
here $\Theta(-E(\varepsilon))$ is Heaviside theta-function. Solving this
equation numerically we obtain the doublon concentration as function of the
model parameters. To study a metal-insulator transition (MIT) mott90 ; edv95 ;
geb97 we apply the gap criterion
$\displaystyle\Delta E=E_{2}(-w)-E_{1}(w)=0.$ (23)
In the point of MIT the polar states (holes and doublons) concentrations
equals zero. Thus, for the non-operators coefficients we have
$\varepsilon^{b}=t_{\bf k}$, $\varepsilon^{h}=0$,
$\tilde{\varepsilon}^{b}=\frac{\tilde{t}_{\bf k}}{2}$,
$\tilde{\varepsilon}^{h}=\frac{\tilde{t}_{\bf k}}{3}$, and for the energy gap
we have the equation
$\displaystyle\Delta E=U-3J_{H}-\tilde{w}-w.$ (24)
Here $w=z|t|(1-\tau_{1})$ and $\tilde{w}=z|t|(1-\tau_{1})(1-2\tau)$ are the
halfbandwidths of the lower and upper subbands, respectively, $z$ is the
number of nearest neighbours to a site, $|t|$ is the magnitude of bare
nearest-neighbour hopping integral,
$\tau_{1},\tau=\frac{t^{\prime}_{ij}}{|t_{ij}|}$ are the correlated hopping
parameters. From the equation (24) one obtains that the critical value of the
intra-cite Coulomb interaction parameter equals the sum of quasiparticle
subbands halfbandwiths.
Analysis of the expression (24) allows explaining the differences of
electrical characteristics (insulator or metallic state realisation) depending
on the correlated hopping strength.
The correlated hopping influence substantially on electrical characteristics
of narrow band material with three-fold orbital degeneracy of the energy
levels. Both the filling of the sites involved into the hopping processes
(through the correlated hopping of the first type) and the neighbor sites
(through the second type correlated hopping), can lead to appearance of the
gap in energy spectrum and stabilization of the insulator state. The energy
gap, however, opens at relatively large increase of correlated hopping
parameters which can not be achieved in a compound by change of external
conditions only. Such critical increase of parameters $\tau_{1}$ and $\tau$
can be realized at doping. A distinct picture is observed at the change of
intra-site Coulomb interaction parameter. At increase of $(U-3J_{H})/w$ over a
critical value (dependent on the correlated hopping strength) the energy gap
occurs and the metal-insulator transition takes place. The critical value for
the partial case of the model when the quasiparticle subbands have the same
widths (in absence of the correlated hopping), is $(U-3J_{H})/w=2$ which
corresponds to the result of works did97 ; did98 for non-degenerated Hubbard
model.
## III Conclusions
Within the variant of triple orbitally degenerate model of the electronic
subsystem of a doped fulleride compound considered above not only the on-site
Coulomb correlations but also additional interactions of principal importance,
namely the correlated hopping, can be introduced and analyzed. The use of
Hubbard X-operators representation appears to be useful to exclude from
consideration the parts of Hilbert space which are irrelevant at particular
band filling. The ground state metal-insulator transition in the triply
degenerate model of partially-filled doped fulleride band takes place at
moderate values of the correlation parameter which in this case is a
combination of on-site Coulomb repulsion energy, Hund’s rule coupling and
electron hopping parameters. The correlated hopping of electrons leads to
further localization of current carriers. The influence of the correlated
hopping is substantial and makes the estimation of the model parameters from
the available spectroscopic data ambiguous. The problem can be resolved by the
additional spectroscopic experiments with use of external pressure. In this
case the reasonable estimates could be obtained using the fact that in
distinction from the on-site parameters, the correlated hopping parameters
must be pressure-dependent. The metal-insulator transition described above can
be realized deg98 ; sac01 in the doped fulleride compound under the external
pressure.
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|
arxiv-papers
| 2012-01-13T07:03:39 |
2024-09-04T02:49:26.297561
|
{
"license": "Public Domain",
"authors": "Yuriy Dovhopyaty, Leonid Didukh, Oleksandr Kramar, Yuriy Skorenkyy,\n Yuriy Drohobitskyy",
"submitter": "Yuriy Skorenkyy",
"url": "https://arxiv.org/abs/1201.2748"
}
|
1201.2925
|
arxiv-papers
| 2012-01-13T19:54:27 |
2024-09-04T02:49:26.310819
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Geetha Manjunatha, M Narasimha Murty, Dinkar Sitaram",
"submitter": "Geetha Manjunath",
"url": "https://arxiv.org/abs/1201.2925"
}
|
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1201.2980
|
# Information algebra system of soft sets
Guan Xuechong guanxc@foxmail.com Li Yongming liyongm@snnu.edu.cn College of
Mathematical Science, Xuzhou Normal University, Xuzhou, 221116, China College
of Computer Science, Shaanxi Normal University, Xi’an, 710062, China
###### Abstract
Information algebra is algebraic structure for local computation and
inference. Given an initial universe set and a parameter set, we show that a
soft set system over them is an information algebra. Moreover, in a soft set
system, the family of all soft sets with a finite parameter subset can form a
compact information algebra.
###### keywords:
soft set , complete lattice , information algebra , compact information
algebra.
††journal:
## 1 Introduction
The information algebra system introduced by Shenoy [1] was inspired by the
formulation of some basic axioms of local computation and inference under
uncertainty [2]. It gives a basic mathematical model for treating
uncertainties in information. Related studies [3, 4, 5] showed that the
framework of information algebras covers many instances from constraint
systems, Bayesian networks, Dempster-Shafer belief functions to relational
algebra, logic and etc. Considering about the feasibility of information
processing with computer, Kohlas [3, 4] presented a special information
algebra with approximation structure called compact information algebra
recently.
On the other hand, Molodtsov [6] initiated a novel concept, which is called
soft set, as a new mathematical tool for dealing with uncertainties [7]. In
fact, a soft set is a parameterized family of subsets of a given universe set.
The way of parameterization in problem solving makes soft set theory
convenient and simple for application. Now it has been applied in several
directions, such as operations research [8, 9], topology [10, 11, 12],
universal algebra [13, 14, 15, 16], especially decision-making [17, 18, 19,
20, 21].
It is thus evident that information algebra theory and soft set theory are
both theoretical research tools for dealing with non-deterministic phenomenon.
To study relationships between them is necessary. In this paper, we are
concerned about the problem that whether there exist the frameworks of
information algebras or even compact information algebras in soft sets. By
choosing some appropriate operators, we construct an information algebra
system of soft sets over an initial universe set and a parameter set. Then we
further prove that, in a soft set system, the family of soft sets with a
finite parameter subset can form a compact information algebra. These
conclusions obtained in this paper demonstrate that soft set systems are also
the instances of information algebras.
## 2 Preliminaries
In this section, first, we present some basic definitions about soft sets and
some notations in lattice theory.
Suppose that $(L,\leq)$ is a partially ordered set and $A\subseteq L$. We
write $\vee A$ and $\wedge A$ for the least upper bound and the greatest lower
bound of $A$ in $L$ respectively if they exist.
Let $L$ be a partially ordered set. If $a\vee b$ and $a\wedge b$ exist for all
$a,b\in L$, then we call $L$ a lattice. If $\vee A$ exists for every subset
$A\subseteq L$, we call $L$ a complete lattice. Clearly, a partially ordered
set $L$ is a complete lattice if, and only if, $L$ has the bottom element and
$\vee A$ exists for all nonempty subset $A\subseteq L$.
A set $A\subseteq L$ is said to be directed, if for all $a,b\in A$, there is a
$c\in A$ such that $a,b\leq c$. For $a,b\in L$, we call $a$ way-below $b$, in
symbols $a\ll b$, if and only if for all directed subsets $X\subseteq L$, if
$\vee X$exists and $b\leq\vee X$, then there exists an $x\in X$ such that
$a\leq x$.
Let $U$ be an initial universe set and $E$ be a set of parameters, which
usually are initial attributes, characteristics, or properties of objects in
the initial universe set. ${\cal P}(U)$ denotes the power set of $U$.
###### Definition 2.1
([6]) A pair $(F,A)$ is called a soft set over $U$, where $F$ is a mapping
given by $F:A\rightarrow{\cal P}(U)$.
Therefore a soft set is a tuple which associates with a set of parameters and
a mapping from the parameter set into the power set of an universe set. In
other words, a soft set over $U$ is a parameterized family of subsets of the
universe $U$. For $\varepsilon\in A$, $F(\varepsilon)$ may be considered as
the set of $\varepsilon$-approximate elements of the soft set $(F,A)$[6].
###### Definition 2.2
([7]) A soft set $(F,A)$ over $U$ is said to be a null soft set, if for all
$e\in A,F(e)=\emptyset$. We write it by $(\emptyset,A)$.
A soft set $(F,A)$ over $U$ is said to be an absolute soft set denoted by
$\tilde{A}$, if for all $e\in A,F(e)=U$.
###### Definition 2.3
([8]) The extended intersection of two soft sets $(F,A)$ and $(G,B)$ over a
common universe $U$ is the soft set $(H,C)$, where $C=A\cup B$, and $\forall
e\in C$,
$H(e)=\left\\{\begin{array}[]{ll}F(e),&{\mbox{if}}\ \ e\in A-B;\\\
G(e),&{\mbox{if}}\ \ e\in B-A;\\\ F(e)\cap G(e),&{\mbox{if}}\ \ e\in A\cap
B.\end{array}\right.$
We write $(F,A)\sqcap_{\varepsilon}(G,B)=(H,C)$.
In this paper we adopt the concept of information algebra given by Kohlas from
[3]. For a full introduction and these abundant examples of information
algebras, please refer to [3, 4, 5].
###### Definition 2.4
([3]) Let $(D,\leq)$ be a lattice. Suppose there are three operations defined
in the tuple $(\Phi,D)$:
1.Labeling $d$: $\Phi\rightarrow D;\phi\mapsto d(\phi)$, where $d(\phi)$ is
called the domain of $\phi$. For an $s\in D$, let $\Phi_{s}$ denote the set of
all valuations with domain $s$.
2.Combination $\otimes$:
$\Phi\times\Phi\rightarrow\Phi;(\phi,\psi)\mapsto\phi\otimes\psi$,
3.Marginalization $\downarrow$: $\Phi\times
D\rightarrow\Phi;(\phi,x)\mapsto\phi^{\downarrow x}$, for $x\leq d(\phi)$.
If the system $(\Phi,D)$ satisfies the following axioms, it is called an
information algebra:
1.Semigroup: $\Phi$ is associative and commutative under combination. For all
$s\in D$, there is a neutral element $e_{s}$ with $d(e_{s})=s$ such that for
all $\phi\in\Phi$ with $d(\phi)=s,e_{s}\otimes\phi=\phi$.
2\. Labeling: For $\phi,\psi\in\Phi$, $d(\phi\otimes\psi)=d(\phi)\vee
d(\psi)$.
3\. Marginalization: For $\phi\in\Phi,x\in D,x\leq d(\phi),d(\phi^{\downarrow
x})=x$.
4\. Transitivity: For $\phi\in\Phi$ and $x\leq y\leq d(\phi),(\phi^{\downarrow
y})^{\downarrow x}=\phi^{\downarrow x}$
5\. Combination: For $\phi,\psi\in\Phi$ with
$d(\phi)=x,d(\psi)=y,(\phi\otimes\psi)^{\downarrow
x}=\phi\otimes\psi^{\downarrow x\wedge y}$.
6\. Stability: For $x,y\in D,x\leq y$, $e_{y}^{\downarrow x}=e_{x}$.
7\. Idempotency: For $\phi\in\Phi$ and $x\in D,x\leq d(\phi)$,
$\phi\otimes\phi^{\downarrow x}=\phi$.
The items putting forward in the definition of information algebra can be seen
as the axiomatic presentations of some basic principles in local computation
and inference. Studies have shown this algebraic structure covers many
instances from belief functions, constraint systems, relational databases, and
possibility theory to relational algebra and logic([5]). For example, each
lattice $L$ is a simply information algebra on a domain set $L$ itself. The
operations are defined as follows:
1\. Labeling $d$: For $x\in L$, $d(x)=x$.
2\. Combination $\otimes$: $x\otimes y=x\vee y$.
3\. Projection $\downarrow$: If $x\leq y$, $x^{\downarrow y}=x\wedge y$.
For an information algebra $(\Phi,D)$, we introduce a order relation as
follows:
$\psi\leq\phi$, if $\psi\otimes\phi=\phi$.
This order relation induced by the operation combination is a partial order on
the set $\Phi$, if $(\Phi,D)$ is an information algebra.
## 3 Information algebra of soft sets
In this section, with these operations of soft sets defined above, we will
construct an information algebra of soft sets. Let $U$ be an initial universe
set and $E$ be a set of parameters. ${\cal S}_{U,E}$(or simply ${\cal S}$ when
this doesn’t lead to confusions) denotes the set of all soft sets $(F,A)$ over
$U$, where $A\subseteq E$, that is,
${\cal S}$=$\\{(F,A):(F,A)\mbox{\ is\ a\ soft\ set\ over\ $U$,\ where
$A\subseteq E$}\\}.$
Three operations are defined as follows:
1\. Labeling $d$: For a soft set $(F,A)$, we define $d((F,A))=A$.
2\. Projection $\downarrow$: If $B\subseteq A$, we define $(F,A)^{\downarrow
B}$ to be a soft set $(G,B)$ such that for all $b\in B$, $G(b)=F(b)$.
3\. Combination $\otimes$: For any two soft sets $(F,A),(G,B)\in{\cal S}$, we
define
$(F,A)\otimes(G,B)=(F,A)\sqcap_{\varepsilon}(G,B).$
We call a quintuple $({\cal S},{\cal
P}(E),d,\sqcap_{\varepsilon},\downarrow)$(abbreviated as $({\cal S},{\cal
P}(E))$ a soft set system over $U$ and $E$. Now we show this system is an
information algebra.
###### Theorem 3.1
The soft set system $({\cal S},{\cal P}(E))$ over $U$ and $E$ is an
information algebra.
Proof. Obviously, ${\cal P}(E)$ is a lattice composed by the domains of soft
sets in ${\cal S}$.
1\. Semigroup: Clearly ${\cal S}$ is commutative with respect to the operation
$\sqcap_{\varepsilon}$. For $A\subseteq E$, the absolute soft set $\tilde{A}$
is the neutral element such that $(F,A)\sqcap_{\varepsilon}\tilde{A}=(F,A)$
for all soft set $(F,A)$ with domain $A$.
Following we show the associative law holds in the set ${\cal S}$. Let
$(F,A),(G,B),(H,C)\in{\cal S}$. We write
$(F,A)\sqcap_{\varepsilon}(G,B)=(Q_{1},A\cup B)$,
$(G,B)\sqcap_{\varepsilon}(H,C)=(Q_{2},B\cup C)$,
$[(F,A)\sqcap_{\varepsilon}(G,B)]\sqcap_{\varepsilon}(H,C)=(Q_{3},A\cup B\cup
C)$,
$(F,A)\sqcap_{\varepsilon}[(G,B)\sqcap_{\varepsilon}(H,C)]=(Q_{4},A\cup B\cup
C)$.
We need to show that $Q_{3}=Q_{4}$. For any an $e\in A\cup B\cup C$, it can be
divided into seven conditions as follows: $e\in(A-B)-C,e\in(B-A)-C,e\in(A\cap
B)-C,e\in C-(A\cup B),e\in(A-B)\cap C,e\in(B-A)\cap C$ and $e\in A\cap B\cap
C$. Here we take the condition of $e\in(A\cap B)-C$ as an example to
illuminate the proof. Assume that $e\in(A\cap B)-C$. Since $(A\cap
B)-C=A\cap(B-C)$, we have $e\in A\cap(B-C)$. Then
$Q_{3}(e)=Q_{1}(e)=F(e)\cap G(e),$
and
$Q_{4}(e)=F(e)\cap Q_{2}(e)=F(e)\cap G(e).$
So $Q_{3}(e)=Q_{4}(e)$. The other conditions are also easy to show. Therefore,
the associative law holds.
2\. According to these related definitions, the proof of the axioms of
labeling, marginalization, transitivity and idempotency are directly.
3\. Stability: For $A\in{\cal P}(E)$, the neutral element with domain $A$ is
the absolute soft set $\tilde{A}$. Furthermore, if $B\subseteq A$, we have
$\tilde{A}^{\downarrow B}=\tilde{B}$. Thus the stability is true.
4\. Combination: For $(F,A),(G,B)\in{\cal S}$, we need to show
$((F,A)\sqcap_{\varepsilon}(G,B))^{\downarrow
S}=(F,A)\sqcap_{\varepsilon}(G,B)^{\downarrow{S\cap B}},$
if $A\subseteq S\subseteq A\cup B$.
In fact, let
$(F,A)\sqcap_{\varepsilon}(G,B)=(H,A\cup B),$
$(F,A)\sqcap_{\varepsilon}(G,B)^{\downarrow{S\cap B}}=(H^{{}^{\prime}},S).$
For all $e\in S$, we have
$H(e)=H^{{}^{\prime}}(e)=\left\\{\begin{array}[]{ll}F(e),&{\mbox{if}}\ \ e\in
S\cap(A-B);\\\ G(e),&{\mbox{if}}\ \ e\in S\cap(B-A);\\\ F(e)\cap
G(e),&{\mbox{if}}\ \ e\in A\cap B.\end{array}\right.$
Then $((F,A)\sqcap_{\varepsilon}(G,B))^{\downarrow
S}=(F,A)\sqcap_{\varepsilon}(G,B)^{\downarrow{S\cap B}}$.
Hence $({\cal S},{\cal P}(E),d,\sqcap_{\varepsilon},\downarrow)$ is an
information algebra.
## 4 Compact information algebra of soft sets
In general, only “finite” information can be treated in computers. Therefore,
a structure called compact information algebra has been proposed by Kohlas.
Its main character is that each information can be approximated by these
“finite” information with a same domain.
###### Definition 4.1
([4]) A system $(\Phi,\Phi_{f},D)$, where $(\Phi,D)$ is an information
algebra, the lattice $D$ has a top element,
$\Phi_{f}=\mathop{\bigcup}\limits_{x\in D}\Phi_{f,x}$
where the sets $\Phi_{f,x}\subseteq\Phi_{x}$ are closed under combination,
contain the neutral element $e_{x}\in\Phi_{f,x}$, and satisfy the following
axioms of convergence and density with respect to the ordering relation $\leq$
induced by the operation combination, is called a compact information algebra.
1\. Convergency: If $X\subseteq\Phi_{f,x}$ is directed, then the supremum
$\vee X$ over $\Phi$ exists and $\vee X\in\Phi_{x}$.
2\. Density: For all $\phi\in\Phi_{x}$,
$\phi=\bigvee\\{\psi\in\Phi_{f,x}:\psi\leq\phi\\}.$
3\. Compactness: If $X\subseteq\Phi_{f,x}$ is a directed set, and
$\phi\in\Phi_{f,x}$ such that $\phi\leq\vee X$ then there exists a $\psi\in X$
such that $\phi\leq\psi$.
###### Lemma 4.1
([4]) If $(\Phi,\Phi_{f},D)$ is a compact information algebra, then
$\phi\in\Phi_{f,x}$ if, and only if $\phi\ll\phi$ in set $\Phi_{x}$.
For convenience, we give an equivalent form for the order relation $\leq$
induced by the operation combination of soft sets.
###### Proposition 4.1
Let the order relation $\leq$ be induced by the operation combination in the
system $({\cal S},{\cal P}(E))$. For two soft sets $(F,A)$ and $(G,B)$ over a
common universe $U$, then $(F,A)\leq(G,B)$ if and only if,
(i) $A\subseteq B$, and
(ii) $\forall\varepsilon\in A$, $G(\varepsilon)\subseteq F(\varepsilon)$.
Proof. We write $(F,A)\sqcap_{\varepsilon}(G,B)=(H,A\cup B)$.
If $(F,A)\leq(G,B)$, then $(H,A\cup B)=(G,B)$. So $A\cup B=B$, that is,
$A\subseteq B$. For all $\varepsilon\in A$, by the definition of the operation
$\sqcap_{\varepsilon}$, we have $H(\varepsilon)=F(\varepsilon)\cap
G(\varepsilon)=G(\varepsilon)$. Then $G(\varepsilon)\subseteq F(\varepsilon)$
for all $\varepsilon\in A$.
The reverse is also obvious.
###### Proposition 4.2
Let $\\{(F_{i},A_{i}):i\in I\\}$ be soft sets over a same universe $U$. Then
$\mathop{\bigvee}\limits_{i\in
I}(F_{i},A_{i})=(H,\mathop{\bigcup}\limits_{i\in I}A_{i}),$
where $H:\mathop{\bigcup}\limits_{i\in I}A_{i}\rightarrow{\cal P}(U)$ is
defined as follows:
$\forall e\in\mathop{\bigcup}\limits_{i\in I}A_{i}$, let $J^{(e)}=\\{i\in
I:e\in A_{i}\\}$, $H(e)=\mathop{\bigcap}\limits_{i\in J^{(e)}}F_{i}(e)$.
Proof. Clearly $(H,\mathop{\bigcup}\limits_{i\in I}A_{i})$ is an upper bound
of $\\{(F_{i},A_{i}):i\in I\\}$. Suppose that $(G,B)$ is another upper bound
of $\\{(F_{i},A_{i}):i\in I\\}$. Thus $\mathop{\bigcup}\limits_{i\in
I}A_{i}\subseteq B$. $\forall e\in\mathop{\bigcup}\limits_{i\in I}A_{i},i\in
J^{(e)}$, we have $G(e)\subseteq F_{i}(e)$. Then
$G(e)\subseteq\mathop{\bigcap}\limits_{i\in J^{(e)}}F_{i}(e)=H(e).$
This proves that $(H,\mathop{\bigcup}\limits_{i\in I}A_{i})\leq(G,B)$. Thus
$\mathop{\bigvee}\limits_{i\in
I}(F_{i},A_{i})=(H,\mathop{\bigcup}\limits_{i\in I}A_{i})$.
###### Proposition 4.3
$({\cal S}_{A},\leq)$ is a complete lattice. The top element is
$(\emptyset,A)$, and the bottom element is $\tilde{A}$. Here ${\cal S}_{A}$ is
the set of all soft sets with domain $A$ in the system $({\cal S},{\cal
P}(E))$.
Proof. For all nonempty subset $\\{(F_{i},A):i\in I\\}\subseteq{\cal S}_{A}$,
by the conclusion of Proposition 4.2, we have
$\mathop{\bigvee}\limits_{i\in I}(F_{i},A)=(H,A)\in{\cal S}_{A},$
where $H:A\rightarrow{\cal P}(U)$ is defined as
$H(e)=\mathop{\bigcap}\limits_{i\in I}F_{i}(e)$ for all $e\in A$. Moreover,
$\tilde{A}$ is the bottom element in the set ${\cal S}_{A}$. Thus ${\cal
S}_{A}$ is a complete lattice.
###### Lemma 4.2
Let $(F,A)$ be a soft set over a universe $U$ and $A$ be a finite subset of
$E$. Then $(F,A)\ll(F,A)$ in ${\cal S}_{A}$ if, and only if $U-F(e)$ is a
finite subset of $U$ for all $e\in A$.
Proof. (1) “if”: Let $\\{(G_{i},A):i\in I\\}$ be a directed set and
$(F,A)\leq\bigvee\limits_{i\in I}(G_{i},A)$. We write $\bigvee\limits_{i\in
I}(G_{i},A)=(G,A)$. $\forall e\in A$, we have $G(e)=\bigcap\limits_{i\in
I}G_{i}(e)\subseteq F(e)$. Then
$U-F(e)\subseteq U-G(e)=\bigcup\limits_{i\in I}(U-G_{i}(e)).$
For any an $x\in U-F(e)$, there is an $i(x)\in I$ such that $x\in
U-G_{i(x)}(e)$. Now we get a finite set $\\{(G_{i(x)},A):x\in U-F(e)\\}$,
because $U-F(e)$ is finite. By the directness of $\\{(G_{i},A):i\in I\\}$,
there exists an $i^{(e)}\in I$ such that $(G_{i(x)},A)\leq(G_{i^{(e)}},A)$ for
all $x\in U-F(e)$. Thus $x\in U-G_{i(x)}(e)\subseteq U-G_{i^{(e)}}(e)$. We
obtain $U-F(e)\subseteq U-G_{i^{(e)}}(e)$, that is, $G_{i^{(e)}}(e)\subseteq
F(e)$.
Since $A$ is a finite set, it implies that $\\{(G_{i^{(e)}},A):e\in A\\}$ is
also finite. By the directness of $\\{(G_{i},A):i\in I\\}$ again, there exists
a $j\in I$ such that $(G_{i^{(e)}},A)\leq(G_{j},A)$ for all $e\in A$. We have
$G_{j}(e)\subseteq G_{i^{(e)}}(e)\subseteq F(e)$ for all $e\in A$. This
implies $(F,A)\leq(G_{j},A)$. Thus $(F,A)\ll(F,A)$ in ${\cal S}_{A}$.
(2) “only if”: For all $e\in A$, $U-F(e)$ can be represented as the supremum
of $\\{B_{i}:i\in I\\}$, where $\\{B_{i}:i\in I\\}$ is a directed family of
all the finite subsets of $U-F(e)$, i.e., $U-F(e)=\bigcup\limits_{i\in
I}B_{i}$. We define a family of soft sets $(H_{i},A)$ as follows:
$H_{i}(\varepsilon)=\left\\{\begin{array}[]{ll}U-B_{i},&{\mbox{if}}\ \
\varepsilon=e;\\\ F(\varepsilon),&{\mbox{otherwise}}.\end{array}\right.$
With respect to the order relation $\leq$, $\\{(H_{i},A):i\in I\\}$ is a
directed subsets of ${\cal S}_{A}$. Also we have $(F,A)=\bigvee\limits_{i\in
I}(H_{i},A)$ by Proposition 4.2. Since $(F,A)\ll(F,A)$ in ${\cal S}_{A}$,
there exists a $k\in I$ such that $(F,A)\leq(H_{k},A)$. Hence $U-F(e)\subseteq
U-H_{k}(e)=B_{k}$. Thus $U-F(e)$ is a finite subset of $U$. This proves what
we have stated.
Let ${\cal S}_{\cal F}\subseteq{\cal S}$ denote the set of all soft sets with
a finite subset of $E$, i.e.,
${\cal S}_{\cal F}=\\{(F,A):(F,A)\mbox{\ is\ a\ soft\ set\ over\
$U$},\mbox{where\ $A$\ is\ a\ finite\ subset\ of\ $E$ }\\}.$
The symbol ${\cal P}_{f}(E)$ denotes the set of all finite subsets of $E$.
Let ${\cal S}_{f,A}=\\{(F,A):\forall e\in A,U-F(e)\mbox{\ is\ a\ finite\
subset\ of\ }U\\}$. We denote
${\cal S}_{f}=\mathop{\bigcup}\limits_{A\in{\cal P}_{f}(E)}{\cal S}_{f,A}.$
###### Theorem 4.1
$({\cal S}_{\cal F},{\cal S}_{f},{\cal P}_{f}(E))$ is a compact information
algebra.
Proof. First, we have $({\cal S}_{\cal F},{\cal P}_{f}(E))$ is an information
algebra. It is similar as the proof of Theorem 3.1. By Proposition 4.3, we
know $({\cal S}_{A},\leq)$ is a complete lattice for all finite subset $A$ of
$E$. Hence the convergency in Definition 4.1 is also true.
By the definition of way-below relation $\ll$ and the conclusion of Lemma 4.2,
the compactness is also clear.
Now we need to show the following equation holds for all finite subset
$A\subseteq E$,
$(F,A)=\vee\\{(G,A)\in{\cal S}_{f,A}:(G,A)\leq(F,A)\\}.$
For all $e\in A$, $U-F(e)$ can be represented as the supremum of
$\\{B_{i}:i\in I^{(e)}\\}$, i.e., $U-F(e)=\bigcup\limits_{i\in I^{(e)}}B_{i}$,
where $\\{B_{i}:i\in I^{(e)}\\}$ is a directed family of all the finite
subsets of $U-F(e)$. We define a family of soft sets $(F_{i},A)$ as follows:
$F_{i}(\varepsilon)=\left\\{\begin{array}[]{ll}U-B_{i},&{\mbox{if}}\ \
\varepsilon=e;\\\ U,&{\mbox{otherwise}}.\end{array}\right.$
Let ${\cal B}=\\{(F_{i},A):i\in I^{(e)},e\in A\\}$. Clearly ${\cal
B}\subseteq{\cal S}_{f,A}$. Meanwhile, for all $(F_{i},A)\in{\cal B}$, we have
$(F_{i},A)\leq(F,A)$. In fact, for all $e\in A$, if $i\in I^{(e)}$, we have
$F(e)=U-\bigcup\limits_{i\in I^{(e)}}B_{i}\subseteq U-B_{i}=F_{i}(e)$.
Otherwise, $F_{i}(e)=U$. Thus $F(e)\subseteq F_{i}(e)$ is true. So
$(F_{i},A)\leq(F,A)$. We write $\bigvee\limits_{(F_{i},A)\in{\cal
B}}(F_{i},A)=(H,A)$. For all $d\in A$, we have
$\displaystyle\begin{array}[]{lll}H(d)&=&\bigcap\limits_{(F_{i},A)\in{\cal
B}}F_{i}(d)\\\ &=&(\bigcap\limits_{i\in
I^{(d)}}F_{i}(d))\cap(\bigcap\limits_{i\in I^{(e)},e\in A,e\not=d}F_{i}(d))\\\
&=&(\bigcap\limits_{i\in I^{(d)}}F_{i}(d))\cap U\\\ &=&\bigcap\limits_{i\in
I^{(d)}}(U-B_{i})\\\ &=&U-\bigcup\limits_{i\in I^{(d)}}B_{i}\\\
&=&F(d).\end{array}$
So we have $F=H$. Therefore,
$\displaystyle\begin{array}[]{lll}(F,A)&=&(H,A)\\\
&=&\bigvee\limits_{(F_{i},A)\in{\cal B}}(F_{i},A)\\\
&\leq&\vee\\{(G,A)\in{\cal S}_{f,A}:(G,A)\leq(F,A)\\}\\\
&\leq&(F,A).\end{array}$
Hence
$(F,A)=\vee\\{(G,A)\in{\cal S}_{f,A}:(G,A)\leq(F,A)\\}.$
According to the proof above, we obtain that $({\cal S}_{\cal F},{\cal
S}_{f},{\cal P}_{f}(E))$ is a compact information algebra.
## 5 Conclusion
In this paper, by defining the operations combination and projection of soft
sets, we obtained the structure of information algebras on the family of all
soft sets over an initial universe set and a parameter set. Therefore, a soft
set system can be subsumed under the specific instances of information algebra
systems. We also gave a model of compact information algebra in a soft set
system. We have shown the family of all soft sets with a finite parameter
subset can form a compact information algebra.
## Acknowledgments
This work is supported by National Science Foundation of China (Grant
No.60873119) and the Higher School Doctoral Subject Foundation of Ministry of
Education of China under Grant 200807180005.
## References
* [1] P.P. Shenoy, A valuation-based language for expert systems, Int. J. Approx. Reason. 3(1989) 383–411.
* [2] J. Kohlas, P.P. Shenoy. Computation in valuation algebras: Algorithms for Uncertainty and Defeasible Reasoning, Handbook of Defeasible Reasoning and Uncertainty Managment Systems, 5, Kluwer Academic Publishers, Dordrecht, 2000.
* [3] J. Kohlas, Information Algebras: Generic Structures for Inference, Springer-Verlag, 2003.
* [4] J. Kohlas, Lecture Notes on The Algebraic Theory of Information, 2010. http://diuf.unifr.ch/drupal/tns/sites/diuf.unifr.ch.drupal.tns/files/file/kohlas/main.pdf.
* [5] J. Kohlas, N. Wilson, Semiring induced valuation algebras: Exact and approximate local computation algorithms, Artif. Intell. 172(2008) 1360-1399.
* [6] D. Molodtsov, Soft set theory-First results, Comput. Math. Appl. 37 (1999) 19-31.
* [7] P.K. Maji, R. Biswas, A.R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) 555-562.
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* [10] M. Shabir, M. Naz, On soft topological spaces, Comput. Math. Appl. 61(2011) 1786-1799.
* [11] B. Tanay, M. Burç Kandemi, Topological structure of fuzzy soft sets, Comput. Math. Appl. 61(2011) 2952-2957.
* [12] W. K. Min, A note on soft topological spaces, Comput. Math. Appl. 62(2011) 3524-3528.
* [13] H. Aktaş, N. Çağman, Soft sets and soft groups, Inform. Sci. 177 (2007) 2726-2735.
* [14] Y.B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl. 56 (2008) 1408-1413.
* [15] Y.B. Jun, C.H. Park, Applications of soft sets in ideal theory of BCK/BCI-algebras, Inform. Sci. 178 (2008) 2466-2475.
* [16] F. Feng, Y.B. Jun, X.Z. Zhao, Soft semirings, Comput. Math. Appl. 56 (2008) 2621-2628.
* [17] D. Chen, E.C.C. Tsang, D.S. Yeung, X. Wang, The parametrization reduction of soft sets and its applications, Comput. Math. Appl. 49 (2005) 757-763.
* [18] A.R. Roy, P.K. Maji, A fuzzy soft set theoretic approach to decision making problems, J. Comput. Appl. Math. 203 (2007) 412-418.
* [19] K. Gong, Z. Xiao, X. Zhang, The bijective soft set with its operations, J. Comput. Appl. Math. 60(2010) 2270-2278.
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|
arxiv-papers
| 2012-01-14T02:51:25 |
2024-09-04T02:49:26.315989
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xuechong Guan and Yongming Li",
"submitter": "Xuechong Guan",
"url": "https://arxiv.org/abs/1201.2980"
}
|
1201.2993
|
# Trudinger-Moser inequalities on the entire Heisenberg group
Yunyan Yang yunyanyang@ruc.edu.cn Department of Mathematics, Renmin
University of China, Beijing 100872, P. R. China
###### Abstract
Continuing our previous work (Cohn, Lam, Lu, Yang, Nonlinear Analysis (2011),
doi: 10.1016 /j.na.2011.09.053), we obtain a class of Trudinger-Moser
inequalities on the entire Heisenberg group, which indicate what the best
constants are. All the existing proofs of similar inequalities on unbounded
domain of the Euclidean space or the Heisenberg group are based on
rearrangement argument. In this note, we propose a new approach to solve this
problem. Specifically we get the global Trudinger-Moser inequality by gluing
local estimates with the help of cut-off functions. Our method still works for
similar problems when the Heisenberg group is replaced by the Eclidean space
or complete noncompact Riemannian manifolds.
###### keywords:
Trudinger-Moser inequality, singular Trudinger-Moser inequality, Adams
inequality
###### MSC:
46E35
††journal: ***
## 1 Introduction
Let $\mathbb{H}^{n}=\mathbb{R}^{2n}\times\mathbb{R}$ be the Heisenberg group
whose group action is defined by
$(x,y,t)\circ({x}^{\prime},{y}^{\prime},{t}^{\prime})=(x+{x}^{\prime},y+{y}^{\prime},t+{t}^{\prime}+2(\langle
y,x^{\prime}\rangle-\langle x,{y}^{\prime}\rangle)),$ (1.1)
where $x,y,x^{\prime},y^{\prime}\in\mathbb{R}^{n}$,
$t,t^{\prime}\in\mathbb{R}$, and $\langle\cdot,\cdot\rangle$ denotes the
standard inner product in $\mathbb{R}^{n}$. Let us denote the parabolic
dilation in $\mathbb{R}^{2n}\times\mathbb{R}$ by $\delta_{\lambda}$, namely,
$\delta_{\lambda}(\xi)=(\lambda x,\lambda y,\lambda^{2}t)$ for any
$\xi=(x,y,t)\in\mathbb{R}^{2n}\times\mathbb{R}$. The Jacobian determinant of
$\delta_{\lambda}$ is $\lambda^{Q}$, where $Q=2n+2$ is the homogeneous
dimension of $\mathbb{H}^{n}$. The following norm
$|\xi|_{h}=\left[\left(\sum_{i=1}^{n}(x_{i}^{2}+y_{i}^{2})\right)^{2}+t^{2}\right]^{\frac{1}{4}}$
(1.2)
is homogeneous of degree one with respect to the dilation $\delta_{\lambda}$.
The associated distance between two points $\xi$ and $\eta$ of
$\mathbb{H}^{n}$ is defined accordingly by
$d_{h}(\xi,\eta)=|\eta^{-1}\circ\xi|_{h},$ (1.3)
where $\eta^{-1}$ denotes the inverse of $\eta$ with respect to the group
action, i.e. $\eta^{-1}=-\eta$. Obviously $d_{h}(\cdot,\cdot)$ is symmetric.
The open ball of radius $r$ centered at $\xi$ is
$B_{h}(\xi,r)=\\{\eta\in\mathbb{H}^{n}:d_{h}(\eta,\xi)<r\\}.$
It is important to note that (see for example Stein [11], Section 5 of Chapter
VIII)
$|B_{h}(\xi,r)|=|B_{h}(0,r)|=|B_{h}(0,1)|r^{Q},$ (1.4)
where $|\cdot|$ denotes the Lebesgue measure. The Lie algebra of
$\mathbb{H}^{n}$ is generated by the left-invariant vector fields
$T=\frac{\partial}{\partial t},\,\,X_{i}=\frac{\partial}{\partial
x_{i}}+2y_{i}\frac{\partial}{\partial t},\,\,Y_{i}=\frac{\partial}{\partial
y_{i}}-2x_{i}\frac{\partial}{\partial t},\,i=1,\cdots,n.$ (1.5)
These generators satisfy the non-commutative formula
$[X_{i},Y_{i}]=-4\delta_{ij}T$. Denote by $|\nabla_{\mathbb{H}^{n}}u|$ the
norm of the sub-elliptic gradient of a smooth function
$u:\mathbb{H}^{n}\rightarrow\mathbb{R}$:
$|\nabla_{\mathbb{H}^{n}}u|=\left(\sum_{i=1}^{n}\left((X_{i}u)^{2}+(Y_{i}u)^{2}\right)\right)^{{1}/{2}}.$
Let $\Omega$ be an open set in $\mathbb{H}^{n}$. We use $W_{0}^{1,p}(\Omega)$
to denote the completion of $C_{0}^{\infty}(\Omega)$ under the norm
$\|u\|_{W_{0}^{1,p}(\Omega)}=\left(\int_{\Omega}\left(|\nabla_{\mathbb{H}^{n}}u|^{p}+|u|^{p}\right)d\xi\right)^{{1}/{p}}.$
(1.6)
In [4], Cohn-Lu proved a Trudinger-Moser inequality on bounded smooth domains
in the Hesenberg group $\mathbb{H}^{n}$. Precisely, there exists some constant
$C_{n}$ depending only on $n$ such that for all bounded smooth domain
$\Omega\subset\mathbb{H}^{n}$, if $u\in W_{0}^{1,Q}(\Omega)$ satisfies
$\|\nabla_{\mathbb{H}^{n}}u\|_{L^{Q}(\Omega)}\leq 1$, then
$\int_{\Omega}e^{\alpha_{Q}|u|^{Q^{\prime}}}d\xi\leq C_{n}|\Omega|,$ (1.7)
where $Q^{\prime}=Q/(Q-1)$, $\alpha_{Q}=Q\sigma_{Q}^{1/(Q-1)}$,
$\sigma_{Q}=\Gamma(\frac{1}{2})\Gamma(n+\frac{1}{2})\omega_{2n-1}/n!$,
$\omega_{2n-1}$ is the surface area of the unit sphere in $\mathbb{R}^{2n}$.
Furthermore, the integrals of all $u\in W_{0}^{1,Q}(\Omega)$ satisfying
$\|\nabla_{\mathbb{H}^{n}}u\|_{L^{Q}(\Omega)}\leq 1$ are not uniformly bounded
if $\alpha_{Q}$ is replaced by any larger number. Recently, Cohn, Lam, Lu and
the author [3] obtained a Trudinger-Moser inequality on the Heisenberg group
$\mathbb{H}^{n}$. Note that $W^{1,Q}(\mathbb{H}^{n})$ is the completion of
$C_{0}^{\infty}(\mathbb{H}^{n})$ under the norm (1.6) with $\Omega$ replaced
by $\mathbb{H}^{n}$. We have the following:
Theorem A ([3]). There exists some constant
$\alpha^{\ast}:0<\alpha^{\ast}\leq\alpha_{Q}$ such that for any pair $\beta$
and $\alpha$ satisfying $0\leq\beta<Q$, $0<\alpha\leq\alpha^{\ast}$, and
$\frac{\alpha}{\alpha^{\ast}}+\frac{\beta}{Q}\leq 1$, there holds
$\sup_{\|u\|_{W^{1,Q}(\mathbb{H}^{n})}\leq
1}\int_{\mathbb{H}^{n}}\frac{1}{|\xi|_{h}^{\beta}}\left\\{e^{\alpha|u|^{Q\,^{\prime}}}-\sum_{k=0}^{Q-2}\frac{\alpha^{k}|u|^{kQ\,^{\prime}}}{k!}\right\\}d\xi<\infty.$
(1.8)
When $\frac{\alpha}{\alpha^{\ast}}+\frac{\beta}{Q}>1$, the integral in
$(\ref{TMH})$ is still finite for any $u\in W^{1,Q}(\mathbb{H}^{n})$, but the
supremum is infinite if further $\frac{\alpha}{\alpha_{Q}}+\frac{\beta}{Q}>1$.
Theorem A is an analogue of (Adimurthi-Yang [1], Theorem 1.1). Earlier works
on this topic (Trudinger-Moser inequalities on unbounded domain of
$\mathbb{R}^{n}$) were done by Cao [2], Panda [9], do Ó [5], Ruf [10], Li-Ruf
[8] and others. The proof of Theorem A is based on symmetrization argument,
radial lemma and the Young inequality. Note that $\alpha^{*}$ in Theorem A is
not explicitly known. A natural question is what the best constant $\alpha$
for (1.8) is. Denote an equivalent norm in $W^{1,Q}(\mathbb{H}^{n})$ by
$\|u\|_{1,\tau}=\left(\int_{\mathbb{H}^{n}}(|\nabla_{\mathbb{H}^{n}}u|^{Q}+\tau|u|^{Q})d\xi\right)^{\frac{1}{Q}}$
(1.9)
for any fixed number $\tau>0$. Our main result is the following:
Theorem 1.1. Let $\tau$ be any positive real number. Let $Q$, $Q^{\prime}$ and
$\alpha_{Q}$ be as in (1.7). For any $\beta:0\leq\beta<Q$ and
$\alpha:0<\alpha<\alpha_{Q}(1-\beta/Q)$, there holds
$\sup_{\|u\|_{1,\tau}\leq
1}\int_{\mathbb{H}^{n}}\frac{1}{|\xi|_{h}^{\beta}}\left\\{e^{\alpha|u|^{Q\,^{\prime}}}-\sum_{k=0}^{Q-2}\frac{\alpha^{k}|u|^{kQ\,^{\prime}}}{k!}\right\\}d\xi<\infty.$
(1.10)
When $\alpha>\alpha_{Q}(1-\beta/Q)$, the above integral is still finite for
any $u\in W^{1,Q}(\mathbb{H}^{n})$, but the supremum is infinite.
Clearly Theorem 1.1 implies that the best constant for the inequality (1.10)
is $\alpha_{Q}(1-\beta/Q)$. But we do not know whether or not (1.10) still
holds when $\alpha=\alpha_{Q}(1-\beta/Q)$. Even so, $(\ref{Ttau})$ gives more
information than (1.8).
According to the author’s knowledge, the existing proofs of Trudinger-Moser
inequalities for unbounded domains are all based on the rearrangement theory
[6]. It is not known that whether or not this technique can be successfully
applied to the Heisenberg group case.
To prove Theorem 1.1, we propose a new approach. The idea can be described as
follows. Firstly, using (1.7), we derive a local Trudinger-Moser inequality,
namely, for any fixed $r>0$ and all $\xi_{0}\in\mathbb{H}^{n}$, there exists
some constant $C$ depending only on $n$, $r$ and $\beta$ such that
$\int_{B_{h}(\xi_{0},r)}\frac{1}{|\xi|_{h}^{\beta}}\left\\{e^{\alpha|u|^{Q\,^{\prime}}}-\sum_{k=0}^{Q-2}\frac{\alpha^{k}|u|^{kQ\,^{\prime}}}{k!}\right\\}d\xi\leq
C\int_{B_{h}(\xi_{0},r)}|\nabla_{\mathbb{H}^{n}}u|^{Q}d\xi$ (1.11)
provided that $0\leq\alpha<\alpha_{Q}(1-\beta/Q)$ and
$\int_{B_{h}(\xi_{0},r)}|\nabla_{\mathbb{H}^{n}}u|^{Q}d\xi\leq 1$. Secondly,
fixing sufficiently large $r>0$, we select a specific sequence of Heisenberg
balls $\\{B_{h}(\xi_{i},r)\\}_{i=1}^{\infty}$ to cover the Heisenberg group
$\mathbb{H}^{n}$. Then we choose appropriate cut-off function $\phi_{i}$ on
each $B_{h}(\xi_{i},r)$. Finally, we obtain (1.10) by gluing all local
estimates (1.11) for $\phi_{i}u$. We remark that our method still works for
similar problems when the Heisenberg group is replaced by the Eclidean space
or complete noncompact Riemannian manifolds. In the Eclidean space case,
$\tau$ can also be arbitrary in (1.10). But in the manifold case, the choice
of $\tau$ may depend on the geometric structure (see [13], Theorem 2.3). As an
easy consequence of Theorem 1.1 (in fact a special case $\beta=0$), the
following corollary holds.
Corollary 1.2. Let $Q=2n+2$. For any $q\geq Q$, $W^{1,Q}(\mathbb{H}^{n})$ is
continuously embedded in $L^{q}(\mathbb{H}^{n})$.
The remaining part of this note is organized as follows. In section 2, we
prove a covering lemma for $\mathbb{H}^{n}$; Cut-off functions are selected
for the subsequent analysis in section 3; The proof of Theorem 1.1 is
completed in section 4.
## 2 A covering lemma for the Heisenberg group
In this section, we will use a sequence of Heisenberg balls with the same
radius to cover the entire Heisenberg group $\mathbb{H}^{n}$. We require these
balls to satisfy the following properties: $(i)$ For any
$\xi\in\mathbb{H}^{n}$, $\xi$ belongs to at most $N$ balls for some constant
integer $N$ which is independent of the base point $\xi$; $(ii)$ If the radius
of those balls becomes appropriately smaller, then they are disjoint.
Firstly, we need to understand the Heisenberg distance between two points of
the Heisenberg group $\mathbb{H}^{n}$. The following two properties are more
or less standard. We prefer to present them by our own way.
Proposition 2.1. Let $\xi$ and $\eta$ be two points of $\mathbb{H}^{n}$. There
holds
$|\eta^{-1}\circ\xi|_{h}\leq 3(|\xi|_{h}+|\eta|_{h}),$
where $|\cdot|_{h}$ is the homogeneous norm defined by (1.2).
Proof. Write $\xi=(x,y,t)$, $\eta=(x^{\prime},y^{\prime},t^{\prime})$. Then
(1.1) gives
$\eta^{-1}\circ\xi=(x-x^{\prime},y-y^{\prime},t-t^{\prime}-2(\langle
y,x^{\prime}\rangle-\langle x,y^{\prime}\rangle)).$
Since
$(|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2})^{1/2}\leq(|x|^{2}+|y|^{2})^{1/2}+(|x^{\prime}|^{2}+|y^{\prime}|^{2})^{1/2}$
and
$\left|2\left(\langle y,x^{\prime}\rangle-\langle
x,y^{\prime}\rangle\right)\right|\leq|x|^{2}+|y|^{2}+|x^{\prime}|^{2}+|y^{\prime}|^{2},$
we have by using the inequality $\sqrt{a+b}\leq\sqrt{a}+\sqrt{b}$ ($a\geq 0$,
$b\geq 0$) repeatedly
$\displaystyle|\eta^{-1}\circ\xi|_{h}$ $\displaystyle=$
$\displaystyle\left[\left(\sum_{i=1}^{n}\left((x_{i}-x_{i}^{\prime})^{2}+(y_{i}-y_{i}^{\prime})^{2}\right)\right)^{2}+(t-t^{\prime}-2\left(\langle
y,x^{\prime}\rangle-\langle
x,y^{\prime}\rangle)\right)^{2}\right]^{\frac{1}{4}}$ $\displaystyle\leq$
$\displaystyle\left(\sum_{i=1}^{n}\left((x_{i}-x_{i}^{\prime})^{2}+(y_{i}-y_{i}^{\prime})^{2}\right)\right)^{\frac{1}{2}}+\left|t-t^{\prime}-2\left(\langle
y,x^{\prime}\rangle-\langle x,y^{\prime}\rangle\right)\right|^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
2\left(\sum_{i=1}^{n}\left(x_{i}^{2}+y_{i}^{2}\right)\right)^{\frac{1}{2}}+2\left(\sum_{i=1}^{n}\left({x_{i}^{\prime}}^{2}+{y_{i}^{\prime}}^{2}\right)\right)^{\frac{1}{2}}+|t|^{\frac{1}{2}}+|t^{\prime}|^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle 3(|\xi|_{h}+|\eta|_{h}).$
$\hfill\Box$
Proposition 2.2. Let $\xi$, $\eta$, $\zeta$ be arbitrary points of
$\mathbb{H}^{n}$. Then we have
$d_{h}(\xi,\eta)\leq 3\left(d_{h}(\xi,\zeta)+d_{h}(\zeta,\eta)\right),$
where $d_{h}(\cdot,\cdot)$ is the distance function defined by (1.3).
Proof. Note that $|\gamma^{-1}|_{h}=|\gamma|_{h}$ for all
$\gamma\in\mathbb{H}^{n}$. It follows from Proposition 2.1 that
$\displaystyle d_{h}(\xi,\eta)$ $\displaystyle=$
$\displaystyle|\eta^{-1}\circ\xi|_{h}$ $\displaystyle=$
$\displaystyle|\eta^{-1}\circ\zeta\circ\zeta^{-1}\circ\xi|_{h}$
$\displaystyle\leq$ $\displaystyle
3(|\eta^{-1}\circ\zeta|_{h}+|\zeta^{-1}\circ\xi|_{h})$ $\displaystyle=$
$\displaystyle 3\left(d_{h}(\zeta,\eta)+d_{h}(\xi,\zeta)\right).$
This gives the desired result. $\hfill\Box$
Secondly, by adapting an argument of (Hebey [7], Lemma 1.6), we obtain the
following useful covering lemma.
Lemma 2.3. Let $\rho>0$ be given. There exists a sequence $(\xi_{i})$ of
points of $\mathbb{H}^{n}$ such that for any $r\geq\rho$:
$(i)$ $\cup_{i}B_{h}(\xi_{i},\rho)=\mathbb{H}^{n}$ and for any $i\not=j$,
$B_{h}(\xi_{i},\rho/6)\cap B_{h}(\xi_{j},\rho/6)=\varnothing$;
$(ii)$ for any $\xi\in\mathbb{H}^{n}$, $\xi$ belongs to at most
$[(24r/\rho)^{Q}]$ balls $B_{h}(\xi_{i},r)$, where $[(24r/\rho)^{Q}]$ denotes
the integral part of $(24r/\rho)^{Q}$.
Proof. Firstly, we claim that there exists a sequence $(\xi_{i})$ of points of
$\mathbb{H}^{n}$ such that
$\cup_{i}B_{h}(\xi_{i},\rho)=\mathbb{H}^{n}\,\,{\rm and}\,\,\forall
i\not=j,B_{h}(\xi_{i},\rho/6)\cap B_{h}(\xi_{j},\rho/6)=\varnothing.$ (2.1)
To see this, we set
$X_{\rho}=\left\\{{\rm sequence}\,\,(\xi_{i})_{i\in
I}:\xi_{i}\in\mathbb{H}^{n},I\,\,{\rm is\,\,countable\,\,and}\,\,\forall
i\not=j,d_{h}(\xi_{i},\xi_{j})\geq\rho\right\\}.$
Then $X_{\rho}$ is partially ordered by inclusion and every element in
$X_{\rho}$ has an upper bound in the sense of inclusion. Hence, by Zorn’s
lemma, $X_{\rho}$ contains a maximal element $(\xi_{i})_{i\in I}$. On one
hand, if $\cup_{i}B_{h}(\xi_{i},\rho)\not=\mathbb{H}^{n}$, then there exists a
point $\xi\in\mathbb{H}^{n}$ such that $d_{h}(\xi_{i},\xi)\geq\rho$ for all
$i\in I$. This contradicts the maximality of $(\xi_{i})_{i\in I}$. Hence
$\cup_{i}B_{h}(\xi_{i},\rho)=\mathbb{H}^{n}$. On the other hand, if
$B_{h}(\xi_{i},\rho/6)\cap B_{h}(\xi_{j},\rho/6)\not=\varnothing$ for some
$i\not=j$, then we can take some $\eta\in B_{h}(\xi_{i},\rho/6)\cap
B_{h}(\xi_{j},\rho/6)$. It follows from Proposition 2.2 that
$\displaystyle d_{h}(\xi_{i},\xi_{j})$ $\displaystyle\leq$ $\displaystyle
3\left(d_{h}(\xi_{i},\eta)+d_{h}(\eta,\xi_{j})\right)$ $\displaystyle<$
$\displaystyle 3\left(\frac{\rho}{6}+\frac{\rho}{6}\right)=\rho.$
This contradicts the fact that $d_{h}(\xi_{i},\xi_{j})\geq\rho$ for any
$i\not=j$. Thus our claim (2.1) holds.
Assume $(\xi_{i})$ satisfies (2.1). For any fixed $r>0$ and
$\xi\in\mathbb{H}^{n}$ we set
$I_{r}(\xi)=\left\\{i\in I:\xi\in B_{h}(\xi_{i},r)\right\\}.$
By (1.4) and Proposition 2.2, we have for $r\geq\rho$
$\displaystyle|B_{h}(\xi,r)|$ $\displaystyle=$ $\displaystyle
4^{-Q}|B_{h}(\xi,4r)|$ $\displaystyle\geq$ $\displaystyle 4^{-Q}\sum_{i\in
I_{r}(\xi)}|B_{h}(\xi_{i},\rho/6)|$ $\displaystyle=$ $\displaystyle
4^{-Q}\,\,{\rm Card}\,\,I_{r}(\xi)\,\,({\rho}/{6})^{Q}|B_{h}(0,1)|,$
where ${\rm Card}\,\,I_{r}(\xi)$ denotes the cardinality of the set
$I_{r}(\xi)$. As a consequence, for $r\geq\rho$ there holds
${\rm Card}\,\,I_{r}(\xi)\leq(24r/\rho)^{Q}.$
This completes the proof of the lemma. $\hfill\Box$
## 3 Cut-off functions on Heisenberg balls
In this section, we will construct cut-off functions on Heisenberg balls. To
do this, we first estimate the gradient of the distance function as follows.
Lemma 3.1. Let $\xi_{0}$ be any fixed point of $\mathbb{H}^{n}$. Define a
function $\rho(\xi)=d_{h}(\xi,\xi_{0})$. Then we have
$|\nabla_{\mathbb{H}^{n}}\rho(\xi)|\leq 1$ for any $\xi\not=\xi_{0}$.
Proof. Write $\xi=(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n},t)$ and
$\xi_{0}=(x_{01},\cdots,x_{0n},y_{01},\cdots,y_{0n},t_{0})$. For any
$\xi\not=\xi_{0}$, we set
$E=\sum_{i=1}^{n}\left((x_{i}-x_{0i})^{2}+(y_{i}-y_{0i})^{2}\right),\quad
F=t-t_{0}-2\sum_{i=1}^{n}(x_{i}y_{0i}-y_{i}x_{0i}).$
Then by (1.1) and (1.3),
$\rho(\xi)=|\xi_{0}^{-1}\circ\xi|_{h}=\left(E^{2}+F^{2}\right)^{1/4}.$
We calculate
$\displaystyle\frac{\partial}{\partial
x_{i}}\rho=\rho^{-3}\left((x_{i}-x_{0i})E-y_{0i}F\right),\quad
2y_{i}\frac{\partial}{\partial t}\rho=\rho^{-3}y_{i}F,$
and then by (1.5),
$X_{i}\rho=\frac{\partial}{\partial x_{i}}\rho+2y_{i}\frac{\partial}{\partial
t}\rho=\rho^{-3}\left((x_{i}-x_{0i})E+(y_{i}-y_{0i})F\right).$
Similarly we have
$\frac{\partial}{\partial
y_{i}}\rho=\rho^{-3}\left((y_{i}-y_{0i})E+x_{0i}F\right)$
and thus by (1.5),
$Y_{i}\rho=\frac{\partial}{\partial y_{i}}\rho-2x_{i}\frac{\partial}{\partial
t}\rho=\rho^{-3}\left((y_{i}-y_{0i})E+(x_{0i}-x_{i})F\right).$
It follows that
$(X_{i}\rho)^{2}+(Y_{i}\rho)^{2}=\rho^{-6}\left((y_{i}-y_{0i})^{2}+(x_{i}-x_{0i})^{2}\right)(E^{2}+F^{2}).$
Note that $E^{2}+F^{2}=\rho^{4}$. We obtain
$\displaystyle|\nabla_{\mathbb{H}^{n}}\rho|$ $\displaystyle=$
$\displaystyle\left(\sum_{i=1}^{n}\left((X_{i}\rho)^{2}+(Y_{i}\rho)^{2}\right)\right)^{1/2}$
$\displaystyle=$ $\displaystyle\rho^{-3}E^{1/2}(E^{2}+F^{2})^{1/2}$
$\displaystyle=$ $\displaystyle\rho^{-1}E^{1/2}\leq 1.$
This completes the proof of the lemma. $\hfill\Box$
Now we construct cut-off functions. Let $\phi:\mathbb{R}\rightarrow\mathbb{R}$
be a smooth function such that $0\leq\phi\leq 1$, $\phi\equiv 1$ on the
interval $[-1,1]$, $\phi\equiv 0$ on $(-\infty,-2)\cup(2,\infty)$, and
$|\phi^{\prime}(t)|\leq 2$ for all $t\in\mathbb{R}$. Let $r>0$ be given.
Define a function on $\mathbb{H}^{n}$ by
$\phi_{0}(\xi)=\phi\left(\frac{d_{h}(\xi,\xi_{0})}{r}\right).$ (3.1)
Then $\phi_{0}$ is a cut-off function supported on the Heisenberg ball
$B_{h}(\xi_{0},2r)$. The estimate of the gradient of $\phi_{0}$ is very
important for the subsequent analysis. Precisely we have the following:
Lemma 3.2. For any fixed $r>0$ and $\xi_{0}\in\mathbb{H}^{n}$, let $\phi_{0}$
be defined by (3.1). Then $\phi_{0}$ is supported in $B_{h}(\xi_{0},2r)$,
$0\leq\phi_{0}\leq 1$, $\phi_{0}\equiv 1$ on $B_{h}(\xi_{0},r)$, and
$|\nabla_{\mathbb{H}^{n}}\phi_{0}(\xi)|\leq 2/r$ for all
$\xi\in\mathbb{H}^{n}$.
Proof. We only need to explain the last assertion, namely
$|\nabla_{\mathbb{H}^{n}}\phi_{0}(\xi)|\leq 2/r$ for all
$\xi\in\mathbb{H}^{n}$. Since $\phi_{0}\equiv 1$ on $B_{h}(\xi_{0},r)$, we
have $\nabla_{\mathbb{H}^{n}}\phi_{0}\equiv 0$ on $B_{h}(\xi_{0},r)$,
particularly $\nabla_{\mathbb{H}^{n}}\phi_{0}(0)=0$. For $\xi\not=\xi_{0}$, a
simple calculation shows
$\nabla_{\mathbb{H}^{n}}\phi_{0}(\xi)=\frac{1}{r}\phi^{\prime}\nabla_{\mathbb{H}^{n}}d_{h}(\xi,\xi_{0}).$
This together with Lemma 3.1 and $|\phi^{\prime}|\leq 2$ concludes the last
assertion. $\hfill\Box$
## 4 Proof of Theorem 1.1
In this section, we will prove Theorem 1.1. For simplicity, we define a smooth
function $\zeta:\mathbb{N}\times\mathbb{R}\rightarrow\mathbb{R}$ by
$\zeta(m,s)=e^{s}-\sum_{k=0}^{m-2}\frac{s^{k}}{k!},\quad\forall m\geq 2.$
(4.1)
As we promised in the introduction, we first derive a local Trudinger-Moser
inequality for the Heisenberg group $\mathbb{H}^{n}$ by using (1.7). Let $Q$,
$Q^{\prime}$ and $\alpha_{Q}$ be given by (1.7). Then we have the following:
Lemma 4.1. Let $r>0$ be given and $\xi_{0}$ be any point of $\mathbb{H}^{n}$.
If $0\leq\beta<Q$, $0\leq\alpha\leq\alpha_{Q}(1-\beta/Q)$, and $w\in
W_{0}^{1,Q}(B_{h}(\xi_{0},r))$ satisfies
$\int_{B_{h}(\xi_{0},r)}|\nabla_{\mathbb{H}^{n}}w|^{Q}d\xi\leq 1$, then there
exists some constant $C$ depending only on $n$, $r$ and $\beta$ such that
$\int_{B_{h}(\xi_{0},r)}\frac{1}{|\xi|_{h}^{\beta}}\zeta(Q,\alpha|w|^{Q^{\prime}})d\xi\leq
C\int_{B_{h}(\xi_{0},r)}|\nabla_{\mathbb{H}^{n}}w|^{Q}d\xi.$ (4.2)
Proof. Using Proposition 2.2, we have that
$|\xi_{0}|_{h}\leq
3(d_{h}(\xi,\xi_{0})+|\xi|_{h}),\quad\forall\xi\in\mathbb{H}.$
If $|\xi_{0}|_{h}>6r$, then for any $\xi\in B_{h}(\xi_{0},r)$ there holds
$|\xi|_{h}\geq\frac{|\xi_{0}|_{h}}{3}-d_{h}(\xi,\xi_{0})>r.$ (4.3)
Let
$\widetilde{w}=w/\|\nabla_{\mathbb{H}^{n}}w\|_{L^{Q}({B}_{h}(\xi_{0},r))}$.
Since $\|\nabla_{\mathbb{H}^{n}}w\|_{L^{Q}({B}_{h}(\xi_{0},r))}\leq 1$ and
$0\leq\alpha\leq\alpha_{Q}(1-\beta/Q)$, we have
$\displaystyle\zeta\left(Q,\alpha|w|^{Q^{\prime}}\right)$ $\displaystyle=$
$\displaystyle\sum_{k=Q-1}^{\infty}\frac{\alpha^{k}|w|^{Q^{\prime}k}}{k!}{}$
(4.4) $\displaystyle=$
$\displaystyle\sum_{k=Q-1}^{\infty}\frac{\alpha^{k}\|\nabla_{\mathbb{H}^{n}}w\|_{L^{Q}({B}_{h}(\xi_{0},r))}^{Q^{\prime}k}|\widetilde{w}|^{Q^{\prime}k}}{k!}{}$
$\displaystyle\leq$
$\displaystyle\|\nabla_{\mathbb{H}^{n}}w\|_{L^{Q}({B}_{h}(\xi_{0},r))}^{Q}\zeta\left(Q,\alpha|\widetilde{w}|^{Q^{\prime}}\right).$
By (1.4) and (1.7),
$\int_{B_{h}(\xi_{0},r)}\zeta\left(Q,\alpha_{Q}|\widetilde{w}|^{Q^{\prime}}\right)d\xi\leq
C_{n}r^{Q}|B_{h}(0,1)|,$
where $C_{n}$ is given by (1.7). Hence when $|\xi_{0}|_{h}>6r$ and
$0\leq\alpha\leq\alpha_{Q}(1-\beta/Q)$, we have by using (4.3) and (4.4),
$\displaystyle\int_{B_{h}(\xi_{0},r)}\frac{1}{|\xi|_{h}^{\beta}}\zeta\left(Q,\alpha|w|^{Q^{\prime}}\right)d\xi$
$\displaystyle\leq$ $\displaystyle
r^{-\beta}\int_{B_{h}(\xi_{0},r)}\zeta\left(Q,\alpha|w|^{Q^{\prime}}\right)d\xi$
$\displaystyle\leq$ $\displaystyle
C_{n}r^{Q-\beta}|B_{h}(0,1)|\int_{B_{h}(\xi_{0},r)}|\nabla_{\mathbb{H}^{n}}w|^{Q}d\xi.$
In the following we assume $|\xi_{0}|_{h}\leq 6r$. If $\xi\in
B_{h}(\xi_{0},r)$, then Proposition 2.2 implies that
$|\xi|_{h}\leq 3(d_{h}(\xi,\xi_{0})+|\xi_{0}|_{h})<21r.$
Hölder’s inequality together with (1.7) implies that there exits some constant
$\widetilde{C}$ depending only on $n$, $r$ and $\beta$ such that
$\int_{B_{h}(\xi_{0},r)}\frac{1}{|\xi|_{h}^{\beta}}\zeta\left(Q,\alpha|\widetilde{w}|^{Q^{\prime}}\right)d\xi\leq\int_{|\xi|_{h}\leq
21r}\frac{1}{|\xi|_{h}^{\beta}}\zeta\left(Q,\alpha|\widetilde{w}|^{Q^{\prime}}\right)d\xi\leq\widetilde{C}.$
It then follows from (4.4) that
$\displaystyle\int_{B_{h}(\xi_{0},r)}\frac{1}{|\xi|_{h}^{\beta}}\zeta\left(Q,\alpha|w|^{Q^{\prime}}\right)d\xi$
$\displaystyle\leq$
$\displaystyle\|\nabla_{\mathbb{H}^{n}}w\|_{L^{Q}({B}_{h}(\xi_{0},r))}^{Q}\int_{B_{h}(\xi_{0},r)}\frac{1}{|\xi|_{h}^{\beta}}\zeta\left(Q,\alpha|\widetilde{w}|^{Q^{\prime}}\right)d\xi$
$\displaystyle\leq$
$\displaystyle\widetilde{C}\int_{B_{h}(\xi_{0},r)}|\nabla_{\mathbb{H}^{n}}w|^{Q}d\xi.$
Hence (4.2) holds. $\hfill\Box$
Proof of Theorem 1.1. Firstly, we prove (1.10). Let $\tau>0$ and
$\alpha:0\leq\alpha<\alpha_{Q}(1-\beta/Q)$ be fixed. Since
$C_{0}^{\infty}(\mathbb{H}^{n})$ is dense in $W^{1,Q}(\mathbb{H}^{n})$ under
the norm (1.9), it suffices to prove (1.10) for all $u\in
C_{0}^{\infty}(\mathbb{H}^{n})$ with
$\int_{\mathbb{H}^{n}}(|\nabla_{\mathbb{H}^{n}}u|^{Q}+\tau|u|^{Q})d\xi\leq 1.$
(4.5)
Assume $u\in C^{\infty}(\mathbb{H}^{n})$ satisfies (4.5). Let $r>0$ be a
sufficiently large number to be determined later. By Lemma 2.3, there exists a
sequence $(\xi_{i})$ of points of $\mathbb{H}^{n}$ such that
$\cup_{i}B_{h}(\xi_{i},r)=\mathbb{H}^{n}\,\,{\rm and}\,\,\forall
i\not=j,\,\,B_{h}(\xi_{i},r/6)\cap B_{h}(\xi_{j},r/6)=\varnothing,$ (4.6)
and for any $\xi\in\mathbb{H}^{n}$,
$\xi\,\,{\rm belongs\,\,to\,\,at\,\,most}\,\,48^{Q}\,\,{\rm
balls}\,\,B_{h}(\xi_{i},2r).$ (4.7)
Let $\phi$ be a smooth function given by (3.1). For each $\xi_{i}$, we set
$\phi_{i}(\xi)=\phi\left(\frac{d_{h}(\xi,\xi_{i})}{r}\right),\quad\forall\xi\in\mathbb{H}^{n}.$
It follows from Lemma 3.2 that $0\leq\phi_{i}\leq 1$, $\phi_{i}\equiv 1$ on
$B_{h}(\xi_{i},r)$, $\phi_{i}\equiv 0$ outside $B_{h}(\xi_{i},2r)$, and
$|\nabla_{\mathbb{H}^{n}}\phi_{i}(\xi)|\leq\frac{2}{r},\,\,\forall\xi\in\mathbb{H}^{n}.$
(4.8)
Clearly $\phi_{i}^{2}u\in W_{0}^{1,Q}\left(B_{h}(\xi_{i},2r)\right)$. Since
$u$ satisfies (4.5), we have that
$\int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}u|^{Q}d\xi\leq 1,\,\,{\rm
and}\,\,\int_{\mathbb{H}^{n}}|u|^{Q}d\xi\leq\frac{1}{\tau}.$
Minkowski inequality together with (4.8) and $0\leq\phi_{i}\leq 1$ leads to
$\displaystyle\left(\int_{B_{h}(\xi_{i},2r)}|\nabla_{\mathbb{H}^{n}}(\phi_{i}^{2}u)|^{Q}d\xi\right)^{1/Q}$
$\displaystyle\leq$
$\displaystyle\left(\int_{B_{h}(\xi_{i},2r)}\phi_{i}^{2Q}|\nabla_{\mathbb{H}^{n}}u|^{Q}d\xi\right)^{1/Q}+\left(\int_{B_{h}(\xi_{i},2r)}|\nabla_{\mathbb{H}^{n}}\phi_{i}^{2}|^{Q}|u|^{Q}d\xi\right)^{1/Q}{}$
(4.9) $\displaystyle\leq$
$\displaystyle\left(\int_{B_{h}(\xi_{i},2r)}|\nabla_{\mathbb{H}^{n}}u|^{Q}d\xi\right)^{1/Q}+\frac{4}{r}\left(\int_{B_{h}(\xi_{i},2r)}|u|^{Q}d\xi\right)^{1/Q}{}$
$\displaystyle\leq$ $\displaystyle 1+\frac{4}{\tau r}.$
Define $\widetilde{u}_{i}={\phi_{i}^{2}u}/(1+\frac{4}{\tau r})$. Then
$\widetilde{u}_{i}\in W_{0}^{1,Q}(B_{h}(\xi_{i},2r))$ and
$\int_{B_{h}(\xi_{i},2r)}|\nabla_{\mathbb{H}^{n}}\widetilde{u}_{i}|^{Q}d\xi\leq
1$. Since $\alpha<\alpha_{Q}(1-\beta/Q)$, we can select $r$ sufficiently large
such that
$\alpha\left(1+\frac{4}{\tau r}\right)^{Q^{\prime}}<\alpha_{Q}(1-\beta/Q).$
This together with Lemma 4.1 implies that there exists some constant $C$
depending only on $n$, $r$ and $\beta$ such that
$\displaystyle\int_{B_{h}(\xi_{i},2r)}\frac{1}{|\xi|_{h}^{\beta}}\zeta\left(Q,\alpha|\phi_{i}^{2}u|^{Q^{\prime}}\right)d\xi$
$\displaystyle=$
$\displaystyle\int_{B_{h}(\xi_{i},2r)}\frac{1}{|\xi|_{h}^{\beta}}\zeta\left(Q,\alpha\left(1+\frac{4}{\tau
r}\right)^{Q^{\prime}}|\widetilde{u}_{i}|^{Q^{\prime}}\right)d\xi{}$ (4.10)
$\displaystyle\leq$ $\displaystyle
C\int_{B_{h}(\xi_{i},2r)}|\nabla_{\mathbb{H}^{n}}\widetilde{u}_{i}|^{Q}d\xi.$
$\displaystyle\leq$ $\displaystyle
C\int_{B_{h}(\xi_{i},2r)}|\nabla_{\mathbb{H}^{n}}(\phi_{i}^{2}u)|^{Q}d\xi.$
Combining (4.6) and (4.10), we obtain
$\displaystyle\int_{\mathbb{H}^{n}}\frac{1}{|\xi|_{h}^{\beta}}\zeta\left(Q,\alpha|u|^{Q^{\prime}}\right)d\xi$
$\displaystyle\leq$
$\displaystyle\sum_{i}\int_{B_{h}(\xi_{i},r)}\frac{1}{|\xi|_{h}^{\beta}}\zeta\left(Q,\alpha|\phi_{i}^{2}u|^{Q^{\prime}}\right)d\xi{}$
(4.11) $\displaystyle\leq$
$\displaystyle\sum_{i}\int_{B_{h}(\xi_{i},2r)}\frac{1}{|\xi|_{h}^{\beta}}\zeta\left(Q,\alpha|\phi_{i}^{2}u|^{Q^{\prime}}\right)d\xi$
$\displaystyle\leq$ $\displaystyle
C\sum_{i}\int_{\mathbb{H}^{n}}|\nabla(\phi_{i}^{2}u)|^{Q}d\xi.$
Using the inequality $|a+b|^{Q}\leq 2^{Q}|a|^{Q}+2^{Q}|b|^{Q}$, $\forall
a,b\in\mathbb{R}$, $0\leq\phi_{i}\leq 1$ and (4.8), we get
$\displaystyle\int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}(\phi_{i}^{2}u)|^{Q}d\xi$
$\displaystyle\leq$ $\displaystyle
2^{Q}\int_{\mathbb{H}^{n}}\left(\phi_{i}^{2Q}|\nabla_{\mathbb{H}^{n}}u|^{Q}+|\nabla_{\mathbb{H}^{n}}\phi_{i}^{2}|^{Q}|u|^{Q}\right)d\xi{}$
$\displaystyle\leq$ $\displaystyle
2^{Q}\int_{\mathbb{H}^{n}}\phi_{i}|\nabla_{\mathbb{H}^{n}}u|^{Q}d\xi+\left(\frac{8}{r}\right)^{Q}\int_{\mathbb{H}^{n}}\phi_{i}|u|^{Q}d\xi.$
In view of (4.7), it then follows that
$\displaystyle\sum_{i}\int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}(\phi_{i}^{2}u)|^{Q}d\xi$
$\displaystyle\leq$ $\displaystyle
2^{Q}\sum_{i}\int_{\mathbb{H}^{n}}\phi_{i}|\nabla_{\mathbb{H}^{n}}u|^{Q}d\xi+\left(\frac{8}{r}\right)^{Q}\sum_{i}\int_{\mathbb{H}^{n}}\phi_{i}|u|^{Q}d\xi$
$\displaystyle\leq$ $\displaystyle
96^{Q}\int_{\mathbb{H}^{n}}|\nabla_{\mathbb{H}^{n}}u|^{Q}d\xi+\left(\frac{384}{r}\right)^{Q}\int_{\mathbb{H}^{n}}|u|^{Q}d\xi.$
This together with (4.11) implies
$\int_{\mathbb{H}^{n}}\frac{1}{|\xi|^{\beta}}\zeta\left(Q,\alpha|u|^{Q^{\prime}}\right)d\xi\leq\widetilde{C}$
for some constant $\widetilde{C}$ depending only on $C$, $Q$, and $r$. Hence
we conclude (1.10).
Secondly, we prove that for any fixed $\beta:0\leq\beta<Q$, $\alpha>0$, and
$u\in W^{1,Q}(\mathbb{H}^{n})$, there holds
$\int_{\mathbb{H}^{n}}\frac{1}{|\xi|_{h}^{\beta}}\zeta\left(Q,\alpha|u|^{Q^{\prime}}\right)d\xi<\infty.$
(4.12)
Since $C_{0}^{\infty}(\mathbb{H}^{n})$ is dense in $W^{1,Q}(\mathbb{H}^{n})$,
we can take some $u_{0}\in C_{0}^{\infty}(\mathbb{H}^{n})$ such that
$\|u-u_{0}\|_{W^{1,Q}(\mathbb{H}^{n})}<\epsilon$, where $\epsilon>0$ is a
small number to be determined later. Set
$w=\frac{u-u_{0}}{\|u-u_{0}\|_{W^{1,Q}(\mathbb{H}^{n})}}.$
Then $\|w\|_{W^{1,Q}(\mathbb{H}^{n})}=1$. We divide the proof of (4.12) into
two cases:
Case $1$. $\beta=0$.
Recall (4.1). By ([12], Lemma 2.2), $\zeta(Q,t)$ is convex with respect to
$t$. Since
$|a+b|^{\gamma}\leq(1+\delta)|a|^{\gamma}+C(\delta,\gamma)|b|^{\gamma}$,
$\forall a,b\in\mathbb{R},\gamma\geq 1,\delta>0$, for some constant
$C(\delta,\gamma)$ depending only on $\delta$ and $\gamma$, we obtain
$\displaystyle\int_{\mathbb{H}^{n}}\zeta\left(Q,\alpha|u|^{Q^{\prime}}\right)d\xi$
$\displaystyle=$
$\displaystyle\int_{\mathbb{H}^{n}}\zeta\left(Q,\alpha|u-u_{0}+u_{0}|^{Q^{\prime}}\right)d\xi$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{H}^{n}}\zeta\left(Q,\alpha(1+\delta)|u-u_{0}|^{Q^{\prime}}+\alpha
C(\delta,Q^{\prime})|u_{0}|^{Q^{\prime}}\right)d\xi$ $\displaystyle\leq$
$\displaystyle\frac{1}{\mu}\int_{\mathbb{H}^{n}}\zeta\left(Q,\mu\alpha(1+\delta)|u-u_{0}|^{Q^{\prime}})\right)d\xi+\frac{1}{\nu}\int_{\mathbb{H}^{n}}\zeta\left(Q,\nu\alpha
C(\delta,Q^{\prime})|u_{0}|^{Q^{\prime}}\right)d\xi$ $\displaystyle\leq$
$\displaystyle\frac{1}{\mu}\int_{\mathbb{H}^{n}}\zeta\left(Q,\mu\alpha(1+\delta)\epsilon^{Q^{\prime}}|w|^{Q^{\prime}})\right)d\xi+\frac{1}{\nu}\int_{\mathbb{H}^{n}}\zeta\left(Q,\nu\alpha
C(\delta,Q^{\prime})|u_{0}|^{Q^{\prime}}\right)d\xi,$
where ${1}/{\mu}+1/\nu=1$, $\mu>1$, $\nu>1$. Now we choose $\epsilon>0$
sufficiently small such that
$\mu\alpha(1+\delta)\epsilon^{Q^{\prime}}<\alpha_{Q}$. By (1.10), there holds
$\int_{\mathbb{H}^{n}}\zeta\left(Q,\mu\alpha(1+\delta)\epsilon^{Q^{\prime}}|w|^{Q^{\prime}})\right)d\xi\leq
C_{1}$
for some constant $C_{1}$ depending only on $n$ and $\tau$. In addition, since
$u_{0}\in C_{0}^{\infty}(\mathbb{H}^{n})$, it is obvious that
$\int_{\mathbb{H}^{n}}\zeta\left(Q,\nu\alpha
C(\delta,Q^{\prime})|u_{0}|^{Q^{\prime}}\right)d\xi<\infty.$
Therefore, we have
$\int_{\mathbb{H}^{n}}\zeta\left(Q,\alpha|u|^{Q^{\prime}}\right)d\xi<\infty.$
Case $2$. $0<\beta<Q$.
Note that
$\displaystyle\int_{\mathbb{H}^{n}}\frac{1}{|\xi|_{h}^{\beta}}\zeta\left(Q,\alpha|u|^{Q^{\prime}}\right)d\xi\leq\int_{|\xi|_{h}\leq
1}\frac{1}{|\xi|_{h}^{\beta}}\zeta\left(Q,\alpha|u|^{Q^{\prime}}\right)d\xi+\int_{\mathbb{H}^{n}}\zeta\left(Q,\alpha|u|^{Q^{\prime}}\right)d\xi.$
This together with Hölder’s inequality and Case $1$ implies (4.12).
Finally, we confirm that for any $\alpha>\alpha_{Q}(1-\beta/Q)$, there holds
$\sup_{\|u\|_{1,\tau}\leq
1}\int_{\mathbb{H}^{n}}\frac{1}{|\xi|_{h}^{\beta}}\zeta\left(Q,\alpha|u|^{Q^{\prime}}\right)d\xi=\infty.$
This is based on calculations of related integrals of the Moser function
sequence. We omit the details but refer the reader to [3]. $\hfill\Box$
Acknowledgements. This work was partly supported by the NSFC 11171347 and the
NCET program 2008-2011.
## References
* [1] Adimurthi, Y. Yang, An interpolation of Hardy inequality and Trudinger-Moser inequality in $\mathbb{R}^{N}$ and its applications, International Mathematics Research Notices 13 (2010) 2394-2426.
* [2] D. Cao, Nontrivial solution of semilinear elliptic equations with critical exponent in $\mathbb{R}^{2}$, Commun. Partial Differential Equations 17 (1992) 407-435.
* [3] W. Cohn, N. Lam, G. Lu, Y. Yang, The Moser-Trudinger inequality in unbounded domains of Heisenberg group and sub-elliptic equations, Nonlinear Analysis (2011), doi: 10.1016/j.na.2011.09.053.
* [4] W. Cohn, G. Lu, Best constants for Moser-Trudinger inequalities on the Heisenberg group, Indiana Univ. Math. J. 50 (2001) 1567-1591.
* [5] J. M. do Ó, $N$-Laplacian equations in $\mathbb{R}^{N}$ with critical growth, Abstr. Appl. Anal. 2 (1997) 301-315.
* [6] G. Hardy, J. Littlewood, G. Polya, Inequalities, Cambridge University Press, 1952.
* [7] E. Hebey, Sobolev spaces on Riemannian maifolds, Lecture notes in mathematics 1635, Springer, 1996.
* [8] Y. Li, B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^{N}$, Ind. Univ. Math. J. 57 (2008) 451-480.
* [9] R. Panda, Nontrivial solution of a quasilinear elliptic equation with critical growth in $\mathbb{R}^{n}$, Proc. Indian Acad. Sci. (Math. Sci.) 105 (1995) 425-444.
* [10] B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^{2}$, J. Funct. Anal. 219 (2005) 340-367.
* [11] E. M. Stein, Harmonic analysis, Princeton University press, 2006.
* [12] Y. Yang, Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space, J. Funct. Anal. 262 (2012) 1679-1704, arXiv: 1106.4622v1.
* [13] Y. Yang, Trudinger-Moser inequalities on complete noncompact Riemannian manifolds, arXiv: 1112.0724v1.
|
arxiv-papers
| 2012-01-14T06:25:12 |
2024-09-04T02:49:26.322006
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yunyan Yang",
"submitter": "Yunyan Yang",
"url": "https://arxiv.org/abs/1201.2993"
}
|
1201.3206
|
# Disordered locality and Lorentz dispersion relations:
an explicit model of quantum foam
Francesco Caravelli fcaravelli@perimeterinstitute.ca Fotini Markopoulou
fmarkopoulou@perimeterinstitute.ca Perimeter Institute for Theoretical
Physics,
Waterloo, Ontario N2L 2Y5 Canada,
and
University of Waterloo, Waterloo, Ontario N2L 3G1, Canada,
and
Max Planck Institute for Gravitational Physics, Albert Einstein Institute,
Am Mühlenberg 1, D-14476 Golm, Germany
###### Abstract
Using the framework of Quantum Graphity, we construct an explicit model of a
quantum foam, a quantum spacetime with spatial non-local links. The states
depend on two parameters: the minimal size of the link and their density with
respect to this length. Macroscopic Lorentz invariance requires that the
quantum superposition of spacetimes is suppressed by the length of these non-
local links. We parametrize this suppression by the distribution of non-local
links lengths in the quantum foam. We discuss the general case and then
analyze two specific natural distributions. Corrections to the Lorentz
dispersion relations are calculated using techniques developed in previous
work.
quantum foam, quantum graphity, quantum gravity, non-locality
###### pacs:
04.60.Pp , 04.60.-m
## I Introduction
A fascinating idea proposed by Wheeler in the early years of Quantum Gravity,
is that, at the Planck scale, geometry may be bumpy due to quantum
fluctuations. This is the quantum foam WheelerFord . While intuitively
natural, this idea is very complicated to put into action. In the present
paper, we will use the framework of Quantum Graphity graphity1 ; graphity2 ;
graphity3 to construct a simple model of quantum foam.
A key feature of a quantum foam is its non-local nature. While non-locality is
undesirable in quantum field theory, the situation in quantum gravity is open.
It is often said that the only way to cure the divergences appearing
perturbatively in quantizations of gravity without introducing new physics
(i.e., string theory or super-symmetric extensions of gravity), is to
introduce some kind of non-locality in the action which smears out Green
functions evaluated on one point only. Until now, ghosts in the theory have
blocked research in this direction (some progress has been achieved recently
in stleomaz ). For the purposes of the present work, it is important to note
that there are two possible types of non-locality which contribute in
different ways. One, violation of microlocality, disappears when the cut-off
is taken to zero, while the other, violation of macrolocality, or disordered
locality, does not marksm . Violations of macrolocality amount to the presence
of what a relativist would call a wormhole lw , a path through spacetime
disallowed in a Lorentzian topology. General relativity allows for such paths
and, in principle, they should be taken into account in a full quantum theory
of gravity. In principle, in order to have traversable wormholes, the common
positive-energy conditions and some other conditions on the throats have to be
satisfied.
On the other hand, in graph-based quantum gravity states, such as in Loop
Quantum Gravity loop , Causets causets or Quantum Graphity revqg , spacetimes
which are not macrolocal are very natural, and violation of macrolocality
appears in the form of non-local links. A first study of the physics of these
non-local links was carried out in marksm ; smochan .
We propose, in the present paper, to use the framework of Quantum Graphity to
provide a concrete implementation of Wheeler’s quantum foam, based on the
assumption that the non-local link can be used to cross from one end to the
other one.
Quantum Graphity models graphity1 ; graphity2 are spin system toy models for
emergent geometry and gravity. They are based on quantum, dynamical graphs
whose adjacency is dynamical: their edges can be on (connected), off
(disconnected), or in a superposition of on and off. We can interpret the
graph as pregeometry (the connectivity of the graph tells us who is
neighbouring whom). A particular graphity model is given by such graph states
evolving under a local Ising-type Hamiltonian. The graphity model of graphity2
, for example, is a toy model for interacting matter and geometry, a Bose-
Hubbard model where the interactions are quantum variables.
In graphity3 , we solved the model of graphity2 in the limit of no
backreaction of the matter on the lattice, and for states with certain
symmetries that are natural for our problem, which we called _rotationally_
invariant graphs. In this case, the problem reduces to an one-dimensional
Hubbard model on a lattice with variable vertex degree and multiple edges
between the same two vertices. The probability density for the matter obeys a
(discrete) differential equation closed in the classical regime. This is a
wave equation in which the vertex degree is related to the local speed of
propagation of probability. This allows an interpretation of the probability
density of particles similar to what is usually considered in analogue gravity
systems: matter inside this analogue system sees a curved spacetime.
We will extend these results we obtained in order to describe a quantum foam:
instead of a classical background state (a single graph), we will consider a
state that is a superposition of many graphs. This amounts to a quantum foam
with a superposition of Planck scale sized non-local links. In our setting,
the intrinsic discreteness of the graph sets the minimum scale. Assuming
foliability of the graph, we can define a metric distance as in graphity3 . We
can then study the effect of the quantumness of the graph on the dispersion
relations.
Quantum Graphity models are lattice models in which the lattice becomes a
quantum object. As in any lattice model, the continuum limit is obtained as in
any other lattice theory, but consider it together for all the states on the
graph.
It is natural to construct graph states in which the largest contribution
comes from the graph with the Lorentz invariant dispersion relations. The
states with non-local links violate macrolocality and give corrections to
Lorentz invariance. We will construct states with a distribution of non-local
links which is suppressed by their combinatorial length. These states resemble
coherent states as considered in Loop Quantum Gravity. In principle, they
could be obtained as correction to the ground state due to a non-zero
temperature bath in Quantum Graphity. The distribution depends on their
density. We will then calculate the effect on the Lorentz dispersion relations
in the continuum limit. The result is, as expected, a non-local differential
equation for the evolution of the particle probability density.
It is reasonable to expect that a non-local link will violate local Lorentz
invariance. A particle can hop through the link and behave like a superluminal
particle. As we will see, the presence of all these shortcuts has an effect on
the local speed of propagation of probability density. Also, we will find that
the probability density acquires a mass which depends on the density of non-
local links. The overall dispersion relation is thus Lorentz invariant and
with a square-positive mass. However, this depends on the distribution and
thus we will study two particular cases. Using the framework of Quantum
Graphity and the techniques developed in graphity3 , we will calculate the
emergent mass and the constants appearing in the effective equation.
This paper is organized as follows. In section II, we summarize the Quantum
Graphity framework and the results of graphity3 . In section III, we show the
effect of a superposition of graphs on the differential equation governing the
time-evolution of the probability density. In section IV, we introduce our
choice of the quantum state of the graph. In section V, we analyze two
particular non-local link distributions and their effect on the dispersion
relations. Conclusions follow.
## II The model
In the following we review the model, as defined and first studied in
graphity2 and the effective geometry encoded in the graph, as obtained in
graphity3 .
### II.1 Bose-Hubbard model on a dynamical lattice
In this section we will introduce briefly the model. For more a more detailed
introduction we refer to the previous papers graphity2 ; graphity3 .
We associate a Hilbert space $\mathscr{H}_{i}$ to the degrees of freedom on
the nodes a graph, with $i$ labelling the nodes. These degrees of freedom
represent matter on the graph and thus can be, in principle, generalized to
other fields. We choose $\mathscr{H}_{i}$ to be the Hilbert space of a
harmonic oscillator. We denote its creation and destruction operators by
$b^{\dagger}_{i}$ and $b_{i}$ respectively, satisfying the usual bosonic
commutators. Our $N_{v}$ physical systems then are $N_{v}$ bosonic modes and
the total Hilbert space of such modes is given by
$\mathscr{H}_{bosons}=\bigotimes_{i=1}^{N_{v}}\mathscr{H}_{i}.$ (1)
If the harmonic oscillators are not interacting, the total Hamiltonian is
trivial:
$\widehat{H}_{v}=\sum_{i=1}^{N_{v}}\widehat{H}_{i}=-\sum_{i=1}^{N_{v}}\mu
b^{\dagger}_{i}b_{i}.$ (2)
The Hamiltonian reads as
$\widehat{H}=\sum_{i}\widehat{H}_{i}+\sum_{{\bf e}\in I}\widehat{h}_{\bf e},$
(3)
where $\widehat{h}_{\bf e}$ is a Hermitian operator on $H_{i}\otimes H_{j}$
representing the interaction between the system $i$ and the system $j$.
We introduce a primitive notion of geometry through the adjacent matrix $A$,
the $N_{v}\times N_{v}$ symmetric matrix defined as
$A_{ij}=\left\\{{\begin{array}[]{ll}1&{\mbox{if $i$ and $j$ are adjacent}}\\\
0&{\mbox{otherwise}}.\end{array}}\right.$ (4)
$A$ defines a graph on $N_{v}$ nodes, with an edge between nodes $i$ and $j$
for every $1$ entry in the matrix. The total Hilbert space for the graph edges
is then
$\mathscr{H}_{graph}=\bigotimes_{\bf e=1}^{N_{v}(N_{v}-1)/2}\mathscr{H}_{\bf
e},$ (5)
with $\mathscr{H}_{\bf e}=Span\\{|0\rangle,|1\rangle\\}$ a qubit representing
on/off links. Therefore, the total Hilbert space of the model is
${\mathscr{H}}=\mathscr{H}_{bosons}\otimes\mathscr{H}_{graph},$ (6)
and a basis state in $H$ has the form
$\displaystyle|\Psi\rangle$ $\displaystyle\equiv$
$\displaystyle|\Psi^{(bosons)}\rangle\otimes|\Psi^{(graph)}\rangle$ (7)
$\displaystyle\equiv$
$\displaystyle|n_{1},...,n_{N_{v}}\rangle\otimes|e_{1},...,e_{\frac{N_{v}(N_{v}-1)}{2}}\rangle.$
(8)
The first factor tells us how many bosons there are at every site $i$, while
the second factor tells us which pairs $(i,j)$ interact.
We note that it is the dynamics of the particles described by
$\widehat{H}_{\textrm{hop}}=-E_{hop}\sum_{i<j}A_{ij}\big{(}\hat{a}_{i}^{\dagger}\hat{a}_{i}+h.c.),$
that gives to the degree of freedom $|e\rangle$ the meaning of geometry and
h.c. denotes the hermitean conjugate.
The hopping amplitude is given by $t$, and therefore all the bosons have the
same speed. Note that, for a larger Hilbert space on the links, we can have
different speeds for the bosons.
As mentioned above, the long-term ambition of these models is to find a
quantum Hamiltonian that is a spin system analogue of gravity. In this spirit,
matter-geometry interaction is desirable as it is a central feature of general
relativity. The above dynamics can be considered as a very simple first step
in that direction.
In the present work, we study the model for a particular class of graphs that
have been conjectured to be analogues of trapped surfaces. We are interested
in the approximation $k\ll t$, which can be seen as the equivalent of ignoring
the backreaction of the matter on the geometry. As in graphity3 , we will
consider an Hamiltonian of the form
$\widehat{H}=\widehat{H}_{v}+\widehat{H}_{\textrm{hop}}.$ (9)
In this case, the total number of particles on the graph is a conserved
charge. $\widehat{H}_{v}$ and $\widehat{H}_{links}$ are constants on fixed
graphs with fixed number of particles. The Hamiltonian is the ordinary Bose-
Hubbard model on a fixed graph, but that graph can be unusual, with sites of
varying connectivity and with more than one edge connecting two sites. Our aim
will then be to study the non-local and quantum corrections to the effective
geometry which can be encoded in the graph, as shown in graphity3 . Even on a
fixed lattice, the Hubbard model is difficult to analyze, with few results in
higher dimensions. It would seem that our problem, propagation on a lattice
with connectivity which varies from site to site is also very difficult.
Fortunately, it turns out that for our purposes it is sufficient to restrict
attention to lattices with certain symmetries and then to restrict to an
effective 1+1 dimensional model.
### II.2 Rotationally invariant graphs and the encoded geometry
Let us present next our definition of rotationally invariant graphs, which
allows to reduce the problem to a 1-dimensional Bose-Hubbard model in the
single particle sector.
A graph $G$ is called $N$-_rotationally invariant_ if there exists an
embedding of $G$ to the plane that is invariant by rotations of an angle
$2\pi/N$. In principle, the edges of the graph, once embedded, can be
overlapping. The main property of the rotationally invariant graphs is that
groups of sub-graphs can be labelled by an integer number $i$. These graphs
can be very far from triangulations, as the rotationally invariant graphs in
Fig. 1 and 2 show.
Figure 1: A planar graph which is rotational invariant. Figure 2: A non-planar
graph which is rotational invariant.
These graphs can be labelled by a set of two integer coordinates,
$(n,\theta_{n})$, where $n$ labels a set of nodes, while $\theta_{n}$ is a
coordinate internal to the subgraph. For convenience we will drop, since now
on, the sub-index $n$ in the $\theta$ coordinates.
We can make use of the coordinates $(n,\theta)$ in order to write the
Hamiltonian defined by a rotationally invariant graph as
$\displaystyle H_{\textrm{rot}}$
$\displaystyle=-\sum_{\theta=0}^{N-1}\sum_{n,n^{\prime}}A_{nn^{\prime}}b_{n\theta}^{\dagger}b_{n^{\prime}\theta}+h.c.$
$\displaystyle-\sum_{\theta=0}^{N-1}\sum_{\varphi=1}^{N-1}\sum_{n,n^{\prime}}B_{n,n^{\prime}}^{(\varphi)}b_{n\theta}^{\dagger}b_{n^{\prime}\theta+\varphi}+h.c.,$
(10)
where $b_{n,\theta}^{\dagger}$ ($b_{n,\theta}$) is the creation (annihilation)
operator at the vertex $(n,\theta)$, $A_{nn^{\prime}}$ is the adjacency matrix
of the graph and $B_{n,n^{\prime}}^{(\varphi)}$ is the adjacency matrix of two
angular sectors at an angular distance $\varphi$ in units of $2\pi/N$.
Let us introduce the rotation operator $\widehat{M}$ defined by
$\displaystyle\widehat{M}b_{n,\theta}=b_{r,\theta+1}\widehat{M}\,$
$\displaystyle\widehat{M}b_{n,\theta}^{\dagger}=b_{r,\theta+1}^{\dagger}\widehat{M}\,.$
(11)
The effect of the operator $\hat{M}$ is particularly easy to understand in the
single particle case:
$\widehat{M}|n,\theta\rangle=\widehat{M}b_{n,\theta}^{\dagger}|0\rangle=b_{n,\theta+1}^{\dagger}\widehat{M}|0\rangle=|n,\theta+1\rangle\,,$
(12)
where we have assumed that the vacuum is invariant under a rotation
$\widehat{M}|0\rangle=|0\rangle$. Note that $\widehat{M}$ is unitary and its
application $N$ times gives the identity,
$\widehat{M}^{N}=\operatorname{\mathds{1}}$. This implies that its eigenvalues
are integer multiples of $2\pi/M$.
Another interesting property of $\widehat{M}$ is that commutes both with the
rotationally invariant Hamiltonians and with the number operator
$\widehat{N}_{p}$,
$[\widehat{H}_{\textrm{rot}},\widehat{M}]=[\widehat{N}_{p},\widehat{M}]=[\widehat{H}_{\textrm{rot}},\widehat{N}_{p}]=0\,.$
(13)
Therefore $\widehat{H}_{\textrm{rot}}$, $\widehat{N}_{p}$, and $\widehat{M}$
form a complete set of commuting observables and the Hamiltonian is diagonal
in blocks of constant $\widehat{M}$ and $\widehat{N}_{p}$. In this sector of
the Hamiltonian, we can reduce the Hamiltonian to:
$\widehat{H}_{0}=\sum_{n=0}^{L-1}f_{n,n+1}\left(|n\rangle\langle
n+1|+|n+1\rangle\langle n|\right)+\sum_{n}\mu_{n}|n\rangle\\!\langle n|\,,$
(14)
with $f_{n,n+1}$ depending on the degree of the graph and $n$ being the label
of the shells we are reducing with the rotational symmetry and $L$ the total
size of the one-dimensional lattice.
### II.3 Restriction of the time-dependent Schrödinger equation to the set of
classical states
Since we want to study the dynamics of a single particle on a fixed graph, it
is only necessary to consider the single particle sector. The one dimensional
Bose-Hubbard model for a single particle reads as in (14), where $f_{n,n+1}$
are the tunneling coefficients between sites $n$ and $n+1$, $\mu_{n}$ is the
chemical potential at the site $n$, and $M$ is the size of the lattice.
In this setup, let us introduce the convex set of classical states
$\mathcal{M}_{C}$, parameterized as
$\widehat{\rho}(t)\equiv\widehat{\rho}\big{(}\Psi(t)\big{)}=\sum_{n=0}^{L-1}\Psi_{n}|n\rangle\\!\langle
n|\,,$ (15)
where $\Psi_{n}$ is the probability of finding the particle at the site $n$.
The states in $\mathcal{M}_{C}$ are classical because the uncertainty in the
position is classical, that is, they represent a particle with an unknown but
well-defined position.
Since our particle is under the effect of a noisy environment, its density
matrix is going to be constantly dephased by the interaction between the
particle and its reservoir. For a more detailed discussion about this
procedure and the connection with the physics of decoherence, we refer to
graphity3 . The dephased state in the position eigenbasis that best
approximates $\rho(t+\Delta t)$ can be easily determined by computing the
double commutator of the previous equation, which was shown to lead to a
closed equation in graphity3 . It obeys the evolution
$\displaystyle\frac{\hbar^{2}}{2}\partial_{t}^{2}\Psi_{n}(t)=$ $\displaystyle
f_{n-1,n}^{2}\left(\Psi_{n+1}(t)+\Psi_{n-1}(t)-2\Psi_{n}(t)\right)$
$\displaystyle+\left(f_{n+1,n}^{2}-f_{n-1,n}^{2}\right)\left(\Psi_{n+1}(t)-\Psi_{n}(t)\right)\,.$
This equation becomes a wave equation in the continuum,
$\partial_{t}^{2}\Psi(x,t)-\partial_{x}\left(c^{2}(x)\partial_{x}\Psi(x,t)\right)=0\,,$
(16)
where
$\frac{1}{c(x)}=\sqrt{\frac{\hbar^{2}}{2f^{2}(x)E_{\textrm{hop}}^{2}}}=\frac{\hbar}{E_{\textrm{hop}}\sqrt{2f^{2}(x)}}\,,$
(17)
and $\Psi(x,t)$ and $f(x)$ are the continuous limit functions of $\Psi_{n}(t)$
and $f_{n,n-1}$ respectively. Eqn (16) is the equation of motion for a scalar
field with a space-dependent refraction index. As it is well known, this
equation in higher dimension is connected with the Gordon metric. In fact, to
the refraction index it is possible to encode a space-time geometry with
spatial curvature and no extrinsic curvature, i.e. a preferred direction of
time. The time direction is the same of the quantum mechanical underlying
model. This equation is the starting point for what we will do in the
following. However, let us first recall how the continuum limit is performed.
### II.4 Dispersion relation and continuum limit
Let us consider in more detail the translationally invariant case in which
$f_{n-1,n}=f$ and $\mu_{n}=\mu$ for all $n$. In this case, the continuous wave
equation (16) becomes
$\partial_{t}^{2}\Psi(x,t)-c^{2}\partial_{x}^{2}\Psi(x,t)=0\,,$ (18)
where $c$ is the speed of propagation.
Let us recall how this limit was performed in graphity3 . Let us first
introduce a discrete Fourier transform in the spatial coordinate and a
continuous Fourier transform in the temporal coordinate, given by
$\Psi_{n}(t)=\frac{1}{\sqrt{L}}\sum_{k=0}^{L-1}\tilde{\Psi}_{k}(t)e^{-\operatorname{\mathrm{i}}\frac{2\pi}{L}nk}\,,$
(19)
and
$\tilde{\Psi}_{k}(t)=Ae^{\operatorname{\mathrm{i}}\omega_{k}t}+Be^{-\operatorname{\mathrm{i}}\omega_{k}t}$.
After a straightforward calculation, we find that the relation between
$\omega_{k}$ and $k$ is given by
$\omega_{k}\ c=\sqrt{2}\ \sqrt{1-\cos\left(\frac{2\pi}{L}k\right)}.$ (20)
Now we can rescale $\omega_{k}\rightarrow\tilde{\omega}_{k}/L$ (or
equivalently $c$) and find that
$\tilde{\omega}_{k}c=L\sqrt{2}\sqrt{1-\cos\left(\frac{2\pi}{L}k\right)},$ (21)
and, therefore,
$\lim_{L\rightarrow\infty}\tilde{\omega}_{k}(L)\approx 2\pi\frac{k}{c}.$
That is, only modes that are slow with respect to the time scale set by $c$
see the continuum. Note that by rescaling the speed of propagation $c$, the
continuum limit can be obtained by a double scaling limit,
$E_{\textrm{hop}}\rightarrow E_{\textrm{hop}}/L$ and $L\rightarrow\infty$ for
lattice size $L$. In this limit, the probability density has a Lorentz
invariant dispersion relation.
## III A non-local state distribution
In this section, we show the effect of having a quantum superposition of graph
in (24) on the equation (II.3).
### III.1 The effect of a quantum superposition of graphs
In order to do the explicit calculation, we will modify the Bose-Hubbard
interaction. Let us consider a one-dimensional Bose-Hubbard of the form,
$\widehat{H}=\sum_{i}A_{i,i-1}(\hat{a}_{i}^{\dagger}\hat{a}_{i}+h.c.)$ (22)
and then consider its generalization, from
$A_{i,j}=\delta_{j,i-1}+\delta_{j,i+1}$, to
$\widehat{A}_{i,j}=\widehat{N}_{ij}$, with
$\widehat{N}_{ij}=\hat{b}^{\dagger}_{ij}\hat{b}_{ij}$ and
$\hat{b}_{ij}$,$\hat{b}^{\dagger}_{ij}$ the ladder operators on the Hilbert
space of the link $ij$. $\widehat{N}_{ij}$ is then the number operator on the
Hilbert space of the graph, as usually considered in Quantum Graphity. This
allows, instead of using fixed classical graphs, fixed quantum graphs, where
the state $|\psi_{graph}\rangle$ is superposition of different graphs. The
full quantum hamiltonian for the system is, as usual, on an Hilbert space of
the form
$|\psi_{total}\rangle=Span\\{|\psi_{graph}\rangle\otimes|\psi_{bosons}\rangle\\}.$
Using this, we want now to repeat the same calculation we performed in the
previous paper, i.e. compute:
$\partial_{t}^{2}\psi_{z}(t)=-i\
Tr\\{[\widehat{H},[\widehat{H},\widehat{\rho}(t)]]\widehat{N}^{\prime}_{z}\\},$
(23)
with $\psi_{n}=\langle\widehat{N}^{\prime}_{n}\rangle$,
$\widehat{N}^{\prime}_{n}$ number operator on the bosons defined on the node
$n$, and $\widehat{\rho}$ the density matrix on the total system. Let us
assume that the graph is not dynamical. We will also to use the Born
approximation, that is,
$\widehat{\rho}(t)\approx\widehat{\rho}_{g}\otimes\widehat{\rho}_{b}(t),$ (24)
with $\widehat{\rho}_{g}$ the density matrix of the graph and
$\widehat{\rho}_{b}(t)$ the density matrix of the bosons. This approximation
allows us to consider a particle disentangled enough from the graph to be
“followed” using the equation (16). It is also a physical requirement, which
accounts for the existence of the particle on its own. In general, we expect
that at long times the full hamiltonian thermalizes to a specific graph,
depending on the parameter of the Hamiltonian which defines the metastable
state. Later on, we will rescale the coupling constant of the hopping
Hamiltonian in order to obtain the continuum limit. Thus, one could think that
this rescaling affects the state of the graph at infinity. However, the
hopping of the bosons allows the graph to thermalize, as it has been shown in
graphity2 . Rescaling this constant, just changes the time it takes for the
system to thermalize, but not the asymptotic state of the graph. As a matter
of fact, we do not know yet a Hamiltonian which gives a specific graph state
asymptotically. However, the results of florian in two dimensions and those
of graphity1 , support the conjecture that, in general, such a Hamiltonian
exists. For the time being, it is fair to say that the ground state of Quantum
Graphity coupled to a thermal bath are rotational invariant graphs konopka .
Thus, these graphs can at least be generated by a known effectively
2d-dimensional model.
Based on these considerations, we conjecture the following graph state,
$|\psi_{graph}\rangle=|\psi_{cl}\rangle+|\psi_{nl}\rangle$ with
$\langle\psi_{cl}|\psi_{nl}\rangle=0$. $|\psi_{nl}\rangle$ is a correction to
the classical graph state $|\psi_{cl}\rangle$ considered in graphity3 that we
will discuss (and construct) in the next section. For the time being, let us
consider the effect of this correction on eqn. (II.3). We have
$\widehat{\rho}_{g}=|\psi_{graph}\rangle\langle\psi_{graph}|$. Thus:
$\widehat{\rho}_{g}=|\psi_{cl}\rangle\langle\psi_{cl}|+|\psi_{nl}\rangle\langle\psi_{nl}|+(|\psi_{nl}\rangle\langle\psi_{cl}|+|\psi_{cl}\rangle\langle\psi_{nl}|).$
(25)
Let us now evaluate these traces. A straightforward calculation shows that,
$\displaystyle-\frac{E_{hop}^{2}}{\hbar^{2}}\partial_{t}^{2}\psi_{n}$
$\displaystyle=$ $\displaystyle Tr\
\\{(\widehat{H}^{2}\widehat{\rho}+\widehat{\rho}\widehat{H}^{2}-2\widehat{H}\widehat{\rho}\widehat{H})\widehat{N}_{z}\\}$
(26) $\displaystyle=$ $\displaystyle
2\sum_{ij,mn}\big{[}Tr\\{\widehat{A}_{ij}\widehat{A}_{mn}\widehat{\rho}_{g}\\}Tr\\{\hat{a}_{i}^{\dagger}\hat{a}_{j}\hat{a}_{m}^{\dagger}\hat{a}_{n}\widehat{\rho}_{b}\widehat{N}_{z}\\}$
$\displaystyle-$ $\displaystyle
Tr\\{\widehat{A}_{ij}\widehat{\rho}_{g}\widehat{A}_{mn}\\}Tr\\{\hat{a}_{i}^{\dagger}\hat{a}_{j}\widehat{\rho}_{b}\hat{a}_{m}^{\dagger}\hat{a}_{n}\widehat{N}_{z}\\}\big{]}.$
We now substitute the equation for $\widehat{\rho}_{g}$, and obtain:
$-\frac{E_{hop}^{2}}{\hbar^{2}}\partial_{t}^{2}\psi_{n}=\tilde{\triangle}\psi_{n}(t)+C_{n}(t),$
with $\tilde{\triangle}\psi_{n}(t)$ is the discrete second derivative and
$C_{n}(t)$ is:
$\displaystyle C_{n}(t)$
$\displaystyle=2\sum_{ij,mn}\big{[}P_{ijmn}Tr\\{\hat{a}_{i}^{\dagger}\hat{a}_{j}\hat{a}_{m}^{\dagger}\hat{a}_{n}\widehat{\rho}_{b}\widehat{N}_{z}\\}$
(27) $\displaystyle-
Q_{ij}Q_{mn}Tr\\{\hat{a}_{i}^{\dagger}\hat{a}_{j}\widehat{\rho}_{b}\hat{a}_{m}^{\dagger}\hat{a}_{n}\widehat{N}_{z}\\}\big{]},$
with:
$P_{ijmn}=\langle\psi_{nl}|\widehat{A}_{ij}\widehat{A}_{mn}|\psi_{nl}\rangle,$
$Q_{ij}=\langle\psi_{nl}|\widehat{A}_{ij}|\psi_{nl}\rangle,$
where we used the orthogonality condition
$\langle\psi_{nl}|\psi_{cl}\rangle=0$.
Our task now is to evaluate these two quantities on different classes of
interesting states.
## IV The choice of the quantum state for the graph
Figure 3: The intuitive picture of non-local links inserted in the graph.
Let us now introduce the states on which we will evaluate the quantities
defined in the previous section, $P_{ijmn}$ and $Q_{ij}$. Motivated by the
fact that we can reduce using the translationally symmetric graphs to one
line, we will restrict our attention to a one-dimensional lattices. These
states will resemble coherent states as considered in Loop Quantum Gravity. In
principle, they could be obtained as correction to the ground state due to a
non-zero temperature bath in Quantum Graphity.
Let us consider first a metric on the classical graph, with $d(i,j)$ the
distance between the nodes of the classical graph $|\psi_{cl}\rangle$, with
all the ordinary properties of distances. On a one-dimensional line this
distance could be, for instance, given by $|i-j|$. Let us then construct
states with non-local links on top. We want to penalize states with too long
non-local links. We then introduce a factor $\rho(i,j)$, which depends on a
distance $d(i,j)$ evaluated on the base graph, assuming that $d(i,j)\geq 0$,
and a parameter $l$ describing how non-local the links are w.r.t. the length
of the graph. Then we define the operator:
$\widehat{T}_{l}=\sum_{i<j}\rho(i,j)\ \hat{a}_{ij}^{\dagger},$ (28)
with
$\sum_{i<j}\rho(i,j)^{2}=1,$ (29)
which ensures that $\rho(i,j)^{2}$ can be interpreted as a classical
probability distribution. When applied to $|\psi_{cl}\rangle$ this operator
generates a superposition of all the possible non-local links which can be
created on $|\psi_{cl}\rangle$, with a factor that with the distance of the
links,
$|\psi^{1}_{nl}\rangle=\widehat{T}_{l}|\psi_{cl}\rangle,$ (30)
and we can imagine to apply this operator several times to create more non-
local links,
$|\psi^{R}_{nl}\rangle=\widehat{\frac{T_{l}^{R}}{R!}}|\psi_{cl}\rangle.$ (31)
The meaning to give to $l$ is thus that of a cut-off in the length of these
non-local links. Note that we can bias the number of links on which we want to
peak the quantum non-local state the same way,
$\widehat{\mathscr{T}}^{K}_{l}=\sum_{s=1}^{\infty}\frac{K^{s}}{s!}\widehat{T}^{s}_{l}=e^{K\widehat{T}_{l}}-1.$
(32)
We see then that we can write the quantum state for the graph in the
convenient form:
$|\psi_{nl}\rangle=\big{[}1+e^{K\widehat{T}_{l}}\big{]}|\psi_{cl}\rangle.$
(33)
This state depends explicitly on two parameters, $l$ and $K$, and on the
classical graph together with its distance. On this state we now want to
evaluate:
$P_{ijmn}=\langle\psi_{nl}|\widehat{A}_{ij}\widehat{A}_{mn}|\psi_{nl}\rangle=\langle\psi_{cl}|\widehat{\mathscr{T}}_{l}^{K{\dagger}}\widehat{A}_{ij}\widehat{A}_{mn}\widehat{\mathscr{T}}_{l}^{K}|\psi_{cl}\rangle,$
(34)
and
$Q_{ij}=\langle\psi_{nl}|\widehat{A}_{ij}|\psi_{nl}\rangle=\langle\psi_{cl}|\widehat{\mathscr{T}}_{l}^{K{\dagger}}\widehat{A}_{ij}\widehat{\mathscr{T}}_{l}^{K}|\psi_{cl}\rangle.$
(35)
Let us then consider first the average. We note that, since $\widehat{A}_{ij}$
acts like a projector, and states with different powers of the
$\widehat{T}_{l}$ operators are orthogonal, we can write:
$Q_{ij}=\sum_{s=1}^{\infty}\frac{K^{2s}}{{s!}^{2}}\langle\psi_{cl}|T^{{\dagger}s}_{l}\widehat{A}_{ij}T^{s}_{l}|\psi_{cl}\rangle.$
(36)
To clarify the idea, let us consider the case in which we add just a link. In
this case, the state is the sum over all possible links which can be created,
with a factor $\rho^{2}(i,j)$.This link can be created in one way only, and so
$\widehat{A}_{ij}$ projects on the only state which can be non-zero. A very
straightforward calculation shows that
$\langle\psi_{cl}|T^{{\dagger}1}_{l}\widehat{A}_{ij}T^{1}_{l}|\psi_{cl}\rangle=2\
\rho^{2}(i,j).$ (37)
For the higher order term, we instead have:
$\langle\psi_{cl}|T^{{\dagger}s}_{l}\widehat{A}_{ij}T^{s}_{l}|\psi_{cl}\rangle=\rho^{2}(i,j)\sum_{i_{1},j_{1},\cdots,i_{s-1},j_{s-1}}\prod_{l=1}^{s-1}\rho^{2}(i_{l},j_{l}).$
(38)
It is easy to see that
$\sum_{i_{1},j_{1},\cdots,i_{s-1},j_{s-1}}\prod_{l=1}^{s}\rho^{2}(i_{l},j_{l})\approx
2^{s}\ s\ (l\ L)^{s},$ (39)
due to the fact that the integration is over the line, while the distribution
has an extension of circa $l$ combinatorial points. The factor $2^{s}$ comes
from the fact that there are 2 points we are summing over and the $s$ factor
from the $s$ sums appearing in $T^{s}_{l}$. Thus, we can write:
$\langle\psi_{cl}|T^{{\dagger}s}_{l}\widehat{A}_{ij}T^{s}_{l}|\psi_{cl}\rangle=c_{s}\
\rho^{2}(i,j)\ 2^{s}\ s\ (L\ l)^{s-1}.$ (40)
In principle, given a distribution, we can calculate this factor from eq.
(39). We will calculate these factors later for two particular distributions.
Plugging eqn. (40) into $Q_{ij}$, we obtain
$Q_{ij}=\sum_{s=1}^{\infty}\frac{K^{2s}}{{s!}^{2}}\ 2^{s}\ s\ (L\
l)^{s-1}\rho^{2}(i,j)c_{s}=\rho^{2}(i,j)R(K,l\ L),$ (41)
with:
$R(K,l\ L)=\sum_{s=1}^{\infty}\frac{K^{2s}}{{s!}^{2}}\ c_{s}\ 2^{s}\ s\ (l\
L)^{s-1},$ (42)
and, therefore,
$Q_{ij}Q_{mn}=\rho^{2}(i,j)\rho^{2}(m,n)R(K,l\ L)^{2}.$ (43)
We can, in fact, do an analogous calculation for $P_{ijmn}$ and find that:
$P_{ijmn}=\rho^{2}(i,j)\rho^{2}(m,n)L(K,l\ L),$ (44)
with:
$L(K,l\ L)=\sum_{s=1}^{\infty}\frac{K^{2s}}{{s!}^{2}}c_{s}(l\ L)^{s-2}\ 2^{s}\
s.$ (45)
Going back to the original problem, we find that the correction to the
discrete Lorentz equation is:
$\displaystyle C_{z}=$ $\displaystyle
2\sum_{ij,mn}\rho^{2}(i,j)\rho^{2}(m,n)\big{[}L(K,l\
L)Tr\\{\hat{a}_{i}^{\dagger}\hat{a}_{j}\hat{a}_{m}^{\dagger}\hat{a}_{n}\widehat{\rho}_{b}\widehat{N}_{z}\\}$
(46) $\displaystyle-R(K,l\
L)^{2}Tr\\{\hat{a}_{i}^{\dagger}\hat{a}_{j}\widehat{\rho}_{b}\hat{a}_{m}^{\dagger}\hat{a}_{n}\widehat{N}_{z}\\}\big{]}.$
If we define:
$S(K,l\ L)=(l\ L)\ R(K,l\ L)=(l\ L)^{2}\ L(K,l\ L),$ (47)
with $S(K,l\ L)=\sum_{s=1}^{\infty}\frac{K^{2s}}{{s!}^{2}}\ c_{s}\ (l\ L)^{s}\
2^{s}\ s$, then we obtain:
$C_{z}(t)=2\sum_{ij,mn}\rho^{2}(i,j)\rho^{2}(m,n)S(K,l\
L)\big{[}Tr\\{\hat{a}_{i}^{\dagger}\hat{a}_{j}\hat{a}_{m}^{\dagger}\hat{a}_{n}\widehat{\rho}_{b}\widehat{N}_{z}\\}-S(K,l\
L)Tr\\{\hat{a}_{i}^{\dagger}\hat{a}_{j}\widehat{\rho}_{b}\hat{a}_{m}^{\dagger}\hat{a}_{n}\widehat{N}_{z}\\}\big{]}.$
(48)
We see that the function $S(K,l\ L)$ depends, as a matter of fact, on
$\xi=K\sqrt{l\ L}$, $S(K,l\ L)\equiv S(\xi)=\sum_{s=1}^{\infty}\ c_{s}\
[\frac{\xi^{s}}{{s!}}]^{2}\ 2^{s}\ s$. A plot of the function $S(K,lL)$ can be
found in Fig. 4.
Figure 4: A plot of the function $S(x)$ which appears in eq. (48).
The traces can be evaluated, as done in (graphity3 ), and the result is:
$C_{z}(t)=2\frac{1}{(l\ L)^{2}}\sum_{j}\rho^{4}(z,j)S(K,l\ L)(\psi_{z}-S(K,l\
L)\psi_{j}).$ (49)
Some comments are now in order. First of all, note that the equation has the
shape of a second derivative. To understand this, we can look at a term of the
form $\sum_{|k|\geq 2}J(k)(\psi_{z}-\psi_{z+k})$. This term can be written as:
$\sum_{k}J(k)\cdots=-J(2)\big{(}\psi_{z+2}-2\psi_{z}+\psi_{z-2}\big{)}-J(3)\big{(}\psi_{z+3}-2\psi_{z}+\psi_{z-3}\big{)}-J(4)\cdots.$
This is a sum of discrete second derivatives with a non-local mass,
$J(k)\big{(}\psi_{z+k}-2\psi_{z}+\psi_{z-k}\big{)}=-J(k)\big{(}\psi_{z+k-1}+\psi_{z-k+1}\big{)}-J(k)\sum_{i=2}^{k-1}\big{(}\psi_{z+i+1}-2\psi_{z+i}+\psi_{z+i-1}\big{)},$
so we expect, in the end, to obtain a mass term out of this equation and, when
we will have rearranged all the terms, we will.
Note that, for the case $c_{s}=1$, $S(\xi)=1$ for $\xi=0.903$, and so
$K=\frac{0.903}{\sqrt{l\ L}}$. We then see that $K^{2}$ plays the role of the
density of non-local links per units of $l\ L$.
To end this section, we have to calculate the norm of this state. This can be
written as:
$|\langle\psi_{nl}|\psi_{nl}\rangle|=\sqrt{1+\langle\psi_{cl}|e^{K\widehat{T}^{\dagger}_{l}}e^{K\widehat{T}_{l}}|\psi_{cl}\rangle^{2}+2\Re\\{\langle\psi_{cl}|e^{K\widehat{T}_{l}}|\psi_{cl}\rangle\\}},$
(50)
which reads,
$|\langle\psi_{nl}|\psi_{nl}\rangle|=\sqrt{1+\big{(}\langle\psi_{cl}|e^{K\widehat{T}^{\dagger}_{l}}e^{K\widehat{T}_{l}}|\psi_{cl}\rangle^{2}-1\big{)}}\\}$
(51)
and substituting for the $\mathscr{T}$ operators, we finally find
$\mathscr{N}=|\langle\psi_{nl}|\psi_{nl}\rangle|=\sqrt{1+\big{(}\sum_{s=1}\frac{K^{2s}}{(s!)^{2}}\sum_{i<j}\rho(i,j)^{2}\big{)}^{2}}=\sqrt{1+\big{(}\sum_{s=1}\frac{K^{2s}}{(s!)^{2}}2^{s}s\big{)}^{2}}=\sqrt{1+S^{2}(K,l\
L)}.$ (52)
We can thus normalize the graph state by dividing by a factor of
$\mathscr{N}$.
## V The modified dispersion relation due to disordered locality
The general case. We will now discuss the continuum limit. As we have seen,
the continuum limit is obtained by rescaling
$E_{hop}\rightarrow\tilde{E}_{hop}/L$ and then sending $L\rightarrow\infty$.
Please note that $E_{hop}$ appears whenever we hop with a particle, so in
these calculations it appears everywhere but in the $\partial_{t}^{2}$ term.
In order to perform the continuum limit, first we have to make sense of the
quantity $(\psi_{z}-S(K,l\ L)\psi_{j})$ at least for the flat case, which we
know correspond to Lorentz from graphity3 . We can add and subtract,
$\displaystyle(S(K,l\ L)-1)\psi_{z}+(S(K,l\ L)\psi_{z}-S(K,l\ L)\psi_{j})=$
$\displaystyle=(S(K,l\ L)-1)\psi_{z}+S(K,l\
L)(\psi_{z}-\psi_{z-1}+\psi_{z-1}+\psi_{z-2}-\cdots-\psi_{j}).$
In the continuum limit this becomes $(S(K,l\ L)-1)\psi(z,t)+S(K,l\
L)\int_{j}^{z}\partial_{\xi}\psi(\xi,t)d\xi$, and thus $C_{z}(t)$ reads:
$C_{z}(t)=\int_{L}dx\ \rho^{4}(z,x)[\frac{(S(K,l\ L)-1)S(K,l\ L)}{(l\
L)^{2}}\psi(z,t)+\frac{S^{2}(K,l\ L)}{(l\
L)^{2}}\int_{x}^{z}\partial_{\xi}\psi(\xi,t)d\xi],$ (53)
which is:
$C_{z}(t)=\psi(z,t)\int_{L}dx\ \rho^{4}(z,x)\frac{(S(K,l\ L)-1)S(K,l\ L)}{(l\
L)^{2}}+\frac{S^{2}(K,l\ L)}{(l\
L)^{2}}\int_{L}\rho^{4}(z,x)\int_{x}^{z}\partial_{\xi}\psi(\xi,t)\ d\xi\ dx.$
(54)
This can be written as:
$C_{z}(t)=\psi(z,t)F(K,l\ L)+O(K,l\
L)\int_{L}\rho^{4}(z,x)\int_{x}^{z}\partial_{\xi}\psi(\xi,t)\ d\xi\ dx,$
with $F(K,l\ L)=\int_{L}dx\rho^{4}(z,x)\frac{(S(K,l\ L)-1)S(K,l\ L)}{(l\
L)^{2}}$,$O(K,l\ L)=\frac{S^{2}(K,l\ L)}{(l\ L)^{2}}$. Please note here that
these steps have been performed naively, though we have an explicit dependence
on $L$ in $S$. It is important to point out that the only way to keep the
function $S(l\ L,K)$ finite is to rescale the quantity
$K^{2}l\approx\frac{\tilde{K}^{2}\tilde{l}}{L}$. To keep the discussion
simple, let us discuss this point at the end of the section. $L$ is the
combinatorial length of the 1-d lattice we are considering, and over which
$\psi(x,t)$ is defined. Thus the equation of motion for the flat case is
given, in the continuum, by:
$[\partial_{t}^{2}-c^{2}\big{(}1+S^{2}(K,l\ L)\big{)}\partial_{z}^{2}-F(K,l\
L)]\psi(z,t)=O(K,l\
L)\int_{L}\rho^{4}(z,x)\int_{x}^{z}\partial_{\xi}\psi(\xi,t)\ d\xi\ dx,$
which is an integro-differential equation for the field integrated over the
line, which shows the strong non-local character of the equation.
We note that there is a contribution to the speed of propagation of the
signal, due to the fact that particle can hop on many more graphs than the
single classical one. This factor contributes with a $c^{2}S^{2}(K,l\ L)$
added to the effective speed $c^{2}$. Let us stress that this contribution is
merely due to the fact that there are many more graphs in the superposition,
and not due to the fact that the particle can hop further: this is kept track
of in the $C_{z}(t)$ term of the equation. Also, we see that $F(l,K)$ becomes
a mass, due to non-locality, while on the r.h.s. there a new term appears. We
can further reduce the equation by evaluating the integrals. It is clear that
in order to have a finite result, which is physically expected, we have to
rescale at this point only $l\approx\tilde{l}/L$, keeping $K^{2}$ independent
from $L$. Said this, we see that the distribution itself, when is well chosen,
becomes a $\delta$ function and therefore the models becomes local again.
Let us now calculate the terms at the leading order in $1/L$, since that is
what we are interested in. The discrete differential equation becomes:
$[\partial_{t}^{2}-c^{2}\big{(}1+S^{2}(K,l\
L)\big{)}\tilde{\partial}_{z}^{2}-\tilde{F}(K,l\ L)]\psi_{z}(t)=-O(K,l\
L)\sum_{x=0}^{L}\rho^{4}(z,x)\psi_{z}(t),$ (55)
where $\tilde{\partial}_{z}^{2}$ is the discrete spatial second derivative.
Using now (19), we see that the dispersion relation for the field becomes:
$\omega_{k}c\big{(}1+S^{2}(K,l\
L)\big{)}=\sqrt{2}\sqrt{1-\cos\big{(}\frac{2\pi}{L}k\big{)}+\tilde{F}(K,l\
L)+\tilde{\rho}^{4}(k)O(K,l\ L)}$
Please note that with this rescaling of $K$, we have that $S(K,l\ L)$ can be
expanded in even powers of $1/L$:
$S(K,l\ L)=2\frac{\tilde{K}^{2}\ l\ L}{L^{2}}+8\frac{\tilde{K}^{2}(l\
L)^{2}}{L^{4}}+\cdots.$
Thus, we see that the superluminal effect, which is, the factor $1+S^{2}(K,l\
L)$, becomes one in the limit $L\rightarrow\infty$; also, in the same limit,
only the part quadratic in $K$ survives. At this point the equation would
become, in the continuum:
$[\partial_{t}^{2}-c^{2}\partial_{z}^{2}-\tilde{F}(K,l\ L)]\psi(z,t)=-O(K,l\
L)\int_{L}\rho^{4}(z,x)\psi(z,t)\ dz$ (56)
with $\tilde{F}(k,l\ L)=F(K,l\ L)+O(K,l\ L)\int_{L}\rho^{4}(z,x)\ dx$. Note
that, while $\tilde{F}$ might seem to be dependent on the point $z$, being
$\tilde{F}$ dependent on $z-x$ and integrated over $x$, it is indeed
independent from it. In particular, if we define $l\ L\equiv\xi$, in the limit
$L\rightarrow\infty$ and with the rescaling of $K$ and $l$, $S(K,l\
L)\rightarrow 2\tilde{K}^{2}\xi$. We see now that the only way to obtain the
continuum dispersion relation by rescaling $c\rightarrow\tilde{c}/L$, as done
for the single-graph state, is to rescale also $K$, with
$K\rightarrow\tilde{K}/L$.
Just as an exercise, we can insert a trivial spatially-constant solution,
which then becomes of the form $\partial_{t}^{2}\psi(t)=R(K,l\ L)\psi(t)$.
where $R(K,l\ L)=F(K,l\ L)+2\ O(K,l\ L)\ \int_{L}\ \rho^{4}(z,x)\ dx$. Note
that this quantity is always positive, so constant solutions are stable. Let
us try to find a generic solution, instead. Let’s do it for the equation:
$[\partial_{t}^{2}-\tilde{c}^{2}\partial_{x}^{2}+\tilde{c}^{2}q]\psi(x,t)=-\tilde{c}^{2}\int_{L}\sigma(z,x)\psi(y,t)\
dy.$ (57)
Since the equation is linear in the field $\psi$, we can solve it by means of
a Fourier transform. We then look at the dispersion relation for the function
$\psi(x,t)$, with $q$ and $P$ generic functions. We can do it by Fourier
transform. In this case, the integral on the right, being a convolution,
becomes just the product of the Fourier transform of the single functions.
Thus we have:
$-\omega^{2}+k^{2}\tilde{c}^{2}+\tilde{c}^{2}q=-\tilde{c}^{2}\sigma(k),$
and we have that:
$\omega=\pm c\sqrt{\tilde{k}^{2}+q+\tilde{\sigma}(k)}.$
Now, of course $\tilde{\sigma}(k)$ depends on the distribution of non-local
links that we inserted in the wavefunction of the graph.
Two specific distributions. Let us consider two specific cases:
* •
$\rho_{1}(x-y)=\pi^{\frac{1}{4}}\ \sqrt{l}\ e^{-\frac{(x-y)^{2}}{2l^{2}}}$;
* •
$\rho_{2}(x-y)=\sqrt{2\ l}\ e^{-\frac{|x-y|}{2l}}$.
In these cases we find, using standard tables of Fourier transforms:
* •
$\tilde{\sigma}_{1}(k)=e^{-\frac{k^{2}}{a}}$;
* •
$\tilde{\sigma}_{2}(k)=\frac{a}{a^{2}+k^{2}}$.
and thus, keeping track of all the factors, we obtain:
$\omega_{1}=\pm\frac{1}{c}\sqrt{\frac{k^{2}}{1+S^{2}(K,l\
L)}+\tilde{F}_{1}(K,l\ L)+O(K,l\ L)e^{-\frac{k^{2}l^{2}}{8}}},$ (58)
and
$\omega_{2}=\pm\frac{1}{c}\sqrt{\frac{k^{2}}{1+S^{2}(K,l\
L)}+\tilde{F}_{2}(K,l\ L)+\frac{2O(K,l\ L)}{\pi}\frac{l^{2}}{l^{2}+k^{2}}},$
(59)
with $\tilde{F}_{1}(K,l\ L)=\sqrt{2}\ O^{2}(K,l\ L)$ and $\tilde{F}_{2}(K,l\
L)=O^{2}(K,l\ L)$, which can be calculated by evaluating
$\int_{L}\rho_{i}^{4}(x-y)dx$. We have that $S^{2}(K,l\
L)=4\tilde{K}^{4}\xi^{2}/L^{4}$ and thus can be neglected with respect to $1$.
Also, since the $c$ contribute with a factor of $L^{2}$ within the square
root, also $S^{2}$ can be neglected, and it contributes only the mass term in
the $L$. Now we note a nice property: both the two distributions go to $0$ for
$k\rightarrow\infty$, that is, at high energy the dispersion relations become
Lorentz again. We see then that the total effect the one of having an
effective scale-dependent mass, which runs from one mass to another one, in
both cases:
$m_{1}(k)=\frac{1}{l\ L}\sqrt{S^{2}(K,l\ L)+S(K,l\
L)e^{-\frac{k^{2}l^{2}}{8}}},$ (60) $m_{2}(k)=\frac{1}{l\ L}\sqrt{S^{2}(K,l\
L)+S(K,l\ L)\frac{2}{\pi}\frac{l^{2}}{l^{2}+k^{2}}}.$ (61)
The masses which are intertwined are given by
$m_{1}(0)=\frac{1}{l\ L}\sqrt{S^{2}(K,l\ L)+S(K,l\ L)},\ \ \ \
m_{1}(\infty)=\frac{S(K,l\ L)}{l\ L},$ (62) $m_{2}(0)=\frac{1}{l\
L}\sqrt{S^{2}(K,l\ L)+\frac{2}{\pi}S(K,l\ L)},\ \ \ \
m_{2}(\infty)=\frac{S(K,l\ L)}{l\ L}.$ (63)
This property, of intertwining two different masses between $k=0$ and
$k=\infty$ is shared by any function which is at least $\mathscr{C}^{1}$. It
is remarkable, instead, that the mass at $k=\infty$ does not depend on the
distribution we inserted at hand. In fact, any $\mathscr{C}^{r}$ distribution
will lead to a Fourier transformed distribution which goes to zero at
$k=\infty$ as $1/k^{r}$ and thus tend to a finite value for the mass. Note
that, if we send $l\rightarrow 0$, as required to have $S$ finite, the
dependence on the scale seems to disappear, leaving a Lorentz dispersion
relation with a mass which depends on the function $S$. However, we have to
remember that, in fact these Fourier distributions come from the discrete
dispersion relation. There, the distributions depend on $2\pi k/L$. If we
define $\tilde{k}L=k$, then we have that the distributions cancel out the
dependence on $L$, leaving exactly (62) and (63) but dependent on this new
momentum $\tilde{k}$. Still, this mass depends on the distribution we have
chosen through $\tilde{\sigma}_{i}(\tilde{k}=0)$ and so it has a valuable
effect. We plot the running of $(l\ L)m_{i}(\tilde{k})$ as a function of
$x=K^{2}\ l\ L$, for the case $L=1$ in Fig. (5) and $L=\infty$ in Fig. (6).
Figure 5: The running of $m_{i}(\tilde{k})$ for $x=0.1$, $l=1$ and $L=1$.
Figure 6: The running of $m_{i}(\tilde{k})$ for $x=0.1$, $l=1$ and $L=\infty$.
We would like to point out that the appearence of a mass which is square-
positive is rather surprising. The physical reason is that, before starting
the calculation, we would have expected that the presence of these non-local
links would have shown a superluminal effect due to the non-local links
themselves. However, the effective speed of propagation is higher because of
the superposition of the graphs and not the non-local links. Indeed, the non-
local links contributed only in the mass, thus the term $C_{z}(t)$ additional
to the differential equation we obtained. Besides, this mass is square-
positive, thus it is an effective mass and not a tachyonic one, which we would
have expected from the presence of non-local links on physical grounds. The
fact that it is square positive comes merely from the fact that the equation
comes from a quantum mechanical average, and thus the terms appear squared.
## VI Conclusions
One of the most striking theoretical consequences of General Relativity is the
existence of wormholes and black holes. While the second is currently
investigated experimentally, less is known about the first. Here we discussed
something which in principle is very similar, quantum states which violate
macro-locality. Besides, it could be that the quantum state of the Universe is
a superposition of spacetimes with non-local links. In the present paper we
considered such a possibility in a toy model constructed using the framework
Quantum Graphity. In order to do so, we had to extend the results of graphity3
to a case in which the quantum state of the background is a superposition of
many graph states. The superposition of these graphs was chosen so that it is
dominated by a graph on which, as we showed in earlier papers, the expectation
values of number operators of the bosons hopping on it satisfy a closed
equation for probability density in the classical regime, i.e. a wave
equation. We extended the formula previously obtained and studied a particular
case: graphs which violate micro- and macro- locality. As discussed, a
violation of macro-locality can be interpreted, within the model, as the
presence of spatial non-local links in the background spacetime. This is a
concrete example of a quantum foam within the framework of Quantum Graphity
graphity1 ; graphity2 . The graph state was chosen on the basis of what we
know from low energy physics, which is that Lorentz invariance is satisfied up
and above the Planck scalePlanckdata . We also used a class of graphs
introduced in graphity3 , rotationally invariant graphs. By exploiting their
symmetry, the problem can be reduced to a 1-dimensional one, i.e. Bose-Hubbard
model on a line with specific couplings depending on the connectivity of the
graph. We thus constructed the states that are corrections to the low-energy
physics by assuming that the non-local links are suppressed by a length
according to a certain distribution. The length is measured by a combinatorial
distance based on the low energy graph and which defines the state. We studied
for the cases $d(x,y)=(x-y)^{2}$ and $d(x,y)=|x-y|$. We found that, in the
continuum limit, there is no superluminal effect on the low-energy physics,
i.e. the speed of propagation is intact. However, there is an appearance of a
mass dependence on the constants of the distribution and that can be
calculated within the model. These masses are square-positive and thus do not
violate the physics of the restricted Lorentz group, i.e., are not tachyonic.
A simple analysis showed that this mass runs with the energy scale and, in
particular, runs to zero at high energy. It is interesting to ask whether a
similar phenomenon happens for the other fields. This analysis suggests the
possibility that a quantum foam could contribute to the mass of a quantum
field. As suggested in marksm and smochan , the possibility of having non-
local link states within Loop Quantum Gravity is very natural. Also, it has
been suggested that these states could contribute to the dark energy puzzle.
The results of the present paper suggests that, as in smochan , the quantum
foam contributes to the mass of fields hopping on such a superposition of
spacetimes. We believe that such possibility needs to be further investigated.
Aknowledgements
Research at Perimeter Institute is supported by the Government of Canada
through Industry Canada and by the Province of Ontario through the Ministry of
Research & Innovation. This research has been made possible by financial
support of the Templeton and Humboldt Foundations.
## References
* (1) J.A. Wheeler, K. Ford, Geons, black holes and quantum foam: a life in physics, W.W. Norton Company, Inc., New York (1998)
* (2) M. Visser, Lorentzian Wormholes: from Einstein to Hawking, American Institute of Physics Press (Woodbury, New York) 1992.
* (3) T. Konopka, F.Markopoulou and L. Smolin, arXiv:hep-th/0611197 ; T. Konopka, F. Markopoulou, S. Severini, Phys. Rev. D 77, 104029 (2008), arXiv:0801.0861; F. Caravelli, F. Markopoulou, Phys. Rev. D 84 024002 (2011), arXiv:1008.1340
* (4) A. Hamma, F. Markopoulou, S. Lloyd, F. Caravelli, S. Severini, K. Markstrom, Phys. Rev. D 81, 104032 (2010), arXiv:0911.5075
* (5) F. Caravelli, A. Hamma, F. Markopoulou, A. Riera, arXiv:1108.2013
* (6) T. Konopka, Phys. Rev. D78 044032 (2008), [arXiv:0805.2283 [hep-th]].
* (7) K.S. Stelle, Phys.Rev. D16 (1977) 953-969; L. Modesto, arXiv:1107.2403; T. Biswas, E. Gerwick, T. Koivist, A. Mazumdar, arXiv:1110.5249;
* (8) F. Markopoulou, L. Smolin, Class.Quant.Grav. 24 (2007) 3813-3824, arXiv:gr-qc/0702044.
* (9) C. Rovelli, Quantum Gravity, Cambridge University Press, Cambridge (2004); A. Ashtekar, Class. Quant. Grav. 21, R53 (2004), arXiv:gr-qc/0404018; T. Thiemann, [gr -qc/0110034]; A. Perez, arXiv:gr-qc/0409061;
* (10) R. Sorkin, Proceedings of the Valdivia Summer School, edited by A. Gomberoff, D. Marolf, arXiv:gr-qc/0309009.
* (11) F. Markopoulou, A. Hamma, New J. Phys. 13:095006 (2011), arXiv:1011.5754;
* (12) C. Prescod-Weinstein, L. Smolin, Phys. Rev. D80 063505 (2009), arXiv:0903.5303.
* (13) F. Conrady, J.Statist.Phys.142:898 (2011), arXiv:1009.3195
* (14) Planck collaboration, Nature Physics 462, 331-334 (2009);
|
arxiv-papers
| 2012-01-16T10:39:44 |
2024-09-04T02:49:26.335484
|
{
"license": "Public Domain",
"authors": "Francesco Caravelli, Fotini Markopoulou",
"submitter": "Francesco Caravelli",
"url": "https://arxiv.org/abs/1201.3206"
}
|
1201.3228
|
# The effect of dynamical quark mass in the calculation of strange quark star
structure
Gholam Hossein Bordbar1,2 111Corresponding author. E-mail:
bordbar@physics.susc.ac.ir and Babak Ziaei1 Department of Physics, Shiraz
University, Shiraz 71454, Iran
and
Research Institute for Astronomy and Astrophysics of Maragha, P.O. Box
55134-441, Maragha 55177-36698, Iran
###### Abstract
We have discussed dynamical behavior of strange quark matter components, in
particular the effects of density dependent quark mass on the equation of
state of strange quark matter. Dynamical masses of quarks have been computed
within Nambu-Jona-Lasinio (NJL) model, then we have done the strange quark
matter calculations employing the MIT bag model with these dynamical masses.
For the sake of comparing dynamical mass interaction with QCD quark-quark
interaction, we have considered the one-gluon-exchange term as the effective
interaction between quarks for MIT bag model. Our dynamical approach
illustrates an improvement for the obtained values of equation of state. We
have also investigated the structure of strange quark star using Tolman-
Oppenheimer-Volkoff (TOV) equations for all applied models. Our results show
that the dynamical mass interaction leads to lower values for the
gravitational mass.
## I Introduction
Strange quark stars (SQS) are the most compact objects with a surface density
$\rho\sim 10^{15}\frac{gr}{cm^{3}}$, which is about fourteen orders of
magnitude greater than the surface density of neutron stars, while their
central density could be up to five times higher than that (Haensel et al.
Haensel2007 (2007); Glendenning Glendenning2000 (2000); Weber Weber1999
(1999)). It was first Itoh (1970) that, even before QCD full development,
proposed SQSs which is made of strange quark matter (SQM). Later, Bodmer
(1971) discussed the fate of an astronomical object collapsing to such a state
of matter.
The quark deconfinement hypothesis is one of the exciting steps in
investigation for the building blocks of matter. Soon after predictions of
quarks in theories and successful laboratory observations, many hadronic
models were developed to describe the probable quark matter proposed at high
energy regimes. In the 1970s, after formulation of QCD, perturbative
calculations of the equations of state of SQM got form, but the area of
validity for these calculations was restricted to very high densities (Collins
& Perry Collins1975 (1975)). The existence of SQSs was also discussed by
Witten (1984), who conjectured that a first order QCD phase transition in the
early universe could concentrate most of the quark excess in dense quark
nuggets. Witten proposed that SQM composed of light quarks is more stable than
nuclei, therefore SQM can be considered as the ground state of matter.
An SQS would be the bulk SQM phase consisting of almost equal numbers of up,
down and strange quarks, plus a small number of electrons to ensure charge
neutrality. A typical electron fraction is less than $10^{-3}$ and decreases
from the surface to the center of an SQS (Haensel et al. Haensel2007 (2007);
Glendenning Glendenning2000 (2000); Weber Weber1999 (1999); Camenzind
Camenzind2007 (2007)). SQM would have a lower charge-to-baryon ratio compared
to the nuclear matter and can show itself in the form of an SQS (Witten
Witten1984 (1984); Alcock et al. Alcock1986 (1986); Haensel et al. Haensel1986
(1986); Kettner et al. Kettner1995 (1995) ). The collapse of a massive star
could lead to the formation of an SQS. An SQS may also be formed from a
neutron star and is denser than the neutron star (Bhattacharyya et al.
Bhattacharyya2006 (2006)). If sufficient additional matter is added to an SQS,
it will collapse into a black hole. Neutron stars with masses of
$1.5-1.8M_{\,\odot}$ with rapid spins are theoretically the best candidates
for conversion to an SQS. An extrapolation based on this indicates that up to
two quark-novae occur in the observable universe each day. In addition, recent
Chandra observations indicate that objects $RXJ185635-3754$ and $3C58$ may
contain SQSs (Prakash et al. Prakash2003 (2003)). Other investigations also
show that the object $SWIFTJ1749.4-2807$ may be an SQS (Yu & Xu Yu2010
(2010)).
The strange quark star, founded from quark matter theory consists of too many
unsolved puzzles which are usually involved in physics of these relativistic
objects. System complexity of these stars prohibit us from considering all
physical and astrophysical properties simultaneously, and it is possible that
some parameters entering the equation of state do not represent specific
physical properties. For example, in MIT bag model, one of the models used in
this paper, when the researchers try to find and fit the bag constant
according to informations achieved from big colliders (Jin & Jenning Jin1997
(1997); Alford et al. Alford1998 (1998); Blaschke et al. Blaschke1999 (1999);
Burgio et al. 2002b ; Begun et al. Begun2011 (2011)), we should keep this
principle as a matter of fact that different parameters like temperature,
electromagnetic intensity, density etc. are important enough on final
interpretation for theoretical calculated bag constant. In this point of view
even constant values of bag pressure no more can be considered purely as the
energy density difference between the perturbative vacuum and the true vacuum.
The role of bag constant for confining quark matter in comparison with gravity
confinement for neutron matter may require more attention when we consider it
for compact stars. Therefore it is better to consider the dynamical properties
of the parameters for investigating of the properties of quarks. Many works
have been done to adapt the theory of bag Model on physics of ultra-dense
matter like using a density dependent bag constant (Burgio et al. 2002a ),
utilizing different values of coupling constants for one gluon exchange (Farhi
& Jaffe Farhi1984 (1984); Berger & Jaffe Berger1987 (1987) ), or instead
considering dynamical mass as effective interaction between particles (Peng et
al. Peng1999 (1999); Shao et al. Shao2011 (2011)).
From perturbative QCD, we know that quarks at ultra high densities
asymptotically interact. One way of considering the interaction is to assume
that quarks exchange one gluon. Therefor we can add a term to equation of
state that is characterized by a coupling constant. But constant values of
this parameter will weaken the power of interaction in lower densities while
in higher densities, it increases it. One method to solve the problem is to
assume a density dependent quark mass as the effective interaction. This
approach was investigated in references (Fowler et al. Fowler1981 (1981);
Chakrabarty et al. Chakrabarty1989 (1989); Chakrabarty Chakrabarty1991 (1991),
Chakrabarty1994 (1994); Benvenuto & Lugones Benvenuto1995 (1995); Lugones &
Benvenuto Lugones1995 (1995)). This has been done by adding a term to the rest
mass which is characterized by a free parameter determined by stability
conditions. They concluded that the density dependent mass is flavor
independent and the applied free parameter has the same meaning as the bag
constant. Then by selecting one value of bag constant for all densities and
flavors, they tried to obtain the equation of state of quark matter (Peng et
al. Peng1999 (1999)). A better approach closer to current work, is to find a
solution for density dependent mass from Nambu-Jona-Lasinio (NJL) method
(Carol Carol2009 (2009)). Carol calculated the equation of state and structure
for hybrid stars within MIT bag model, while numerical values of density
dependent mass entering in the energy equation had been obtained from
dynamical calculations of mass in NJL model. These numerical values were
entered directly in the pressure equation without considering density
dependency. Quark masses and NJL constants were also approximate values. The
bag constant in that work was density independent; therefore in addition to
previous known problems of constant values for this parameter (Baldo et al.
Baldo2006 (2006); Alford & Reddy Alford2003 (2003); Alford et al. Alford2005
(2005)), it misinterprets the meaning of the effective interaction in some
densities.
In Our previous work we considered a hot strange star just after the collapse
of a supernova (Bordbar et al. 2011a ), at finite temperature with a density
dependent bag constant. The calculations for the structure properties of the
strange star at different temperature indicates that it’s maximum mass
decreases by increasing the temperature. In another work (Bordbar & Peivand
2011b ), we concentrated on the calculation of a bulk of spin polarized SQM at
zero temperature in the presence of a strong magnetic field. We computed
structure properties of this system and found that the presence of a magnetic
field leads to a more stable SQS when compared to the structure properties of
an unpolarized SQS. In present paper, we investigate the quark matter equation
of state and the strange quark star structure following Carroll (Carol
Carol2009 (2009)). We base our calculations on MIT bag model, and after
following NJL formalism we extrapolate a density dependent equation from
numerical values of dynamical mass obtained using NJL method. In Sec. II, the
required equations for the MIT bag model are written, the same has been done
for NJL model. In Sec. II.3, we describe the formalism applied in this
article, and after solving TOV equations in Sec. III, we calculates SQS
structure for our method.
## II Calculation of equation of state for SQM
In this section, we calculate the equation of state of strange quark matter
(SQM) using MIT and NJL methods as well as MIT method with the dynamical mass.
At first, we introduce these three models in three separate sections, then we
give our results for the energy and the equation of state of SQM in sec. II.4.
### II.1 The MIT Bag Model
Total energy of a bulk of deconfined up ($u$), down ($d$) and strange ($s$)
quarks within MIT bag model is as follows (Witten Witten1984 (1984); Farhi &
Jaffe Farhi1984 (1984); Baym Baym1985 (1985); Baym et al. Baym1985 (1985);
Berger & Jaffe Berger1987 (1987); Glendenning Glendenning1990 (1990); Maruyama
et al. Maruyama2007 (2007)):
$\displaystyle\varepsilon=\varepsilon_{u}+\varepsilon_{d}+\varepsilon_{s}+B.$
(1)
In Eq. (1), $B$ is the bag constant, and
$\displaystyle\varepsilon_{f}\left(\rho_{f}\right)=\frac{3m_{f}\,^{4}}{8\pi^{2}}\left[x_{f}\left(2x_{f}^{2}+1\right)\left(\sqrt{1+x_{f}^{2}}\right)-arcsinh\
x_{f}\right]$
$\displaystyle-\alpha_{c}\,\frac{m_{f}\,^{4}}{\pi^{3}}\left[x_{f}^{4}-\frac{3}{2}\left[x_{f}\left(\sqrt{1+x_{f}^{2}}\right)-arcsinh\
x_{f}\right]^{2}\right],$ (2)
where $f$ denotes the flavor of the relevant quark, $\alpha_{c}$ is QCD
coupling constant and the following term demonstrates the one-gluon-exchange
interaction. In above equation, $x_{f}$ is defined as follows,
$x_{f}=k_{F}\,^{\left(f\right)}/m_{f},$ (3)
where the Fermi momentum $k_{F}\,^{(f)}$ is given by
$k_{F}\,^{(f)}=\left(\rho_{f}\,\pi^{2}\right)^{1/3}$ (4)
For the bag constant ($B$), we use a density dependent Gaussian
parametrization (Burgio et al. 2002a ; Baldo et al. Baldo2006 (2006)):
$B\left(\rho\right)=B_{\infty}+\left(B_{0}-B_{\infty}\right)\exp[-\beta\left(\rho/\rho_{0}\right)^{2}]$
(5)
with
$B_{\infty}=B\left(\rho=\infty\right)=8.99\>MeV/fm^{3},B_{0}=B\left(\rho=0\right)=400\>MeV/fm^{3}$
and $\beta=0.17$. In SQM, the beta-equilibrium and charge neutrality
conditions lead to the following relation for the number density of quarks,
$\rho=\rho_{u}=\rho_{d}=\rho_{s}$ (6)
From the total energy, we can obtain the equation of state of SQM using the
following relation,
$P(\rho)=\rho\frac{\partial\varepsilon}{\partial\rho}-\varepsilon.$ (7)
### II.2 The Nambu-Jona-Lasinio Model
Here we give a brief introduction regarding the calculations in the Nambu-
Jona-Lasinio (NJL) method. For NJL model, we use a common three flavor
lagrangian adopted from (Rehberg et al. Rehberg1996 (1996)) which preserves
chiral symmetry of QCD,
$\displaystyle{\mathcal{L}}=\bar{q}\left(i\gamma^{\mu}\partial_{\mu}-\hat{m_{0}}\right)q+G{\textstyle{\displaystyle\sum_{k=0}^{8}\left[\left(\bar{q}\lambda_{k}q\right)^{2}+\left(\bar{q}i\gamma_{5}\lambda_{k}q\right)^{2}\right]-}}$
(8) $\displaystyle
K\left[det_{f}\left(\bar{q}\left(1+\gamma_{5}\right)q\right)+det_{f}\left(\bar{q}\left(1-\gamma_{5}\right)q\right)\right].$
In adopted lagrangian, $q$ denotes quark field with three flavors $u$, $d$ and
$s$, and three colors. $\hat{m_{0}}=diag(m_{0}^{u},\,m_{0}^{d},\,m_{0}^{s})$
is a $3\times 3$ matrix in flavor space. And $\lambda_{k}$ ( $0\leq k\leq 8$ )
are the $U(3)$ flavor matrices. We restrict ourselves to the isospinsymmetric
case, $m_{0}^{u}=m_{0}^{d}$. We have picked up the parameters from references
(Kunihiro Kunihiro1989 (1989); Ruivo et al. Ruivo1999 (1999); Buballa & Oertel
Buballa1999 (1999)) which are fitted to the pion mass, the pion decay
constant, the kaon mass and the quark condensates.
NJL model is an unrenormalizable method with divergent integrations. To
prevent the divergence, we need to introduce some breaking points for upper
limit of integrals which satisfy the physics ranges of our problem. It is
usually done by choosing a proper cut-off. In present paper, the adopted cut-
off is named Ultra-violet cut-off that indicates restoring of chiral symmetry
breaking, $\Lambda=602.3\ MeV$. $G$ and $K$ are coupling strengths that read,
$G\Lambda^{2}=1.835,\>K\Lambda^{5}=12.36$. The rest mass of $s$ quark is
$m_{0}^{s}=140.7\ MeV$, and there is $m_{0}^{u}=m_{0}^{d}=5.5\ MeV$ for $u$
and $d$ quarks. The baryon number density is given by
$\rho_{B}=\frac{1}{3}n_{B}=\frac{1}{3}\left(n_{u}+n_{d}+n_{s}\right),$ (9)
where $n_{i}=\left\langle q{}_{i}^{\dagger}q_{i}\right\rangle$. Within mean
field approximation, the dynamical mass is calculated by the following gap
equation,
$m_{i}=m_{0}^{i}-4G\left\langle\bar{q_{i}}q_{i}\right\rangle+2K\left\langle\bar{q_{j}}q_{j}\right\rangle\left\langle\bar{q_{k}}q_{k}\right\rangle.$
(10)
In the above equation, we need to calculate permutation of all quark flavors.
The quark condensate in Eq. (10) reads
$\left\langle\bar{q_{i}}q_{i}\right\rangle=-\frac{3}{\pi^{2}}{\int}_{P_{Fi}}^{\Lambda}P^{2}dp\frac{m_{i}}{\sqrt{m_{i}^{2}+p^{2}}},$
(11)
and $P_{Fi}$, Fermi momentum of quark $i$, is obtained from the following
relation,
$P{}_{Fi}=\left(\pi^{2}n_{i}\right)^{\frac{1}{3}}.$ (12)
Equations (10) and (11) have self consistent solutions. It means that for a
given number density, $n_{i}$, we should calculate quark condensate and
substituting the corresponding value in Eq. (10) to reach a consistent result
of the dynamical mass after doing the iteration process. In Fig. 1, we have
plotted the results of density dependent mass for $u$, $d$ and $s$ quarks as a
function of density. As it is clear from Fig. 1, quark masses vary from
current masses ($5.5MeV$ for $u$ and $d$ quarks, and $140.7MeV$ for $s$ quark)
at high densities to constituent mass at near zero densities ($368.7MeV$ for
$u$ and $d$ quarks, and $550MeV$ for $s$ quark).
The solution via mean field approximation forces us to stabilize equations by
diminishing energy density and pressure in vacuum. This is satisfied by
entering a parameter which has the same meaning of bag constant in MIT bag
model (Buballa & Oertel Buballa1999 (1999)):
$\displaystyle
B=\sum_{i=u,d,s}\left(\frac{3}{\pi^{2}}{\int}_{0}^{\Lambda}\;p^{2}dp\left(\sqrt{p^{2}+m_{i}^{2}}-\sqrt{p^{2}+{m^{i}}_{0}^{2}}\right)-2{G\left\langle\bar{q_{i}}q_{i}\right\rangle}^{2}\right)$
$\displaystyle+\;4K\left\langle\bar{u}u\right\rangle\left\langle\bar{d}d\right\rangle\left\langle\bar{s}s\right\rangle$
(13)
Now we can calculate the equation of state of SQM in NJL model,
$p=-\varepsilon+\sum_{i=u,d,s}n_{i}\sqrt{P_{F}{}_{i}^{2}+m_{i}^{2}},$ (14)
where
$\varepsilon=\sum_{i=u,d,s}\frac{3}{\pi^{2}}{\int}_{0}^{P_{Fi}}p^{2}dp\sqrt{p^{2}+m_{i}}-(B-B_{0}).$
(15)
Parameter $B$ is the bag pressure, which is explained by Buballa (2005), and
is a dynamical consequence of the mean field solution, not a parameter
inserted by hand, as was done in MIT bag model. It is shown in Fig. 1, matter
in NJL method acquires dynamical mass in nonzero baryon densities, but in MIT
bag model, the given mass remains constant for all densities. Consequently,
this will lead to dissimilar chiral symmetry behavior as density changes. In
NJL model, since quarks acquire dynamical mass, the chiral symmetry
spontaneously breaks in lower densities, while in MIT bag model, it will
happen physically when quarks change their directions by hitting the bag (what
is not considered theoretically in ordinary MIT bag model). The bag constant
versus density is presented in Fig. 2 for our used models. It is apparent from
Fig. 2 that chiral symmetry in our calculations is fully restored in the
densities greater than $\rho\simeq 2.5\ fm^{-3}$. It is also important to
mention that vacuum in MIT bag model is totally free of particles (flow of
particle’s wave function is restricted by the confinement), while in NJL model
no confinement is produced. In other word, the vacuum in NJL model is made of
paired quasi-quarks that lower the energy density of particles in comparison
to MIT bag model. From the above discussions, it seems reasonable to add an
effective bag constant to energy equation (Buballa Buballa2005 (2005)),
$\displaystyle B_{0}=B\mid_{n_{u}=n_{d}=n_{s}=0},$ $\displaystyle
B_{eff}=B-B_{0}.$ (16)
From Fig. 2, it seems that the effective bag constant diminishes at zero
density. Then the correct interpretation for the effective bag constant is the
energy per volume needed to fully break quark-antiquark pairs in order to
completely restore chiral symmetry at ultra high densities. Even the maximum
value of dynamical NJL bag constant is smaller than that of MIT’s one, because
it reduces the energy per particle due to quark-antiquark pairing at lower
densities (Buballa Buballa2005 (2005)). Fig. 2 shows that the decreasing rate
of MIT bag constant is higher than that of NJL. This indicates that MIT bag
model does gross approximation over physics of matter in middle and higher
densities $(\rho>0.8\ fm^{-3})$. Therefore, the density dependent bag constant
should be corrected by another higher density sensitive parameter. This could
not be achieved by a one gluon exchange term that considers the interaction
with a constant strength in all energy regimes. Fig. 2 indicates that at the
density $\rho\simeq 0.45\ fm^{-3}$, there is a cross point for the effective
bag constant of NJL model and the bag constant of MIT model. As it is
mentioned in above discussions, the bag pressure is the energy needed to
confine particles where effective bag constant is energy needed to destabilize
quark-antiquark pairs. Now, we can suggest that the hadron-quark phase
transition can takes place at the density $\rho\simeq 0.45\ fm^{-3}$. This is
in good agreement with the results of others (Heinz Heinz2001 (2001); Heinz &
Jacob Heinz2000 (2000)).
### II.3 MIT bag model with dynamical mass
In MIT bag model with dynamical mass, we consider the effect of dynamical
behavior of the quark mass in calculating the equation of state of SQM within
MIT bag model using NJL numerical mass results. In fact, we use the dynamical
masses (Fig. 1) for $u$, $d$ and $s$ quarks in Eq. (II.1) instead of their
fixed values.
### II.4 Our results for the energy and equation of state of SQM
To distinguish numerous outcomes, we present the results of our calculations
in three following models;
* •
Model 1: MIT model by a density dependent bag constant and one gluon-exchange
$(\alpha_{c}=0,\,0.16,\,0.5)$ as effective interaction.
* •
Model 2: NJL model.
* •
Model 3: MIT bag model by a density dependent bag constant, dynamical mass and
one gluon-exchange $(\alpha_{c}=0,\,0.16,\,0.5)$ as effective interaction.
Our results for the energy of SQM versus density calculated with above models
have been plotted in Fig. 3. We see that for both MIT based calculations
(models 1 and 3), at lower densities $(\rho<0.5\ fm^{-3})$, the energy of SQM
suddenly increases as the density decreases. This shows the concept of
confinement (Buballa Buballa2005 (2005)). For these two models, we also see
that the energy of SQM gets to a minimum, then increases with a small rate.
Fig. 3 shows that for model 1 and model 3, the energies of different coupling
constants are nearly identical for densities $\rho<0.5fm^{-3}$. However, they
have a substantial difference as the density increases. We can see that at
lower densities $(\rho<0.7\ fm^{-3})$, the results of model 3 is considerably
different from those of model 1\. While this difference becomes small as
density increases, specially for lower values of coupling constant, due to
asymptotic freedoms which is the simple MIT bag model without interaction.
From Fig. 3, it is seen that the energy of SQM in model 2 (NJL model) has
finite values even at low densities showing no confinement. We also see that
the energy of SQM from model 3 with smaller values of coupling constant is
lower than that of model 2 for $\rho>0.7\ fm^{-3}$ indicating a more stable
state of quark matter at these densities. However, at very high densities, the
difference between the results of these two models becomes negligible.
In Fig. 4, our results for the pressure of SQM have been plotted versus
density. It can be found that for MIT bag model, the higher values of coupling
constant leads to the stiffer equation of state for SQM. Fig. 4 shows that by
considering a dynamical mass for the quarks (density dependent mass) in MIT
model, we get the lower values for the pressure of SQM. For $\alpha_{c}=0.0$,
we see that the result of model 3 for the equation of state of SQM is nearly
identical with that of model 1. It can be seen that for $\rho>0.6\ fm^{-3}$,
our results for the pressure of SQM calculated by NJL model are nearly
identical with those of model 3 and model 1 for $\alpha_{c}=0.0$, while at
lower densities, there is a considerable difference between them.
In order to investigate the quark matter stability, the energy of SQM versus
pressure has been plotted in Fig. 5. It is clearly seen that at zero pressure,
the MIT bag model with $\alpha_{c}=0$ leads to the lowest value for the energy
of SQM ($950MeV\ fm^{-3}$) compared to other models. This value is comparable
with the result for the binding energy per particle of ${}^{56}Fe$ ($930MeV\
fm^{-3}$) (Witten Witten1984 (1984)). This indicates that among different
models used in this work, MIT model with $\alpha_{c}=0$ shows the most stable
state of SQM.
## III Calculation of strange quark star structure
The gravitational mass ($M$) and radius ($R$) of compact stars are of special
interest in astrophysics. In this section, we calculate the structural
properties of a strange quark star for our three models. Using the equation of
state of strange quark matter for the models applied in this work, we can
obtain $M$ and $R$ by numerically integrating the general relativistic
equations of hydrostatic equilibrium, the Tolman-Oppenheimer-Volkoff (TOV)
equations, which are as follows (Shapiro & Teukolsky Shapiro1983 (1983)),
$\frac{dm}{dr}=4\pi r^{2}\varepsilon\left(r\right),$ (17)
$\frac{dp}{dr}=-\frac{Gm\left(r\right)\varepsilon\left(r\right)}{r^{2}}\left(1+\frac{p\left(r\right)}{\varepsilon\left(r\right)c^{2}}\right)\left(1+\frac{4\pi
r^{3}p\left(r\right)}{m\left(r\right)c^{2}}\right)\left(1-\frac{2Gm\left(r\right)}{c^{2}r}\right)^{-1},$
(18)
where $\varepsilon\left(r\right)$ is the energy density, $G$ is the
gravitational constant, and
$m(r)={\int}_{0}^{r}4\pi\acute{r}^{2}\,\varepsilon(\acute{r})d\acute{r}$ (19)
has the interpretation of the mass inside radius $r$. By selecting a central
energy density $\varepsilon_{c}$, under the boundary conditions $P(0)=P_{c}$
and m(0)=0, we integrate the TOV equation outwards to a radius $r=R$, at which
$P$ vanishes.
In Fig. 6, we have presented our results for the gravitational mass of SQS
versus the central energy density. Fig. 6 shows that at low energy densities,
the gravitational mass increases rapidly by increasing the energy density, and
it finally reaches to a limiting value (maximum gravitational mass) at higher
energy densities. It is seen that the increasing rate of mass for Model 3 with
higher values of coupling constant is substantially higher than those of other
models. Table 1 summarizes maximum gravitational masses of different applied
models and the corresponding radii. As it seen from Table 1, we can conclude
that using dynamical mass in energy equation and equation of state of SQM
reduces the calculated maximum mass. This is in a good agreement with many
observational data obtained from low mass compact stars (Zhang Zhang2007
(2007)). It is interesting that in spite of considering dynamical mass as the
effective interaction in MIT bag model (model 3 with $\alpha_{c}=0$), we find
the smaller SQS maximum mass in comparison to MIT bag model (model 1) even
without interaction ($\alpha_{c}=0$). As it is obvious from Table 1, for
models 1 and 3, the calculated maximum mass increases as strong coupling
constant increases. This behavior demonstrates that ultra massive SQS with
masses greater than $M=1.05M_{\odot}$ are stars which are composed of highly
interacting strange quark matter. We note that some studies indicate that
there exist a big uncertainty about mass and radius of ultra massive stars
with $M>1.9M_{\odot}$ (Lattimer & Prakash Lattimer2010 (2010)). These studies
showed that the observed data of mass and radius for these stars, which
commonly belong to X-ray stars, were wrongly calculated and the calculations
were revised to the smaller values for mass and radius. The best example is
$PSR\>J\,0751+1807$ pulsar that initially was supposed to have a mass of
$M=2.2\pm 0.2M_{\odot}$ but recently revised to $M=1.26M_{\odot}$ (Lattimer &
Prakash Lattimer2010 (2010)).
We have also plotted the gravitational mass of SQS versus radius for our three
models in Fig. 7. It is seen that for all models, the mass increases by
increasing the radius, but with different increasing rates for different
models. Fig. 7 shows that for a given value of radius, the dynamical model
(model 3) gives the smaller mass with respect to that of MIT bag model (model
1); however, for $\alpha_{c}=0$, it is close to the result of NJL model (model
2).
## Acknowledgements
This work has been supported by Research Institute for Astronomy and
Astrophysics of Maragha. We wish to thank Shiraz University Research Council.
## References
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Table 1: Maximum gravitational mass $(M_{max})$ and the corresponding radius $(R)$ for different applied models. | $M_{max}\left(M_{\,\odot}\right)$ | $R\left(km\right)$
---|---|---
Model 1; $\alpha_{c}=0$ | 1.43 | 7.61
Model 1; $\alpha_{c}=0.16$ | 1.73 | 8.17
Model 1; $\alpha_{c}=0.5$ | 2.6 | 10.6
Model 2 | 0.98 | 5.59
Model 3; $\alpha_{c}=0$ | 1.05 | 6.03
Model 3; $\alpha_{c}=0.16$ | 1.65 | 6.98
Model 3; $\alpha_{c}=0.5$ | 2.3 | 8.69
Figure 1: Density dependent mass ($m$) versus density ($\rho$) obtained from
dynamical NJL model.
Figure 2: Bag constant as a function of density for NJL and MIT models.
Figure 3: The energy per baryon versus density for models 1 and 2 (a), and
model 3 (b).
Figure 4: Pressure as a function of density for models 1 and 2 (a), and model
3 (b).
Figure 5: Energy per particle versus pressure for models 1 and 2 (a), and
model 3 (b).
Figure 6: Gravitational mass $(M)$ in unit of solar mass $(M_{sun})$ versus
central energy density $(\varepsilon_{c})$ for models 1 and 2 (a), and model 3
(b).
Figure 7: Gravitational mass $(M)$ in unit of solar mass $(M_{sun})$ versus
radius $(R)$ for models 1 and 2 (a), and model 3 (b).
|
arxiv-papers
| 2012-01-16T12:01:16 |
2024-09-04T02:49:26.345054
|
{
"license": "Public Domain",
"authors": "G. H. Bordbar and B. Ziaei",
"submitter": "Gholam Hossein Bordbar",
"url": "https://arxiv.org/abs/1201.3228"
}
|
1201.3247
|
11institutetext: CERN, Geneva, Switzerland
# RF measurements I: signal receiving techniques
F. Caspers
###### Abstract
For the characterization of components, systems and signals in the RF and
microwave range, several dedicated instruments are in use. In this paper the
fundamentals of the RF-signal sampling technique, which has found widespread
applications in ‘digital’ oscilloscopes and sampling scopes, are discussed.
The key element in these front-ends is the Schottky diode which can be used
either as an RF mixer or as a single sampler. The spectrum analyser has become
an absolutely indispensable tool for RF signal analysis. Here the front-end is
the RF mixer as the RF section of modern spectrum analysers has a rather
complex architecture. The reasons for this complexity and certain working
principles as well as limitations are discussed. In addition, an overview of
the development of scalar and vector signal analysers is given. For the
determination of the noise temperature of a one-port and the noise figure of a
two-port, basic concepts and relations are shown. A brief discussion of
commonly used noise measurement techniques concludes the paper.
## 0.1 Introduction
In the early days of RF engineering the available instrumentation for
measurements was rather limited. Besides elements acting on the heat developed
by RF power (bimetal contacts and resistors with very high temperature
coefficient) only point/contact diodes, and to some extent vacuum tubes, were
available as signal detectors. For several decades the slotted measurement
line [1] was the most used instrument for measuring impedances and complex
reflection coefficients. Around 1960 the tedious work with such coaxial and
waveguide measurement lines became considerably simplified with the
availability of the vector network analyser. At the same time the first
sampling oscilloscopes with 1 GHz bandwidth arrived on the market. This was
possible due to progress in solid-state (semiconductor) technology and
advances in microwave elements (microstrip lines). Reliable, stable, and
easily controllable microwave sources are the backbone of spectrum and network
analysers as well as sensitive (low noise) receivers. This paper will only
treat signal receiving devices such as spectrum analysers and oscilloscopes.
For an overview of network analysis tools see RF measurements II: network
analysis.
## 0.2 Basic elements and concepts
Before discussing several measurement devices, a brief overview of the most
important components in such devices and some basic concepts are presented.
### 0.2.1 Decibel
Since the unit dB is frequently used in RF engineering a short introduction
and definition of terms is given here. The decibel is the unit used to express
relative differences in signal power. It is expressed as the base 10 logarithm
of the ratio of the powers of two signals:
$P\text{[dB]}=10\cdot\text{log}(P/P_{0})\hskip 5.69046pt.$ (1)
It is also common to express the signal amplitude in dB. Since power is
proportional to the square of a signal’s amplitude, the voltage in dB is
expressed as follows:
$V\text{[dB]}=20\cdot\text{log}(V/V_{0})\hskip 5.69046pt.$ (2)
In Eqs. (1) and (2) $P_{0}$ and $V_{0}$ are the reference power and voltage,
respectively. A given value in dB is the same for power ratios as for voltage
ratios. Please note that there are no ‘power dB’ or ‘voltage dB’ as dB values
always express a ratio.
Conversely, the absolute power and voltage can be obtained from dB values by
$\displaystyle P=P_{0}\cdot 10^{\frac{P\text{[dB]}}{10}}\hskip 5.69046pt,$ (3)
$\displaystyle V=V_{0}\cdot 10^{\frac{V\text{[dB]}}{20}}\hskip 5.69046pt.$ (4)
Logarithms are useful as the unit of measurement because
1. 1.
signal power tends to span several orders of magnitude and
2. 2.
signal attenuation losses and gains can be expressed in terms of subtraction
and addition.
Table 1 helps to indicate the order of magnitude associated with dB.
Table 1: Overview of dB key values and their conversion into power and voltage ratios. | Power ratio | Voltage ratio
---|---|---
$-$20 dB | 0.01 | 0.1
$-$10 dB | 0.1 | 0.32
$-$3 dB | 0.50 | 0.71
$-$1 dB | 0.74 | 0.89
0 dB | 1 | 1
1 dB | 1.26 | 1.12
3 dB | 2.00 | 1.41
10 dB | 10 | 3.16
20 dB | 100 | 10
$n\cdot$10 dB | 10n | 10${}^{n\text{/2}}$
Frequently dB values are expressed using a special reference level and not SI
units. Strictly speaking, the reference value should be included in
parentheses when giving a dB value, e.g., +3 dB (1 W) indicates 3 dB at
$P_{0}$ = 1 watt, thus 2 W. However, it is more common to add some typical
reference values as letters after the unit, for instance, dBm defines dB using
a reference level of $P_{0}$ = 1 mW. Thus, 0 dBm correspond to $-$30 dBW,
where dBW indicates a reference level of $P_{0}$ = 1 W. Often a reference
impedance of 50 $\Omega$ is assumed. Other common units are
* •
dBmV for the small voltages with $V_{0}$ = 1 mV and
* •
dBmV/m for the electric field strength radiated from an antenna with reference
field strength $E_{0}$ = 1 mV/m
### 0.2.2 The RF diode
One of the most important elements inside all sophisticated measurement
devices is the fast RF diode or Schottky diode. The basic metal–semiconductor
junction has an intrinsically very fast switching time of well below a
picosecond, provided that the geometric size and hence the junction
capacitance of the diode is small enough. However, this unavoidable and
voltage dependent junction capacity will lead to limitations of the maximum
operating frequency.
The equivalent circuit of such a diode is depicted in Fig. 1 and
VRF inVideo outRF bypasscapacitordiode impedance50 Figure 1: The equivalent
circuit of a diode
an example of a commonly used Schottky diode can be seen in Fig. 2.
Figure 2: A commonly used Schottky diode. The RF input of this detector diode
is on the left and the video output on the right (figure courtesy Agilent)
One of the most important properties of any diode is its characteristic which
is the relation of current as a function of voltage. This relation is
described by the Richardson equation [2]:
$I=AA_{\text{RC}}T^{2}\text{exp}\left(-\frac{q\phi_{\text{B}}}{kT}\right)\left[\text{exp}\left(\frac{qV_{\text{J}}}{kT}\right)-\text{M}\right]\hskip
5.69046pt,$ (5)
where $A$ is the area in cm2, $A_{\text{RC}}$ the modified Richardson
constant, $k$ Boltzmann’s constant, $T$ the absolute temperature,
$\phi_{\text{B}}$ the barrier height in volts, $V_{\text{J}}$ the external
Voltage across the depletion layer, M the avalanche multiplication factor and
$I$ the diode current.
This relation is depicted graphically for two diodes in Fig. 3.
I50 $\mu$A/divTypicalLBSDTypicalSchottkyDiodeV50 mV/div Figure 3: Current as a
function of voltage for different diode types (LBSD = low barrier Schottky
diode)
As can be seen, the diode is not an ideal commutator (Fig. 4) for small
signals. Note that it is not possible to apply big signals, since this kind of
diode would burn out.
Threshold voltage VoltageCurrent Figure 4: The current–voltage relation of an
ideal commutator with threshold voltage
However, there exist rather large power versions of Schottky diodes which can
stand more than 9 kV and several 10 A but they are not suitable in microwave
applications due to their large junction capacity.
The Richardson equation can be roughly approximated by a simpler equation [2]:
$I=I_{\text{s}}\left[\text{exp}\left(\frac{V_{\text{J}}}{0.028}\right)-1\right]\hskip
5.69046pt.$ (6)
This approximation can be used to show that the RF rectification is linked to
the second derivation (curvature) of the diode characteristic.
If the DC current is held constant by a current regulator or a large resistor
assuming external DC bias111Most diodes do not need an external bias, since
they have a DC return self-bias., then the total junction current, including
RF is
$I=I_{0}=\text{i}_{0}\text{\,cos\,}\omega t$ (7)
and hence the current–voltage relation can be written as
$V_{\text{J}}=0.028\text{ln}\left(\frac{I_{\text{S}}+I_{0}+\text{icos\,}\omega
t}{I_{\text{S}}}\right)=0.028\text{ln}\left(\frac{I_{0}+I_{\text{S}}}{I_{\text{S}}}\right)+0.028\text{ln}\left(\frac{\text{icos\,}\omega
t}{I_{0}+I_{\text{S}}}\right)\hskip 5.69046pt.$ (8)
If the RF current $I$ is small enough, the second term can be approximated by
Taylor expansion:
$V_{\text{J}}\approx
0.028\text{ln}\left(\frac{I_{0}+I_{\text{S}}}{I_{\text{S}}}\right)+0.028\left[\frac{\text{icos\,}\omega
t}{I_{0}+I_{\text{S}}}-\frac{\text{i}^{2}\text{cos}^{2}\omega
t}{2(I_{0}+I_{\text{S}})^{2}}+\ldots\right]=V_{\text{DC}}+V_{\text{J}}\text{\,cos\,}\omega
t+\text{higherorderterms}$ (9)
With the identity cos${}^{2}=0.5$, the DC and the RF voltages are given by
$V_{\text{J}}=\frac{0.028}{I_{0}+I_{\text{S}}}\text{i}=R_{\text{S}}\text{i\hskip
5.69046ptand\hskip
5.69046pt}V_{\text{DC}}=0.028\text{ln}\left(1+\frac{I_{0}}{I_{\text{S}}}\right)-\frac{0.028^{2}}{4(I_{0}+I_{\text{S}})^{2}}=V_{0}-\frac{V_{\text{J}}^{2}}{0.112}\hskip
5.69046pt.$ (10)
The region where the output voltage is proportional to the input power is
called the square-law region (Fig. 5).
[algebraic=true,linecolor=olive]911-0.15*x*x+3.25*x-10.6[algebraic=true,linecolor=olive]911-0.15*x*x+3.25*x-10.6without
loadsquare law loadedLBSDLBSD0-10-20-30-40-500.0050.050.55.050500Input power
[dBm]Output power [mV] Figure 5: Relation between input power and output
voltage
In this region the input power is proportional to the square of the input
voltage and the output signal is proportional to the input power, hence the
name square-law region.
The transition between the linear region and the square-law region is
typically between $-$10 and $-$20 dB (Fig. 5).
There are fundamental limitations when using diodes as detectors. The output
signal of a diode (essentially DC or modulated DC if the RF is amplitude
modulated) does not contain a phase information. In addition, the sensitivity
of a diode restricts the input level range to about $-$60 dBm at best which is
not sufficient for many applications.
The minimum detectable power level of an RF diode is specified by the
‘tangential sensitivity’ which typically amounts to $-$50 to $-$55 dBm for 10
MHz video bandwidth at the detector output [3].
To avoid these limitations, another method of operating such diodes is needed.
### 0.2.3 Mixer
For the detection of very small RF signals a device that has a linear response
over the full range (from 0 dBm ( = 1mW) down to thermal noise = $-$174 dBm/Hz
= 4$\cdot$10-21 W/Hz) is preferred. An RF mixer provides these features using
1, 2, or 4 diodes in different configurations (Fig. 6). A mixer is essentially
a multiplier with a very high dynamic range implementing the function
$f_{1}(t)f_{2}(t)\text{\hskip 5.69046ptwith}f_{1}(t)=\text{RFsignal\hskip
5.69046ptand}f_{2}(t)=\text{LOsignal}\hskip 5.69046pt,$ (11)
or more explicitly for two signals with amplitude $a_{i}$ and frequency
$f_{i}$ ($i=1,2$):
$a_{1}\text{\,cos}(2\pi f_{1}t+\varphi)\cdot a_{2}\text{\,cos}(2\pi
f_{2}t)=\frac{1}{2}a_{1}a_{2}\left[\text{cos}((f_{1}+f_{2})t+\varphi)+\text{cos}((f_{1}-f_{2})t+\varphi)\right]\hskip
5.69046pt.$ (12)
Thus we obtain a response at the IF (intermediate frequency) port that is at
the sum and difference frequency of the LO (local oscillator $=f_{1}$) and RF
($=f_{2}$) signals.
Examples of different mixer configurations are shown in Fig. 6.
Figure 6: Examples of different mixer configurations
As can be seen from Fig. 6, the mixer uses diodes to multiply the two ingoing
signals. These diodes function as a switch, opening different circuits with
the frequency of the LO signal (Fig. 7).
[algebraic=true]00.60.1*sin(10.4719755*x)LO[algebraic=true]00.60.1*sin(10.4719755*x)
Figure 7: Two circuit configurations interchanging with the frequency of the
LO where the switches represent the diodes
The response of a mixer in time domain is depicted in Fig. 8.
[algebraic=true,linecolor=myred]06.283185sin(2*x)RFLO[algebraic=true,linewidth=0.01,linestyle=dashed]06.283185sin(2*x)[algebraic=true,linewidth=0.01,linestyle=dashed]06.283185sin(2*x
+
3.1415926)[algebraic=true,linecolor=orange]01sin(2*x)[algebraic=true,linecolor=orange]23sin(2*x)[algebraic=true,linecolor=orange]45sin(2*x)[algebraic=true,linecolor=orange]66.283185sin(2*x)[algebraic=true,linecolor=orange]12sin(2*x
+ 3.1415926)[algebraic=true,linecolor=orange]34sin(2*x +
3.1415926)[algebraic=true,linecolor=orange]56sin(2*x + 3.1415926)IF Figure 8:
Time domain response of a mixer
The output signal is always in the ‘linear range’ provided that the mixer is
not in saturation with respect to the RF input signal. Note that for the LO
signal the mixer should always be in saturation to make sure that the diodes
work as a nearly ideal switch. The phase of the RF signal is conserved in the
output signal available form the RF output.
### 0.2.4 Amplifier
A linear amplifier augments the input signal by a factor which is usually
indicated in decibel. The ratio between the output and the input signal is
called the transfer function and its magnitude—the voltage gain $G$—is
measured in dB and given as
$G[\text{dB}]=20\cdot\frac{\text{V}_{\text{RFout}}}{\text{V}_{\text{RFin}}}\text{\hskip
5.69046ptor\hskip
5.69046pt}\frac{\text{V}_{\text{RFout}}}{\text{V}_{\text{RFin}}}=20\cdot\text{log}G[\text{lin}]\hskip
5.69046pt.$ (13)
The circuit symbol of an amplifier is shown in Fig. 9 together with its
S-matrix.
12S = $\left(\begin{array}[]{cc}0&0\\\ G&0\end{array}\right)$ Figure 9:
Circuit symbol an S-matrix of an ideal amplifier
The bandwidth of an amplifier specifies the frequency range where it is
usually operated. This frequency range is defined by the $-$3 dB points222The
$-$3 dB points are the points left and right of a reference value (e.g., a
local maximum of a curve) that are 3 dB lower than the reference. with respect
to its maximum or nominal transmission gain.
In an ideal amplifier the output signal would be proportional to the input
signal. However, a real amplifier is nonlinear, such that for larger signals
the transfer characteristic deviates from its linear properties valid for
small signal amplification. When increasing the output power of an amplifier,
a point is reached where the small signal gain becomes reduced by 1 dB (Fig.
10).
Figure 10: Example for the 1dB compression point [4]
This output power level defines the 1 dB compression point, which is an
important measure of quality for any amplifier (low level as well as high
power).
The transfer characteristic of an amplifier can be described in terms which
are commonly used for RF engineering, i.e., the S-matrix (for further details
see the paper on S-matrices of this School). As implicitly contained in the
S-matrix, the amplitude and phase information of any spectral component are
preserved when passing through an ideal amplifier. For a real amplifier the
element $G=\text{S}_{21}$ (transmission from port 1 to port 2) is not a
constant but a complex function of frequency. Also the elements S11 and S22
are not 0 in reality.
### 0.2.5 Interception points of nonlinear devices
Important characteristics of nonlinear devices are the interception points.
Here only a brief overview will be given. For further information the reader
is referred to Ref. [4].
One of the most relevant interception points is the interception point of 3rd
order (IP3 point). Its importance derives from its straightforward
determination, plotting the input versus the output power in logarithmic scale
(Fig. 10). The IP3 point is usually not measured directly, but extrapolated
from measurement data at much smaller power levels in order to avoid overload
and damage of the device under test (DUT). If two signals
$(f_{1},f_{2}>f_{1})$ which are closely spaced by $\Delta f$ in frequency are
simultaneously applied to the DUT, the intermodulation products appear at +
$\Delta f$ above $f_{2}$ and at $-$ $\Delta f$ below $f_{1}$.
The transfer functions or weakly nonlinear devices can be approximated by
Taylor expansion. Using $n$ higher order terms on one hand and plotting them
together with an ideal linear device in logarithmic scale leads to two lines
with different slopes
($x^{n}\stackrel{{\scriptstyle\text{log}}}{{\rightarrow}}n\cdot\text{log}x$).
Their intersection point is the intercept point of $n$th order. These points
provide important information concerning the quality of nonlinear devices.
In this context, the aforementioned 1 dB compression point of an amplifier is
the intercept point of first order.
Similar characterization techniques can also be applied with mixers which with
respect to the LO signal cannot be considered a weakly nonlinear device.
### 0.2.6 The superheterodyne concept
The word superheterodyne is composed of three parts: super (Latin: over),
$\epsilon\tau\epsilon\rho\omega$ (hetero, Greek: different) and
$\delta\upsilon\nu\alpha\mu\iota\sigma$ (dynamis, Greek: force) and can be
translated as two forces superimposed333The direct translation (roughly) would
be: Another force becomes superimposed.. Different abbrevations exist for the
superheterodyne concept. In the US it is often referred to by the simple word
‘heterodyne’ and in Germany one can find the terms ‘super’ or ‘superhet’. The
‘weak’ incident signal is subjected to nonlinear superposition (i.e., mixing
or multiplication) with a ‘strong’ sine wave from a local oscillator. At the
mixer output we then get the sum and difference frequencies of the signal and
local oscillator. The LO signal can be tuned such that the output signal is
always at the same frequency or in a very narrow frequency band. Therefore a
fixed frequency bandpass with excellent transfer characteristics can be used
which is cheaper and easier than a variable bandpass with the same
performance. A well-known application of this principle is any simple radio
receiver (Fig. 11).
BPRF amplifierMixerLocal oscillator (often locked to a quarz)Bandpass filterIF
amplifierDemodulatorAudio amplifier Figure 11: Schematic drawing of a
superheterodyne receiver
## 0.3 Oscilloscope
An oscilloscope is typically used for acquisition, display, and measurement of
signals in time domain. The bandwidth of real-time oscilloscopes is limited in
most cases to 10 GHz. For higher bandwidth on repetitive signals the sampling
technique has been in use since about 1960. One of the many interesting
features of modern oscilloscopes is that they can change the sampling rate
through the sweep in a programmed manner. This can be very helpful for
detailed analysis in certain time windows. Typical sampling rates are between
a factor 2.5 and 4 of the maximum frequency (according to the Nyquist theorem
a real-time minimum sampling rate of twice the maximum frequency
$f_{\text{max}}$ is required).
Sequential sampling (Fig. 12) requires a pre-trigger (required to open the
sampling gate) and permits a non-real-time bandwidth of more than 100 GHz with
modern scopes.
[algebraic=true,linestyle=dashed]2.53.5-7.2*x*x*x+64.8*x*x-189*x+182
[algebraic=true,linestyle=dashed]2.53.57.2*x*x*x-64.8*x*x+189*x-182
Figure 12: Illustration of sequential sampling
Random sampling (rarely used these days, Fig. 13) was developed about 40 years
ago (around 1970) for the case where no pre-trigger was available and relying
on a strictly periodic signal to predict a pre-trigger from the measured
periodicity.
[algebraic=true,linestyle=dashed]2.53.5-7.2*x*x*x+64.8*x*x-189*x+182
[algebraic=true,linestyle=dashed]2.53.57.2*x*x*x-64.8*x*x+189*x-182
Figure 13: Illustration of random sampling
Sampling is discussed in more detail in the following. Consider a bandwidth-
limited time function s($t$) and its Fourier transform S($f$). The signal
s($t$) is sampled (multiplied) by a series of equidistant $\delta$-pulses
p($t$) [5]:
$\text{p}(t)=\sum^{+\infty}_{n=-\infty}{\delta(t-nT_{\text{s}})=III(t/T_{\text{s}})}$
(14)
where the symbol $III$ is derived from the Russian letter III and is
pronounced ‘sha’. It represents a series of $\delta$-pulses.
The sampled time functions s${}_{\text{s}}(t)$ is
$\displaystyle\text{s}_{\text{s}}(t)=\text{s}(t)\text{p}(t)=\text{s}(t)III(t/T_{\text{s}})$
$\displaystyle\text{S}_{\text{s}}(f)=\text{S}(f)*\frac{1}{T_{\text{s}}}III(T_{\text{s}}f)$
(15)
$\displaystyle\text{S}_{\text{s}}(f)=\frac{1}{T_{\text{s}}}\sum^{+\infty}_{n=-\infty}{\text{S}(f-mF)}\text{with}F=\frac{1}{T_{\text{s}}}\hskip
5.69046pt.$ (16)
Note that the spectrum is repeated periodically by the sampling process. For
proper reconstruction, one ensures that overlapping as in Fig. 14 does not
occur.
$\left|\text{S}(f)\right|$$\left|\text{S}_{\text{s}}(f)\right|$$f_{\text{g}}$–$f_{\text{g}}$+$f_{\text{g}}$–$\frac{1}{T_{\text{s}}}$+$\frac{1}{T_{\text{s}}}$$\frac{\text{s}(f)}{T_{\text{s}}}$
Figure 14: Periodically repeated component of the Fourier Transform of
s${}_{\text{s}}(t)$ [5]
If the spectra overlap as in Fig. 14 we have undersampling, the sampling rate
is too low. If big gaps occur between the spectra (Fig. 15) we have
oversampling, the sampling rate is too high.
$f$LOW PASS
H${}_{\text{LP}}(f)$$\left|\text{S}_{\text{s}}(f)\right|$–$f_{\text{g}}$+$f_{\text{g}}$–$1/T_{\text{s}}$+$1/T_{\text{s}}$
Figure 15: Reconstruction of S$(f)$ via ideal lowpass from S${}_{\text{s}}(f)$
(slightly oversampled)
But this scheme applies in most cases. In the limit we arrive at a Nyquist
rate of $1/T_{\text{s}}=2f_{\text{g}}=F$.
The rules mentioned above are of great importance for all ‘digital’
oscilloscopes. The performance (conversion time, resolution) of the input ADC
(analog–digital converter) is the key element for single-shot rise time. With
several ADCs in time-multiplex one obtains these days 8-bit vertical
resolution at 20 GSa/s = 10 GHz bandwidth.
Another way to look at the sampling theorem (Nyquist) is to consider the
sampling gate as a harmonic mixer (Fig. 16).
[algebraic=true]00.60.1*sin(10.4719755*x)s($t$)$R_{i}$q($t$)$C_{\text{S}}$$R_{\text{L}}$
Figure 16: Sampling gate as harmonic mixer; C${}_{\text{s}}$ = sampling
capacitor [6]
This is basically a nonlinear element (e.g., a diode) that gives product terms
of two signals superimposed on its nonlinear characteristics.
The switch in Fig. 16 may be considered as a periodically varying resistor
R($t$) actuated by q($t$). If q($t$) is not exactly a $\delta$-function then
the higher harmonics decrease with $f$ and the spectral density becomes
smaller at high frequencies.
For periodic signals one may apply a special sampling scheme. With each signal
event the sampling time is moved by a small fraction $\Delta t$ along the
signal to be measured (Fig. 17).
[algebraic=true]0.451.5sin(6*x + 3.14192)[algebraic=true]0.451.5sin(6*x +
3.14192)[algebraic=true]0.451.5sin(6*x +
3.14192)[algebraic=true]0.451.5sin(6*x +
3.14192)[algebraic=true]0.451.5sin(6*x +
3.14192)[algebraic=true,linestyle=dashed]4.5150.6*sin(0.6*x + 3.14192)$\Delta
t$2 $\Delta t$3 $\Delta t$4 $\Delta t$ Figure 17: Signal reconstruction with
sampling shift by $\Delta t$ per pulse [7]
The highest possible signal frequency for this sequential sampling is linked
to the width of the sampling pulse. This sampling or gating pulse should be as
short as possible otherwise signal averaging during the ‘gate-open’ period
would take place.
The sampling pulse is often generated by step-recovery diodes (snap-off
diodes) which change their conductivity very rapidly between the conducting
and non-conducting state. The actual switch (Schottky diode) becomes
conductive during the gate pulse and charges a capacitor (sample and hold
circuit) but not to the full signal voltage. Assuming a time constant
$R_{i}C_{\text{s}}$ much bigger than the ‘open’ time of the sampling gate, we
obtain approximately (Fig. 18)
$i_{\text{c}}(t)=\frac{\text{s}(t)}{R_{i}+R_{\text{d}}(t)}\hskip 5.69046pt.$
(17)
[algebraic=true]00.60.1*sin(10.4719755*x)s($t$)$R_{i}$q($t$)$C_{\text{S}}$u${}_{\text{C}}(t)$[algebraic=true]00.60.1*sin(10.4719755*x)s($t$)$R_{\text{d}}(t)$$C_{\text{S}}$$R_{i}$
Figure 18: Equivalent circuit for the sample-and-hold element [6]
After the sampling process we have [6]
$u_{\text{c}}(t)=\frac{1}{C_{\text{s}}}\int^{+\infty}_{-\infty}i_{\text{c}}(t)\text{d}t=\int^{+\infty}_{-\infty}\text{s}(t)\frac{1}{C_{\text{s}}(R_{i}+R_{\text{d}}(t))}\hskip
5.69046pt,$ (18)
with the control signal for the Schottky diode being
$\text{q}(t)=\frac{1}{C_{\text{s}}(R_{i}+R_{\text{d}}(t))}\hskip 5.69046pt.$
(19)
The control or switching signal is moved by $\tau$ or n$\Delta t$ (Fig. 21)
with respect to the signal to be sampled s($t$):
$u_{\text{c}}(\tau)=\int^{+\infty}_{-\infty}{\text{s}(t)\text{q}(t-\tau)\text{d}t}\hskip
5.69046pt.$ (20)
Note that the time constant $R_{i}C_{\text{s}}$ is much bigger than the length
of q$(t)$. $C_{\text{s}}$ is only charged to a fraction of s$(t)$ (Fig. 19).
tA [V]s($t$)q($t-\tau$) Figure 19: Sampling with finite-width sampling pulse
The sampling efficiency $\eta$ is defined as
$\eta=\frac{u_{\text{c}}(\tau)}{\text{s}(\tau)}\hskip 5.69046pt.$ (21)
In order to circumvent the problem of poor sampling efficiency a feedback loop
technique (integrator) can be used. This integrator amplifies the voltage step
on the sampling capacitor, after the sampling gate is closed, exactly by a
factor $1/\eta$. If the sampling gate has not moved with respect to the
trigger, the sampling capacitor is already charged to the correct voltage
u${}_{c}(\tau)$ and there is no change. Otherwise the change in uc just
amounts to the change in signal voltage.
The sampling gate is interesting from a technological point of view. As
aperture times (Fig. 19) may be of the order of 10 ps, MIC (Microwave
Integrated Circuit) technology has been used for many years. Today, the latest
generation of sampling heads (50 GHz) is even one step further with MMIC
(Monolithic Microwave Integrated Circuits) technology.
In MIC technology the sampling pulse is applied to a slotline in the ground-
plane metallization of a microstrip substrate (Fig. 20).
Figure 20: Sampling circuits [7]
This slot line has a length of some 10 mm and is shorted at both ends. With a
voltage across the slotline the fast Schottky diodes open and connect the
microstrip line via a through hole to the sampling capacitor $C_{\text{s}}$.
Owing to the particular topology of the circuit the signal line (microstrip)
is decoupled from the sampling pulse line over a wide frequency range (Fig.
20).
To move the sampling pulse by $\Delta t$ for each event requires a pre-trigger
(several 10 ns ahead), to start a fast-ramp generator. The intersection
(comparator) of the ramp generator output with a staircase-like reference
voltage defines the sampling time and $\Delta t$ (Fig. 21).
$\Delta t$2 $\Delta t$3 $\Delta t$4 $\Delta t$5 $\Delta t$FASTSWEEPREF Figure
21: Timing of sampling pulses [7]
The delay required for the pre-trigger has been a significant problem for many
applications, since it may be as large as 70 ns on certain (older)
instruments. A 70 ns delay-line leads to considerable signal distortions
especially for the high-frequency components.
To avoid the delay for the pre-trigger a technique named ‘random sampling’ was
developed about 45 years ago. It requires a strictly periodic signal rather
than just the repetitive one for sequential sampling (Fig. 21). By measuring
the (constant) repetition frequency of this strictly periodic signal, a
prediction of the next pulse arrival time can be given in order to generate a
trigger. Today there is little interest in random sampling, as pre-trigger
delays are drastically reduced (12 ns). There are also problems with jitter,
and random sampling needs repetition rates of serveral kHz [7].
Features of modern sampling scopes are summarized in Table 2.
Table 2: Features of modern sampling scopes Rise time: 7 ps | $\approx$ 50 GHz
---|---
Jitter | $\approx$ 1.5 ps
Static operation possible, no minimum repetition rate required. |
Optical sampling (mode-locked laser), 1 ps rise time | $\approx$ 350 GHz
## 0.4 Spectrum analyser
Radio-frequency spectrum analysers can be found in virtually every control-
room of a modern particle accelerator. They are used for many aspects of beam
diagnostics including Schottky signal acquisition and RF observation. A
spectrum analyser is in principle very similar to a common superheterodyne
broadcast receiver, except for the requirements of choice of functions and
change of parameters. It sweeps automatically through a specified frequency
range which corresponds to an automatic turning of the nob on a radio. The
signal is then displayed in the amplitude/frequency plane. Thirty years ago,
instruments were set manually and had some sort of analog or CRT (cathode ray
tube) display. Nowadays, with the availability of cheap and powerful digital
electronics for control and data processing, nearly all instruments can be
remotely controlled. The microprocessor permits fast and reliable setting of
the instrument and reading of the measured values. Extensive data treatment
for error correction, complex calibration routines, and self tests are a great
improvement. However, the user of such a sophisticated system may not always
be aware what is really going on in the analog section before all data are
digitized. The basis of these analog sections are discussed now.
In general there are two types of spectrum analyser:
* •
Scalar spectrum analysers (SA) and
* •
Vector spectrum analysers (VSA).
The SA provides only information of the amplitude of an ingoing signal, while
the VSA provides the phase as well.
### 0.4.1 Scalar spectrum analysers
A common oscilloscope displays a signal in the amplitude-time plane (time
domain). The SA follows another approach and displays it in frequency domain.
One of the major advantages of the frequency-domain display is the sensitivity
to periodic perturbations. For example, 5% distortion is already difficult to
see in the time domain but in the frequency domain the sensitivity to such
‘sidelines’ (Fig. 22) is very high ($-$120 dB below the main line).
[algebraic=true,linecolor=orange]040.06*sin(5*x)[algebraic=true,linecolor=orange]040.06*sin(5*x)A
[V]A [dB]tt2% AMPLITUDE MODULATION2% AM IN FREQUENCY DOMAIN– 400 Figure 22:
Example of amplitude modulation in time and frequency domain
A very faint amplitude modulation (AM) of 10${}^{-\text{12}}$ (power) on some
sinusoidal signals would be completely invisible on the time trace, but can be
displayed as two sidelines 120 dB below the carrier in the frequency domain
[8].
We will now consider only serial processing or swept tuned analysers (Fig.
23).
SIGNALTUNABLEBANDPASSAMPLITUDEDETECTORDISPLAY Figure 23: A tunable bandpass as
a simple spectrum analyser (SA)
The easiest way to design a swept tuned spectrum analyser is by using a
tunable bandpass. This may be an LC circuit, or a YIG filter (YIG = Yttrium-
Iron-Garnet) beyond 1 GHz. The LC filter exhibits poor tuning, stability and
resolution. YIG filters are used in the microwave range (as preselector) and
for YIG oscillators. Their tuning range is about one decade, with Q-values
exceeding 1000.
For much better performance the superheterodyne principle can be applied (Fig.
11).
SIGNALINPUTSWITCHABLEATTENUATORLOW PASSMIXERIF FILTERSAW
TOOTHGENERATORTUNABLEOSCILLATORLOIFAMPLIFIERDISPLAYVIDEOAMPLIFIERVIDEO
FILTERLOW PASSAMPLITUDEDETECTOR Figure 24: Block diagram of a spectrum
analyser
As already mentioned, the nonlinear element (four-diode mixer, double-balanced
mixer) delivers mixing products as
$f_{\text{s}}=f_{\text{LO}}\pm f_{\text{IF}}\hskip 5.69046pt.$ (22)
Assuming a signal range from 0 to 1 GHz for the spectrum analyser depicted in
Fig. 24 and $f_{\text{LO}}$ between 2 and 3 GHz we get the frequency chart
shown in Fig. 25.
1234523 GHZf${}_{\text{LO}}$fS(+)(–) Figure 25: Frequency chart of the SA of
Fig. 24, intermediate frequency = 2 GHz
Obviously, for a wide input frequency range without image response we need a
sufficiently high intermediate frequency. A similar situation occurs for AM-
and FM-broadcast receivers (AM-IF = 455 kHz, FM-IF = 10.7 MHz). But for a high
intermediate frequency (e.g., 2 GHz) a stable narrow-band IF filter is
difficult to construct which is why most SAs and high quality receivers use
more than one IF. Certain SAs have four different LOs, some fixed, some
tunable. For a large tuning range the first, and for a fine tune (e.g., 20
kHz) the third LO is tuned.
Multiple mixing is necessary when going to a lower intermediate frequency
(required when using high-Q quartz filters) for good image response
suppression of the mixers.
It can be shown that the frequency of the $n$-th LO must be higher than the
(say) 80 dB bandwidth of the ($n$ $-$ 1)th IF-band filter. A disadvantage of
multiple mixing is the possible generation of intermodulation lines if
amplitude levels in the conversion chain are not carefully controlled.
The requirements of a modern SA with respect to frequency are
* •
high resolution
* •
high stability (drift, phase noise)
* •
wide tuning range
* •
no ambiguities,
and with respect to amplitude response are
* •
large dynamic range (100 dB)
* •
calibrated, stable amplitude response
* •
low internal distortions.
It should be mentioned that the size of the smallest IF-bandpass filter width
$\Delta f$ has an important influence on the maximum sweep rate (or step-width
and -rate when using a synthesizer)
$\frac{\text{d}f}{\text{d}t}<(\Delta f)^{2}\hskip 5.69046pt.$ (23)
In other words, the signal frequency has to remain at $\Delta T=1/\Delta f$
within the bandwidth $\Delta f$.
On many instruments the proper relation between $\Delta f$ and the sweep rate
is automatically set to the optimum value for the highest possible sweep
speed, but it can always be altered manually (setting of the resolution
bandwidth).
Certain SAs do not use a sinusoidal LO signal but, rather, periodic short
pulses or a comb spectrum (harmonic mixer). This is very closely related to a
sampling scope, except that the spacing of the comb lines is different
$f_{\text{s}}=Nf_{\text{LO}}\pm f_{\text{IF}}\hskip 14.22636ptn=1,2,3,...$
(24)
A single, constant input-frequency line may appear several times on the
display. This difficulty (multiple response) was a particular problem with
older instruments. Certain modulation and sweep modes permit the
identification and rejection of these ‘ghost’ signals. On modern spectrum
analysers the problem does not occur, except at frequencies beyond 60 GHz,
when a tracking YIG filter may need to be installed.
Caution is advised when applying, but not necessarily displaying, two or more
strong (> 10 dBm) signals to the input. Intermodulation 3rd-order products may
appear (from the first mixer or amplifier) and could lead to misinterpretation
of the signals to be analysed.
SAs usually have a rather poor noise figure of 20–40 dB as they often do not
use preamplifiers in front of the first mixer (dynamic range, linearity). But
with a good preamplifier the noise figure can be reduced to almost that of the
preamplifier. This configuration permits amplifier noise figure measurements
to be made with reasonable precision of about 0.5 dB. The input of the
amplifier to be tested is connected to a hot and a cold termination and the
corresponding two traces on the SA display are evaluated [9, 10, 11, 12, 13].
Spectrum analysers can also be used to directly measure the phase noise of an
oscillator provided that the LO phase noise in the SA is much lower than that
of the device under test [9]. For higher resolution, set-ups with delay lines
and additional mixers (SA at low frequencies or FFT) are advised.
## 0.5 Vector spectrum and FFT analyser
The modern vector spectrum analyser (VSA) is essentially a combination of a
two-channel digital oscilloscope and a spectrum analyser FFT display. The
incoming signal gets down-mixed, bandpass (BP) filtered and passes an ADC
(generalized Nyquist for BP signals; $f_{\text{sample}}$ = 2 BW). A schematic
drawing of a modern VSA can be seen in Fig. 26.
Figure 26: Block diagram of a vector spectrum analyser
The digitized time trace then is split into an I (in phase) and Q (quadrature,
90 degree offset) component with respect to the phase of some reference
oscillator. Without this reference, the term vector is meaningless for a
spectral component.
One of the great advantages is that a VSA can easily separate AM and FM
components.
An example of vector spectrum analyser display and performance is given in
Fig. 27 and Fig. 28. Both figures were obtained during measurements of the
electron cloud in the CERN SPS.
Figure 27: Single-sweep FFT display similar to a very slow scan on a swept
spectrum analyser Figure 28: Spectrogram display containing about 200 traces
as shown on the left side in colour coding. Time runs from top to bottom
## 0.6 Noise basics
The concept of ‘noise’ was applied originally to the type of audible sound
caused by statistical variations of the air pressure with a wide flat spectrum
(white noise). It is now also applied to electrical signals, the noise ‘floor’
determining the lower limit of signal transmission. Typical noise sources are:
Brownian movement of charges (thermal noise), variations of the number of
charges involved in the conduction (flicker noise), and quantum effects
(Schottky noise, shot noise). Thermal noise is only emitted by structures with
electromagnetic losses which, by reciprocity, also absorb power. Pure
reactances do not emit noise (emissivity = 0).
Different categories of noise can be defined:
* •
white, which has a flat spectrum,
* •
pink, being low-pass filtered, and
* •
blue, being high-pass filtered.
In addition to the spectral distribution, the amplitude density distribution
is also required in order to characterize a stochastic signal. For signals
coming from very many independent sources, the amplitude density has a
Gaussian distribution. The noise power density delivered to a load by a black
body is given by Planck’s formula:
$\frac{N_{\text{L}}}{\Delta f}=hf\left(\text{e}^{hf/kT}-1\right)^{-1}\hskip
5.69046pt,$ (25)
where $N_{\text{L}}$ is the noisepower delivered to a load, $h=6.625\cdot
10^{-34}$ Js the Planck constant and $k=1.38056\cdot 10^{-23}$ J/K Boltzmann’s
constant.
Equation (25) indicates constant noise power density up to about 120 GHz (at
290 K) with 1% error. Beyond, the power density decays and there is no
‘ultraviolet catastrophe’, i. e., the total noise power is finite.
The radiated power density of a black body is given as
$W_{\text{r}}(f,T)=\frac{hf^{3}}{c^{2}\left[\text{e}^{hf/kT}-1\right]}\hskip
5.69046pt.$ (26)
For $hf<<kT$ the Rayleigh–Jeans approximation of Eq. (25) holds:
$N_{\text{L}}=kT\Delta f\hskip 5.69046pt,$ (27)
where in this case $N_{\text{L}}$ is the power delivered to a matched load.
The no-load noise voltage $u(t)$ of a resistor $R$ is given as
$\overline{u^{2}(t)}=4kTR\Delta f$ (28)
and the short-circuit current $i(t)$ by
$\overline{i^{2}(t)}=4\frac{kT\Delta f}{R}=4kTG\Delta f\hskip 5.69046pt,$ (29)
where $u(t)$ and $i(t)$ are stochastic signals and $G$ is $1/R$. The linear
average $\overline{u(t)},\overline{i(t)}$ vanishes. Of special importance is
the quadratic average $\overline{u^{2}(t)},\overline{i^{2}(t)}$.
The available power (which is independent of $R$) is given by (Fig. 29)
$\frac{\overline{u^{2}(t)}}{4R}=kT\Delta f\hskip 5.69046pt.$ (30)
[algebraic=true]00.60.1*sin(10.4719755*x)R1 = noiseless resistorR2 = noiseless
loadW${}_{\text{u}}=4kT\text{R}_{1}$ Figure 29: Equivalent circuit for a noisy
resistor $R_{1}$ terminated by a noisless load $R_{2}$
We define a spectral density function [9]
$\displaystyle W_{\text{u}}(f)$ $\displaystyle=$ $\displaystyle 4kTR$
$\displaystyle W_{\text{i}}(f)$ $\displaystyle=$ $\displaystyle 4kTG$ (31)
$\displaystyle\overline{u^{2}(t)}$ $\displaystyle=$
$\displaystyle\int_{f_{1}}^{f_{2}}W_{\text{u}}(f)\text{d}f\hskip 5.69046pt.$
A noisy resistor may be composed of many elements (resistive network). In
general, it is made from many carbon grains which have homogeneous
temperatures. But if we consider a network of resistors with different
temperatures and hence with an inhomogeneous temperature distribution (Fig.
30)
$R_{i},W_{\text{u}}$$R_{1},T_{1}$$W^{\prime}_{\text{u}1}$$W^{\prime}_{\text{u}2}$$W^{\prime}_{\text{u}3}$$R_{2},T_{2}$$R_{2},T_{2}$
Figure 30: Noisy one-port with resistors at different temperatures [14, 9]
the spectral density function changes to
$\displaystyle
W_{\text{u}}=\sum_{j}W_{\text{u}j}=4kT_{\text{n}}R_{\text{i}}\hskip
5.69046pt,$ (32) $\displaystyle T_{\text{n}}=\sum_{j}\beta_{j}T_{j}\hskip
5.69046pt,$ (33)
where $W_{\text{u}j}$ are the noise sources (Fig. 31), $T_{\text{n}}$ is the
total noise temperature, $R_{\text{i}}$ the total input impedance, and
$\beta_{j}$ are coefficients indicating the fractional part of the input power
dissipated in the resistor $R_{j}$. It is assumed that the $W_{\text{u}j}$ are
uncorrelated for reasons of simplicity.
(2,2.4)(7,2.4)$W_{\text{u}}$$R_{i}$$R_{i}$$W_{\text{u}}$$W_{\text{u}1}$$W_{\text{u}2}$$W_{\text{u}i}$
Figure 31: Equivalent sources for the circuit of Fig. 30.
The relative contribution ($\beta_{j}$) of a lossy element to the total noise
temperature is equal to the relative dissipated power multiplied by its
temperature:
$T_{\text{n}}=\beta_{1}T_{1}+\beta_{2}T_{2}+\beta_{3}T_{3}+\cdots$ (34)
A nice example is the noise temperature of a satellite receiver, which is
nothing else than a directional antenna. The noise temperature of free space
amounts roughly to 3 K. The losses in the atmosphere, which is an air layer of
10 to 20 km length, cause a noise temperature at the antenna output of about
10 to 50 K. This is well below room temperature of 290 K.
So far only pure resistors have been considered. Looking at complex
impedances, it can be seen that losses from dissipation occur in $Re$(Z) only.
The available noise power is independent of the magnitude of Re($Z$) with
Re($Z$) > 0\. For Figs. 30 and 31, Eq. (33) still applies, except that
$R_{\text{i}}$ is replaced by $Re$(Z${}_{\text{i}}$). However, it must be
remembered that in complex impedance networks the spectral power density
$W_{\text{u}}$ becomes frequency dependent [14].
The rules mentioned above apply to passive structures. A forward-biased
Schottky diode (external power supply) has a noise temperature of about
$T_{0}$/2 + 10%. A biased Schottky diode is not in thermodynamic equilibrium
and only half of the carriers contribute to the noise [9]. But it represents a
real 50 $\Omega$ resistor when properly forward biased. For transistors and in
particular field-effect transistors (FETs), the physical mechanisms are
somewhat more complicated. Noise temperatures of 50 K have been observed on a
FET transistor at 290 K physical temperature.
## 0.7 Noise-figure measurement with the spectrum analyser
Consider an ideal amplifier (noiseless) terminated at its input (and output)
with a load at 290 K with an available power gain ($G_{\text{a}}$). We measure
at the output [10, 15]:
$P_{\text{a}}=kT_{0}\Delta fG_{\text{a}}\hskip 5.69046pt.$ (35)
For $T_{0}$ = 290 K (or often 300 K) we obtain $kT_{0}$ = $-$174 dBm/Hz
($-$dBm = decibel below 1 mW). At the input we have for some signal Si a
certain signal/noise ratio S${}_{i}/\text{N}_{i}$ and at the output
S${}_{0}/\text{N}_{0}$. For an ideal (= noiseless) amplifier
S${}_{i}/\text{N}_{i}$ is equal to S${}_{0}/\text{N}_{0}$, i. e., the signal
and noise levels are both shifted by the same amount. This gives the
definition of the noise figure $F$:
$F=\frac{\text{S}_{i}/\text{N}_{i}}{\text{S}_{0}/\text{N}_{0}}\hskip
5.69046pt.$ (36)
The ideal amplifier has $F$ = 1 or $F$ = 0 dB and the noise temperature of
this amplifier is 0 K. The real amplifier adds some noise which leads to a
decrease in $\text{S}_{0}/\text{N}_{0}$ due to the noise added (=
N${}_{\text{ad}}$):
$F=\frac{\text{N}_{\text{ad}}+kT_{0}\Delta fG_{\text{a}}}{kT_{0}\Delta
fG_{\text{a}}}\hskip 5.69046pt.$ (37)
For a linear system the minimum noise figure amounts to $F_{\text{min}}$ = 1
or 0 dB. However, for nonlinear systems one may define noise figures $F$ < 1\.
Now assume a source with variable noise temperature connected to the input and
measure the linear relation between amplifier output power and input
termination noise temperature ($T_{\text{s}}$ = $T_{\text{source}}$).
In a similar way a factor $Y$ can be defined:
$\displaystyle Y=\frac{T_{\text{e}}+T_{\text{H}}}{T_{\text{e}}+T_{\text{C}}}$
$\displaystyle T_{\text{e}}=\frac{T_{\text{H}}-YT_{\text{C}}}{Y-1}$
$\displaystyle
F=\frac{\left[T_{(\text{H}}/290)-1\right]-Y\left[(T_{\text{C}}/290)-1\right]}{Y-1}\hskip
5.69046pt,$ (38)
where $T_{\text{e}}$ is the effective input noise temperature (see Fig. 32)
and $T_{\text{H}}$ and $T_{\text{C}}$ are the noise temperatures of a hot or
cold input termination.
P${}_{\text{OUT}}$TS [K]Source temperatureNOISE FREE++Slope $\approx
k\text{G}_{\text{a}}\Delta$fNoise freeT0 = 300 KT${}_{\text{hot}}$ = 12000 K
(Drawing not to scale) Figure 32: Relation between source noise temperature
$T_{\text{s}}$ and output power $P_{\text{out}}$ for an ideal (noise free) and
a real amplifier [10, 15]
To find the two points on the straight line of Fig. 32 one may switch between
two input terminations at 373 K (100∘ C) and 77 K. For precise reading of RF
power, calibrated piston attenuators in the IF path (Intermediate Frequency
Superheterodyne Receiver) are in use. This is the hot/cold method. The
difference between the $Y$-factor and the hot/cold method is that for the
latter the input of the amplifier becomes physically connected to resistors at
different temperatures (77, 373 K). For the $Y$-factor, the noise temperature
of the input termination is varied by electronic means between 300 K and 12
000 K (physical temperature always around 300 K).
As a variant of the 3-dB method with a controllable noise source, the excess
noise temperature definition ($T_{\text{ex}}=T_{\text{H}}-T_{\text{C}}$) is
often applied. A switchable 3-dB attenuator at the output of the amplifier
just cancels the increase in noise power from $T_{\text{H}}-T_{\text{C}}$.
Thus the influence of nonlinearities of the power meter is eliminated. To
measure the noise of one port one may also use a calibrated spectrum analyser.
However, spectrum analysers have high noise figures (20–40 dB) and the use of
a low-noise preamplifier is recommended. This ‘total power radiometer’ [11] is
not very sensitive but often sufficient, e. g., for observation of the
Schottky noise of a charged particle beam. Note that the spectrum analyser may
also be used for two-port noise figure measurements. An improvement of this
‘total power radiometer’ is the ‘Dicke Radiometer’ [11]. It uses a 1 kHz
switch between the unknown one port and a controllable reference source. The
reference source is made equal to the unknown via a feedback loop, and one
obtains a resolution of about 0.2 K. Unfortunately, switch spikes sometimes
appear. Nowadays, switch-free correlation radiometers with the same
performance are available [6].
The noise figure of a cascade of amplifiers is [14, 9, 10, 15, 6]
$F_{\text{total}}=F_{1}+\frac{F_{2}-1}{G_{\text{a}1}}+\frac{F_{3}-1}{G_{\text{a}1}G_{\text{a}2}}+\cdots$
(39)
As can be seen from Eq. (39) the first amplifier in a cascade has a very
important effect on the total noise figure, provided $G_{\text{a}1}$ is not
too small and F2 is not too large. In order to select the best amplifier from
a number of different units to be cascaded, one can use the noise measure $M$:
$M=\frac{F-1}{1-(1/G_{\text{a}})}\hskip 5.69046pt.$ (40)
The amplifier with the smallest M has to be the first in the cascade [15].
## References
* [1] F. Caspers, RF engineering basic concepts: the Smith chart, CAS proceedings, 2010.
* [2] G. D. Vendelin, A. M. Pavio and U. L. Rohde, Microwave Circuit Design Using Linear and Nonlinear Techniques, second edition, Wiley-Interscience, 2005, ISBN-10 0-471-41479-4.
* [3] M. Thumm, W. Wiesbeck and S. Kern, Hochfrequenzmesstechnik, B.G. Teubner Stuttgart $\cdot$ Leipzig, 1998, ISBN 3-519-16360-8
* [4] R. A. Witte, Spectrum and Network Measurements, Prentice-Hall Inc., 1991, ISBN 0-13-826959-9
* [5] H. D. Lueke, Signaluebertragung, Springer, 1975, ISBN 3-540-07125-3.
* [6] B. Schiek, Messysteme der Hochfrequenztechnik, Hiithig 1984, ISBN 3-7785-1045-2.
* [7] K. Lipinski, Moderne Oszillographen, VDE, 1974.
* [8] W. O. Schleifer, Hochfrequenz und Mikrowellenmesstechnik in der Praxis, Hiithig, 1981, ISBN 3-7785-0675-7.
* [9] B. Schiek and H. J. Sieveris, Rauschen im Hochfrequenzschaltungen, Hiithig, 1984, ISBN 3-7785-2007-5.
* [10] P. C. L. Yip, High Frequency Circuit Design and Measurement, Chapman and Hall, 1990, ISBN 0-412-34160-3.
* [11] G. Evans and C. W. McLeisch, RF-Radiometer Handbook, Artech, 1977, ISBN 0-89006-055-X.
* [12] F. R. Connor, Noise, Edward Arnold, 1973, ISBN 0-7131-3306-6.
* [13] F. Landstorfer and H. Graf, Rauschprobleme der Nachrichtentechnik, Oldenbourg, 1981, ISBN 3-486-24681-X.
* [14] O. Zinke and H. Brunswig, Lehrbuch der Hochfrequenztechnik, Zweiter Band, Springer, 1974, ISBN 3-540-06245-9.
* [15] Fundamentals of RF and microwave noise figure measurements, Hewlett-Packard Application Note 57-1.
|
arxiv-papers
| 2012-01-16T13:14:40 |
2024-09-04T02:49:26.352716
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. Caspers (CERN)",
"submitter": "Scientific Information Service CERN",
"url": "https://arxiv.org/abs/1201.3247"
}
|
1201.3363
|
# Testing Yukawa-like potentials from $f(R)$-gravity in elliptical galaxies
N.R. Napolitano11affiliation: INAF – Osservatorio Astronomico di Capodimonte,
Salita Moiariello, 16, 80131 - Napoli, Italy , S. Capozziello22affiliation:
Dipartimento di Scienze Fisiche, Università di Napoli “Federico II”, Napoli,
Italy 33affiliation: Istituto Nazionale di Fisica Nucleare, Sez. di Napoli,
Italy , A.J. Romanowsky44affiliation: UCO/Lick Observatory, University of
California, Santa Cruz, CA 95064, USA , M. Capaccioli22affiliation:
Dipartimento di Scienze Fisiche, Università di Napoli “Federico II”, Napoli,
Italy , C. Tortora55affiliation: Universit$\rm\ddot{a}$t Z$\rm\ddot{u}$rich,
Institut f$\rm\ddot{u}$r Theoretische Physik, Winterthurerstrasse 190,
CH-8057, Z$\rm\ddot{u}$rich, Switzerland napolita@na.astro.it
###### Abstract
We present the first analysis of extended stellar kinematics of elliptical
galaxies where a Yukawa–like correction to the Newtonian gravitational
potential derived from $f(R)$–gravity is considered as an alternative to dark
matter. In this framework, we model long-slit data and planetary nebulae data
out to 7 $R_{\rm eff}$ of three galaxies with either decreasing or flat
dispersion profiles. We use the corrected Newtonian potential in a
dispersion–kurtosis Jeans analysis to account for the mass–anisotropy
degeneracy. We find that these modified potentials are able to fit nicely all
three elliptical galaxies and the anisotropy distribution is consistent with
that estimated if a dark halo is considered. The parameter which measures the
“strength” of the Yukawa–like correction is, on average, smaller than the one
found previously in spiral galaxies and correlates both with the scale length
of the Yukawa–like term and the orbital anisotropy.
###### Subject headings:
galaxies : kinematics and dynamics – galaxies : general – galaxies :
elliptical and lenticular – cosmology: theory
## 1\. Introduction
The “concordance” $\Lambda$CDM cosmological model, which includes some unseen
Cold Dark Matter (DM) and a cosmological constant ($\Lambda$) acting as a
repulsive form of Dark Energy (DE), has been remarkably successful in
explaining the formation and evolution of cosmological structures at different
scales (e.g., Springel et al. 2006).
However, at cosmological scales, the cosmological constant as “vacuum state”
of the gravitational field is about 120 orders of magnitude smaller than the
value predicted by any quantum gravity theory (Weinberg 1989) and comparable
to the matter density (coincidence problem), even if they evolved decoupled in
the history of the universe.
In addition, looking at the galaxy scales there are a few critical issues yet
to be solved, which are giving hard time to the whole $\Lambda$CDM framework.
Since the discovery of the flat rotation curves of spiral systems, galaxies
have been the most critical laboratory to investigate the gravitational
effects of the DM halos, to be compared against the expectation of the
cosmological simulations (Navarro et al. 1997, NFW hereafter, Burkert 1995,
Navarro et al. 2010). Here, the $\Lambda$CDM model is not able to fully
explain the shallow central density profile of spiral and dwarf galaxies
(Gilmore et al., 2007; Salucci et al., 2007; Kuzio de Naray et al., 2008).
Early-type galaxies (ETGs hereafter) have been proven only recently to be
consistent with $\Lambda$CDM predictions (and WMAP5 cosmological parameters,
e.g. Komatsu et al. 2009), from their centers (Napolitano et al. 2010) to
their peripheries (Napolitano et al. 2011, N+11 hereafter), although there are
also diverging results showing that ETGs in some cases have too high (Buote et
al. 2007) or too low (e.g. Mandelbaum et al. 2008) concentrations.
This very uncertain context has been a fertile soil for alternative approaches
to the so-called “missing mass”. The basic approach is that the Newtonian
Theory of Gravity, which has been tested only in the Solar System, might be
inaccurate on larger (galaxies and galaxy clusters) scales. The most popular
theory investigated so far, the Modified Newtonian dynamics (MOND) proposed by
Milgrom (1983), is based on phenomenological modifications of Newton dynamics
in order to explain the flat rotation curves of spiral galaxies, and passed a
number of observational tests (Ferreira & Starkman 2009), included ETG
kinematics (Milgrom & Sanders 2003; Tiret et al. 2007; Kroupa et al. 2010;
Cardone et al. 2010; Richtler et al. 2011). Only lately it has been derived in
a cosmological context (Bekenstein, 2004).
A new approach, motivated from cosmology and quantum field theories on a
curved space-time, has been proposed to study the gravitational interaction:
the Extended Theories of Gravity (Capozziello 2002; Capozziello & Faraoni
2011). In particular, the so called $f(R)-$gravity seem to have passed
different observational tests like spiral galaxies’ rotation curves, X-ray
emission of galaxy clusters and cosmic acceleration (see e.g. Capozziello et
al. 2007a, C+07 hereafter, Capozziello et al. 2009, C+09 hereafter,
Capozziello et al. 2008). This approach is based on a straightforward
generalization of Einstein theory where the gravitational action (the Hilbert-
Einstein action) is assumed to be linear in the Ricci curvature scalar $R$. In
the case of $f(R)$-gravity, one assumes a generic function $f$ of the Ricci
scalar $R$ (in particular analytic functions) and asks for a theory of gravity
having suitable behaviors at small and large scale lengths.
As shown in Capozziello et al. (2009), analytic $f(R)$-models give rise, in
general, to Yukawa–like corrections to the Newtonian potentials in the weak
field limit approximation (see also Lubini et al. 2011). The correction
introduces a new gravitational scale, besides the standard Schwarzschild one,
depending on the dynamical structure of the self-gravitating system.
Here we want to test these Yukawa-like gravitational potentials against a
sample of elliptical galaxies. This approach has been proposed earlier, in a
phenomenological scheme for anti–gravity, to model flat rotation curves of
spiral galaxies (Sanders, 1984), and recently, in $f(R)$ theories to model
disk galaxies combined with NFW haloes (see Cardone & Capozziello 2011). The
test we are proposing at galaxy scales is crucial: reproducing kinematics and
then dynamics of these very different classes of astrophysical systems in the
realm of the same paradigm is needed to test these new gravitational theory as
an alternative to DM which has not been definitely found out at fundamental
level.
The layout of the paper is the following. In §2, we sketch the main
ingredients of $f(R)$-gravity deriving, in the weak field limit, the Yukawa-
like corrected gravitational potential. §3 is devoted to the high-order Jeans
analysis suitable for ellipticals. The dispersion-kurtosis fitting and the
data sample are presented in §4. Discussion and conclusions are in §5.
## 2\. Post- Newtonian potentials from $f(R)$-gravity
We are interested in testing a class of modified potentials which naturally
arise in post-Newtonian approximation of $f(R)-$gravity for which no
particular choice of the Lagrangian has been provided.
The starting point is a general gravity action of the form
$\displaystyle\mathcal{A}\,=\,\int
d^{4}x\sqrt{-g}\biggl{[}f(R)+\mathcal{X}\mathcal{L}_{m}\biggr{]}\,,$ (1)
where $f(R)$ is an analytic function of Ricci scalar, $g$ is the determinant
of the metric $g_{\mu\nu}$, ${\displaystyle\mathcal{X}=\frac{16\pi G}{c^{4}}}$
is the gravitational coupling constant, and $\mathcal{L}_{m}$ describes the
standard fluid-matter Lagrangian. Such an action is the straightforward
generalization of the Hilbert-Einstein action obtained as soon as $f(R)=R$.
In Capozziello et al. (2009, and reference therein) it has been shown that if
one solves the field equations in the weak field limit under the general
assumption of an analytic Taylor expandable $f(R)$ functions of the form
$\displaystyle f(R)\simeq f_{0}+f_{1}R+f_{2}R^{2}+f_{3}R^{3}+...\,$ (2)
the following gravitational potential arises
$\displaystyle\Phi\,=\,-\left(\frac{GM}{f_{1}r}+\frac{L\delta_{1}(t)~{}e^{-\frac{r}{L}}}{6~{}r}\right),$
(3)
where $L\doteq\displaystyle-\frac{6f_{2}}{f_{1}}$, $f_{1}$ and $f_{2}$ are the
expansion coefficients obtained by Taylor expansion. We note that the $L$
parameter is related to the effective mass
$m=(-3/L^{2})^{-1/2}=(2f_{2}/f_{1})^{1/2}$ and can be interpreted also as an
effective length.
From Eq. 3, the standard Newton potential is recovered only in the particular
case $f(R)=R$. Furthermore, the parameters $f_{1}$ and $f_{2}$ and the
function $\delta_{1}$ represent the deviations with respect the standard
Newton potential. On the Solar system scale, it has been shown that
Yukawa–like deviation from the pure Newtonian potential are not in
contradiction with classical tests of General Relativity (see e.g. Capozziello
& Tsujikawa 2008), thanks to the so-called Chameleon mechanism (Khoury &
Weltman 2004). In particular, $f_{1}$ and $f_{2}$ parameters are expected to
allow the regular Newtonian potential, while at larger scales they can assume
non-trivial values (e.g. $f_{1}\neq 1,\,\delta_{1}(t)\neq 0,\,\xi\neq 1$, see
Capozziello et al. 2007b, Capozziello et al. 2009).
Eq. (3) can be recast as
$\Phi(r)=-\frac{GM}{(1+\delta)r}\left(1+\delta e^{-\frac{r}{L}}\right)\,,$ (4)
where the first term is the Newtonian–like part of the potential associated to
baryonic point–like mass $M/(1+\delta)$ (no DM) and the second term is a
modification of the gravity including a “scale length”, $L$ associated to the
above coefficient of the Taylor expansion. If $\delta=0$ the Newtonian
potential is recovered. Comparing Eqs. 3 and 4, we obtain that
$1+\delta=f_{1}$, and $\delta$ is related to $\delta_{1}(t)$ through
$\delta_{1}=-\frac{6GM}{L^{2}}\frac{\delta}{1+\delta}$ (5)
where $6GM/L^{2}$ and $\delta_{1}$ can be assumed quasi-constant. From Eq. 5,
it turns out that $L\propto\sqrt{-\delta/(1+\delta)}$. Due to the arbitrarity
of $\delta_{1}(t)$, the actual value of the $\delta$ parameter can assume any
values, however, in order to have a Yukawa potential with a non imaginary
exponent (i.e. $L$ must be real) it is required that $\xi<0$ or $-1<\delta<0$.
As comparison, Sanders (1984) adopted the same potential as in Eq. 4 under the
assumption of anti–gravity generated by massive particles (of mass $m_{0}$)
carrying the additional gravitational force. In this case a typical scale
length would naturally arise ($L=h/m_{0}c$ being a Compton length) and a
$-1<\delta<0$ would provide a repulsive term to the Newtonian–like term
producing flat rotation curves at $r\gg L$ as observed in spiral galaxies. In
particular, for a small sample of spiral systems Sanders (1984) found
$-0.95\mathrel{\hbox to0.0pt{\lower 3.5pt\hbox{\hskip 0.5pt$\sim$}\hss}\raise
0.5pt\hbox{$<$}}\delta\mathrel{\hbox to0.0pt{\lower 3.5pt\hbox{\hskip
0.5pt$\sim$}\hss}\raise 0.5pt\hbox{$<$}}-0.92$.
Figure 1.— Circular velocity produced by the modified potential in Eq. 4 for
the two galaxies N4494 (top) and N4374 (bottom). In both cases the $M/L_{*}$
has been fixed to some fiducial value (as expected from stellar population
models and Kroupa 2001 IMF): $M/L_{*}=4.3\Upsilon_{\odot,B}$ for NGC 4494 and
$M/L_{*}=5.5\Upsilon_{\odot,V}$ for NGC 4374. The potential parameters adopted
are: $L=250^{\prime\prime}$ and $\delta$=0, -0.65, -0.8, -0.9 (lighter to
darker solid lines) and $L=180^{\prime\prime}$ and $\delta$=-0.8 (dashed
lines). The dotted line is a case with positive coefficient of the Yukawa–like
term and $L=5000^{\prime\prime}$ which illustrates that positive $\delta$
cannot produce flat circular velocity curves. Finally some reference
Navarro–Frenk–White (NFW) models are shown as dot-dashed lines.
Here we want to test the modified potential as in Eq. 4 in elliptical galaxies
and check whether it is able to provide a reasonable match to their kinematics
and how the model parameters compare with the results obtained from spiral
systems. We construct equilibrium models based on the solution of the radial
Jeans equation (see §3) to interpret the kinematics of planetary nebulae (PNe,
see Napolitano et al. 2002, 2005; Romanowsky et al. 2003; Coccato et al 2009)
which are the only stellar–like tracers for galaxy dynamics available in ETGs
out to $\sim 5-10$ effective radii ($R_{\rm eff}$).
We will use the inner long slit data and the extended PN kinematics for three
galaxies which have published dynamical analyses within DM halo framework: NGC
3379 (Douglas et al. 2007; De Lorenzi et al. 2009, DL+09 hereafter), NGC 4494
(Napolitano et al. 2009, N+09), NGC 4374 (N+11). The decreasing velocity
dispersion profiles of the first two galaxies have been modeled with an
intermediate mass halo, $\log M_{\rm vir}\sim 12-12.2M_{\odot}$, with
concentration $c_{\rm vir}=6-8$ and a fair amount of radial anisotropy in the
outer regions. For NGC 4374, having a rather flat dispersion profile, a more
massive (adiabatically contracted) halo with $\log M_{\rm vir}\sim
13.4M_{\odot}$ and $c_{\rm vir}\sim 7$ was required with a negligible amount
of anisotropy in the outer regions. These models turned out to be in fair
agreement with the expectation of WMAP5 $c_{\rm vir}-M_{\rm vir}$ relation
(N+11) and with a Kroupa (2001) IMF, making this sample particularly suitable
for a comparison with an alternative theory of gravity with no-DM as we want
to propose here.
Before we go on with detailed stellar dynamics, we show in Fig. 1 the circular
velocity of the modified potential as a function of the potential parameters
$L$ and $\delta$ for NGC 4494 and NGC 4374. As for the spiral galaxies,
negative values of the $\delta$ parameter make the circular velocity more and
more flat also reproducing the typical dip (e.g. NGC 4374) of the circular
velocity found for the DM models (dot–dashed curves) of the most massive
systems. On the contrary, positive $\delta$ values cannot produce flat
circular velocity curves (see Fig. 1).
Figure 2.— Dispersion in km s-1 (top) and kurtosis (bottom) fit of the galaxy
sample for the different $f(R)$ parameter sets: the anisotropic solution
(solid lines) is compared with the isotropic case (dashed line – for NGC 4374
and NGC 4494 this is almost indistinguishable from the anisotropic case). From
the left, NGC 4494, NGC 3379 and NGC 4374 are shown with DM models as gray
lines from N+09, DL+09 (no kurtosis is provided), and N+11 respectively.
### 2.1. A consistency check with galaxy scaling relations
To conclude the inspection of the modified potential as in Eq. 4 here we want
to show that, beside flat rotation curves, this also naturally accounts for
fundamental scaling relations of galaxies: the Tully-Fisher (TF) relation for
spirals and Faber-Jackson (FJ) relation for ETGs.
Both relations connect the total mass $M$ of galaxies with some characteristic
velocity defining the kinetic energy of the systems (i.e. the maximum rotation
velocity, $v_{\rm max}$, for spirals and the central velocity dispersion,
$\sigma_{0}$, for ETGs). In either cases the kinematical quantities involved
are proportional to the circular velocity of the systems through some
“structure” constant, thus the arguments below apply to galaxies in general.
Although the point–like version of the potential implies that the circular
velocity $v_{c}$ scales with mass as $M\sim v_{\rm c,max}^{2}$ (as pointed out
by Sanders 1984), if one derives the circular velocity for an extended galaxy
this can be generalized as
$v_{c}^{2}(r)=(GM_{\rm tot}/r_{*})\times f(r/r_{*};\delta,L/r_{*})$ (6)
where $r_{*}$ is a characteristic radius (e.g. the disk length for spirals or
the effective radius encircling half of the galaxy light for ETGs),
$f(r/r_{*};~{}\delta,~{}L/r_{*})$ is a generic function which includes the
radial dependence of the enclosed mass and the above Yukawa-like term. This
function is defined such as, for $\delta=0$, it gives $v_{c}^{2}(r)=GM(r)/r$
as the usual Newtonian expression. It is easy to show that if galaxies are
homologous the maximum of $v_{c}$ is reached at the same $r/r_{*}$, for a
given $\delta$ and $L/r_{*}$ and this maximum can be written as
$v_{\rm c,max}^{2}={\rm K}M_{\rm tot}/r_{*}$ (7)
where the constant $K$ depends on the set of parameters
{$\delta,~{}L,~{}r_{*}$} adopted. In Eq. 7, though, $M_{\rm tot}$ and $r_{*}$
are linked by the size–mass relation which is generally written as
$r_{*}\propto M_{\rm tot}^{\alpha}$, from which Eq. 7 can be written as
$v_{\rm c,max}^{2}\propto M_{\rm tot}^{1-\alpha}.$ (8)
The size-mass relation of spiral galaxies can be found in Persic et al. (1996,
see also ) to be $r_{*}\propto M_{\rm tot}^{0.4}$, while it is $r_{*}\propto
M_{\rm tot}^{0.6}$ for ETGs (e.g. Shen et al. 2003, Napolitano et al. 2005).
This would give a TF slope of $3.33$ and FJ slope of $5$ which are both in the
range of the observed relations (see e.g. McGaugh 2005 and Nigoche-Netro et
al. 2010 respectively) with the remaining discrepancy being mainly due to the
conversion factor to the observed quantities and non homologies.
We finally remark that the TF relation has been found not to be conflicting
with $f(R)$ potentials in Capozziello et al. (2006), although the potentials
from $f(R)\propto R^{n}$ adopted there are just a series expansion of the
Yukawa–like potential coming out from a more general polynomial f(R) as in Eq.
3.
## 3\. High-order Jeans analysis
From the model point of view, the problem of fitting a modified potential as
in Eq. 4 (which is formally self-consistent since the source of the potential
is the only mass of the dynamical tracers, i.e. stars) implies the same kind
of degeneracies between the anisotropy parameter,
$\beta=1-\sigma_{\theta}^{2}/\sigma_{r}^{2}$ (where $\sigma_{\theta}$ and
$\sigma_{r}$ are the azimuthal and radial dispersion components in spherical
coordinates), and the non–Newtonian part of the potential (characterized by
two parameters like typical dark haloes) in a similar way of the classical
mass-anisotropy degeneracy. We have shown (N+09, N+11) that these degeneracies
can be alleviated via higher-order Jeans equations including in the dynamical
models both the dispersion111For the slow–rotating models we use the velocity,
$v_{\rm rms}=\sqrt{v^{2}+\sigma^{2}}$ as a measure of the velocity dispersion.
($\sigma_{\rm p}$) and the kurtosis ($\kappa$) profiles of the tracers.
In the following, we will use the assumption of spherical symmetry since
galaxies in the sample are all E0–E1 for which, if one exclude the singular
chance that they are all flattened systems seen face–on (see discussion in
Sect. 8.1. of Douglas et al. 2007), the spherical approximation is good at 10%
(Kronawitter et al. 2000)222 The effect of non–spherical models is outside the
scope of this paper, but details for NGC 3379 and NGC 4494 can be found in
DL+09 and N+09.. Under spherical assumption, no-rotation and $\beta=\rm const$
(corresponding to the family of distribution functions
$f(E,L)=f_{0}L^{-2\beta}$, see Łokas 2002 and references therein333 Here,
there is the caveat that the solution of Jeans Equations does not ensure that
the final distribution function is non negative and thus fully physical (see
e.g., An & Evans 2006)., the 2-nd and 4-th moment radial equations can be
compactly written as:
$s(r)=r^{-2\beta}\int_{r}^{\infty}x^{2\beta}H(x)dx$ (9)
where $s(r)=\\{\rho\sigma_{r}^{2};\rho\overline{v_{r}^{4}}\\}$, $\beta$ is the
anisotropy parameter, and
$H(r)=\left\\{\rho\frac{d\Phi}{dr};3\rho\frac{d\Phi}{dr}\overline{v_{r}^{2}}\right\\}$
respectively for the dispersion and kurtosis equations, being the latter
$\kappa(r)={\overline{v_{r}^{4}}}/\sigma_{r}^{4}$. In the same equations,
$\Phi(r)$ is the spherical extended source version of the point–like potential
as in Eq. 4444This is obtained assuming the onion shell approximation:
$\Phi(r)=\int_{0}^{r}\int_{0}^{2\pi}\int_{0}^{\pi}\phi(r)~{}r^{2}\sin\theta~{}d\theta
d\varphi dr$, see also Eq. 18 of C+09. and $\rho(r)$ is the 3D density of the
tracer obtained by multiplying the deprojection of the stellar surface
brightness profile, $j_{\star}(r)$, by some constant stellar mass-to-light
ratio, $M/L_{\star}$.
This $M/L_{\star}=const$ might be a strong assumption to check further in a
separate paper as it neglects the presence of stellar population gradients
(see e.g. Tortora et al. 2010). However, colour (and $M/L$) gradients are
generally stronger within $R_{\rm eff}$ (see e.g. Tortora et al 2010) and
might mainly drive the best fit in the central regions, while they are
possibly shallower outside (e.g., Tamura & Ohta 2003) where the $f(R)$
parameters should be better constrained. In the following, $j_{\star}(r)$ is
derived by photometry presented in previous dynamical studies (i.e. DL+09,
N+09, N+11 for NGC 3379, NGC 4494 and NGC 4374 respectively).
Eqs. 9 are the ones interested by the potential modification and include four
free parameters to be best-fitted: the $f(R)$ parameters {$\delta,L$}, the
“dynamically inferred” stellar mass-to-light ratio $M/L_{\star}$ and the
constant anisotropy $\beta$ (see also §4). The solutions of Eqs. 9 on a
regular grid in the parameter space are then projected to match the observed
line-of-sight kinematical profile via ordinary Abel integrals (see N+09 for
details).
As mentioned earlier, Eqs. 9 are written under the assumption of a constant
$\beta$ with radius, which provides a average global anisotropy distribution
over all the galaxy. As seen in previous analyses (e.g. N+09, DL+09 and N+11),
it is likely that this might not be a fair assumption, as $\beta$ turns out to
be constant somewhere in the outer regions, but strongly varying in the
central radii. In this preliminary test we will skip this implementation of
the models since we expect this to possibly improve the fit to the data in the
central part only, where we do not expect the overall dynamics of being
strongly ruled by the $f(R)$ potential, whose parameters are the main focus of
this work. Furthermore, we have shown previously (see e.g. N+09 and N+11) that
the assumption of the constant or radial varying anisotropy did not strongly
affect the determination of the other important parameter, the dynamically
based stellar $M/L$. In the following we will take the $\beta=$const as fair
estimate of the average galaxy anisotropy.
## 4\. Dispersion-kurtosis fitting
In Fig. 2 we show the dispersion and kurtosis profiles of the three galaxies
with the $f(R)$ models superimposed (solid lines). The fitting procedure is
based on the simultaneous $\chi^{2}$ minimization of the dispersion and
kurtosis profiles over a regular grid in the parameter space. The best–fit
parameters are summarised in Table 1 together with some info of the galaxy
sample.
Overall the agreement of the model curves with data is remarkably good and it
is comparable with models obtained with DM modeling (gray lines in Fig. 2).
In all cases, the $f(R)$ models allow to accommodate a constant orbital
anisotropy $\beta$ which is very close to the estimates from the DM models
(see e.g. Table 1555For NGC 4374 only to be nicely fitted at all radii we
needed to include some radial anisotropy in the very central regions,
following the $\beta(r)$ distribution adopted in N+11 (see Eq. 5, whith best
fit $r_{a}=22.5$).). This is mainly guaranteed by the fit to the $\kappa(r)$
which does not respond much differently to the modified potential with respect
the DM models. Thus, an important result of the analysis is that the orbital
anisotropy is fairly stable to the change of the galaxy potential. In
particular, the use of the kurtosis profiles has allowed us to solve the
degeneracy of the models and favor the anisotropic solutions for NGC 3379 and
NGC 4494 (NGC 4374 being almost isotropic everywhere). Although the isotropy
solutions provide also a good fit for the dispersion profile only (see e.g.
the dashed lines in Fig. 2), they not correctly match the observed $\kappa$.
This produces a significantly worse total $\chi^{2}$/dof (NGC 3379: 45/26;
NGC4374: 35/40; NGC 4494: 27/44) with respect to the best–fit in the Table 1,
although still close to $\chi^{2}$/dof$~{}\sim 1$ mainly because of the large
error bars.
Table 1Model parameters for the $f(R)$ potential.
Galaxy | Mag (band) | $R_{\rm eff}$ | $M/L_{\star}$ | $L$ | $\delta$ | $\beta$ | $\chi^{2}$/dof
---|---|---|---|---|---|---|---
NGC3379 | -19.8(B) | 2.2 | 6 (7) | 6 | -0.75 | 0.5($<$0.8) | 14/25
NGC4374 | -21.3(V) | 3.4 | 6 (6) | 24 | -0.88 | 0.01(0.01) | 14/39
NGC4494 | -20.5(B) | 6.1 | 3 (4) | 20 | -0.79 | 0.5(0.5) | 18/43
NOTES – Galaxy ID, total magnitude, effective radius and model parameters for
the unified solution. DM–based estimates for $M/L_{\star}$ and $\beta$ (NGC
3379: DL+09; NGC 4374: N+11; NGC 4494: N+09) are shown in parentheses for
comparison. $M/L_{\star}$ are in solar units, $R_{\rm eff}$ and $L$ in kpc.
Typical errors on $M/L_{\star}$ are of the order of 0.2$M/L_{\odot}$ and on
$\beta$ of 0.2 (see also Fig. 3). The small $\chi^{2}$ values are mainly due
to the large data error bars.
Finally, the best fit $M/L_{\star}$ in Table 1 are very similar to the values
found for DM models (reported between brackets) in all cases, generally
consistent with a Kroupa (2001) IMF.
Looking at the $f(R)$ parameters, in Fig. 3 we show the marginalized
confidence contours of the main two potential parameters for the three
galaxies. As also reported in Table 1, the $\delta$ parameter has a mean value
$\delta=-0.81\pm 0.07$ which is inconsistent with the one previously found for
spiral galaxies (e.g. Sanders 1984, also shown in Fig. 3). On the contrary,
$\delta$ seems nicely correlated with the other potential parameter, $L$, as
expected from Eq. 5. In the same figure the correlation is supported by the
tentative fit into the $\delta-L$ plane (whether or not the spiral galaxy
sample is included in the fit), although the sample is too small to drive any
firm conclusion.
Interestingly, there seems to be a possible increasing trend of $\delta$ with
the orbital anisotropy: this is also shown in Fig. 3 where we have added the
fiducial value obtained for the spiral sample (having assumed a reference
$\beta=-1$ for fiducial tangential anisotropy for late-type systems, see e.g.
Battaglia et al. 2005). This evidence leaves room for an interpretation of
$\delta$ and the physics of the galaxy collapse (e.g. the spherical infall
model, Gunn & Gott 1972; Gunn 1977).
In fact, as discussed in §2, $\delta$ is linked to $\delta_{1}$, which is an
arbitrary function that comes out because the field equations in the post-
Newtonian approximation, depending only on the radial coordinate. From a
physical point of view, such a function could be related to second order
effects related to anisotropies and non-homogeneities which could trigger the
formation and the evolution of stellar systems. To take into account such a
situation, one should perform the post-Newtonian limit of the theory not only
in the simple hypothesis of homogeneous spherical symmetry (Schwarzschild
solution) but also considering more realistic situation as
Lema$\hat{i}$tre–Tolman–Bondi solutions (see e.g. Herrera et al. 2010).
Figure 3.— Top: 1- and 2-$\sigma$ confidence levels in the $\delta-L$ space
marginalized over $M/L_{\star}$ and $\beta$ (see also Table 1). Spiral galaxy
results from Sanders (1984) are shown as empty triangle with error bars. Solid
(dashed) curve shows the tentative best-fit to the data including (excluding)
the spiral galaxies and assuming a $L\propto\sqrt{\delta/(1+\delta)}$
correlation as expected from Eq. 5. Bottom: the anisotropy and the $\delta$
parameters turn out to be correlated for the elliptical sample (full squares).
This correlation seems to include also the spiral sample cumulatively shown as
the empty triangle (here we have assumed $\beta=-1.0\pm 0.5$ as a fiducial
value for spiral galaxies to draw a semi-quantitative trend across galaxy
types).
## 5\. Discussion and Conclusions
There is a growing attention to alternative model to the $\Lambda$CDM paradigm
as the latter is still suffering some discrepancies at the galaxy scales and,
most importantly, is based on the assumption of the existence of two
ingredients (DM and DE) whose nature is still unknown.
Different attempts have tried to circumvent the problem by introducing a
modified dynamics, e.g. with the MOND theory (see Sanders & McGaugh 2002,
Swaters et al. 2010, Cardone et al. 2010), but this seems still needing some
DM at least to cluster scales which might be still consistent with the
primordial nucleosynthesis (e.g. via high energy neutrinos, Angus et al. 2010)
and does not provide an explanation for the DE.
Lately $f(R)$–gravity models have made their step out as a natural explanation
for the two dark ingredients of the Universe assuming that they are related to
the fact that gravitational interaction could present further degrees of
freedom whose dynamical effects emerge at large scales (Capozziello & Faraoni
2011). In this paper we have checked the Yukawa-like modification to the
Newtonian potential obtained as post-Newtonian approximation of $f(R)-$gravity
for which no particular choice of the Lagrangian has been provided, with the
only assumption that $f(R)$ is analytic function.
We have used a combination of long-slit spectroscopy and planetary nebulae
kinematics out to $\sim$7 $R_{\rm eff}$ in three systems (NGC 3379, NGC 4374,
NGC 4494) for which $\Lambda$CDM models turned out to be fairly consistent
with WMAP5 measurements (see N+11 for a discussion).
Due to the small galaxy sample, the spirit of this analysis has been to check
whether $1)$ the modified potential introduced by the $f(R)$-gravity allowed a
fit to the galaxy kinematics comparable to the DM models; $2)$ the three
galaxies returned a parameter $\delta$ which is comparable with spiral
galaxies (Sanders 1984).
We have found that the modified potentials allow to nicely model the three
galaxies with a distribution of the $\delta$ parameters which turned out to be
inconsistent with the results found in spiral systems. We have shown some
hints that $\delta$ might be correlated with the galaxy anisotropy, $\beta$,
and the scale parameter, $L$, with elliptical and spiral galaxies following
the same pattern.
This evidence can have interesting implication on the ability of the theory to
make predictions on the internal structure of the gravitating systems after
their spherical collapse (e.g. Gunn 1977) which has to be confirmed on a
larger galaxy sample which we expect to do in a near future.
Despite of some simplifications on the model adopted (e.g. constant $M/L$ and
anisotropy across the galaxy) and the degeneracies between the model
parameters, the results are very encouraging. The fit to the data is very good
in all cases and both the stellar $M/L$ (with Kroupa IMF generally favored)
and orbital anisotropy turn out to be similar to the one estimated if a dark
halo is considered.
Getting a modified gravity to work self-consistently for all gravitating
systems in general, and all galaxy families in particular, is a very non-
trivial challenge that has foiled other theories (e.g. MOND).
We thank the anonymous referee for constructive comments which allowed us to
significantly improve the paper. AJR was supported by National Science
Foundation Grants AST-0808099 and AST-0909237. CT was supported by the Swiss
National Science Foundation.
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|
arxiv-papers
| 2012-01-16T21:00:04 |
2024-09-04T02:49:26.364432
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N. R. Napolitano, S. Capozziello, A. J. Romanowsky, M. Capaccioli, C.\n Tortora",
"submitter": "Nicola R. Napolitano",
"url": "https://arxiv.org/abs/1201.3363"
}
|
1201.3409
|
# Nonlocal symmetries related to Bäcklund transformation and their
applications
S Y Lou1,3, Xiaorui Hu2,1 and Yong Chen2,1 1Faculty of Science, Ningbo
University, Ningbo, Zhejiang 315211, People’s Republic of China 2Shanghai Key
Laboratory of Trustworthy Computing, East China Normal University, Shanghai
200062, China 3Department of Physics, Shanghai Jiao Tong University, Shanghai
200240, China lousenyue@nbu.edu.cn, ychen@sei.ecnu.edu.cn
###### Abstract
Starting from nonlocal symmetries related to Bäcklund transformation (BT),
many interesting results can be obtained. Taking the well known potential KdV
(pKdV) equation as an example, a new type of nonlocal symmetry in elegant and
compact form which comes from BT is presented and used to make researches in
the following three subjects: two sets of negative pKdV hierarchies and their
corresponding bilinear forms are constructed; the nonlocal symmetry is
localized by introduction of suitable and simple auxiliary dependent variables
to generate new solutions from old ones and to consider some novel group
invariant solutions; some other models both in finite dimensions and infinite
dimensions are generated by comprising the original BT and evolution under new
nonlocal symmetry. The finite-dimensional models are completely integrable in
Liouville sense, which are shown equivalent to the results given through the
nonlinearization method for Lax pair.
###### pacs:
02.30.Ik, 11.30.Na, 04.20.Jb
## 1 Introduction
With the development of integrable systems and solion theory, symmetries [1,
2, 3] play the more and more important role in nonlinear mathematical physics.
Thanks to the classical or nonclassical Lie group method, Lie point symmetries
of a differential system can be obtained, from which one can transform given
solutions to new ones via finite transformation and construct group invariant
solutions by similarity reductions. However, little importance is attached to
the existence and applications of nonlocal symmetries [2, 3]. Firstly, seeking
for nonlocal symmetries in itself is a difficult work to perform. One of our
authors (Lou) has made some efforts to get infinite many nonlocal symmetries
by inverse recursion operators [4, 5] the conformal invariant form (Schwartz
form) [6] and Darboux transformation [7, 8]. Moreover, it appears that the
nonlocal symmetries are rarely used to construct explicit solutions since the
finite symmetry transformations and similarity reductions can not be directly
calculated under the nonlocal symmetries. Naturally, it is necessary to
inquire as to whether nonlocal symmetries can be transformed to local ones.
The introduction of potential [3] and pseudopotential type symmetries [9, 10,
11] which possesses close prolongation extends the applicability of symmetry
methods to obtain solutions of differential equations (DEs). In that context,
the original given equation(s) can be embedded in some prolonged systems.
Hence, these nonlocal symmetries with close prolongation are anticipated [12,
13, 14].
On the other hand, to find new integrable models is another important
application of symmetry study. A systematic approach have been developed by
Cao [15, 16, 17] to find finite-dimensional integrable systems by the
nonlinearization of Lax pair under certain constraints between potentials and
eigenfunctions. Especially in the study of (1+1)-dimensional soliton
equations, various new kinds of confocal involutive systems are constructed by
the approach of nonlinearization of eigenvalue problems or constrained flows
[18, 19]. It has also been pointed that by restricting a symmetry constraint
to the Lax pair of soliton equation, one can not only obtain the lower
dimensional integrable models from higher ones, but also embed the lower ones
into higher dimensional integrable models [6, 8, 20]. Here, alternatively, we
are inspired to act the given nonlocal symmetry on the Bäcklund transformation
(BT) instead of Lax pair to generate some other new systems via symmetry
constraint method. The related work may be adventurous but full of enormous
interest.
In this paper, taking the well known potential KdV equation (pKdV) for a
special example, we will study the nonlocal symmetry defined by BT. Since the
BT reveals a finite transformation between two exact solutions of DEs, it must
hint some symmetry. For pKdV equation, a new class of nonlocal symmetries are
derived from its BT, which may give more interesting applications than those
nonlocal symmetries only including potentials and pseudopotentials. The
prolongation of the new nonlocal symmetries are found close after extending
pKdV equation to an auxiliary system with four dependent variables. The finite
symmetry transformation and similarity reductions are computed to give exact
solutions of KdV equation. What we want to mention is the process can once
lead to two exact solutions from one given result due to the Bäcklund
transformation. Moreover, for the pKdV equation, some other models both in
finite dimensions and infinite dimensions are obtained. The finite-dimensional
systems obtained here are found equivalent to the results given by Cao [16],
which have been verified completely integrable in Liouville sense. This
discovery confirms that these obtained infinite-dimensional models should have
many nice integrable properties, which needs our further study.
The paper is organized as follows. In section II, we present a detailed
description about the new nonlocal symmetry with BT of pKdV equation. Two
kinds of flow equations corresponding to the given nonlocal symmetry, i.e. the
negative pKdV hierarchies, are obtained and their corresponding bilinear forms
are also given out. In section III, we extend the nonlocal symmetry to be
equivalent to a Lie point symmetry of a auxiliary prolonged system admitting
pKdV equation and its BT. Then the finite symmetry transformation and
similarity reductions are made to produce exact solutions of pKdV and then KdV
equation. Section IV is devoted to constructing various integrable systems by
means of symmetry constraint method. Conclusions and discussions are given in
Section V.
## 2 Nonlocal symmetries and flow equations related to BT
### 2.1 BT for the pKdV equation
The well-known KdV equation reads
$\displaystyle\omega_{t}+\omega_{xxx}-6\omega\omega_{x}=0,$ (1)
where subscripts $x$ and $t$ denote partial differentiation. For convenience
to deal analytically with a potential function $u$, introduced by setting
$\omega=u_{x}$, it follows from equation (1) that $u$ would satisfy the
equation
$\displaystyle u_{t}+u_{xxx}-3u^{2}_{x}=0,$ (2)
which is called potential KdV (pKdV) equation.
For equation (2), there exists the following BT [21]
$\displaystyle u_{x}+u_{1,x}=-2\lambda+\frac{(u-u_{1})^{2}}{2},$ (3)
$\displaystyle
u_{t}+u_{1,t}=2u^{2}_{x}+2u^{2}_{1,x}+2u_{x}u_{1,x}-(u-u_{1})(u_{xx}-u_{1,xx})$
(4)
with $\lambda$ being arbitrary parameter.
Equations (3) and (4) show that if $u$ is a solution of equation (2), so is
$u_{1}$, that is to say, they represent a finite symmetry transformation
between two exact solutions of equation (2).
On the other hand, equations (3) and (4) can also be viewed as a nonlinear Lax
pair of equation (2). For
$\displaystyle u_{1,x}=-u_{x}-2\lambda+\frac{(u-u_{1})^{2}}{2},$ (5)
$\displaystyle
u_{1,t}=-u_{t}+2u^{2}_{x}+2u^{2}_{1,x}+2u_{x}u_{1,x}-(u-u_{1})(u_{xx}-u_{1,xx}),$
(6)
its compatibility condition $u_{1,xt}=u_{1,tx}$ is exactly equation (2). In
fact, both equation (3) and equation (4) hint that they are all Riccati type
equations about $u$ or $u_{1}$, which can be linearized by the well known
Cole-Hopf transformation
$\displaystyle u=-2\frac{\psi_{x}}{\psi},\qquad{\rm{or}}\qquad
u_{1}=-2\frac{\psi_{1,x}}{\psi_{1}}.$ (7)
Moreover, by virtue of the dependent variables transformation (7), one can
convert equation (2) into the following bilinear form
$\displaystyle(D^{4}_{x}+D_{x}D_{t})\psi\cdot\psi=0,$ (8)
meanwhile it leads equations (3) and (4) to
$\displaystyle(D^{2}_{x}-\lambda)\psi\cdot\psi_{1}=0,$ (9)
$\displaystyle(D_{t}+D^{3}_{x}+3\lambda D_{x})\psi\cdot\psi_{1}=0,$ (10)
where the Hirota’s bilinear operator $D^{m}_{x}D^{n}_{t}$ is defined by
$D^{m}_{x}D^{n}_{t}a\cdot b=\left.\left(\frac{\partial}{\partial
x}-\frac{\partial}{\partial
x^{\prime}}\right)^{m}\left(\frac{\partial}{\partial
t}-\frac{\partial}{\partial
t^{\prime}}\right)^{n}a(x,t)b(x^{\prime},t^{\prime})\right|_{x^{\prime}=x,t^{\prime}=t}.$
### 2.2 The nonlocal symmetry from Bäcklund transformation
For equation (2) with its BT (3) and (4), considering the invariant property
under
$\lambda\rightarrow\lambda+\epsilon\delta,\qquad u\rightarrow
u+\epsilon\sigma,\qquad u_{1}\rightarrow u_{1}+\epsilon\sigma^{\prime},$
we may find substantial possible nonlocal symmetries and a special case is
presented and studied as follows.
_Proposition 1._ The pKdV equation (2) has a new type of nonlocal symmetry
given by
$\displaystyle\sigma=\exp(\int{u-u_{1}}\rmd x),$ (11)
where $u$ and $u_{1}$ satisfy BT (3) and (4). That means $\sigma$ given by
(11) satisfies the following symmetry equation
$\displaystyle\sigma_{t}+\sigma_{xxx}-6u_{x}\sigma_{x}=0.$ (12)
_Proof:_ By direct calculation.
On the other hand, we let the bilinear pKdV equation (8) be invariant under
the transformation $\psi\rightarrow\psi+\epsilon\sigma_{\psi}$, which produces
the corresponding symmetry equation
$\displaystyle(D^{4}_{x}+D_{x}D_{t})\sigma_{\psi}\cdot\psi=0.$ (13)
The Cole-Hopf transformation $u=-2\frac{\psi_{x}}{\psi}$ between equation (2)
and its bilinear equation (8) determines a symmetry transformation for
$\sigma$ and $\sigma_{\psi}$, saying
$\displaystyle\sigma=\frac{2\psi_{x}\sigma_{\psi}}{\psi^{2}}-\frac{2\sigma_{\psi,x}}{\psi}.$
(14)
Taking equations (11) and (7) into equation (14), we obtain a class of
nonlocal symmetry for equation (8)
$\displaystyle\sigma_{\psi}=-\frac{\psi}{2}\int\frac{\psi^{2}_{1}}{\psi^{2}}dx.$
(15)
Correspondingly, it gives the following proposition for equation (8).
_Proposition 2._ The bilinear pKdV equation (8) has the nonlocal symmetry
expressed by (15), where $\psi$ and $\psi_{1}$ satisfy bilinear BT (9) and
(10).
_Proof:_ One can directly check that $\sigma_{\psi}$ given by (15) satisfies
symmetry equation (13) under the consideration (9) and (10).
### 2.3 Two sets of negative pKdV hierarchies
The existence of infinitely many symmetries leads to the the existence of
integrable hierarchies and with the help of infinitely many nonlocal
symmetries, one can extend the original system to its negative hierarchies
[22, 23]. Here, starting from the nonlocal symmetry (11) related to BT of
equation (2), we would like to present two sets of negative pKdV hierarchies
and their corresponding bilinear forms are also constructed only by the
transformation (7).
_Case 1._ The first kind of negative pKdV hierarchy can be obtained, reading
$\displaystyle u_{t_{-N}}=-\sum^{N}_{i=1}\exp(\int{u-u_{i}}\rmd x),$ (16)
$\displaystyle u_{x}+u_{i,x}=-2\lambda_{i}+\frac{(u-u_{i})^{2}}{2},\quad
i=1,2,...,N,$ (17)
where $\lambda_{i}$ is arbitrary constant.
In particular, when $N=1$, one has the first equation of negative pKdV
hierarchy, namely
$\displaystyle
2u_{xxt}u_{t}-4u_{x}u^{2}_{t}-u^{2}_{xt}-4\lambda_{1}u^{2}_{t}=0.$ (18)
Here we have instead $t_{-1}$ with $t$ for simplicity. It is well known that
the first negative flow in the KdV hierarchy is linked to the Camassa-Holm
equation via a hodograph transformation [24] or can be reduced to the sinh-
Gordon/sine-Gordon/Liouville equations [25]. Here we transform equation (18)
into sine-Gordon and Liouville equations.
In fact, by setting $\beta\equiv\beta(x,t)=-u_{t}$, we can rewrite equation
(18) in the form
$\displaystyle\beta_{x}=\left(-\frac{\beta_{xx}}{2\beta}+\frac{\beta^{2}_{x}}{4\beta^{2}}\right)_{t},$
(19)
which can be integrated once with respect to $x$ to give
$\displaystyle\beta(\ln{\beta})_{xt}+\beta^{2}=\beta_{0}(t),$ (20)
where $\beta_{0}(t)$ is an arbitrary function of $t$.
As it is reported in Ref.[24], for non-zero $\beta_{0}(t)$, one can rescale
$\beta$ to $\sqrt{\beta_{0}(t)}\beta$, redefine $t$ as $t/\sqrt{\beta_{0}(t)}$
and set $\beta=\exp({i\eta})$ to give the sine-Gordon equation
$\displaystyle\eta_{xt}=\sin{\eta},$ (21)
while for $\beta_{0}(t)=0$, by setting $\beta=-\exp({\eta})$, equation (20)
becomes the Liouville equation
$\displaystyle\eta_{xt}=\rm{e}^{\eta}.$ (22)
_Remark 1._ The quantity
$-\frac{\beta_{xx}}{2\beta}+\frac{\beta^{2}_{x}}{4\beta^{2}}\equiv A$ in the
right hand side of equation (19) can be given in terms of a Miura
transformation
$\displaystyle A=-\theta_{x}-\theta^{2},\quad\theta=\frac{\beta_{x}}{2\beta}.$
(23)
Furthermore, by virtue of the dependent variable transformation
$\displaystyle u=-2\frac{\psi_{x}}{\psi},\qquad
u_{i}=-2\frac{\psi_{i,x}}{\psi_{i}},\qquad(i=1,2,...,N)$ (24)
the negative pKdV hierarchy (16)-(17) is directly transformed into its
bilinear form
$\displaystyle D_{x}D_{t_{-N}}\psi\cdot\psi=\sum^{N}_{i=1}\psi^{2}_{i},$ (25)
$\displaystyle(D^{2}_{x}-\lambda_{i})\psi\cdot\psi_{i}=0,\quad i=1,2,...,N.$
(26)
_Case 2._ For the nonlocal symmetry (11) being dependent with parameter
$\lambda$, we may derive the second kind of negative pKdV hierarchy by
expanding the dependent variable in power series of $\lambda$. In this case,
we have
$\displaystyle\left.u_{t_{-N}}=-\frac{1}{N!}\left(\frac{\partial^{N}\exp(\int{u-u_{1}}\rmd
x)}{\partial\lambda^{N}}\right)\right|_{\lambda=0},$ (27) $\displaystyle
u_{x}+u_{1,x}=-2\lambda+\frac{(u-u_{1})^{2}}{2}.$ (28)
Under the transformation $u=-\frac{2\psi_{x}}{\psi}$ and
$u_{1}=-\frac{2\psi_{1,x}}{\psi_{1}}$, the negative pKdV hierarchy (27)-(28)
becomes
$\displaystyle\left.D_{x}D_{t_{-N}}\psi\cdot\psi=\frac{1}{N!}\left(\frac{\partial^{N}\psi^{2}_{1}}{\partial\lambda^{N}}\right)\right|_{\lambda=0},$
(29) $\displaystyle(D^{2}_{x}-\lambda)\psi\cdot\psi_{1}=0.$ (30)
Let $\psi_{1}=\psi_{1}(\lambda)$ have a formal series form
$\displaystyle\psi_{1}=\sum^{\infty}_{i=0}\bar{\psi}_{i}\lambda^{i},$ (31)
where $\bar{\psi}_{i}$ is $\lambda$ independent. Then (29)-(30) can be
rewritten as
$\displaystyle
D_{x}D_{t_{-N}}\psi\cdot\psi=\sum^{N}_{k=0}\bar{\psi}_{k}\bar{\psi}_{N-k},$
(32) $\displaystyle
D^{2}_{x}\psi\cdot\bar{\psi}_{k}=\psi\bar{\psi}_{k-1}\qquad(k=0,1,...,N)$ (33)
with $\bar{\psi}_{-1}=0.$
The negative pKdV hierarchy in bilinear form (32)–(33) is just the special
situation of the bilinear negative KP hierarchy for $\partial_{y}=0$ in
Ref.[22]. From this observation, we have the following remark:
Remark 2. The second negative pKdV hierarchy shown by (27)-(28) is a potential
form of a known negative KdV hierarchy given by other methods, say, the
inverse recursion operator [4], Lax operator [23], and the Guthrie’s approach
[26].
## 3 Localization of the nonlocal symmetries
We know that the Lie point symmetries [2, 3] can be applied to construct
finite symmetry transformation and group invariant solutions for DEs, whereas
the calculations are invalid for the nonlocal symmetries. So it is anticipant
to turn the nonlocal symmetries into local ones, especially into Lie point
symmetries. In order to make the nonlocal symmetry localized, one may extend
the original system to a closed prolonged system by introducing some
additional dependable variables [12, 13, 14] to eliminate integration and
differentiation.
Fortunately, starting from the nonlocal symmetry (11), the prolongation is
found to be closed when another two dependent variables $v\equiv v(x,t)$ and
$g\equiv g(x,t)$ are introduced by
$\displaystyle\eqalign{v_{x}=u-u_{1},\qquad
v_{t}=2(u-u_{1})(u_{x}-2\lambda)-2u_{xx},\cr g_{x}=\rme^{v},\qquad\qquad
g_{t}=-e^{v}[2u_{x}+8\lambda-(u-u_{1})^{2}].}$ (36)
Now the prolonged equations (2), (3), (4) and (36) contain four dependable
variables $u$, $u_{1}$, $v$ and $g$, whose corresponding symmetries are
$\displaystyle\sigma_{u}=\rme^{v},\quad\sigma_{u_{1}}=0,\quad\sigma_{v}=g,\quad\sigma_{g}=\frac{1}{2}g^{2}.$
(37)
_Remark 3._ What is more interesting here is that the symmetry $\sigma^{g}$
shown in (37) implies the auxiliary dependent variable $g$ satisfies
$g_{t}=\\{g;x\\}g_{x}+6\lambda g_{x},\
\\{g;x\\}\equiv\frac{g_{xxx}}{g_{x}}-\frac{3}{2}\frac{g_{xx}^{2}}{g_{x}^{2}},$
(38)
which is just the Schwartz form of the KdV (SKdV) equation (1). This may
provide us with a new way to seek for the Schwartz forms of DEs, especially
for the discrete integrable models, without using Painlevé analysis.
Due to (37), the symmetry vector of the prolonged system has the form
$\displaystyle V=\rme^{v}\frac{\partial}{\partial u}+0\frac{\partial}{\partial
u_{1}}+g\frac{\partial}{\partial v}+\frac{1}{2}g^{2}\frac{\partial}{\partial
g}.$ (39)
Then, by solving the following initial value problem
$\displaystyle\eqalign{\frac{d\bar{u}}{d\epsilon}=\rme^{\bar{v}},\quad\frac{d\bar{u}_{1}}{d\epsilon}=0,\quad\frac{d\bar{v}}{d\epsilon}=\bar{g},\quad\frac{d\bar{g}}{d\epsilon}=\frac{1}{2}\bar{g}^{2},\cr\bar{u}|_{\epsilon=0}=u,\quad\bar{u}_{1}|_{\epsilon=0}=u_{1},\quad\bar{v}|_{\epsilon=0}=v,\quad\bar{g}|_{\epsilon=0}=g,}$
(42)
the finite transformation can be written out as follows
$\displaystyle\bar{u}=u+\frac{2\epsilon}{2-\epsilon
g}\rme^{v},\quad\bar{u}_{1}=u_{1},\quad\bar{v}=v+2\ln\frac{2}{2-\epsilon
g},\quad\bar{g}=\frac{2}{2-\epsilon g}g.$ (43)
_Remark 4._ The original BT (3) and (4) in itself suggests a finite
transformation from one solution $u$ to another one $u_{1}$ and then the new
BT (43) obtained via (11) will arrive at a third solution $\bar{u}$. Actually,
the finite transformation (43) is just the so-called Levi transformation [27].
The result of this paper shows the fact that two kinds of BT possess the same
infinitesimal form (11).
Now by force of the finite transformation (43), one can get new solution from
any initial solution. For example, it is easy to solve an initial solution of
prolonged equation system (2), (3), (4) and (36), namely
$\displaystyle\eqalign{u=c,\qquad u_{1}=c+2\sqrt{\lambda}\tanh{\zeta},\qquad
v=-\ln(\tanh^{2}{\zeta}-1),\cr
g=\frac{\sinh(2\zeta)}{4\sqrt{\lambda}}-\frac{x}{2}+6\lambda
t+c_{0},\qquad\zeta=\sqrt{\lambda}(-x+4\lambda t),}$ (46)
Where $\lambda$, $c$ and $c_{0}$ are three arbitrary constants.
Starting from this original solution (46), a new solution of equation (2) can
be presented immediately from (43):
$\displaystyle\bar{u}=c-\frac{8\sqrt{\lambda}\epsilon\cosh^{2}{\zeta}}{8\sqrt{\lambda}-\epsilon[\sinh(2\zeta)-2\sqrt{\lambda}(x-12\lambda
t-2c_{0})]},$ (47)
which then gives the corresponding solution of KdV equation
$\displaystyle\bar{\omega}=\bar{u}_{x}=16\lambda\epsilon\cdot\frac{[\cosh(2\zeta)+1]\epsilon+\sqrt{\lambda}\sinh(2\zeta)[4+\epsilon(x-12\lambda
t-2c_{0})]}{[8\sqrt{\lambda}-\epsilon(\sinh(2\zeta)-2\sqrt{\lambda}(x-12\lambda
t-2c_{0}))]^{2}}$ (48)
with $\zeta=\sqrt{\lambda}(-x+4\lambda t)$.
Besides obtaining new solutions from old ones, symmetries can be applied to
get special solutions that are invariant under the symmetry transformations by
reducing dimensions of a partial differential equation. To find more
similarity reductions of equation (2), we will study Lie point symmetries of
the whole prolonged equation system instead of the single equation (2).
Suppose equations (2), (3), (4) and (36) be invariant under the infinitesimal
transformations
$\displaystyle u\rightarrow u+\epsilon\sigma,\qquad u_{1}\rightarrow
u_{1}+\epsilon\sigma_{1},\qquad v\rightarrow v+\epsilon\sigma_{2},\qquad
g\rightarrow g+\epsilon\sigma_{3},$
with
$\displaystyle\eqalign{\sigma=X(x,t,u,u_{1},v,g)u_{x}+T(x,t,u,u_{1},v,g)u_{t}-U(x,t,u,u_{1},v,g),\cr\sigma_{1}=X(x,t,u,u_{1},v,g)u_{1,x}+T(x,t,u,u_{1},v,g)u_{1,t}-U_{1}(x,t,u,u_{1},v,g),\cr\sigma_{2}=X(x,t,u,u_{1},v,g)v_{x}+T(x,t,u,u_{1},v,g)v_{t}-V(x,t,u,u_{1},v,g),\cr\sigma_{3}=X(x,t,u,u_{1},v,g)g_{x}+T(x,t,u,u_{1},v,g)g_{t}-G(x,t,u,u_{1},v,g).}$
(53)
Then substituting the expressions (53) into the symmetry equations of
equations (2), (3), (4) and (36)
$\displaystyle\eqalign{\sigma_{t}+\sigma_{xxx}-6u_{x}\sigma_{x}=0,\cr\sigma_{1,x}+\sigma_{x}-(\sigma-\sigma_{1})(u-u_{1})=0,\cr\sigma_{1t}-\sigma_{xxx}+2(u-u_{1})\sigma_{xx}+2(\sigma-\sigma_{1})u_{xx}-[4\lambda+(u-u_{1})^{2}-2u_{x}]\sigma_{x}\\\
+2(\sigma-\sigma_{1})(u-u_{1})(2\lambda-
u_{x})=0,\cr\sigma_{2,x}-\sigma+\sigma_{1}=0,\cr\sigma_{2t}+2\sigma_{xx}+2(u_{1}-u)\sigma_{x}+2(\sigma_{1}-\sigma)(u_{x}-2\lambda)=0,\cr\sigma_{3,x}-\rme^{v}\sigma_{2}=0,\cr\sigma_{3t}+2\rme^{v}[\sigma_{x}+(u_{1}-u)(\sigma-\sigma_{1})-\frac{1}{2}(u-u_{1})^{2}\sigma_{2}+(4\lambda+u_{x})\sigma_{2}]=0,}$
(62)
and collecting together the coefficients of partial derivatives of dependent
variables, it yields a system of overdetermined linear equations for the
infinitesimals $X$, $T$, $U$, $U_{1}$, $V$ and $G$, which can be solved by
virtue of _Maple_ to give
$\displaystyle\eqalign{X(x,t,u,u_{1},v,g)=c_{1}(x+12\lambda t)+c_{5},\cr
T(x,t,u,u_{1},v,g)=3c_{1}t+c_{2},\cr U(x,t,u,u_{1},v,g)=-c_{1}(2\lambda
x+u)+2c_{4}\rme^{v}+c_{3},\cr U_{1}(x,t,u,u_{1},v,g)=-c_{1}(2\lambda
x+u_{1})+c_{3},\cr V(x,t,u,u_{1},v,g)=-c_{1}+2c_{4}g+c_{6},\cr
G(x,t,u,u_{1},v,g)=c_{4}g^{2}+c_{6}g+c_{7},}$ (69)
where $c_{i}(i=1...7)$ are seven arbitrary constants. When
$c_{1}=c_{2}=c_{3}=c_{5}=c_{6}=c_{7}=0$, the reduced symmetry is just (37).
To give the group invariant solutions, we would like to solve symmetry
constraint conditions $\sigma=0$ and $\sigma_{i}=0(i=1,2,3)$ defined by (53)
with (69), which is equivalent to solve the following characteristic equation
$\displaystyle\eqalign{\frac{\rmd x}{c_{1}(x+12\lambda t)+c_{5}}=\frac{\rmd
t}{3c_{1}t+c_{2}}=\frac{\rmd u}{-c_{1}(2\lambda
x+u)+2c_{4}\rme^{v}+c_{3}}\cr=\frac{\rmd u_{1}}{-c_{1}(2\lambda
x+u_{1})+c_{3}}=\frac{\rmd v}{-c_{1}+2c_{4}g+c_{6}}=\frac{\rmd
g}{c_{4}g^{2}+c_{6}g+c_{7}}.}$ (72)
Two nontrivial similar reductions under consideration $c_{4}\neq 0$ are
presented and substantial group invariant solutions are found in the follows.
_Case 1:_ $c_{1}\neq 0$ and $c^{2}_{6}-4c_{4}c_{7}\neq 0$.
Without loss of generality, we let $c_{1}\equiv 1$. For simplicity, we
introduce arbitrary constants $a_{4}$ and $a_{7}$ to replace $c_{4}$ and
$c_{7}$ by $a^{2}_{4}=c^{2}_{6}-4c_{4}c_{7}$ and $a_{7}=-a^{2}_{4}/(16c_{4})$,
then after solving equation (72), we have
$\displaystyle\eqalign{u=-\lambda
x+3\lambda^{2}t+c_{3}+c_{5}\lambda-3c_{2}\lambda^{2}+(3t+c_{2})^{-\frac{1}{3}}[U(\xi)\\\
\qquad-\frac{a_{4}}{4a_{7}}\exp(V(\xi)-G(\xi))\tanh B],\cr u_{1}=-\lambda
x+3\lambda^{2}t+c_{3}+c_{5}\lambda-3c_{2}\lambda^{2}+\frac{U_{1}(\xi)}{(3t+c_{2})^{\frac{1}{3}}},\cr
v=-\frac{1}{3}\ln(3t+c_{2})-G(\xi)+V(\xi)-2\ln{\cosh B},\cr
g=\frac{8a_{7}}{a_{4}}[\tanh B+\frac{c_{6}}{a_{4}}]}$ (78)
with $B=a_{4}(3G(\xi)+\ln(3c_{1}t+c_{2}))/6$ and $\xi=(x-6\lambda
t+c_{5}-6c_{2}\lambda)/(3t+c_{2})^{\frac{1}{3}}.$
Here, $U(\xi)$, $U_{1}(\xi)$, $V(\xi)$, $G(\xi)$ and $\xi$ represent five
group invariants and substituting (78) into the prolonged equations system
gives the following reduced equations
$\displaystyle H_{\xi\xi}=\frac{1}{2}\frac{H^{2}_{\xi}}{H}+4a_{7}H^{2}-\xi
H-\frac{a^{2}_{4}}{32a^{2}_{7}H},$ (79)
$\displaystyle\eqalign{U_{1}(\xi)=\frac{a_{7}H^{2}_{\xi}}{H}-4a^{2}_{7}H^{2}+2a_{7}\xi
H-\frac{\xi^{2}}{4}-\frac{a^{2}_{4}}{16a_{7}H}\cr
U(\xi)=U_{1}(\xi)-\frac{H_{\xi}}{H},\quad V(\xi)=G(\xi)-\ln(H),\quad
G_{\xi}(\xi)=\frac{1}{4a_{7}H}}$ (82)
with $H\equiv H(\xi).$ One can see that whence $H$ is solved from equation
(79), two new group invariant solutions $u$ and $u_{1}$ of equation (2) would
be immediately obtained through equations (78) and (82).
Moreover, by making a further transformation [28]
$\displaystyle H(\xi)=\frac{1}{2a_{7}}(P_{\xi}+P^{2}+\frac{\xi}{2}),\qquad
P\equiv P(\xi),$ (83)
equation (79) can be converted into the second Painlevé equation
${\rm{P}_{II}}$, reading
$\displaystyle P_{\xi\xi}=2P^{3}+\xi P+\alpha,$ (84)
with $\alpha=-(a_{4}+1)/2$. Now, every known solution of ${\rm{P}_{II}}$ (84)
will generate two new group invariant solutions of equation (2), and then two
new solutions of KdV equation (1) denoted as $\omega_{1}$ and $\omega_{2}$ can
be given directly after one derivative with respect to $x$ for $u_{1}$ and $u$
$\displaystyle\omega_{1}=\frac{1}{(3t+c_{2})^{\frac{2}{3}}}(P_{\xi}+P^{2})-\lambda,$
(85)
$\displaystyle\omega_{2}=\frac{1}{(3t+c_{2})^{\frac{2}{3}}}[-\frac{a^{2}_{4}}{2F^{2}}{\rm{sech}}^{2}{R_{1}}+(\frac{2a_{4}P}{F}-\frac{a^{2}_{4}}{F^{2}})\tanh{R_{1}}+\frac{2a_{4}P}{F}+P_{\xi}-P^{2}]+\lambda,$
(86)
where
$\displaystyle F\equiv F(\xi)=2P_{\xi}+2P^{2}+\xi,\quad
R_{1}=\frac{1}{6}a_{4}[\ln(3t+c_{2})+3G(\xi)],\quad
G_{\xi}(\xi)=\frac{1}{2P_{\xi}+2P^{2}+\xi},$
and $P$ satisfies ${\rm{P}_{II}}$ (84) with $\alpha=-(a_{4}+1)/2$.
It is known that the generic solutions of ${\rm{P}_{II}}$ are meromorphic
functions and more information about ${\rm{P}_{II}}$ is provided in Ref.[29],
saying: (1) For every $\alpha=N\in Z$, there exists a unique rational solution
of ${\rm{P}_{II}}$; (2) For every $\alpha=N+\frac{1}{2}$, with $N\in Z$, there
exists a unique one-parameter family of classical solutions which are
expressible in terms of Airy functions; (3) For all other values of $\alpha$,
the solution of ${\rm{P}_{II}}$ is transcendental.
For example, when $\alpha=1$ $(a_{4}=-3)$, ${\rm{P}_{II}}$ (84) possesses a
simple rational solution $P(\xi)=-{1}/{\xi}$, which leads the solutions (85)
and (86) to
$\displaystyle\tilde{\omega}_{1}=\frac{2}{(x-6\lambda
t+c_{5}-6c_{2}\lambda)^{2}}-\lambda,$ (87)
and
$\displaystyle\tilde{\omega}_{2}=-[x^{6}-36tx^{5}+(540t^{2}-6)x^{4}-(4320t^{3}-168t-2)x^{3}+36t(540t^{3}-48t-1)x^{2}$
$\displaystyle\qquad-(46656t^{5}-7776t^{3}-216t^{2}-144t-12)x+46656t^{6}-12960t^{4}-432t^{3}$
$\displaystyle\qquad-720t^{2}-48t+1]/[x^{3}-18x^{2}t+108xt^{2}-(6t+1)(36t^{2}-6t-1)]^{2}$
(88)
In the formulation (88), we have made $c_{2}=0$, $c_{5}=0$ and $\lambda=1$
because the original expression is much too complicated. The simple rational
solutions of PII will yield abundant rational solutions of KdV equation.
When $\alpha=\frac{1}{2}$ $(a_{4}=-2)$, ${\rm{P}_{II}}$ (84) has a solution
expressed by $Airy$ function
$\displaystyle
P(\xi)=2^{-\frac{1}{3}}\frac{3{\rm{Ai}}(1,-2^{-\frac{1}{3}}\xi)-\sqrt{3}{\rm{Bi}}(1,-2^{-\frac{1}{3}}\xi)}{3{\rm{Ai}}(-2^{-\frac{1}{3}}\xi)-\sqrt{3}{\rm{Bi}}(-2^{-\frac{1}{3}}\xi)}.$
(89)
For simplicity, we convert equation (89) into the equivalent form
$\displaystyle
P(\xi)=\frac{\sqrt{2}\xi^{\frac{3}{2}}{\rm{J}}({\frac{4}{3}},\frac{\sqrt{2}}{3}\xi^{\frac{3}{2}})-2{\rm{J}}({\frac{1}{3}},\frac{\sqrt{2}}{3}\xi^{\frac{3}{2}})}{\xi{\rm{J}}({\frac{1}{3}},\frac{\sqrt{2}}{3}\xi^{\frac{3}{2}})},$
(90)
where ${\rm{J}}(n,\xi)$ is the first kind of Bessel function. Substituting
(90) into (85) and (86) with $c_{2}=c_{5}=0$ and $\lambda=1$ (or else the
formulae are too long to written down here), two exact solutions of KdV
equation are obtained as follows:
$\displaystyle\omega_{1}^{\prime}=\frac{x-12t}{6t}+\frac{x-6t}{3t}\frac{{\rm{J}}^{2}_{2}}{{\rm{J}}^{2}_{1}},$
(91) $\displaystyle\omega_{2}^{\prime}=-\Psi/\Omega$ (92)
with
$\displaystyle\Psi=32\sqrt{t}(x-6t)^{4}[(x(x-6t)^{2}-12t){\rm{J}}^{6}_{1}+x(x-6t)^{2}{\rm{J}}^{6}_{2}]+128\sqrt{6}t(x-6t)^{\frac{11}{2}}\cdot({\rm{J}}^{5}_{1}{\rm{J}}_{2}$
$\displaystyle\qquad+{\rm{J}}^{5}_{2}{\rm{J}}_{1})+96\sqrt{t}(x-6t)^{4}[(x(x-6t)^{2}-4t){\rm{J}}^{4}_{1}{\rm{J}}^{2}_{2}+x(x-6t)^{2}{\rm{J}}^{4}_{2}{\rm{J}}^{2}_{1}]-72\cdot
2^{\frac{1}{3}}\sqrt{t}$
$\displaystyle\qquad\cdot(x-6t)^{2}\cdot[(x(x-6t)^{2}-4t)\cdot{\rm{J}}^{4}_{1}+x(x-6t)^{2}{\rm{J}}^{4}_{2}]+256\sqrt{6}t(x-6t)^{\frac{11}{2}}{\rm{J}}^{3}_{1}{\rm{J}}^{3}_{2}$
$\displaystyle\qquad-192\sqrt{3}\cdot
2^{\frac{5}{6}}t(x-6t)^{\frac{7}{2}}\cdot({\rm{J}}^{3}_{1}{\rm{J}}_{2}+{\rm{J}}^{3}_{2}{\rm{J}}_{1})+54\cdot
2^{\frac{2}{3}}\cdot x\sqrt{t}(x-6t)^{2}({\rm{J}}^{2}_{1}+{\rm{J}}^{2}_{2})$
$\displaystyle\qquad-144\cdot
2^{\frac{1}{3}}x\sqrt{t}(x-6t)^{4}{\rm{J}}^{2}_{1}{\rm{J}}^{2}_{2}+144\sqrt{3}\cdot
2^{\frac{1}{6}}t(x-6t)^{\frac{3}{2}}{\rm{J}}_{1}{\rm{J}}_{2}-27x\sqrt{t},$
and
$\displaystyle\Omega=6t^{\frac{3}{2}}[2^{\frac{5}{3}}(x-6t)^{2}({\rm{J}}^{2}_{1}+{\rm{J}}^{2}_{2})-3][8\cdot
2^{\frac{1}{3}}(x-6t)^{4}({\rm{J}}^{2}_{1}-{\rm{J}}^{2}_{2})^{2}-12\cdot
2^{\frac{2}{3}}(x-6t)^{2}({\rm{J}}^{2}_{1}+{\rm{J}}^{2}_{2})+9],$
where we denote
${\rm{J}}_{1}={\rm{J}}(\frac{1}{3},\frac{\sqrt{6}}{9}\frac{(x-6t)^{\frac{3}{2}}}{\sqrt{t}}),\quad{\rm{J}}_{2}={\rm{J}}(-\frac{2}{3},\frac{\sqrt{6}}{9}\frac{(x-6t)^{\frac{3}{2}}}{\sqrt{t}})$.
Then continue to do the same, sequences of rational solutions and Bessel
(Airy) function solutions for KdV equation will be easily constructed.
Furthermore, by selecting suitable parameters in this kind of similar
reduction, we may discover more unknown exact solutions among interaction
solitons and Painlevé waves of KdV equation.
_Case 2:_ $c_{1}=0$ and $c_{2}\neq 0$.
Firstly, it is convenient to replace $c_{4}$ and $c_{5}$ with $a_{4}$ and $k$
by $a^{2}_{4}=c^{2}_{6}-4c_{4}c_{7}$ and $k={c_{5}}/{c_{2}}$, and it follows
the results from equation (72), saying
$\displaystyle\eqalign{u=\frac{c_{3}}{c_{2}}t+U(z)+\frac{c_{3}}{c_{2}}G(z)-\frac{(a^{2}_{4}-c^{2}_{6})}{a_{4}c_{7}}\rm{e}^{V(z)}\tanh[\frac{a_{4}(t+G(z))}{2c_{2}}],\cr
u_{1}=\frac{c_{3}}{c_{2}}t+U_{1}(z),\cr
g=\frac{2c_{7}}{a^{2}_{4}-c^{2}_{6}}[c_{6}+a_{4}\tanh(\frac{a_{4}(t+G(z))}{2c_{2}})],\cr
v=V(z)-2\ln{\cosh[\frac{a_{4}(t+G(z))}{2c_{2}}]}}$ (97)
with $z=x-kt$. Substituting (97) into equations (2), (3), (4) and (36) and
redefining the parameters for the sake of simplicity, we notice that the new
group invariants $U(z)$, $U_{1}(z)$, $V(z)$ and $G(z)$ are subject to
$\displaystyle W^{2}_{z}-a^{2}_{2}W^{4}-a_{3}W^{3}+a_{5}W^{2}-a_{7}W=0,$ (98)
$\displaystyle\eqalign{U_{1z}(z)=\frac{a_{7}}{2W}-\lambda-\frac{a_{5}}{4},\cr
U(z)=U_{1}(z)+\frac{W_{z}}{W}+(3\lambda^{2}+\frac{\lambda
a_{5}}{2}-\frac{a^{2}_{5}}{16}+\frac{a_{3}a_{7}}{4})G,\cr V(z)=\ln(W),\cr
G_{z}(z)=\frac{W}{a_{7}}}$ (103)
with $W\equiv W(z)$ and $a_{2}=\frac{a^{2}_{4}-c^{2}_{6}}{a_{4}c_{7}}$,
$a_{3}=\frac{(a^{2}_{4}-c^{2}_{6})(c_{2}k^{2}+48c_{2}\lambda^{2}-16c_{2}k\lambda-4c_{3})}{a^{2}_{4}c_{7}}$,
$a_{5}=2k-12\lambda$,
$a_{7}=\frac{a^{2}_{4}c_{7}}{c_{2}(a^{2}_{4}-c^{2}_{6})}$.
Remark 5. The case $c_{1}=0$ here is interesting. From equation (98), we know
that $W$ can be expressed as an elliptic integration and can be expressed by
means of Jaccobi elliptic functions. Whence $W$ is fixed from (98), all the
other quantities are given simply given by differentiation or integration. The
first equation of (97) implies the important byproduct, the explicit exact
interaction between cnoidal periodic wave and kink soliton.
A simple example of this case can be obtained by using the simplest Jacobi
Elliptic function expansion method which leads to
$\displaystyle
W(z)=\frac{a_{3}}{4a^{2}_{2}}[{\rm{sn}}(\frac{a_{3}z}{4a_{2}n},n)-1]$ (104)
with the constraint conditions
$a_{5}=\frac{a^{2}_{3}(1-5n^{2})}{16n^{2}a^{2}_{2}}$ and
$a_{7}=\frac{a^{2}_{3}(n^{2}-1)}{32n^{2}a^{4}_{2}}$ in equation (98), where
$n$ is the modulus of the Jacobian elliptic function $\rm{sn}$.
After solving equation (103) with the given solution (104) and taking the
results into (97), two exact solutions of the KdV equation are obtained
$\displaystyle\omega_{3}=\frac{a^{2}_{3}(1-n^{2})}{16a^{2}_{2}n^{2}(Y+1)}+\frac{a^{2}_{3}(5n^{2}-1)}{64a^{2}_{2}n^{2}}-\lambda,$
(105)
$\displaystyle\omega_{4}=\frac{a^{2}_{3}(Y+1)^{2}}{32a^{2}_{2}}\tanh^{2}{R_{2}}+\frac{a^{2}_{3}\sqrt{Y^{2}-1}\sqrt{n^{2}Y^{2}-1}}{16na^{2}_{2}}\tanh{R_{2}}+\frac{2a^{2}_{3}(Y^{2}-2Y)}{a^{2}_{2}}$
$\displaystyle-\frac{a^{2}_{3}(n^{2}+1)}{n^{2}a^{2}_{2}}-64\lambda,$ (106)
where we have
$R_{2}=\frac{a^{3}_{3}(n^{2}-1)(t+a_{6})}{64a^{3}_{2}n^{2}}+\frac{1}{4}\ln\left({\frac{2n}{2n^{2}Y^{2}+2n\sqrt{Y^{2}-1}\sqrt{n^{2}Y^{2}-1}-n^{2}-1}}\right)$
$+\frac{1}{2}n\int^{Y}_{0}{\frac{1}{\sqrt{1-t^{2}}\sqrt{1-n^{2}t^{2}}}}dt,\quad
Y=\rm{sn}\left(\frac{a_{3}(192\lambda
a^{2}_{2}\emph{n}^{2}-5a^{2}_{3}\emph{n}^{2}+a^{2}_{3})\emph{t}}{128\emph{n}^{3}a^{3}_{2}}-\frac{a_{3}\emph{x}}{4a_{2}\emph{n}},\emph{n}\right),\quad$
and $a_{2}$, $a_{3}$, $a_{6}$ and $\lambda$ are four arbitrary constants.
## 4 Integrable models from nonlocal symmetry with Bäcklund transfor- mation
To find new integrable models is another important application of the symmetry
study. Symmetry constraint method is one of the most powerful tools to give
out new integrable models from known ones. Especially, casting symmetry
constraint condition to Lax pair of soliton equations, one can obtain many
other integrable models. In this section, we would like to combine the
nonlocal symmetry with BT of pKdV equation to give some integrable models both
in lower and higher dimensions.
Let every pair $(u,u_{i})$ ($i=1,2,...,N$) satisfy the following BT
$\displaystyle u_{x}+u_{i,x}=-2\lambda_{i}+\frac{(u-u_{i})^{2}}{2},$ (107)
$\displaystyle
u_{t}+u_{i,t}=2u^{2}_{x}+2u^{2}_{i,x}+2u_{x}u_{i,x}-(u-u_{i})(u_{xx}-u_{i,xx}),$
(108)
and the corresponding nonlocal symmetry of $u$ reads
$\sigma^{i}=\exp(\int{u-u_{i}}\rmd x)$ for $i=1,2,...,N$.
### 4.1 Finite-dimensional integrable systems
In general, every one symmetry of a higher dimensional model can lead the
original one to its lower form. Now, considering
$\displaystyle u_{x}=\sum_{i=1}^{N}a_{i}\exp(\int{u-u_{i}}\rmd x)$ (109)
as a generalized symmetry constraint condition and acting it on the $x$-part
of the BT (3), we firstly give the finite dimensional $(N+1)$-component
integro-differential system
$\displaystyle\eqalign{u_{x}=\sum_{i=1}^{N}{a_{i}\exp(\int{u-u_{i}}\rmd
x)},\cr u_{x}+u_{i,x}=-2\lambda_{i}+\frac{(u-u_{i})^{2}}{2},\quad
i=1,2,...,N,}$ (112)
where every $a_{i}$ and $\lambda_{i}$ are arbitrary constants. For further
simplification, making $u_{i}=u-(\ln{w_{ix}})_{x}$, then the constraint
condition (109) becomes
$\displaystyle u=\sum_{m=1}^{N}a_{m}w_{m},$ (113)
which transforms (112) into the N-component differential system
$\displaystyle
2w_{ixxx}w_{ix}-4(\sum^{N}_{m=1}a_{m}w_{mx})w^{2}_{ix}-w^{2}_{ixx}-4\lambda_{i}w^{2}_{ix}=0,\qquad
i=1,2,...,N.$ (114)
Taking $s_{i}=w_{ix}$, we rewrite equation (114) as
$\displaystyle
2s_{ixx}s_{i}-4(\sum^{N}_{m=1}a_{m}s_{m})s^{2}_{i}-s^{2}_{ix}-4\lambda_{i}s^{2}_{i}=0,\quad
i=1,2,...,N.$ (115)
Making $s_{i}=b_{i}q^{2}_{i}$, equation (115) is equivalent to the downward
integrable system
$\displaystyle
q_{ixx}-(\sum^{N}_{m=1}c_{m}q^{2}_{m})q_{i}-\lambda_{i}q_{i}=0,\quad
i=1,2,...,N$ (116)
with $c_{i}=a_{i}b_{i}$ being arbitrary constant.
On the other hand, by the same symmetry constraint (109) and the $t$-part of
the BT (4), we can construct another set of integrable system
$\displaystyle\eqalign{u_{x}=\sum_{i=1}^{N}{a_{i}\exp(\int{u-u_{i}}\rmd
x)},\cr
u_{t}+u_{i,t}=2u^{2}_{x}+2u^{2}_{i,x}+2u_{x}u_{i,x}-(u-u_{i})(u_{xx}-u_{i,xx}),\quad
i=1,2,...,N.}$ (119)
Considering the similarity transformations of dependent variables done in
$x$-part, after a series of tedious substitutions, equation (119) becomes
$\displaystyle
q_{i}q_{ixt}-q_{it}q_{ix}+(2\sum_{m=1}^{N}c_{m}q^{2}_{m}-4\lambda_{i})q^{2}_{ix}-4q_{i}q_{ix}\sum_{m=1}^{N}c_{m}q_{m}q_{mx}-2q^{2}_{i}(\sum_{m=1}^{N}c_{m}q^{2}_{m})^{2}$
$\displaystyle+2\lambda_{i}q^{2}_{i}\sum_{m=1}^{N}c_{m}q^{2}_{m}+4\lambda^{2}_{i}q^{2}_{i}+2q^{2}_{i}\sum_{m=1}^{N}c_{m}(q^{2}_{mx}+q_{m}q_{mxx})=0,\qquad
i=1,2,...,N.$ (120)
Taking equation (116) into account, equation (120) can be integrated once
about $x$ to give $N$-component integrable system
$\displaystyle
q_{it}=-2\sum_{m=1}^{N}c_{m}q_{m}q_{mx}q_{i}+2\sum_{m=1}^{N}c_{m}q^{2}_{m}q_{ix}-4\lambda_{i}q_{ix},\qquad
i=1,2,...,N.$ (121)
In fact, equations (116) and (121) are essentially the canonical equation
$(F_{0})$ and $(F_{1})$ respectively [16], saying
$\displaystyle(F_{0}):\qquad q_{ix}=p_{i},\qquad
p_{ix}=(\sum_{m=1}^{N}c_{m}q^{2}_{m})q_{i}+\lambda_{i}q_{i}.$ (122)
$(F_{1}):\left\\{\begin{array}[]{rl}q_{it}&=-2(\sum_{m=1}^{N}c_{m}p_{m}q_{m})q_{i}+2(\sum_{m=1}^{N}c_{m}q^{2}_{m})p_{i}-4\lambda_{i}p_{i},\\\
p_{it}&=2(\sum_{m=1}^{N}c_{m}p_{m}q_{m})p_{i}-2(\sum_{m=1}^{N}c_{m}p^{2}_{m})q_{i}-4\lambda^{2}_{i}q_{i}\\\
&-2\lambda_{i}(\sum_{m=1}^{N}c_{m}q^{2}_{m})q_{i}-2(\sum_{m=1}^{N}\lambda_{m}c_{m}q^{2}_{m})q_{i}.\end{array}\right.$
(123)
It should be stressed here that the finite integrable systems (116) and (121)
reobtained via this way are just the remarkable results given by Cao in
Ref.[16] through the nonlinearization method, both of which have been proved
completely integrable in Liouville sense. Thanks to these finite integrable
systems (122) and (123), the original high dimensional KdV equation would be
solved.
### 4.2 Infinite-dimensional integrable systems
For getting some higher dimensional integrable models, one may introduce some
internal parameters [6, 8, 20]. Here, we would like to use the internal
parameter dependent symmetry constraints on BT to construct two sets of
infinite-dimensional integrable systems.
It is obvious that equation (2) is invariant under the internal parameter
translation, say $y$ translation, so we can view
$\displaystyle u_{y}=\sum_{i=1}^{N}a_{i}\exp(\int{u-u_{i}}\rmd x)$ (124)
as a new symmetry constraint condition.
Firstly, imposing (124) on the $x$-part of the BT (3) yields a
(1+1)-dimensional $(N+1)$-component integro-differential system
$\displaystyle\eqalign{u_{y}=\sum_{i=1}^{N}a_{i}\exp(\int{u-u_{i}}\rmd x),\cr
u_{x}+u_{i,x}=-2\lambda_{i}+\frac{(u-u_{i})^{2}}{2},\qquad i=1,2,...,N,}$
(127)
where $\lambda_{i}$ and $a_{i}$ ($i=1,2,...,N$) are constants.
By the transformation $u_{i}=u-(\ln{\phi_{iy}})_{x}$, equation (124) becomes
$\displaystyle u=\sum^{N}_{i=1}a_{i}\phi_{i},$ (128)
which then converts (127) into the following (1+1)-dimensional N-component
differential system
$\displaystyle
2\phi_{ixxy}\phi_{iy}-4\left(\sum^{N}_{m=1}a_{m}\phi_{mx}\right)\phi^{2}_{iy}-\phi^{2}_{ixy}-4\lambda_{i}\phi^{2}_{iy}=0,\quad
i=1,2,...,N.$ (129)
Alternatively, combining the constraint condition (124) with the $t$-part of
the BT (4) will produce a (1+2)-dimensional system about $x$, $y$ and $t$,
reading
$\displaystyle\eqalign{u_{y}=\sum_{i=1}^{N}a_{i}\exp(\int{u-u_{i}}\rmd x),\cr
u_{t}+u_{i,t}=2u^{2}_{x}+2u^{2}_{i,x}+2u_{x}u_{i,x}-(u-u_{i})(u_{xx}-u_{i,xx}),\quad
i=1,2,...,N.}$ (132)
Using the same transformation and equation (129), equation (132) is
transformed into the following $N$-component system
$\displaystyle\phi_{ixyt}\phi_{iy}-\phi_{ixy}\phi_{iyt}-2(\sum^{N}_{m=1}a_{m}\phi_{mxx})\phi_{iy}\phi_{ixy}+(\sum^{N}_{m=1}a_{m}\phi_{mx}-2\lambda_{i})\phi^{2}_{ixy}$
$\displaystyle+2[(\sum^{N}_{m=1}a_{m}\phi_{mx})^{2}+2\lambda_{i}\sum^{N}_{m=1}a_{m}\phi_{mx}-\sum^{N}_{m=1}a_{m}\phi_{mt}+4\lambda^{2}_{i}]\phi^{2}_{iy}=0.$
(133)
It should be noted that the integrability of the infinite-dimensional systems
(129) and (133) obtained in this way is not quite clear. The finite-
dimensional models obtained here are completely integrable, that strongly
suggests these infinite-dimensional models should have many nice integrable
properties. It will be of much interest to investigate the integrability of
these models in the further work.
## 5 Conclusion and discussions
In this paper, we have shown that combining nonlocal symmetries with BTs can
result in many diverse applications. The main new progresses made in this
paper in the general aspect of integrable systems are:
(i). The BTs are used to find nonlocal symmetries;
(ii). Different types of BTs may possess same infinitesimal forms and then new
types of BTs may be obtained from old ones;
(iii). New integrable (negative) hierarchies can be obtained from nonlocal
symmetries related to BTs;
(iv). New finite dimensional integrable systems can be obtained from BTs and
related symmetry constraints and reductions. And then the original high
dimensional model can be solved from lower dimensional ones because of the
existence of nonlocal symmetries depending on BTs ;
(v). The exact interaction solutions among solitons and other complicated
waves including periodic cnoidal waves and Painlevé waves are revealed which
have not yet found for any integrable models because it is difficult to solve
the original BT (or Darboux transformation) problem if the original seed
solutions are taken as the cnoidal or Painlevé waves;
(vi). The localization procedure results in a new way to find Schwartz form of
the original model which is obtained usually via Painlevé analysis for the
continuous integrable systems. The method may provide a potential method to
transform a discrete integrable systems to Schwartz forms because usually the
BTs of discrete integrable models are known.
The above progresses are realized especially for potential KdV (pKdV)
equation. For pKdV equation, it possesses a new class of nonlocal symmetry
resulting from its BT. Since this BT is of Riccati type, more information
about its bilinear forms is learned via the Cole-Hopf transformation. Based on
the new nonlocal symmetry with internal parameters, we construct two sets of
negative pKdV hierarchies and fulfill their corresponding bilinear forms.
In order to extend applicability of nonlocal symmetry to obtain explicit
solutions of KdV equation, we introduce another two auxiliary variables $v$
and $g$ to form a prolonged system with $u$ and $u_{1}$, so that the original
nonlocal symmetry can be transformed to a Lie point symmetry of the new
equations system. Then what follows naturally are Lie-Bäcklund transformation
and two kinds of novel similarity reductions. By virtue of two kinds of BTs,
the solitary wave solutions of KdV equation are obtained through the
transformations of the trivial solutions. Concerning the complete Lie point
symmetries of the prolonged system, we achieve rich group invariant solutions
including rational solution hierarchy, Bessel function solution hierarchy and
periodic function solutions.
The nonlocal symmetry has also been devoted to construct various new
integrable systems by symmetry constraint method. Applying nonlocal symmetry
on the BT of pKdV equation, finite-dimensional integrable systems are given,
which are found equivalent to the excellent work done by Cao [16]. Moreover,
the introduction of an internal parameter as new independent variable helps us
to build two sets of infinite-dimensional models.
We believe that both the negative pKdV hierarchies and two sets of infinite-
dimensional models obtained in the paper should have many nice integrable
properties. For the completely integrable finite-dimensional models, one may
consider their algebraic geometry solutions to achieve related solutions of
KdV equation. For the localization of nonlocal symmetries, it still remains
unclear what kind of nonlocal symmetries must have close prolongations and can
be applied to construct exact solutions. It is quite reasonable that these
matters merit our further study.
## Acknowledgments
The authors are indebt to thank very much for the referees’ comments and
suggestions and the helpful discussions with Profs. X B Hu, Q P Liu, E G Fan,
C W Cao and X Y Tang. This work is supported by the National Natural Science
Foundation of China (Grant Nos. 11075055, 11175092, 61021004, 10735030),
Shanghai Leading Academic Discipline Project (No. B412) and K C Wang Magna
Fund in Ningbo University.
## References
## References
* [1] Rogers C and Shadwick W F 1982 Bäcklund Transformation and Their Applications (New York)
* [2] Olver P J 1993 Applications of Lie Groups to Differential Equations (New York: Springer)
* [3] Bluman G W and Kumei S 1989 Symmetries and Differential Equations (Berlin:Springer)
* [4] Lou S Y 1993 Phys. Lett. B 302 261
Lou S Y 1993 Int. J. Mod. Phys. A 3A 531
Lou S Y 1994 J. Math. Phys. 35 2390
* [5] Lou S Y 1993 J. Phy. A: Math. Gen. 26 L789
Lou S Y 1993 Phys. Lett. A 181 13
Lou S Y 1994 Phys. Lett. A 187 239
Lou S Y 1994 Solitons, Chaos and Fractals 4 1961
Ruan H Y and Lou S Y 1993 J. Phys. Soc. Japan 62 1917
Han P and Lou S Y 1994 Acta Phys. Sinica 43 1042
Lou S Y and Chen W Z 1993 Phys. Lett. A 179 271
* [6] Lou S Y 1997 J. Phys. A: Math. Phys. 30 4803
* [7] Lou S Y and Hu X B 1997 J. Phys. A: Math. Gen. 30 L95
* [8] Lou S Y and Hu X B 1997 J. Math. Phys. 38 6401
* [9] Edelen D G 1980 Isovector Methods for Equations of Balance (Alphen aam den Rijn: Sijthoff and Noordhoff)
* [10] Krasil’shchik I S and Vinogradov A M 1984 Acta Appl. Math. 2 79
* [11] Krasil’shchik I S and Vinogradov A M 1989 Aata Appl. Math. 15 161
* [12] Galas F 1992 J. Phys. A 25 L981
* [13] Lou S Y and Hu X B 1993 Chin. Phys. Lett. 10 577
* [14] Lou S Y, Ruan H Y, Chen W Z, Wang Z L and Chen L L 1994 Chin. Phys. Lett. 11 593
* [15] Cao C W 1990 Science in china (Series A) 33 528
* [16] Cao C W Acta Mathematica Sinica 1991 7 216
* [17] Cao C W, Wu Y T and Geng X G 1999 J. Math. Phys. 40 3948
* [18] Antonowicz M and Rauch-Wojciechowski S 1991 J. Phys. A 24 5043
* [19] Antonowicz M and Rauch-Wojciechowski S 1992 J. Math. Phys. 33 2115
* [20] Lou S Y 1997 Commun. Theor. Phys. 27 249
* [21] Wahlquist H D and Estabrook F B 1973 Phys. Rev. Lett. 31 1386
* [22] Hu X B, Lou S Y and Qian X M 2009 Stud. Appl. Math. 122 305
* [23] Lou S Y 1998 Physica Scripta 57 481
* [24] Hone1 A N and Wang J W 2003 Inverse Problems 19 129
* [25] Verosky J M 1991 J. Math. Phys. 32 1733
* [26] Guthrie G A 1993 J. Phys. A: Math. Gen. 26 L905
Guthrie G A and Hickman M S 1993 J. Math. Phys. 26 193
Guthrie G A 1994 Phys. R. Soc. Lond. A 446 107
* [27] Levi D 1988 Inverse Problems 4 165
* [28] Ince E L 1956 Ordinary Differential Equations (New York)
* [29] Umemura H and Watanabe H 1997 Nagoya Math. J. 148 151
|
arxiv-papers
| 2012-01-17T01:43:28 |
2024-09-04T02:49:26.373523
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. Y. Lou, Xiaorui Hu and Yong Chen",
"submitter": "Yong Chen",
"url": "https://arxiv.org/abs/1201.3409"
}
|
1201.3413
|
# A unified constraint on the Lorentz invariance violation from both short and
long GRBs
Zhe Chang changz@ihep.ac.cn Yunguo Jiang jiangyg@ihep.ac.cn Hai-Nan Lin
linhn@ihep.ac.cn Institute of High Energy Physics, Chinese Academy of
Sciences, 100049 Beijing, China Theoretical Physics Center for Science
Facilities, Chinese Academy of Sciences, 100049 Beijing, China
(August 27, 2024)
###### Abstract
Possible Lorentz invariance violation (LIV) has been investigated for a long
time based on observations of GRBs. These arguments relied on the assumption
that photons with different energy are emitted at the same place and time. In
this work, we try to take account of the intrinsic time delay $\Delta t_{\rm
int}$ between emissions of low and high energy photons by using the magnetic
jet model. The possible LIV effects are discussed in a unified scenario both
for long and short Fermi-detected GRBs. This leads to a unique quantum gravity
energy scale $M_{1}c^{2}\sim 1.0\times 10^{20}$ GeV respecting the linear
dispersion relation.
###### keywords:
gamma-ray burst; Lorentz invariance violation; magnetic jet model
## 1 Introduction
Lorentz invariance is one of the most important cornerstones of modern
physics. Recently, the OPERA collaboration reported that the GeV neutrinos
propagate faster than the speed of light [1]. Although other independent
experiments should be performed to verify the superluminal phenomenon, it is
valuable to ask whether Lorentz invariance violation (LIV) happens in high
energy scale. A favored way to test LIV is to study the most explosive events
in the present universe: gamma-ray bursts (GRBs). The Fermi satellite has
observed several GRBs with photon energy $>100$ MeV in recent years. Fermi
carries two instruments: the Gamma-Ray Burst Monitor (GBM) and the Large Area
Telescope (LAT), which detect the energy band $8{\rm KeV}-40{\rm MeV}$ and
$30{\rm MeV}-300{\rm GeV}$, respectively. An interesting feature of the
observation is that GeV photons arrive several seconds later than MeV photons
[2, 3, 4, 5].
One possible explanation is given by quantum gravity effects. Some quantum
gravity theories predict that high energy photons may interact with the foamy
structure of the space-time, thus photons with different energy propagate with
different velocities [6, 7, 8, 9]. Such effects can be accumulated after
photons travel a cosmological distance. In these theories, high energy photons
are subluminal. But there are still some other theories which show that high
energy photons can be superluminal. For example, LIV can also be induced from
the geometry of the space-time itself, such as the Finsler geometry [10, 11,
12]. One can expect that the velocity of photon also depends on the energy,
and may be superluminal.
A straight forward way to test LIV is studying the individual GRB, several
papers have discussed the upper limits of variations of the light speed [13,
14]. Ellis et al. have proposed a data fitting procedure to test LIV effects
[9, 15, 16, 17]. The linear fitting function is expressed as $\Delta t_{\rm
obs}/(1+z)=a_{\rm LIV}K(z)+b$, where $K(z)$ is a non-linear function of the
redshift measuring the cosmological distance, and $b$ represents the ignorance
of the intrinsic time lag. A statistical ensemble of GRBs was used to fit
values of $a_{\rm LIV}$ and $b$, and no strong evidence of LIV was found. The
explicit form of $K(z)$ depends on the cosmological model. Biesiada and
Piórkowska applied this procedure to various cosmological models [18]. Shao et
al. used this method to discuss four Fermi-detected GRBs [17], which we will
discuss in the present work. Nevertheless, all these investigations
concentrated on the time lag induced by LIV, the intrinsic time lag which
depends on the emission mechanism of GRBs was not considered.
On the other hand, without considering LIV effects, several mainstream GRB
models were proposed to explain the delayed arrival of GeV photons [19, 20,
22, 21]. Mészáros and Rees presented a magnetic dominated jet model to explain
this phenomenon. The MeV photons can escape the plasma when their optical
depth decreases to unity at the photosphere radius. While the GeV photons are
produced by the nuclear collision between protons and neutrons, which happens
at a large radius (compared to the photosphere radius) [20]. Duran and Kumar
considered that photons are emitted by electrons via the synchrotron
radiation, it consumes more time for electrons to be accelerated in order to
radiate GeV photons [21]. Bo$\check{\rm s}$njak and Kumar proposed the
magnetic jet model, the time delay depends linearly on the distance where the
jet is launched [22].
In this work, we argue that the observed time lag for two photons with energy
$E_{\rm high}$ and $E_{\rm low}$ consists of two parts,
$\displaystyle\Delta t_{\rm obs}=\Delta t_{\rm LIV}+\Delta t_{\rm int},$ (1)
where $\Delta t_{\rm int}$ denotes the intrinsic emission time delay, and
$\Delta t_{\rm LIV}$ represents the flying time difference caused by LIV
effects. In section 2, we take use of the magnetic jet model in Ref.[22] to
estimate $\Delta t_{\rm int}$. The LIV induced time lag $\Delta t_{\rm LIV}$
is given, and the quantum gravity energy scale is discussed in Sec. 3.
Finally, the discussion and conclusions are given in Sec. 4.
## 2 Magnetic jet model
In the magnetic jet model, photons with energy less than $10$ MeV can escape
when the jet radius is beyond the Thomson photosphere radius, i.e., the
optical depth for low energy photons is $\tau_{T}\sim 1$. However, GeV photons
will be converted to electron-positron pairs at this radius, and can escape
later when the pair-production optical depth $\tau_{\gamma\gamma}(E)$ drops
below unity.
The bulk Lorentz factor of an expanding spherical fireball increases with the
radius roughly as $\Gamma\propto r$, until reaching a saturate radius $r_{s}$
where the Lorentz factor is saturated [23]. However, for an expanding jet with
small ejecting angle, the effective dynamical dimension is one. The bulk
Lorentz factor increases with the radius roughly as [22]
$\displaystyle\Gamma(r)\approx\begin{cases}(r/r_{0})^{1/3}\quad\ &\ {\rm
for}\quad r_{0}\lesssim r\lesssim r_{s},\\\ \eta\quad\ &\ {\rm for}\quad
r\gtrsim r_{s},\\\ \end{cases}$ (2)
where $r_{0}\approx 10^{7}$ cm is the base of the outflow, which represents
the distance from the central engine where the jet is launched. $\eta$ is the
final bulk Lorentz factor of the jet.
The optical depth for the photon-electron scattering is defined as
$\tau_{T}(r)=\int_{r}^{\infty}\frac{dr^{\prime}}{2\Gamma^{2}}\sigma_{T}n\Gamma,$
(3)
where $\sigma_{T}=e^{4}/(6\pi\varepsilon_{0}^{2}c^{4}m_{e}^{2})\approx
6.65\times 10^{-25}$ cm2 is the Thomson scattering cross-section. The baryon
number density in the observer frame is $n\simeq L/4\pi r^{2}m_{p}\Gamma
c^{3}\sigma_{0}$, where $L$ is the isotropic luminosity, $m_{p}$ is the mass
of protons, $\sigma_{0}\equiv\Gamma(r_{0})[1+\sigma(r_{0})]$, and
$\sigma(r_{0})$ is the initial ratio of the magnetic and baryon energy
densities. When $\tau_{T}(r)=1$, the Thomson photon sphere radius is
$\frac{r_{p}}{r_{0}}\approx 1.36\times
10^{5}L_{52}^{3/5}\sigma_{0,3}^{-3/5}r_{0,7}^{-3/5},$ (4)
where $L_{52}\equiv L/10^{52}\,\,{\rm erg}\cdot{\rm s}^{-1}$,
$\sigma_{0,3}\equiv\sigma_{0}/10^{3}$, and $r_{0,7}\equiv r_{0}/10^{7}$ cm. We
use cgs units for numerical values here and after.
The optical depth for a photon of energy $E_{0}$ to be converted to $e^{\pm}$
while traveling through the jet at a radius $r$, is given by [22]
$\tau_{\pm}(E_{0},r)=\bigg{(}\frac{\beta-2}{\beta-1}\bigg{)}\frac{\sigma_{\gamma\gamma}}{4\pi
r\Gamma^{2}}\frac{L_{>p}}{(1+z)^{3-2\beta}E_{p}c}\bigg{[}\frac{E_{p}E_{0}}{\Gamma^{2}m_{e}^{2}c^{4}}\bigg{]}^{\beta-1},$
(5)
where $\sigma_{\gamma\gamma}=6\times 10^{-26}$ cm2 is the cross-section for
photons producing $e^{\pm}$ just above the threshold energy, $E_{p}$ is the
peak energy of the $\nu F_{\nu}$ spectrum, $L_{>p}$ is the frequency-
integrated luminosity above $E_{p}$, and $\beta\approx 2.2$ is the photon
index of the spectrum above $E_{p}$. Setting $\tau_{\pm}=1$, the pair-
production photonsphere radius is given by
$\frac{r_{\gamma\gamma}(E_{0})}{r_{0}}\approx 4.13\times
10^{6}L_{>p,52}^{0.41}E_{p,-6}^{0.08}E_{0,-4}^{0.49}r_{0,7}^{-0.41}(1+z)^{0.57},$
(6)
where $E_{p,-6}=E_{p}/{\rm MeV}$, and $E_{0,-4}=E_{0}/100{\rm MeV}$.
In the observer frame, the relative time delay between MeV photons and GeV
photons equates to the time for the jet to propagate from $r_{p}$ to
$r_{\gamma\gamma}$ [22]
$\Delta
t=\frac{3r_{0}(1+z)}{2c}\bigg{[}\bigg{(}\frac{r_{\gamma\gamma}(E_{0})}{r_{0}}\bigg{)}^{1/3}-\bigg{(}\frac{r_{p}}{r_{0}}\bigg{)}^{1/3}\bigg{]}.$
(7)
Taking use of Eq.(7) together with Eq.(4) and Eq.(6), one can calculate the
time delay for the arrival of GeV and 100 MeV photons relative to MeV photons.
Thus, the intrinsic time delay is $\Delta t_{\rm int}=\Delta t(E_{\rm
high})-\Delta t(E_{\rm low})$. The observed values of $E_{p}$, $E_{\rm
ios,54}$, T90 and $z$ for four $Fermi$-detected GRBs are taken from [22], and
are listed in Table 1.
GRB | Ep | Eios,54 | T90 | z
---|---|---|---|---
| keV | erg | s |
080916c | 424 | 8.8 | 66 | 4.35
090510 | 3900 | 0.11 | 0.6 | 0.90
090902b | 726 | 3.7 | 22 | 1.82
090926 | 259 | 2.2 | 13 | 2.11
Table 1: The observed parameters of four $Fermi$-detected GRBs. $E_{p}$ is the
energy at the peak of $\nu f_{\nu}$ spectrum. $E_{\rm ios,54}$ is the
isotropic equivalent energy in unit of $10^{54}$ ergs. T90 is the GRB duration
which $90$% of the counts are above background. $z$ is the GRB redshift [22].
## 3 Test of LIV effects
As is mentioned above, some quantum gravity theories predict that photons with
different wavelength propagate in different speed [6, 7, 8, 13, 18, 24]. The
non-trivial space-time structure may affect the propagation of photons, so
high energy photons may arrive later than low energy ones. Many works have
studied LIV effects of the high energy photons in this direction [9, 15, 25,
26, 27, 28].
Consider two photons emitted at the same time and place, within the LIV
phenomenology, the arrival time delay between them can be written as [25, 27]
$\displaystyle\Delta t_{\rm LIV}=\frac{1+n}{2c}\big{(}\frac{\Delta
E}{M_{n}c^{2}}\big{)}^{n}D_{n},$ (8)
where $n=1$ or $n=2$ denotes the linear or quadratic correction to the
dispersion relation, and $D_{n}$ is defined to be [25, 27]
$\displaystyle
D_{n}\equiv\frac{c}{H_{0}}\int_{0}^{z}\frac{(1+z^{\prime})^{n}dz^{\prime}}{\sqrt{\Omega_{M}(1+z^{\prime})^{3}+\Omega_{\Lambda}}},$
(9)
where $H_{0}\simeq 72$ km sec-1 Mpc-1 is the Hubble constant, $\Omega_{M}$ and
$\Omega_{\Lambda}$ are the present values of the matter density and
cosmological constant density, respectively. In the standard cosmological
model, ($\Omega_{M},\Omega_{\Lambda}$) are given by observations as
($0.3,0.7$) [27]. For $n=1$, the time delay depends linearly on the variation
of energy, which we will consider in the follow. In this case, the effective
LIV energy scale is
$\displaystyle M_{1}c^{2}=\frac{\Delta ED_{1}}{c\Delta t_{\rm LIV}}.$ (10)
In Ref.[17], Shao et al. took use of 4 GRBs the same as in Table 1 to predict
LIV effects, the observed time delay $\Delta t_{\rm obs}/(1+z)$ vs. $K(z)$ was
plotted, where $K(z)$ is defined as
$\displaystyle K(z)\equiv\frac{\Delta E}{(1+z)}\frac{D_{1}}{c}.$ (11)
In their plot, the three long bursts were found to be near one line. However,
the short burst was not fitted well by the same line. The intercept of the
line was interpreted as $\Delta t_{\rm int}/(1+z)$, and the slope of the line
can be interpreted as $1/M_{1}c^{2}$. $\Delta t_{\rm int}/(1+z)$ was found to
be negative, which means that high energy photons are emitted earlier than low
energy ones. This conflicts with standard GRB models. Taking into account of
the four GRBs, the quantum gravity energy scale is estimated to be
$M_{1}c^{2}\sim 2\times 10^{17}$ GeV for the linear energy dependence. In
their work, the intrinsic time delay is assumed to be the same for three long
bursts. Besides, their result strongly depends on the artificial choices. For
example, if the 33.4 GeV photon is replaced by the 11.16 GeV photon in GRB
090902b, the three long bursts can not be fitted well by one line.
In the follow, we first calculate $\Delta t_{\rm int}$ by using Eq.(7). Then,
combining with the observation data, we give $\Delta t_{\rm LIV}$ by the
relation given in Eq.(1). In Ref.[25], the authors used a statistical method
to determine the observed time difference $\Delta t_{\rm obs}$ between photons
with energy $E_{\rm low}$ and $E_{\rm high}$. In our present work, $E_{\rm
low}$ is taken to be 100 MeV, since the onset of $100$ MeV photons can be read
directly from the data of the LAT monitor. $E_{\rm high}$ is the energy of the
most energetic photon in each GRB. One exception is that the second energetic
photon with $E_{\rm high}=11.16$ GeV in GRB 090902b is chosen. The most
energetic $33.4$ GeV photon arriving at $82$ s is excluded due to its
incoincidence with the main burst. In Ref.[17], Shao et al. selected this
event to estimate LIV effects without considering the central engines and
emission mechanism. In the magnetic jet model, if we believe that the delayed
GeV photons are due to the high optical depth, the arrival time of the $33.4$
GeV photon should be 3 s later than the 11.16 GeV photon. However, the
observed time interval $70$ s is far beyond the model’s prediction. This
photon may be due to the inelastic collision of protons and neutrons [19, 20],
and it is quite possible that this individual event happens when the jet
encounter the interstellar medium.
With Eq.(7), we can estimate $\Delta t_{\rm int}\simeq 0.06$ s for GRB 090510,
if $r_{0}=10^{6}$ cm. Then $\Delta t_{\rm LIV}\simeq 0.14$ s, and
$M_{1}c^{2}\sim 9.73\times 10^{19}$ GeV, which is about 8 times of the Planck
energy $E_{\rm Planck}\sim 10^{19}$ GeV [4]. However, if we increase $r_{0}$
to $10^{7}$ cm, then $\Delta t_{\rm int}\simeq 0.46$ s and $\Delta t_{\rm
LIV}\simeq-0.26$ s. In this case, high energy photons become superluminal,
which is against the argument of quantum gravity theories. It is a reasonable
assumption that $r_{0}\simeq 10^{6}$ cm for GRB 090510, because this is a
short burst and its radius should be small than long bursts.
GRB | $E_{\rm low}$ | $E_{\rm high}$ | $\Delta t_{\rm obs}$ | $\Delta t_{\rm LIV}$ | $K(z)$ | $M_{1}c^{2}$
---|---|---|---|---|---|---
| MeV | GeV | s | s | s$\cdot$GeV | GeV
080916c | 100 | 13.22 | 12.94 | 0.24 | 4.50 $\times 10^{18}$ | 10.02 $\times 10^{19}$
090510 | 100 | 31 | 0.20 | 0.14 | 7.02 $\times 10^{18}$ | 9.73 $\times 10^{19}$
090902b | 100 | 11.16 | 9.5 | 0.10 | 3.38 $\times 10^{18}$ | 9.94 $\times 10^{19}$
090926 | 100 | 19.6 | 21.5 | 0.20 | 6.20 $\times 10^{18}$ | 9.59 $\times 10^{19}$
Table 2: The LIV induced time delay $\Delta t_{\rm LIV}$ and quantum gravity
energy scale $M_{1}c^{2}$ derived from four Fermi-detected GRBs. $\Delta
t_{\rm obs}$ is collected from Ref.[2, 3, 4, 5]. $\Delta t_{\rm LIV}=\Delta
t_{\rm obs}-\Delta t_{\rm int}$, where $\Delta t_{\rm int}$ is calculated by
Eq.(4), Eq.(6) and Eq.(7). The value of $\sigma_{0,3}$ in each GRB is
approximately the bulk Lorentz factor of the jet in unit of $10^{3}$ and is
taken as $\sigma_{0,3}\sim 1$ [22]. $r_{0,7}$ is chosen as 16.7, 0.1, 28.7 and
55.0 for GRB 080916c, GRB 090510, GRB 090902b and GRB 090926, respectively.
With the above criteria, we try to use the line fitting method to predict the
LIV effects. In principle, if the linear dispersion relation holds, the
$\Delta t_{\rm LIV}/(1+z)$ vs. $K(z)$ plot should be a zero-intercept line,
whose slope is the inverse of quantum gravity energy scale, i.e.,
$1/M_{1}c^{2}$. By choosing $r_{0}$ of each burst properly, the limits of LIV
effects for both short and long bursts can be unified respecting the linear
dispersion relation. The four GRB points can be fitted well by one line, if we
choose $r_{0,7}=$ 16.7, 0.1, 28.7 and 55.0 for GRB 080916c, GRB 090510, GRB
090902b and GRB 090926, respectively. In this case, LIV effects are calculated
in Table 2, and the $\Delta t_{\rm LIV}/(1+z)$ vs. $K(z)$ plot is given in
Fig.1. The values of $r_{0}$, which indicate the active scale of central
engines, are reasonable. The inverse of the slope gives $M_{1}c^{2}\sim
1.0\times 10^{20}$ GeV, which is roughly the same result of the GRB 090510.
The energy scale of the modified photon dispersion relation is one order of
magnitude higher than the conventional Planck scale. This may suggest that the
quantum gravity scale may be more subtle than one naively thinks, because this
quantity is model dependent. For instance, the effective quantum energy scale
depends on the density of $D$-particles in the D-foam model [7, 8]. It is an
interesting future work to combine both the quantum gravity model and the GRB
model together to study the LIV effects. Supposing LIV effects are strongly
suppressed, which is the assumption taken by the mainstream GRB models, the
observed time delay can well predict the value of $r_{0}$ by using the
magnetic jet model, since the tuning of $r_{0}$ is sensitive to the fitting.
Figure 1: The plot of $\Delta t_{\rm LIV}/(1+z)$ vs. $K(z)$ for four Fermi-
detected GRBs.
## 4 Discussion and conclusion
From Eq.(7), one can infer that the time lag has approximate linear relation
with the initial radius ($\propto r_{0}^{0.86}$) and the redshift
($\propto(1+z)^{1.19}$). It has a weak dependence on the photon energy
$E_{0}$, the peak energy $E_{p}$, and the peak luminosity $L_{>p}$, which are
$\Delta t_{\rm int}\propto E_{0}^{0.17},E_{p}^{0.05},$ and $L_{>p}^{0.14}$,
respectively. In addition, the time lag is hardly dependent on $\sigma_{0}$.
Another model proposed to explain the delayed GeV photons was given by
Mészáros and Rees. In this model, the time lag also depends linearly on
$r_{0}$ (See Eq.(19) in [20]). These GRB models also predict the spectrum, so
they can be verified by the observed spectrum of the whole energy band.
As mentioned above, in Ellis et al.’s proposal, the linear fitting function is
written as $\Delta t_{\rm obs}/(1+z)=a_{\rm LIV}K(z)+b$. In our case,
$b\sim\Delta t_{\rm int}/(1+z)$ is not a constant, and we can estimate it
approximately as
$\displaystyle b\simeq
0.08\,r_{0.7}^{0.86}L_{>p,52}^{0.14}E_{p,-6}^{0.03}E_{0,-4}^{0.16}(1+z)^{0.19},$
(12)
which roughly agrees with Ellis et al.’s result $b\sim 10^{-2}$ [15]. Since
$b$ weakly depends on the redshift, it can be regarded as a distance-
independent quantity [17]. However, $b$ strongly depends on $r_{0}$, so the
long and the short bursts will lead to quite different values of $b$. If the
long bursts have roughly the same $r_{0}$, $b$ can be considered as a
constant, and the linear relation between $\Delta t_{\rm obs}/(1+z)$ and
$K(z)$ holds.
The analysis above gives us the hint that, if we want to consider source
effects of the GRBs, long bursts with small redshifts are preferred. Due to
the short cosmological distance, quantum gravity effects can be attenuated.
With the purpose of enhancing quantum gravity effects, short GRB bursts with
high redshifts should be selected. Therefore, the future observations on short
GRBs will improve the test of LIV effects. If we use an ensemble of GRBs with
both long and short bursts, the fitting function method is not convincing. The
intrinsic time delay plays an important role. Better knowledge of the
intrinsic property of the source will help us to improve the test of LIV
effects. Inversely, better understanding of quantum gravity can help us to
predict the parameters in GRB models.
In this work, we discussed LIV effects by making use of the magnetic jet
model. GeV photons are emitted later than MeV photons, due to their different
optical depths. This physical ingredient should be included in probe of LIV
effects. The neglect of the photon emission mechanism may lead to misleading
results. The constraints of LIV effects can be unified for both long and short
bursts. The calculation of the linear energy dependence of dispersion relation
gives $M_{1}c^{2}\sim 1.0\times 10^{20}$ GeV. Although the magnetic jet model
itself should be tested by further investigations, the analysis of the
intrinsic time delay is important when we study the photons from the
astrophysical sources to test LIV effects.
## Acknowledgments
We are grateful to M. H. Li, X. Li and S. Wang for useful discussion. This
work has been funded in part by the National Natural Science Fund of China
under Grant No. 10875129 and No. 11075166.
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|
arxiv-papers
| 2012-01-17T02:18:32 |
2024-09-04T02:49:26.381313
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhe Chang, Yunguo Jiang, and Hai-Nan Lin",
"submitter": "Yunguo Jiang",
"url": "https://arxiv.org/abs/1201.3413"
}
|
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